Advances in Geophysics: Issues in Atmospheric and Oceanic Modeling, Part B : Weather Dynamics

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ADVANCES IN

GEOPHYSICS

VOLUME 28 Issues in Atmospheric and Oceanic Modeling

Part B

Weather Dynamics

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Advances in

GEOPHYSICS VOLUME 28 Issues in Atmospheric and Oceanic Modeling

Part B Weather Dynamics

Edited by

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

Volume Editor

SYUKURO MANABE Geophysical Nuid Dynamics LaboratorylNOA.4 Princefon Universify Princeton, New J e w

1985

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

Orlando San Diego New York Austin London Montreal Sydney Tokyo Toronto

Dedicated to Joseph Smagorinsky on his retirement from the Geophysical Fluid Dynamics Laboratory of NOAA, January 31,1983.

COPYRIGHT @ 1985 BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMlTTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS,INC. Orlando, Florida 32887

United Kingdom Edition published by

ACADEMIC PRESS INC. (LONDON) LTD 24-28 Oval Road. London NWI 7DX

LIBRARY OF CONGRESS CATALOG

CARD

ISBN 0-1 2-01 8849-X ISBN 0-1 2-000004-0 (paperback) PRINTED IN THE UNITED STATES OF AMERICA

85868788

9 8 7 6 5 4 3 2 1

NUMBER:52-1 2266

CONTENTS CONTRIBUTORS.....................................................

ix

........................................................ PREFACE ..........................................................

xi xv

FORE WORD

Part I. Numerical Weather Prediction Mediurn-Range Forecasting at the ECMWF LENNART BENGTSSON 1 . Introduction .................................................... 2 . The Physical and Mathematical Basis for Medium-Range Forecasting . 3 . Numerical Methods and Modeling Technique ....................... 4 . Observations. Their Use and Importance ........................... 5. Operational Application and Results ............................... 6. Problems and Prospects in Numerical Weather Prediction . . . . . . . . . . . . 7. Concluding Remarks ............................................. References ......................................................

3 5

7 15 24

35 50 51

Extended Range Forecasting K . MIYAKODA A N D J . SIRUTIS 1 . Introduction .................................................... 2 . An Evolution of IO-Day Forecast Performance ...................... 3. Examples of Monthly Forecasts ................................... 4 . A Projection of Seasonal Forecasts ................................ 5. Postscript ....................................................... References ......................................................

55 56 65 79 82 83

Predictability J . SHUKLA 1 . Introduction .................................................... 2 . Classical Predictability Studies .................................... 3 . Predictability of Space-Time Averages ............................. 4 . Some Outstanding Problems ...................................... 5 . Concluding Remarks ............................................. References ...................................................... V

87 89 110 116

119 121

vi

CONTENTS

Data Assimilation W . BOURKE.R . SEAMAN. A N D K . PURI 1 . Introduction .................................................... 2 . Evolution of Assimilation and the FGGE ........................... 3. Components of Four-Dimensional Assimilation Systems ............. 4 . Characteristics of Some Current Assimilation Schemes .............. 5. Role of Four-Dimensional Assimilation in Research and Operations ... 6. Conclusion ...................................................... References ......................................................

123 125 131 138 I43 151 151

Part I I . Mesoscale Dynamics Predictability of Mesoscale Atmospheric Motions RICHARD A . ANTHES. YING-HWAK U O . DAVIDP. BAUMHEFNER. RONALDM . ERRICO.A N D THOMAS W . BETTGE

I . Introduction ...................................................

159

2 . Classic Predictability Experiments and Their Relationship

to Mesoscale Predictability ...................................... 3. Preliminary Predictability Study with a Mesoscale Model ............ 4 . Discussion and Comparison with a Predictability Study Usinga Global Model ............................................ 5. Summary and Conclusions ........................................ References ......................................................

164 165

195 198 200

Thermal and Orographic Mesoscale Atmospheric SystemsAn Essay ROGERA . PIELKE 1. 2. 3. 4.

Introduction .................................................... Summary of Major Research Accomplishments ..................... Research Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eventual Goals .................................................. References ......................................................

203 204 216 218 219

Advances in the Theory of Atmospheric Fronts I . ORLANSKI. B . Ross. L . POLINSKY. A N D R . SHAGINAW

1. Introduction .................................................... 2 . Baroclinic Waves and Fronts .....................................

223 225

CONTENTS

3 . Mature Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . What Observed Features Can Be Explained by Theory? ............. 5 . What Other Processes Are Important in Frontogenesis? . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 229 233 237 250

Part Ill. Tropical Dynamics Numerical Modeling of Tropical Cyclones YOSHIOKU R IHA R A 1 . Introduction .................................................... 2 . Numerical Models of Humcanes .................................. 3 . Numerical Simulation of Tropical Cyclones ......................... 4 . Some Challenging Issues in the Future ............................. Appendix . GFDL Hurricane Models ............................... References ......................................................

255 256 261 276 278 280

Numerical Weather Prediction in Low Latitudes T . N . KRISHNAMURTI 1. 2. 3. 4. 5. 6.

Introduction .................................................... Initialization: Dynamic, Normal Mode. and Physical . . . . . . . . . . . . . . . . . Parameterization of Physical Processes ............................. Medium-Range Prediction of Monsoon Disturbances . . . . . . . . . . . . . . . . . On the Prediction of the Quasi-Stationary Component . . . . . . . . . . . . . . . Scope of Future Research .................... . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 291 306 314 323 330 331

Part IV . Turbulence and Convection Sub-Grid-Scale Turbulence Modeling J . W . DEARDORFF 1 . Introduction: The Need for Grid-Scale Reynolds Averaging . . . . . . . . . . 2 . The Effect of Grid-Volume Reynolds Averaging ..................... 3 . The Sub-Grid-Scale Eddy Coefficient .............................. 4 . Recent Developments ............................................ 5. Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 338 340 341 341 342

viii

CONTENTS

Ensemble Average. Turbulence Closure GEORGE L . MELLOR 1. Introduction .................................................... 2 . The Turbulence Macroscale and Turbulence Closure ................ 3 . Averaging Distance for Measurements in the Atmosphere and Oceans and for Numerical Models ........................................ 4 . Numerical Modeling Applications and Horizontal Diffusion .......... 5 . Concluding Remarks ............................................. References ......................................................

345 347 354 355 356 357

The Planetary Boundary Layer H . A . PANOFSKY 1. General Characteristics........................................... 2 . The Equations in the PBL ........................................ 3 . The Surface Layer ............................................... 4 . First- and Second-Order Closures ................................. 5 . Boundary-Layer Models .......................................... 6 . Boundary-Layer Parameterization ................................. References ....................................................

359 364 367 377 380 383 383

Modeling Studies of Convection YOSHl OGURA

1. Introduction .................................................... 2 . Benard-Rayleigh Convection ..................................... 3 . Complexity of Convection in the Atmosphere ....................... 4 . Shallow Moist Convection ........................................ 5 . Deep Moist Convection .......................................... 6. Feedback Effects of Cumulus Clouds on Larger-Scale Environments . . 7 . Concluding Remarks ............................................. References ......................................................

INDEX

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

387 388 391 392 397 408 413 416 423

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.

RICHARD A. ANTHES,National Center for Atmospheric Research, Boulder, Colorado 80307 (159) DAVIDP. BAUMHEFNER, National Center for Atmospheric Research, Boulder, Colorado 80307 ( 1 59) LENNARTBENGTSSON, European Centre for Medium Range Weather Forecasts, Reading RG2 9AX, England (3) THOMAS W. BETTGE,National Center for Atmospheric Research, Boulder, Colorado 80307 (1 59) W. BOURKE,Bureau of Meteorology Research Centre, Melbourne 3001, Australia (123)

J. W. DEARDORFF, Department of Atmospheric Sciences, Oregon State University, Corvallis, Oregon 97331 (337) RONALDM. ERRICO, National Center for Atmospheric Research, Boulder, Colorado 80307 (159) T . N . KRISHNAMURTI, Department of Meteorology, Florida State University, Tallahassee, Florida 32306 (283)

YING-HWAKuo, National Center for Atmospheric Research, Boulder, Colorado 80307 (159) YOSHIOKURIHARA, Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (255) GEORGEL. MELLOR,Geophysical Fluid Dynamics Program, Princeton University, Princeton, New Jersey 08542 (345) K. MIYAKODA, Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (55) YOSHIOGURA,Department of Atmospheric Sciences, University of Zllinois, Urbana, Illinois 61801 (387) I. ORLANSKI, Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (223) ix

CONTRIBUTORS

X

H. A. PANOFSKY,* Department of Meteorology, Pennsylvania State University, University Park, Pennsylvania 16802 (359) ROGERA. PIELKE,Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado 80523 (203)

L. POLINSKY, Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (223) K . PURI,Bureau of Meteorology Research Centre, Melbourne, 3001, Australia (123) B . Ross, Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (223)

R. SEAMAN, Bureau of Meteorology Research Centre, Melbourne, 3001, Australia (123)

R. SHAGINAW, Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (223) J . S H U K L A ,Center for Ocean-Land-Atmosphere Interactions, Department of Meteorology, University of Maryland, College Park, Maryland 20742 (87)

J. SIRUTIS,Geophysical Fluid Dynamics LaboratorylNOAA, Princeton University, Princeton, New Jersey 08542 (55)

* Present address: Scripps Institute of Oceanography, University of California at San Diego, La Jolla, California 92093.

FOREWORD A general scientific revolution was taking place in the mid- 1940s, no small part of which was being played by research meteorologists. At the successful urging of John von Neumann, a computer of incomparable capacity for its time was being built, with improved weather forecasting as one of its main justifications. Within a few short years, the practice of theoretical meteorological research would be transformed from one characterized only by highly individualistic efforts to one in which a major role would be played by teams of co-workers using commonly shared highspeed computers to model and experiment with the workings of the atmosphere and its larger climatic system setting. It was Joseph Smagorinsky who emerged as one of the pioneer, prototypical leaders of the new mode of research engendered by this revolution. Already an accomplished individual researcher who had made significant contributions to dynamical meteorology, especially regarding the effects of heating and moisture fields on weather and climate patterns, Smagorinsky realized early that the ultimate nonlinear theory of the detailed climatic state of the atmosphere and oceans would rest on largescale numerical modeling, and he set about to create an institutional structure for accomplishing the long-term process of model building that would be necessary. In essence, Smagorinsky inaugurated a new style of creative scientific endeavor in which the contributions of a group of outstanding theorists and computer specialists are “orchestrated” to produce new, otherwise unobtainable, results and insights relevant to extremely complex problems. This organic model-building process is now more alive than ever, with major general circulation modeling (GCM) groups active at many locations. These parallel efforts are, in fact, necessary to provide the only source of “reproducibility” possible in this kind of work; unlike the classic “pencil-and-paper” theories, no single individual can replicate the reported results of a GCM experiment. It is now widely recognized that the GCM solutions represent the most complete deductive explanation obtainable of the statistical equilibrium state of the atmosphere and its lower surface boundary layer. As such, they provide the basis for our most credible predictions of the climatic consequences of prescribed external changes and of prescribed changes in some of the slower response parts of the climatic system. In addition, the numerical models are just beginning to come to the fore as an experimental tool used to understand the dynamics of the atmosphere and oceans to the degree necessary to parameterize the time-average fluxes of mass, momentum, and energy by the synoptic-scale motions. This role is bound to increase in the future. xi

xii

FOREWORD

Under the leadership of Joseph Smagorinsky , the Geophysical Fluid Dynamics Laboratory (GFDL) became a pacesetter in all these areas. It is also true that it is to the GCM that we must continually turn to obtain information about the likely consequences of the actions of humanity on the global environment. Our understanding of the consequences of increasing CO2 on climatic conditions largely derive from GCMs. Questions about the possible decrease of the ozone content of the stratosphere as a result of the release of chemicals in that region of the Earth’s atmospheric envelope are illuminated by computations with GCMs. Our concerns with the effects of a nuclear exchange on the general circulation of the atmosphere can only be treated through the computations of general circulation models. The ability to extend the time range of weather forecasts is fundamentally dependent on the refinement of general circulation models and the acquisition of the necessary observational data required for their applications. In recent years in the United States, Europe, and other regions of the world, significant advances have been made in longrange weather forecasting as a result of the development of these models. Smagorinsky was not only a creative scientist and a leader of teams of scientists, he was a remarkably able administrator, at home within the administrative currents of government organizations as well as within the intellectual give and take of a scientific laboratory. The history of the Geophysical Fluid Dynamics Laboratory is indeed the reflection of his career. After leaving Princeton and returning to the Weather Bureau in Washington, D.C., he became the focus for the predecessor laboratory of the Geophysical Fluid Dynamics Laboratory. The General Circulation Laboratory of the Weather Bureau was housed in a storefront building on Pennsylvania Avenue in the shadow of the U.S. Capitol. On display there were the computers of whose basic necessity Smagorinsky had been able to convince governmental officials. People walked back and forth observing the curious phenomenon of scientists placing tapes on machines and deciphering the incredible volume of numbers produced. It was clear that a storefront building on Pennsylvania Avenue, however choice the location, was not the best environment for Smagorinsky’s dreams. He wanted a laboratory on a university campus, in which the intellectual stimulation of the scientific community in a university could be melded with the resources and facilities provided by the government. It would be a place where scientists from his laboratory, cooperating with others invited from all parts of the world, could work in a stimulating environment totally focused on the key problems of geophysical fluid dynamics. It was a mark of his administrative ability that he persuaded the Weather Bureau to move the General Circulation Laboratory to the

FOREWORD

xiii

campus at Princeton University to be housed in a building exclusively devoted to geophysical fluid dynamics research. From its inception the laboratory bore the imprint of Smagorinsky’s insistence on high standards in every aspect of work. It became and continues to be a mecca for scientists from many countries for the study of numerical approaches to a variety of atmospheric and oceanographic problems. Perhaps no single laboratory has had the worldwide influence of Smagorinsky’s GFDL. It became a model for many other groups and countries throughout the world. It has been our great pleasure to observe and admire “Smag’s” efforts and the monumental achievements of the GFDL. We have enjoyed and benefited greatly from our long years of association with him as friends and colleagues. It is a privilege now to introduce this Festschrift volume dedicated to him. Rich in excellent review papers and ably edited by Syukuro Manabe and his GFDL colleagues, the contents give ample testimony to Joe Smagorinsky’s seminal influence on theoretical meteorology and its practitioners.

BARRYSALTZMAN ROBERTM. WHITE

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PREFACE This book is dedicated to Joseph Smagorinsky in commemoration of his retirement as Director of the Geophysical Fluid Dynamics Laboratory (GFDL) of the National Oceanic and Atmospheric Administration (NOAA) and to celebrate his lifelong achievements in the atmospheric and oceanic sciences. Joe Smagorinsky pioneered the development of mathematical models of the atmosphere, beginning in the 1950s with his construction of a general circulation model of the atmosphere based on the primitive equations and continuing over the next two decades with the evolution of more comprehensive models. In addition, he promoted the application of the modeling technique to a wide variety of phenomena, both large and small in scale, basic and applied in form, and to a wide variety of systems, both oceanic and atmospheric, terrestrial and extraterrestrial. From the beginning he recognized the enormous potential of the computer for the atmospheric and oceanic sciences and throughout his term as Director of the GFDL, he succeeded in obtaining the most advanced machines made or conceived of for his laboratory’s use and in creating a research environment that optimized their potential. Joe Smagorinsky also made a major contribution in the planning and execution of the Global Atmospheric Research Program (GARP), first as a member and then as the Chairman of the Joint Organizing Committee of the World Meteorological Organization (WMO) and the International Council of Scientific Union (ICSU). His imaginative leadership and absolute commitment were indispensable to the success of GARP and its First Global GARP Experiment (FGGE). His scientific expertise was invaluable in the design and realization of GARP’s major modeling studies of atmospheric variations on both the weather and climate scales. The main objective of this book is to assess the current state of atmospheric and oceanic modeling and of related theoretical and diagnostic studies and to pinpoint outstanding problems of immediate and future concern. A wide range of topics is discussed; for convenience, the book is split into two parts according to scale. The first part, “Climate Dynamics,” contains articles on climate, the middle atmosphere, planetary atmospheres, and ocean dynamics. The second part, “Weather Dynamics,” covers numerical weather prediction, mesoscale dynamics, tropical dynamics, turbulence, and convection. Each essay presents a topic in historical perspective, discussing its present status and assessing the future prospect for research. Another objective of the book is to give some measure of the progress made during the past 30 years and of Joe’s role in that achievement. The xv

xvi

PREFACE

authors were thus chosen from among his friends and students, both at the GFDL and elsewhere, and were encouraged-mainly because of space limitations-to present their own views of their topic rather than to attempt a comprehensive review. We regret any omissions or bias that this selectivity has caused and beg the forebearance of the many people who have contributed to the science and of the many friends of Joe who could not be included in these volumes. It is hoped that this book will give some indication of the enormous power and future potential of mathematical modeling and that it will stimulate further interest in its application to atmospheric and oceanic problems. If we achieve this aim, we will have begun to repay the great debt we owe to Joe Smagorinsky. The publication of this book would not have been possible without the help and enthusiasm of the associate editors-K. Bryan, Y. Kurihara, J. D. Mahlman, G. L. Mellor, K. Miyakoda, A. H. Oort, 1. Orlanski, G. Philander, and G. P. Williams-who participated in planning the book and editing its various sections. Barry Saltzman, the regular editor of Advances in Geophysics gave me the privilege of serving as the guest editor of this two-part volume. His thoughtful guidance and assistance made this book possible. Robert White, who gave his wholehearted support to the GFDL during his term as administrator of the NOAA, kindly agreed to write the Foreword in collaboration with Barry Saltzman. J. D. Mahlman, the current director of the GFDL has given us constant encouragement and support and has made the resources of the GFDL available for this large effort. Finally, it is a great pleasure to acknowledge all those, especially the authors, who set aside many other pressing commitments to contribute with insight and enthusiasm to these two volumes. SYUKURO MANABE

Part I

NUMERICAL WEATHER PREDICTION

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Part I

NUMERICAL WEATHER PREDICTION

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MEDIUM-RANGE FORECASTING AT THE ECMWF LENNART BENGTSSON European Cenlre f o r Medium Range Weather Forecasts Reading, England I . Introduction . . . . . . . . . . . . . . . . . . . . . . 2. The Physical and Mathematical Basis for Medium-Range Forecasting 3. Numerical Methods and Modeling Technique. . . . . . . . . . 3.1. Basic Equations . . . . . . . . . . . . . . . . . . . 3.2. Numerical Formulation . . . . . . . . . . . . . . . . 3.3. Radiation and Clouds . . . . . . . . . . . . . . . . . 3.4. The Planetary Boundary Layer . . . . . . . . . . . . . 3.5. Moist Convection . . . . . . . . . . . . . . . . . . 3.6. Nonconvective Parameterization. . . . . . . . . . . . . 3.7. Surface Values . . . . . . . . . . . . . . . . . . . 4. Observations, Their Use and Importance . . . . . . . . . . . 5. Operational Application and Results . . . . . . . . . . . . . 5.1. Northern Hemisphere Forecasts . . . . . . . . . . . . . 5.2. Tropical and Southern Hemisphere Forecasts. . . . . . . . 6. Problems and Prospects in Numerical Weather Prediction. . . . . 7. Concluding Remarks . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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51

1 . INTRODUCTION

A very substantial improvement has taken place in numerical weather prediction since the very first numerical forecasts were made almost 35 years ago. These improvements are essentially due to much better and more realistic models but also to the considerable increase in the number of meteorological observations that have taken place over the period. The improvement is manifested in two ways, more accurate short-range forecasts and a substantial extension in time of useful predictive skill. Today 4-day forecasts at 500 mb are now as accurate as the 1-day forecasts produced in the early 1950s; see Table I. The division of weather forecasts into short-, medium-, and long-range forecasts is both logical and practical. In the introduction to a conference on the dynamics of climate held at Princeton in 1955 (Pfeffer, 1960), von Neumann outlined an overall strategy in atmospheric modeling and prediction. He considered that the prediction problem could conveniently be divided into three different categories depending on the time scale of the forecast. In the first was the short-range prediction of motions that are determined mainly by the initial state of the atmosphere. The second 3 ADVANCES I N GEOPHYSICS, V O L U M F

28B

Copyright 0 1985 by Acddemic Pre\\. Inc All rightq of reproduction in dny form re\erved

4

LENNART BENGTSSON ERRORFOR 500-MB FORECAST OVER EUROPE TABLEI. STANDARD DEVIATION

Forecast time

November 1951-April 1954 (24 cases) barotropic model" ~~

24 hr 48 hr 72 hr 96 hr

January 1981 barotropic modelb

January 1981 ECMWF'

January1984 ECMWFd

47 m 97 m 151 m

22 m 41 m 62 m 85 m

21 m 38 m 57 m

~

76 m

75 m

* Results of 24 operational and quasi-operational 24-hr predictions by the barotropic model (data from University of Stockholm, 1954). Barotropic forecast for the Northern Hemisphere during January 1981. Initial state taken from the operational ECMWF analyses. ECMWF operational forecast for January 1981. d ECMWF operational forecast for January 1984.

comprised much longer-term predications of characteristics of the motion that are largely independent of the initial atmospheric conditions and thus included the problem of climate simulation. Third, between these two extremes there was another category of prediction for which it was necessary to consider the details both of the initial state and of the external forcing that determined the forecast equilibrium. The logical approach was to attack those problems in the order in which they are listed. This approach has, in fact, been followed, and in this chapter we will review the progress and development that have taken place in medium-range forecasting. In this we will define medium range as covering the time interval from 2 days to 2 weeks. This is much lower than the time span that von Neumann had in mind, but his suggested approach to the order of doing things in tackling the problem is essentially valid and intact. The initial experiments in medium-range forecasting came at the end of the 1960s when a series of 2-week hemispheric predictions were carried out at the GFDL at Princeton University (Miyakoda et al., 1972). These experiments were of epoch-making importance and is just one of many examples of the fundamental role of the GFDL in atmospheric modeling and prediction. The forecast experiments into the medium range at the GFDL were encouraging and served an important catalytic function. They stimulated the planning and later the successful implementation of the First GARP (Global Atmosphere Research Programme) Global Experiment (FGGE) in 1979 and they constituted a very important impetus for the setting up of the European Centre for Medium Range Weather Forecasts (ECMWF). Operational production of forecasts on a

MEDIUM-RANGE FORECASTING AT THE ECMWF

5

daily basis for a period up to 10 days ahead started at the ECMWF on 1 August 1979. In this chapter we will discuss the physical and mathematical basis for medium-range forecasting, the numerical methods, and the modeling technique, as well as the use and importance of observations. Operational application and results at the ECMWF obtained over more than four years will be presented and analyzed. Concentration on the performance of just one forecasting center may seem inappropriate in a publication such as this, but the forecast results to be discussed are, at the time of writing, unique in providing a regular record of more than 1500 experiments in atmospheric predictability on a time scale as long as 10 days.

2. THE PHYSICAL AND MATHEMATICAL BASIS FOR MEDIUM-RANGE FORECASTING Numerical weather prediction and general circulation modeling grew from similar origins in the early 1950s, but modeling for the two applications differed widely. Weather forecasting was concentrated on a time range of up to at most a few days ahead over which period close attention to the slowly acting physical processes was not of paramount interest. It was also subject to operational time constraints that necessitated the use of less than hemispheric domains. However, as was recognized at the outset of numerical modeling and more specifically by Miyakoda et al. (1972), the climatological balance of the forecast model may become of importance after several days of prediction. Moreover, the extension of the prediction time necessitated a gradual extension of the integration domain to the whole globe. This has been confirmed in practice (Hollingsworth et al., 1980; Bengtsson and Simmons, 1983) and means that the modeling problem in medium-range forecasts is very similar to that of studying climate processes on short-time scales (less than a few months). Very recently the World Climate Research Programme (WCRP) has organized its work into three streams defined by the time scale of the characteristic climatic processes. The modeling effort for the first of these streams, which is concerned with short-term climate processes, is very similar to the modeling in medium-range forecasting. In fact, short-term (50-day) climate simulations play a very important role at the ECMWF in the testing of alternative formulations of the model. Thus a clear distinction can no longer be drawn between numerical models for climate studies and for medium-range weather prediction, and we may anticipate that further improvements in model

MEDIUM-RANGE FORECASTING AT THE ECMWF

25

(b)

FIG.9. (Continued)

spectral model, described in Section 3, replaced the grid-point model. At the same time, a new representation of orography, including a special treatment of sub-grid-scale processes, was introduced. However, as can be seen from Fig. 10, a considerable improvement of the predictive skill on a monthly basis has taken place even with the ECMWF’s first forecasting system. This improvement, which is characteristic of a new forecasting system when it is introduced, is partly due to a series of refinements of the data analysis that have been implemented in the light of operational experience. Incorporation of physical forcing in the initialization has particularly affected the forecasts at lower latitudes.

40

LENNART BENGTSSON

0 1 , 0

1

I

1

,

I

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,

I

I

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,

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5

6

7

8

9

10

Days

' m-4-0

0

1

2

3

4

5 Days

6

7

8

Q

10

FIG. 16. (a) Height anomaly correlation scores for wave numbers 1-3, 4-9, 10-20. ECMWF Northern Hemisphere (20-82.5"N) forecasts, January 1982. (b) Height anomaly correlation scores for wave numbers 1-3,4-9, 10-20. ECMWF Northern Hemisphere (2082.Y") forecasts, January 1983.

the atmosphere that is possibly on the order of a few weeks. While this is the case for instantaneous weather patterns, there are clear indications [e.g., Shukla (1984), Miyakoda et al. (1983)l that the predictability for time averages (5, 10, or 30 days) is longer, in particular in the tropics. However, we will not consider this aspect in this study.

250

I . ORLANSKI, B. ROSS. L. POLINSKY, AND R. SHAGINAW

1984). While we have developed a good understanding of the main dynamics of frontal systems, there remain many opportunities for future advancement in these new areas. We have so far only scratched the surface in our understanding of how this important phenomenon, the front, interacts with other atmospheric systems.

ACKNOWLEDGMENTS The authors, who were the members of the Mesoscale Group at the GFDL during the late period of Joseph Smagorinsky’s tenure as director of the GFDL, would like to express our deep appreciation to him for his constant encouragement, support, and interest in mesoscale research. Also, regarding this chapter, the authors thank Professor William Blumen for his helpful review, Professor Yoshio Ogura for providing the photograph for Fig. 5 , and Professor Frederick Sanders, Dr. Isaac Held, and Dr. Daniel Keyser for providing useful comments. Also, we appreciate the fine work done by Mrs. Joan Pege, Messrs. John Conner, Philip Tunison. Jeffrey Varanyak, and Michael Zadworney during the preparation of this paper.

REFERENCES Anthes, R . A., Kuo, Y.-H., Benjamin, S. G., and Li, Y.-F. (1982). The evolution of the mesoscale environment of severe local storms: Preliminary model results. M o n . Weather Rev. 110, 1187-1213. Baines, P. G . (1980). The dynamics of the Southerly Buster. Aust. Mefeorol. Mag. 28, 175- 199. Bjerknes, J. (1919). On the structure of moving cyclones. Geophys. Noru. 1, No. 2. Bjerknes, J. (1937). Theorie der aussertropischen Zyklonenbildung. Mefeorol. Z . 54, 462-466. Bjerknes, J., and Solberg, H. (1921). Meteorological conditions for the formation of rain. Geophys. Noru. 2, No. 3, 1-61. Blumen, W. (1980). A comparison between the Hoskins-Bretherton model of frontogenesis and the analysis of an intense surface frontal zone. J . A f m o s . Sci. 37, 64-77. Blumen, W. (1981). The geostrophic coordinate transformation. J . Atmos. Sci. 38, 1100-1105. Boyd, J. P. (1980). The nonlinear equatorial Kelvin wave. J . PhyA. Oceanogr. 10, 1-11. Charney, J. (1947). The dynamics of long waves in a baroclinic westerly current. J . Mefeorol. 4, 35-162. Charney, J. (1975). Jacob Bjerknes-An Appreciation, in “Selected Papers of Jacob Aall Bonnevie Bjerknes” (M. G. Wurtele, ed.), pp. 11-13. Western Periodicals Company, North Hollywood, California. Decker, M. T. (1984). Observation of low-level frontal passages with microwave radiometers. In “Analysis of Some Cloud and Frontal Events, Recorded during the Boulder Upslope Cloud Observation (BOUCE) of 1982.” (E. E. Gossard, ed.), pp. 27-32. March 1984, NTIS #PB84-179662. Eady, E. (1949). Long waves and cyclonic waves. Tellus 1, No. 3, 33-52.

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Eliasen, E. (1960). On the initial development of frontal waves. Publ. Dan. Meteorol. Inst. No. 13. Eliassen, A. (1959). On the formation of fronts in the atmosphere. In “The Atmosphere and the Sea in Motion” (B. Bolin, ed.), pp. 277-287. Rockefeller Inst. Press, New York. Eliassen, A. (1962). On the vertical circulation in frontal zones. Geophys. Norv. 24, 147-160. Hoskins, B. J. (1982). The mathematical theory of frontogenesis. Annu. Rev. Fhid Mech. 14, 131-151. Hoskins, B. J., and Bretherton, F. P. (1972). Atmospheric frontogenesis models: Mathematical formulation and solution. J . Atmos. Sci. 29, 11-37. Hoskins, B. J., and Heckley, W. A. (1981). Cold and warm fronts in baroclinic waves. Q. J . R . Meteorol. Soc. 107, 79-90. Hoskins, B. J., and West, N . V. (1979). Baroclinic waves and frontogenesis. Part 11. Uniform potential vorticity jet flows-cold and warm fronts. J . Atmos. Sci. 36, 1663-1680. Houghton, D. D. (1969). Effect of rotation on the formation of hydraulic jumps. JGR, 1. Geophys. Res. 74, 1351-1360. Keyser, D., and Anthes, R. A. (1982). The influence of planetary boundary layer physics on frontal structure in the Hoskins-Bretherton horizontal shear model. J. Atmos. Sci. 39, 1783- 1802. Kotschin, N. (1932). Uber die Stabilitat von Margulesschen Diskontinuitatsflachen. Beitr. Phys. Atmos. 18, 129-164. Margules, M. (1906). Uber Temperaturschichtung in stationar bewegter und ruhender h f t . Hann-Band. Meteorol. Z., pp. 243-254. Miller, J. E. (1948). On the concept of frontogenesis. J . Meteorol. 5 , 169-171. Ogura, Y.,and Portis, D. (1982). Structure of the cold front observed in SESAME-AVE 111 and its comparison with the Hoskins-Bretherton frontogenesis model. J . Atmos. Sci. 39, 2773-2792. Orlanski, I. (1968). Instability of frontal waves. J . Atmos. Sci. 25, 178-200. Orlanski, I., and Polinsky, L. J. (1983). Ocean response to mesoscale atmospheric forcing. Tellus 35A, 296-323. Orlanski, I., and Polinsky, L. J. (1984). Predictability of mesoscale phenomena. Proc. I n t . Symp. Nowcasting, 2nd, 1984. Norrkbping, Sweden, pp. 271-280. Orlanski, I., and Ross, B. B. (1977). The circulation associated with a cold front. Part 1. Dry case. J . Atmos. Sci. 34, 1619-1633. Orlanski, I., and Ross, B. B. (1984). The evolution of a cold front. Part 11. Mesoscale dynamics. J . Atmos. Sci. 41, 1669-1703. Palmen, E., and Newton, C. W. (1969). “Atmospheric Circulation Systems: Their Structure and Physical Interpretation (Int. Geophys. Ser., Vol. 13). Academic Press, New York. Pedlosky, J. (1979). “Geophysical Fluid Dynamics.” Springer, New York. Pettersen, S . (1956). “Weather Analysis and Forecasting,” 2nd ed., Vol. I. McGraw-Hill. New York. Phillips, N. A. (1956). The general circulation of the atmosphere: A numerical experiment. Q. J . R . Meteorol. Soc. 82, 123-164. Rao, G. V. (1966). On the influence of fields of motion, baroclinicity, and latent heart source on frontogenesis. J. Appl. Meteorol. 5 , 377-387. Ross, B. B., and Orlanski, I. (1978). The circulation associated with the cold front. Part 11. Moist Case. J. Atmos. Sci. 35, 445-465. Ross, B. B., and Orlanski, I. (1982). The evolution of a cold front. Part I. Numerical simulation. J. Atmos. Sci. 39, 296-327.

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Sanders, F. (1955). An investigation of the structure and dynamics of an intense surface frontal zone. J. Mefcorol. 12, 542-552. Sawyer, J. S. (1952). Dynamical aspects of some simple frontal models. Q. J. R. Meteorol. SOC.78, 170-178. Sawyer, J.S. (1956). The vertical circulation at meteorological fronts and its relation to frontogenesis. Proc. R. SOC. London, Ser. A 234, 346-362. Solberg, H. (1928). Integrationen des atmosphanschen Storungsgleichungen. Geophys. Norv.5, No. 9. Williams, R. T. (1967). Atmospheric frontogenesis: A numerical experiment. J. Atmos. Sci. 24, 627-641. Williams, R. T. (1974). Numerical simulation of steady-state fronts. J. A m o s . Sci. 31, 12861296. Witham, G. B. (1974). “Linear and Non Linear Waves.” Wiley, New York.

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TROPICAL DYNAMICS

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NUMERICAL MODELING OF TROPICAL CYCLONES YOSHIOKURIHARA Geophysical Fluid Dynamics Laboratory1 NOAA Princeton University Princeton, N e w Jersey 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Numerical Models o f Hurricanes . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Modeling Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. GFDL Hurricane Models . . . . . . . . . . . . . . . . . . . . . . . . . 3. Numerical Simulation of Tropical Cyclones . . . . . . . . . . . . . . . . . . . . 3.1. Tropical Storm Genesis. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Intensification of Tropical Storms . . . . . . . . . . . . . . . . . . . . . . 3.3. Structure of Hurricanes. . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Spiral Bands and Comma Vortices . . . . . . . . . . . . . . . . . . . . . 3.5. Landfall of Hurricanes . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Some Challenging Issues in the Future. . . . . . . . . . . . . . . . . . . . . . 4.1. Improvement of Numerical Models . . . . . . . . . . . . . . . . . . . . . 4.2. Basic Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Prediction of Tropical Cyclones . . . . . . . . . . . . . . . . . . . . . . Appendix. GFDL Hurricane Models. . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. INTRODUCTION A tropical cyclone is an intense, wet vortex that develops over the warm ocean in the tropics. Its life cycle begins at the formation of a tropical depression from an easterly wave, in the intertropical convergence zone, or from cloud clusters. When a tropical depression develops into a tropical cyclone, with maximum sustained surface wind larger than -17 m SKI,it is called a tropical storm. The more intense second stage (the surface wind larger than -33 m s-I), though not always reached, is called a hurricane, typhoon, or cyclone, depending on the location where it is observed. A tropical cyclone generally moves a long distance, sometimes in a meandering track. The life cycle of a tropical cyclone ends either through its decay, often after the landfall, or its transformation into an extratropical system. Statistics (Gray, 1981) indicate that about 80 tropical cyclones are observed on the globe per year where the maximum sustained (1-min mean) surface wind exceed 20-25 m s-I. Nearly one-half to two-thirds of these cyclones eventually reach hurricane strength. In these violent vortices, s-I, which is the relative vorticity near the center is on the order of 255 ADVANCES I N GEOPHYSICS, VOLUME

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-100 times the value at the formative stage; the corresponding rotation rate is 10 times faster than the rotation rate of the Earth. In an extreme case such as super typhoon Tip, the minimum surface pressure dropped to 870 mb with an estimated maximum surface wind of 85 m s-l reported on October 12, 1979. Tropical cyclones make an enormous impact in various ways on the lives of the people in the affected areas. A large number of investigators in the past and present have been attracted by the behavior of tropical cyclones. They studied the characteristics of these intense vortices through observational analysis, theoretical study, and numerical modeling. In the review articles and books [e.g., Anthes (1982), Gray (1981), Ooyama (1982)], one can see the advancement of our knowledge on tropical cyclones and at the same time find some outstanding problems that remain. At the Geophysical Fluid Dynamics Laboratory (GFDL), National Oceanic and Atmospheric Administration (NOAA), the Hurricane Dynamics Project was established in 1970 after a research effort had developed over several years concerned with tropical meteorology. The formal decision to organize such a project was made by Joseph Smagorinsky, then director of the GFDL, under strong support of Robert White, then the administrator of the NOAA. In this chapter, the author intends to document the accomplishment of this project by presenting the obtained results on selected topics. Although the interpretation of the results carries this author’s subjectivity, it is hoped that the readers will learn what the numerical modeling effort has added to our understanding of hurricanes. The goal of the hurricane research at GFDL is twofold: (1) to understand basic mechanisms of the evolution of a hurricane during its whole life cycle and (2) to investigate the capability of numerical models in the hurricane prediction. Research with the former objective may be characterized as an analysis of particular phenomena or processes and the latter as a synthesis of knowledge on a variety of disciplines. Although the two goals are set, each piece of research work actually serves for both purposes; the degree of contribution to each objective is dependent on the main theme of the work.

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2. NUMERICAL MODELSOF HURRICANES 2.1. Modeling Problems

A tropical cyclone is a system that is surrounded and strongly influenced by its larger-scale environment. On the other hand, it contains

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various small-scale structures, and the convective activities within the system are essential for the development and maintenance of the system. Indeed, interaction among the synoptic scale, the scale of the hurricane vortex, and the scale of cumulus convection is a unique feature of hurricane dynamics. Another important aspect of hurricane dynamics is that it is strongly connected to the boundary-layer physics, including the interaction at the sea or land surface. In constructing a numerical simulation model to study hurricanes, we have to carefully consider the above conditions. Then, we inevitably face a difficult problem, i.e., how to treat effects of the phenomena that cannot explicitly be simulated in a model and yet cannot be ignored. Suppose that we use the primitive-equations system to calculate the time changes of the wind, temperature, surface pressure, and mixing ratio of the water vapor in a hurricane model. Because of the choice of the governing equations, we cannot exactly describe the convective-scale processes for which the nonhydrostatic equations system are more appropriate. In addition to such a scale limit due to the governing equations used, the model's capability to resolve a phenomenon is also limited by the spatial resolution of the numerical model. We presume that a model with a horizontal grid distance of d can only resolve a vortex of diameter larger than approximately 3d or a wave of wavelength larger than 6d. Figure 1 is a schematic diagram showing predictability of a hurricane model. The time elapsed after the start of the time integration of the model is taken along the abscissa, and the space scales involved in the model are km

2000

-

deterministic

0

-0

E 0

200

"0 ._ L

L

a

L

I

t=O

1 day

I

time

2 days

FIG.I . Treatment of different spatial scales in the time integration of a numerical model.

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indicated on the ordinate. The scale L 1 indicates a typical scale of moist convection for which a direct application of the primitive equations system is not valid. The scale L2 (L2 > L 1 )is the smallest resolvable scale of the model; i.e., Lz = -3d. An important feature of the time integration of a hurricane model is that the effects of the phenomena with the scale less than L1 have to be incorporated into the model by some means in the course of the integration. Ordinarily, such effects as those due to the moist convection, eddy diffusion, and surface exchange of heat, moisture, and momentum are estimated by a scheme of parameterization. With the inclusion of the parameterized effects, the meteorological fields resolved by the model are modified at each step of integration. As the time integration proceeds, small-scale fields initially in the resolvable range are likely to become undeterministic because these fields are usually less persistent and more sensitive to the behavior of individual convective elements as compared with larger-scale fields. The dotted line in Fig. 1 represents the predictability limit, or the smallest deterministic scale, as a function of the integration period. The fields corresponding to the scales between the above dotted line and line L2 are probably meaningful only in a probabilistic sense. We note that there generally exists a gap between the scales L 1 and L z , It is difficult to estimate the effects of the mesoscale phenomena belonging to this gap. The line L2 can be lowered down to the level of L1in the limit by increasing the spatial resolution of the model. 2.2. GFDL Hurricane Models The hurricane simulation models constructed so far at the GFDL are governed by the primitive equations. Components of this equation system, and the connections among them are illustrated in Fig. 2 . A brief RADIATIVE TRANSFER

CONDENSATION OF WATER VAPOR

r c

,

1

CONDITION AT EARTH'S SURFACE

FIG.2. System of governing equations and the interrelationship among the components. [From Kurihara and Tuleya (1981). From Monthly Weuther Review, copyright 1981 by the American Meteorological Society .]

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description is given about the parameterization schemes used in the currently working models. The literature references that contain detailed descriptions of the model physics and numerical schemes used are summarized in the Appendix. In our model, the vertical fluxes of the momentum, the sensible heat, and the moisture in the surface constant flux layer and their exchange at the sea or land surface are estimated by using the Monin-Obukhov-type formulation. The roughness length at the surface is given by Charnock’s relation over the sea, while it is specified over the land. The jump of the temperature and the mixing ratio at the interfacial layer is allowed for by the use of Sasamori’s scheme (1970). The vertical mixing of momentum, temperature, and moisture above the constant flux layer is treated by the level-2 formulas of the turbulence closure model derived by Mellor and Yamada (1974). The turbulent mixing becomes strong under the unstable stability condition. The horizontal mixing due to the turbulence is parameterized by the nonlinear viscosity scheme given by Smagorinsky (1963). The effects of the ensemble moist convection are incorporated into the model through a method of moist convective adjustment (Kurihara, 1973). In the model, moist convection takes place if a hypothetical cloud can develop while entraining the surrounding environmental air. The size of the cloud used in the entrainment formulation is dependent on the local relative humidity. The adjustment of the vertical profiles of temperature and moisture is made through condensation and the vertical transport of heat and moisture, with the sum of the total potential and latent energy conserved. This is done so that the ultimate state is just marginal for cloud development. In our scheme, it is assumed that the adjustment to the ultimate state is achieved not instantaneously but in a certain relaxation time. The model atmosphere is divided into 1 1 layers, each containing a representative level. Positions of these levels in the CT coordinate, where u is the pressure normalized by the surface value, and their approximate heights are listed in Table I. The lowest three or four layers are in the planetary boundary layer, i.e., an important region in the hurricane dynamics. The horizontal spacing between the grid points should be small TABLEI . T H E(r LEVELSA N D APPROXIMATE HEIGHTS

Level J

I

2

3

4

5

6

7

8

9

10

II

0.031

0.120

0.215

0.335

0.500

0.665

0.800

0.895

0.950

0.977

0.992

8.33

5.48

3.31

1.84

0.926

0.435

0.196

0.068

Height

(km) 23.6

14.9

11.2

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sional moist models with different degrees of sophistication in cloud physics parameterizations. Notably, Takeda was successful in showing that a model convective cell attained a quasi-steady state when the jetlike vertical profile of the ambient wind (i.e., reverse shear) was assumed. Hane also showed that his thunderstorm cell was long-lived when the ambient wind has low-level shear and no shear at middle and upper levels. Meanwhile theoretical investigations were advanced for the structure of steady-state, inviscid, two-dimensional convective circulations in the sheared environment by Moncrieff (1978, 1981) and Moncrieff and Green (1972). They were able to drive several conserved quantities. One of their conclusions is that steady convection cannot exist under assumptions of constant shear and deep downdrafts, consistent with the preceding simulation results. Further, radar observations have accumulated evidence that shows that generation of new convective cells at boundaries of cold outflows formed by downdrafts is an integral component of structures and movements of multicell storms [e.g., Chalon et al. (1976) and Fankhauser (1976)l. Formation of new cells was simulated in the three-dimensional models of Miller (1978), Thorpe and Miller (1978), Clark (1979), Tripoli and Cotton (1980), and Wilhelmson and Chen (1982). All these experiences in numerical studies and knowledge gained by theoretical and observational work have culminated in two numerical models that were able to simulate qualitatively some aspects of the observed squall lines even in the two-dimensional framework (Thorpe et al., 1982; Soong and Chen, 1984). First let us look at the observed air motion in and around a squall line that developed over central Oklahoma (Ogura and Liou, 1980). The air motion shown in Fig. 5 was determined by compositing rawinsonde data gathered from the dense mesoscale network, thus representing the stormscale rather than the cell-scale airflow. The streamlines of Fig. 5 show a familiar upshear slope of the updraft. It further shows the dual updraft structure: the right-side circulation involves overturning, whereas the left-side does not. For this reason, Thorpe et al. (1982) labeled these two updrafts “overturning” and “jump-type,” respectively. However, this squall line accompanied an anvil only in the rear of the leading edge of the line. Heavy precipitation was confined in a narrow zone near the leading edge of the line, followed by weak precipitation from the trailing anvil. This feature is similar to those analyzed by Sanders and Paine (1975), Sanders and Emanuel (1977), and Houze and Smull(1982). Figure 5 also shows a strong front-to-rear flow located about 40 km behind the leading edge of the line at the 500-600-mb levels. A similar enhanced flow was also found in a tropical cloud cluster (see Fig. 7 later in this chapter). Analyzing the aircraft data gathered for a convective line

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x (krn)

FIG.5 . Streamlines in a plane normal to an Oklahoma squall line. Solid arrows indicate the positions of the local maxima of wind compomnt normal to the squall line relative to the propagating squall line. Negative and positive values indicate inflows and outflows, respectively, in units of meters per second. Open arrows are for the vertical velocity. Negative and positive values indicate upward and downward motion, respectively, in units of W 3mb s-I; x represents distance ahead of the leading edge of the squall line. [From Ogura and Liou (1980). From Journal of Atmospheric Science, copyright 1980 by the American Meteorological Society.]

during the GARP (Global Atmospheric Research Programme) Atlantic Tropical Experiment (GATE), LeMone (1983) showed that a mesolow was located in the rear half of the leading convective region. The enhanced flow at middle levels may have been caused by the front-to-rear pressure gradient force associated with this mesolow. Another interpretation is that this enhanced flow is the reflection of the cloud top outflow of middle-level clouds that develop above the surface gust front. In the case under consideration, this enhanced front-to-rear middle-level flow meets with air entering into the storm from the rear. The resulting convergence (and consequently the ascending motion) sustains the wide anvil in the rear. A similar air flow at midlevels was also obtained by Gamache and Houze (1982) and Johnson and Kriete (1982) for tropical cloud systems. [See also Houze and Betts (1981) for earlier references.]

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Thorpe et al. (1982) performed two-dimensional simulations of moist convection for various ambient wind profiles. They discussed in detail one of the cases in which the wind profile had a constant shear up to the 2.5-km height and no shear above this level. Quasi-steady convection was produced. Numerous shallow- to middle-level cells were continuously generated behind the leading edge of the line, followed by a major updraft. Figure 6 shows the result of their simulation averaged over 90 to 170 min. Obviously, some observed features described earlier are well simulated qualitatively. Figure 7 is a schematic diagram showing a slowly moving tropical convective band observed in GATE (Zipser et al., 1981). Figure 8 is a twodimensional numerical counterpart (Soong and Chen, 1984) in which a simple ice-phase parameterization scheme suggested by Cotton et al. (1981) was incorporated. Both Figs. 7 and 8 show that new cells systematically form on the leading edge of the line. As the cells mature, they become the main cells of the line. Dissipating cells occur to the rear of the mature cells. The result shown in Fig. 8 was obtained for the vertical profile shown in Fig. 8. From these numerical results, it is now apparent why the particular vertical profiles of the ambient wind employed in these experiments are favorable for sustaining quasi-steady squall lines. A low-level shear represents a low-level inflow relative to the propagating squall line. New cells form at the location where the inflow meets with the gust front formed by the downdraft of the old cells. Once a new cell forms, it develops most vigorously when the shear is weak at middle and upper levels. Unless the cell develops vigorously, there would be no strong downdraft and outflow of cool air to generate new cells. This will bring up a question about the propagation speed of squall lines. In Soong and Chen's experiment, the leading edge of the line moved to the west (left of the diagram, Fig. 8) with the speed of 4 m s-l without the ice phase and 7 m s-I with the ice phase during the last hour of their simulation (2-3 hr). It is not immediately clear why the inclusion of the ice phase resulted in such a large difference in the propagation speed. The point here is that since the ambient wind is westerly, the air entrains into the cloud system at all altitudes, as is the case in many tropical squall lines. One of the characteristics of tropical squall lines is that they propagate fast. Typical propagation speed of squall lines over the GATE area is 15 m s-' (Aspliden et al., 1976). The speed of the squall lines documented by Fortune (1980) was as large as 15-25 m s-'. It remains unanswered whether successive formation of cells at the gust front alone can explain this high velocity. In an experiment related to this problem, Wilhelmson and Chen (1982)

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b*

-2 0

4

HORIZONTAL DISTANCE (km)

FIG.8. Cross section of a two-dimensional model squall line at 3 hr. The stippled region is cloud; solid and dashed lines are for mixing ratios of cloud droplets and cloud ice with contour intervals at 1 g kg-I; vertical thin lines indicate heavy precipitation areas (mixing ratios of rain water or graupel greater than 1 g kg-I); and outside scalloped contour outlines boundaries of the entire precipitating area. [From Soong and Chen (1984).]

were able to simulate successive formation of five discrete new cells in agreement with the observations of the well-documented Raymer storm that occurred on 9 July 1973. However, the model cells formed at intervals of 30 min, whereas the observed cells formed at 15-min intervals. This interval of 30 min was needed for each model cell to develop, become mature, generate a downdraft, and finally stimulate formation of a new cell. How then could the real Raymer storm generate discrete radar echoes at 15-min intervals? In Clark’s experiment (1979), new cells formed ahead of the gust front. Clark speculated that formation of new cells was partly due to gravity waves set off by an upper-level downdraft. No detailed discussion was given, however. In Wilhelmson and Chen’s experiment, no appreciable velocities occurred ahead of the gust front that would suggest the presence of gravity waves. Speaking of gravity waves, another theory of propagation of squall lines was proposed by Raymond (1975, 1976) and extended by Silva-Dias (1979). This theory considers the entire storm as a gravity wave forced by “wave CISK.” [See Lilly (1979) for further discussion.] Up to this point, we have discussed mainly two-dimensional simulations of squall lines. This implies that the vertical wind shears in these experiments are unidirectional. Since the mid-l970s, three-dimensional modeling studies of severe storms have been greatly advanced. This subject is discussed in a separate chapter in this book. For completeness of our discussion, we note that the directional shear at low levels is found to

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be critically important in determining the structure and evolution of severe storms.

5.4. Life Cycle of Mesoscale Convective Systems Mesoscale convective systems generally experience life cycles. As an example, Fig. 9 shows the life cycle of a tropical rainband observed during GATE in terms of vertical velocity w estimated from the rawinsonde wind observations. Since the average separation between the neighboring observing ships was a few hundred kilometers in this case, w in Fig. 9 should be regarded as the vertical velocity averaged over an area encompassing the rainband. The similar evolution of the w field was also observed in a mid-latitude convective line (Ogura and Chen, 1977), tropical cloud clusters (Frank, 1978; Betts, 1978; Ogura et al., 1979; Sikdar and Hentz, 1980), easterly waves (Reed et al., 1977; Thompson et al., 1979; Chen and Ogura, 1982), and in mid-latitude mesoscale convective complexes (Maddox, 1980, 1983). Outstanding features of the evolution of the w field common in almost

TlME (GMT)

FIG.9. Time-pressure section of the vertical p velocity w at the center of the rainband that occurred on 12 August 1974 over the eastern Atlantic; units in 10 mb S-I. [From Ogura et a / . (1979). From Monthly W m t h c r Reuiew. copyright 1979 by the American Meteorological Society.]

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407

all of the studies are as follows: (1) Prior to the onset of the organized rainband, low-level ascending motion was present, whereas subsidence was present at middle and upper levels. This low-level ascending motion triggers overturning of the conditionally unstable atmosphere. (2) As deep clouds developed, o increased in intensity and the height of the local maximum of o also increased. (3) The decaying stage was characterized by the marked weakening of the ascending motion at low levels, followed by the development of descending motion, while the upper-tropospheric upward motion remained substantial. Up to the present, much of the attention of mesometeorologists has been focused on the initiation and development processes of MCSs and less attention has been paid to the decaying stage in both observational and theoretical studies. Consequently, physical processes responsible for the dissipation of MCSs are less well understood. The decaying stage should be as important as other stages of the life cycle because it determines the longevity of MCSs. For practical purposes, Zipser (1982) noted that knowledge of the life cycle of MCSs will be helpful to increase the accuracy of nowcasting beyond the simple extrapolation. An easy answer to the question why MCSs dissipate would be that MCSs dissipate when they propagate to a region in which the environmental conditions are not favorable to sustain them. Indeed, the model squall lines shown in Figs. 6 and 8 organized themselves so that they would keep propagating forever. However, the cloud clusters sampled by the GATE ship array were found to be predominantly nonsquall clusters. They have lifetimes of a day or less (Martin, 1975; Martin and Schreiner, 1981). Occasionally clusters developed strong rotational flow and evolved into longer-lived tropical storms or hurricanes. Why then are the lifetimes of the majority of nonsquall clusters limited to such short periods in spite of the fact that conditionally unstable layers extend over very large portions of the tropical oceans? Soong and Ogura (1982) simulated the evolution of a cloud cluster with resolvable convective cells with partial success. The limitation of their experiment, which was two-dimensional, was that the domain of their model cluster was too narrow to adequately treat the dissipating stage. Nonetheless, the dissipation stage appeared to occur in the model when the supply of water vapor, by moisture flux from the ocean surface and by horizontal advection associated with an induced mesoscale circulation in the lower troposphere, was not sufficient to compensate the drying effect of deep clouds. Simpson and van Helvoirt (1980) discussed the interactions between the clouds and subcloud layer in GATE by means of a three-dimensional cumulus model. Cumulus downdrafts were postulated to be the main interaction mechanism on the scales of interest.

408

YOSHl OGURA

Returning to the w field at the mature stage shown in Fig. 9, Ogura (1982) noted that the large w with its local maximum located in the upper troposphere should be interpreted as a mere reflection of a group of growing or already well-developed deep convective clouds, rather than that large-scale upper-tropospheric upward motion forced development of deep convection. We noted earlier that during the decaying stage, the upper-tropospheric upward motion remained substantial but decreased markedly in the lower troposphere. This upward motion may be interpreted as a reflection of a mesoscale circulation that developed concurrently with, or as a result of, latent heating (see Section 6). This occurs in the tropics (Frank, 1978; Leary, 1979; Ogura et al., 1979) as well as in the mid-latitudes (Maddox, 1980; Bosart and Sanders, 1981). In other words, there was a transition of horizontal scales in upward motion, from cloud scale to mesoscale during the developing stage through the decaying stage. This interpretation is consistent with the life cycle of MCSs as viewed by radar observations; i.e., a group of isolated cloud radar echoes observed in the formative stage of nonsquall cluster transforms into stratiform precipitation areas in the dissipative stage. During the dissipating stage the contribution to vertical motion by boundary layer-rooted convective-scale (1-10-km) clouds is comparatively small (Houze et al., 1981; Leary and Houze, 1979). If the preceding interpretations are correct, it is not surprising to find, as Yanai et al. (1976) actually did for tropical Pacific convective systems, that deep convection is best correlated to upper-tropospheric vertical motions. This implies, however, that this high correlation has no use for predicting the occurrence of deep convection simply because the uppertropospheric upward motion is a consequence, not a cause, of deep convection. In order to avoid misunderstanding, it is remarked that synoptic scale w, though small in magnitude, can destabilize the atmosphere and produce favorable conditions for the enhancement of convection. This may have made some contribution to the high correlation between upper-tropospheric vertical motion and deep convection mentioned before. However, this author believes that this contribution is small statistically in the tropics. 6. FEEDBACK EFFECTSOF CUMULUS CLOUDS ON LARGER-SCALE ENVIRONMENTS Another area in which the modeling study of atmospheric convection has proved useful is in the study of the feedback effects of a group of

MODELING STUDIES OF CONVECTION

409

cumulus clouds on the large-scale environment in which the group of clouds is embedded. Processes through which cumulus clouds modify their environment include condensation of water vapor; evaporation from liquid and ice particles; vertical transport of various variables such as heat, moisture, momentum and vorticity by cloud updrafts and downdrafts. Thus this study is intimately related to laying the scientific foundation for cumulus parameterization. The full discussion of the cumulus parameterization problem is beyond the scope of this article. The reader is referred to the latest review article by Frank (1983). Soong and Ogura (1980) developed a model that they called a cumulus ensemble model. Basically, it is a nonhydrostatic anelastic moist cloud model. However, unlike many other cloud models that are aimed at simulating isolated convective clouds [models of Hill (1974) and Yau and Michaud (1982) are exceptions], the cloud-ensemble model allows several convective clouds to develop simultaneously inside the model domain. The objective of this model has been to investigate statistical properties of cumulus clouds that occur as a result of imposed large-scale forcing. In this respect this model may be regarded as an ultrafine mesh, ultralimited area model nested in a fine mesh, limited-area prediction model. The model was first applied to shallow moist convection by Soong and Ogura (1980) and to deep moist convection by Soong and Tao (1980). Tao (1983) and Soong and Tao (1984) extended their research in several aspects including a discussion of vertical transport of momentum by cumulus clouds. In general, the heating and drying effects of clouds may be expressed as (Yanai et al., 1973) (6.la)

(6.2a) (6.2b) , the radiawhere T is the nondimensional pressure given by ( p l P ) R ' " ~Qr tive cooling rate, De and D , the diffusion effects by microscale turbulence on the 6 and qu,respectively, c the rate of condensation of water vapor, and e the rate of evaporation from liquid drops and other notations are conventional. According to the result of Soong and Tao (1980), the effects of De and D, are negligibly small compared to other terms and will not be

6

LENNART BENGTSSON

design will result not only from comparison of the climate simulations produced by different models but also from studies of the growth of climate error in forecast experiments. The fundamental process driving the Earth’s atmosphere is the heating by incoming short-wave solar radiation and the cooling by long-wave radiation to space. The heating is strongest at the tropical latitudes, while cooling predominates at the polar latitudes of at least the winter hemisphere. The bulk of the net incoming solar radiation is absorbed not by the atmosphere, but by the underlying surface. The evaporation of moisture and the heating of the surface lead, however, to much of this energy being transferred to the atmosphere as latent heat and, to a lesser extent, as sensible heat. Thus the dominant direct heating of the atmosphere is found to be the latent heat release associated with deep tropical convection. The atmosphere’s meridional energy transfer across middle latitudes is accomplished largely by transient weather systems with a time scale on the order of days, which themselves develop due to “baroclinic” instability (Charney, 1947; Eady , 1949) of the predominantly zonal flow set up by the differential radiative heating. This flow is further perturbed, especially in the Northern Hemisphere, by orographic effects and land-sea contrasts (Charney and Eliassen, 1949; Smagorinsky, 1953). These have a strong local influence and on a larger scale lead to a concentration of baroclinic transfer along the storm tracks of the North Pacific and Atlantic Oceans (Blackmon et al., 1977). Lower-frequency variability, of which the quasi-stationary “blocking” anticyclones are just one manifestation, also occurs both in the extratropics and in tropical circulation systems. It may arise either from slowly varying boundary conditions such as sea-surface temperature (e.g., Bjerknes, 1966, 1969; Namias, 1969) or from variability in the dynamical response to a fixed external forcing (e.g., Charney and DeVore, 1979; Wiin-Nielsen, 1979; Simmons e l al., 1983). Many other processes are important in determining the detailed behavior of the atmosphere on time scales of more than a few days, as illustrated schematically in Fig. 1 . First, turbulent vertical transfers of heat, moisture, and momentum influence all larger scales of motion and are determined both by the nature of the underlying surface and by the nature of the larger-scale flow itself. The type of surface, as characterized by its albedo, also determines the proportion of incoming solar radiation that is reflected back toward space. Latent heat release not only helps drive the mean circulation but can also be a significant component of synoptic-scale systems, in middle latitudes as well as the tropics. The related clouds play an important role in reflecting incoming short-wave radiation and in absorbing and emitting long-wave components.

MEDIUM-RANGE FORECASTING AT THE ECMWF

7

3. NUMERICAL METHODSAND MODELING TECHNIQUE 3.1. Basic Equations Models used for medium-range forecasting are normally based on a set of equations known as the primitive equations. As discussed by Phillips ( 1973), the governing dynamic, thermodynamic, and conservation equations are mapped to a spherical geometry, and the reduced set of primitive equations is obtained by assuming the height scale of the motion to be small compared with its horizontal-

8

LENNART BENGTSSON

length scale, an acceptable approximation for horizontal scales upward of tens of kilometers. The basic predicted variables are the horizontal wind components of u and u , temperature T , water vapor as represented usually by the specific humidity q, and surface pressure p s . We illustrate the form of the primitive equations using the vertical coordinate system in most common use, namely, the “sigma” coordinate system proposed by Phillips (1957), for which the vertical coordinate o is given by Cr =

PIPS

(3.1)

where p is pressure. In this case the equations become

Momentum

OY +fk X V v+ + R d T V h p , = P, + K , Dt

(3.2)

Thermodynamic (3.3)

Moisture conservation Dq = P, Dt

+ K,

(3.4)

Mass conservation

p& + PS(V v + ”)= 0 Dt au *

(3.5)

Hydrostatic

Here t is time and DIDt denotes the rate of change moving with a fluid particle, which in u coordinates takes the form

Here v is the horizontal velocity vector, v = (u, u, 0 ), and V is the two-dimensional gradient operator on a surface of constant o;f is the Coriolis parameter; k is the unit vertical vector; 4 is the geopotential; Rd is the gas constant for dry air; cpd is the specific heat of dry air at constant pressure; P, denotes the rate of change of variable X due to the

MEDIUM-RANGE FORECASTING A T THE ECMWF

9

parameterized processes of radiation, convection, turbulent vertical mixing, and large-scale precipitation. Finally, K , represents the rate of change of X due to the explicit horizontal smoothing that is usually included in models to prevent an unrealistic growth of the smallest resolved scales. The latter term would ideally be regarded as representing the influence of unresolved scales of motion on the explicitly predicted scales and treated as part of the parameterization. In practice, since the smallest scales in a model are inevitably subject to numerical misrepresentation, it is common to choose empirically a computationally convenient form of smoothing and thereby eliminate unrepresentative noise. A predictive equation for surface pressure is obtained by integrating Eq. (3.5) from u = 0 to u = 1, using the boundary conditions u = 0 at u = O a n d u = 1:

* at

=

-lo'V -

(p,v) dm

(3.8)

Vertical velocities are not explicitly predicted, but they, too, can be deduced from Eq. (3.5). The sigma- and pressure-coordinate forms u and o are given by

and (3.10) The form of the primitive equations already given neglects the local mass of water vapor compared with that of dry air, an approximation that can introduce a small, but not always negligible, error in moist tropical regions. It is thus common to use a more accurate form in which the temperature appearing in Eqs. (3.2) and (3.6) is replaced by the virtual temperature Tv, which is defined by

where R, is the gas constant for water vapor. In addition, the second term on the left-hand side of Eq. (3.3) may be multiplied by a factor

10

LENNART BENGTSSON

where cpv is the specific heat of water vapor at constant pressure. As parameterization schemes become more complex, additional equations may be added to this basic set. Sundqvist (1981) has developed a formulation for a separate prediction of liquid water. A type of boundary-layer parameterization that uses the turbulent kinetic energy as a predicted variable has been tested by Miyakoda and Sirutis (1977).

3.2. Numerical Formulation It will not be possible here to give a comprehensive and systematic presentation of models used in medium-range forecasting and large-scale modeling. For a more detailed review reference is made to Simmons and Bengtsson (1984). Such a review shows that models differ more in details than in overall design. Today's models are more generally designed than those of yesterday and new ideas developed in one research group are often tested and evaluated in other modeling groups. The intensive international cooperation within the GARP and the WCRP is playing a very important role. Since we essentially will present results obtained

-Ah-

FIG.2. The horizontal distribution of variables in the ECMWF grid-point model. In the current operational system, AX = A0 = 1.875".

11

MEDIUM-RANGE FORECASTING AT THE ECMWF

from ECMWF models, we will here summarize the main features of the ECMWF’s operational models. The first model developed at the ECMWF was a grid-point model. The finite-difference scheme was based on a staggered grid of variables known as the C-grid (Arakawa and Lamb, 1977) (Fig. 2). This model conserves potential enstrophy during vorticity advection by horizontal flow (Sadourny, 1975; Burridge, 1979), an integral constraint that was found to be of importance in long-term integrations. The model had 15 vertical levels and a horizontal resolution of 1.875” IatitudeAongitude. After 3.5 years of operational use, the grid-point model was replaced in April 1983 by a spectral transform model. A substantial evaluation (Girard and Jarraud, 1982) demonstrated that the spectral transform model gave more accurate forecasts for the same computational cost. The new model also uses a generalized vertical coordinate that is gradually transformed from u surfaces in the lower troposphere to p surfaces in the stratosphere (Fig. 3). The mathematical formulation of the spectral transform technique adopted for the model largely follows the approach of the first multilevel spectral model of Bourke (19721, Hoskins and Simmons (1975), and Baede

Pressure (rnb)

75 128 185 250

325 407 496 584 677 769 846 910 955

983 996

4=& FIG.3. Vertical distribution of variable in the ECMWF spectral model. The pressure levels given at the right side of the figure show the levels at which variables are represented in the 16-level resolution of the model, assuming a surface pressure of 1000 mb.

12

LENNART BENGTSSON

et al. (1979) and hence uses vorticity and divergence as predictive variables. These equations are easily derived from Eq. (3.2). The predicted variables are represented in terms of truncated expansions of spherical harmonics M

X(A,e,u,t) =

N(m)

C C m=-M

X;(u,t)p;(sin

t9)e;mh

(3.13)

n=lm/

where X is any variable, 0 is latitude, A is longitude, and the p ; are the associated Legendre polynomials. The code is written in a very flexible way and different resolutions and representations (triangular-or rhomboidal-truncation) can be implemented (Fig. 4). The truncation adopted operationally is triangular at wave number 63. The computations in physical space are performed on a grid having 192 points distributed regularly around each of 96 Gaussian latitudes. The resolution in the physical space is thus practically identical to the grid-point model. The associated parameterization scheme (Tiedtke et al., 1979) described the processes thought to be of importance in the medium range. These include a full hydrological cycle, a comparatively detailed representation of turbulence fluxes, and an interaction between radiation and model-generated clouds.

FIG.4. The region of wave-number space (shaded) for which spectral components are retained in triangular truncation.

MEDIUM-RANGE FORECASTING AT T H E ECMWF

13

3.3. Radiation and Clouds Radiation is parameterized with emphasis on those processes that are most likely to affect the atmospheric flow in medium-range time scales, and consequently cloud-aerosol effects are given high priority. In the present version of the radiation code the gaseous absorption is considered as a perturbation term against other effects, such as Raleigh scattering and scattering and absorption by clouds and aerosols. Radiative fluxes are calculated for five spectral intervals [two in the short-wave part of the spectrum (solar radiation) and three in the long-wave part of the spectrum (terrestrial radiation)]. The effects of water vapor, ozone, C 0 2 , and selected aerosols are included. For further details, see Geleyn and Hollingsworth (1979). The cloud cover in the model is specified as a function of the grid-scale relative humidity and height according to empirical relations (Geleyn et al., 1982). Frontal cloud systems are usually well predicted, while thin stratiform clouds in the boundary layer are not. The surface albedo used for the solar radiation has been calculated using NOAA-ERB data (Winston and Krueger, 1978)from June 1975 to May 1976. Surface albedo is modified by snow on the ground (analyzed daily and hence prescribed in the initial state) and is also changed during the course of the integration (falling and melting of snow). The most serious deficiency of the present scheme occurs in the presence of small cloud amounts and aerosols, the effects of which are overestimated. An alternative solution based on the principle of exponential sum fitting (ESFT) (Wiscombe and Evans, 1977) is presently being evaluated. The ESFT scheme reduces slightly the tendency of mid-tropospheric cooling associated with the present operational scheme. 3.4. The Planetary Boundary Layer The calculation of boundary-layer fluxes is based on the MoninObukov similarity theory, which assumes that the gradients of wind and internal energy are universal functions of a stability parameter (to be defined) or empirically determined. Above the lowest level, turbulent vertical fluxes are represented as the product of eddy diffusivities and the vertical gradients of the explicitly resolved fields. The diffusivity depends on the local stability as given by the Richardson number and the roughness length and uses a mixing length that does not vanish in the free atmosphere. An analytic form of the stability function (one for stable and another for unstable stratification) is determined from data (Louis, 1979).

14

LENNART BENGTSSON

The normalized drag coefficients are determined in such a way that thecoefficient for heat is less than that of momentum for positive Richardson numbers (stable flow) and the opposite for negative Richardson numbers (unstable flow). Roughness length varies over land, where it depends on the sub-grid-scale orography. Over sea the roughness length is given by the Charnock formula. 3.5. Moist Conuection The cumulus convection scheme used in the operational model is based on Kuo (1974) whereby the occurrence of convection depends not only on the existence of an unstable lapse rate, but also on a net convergence of moisture caused by large-scale flow and surface fluxes. A specific parameter b is defined that determines the fraction of moisture that is used for condensation in the cloud and hence is used for heating the atmosphere. The remaining part 1 - b is used for moistening the environmental air. The specification of b that is used in the ECMWF model follows an approach suggested by Anthes (1977). The parameter b is linearly dependent on the mean relative humidity of the environment such that the moistening is larger in dry air than in moist air. A very small release of latent heat therefore takes place if the atmosphere is dry. Convection can start either from the planetary boundary layer or from any level in the free atmosphere. Evaporation from convective rain is included and follows the proposal by Kessler (1969). Numerical experiments in the tropics have shown a great sensitivity to the specific value of the parameter b. Substantial research efforts in developing and testing different parameterization schemes for deep and shallow convection, including the Arakawa-Schubert scheme (1974), is taking place at the ECMWF. 3.6. Nonconvective Parameterization

The treatment of nonconvective precipitation appears to be one of the simpler elements of the overall parameterization scheme. It is computed after the computation of other dynamical and physical processes that change temperature and water vapor content, and it generally allows for condensation with associated latent heat release of sufficient vapor to keep the relative humidity below a fixed threshold value. The threshold value is set to 100%.

MEDIUM-RANGE FORECASTING AT THE ECMWF

1s

3.7. Surface Values Changes are computed in a number of land-surface characteristics [temperature, moisture, and albedo (snow)] used for the calculation of surface heat, moisture, and momentum fluxes. The initial values for snow are analyzed using synoptic observations, and soil moisture is estimated from observations of precipitation. Precipitation as predicted by the model is classified as rain or snow according to a low-level temperature criterion, and snowmelt (which also influences the surface temperature) and run-off are included as well as changes due to evaporation. The surface roughness varies according to the values of the orography, vegetation, and urbanization and over sea according to the surface wind stress. Sea-surface temperature is analyzed but kept constant during the course of the operational forecasts. For longer integrations, sea-surface temperature and deep soil moisture varies according to climate or as otherwise prescribed. 4.

OBSERVATIONS, THEIR USEAND

IMPORTANCE

As the forecasts are extended in time, the area of influence increases. Forecasts beyond S to 7 days have an area of influence larger than one hemisphere. During certain synoptic events there is a considerable interhemispheric exchange. The Indian monsoon is one example; other, shorter and less regular episodes were observed during the FGGE. For these reasons, global models are used in medium-range forecasts and the requirements for observations to initialize these models encompasses the whole global atmosphere as well as certain aspects of the surface of the Earth. Present observing systems cannot satisfy the observational requirements that ideally would be needed in every grid point of the model consisting of the basic parameters: horizontal wind, temperature, moisture, and surface pressure. Additionally, sea-surface temperature, soil moisture, and snow observations are required. Fortunately, this overwhelming requirement can be relaxed due to the strong dynamical and physical coupling between meteorological observations in space and time. This was clearly stated by Smagorinsky (1969), who demonstrated that a dynamical model was able to reconstitute the humidity field in most of its details within a day or two in an experiment in which humidity initially was given as a zonal mean. Due to a programming error, the ECMWF model ran operationally for a period of

16

LENNART BENGTSSON

several months in 1980 without using humidity observations. This was not noticed since the model was able to generate a moisture field and produce realistic precipitation forecasts. It should be stressed, however, that this essentially only holds in the extratropics when the dynamical forcing is dominating. In the tropics, humidity observations often play a crucial role and, as has been demonstrated by Krishnamurti et al. (1983), are essential for an accurate prediction of tropical disturbances. Numerical experiments (e.g., Hollingsworth et al., 1985; Arpe et al., 1985) have shown that an accurate specification of the wind-and-mass field in the baroclinic zones is crucial for forecast quality, while a detailed analysis of the boundary layer appears to be less important. The explanation is again that the models are able to generate missing information that indicates a considerable degree of redundancy among the large-scale variables in the atmosphere (if they can be regarded as variables in a four-dimensional field). The assimilation of observations in four dimensions using a comprehensive numerical model as a vehicle is therefore a necessary requirement and the development of such systems in the past 10 years has played a very important part in the improvements that have taken place in medium-range forecasts. Moreover, asynoptic data, such as observations from satellites and aircraft, cannot be properly used without four-dimensional data assimilation. The Global Weather Experiment in 1979 had as one of its main objectives the development of a technique for using asynoptic observations with the ultimate purpose of designing an optimum composite meteorological observing system for routine numerical weather prediction. In this section we shall discuss how the data from different observing systems are being used and define their relative importance for medium-range forecasts. The meteorological observing systems have gone through a rapid development in recent years stimulated by the FGGE experiment. Figure 5 shows a typical data distribution for each of the individual observing systems. The present (1984) observational distribution is slightly worse; this is particularly the case for aircraft observations and ocean-drifting buoys. Comprehensive data-assimilation systems have been developed and have been used to assimilate FGGE data (Bengtsson et al., 1982; Stern, 1982). Similar systems are also gradually being implemented for operational numerical weather prediction. Although a fully continuous data-assimilation system is practically possible, there are still advantages to arranging data in time slots. The ECMWF data-assimilation system organizes the data in 6-hr time windows. Successful experiments have been carried out by using a 3-hr time window that for practical purposes can be regarded as a continuous data assimilation. This scheme is outlined

MEDIUM-RANGE FORECASTING AT THE ECMWF

17

in Fig. 6. The global atmosphere is analyzed at 15 levels, 1000-30 mb, four times a day. A data-assimilation step consists of a first guess, a 6-hr forecast from the previous initial state, an analysis procedure using all available observations in a time interval of 5 3 hr, and finally an initialization using the so-called nonlinear normal mode. After the initialization, integration continues for another 6 hr, whereafter the process is repeated. A numerical forecast of any length can be started from any of these initial states. The analysis method is an extension of optimal interpolation (Gandin, 1963) to a multivariate, three-dimensional interpolation of deviations of observations from a forecast field (Lorenc, 1981). This technique allows the consistent use of observations with different error characteristics in their statistical structure and takes into full account the irregular distribution of these observations. Optimum interpolation is hence a convenient procedure to analyze the mixture of different observing systems that we have today. In its present form the analyzed correction of the forecasts (first guess) are locally nondivergent and approximately geostrophic (the geostrophic relationship gradually relaxed toward the equator). If analyzed data are used directly as initial conditions for a forecast, imbalances (even very minor ones) between the mass and wind fields will cause the forecast to be contaminated by spurious, high-frequency, gravity-wave oscillations of much larger amplitudes than are observed in the real atmosphere (Fig. 7). Although these oscillations tend to die away slowly due to various dissipation mechanisms in the model, they may be quite detrimental to the analysis cycle in that the 6-hr forecast is used as a first-guess field for the next analysis. The synoptic changes over the 6-hr period may be swamped by spurious changes due to these oscillations with the consequence that the next analysis time with good data may be rejected as being too different from the first-guess field. For this reason, an initialization step is performed between the analysis and the forecast in order to eliminate these spurious oscillations. The normal modes of the forecast model are linearized with respect to a basic state characterized by an atmosphere at rest and in hydrostatic equilibrium. For each level and for each grid point there are three modes-one Rossby mode and two gravity modes-where only the Rossby modes are observed in the atmosphere with significant amplitude. For a linear system, gravity modes can easily be eliminated by simply projecting the state of the model on these three modes and setting the amplitudes for the gravity waves equal to zero. However, this does not work in the nonlinear case. In this case, the ECMWF has implemented a proposal by Machenhauer (1977) according to which, by means of an

B

r 6

18

5

c

.. ..

6

6

.>

...

e

\o 0.

.

A

A 0 A’ A

A A

A

A

071

A

&aw

0

oBaAo

w_n

..... . .,. .. ,' . . . . ,... .; :... . . . ... ... .. .. ,.. . . . .

. . . . "... ,, ....... . . .. .

A

.*.

4

..d

.

A

’

*

0

. ... .... . . .-. ... . . . . I. I .

-

A s

. . .. . . . .

FIG.5. Coverage of ECMWF FGGE 11-b data. Observation time is 0901 to 1500 GMT on 14 November 1979.

20 DATA OBSERVATION TIME

(Approximate time of opcratiinal suite runs)

LENNART BENGTSSON

I

15012100

t

(17bO)

I

2101-0300

t

(1800)

1

0301-0900

t

(1900,

I

0901-1500

t

FORECAST 1200t6H YT 1800

FIG.6. The ECMWF operational data-assimilation forecast cycle, valid February 1984.

iterative procedure, gravity-mode oscillations are removed by setting the initial time derivatives of the gravity-mode coefficients equal to zero. Convergence is rapid and two iterations are perfectly adequate. In the current version only the first five vertical modes are initialized. The higher modes have very low frequencies and thus do not contribute to the problem of spurious, high-frequency oscillations. The nonlinear forcing terms are computed by running the model itself for one time step at each iteration. Although in principle the nonlinear forcing can include the contribution from physics as well as dynamics, in practice this leads to the immediate divergence of the iteration process. Therefore, an estimate of the physical forcing obtained by averaging a 2-hr integration from an uninitialized analysis is used, independent of the iteration count. The incorporation of the “physical” forcing in the initialization has improved the forecasts in the tropics. Initial errors due to lack of observations or erroneous data are propagated downstream under amplification. If the error patterns have an organized structure and are embedded in an unstable flow, the error can grow fast. Figure 8 shows how an initial error, as obtained as a difference between two independent analyses, affects a wider and wider area as the forecasts are extended in time. Figure 8 illustrates the importance of reducing large initial errors that can occur in areas where there is unsatisfactory data coverage. In the absence of satellite data, such areas are common over large parts of the tropics and the Southern Hemisphere, but also over the large ocean area of the extratropical Northern Hemisphere. Observing system experiments carried out at the ECMWF have demonstrated that satellite and aircraft data together with surface pressure observations only can provide a realistic analysis of the

MEDIUM-RANGE FORECASTING AT THE ECMWF

21

FIG.7. The effect of initialization for the surface-pressure forecast at a grid point located at (A, +) = 0", 41.5". Solid line, forecast from uninitialized analysis; dotted line, forecast from normal-mode initialized analysis.

atmosphere. Figure 9 shows 500-mb height fields for OOZ, 16 November 1979. In Fig. 9a only FGGE data have been used, and in the Fig. 9b only satellite, aircraft, and surface pressure observations have been used. The rms difference is less than 25 m. Table I1 summarizes the results of forecast experiments from different observing systems. The table clearly demonstrates the importance of comprehensive global observations on medium-range forecasting skill. Medium-range forecasts are sensitive to the initial specification of soil moisture and sea-surface temperature, particularly in the tropics. Forecast sensitivity to initial soil moisture specifications was demonstrated by Rowntree and Bolton (1978) for the European area (during summer) and by Bengtsson (1983) for the United States. It was found that a low initial value changes very slowly. Convective precipitation was very much reduced. The effect of snow cover has not yet been evaluated on the medium-range time scale, but will presumably be of local or regional importance. The ECMWF introduced analyzed

22

1207

W N

FIG.8. Growth rate of initial errors. Differences in the 300-mb height field between forecasts with the ECMWF model using the ECMWF and NMC analyses valid OOGMT 18 February 1979. Contour interval, dam. Dashed lines, negative. Solid lines, positive.

24

LENNART BENGTSSON

(a)

FIG. 9. 500-mb height analysis for OOZ 16 November 1979 (a) is based on all FGGE observations and (b) on a “space-based observing system.”

values of snow and soil moisture (estimated from synoptic observation) in the operational forecast at the end of 1983. A minor improvement could be found in the forecast skill.

5.

OPERATIONAL

APPLICATION AND RESULTS

In assessing the results to be presented in this section it should be borne in mind that the ECMWF forecasting system has undergone a number of changes. The greatest change took place at the end of April 1983 when the

26

LENNART BENGTSSON

TABLE11.

USEFUL

PREDICTIVE SKILL IN DAYSFOR THREEDIFFERENT OBSERVING SYSTEMS' Northern Hemisphere 20-82.5"N 1000-200 mb

Southern Hemisphere 20-67.5"s 1000-200 mb

Tropics 35OS-32"N 850-200 mb

7.0

5.5

3.1

5.5

3.5

2.1

5.7

4.7

2.5

~

All FGGE data No satellite or aircraft data No conventional upper air data

0 As a limit for useful predictive skill for the Northern and Southern Hemispheres, we have used an anomaly correlation for the geopotential height of 60%. For the tropics we have used an rms wind error of 7 m sec-I. Although these values may have been selected somewhat arbitrarily, they nevertheless clearly indicate the effect of better observation on the forecasts. The numbers are averaged values for seven forecasts during November 1979 (Bengtsson, 1983).

Model changes have been few. Mean adjustment of the physical parameterization were introduced early in the period, and a more efficient and realistic horizontal diffusion formulation has also been incorporated. A change to a more realistic representation of the orography and coastlines was made in April 1981. Assessment of medium-range forecasts can, in principle, be carried out in three different ways. The first is assessment by objective measures, and for the convenience of this chapter we will concentrate our attention on just one such measure. The second is by subjective evaluation by the meteorologists of the Member States, and the third is by examination of the physical consistency of the forecast, such as the budget of heat and momentum. A question that arises with objective scoring of forecasts is which scores constitute a limit beyond which a forecast is no longer useful. There is no unique answer to such a question, since the level of accuracy required for a forecast to be useful depends very much on the purpose for which the forecast is to be used. At the extreme, a positive anomaly correlation may be an indication that a forecast possesses some skill, but in general a more restrictive criterion is used. Long series of intercomparisons against subjective evaluations indicate that the limiting value is 50-60%. It should be noted, however, that such criteria assume a forecast for one particular instant to be judged against the analysis for precisely that time. A forecast that is poor in synoptic detail or timing may score badly while still providing useful guidance in the overall change of the weather type.

MEDIUM-RANGE FORECASTING AT THE ECMWF

27

IV I II 111 1v I II 111 IV I I1 Ill IV I II 111 IV 1979 1980 1981 1982 1983

FIG. 10. A measure of the skill of the ECMWF Northern Hemisphere forecasts for September 1979 to December 1983. The number of days of predictability is derived from monthly means of daily averages of the anomaly correlations and standard deviation of the errors of geopotential height and temperature forecasts for levels 1000 to 200 mb (height) and 850 to 200 mb (temperature). Solid line, monthly mean values; dashed line, 12-month running mean.

5.1. Northern Hemisphere Forecasts The predictive skill of the model varies considerably in space and time. An idea of the variation in predictability with spatial scale may be gained from Fig. 1la, which shows anomaly correlation for three separate groups of zonal wave numbers for the 500-mb height field. The spectral decomposition demonstrates that larger scales are more accurately forecast in contrast to an earlier experience reviewed by Leith (1 978), when medium scales were reported to be better predicted than the planetary scales. A poorer forecast of shorter synoptic scales is represented here by zonal wave numbers 10 to 20. The latter result may be associated with erroneous timing of intensities of individual weather events within an overall weather situation, which is better predicted, but examples may also be found in which the erroneous forecast of a small-scale feature is followed by a deterioration of the forecast over a much larger area. The spatial variability of predictive skill has also been examined by comparing objective measures over more limited areas. During the wintertime the scores over eastern Asia and North America are often

28

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m=1- 3

m=4-9

1

0

2

5

4

3

7

6

8

9

10

Days 100

1000 mb

1000 mb

1

I

I

(

/

,

I

I

I

l

2

3

4

6

0

7

8

9

10

Days

FIG. 11. (a) Height anomaly correlation scores for wave numbers 1-3, 4-9, and 10-20. ECMWF Northern Hemisphere (20-82.5"N) forecasts, December 1983. (b) Height anomaly correlation scores, ECMWF 50-, 500-, and 1000-mb Northern Hemisphere (20-82.Y") forecasts, December 1983, and persistence scores for lo00 mb.

higher than those for the European area. This is very likely not related to data but to systematic model height errors that are particularly large over the European area. We shall comment on this in Section 6. The predictive skill also varies considerably by height. The lower troposphere is strongly influenced by small-scale features, while the

MEDIUM-RANGE FORECASTING AT THE ECMWF

29

upper troposphere and lower stratosphere are dominated by long waves. Figure l l b shows the anomaly correlation of 50-mb, 500-mb, and 1000-mb height fields calculated for most of the extratropical Northern Hemisphere and averaged over all operational forecasts from December 1983. This shows that 50 mb provide the most accurate forecasts and also that the forecast at 500 mb is generally more accurate than at 1000 mb, a result that is in agreement with synoptic assessment. Useful predictive skill at 500 mb is noticed up to around day 6 and 7. The high predictive skill, at 50 mb during wintertime is interesting to note. Bengtsson et al. (1982) have demonstrated an excellent 10-day prediction of the splitting of the polar night vortex into two separate vortices that took place in February 1979. As can be seen from Fig. 10, there are considerable variations in the temporal variability of the forecasts with a minimum in the summer and a maximum in the winter. In addition, there is a significant interannual variation in the predictive skill, as well as considerable variations within the month. Figure 12 shows examples of two 7-day forecasts for November 1983; a month during which the forecasts varied considerably between the first and second part (see graph). Two forecasts have been selected from these two episodes to illustrate a good forecast (see 1-3) versus a bad forecast (see 4-6), respectively. Figure 13 shows the daily anomaly correlation scores for the first three months of 1982 and 1983, at 500 mb. As can be seen from the figure, there are considerable variations in the prediction skill, and the spells of high skill dominate the second winter. It should be added that only minor changes were made to the model and the data-assimilation system over the period in question. Figure 14 shows for January 1982 the average analyzed for the TOO-mb height field, the ensemble 10-day forecast, and the corresponding deviation from climate. Figure 15 shows the same maps for January 1983. It is clearly seen that the prediction of the stationary component (monthly averages) was very good in 1983 but very bad in 1982. An inspection of the predictive skill of the stationary component over a two-year period shows relatively large variation from year to year with no particular seasonal variation. January 1982 was, in that context, distinctly worse than the other months. Figure 16 shows the anomaly correlation for the same three groups of zonal wave number as used in Fig. lla. By comparing the two months, the low predictive skill in January 1982 was shown to be very much related to lower than normal scores for the planetary waves. The scores for the short waves (zonal wave numbers 10-20), on the other hand, are essentially the same for the two months. The fact that numerical models show periods of high and low predictive

30

LENNART BENGTSSON

skill lasting one to several weeks appears to be typical of other operational models (Bengtsson and Lange, 1982; Lange and Hellsten, 1983) for forecasts up to 3 days. Some of the cases of high predictive skill have been discussed by Bengtsson and Simmons (1983). The model correctly maintains a highly anomalous flow, and on other occasions large changes such as the development of blocking highs or breaking down of blocks occurs. Presently there are no convincing arguments as to whether periods of high predictive skills reflect a high internal predictability of a particular flow pattern or whether the model can handle certain weather types better than others. The author has the view, based on more heuristic and simplistic arguments, that the latter is the more likely explanation. 5.2. Tropical and Southern Hemisphere Forecasts Tropical forecasts have not been evaluated as extensively or objectively as those for the extratropical Northern Hemisphere, but there is no doubt that at present their accuracy and usefulness is substantially lower. Objective verification indicates a limited short-range predictive skill in the middle and upper troposphere. Thus for the belt from 18"N to 18"s the root-mean-square error (rmse) of the 2-day forecast vector wind

09 08

07

n

E 06

805 In

04 03

02 01

1

5

10

15

20

25

30

DAY

FIG.12. Graph showing anomaly correlation of 500-mb height scores of ECMWF 3-day (D+3) and 7-day (D+7) forecasts for each day of November 1983. Note the high scores of the D+7 forecasts around 1 1 November (see 1-3) and the low scores of the forecasts around 28 November (see 4-6). 500-mb analysis of 1 1 November 1983 (l), D+7 forecast valid 18 November 1983 (2), and verifying analysis of 18 November 1983 (3). 500-mb analysis of 28 November 1983 (4),D+7 forecast valid 5 December 1983 (51, and verifying analysis of 5 December 1983 (6).

MEDIUM-RANGE FORECASTING AT THE ECMWF IWE

WE

FIG. 12. (Continued)

31

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LENNART BENGTSSON

MEDIUM-RANGE FORECASTING AT THE ECMWF

33

9

.

1.0 0.8 0

Dt3

D+7

0.3 0.2 (b) 0.1

-

FIE. 13. Anomaly correlationof 500-mb height scores of ECMWF 3-day (D+3) and 7-day (D+7) forecasts for the period (a) I January to 31 March 1982, and (b) 1983. Note the long periods of relatively high D+7 scores in the 1983 scores.

MEDIUM-RANGE FORECASTING AT THE ECMWF

35

(averaged for 1983) is 4.8 m-l at 500 mb compared with 6.8 m s-l for persistence. The indication from both subjective and objective assessments of tropical forecasts is that there are serious deficiencies in the parameterization of convection, and a substantial effort to understand and correct these deficiencies is currently being made. Forecast experiments reported by Bengtsson and Simmons (1983) have shown a great sensitivity to the parameterization of moist convection. These forecasts experiments, which used FGGE rather than operational data, cover a period that started 1 1 June 1979, a day marked by the onset (rather later than normal) of the southwest monsoon. The two forecasts differ only in their parameterization of convection, one using the scheme of Kuo (1974) adopted for operational forecasting and the other the Arakawa and Schubert (1974) scheme. The latter evidently produces a quite different forecast and, in fact, a much better representation of the development of the strong monsoon flow over the Arabian Sea. Just one experiment cannot, of course, be used to draw firm conclusions as to which of the convection schemes is the better (and, indeed, the extratropical forecasts in this case were slightly better using the Kuo scheme), but the sensitivity of the forecast is worth noting. Other studies have, in addition, shown sensitivity to the prescription of soil moisture and orography that resulted in a general agreement with those found elsewhere (Rowntree, 1978). Overall, it seems that the tropical forecasts respond more quickly and acutely to defects in the model than do forecasts at middle and high latitudes. Assessment of the forecasts for the Southern Hemisphere have also been less detailed than those for the Northern Hemisphere. Comparing anomaly correlations for 18-78"N with those for 18-67.5"s shows that the 60% value is reached about 1.5 days earlier than for the Northern Hemisphere (Table 11).The lower values found in the latter case are not surprising in view of the sparsity of Southern Hemispheric data. Generally, more accurate forecasts for this hemisphere have been reported during the enhanced data coverage of the FGGE year (Bengtsson, 1983). The FGGE data, in particular the satellite data, have also had a substantial effect on tropical forecasts. AND PROSPECTS IN NUMERICAL WEATHERPREDICTION 6. PROBLEMS

It is reasonably well established that prediction of instantaneous weather patterns at sufficiently long range is impossible, and there are clear indications that there is an inherent limitation in the predictability of

12rH

90%

60'W

90%

60%

FIG.14. January 1982 500-mb height as observed (a) and as predicted (b) in ECMWF 10day forecasts and January 1982 500-mb height anomaly as observed (c) and as predicted (d). 36

3.09

%06

Mt

3.09

Lo6

3IK1

FIG.15. January 1983 500-mb height as observed (a) and predicted (b) in ECMWF 10-day forecasts and January 1983 50-mb height anomaly as observed (c) and as predicted (d). 38

FIG. IS. (Conrinrrrd) 39

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0

5

41

10

Days

FIG. 17. The global root-mean-square error of 500-mb height in meters (line 1) for all operational forecasts from 1 December 1980 to 10 March 1981. According to the estimate by Lorenz (1982) line 2 would result from the use of an ideal model, even with no improvement in the observing system.

Lorenz (1982), in a study comparing the inherent error growth of the ECMWF model with the forecast error growth, has demonstrated that there is a scope for extending the range of useful forecasts by 3 to 4 days even with today’s incomplete and inaccurate observations (Fig. 17). More accurate initial states will naturally extend predictive skill further, and it appears, according to Lorenz, that a reduction of the initial error by half will extend predictive skill by about 2 days and presumably another reduction by half can give another 2 days. The improvements in medium-range forecasts are likely to fall into the largest scales of motion, where the predictions have considerable errors of a systematic nature (Bengtsson and Simmons, 1983). These errors are revealed by averaging forecasts over a number of cases, preferably for one month. The errors characteristically grow in amplitude throughout

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the forecast period, and their general similarity to errors in the model climatology, revealed by integrations over extended periods, indicates that these errors represent a gradual drift from the climate of the atmosphere to that of the model. The rate of this drift is found to vary from case to case, but the overall error associated with it appears to be independent of the initial data. The model shows an average cooling of about 1.5"C in 10 days. In extended model integrations this cooling continues for another 50 days and reaches an equilibrium value of about 3 to 4°C. No further cooling takes place hereafter and appears to be almost independent of resolution; a T21 version of the model has been integrated for 10 years without any further cooling. There is an area of maximum cooling in the mid-troposphere around 500 mb and another one in the stratosphere. Figure 18 shows four ensemble temperature profiles observed and predicted at the latitudes O", 30"N, 55"N, and 80"N. This tropospheric cooling has been noted for other models as well but can be sensitive to a number of aspects of model design, including not only the parameterization of diabatic processes but also such features as the prescription of the orography (Wallace et al., 1983) and the amount of horizontal smoothing of model fields (Girard and Jarraud, 1982), both of which may influence precipitation and associated latent heat release. The increased flux of sensible heat, associated with the incorporation of a diurnal radiation cycle (not yet included in the operational model) reduces the overall cooling by about 25%. Temperature errors become more pronounced at upper levels, where there is a general tendency for the stratosphere to be too cold, particularly near the winter pole. Associated with this, the stratospheric polar-night jet is generally too strong and is often insufficiently separated from the main subtropical jet. Whether this is related to the treatment of radiative fluxes in the model or due to dynamical processes has not yet been demonstrated. The model also shows associated deficiencies in the zonally averaged flow. At the surface, the middle-latitude westerlies are generally found to be a few meters per second stronger than those observed in the Northern Hemisphere. It is interesting to note that this discrepancy becomes more pronounced when the horizontal resolution is increased (Manabe et al., 1979; Cubasch, 1981). Conversely, a higher resolution (T40* or more) generally yields a better simulation of the stronger surface westerlies of the Southern Hemisphere. In the upper troposphere, westerly maxima are usually found to be too strong (by a few meters per second) and displaced * T40 triangular spectral truncation at wave number 40.

MEDIUM-RANGE FORECASTING AT THE ECMWF

43

slightly poleward and upward. Tropical easterlies tend to be underestimated near the tropopause and overestimated at higher stratospheric levels. More interesting details of the systematic errors are revealed by an inspection of their geographical distribution. Maps of the day-10 temperature error at 500 and 850 mb and at the 500- and 1000-mb height error are presented as Fig. 19. Looking first at the height field we see very similar error patterns at 1000 and 500 mb with distinct centers of low pressure over the northeastern Atlantic Ocean/Scandinavia and the northern Pacific Ocean. The amplitude of the error increases slightly with height. Consistent with this, areas of too low temperature tend to coincide with the areas of too low pressure, particularly at 500 mb. Elsewhere the 500-mb temperature error is small, but at 850 mb, regions substantially too warm in temperature are evident over eastern Siberia and northern Canada. The general distribution of temperature error implies areas of too low static stability that are likely to be the cause of a slightly erroneous structure of the baroclinic waves. Progress in the understanding of the slow-acting physical processes at large scales and a better description of orographic forcing are likely to improve the predictability of the large-scale features. As demonstrated by Wallace et af. (1983), the incorporation of an improved orographic representation by using a so-called envelope orography has led to a more accurate prediction of large-scale flow, at least during the winter. A better understanding and modeling of the large-scale stationary forcing is expected to greatly benefit forecasts in the Tropics. As has been demonstrated by Krishnamurti et al. (1983), it is of great importance to have an accurate initial description of the diabatic forcing. If this is initially wrong due to lack of relevant observation or to an unsatisfactory initialization procedure, large errors develop quickly in the model. The ECMWF model has problems of this kind over the African continent. The incorporation of satellite cloud observations as well as observations of precipitation in the initialization procedure, combined with a diurnal cycle, are expected to reduce these errors significantly. A better handling of the modeling problems in the tropics is crucial for the improvement of medium-range forecasts and a further extension of useful predictive skill at higher latitudes. In order to demonstrate this, Haseler (1982) has carried out a series of numerical experiments with the ECMWF model in which analyzed data were inserted at a boundary along 20”N in order to simulate a “correct” tropical forecast. As can be seen from Fig. 20, there is a considerable difference in the forecast far downstream, and after 6 days the circulation over the European area is markedly influenced. A detailed analysis revealed in this case that the

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FIG.18. Vertical temperature profiles of ECMWF 10-day forecast (dotted line) and the verifying analyses (solid line) at the Equator (a), 55"N (b), 30"N (c), and 80"N (d).

I

45

MEDIUM-RANGE FORECASTING AT THE ECMWF -80

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-10

o

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20

b

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8

46

E

FIG. 19. Difference fields between ECMWF 10-day forecasts and the corresponding analysis for winters of 1981/1982 and 198?/1983. ( I ) 500-mb height, contour interval, dam. (2) 500-mb temperature, contour interval in kelvin. (3) 1000-mb height. (4) 850-mb temperature, same contour intervals.

48

W P

FIG.20. 500-mb height analyses for 14 November 1979 (1) and for 20 November 1979 (2). The 6-day operational forecast from 14 November 1979 (3), and the 6-day forecast (4)that was obtained when the forecast was relaxed toward 6-hr FGGE analyses in the tropics with pure analyzed values in the band 15"N-l5"S, pure forecast values to the north of 25"N and 25"s and smoothly mixed values in the intermediate zones.

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interaction between the Hadley cell circulation over Central American and an extratropical depression moving eastward over the United States interacted in a crucial way that very much affected the downstream development.

7. CONCLUDING REMARKS Miyakoda et al. (1972), who carried out the first medium-range forecasts, suggested that perhaps the results of their first comprehensive trial of 2-week predictions might be taken as a benchmark for future comparisons. Their ensemble mean anomaly correlations for the extratropical Northern Hemisphere were based on 12 January cases taken from the years 1964 to 1969 that reached values of 80% after about 2 days of the forecast period for the 500-mb height. The corresponding time for the 60% correlation was about 3.5 days. The average of the operational ECMWF forecasts for the winter 1982/1983 was 4.5 days for the 80% correlation and 6.5 days for the 60% level. These comparisons indicate that a substantial improvement has taken place over the past 10 to 15 years in our ability to predict at least the larger scales of motion. These improvements are very much a reflection of model development and the considerable increase in meteorological observations that have taken place over this period. This has been demonstrated by observing system experiments using the FGGE data [e.g., Bengtsson (1983)l. In particular, observations from satellites and aircraft have had a considerable impact on forecasts in the medium range (2-14 days) (Table 11). Furthermore, observations are now better utilized due to improved analysis methods, more accurate initialization, and a more accurate and consistent use of the prediction model to provide the first guess. The ongoing intense development in computer technology makes it possible that we may, toward the end of this decade, have computers with a processing speed more than 10 times faster than today’s supercomputer. This will make it possible to use global models with a horizontal resolution on the order of 50 km that will be able to resolve and describe the evolution of lee-wave cyclones and possibly tropical cyclones. To what extent such a resolution is realistic for improving forecasts in the medium range and for extending the limit of useful predictive skill in general remains to be demonstrated. The other necessary condition for better medium-range forecasts is improvement of global observing systems, in particular in the tropics and the Southern Hemisphere. For practical and economical reasons, we must rely heavily on satellite observations, substantial research and

MEDIUM-RANGE FORECASTING AT THE ECMWF

51

development must take place to use these observations more efficiently in the data-assimilation systems and for the instrumental designers to develop better sensors. Closer cooperation between experts on satellite instruments and retrieval procedures and the numerical modelers is essential. For operational reasons, more efficient means for the global exchange of data in real-time are required. There does not yet seem to be a realistic alternative to the “brute force approach,” and future models are likely to be improved through the systematic and meticulous development of all aspects of the forecasting system. An alternative approach can possibly be considered when the forecasts are extended beyond the predictability of the individual weather systems. As has been shown by Miyakoda and Chao (1982) and Shukla (1984), there are indications that useful predictions of time averages can be extended even further, perhaps up to a month or more. The fact that considerable anomalies can exist for this length of time and that present GCM’s can simulate such long-lasting anomalies (Volmer et al., 1983) gives us hope that we may be successful. It seems that the most logical approach to such long integrations would be to apply a simple kind of Monte Carlo prediction using a sequence of initial states, say over a few days, to produce an ensemble of predictions. The ECMWF has benefited tremendously from the achievements of the GFDL and the very generous assistance provided by its former director, Dr. J. Smagorinsky, and his staff. In the early years of the ECMWF we were given full access to the GCM developed at the GFDL and the necessary training in how to use the model. This help and encouragement were of great importance and are an example of J. Smagorinsky’s contribution to meteorology and international cooperation.

ACKNOWLEDGMENTS The author wishes to acknowledge the assistance of Dr. J. A. Woods in preparing this paper. The work discussed represents the efforts of many staff members and visiting scientists at the ECMWF as well as other colleagues. The author would like to express his gratitude for their contribution.

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Arakawa, A., and Schubert, W. H. (1974). Interaction of a cumulus cloud ensemble with the large-scale environment. J. Atmos. Sci. 31, 674-701. Arpe, K., Hollingsworth, A., Tracton, M. S., Lorenc, A. C., Uppala, S., and Kallberg, P. (1985). The response of numerical weather prediction systems to FGGE level IIB data. Part 11. Forecast verification in implications for predictability. Q. J . Meteorol. SOC.111, 67- 102. Baede, A. P. M., Jarraud, M., and Cubasch, U. (1979). Adiabatic formulation of ECMWF’s spectral model. ECMWF Tech. Rep. No. 15, pp. 1-40. Bengtsson, L. (1983). Observational requirements for long-range forecasts. In “Collection of Position Papers Presented at the WMO-CASIJSC Expert Study Meeting on Longrange Forecasting, Princeton, December, 1982, LRF Publ. Ser. No. I , pp. 219-229. WMO, Geneva. Bengtsson, L., and Lange, A. (1982). Results of the WMOKAS Numerical Weather Prediction Data Study and Intercomparison Project for Forecasts for the Northern Hemisphere in 1979-80. WMO, Geneva. Bengtsson, L., and Simmons, A. J. (1983). Medium range weather prediction-operational experience at ECMWF. In “Large-Scale Dynamical Processes in the Atmosphere” (B. J. Hoskins and R. P. Pearce, eds.), pp. 337-363. Academic Press, New York. Bengtsson, L., Kanamitsu, M., KUlberg, P., and Uppala, S. (1982). FGGE 4-dimensional data assimilation. Bull. Am. Meteorol. SOC.63, 29-33. Bjerknes, J. (1966). A possible response of the atmospheric Hadley circulation to equatorial anomalies of ocean temperature. Tellus 18, 820-829. Bjerknes, J. (1969). Atmospheric teleconnections from the equatorial Pacific. Mon. Weather Rev. 97, 163-172. 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. Bourke, W. (1972). An efficient one-level primitive-equation spectral model. Man. Weather Rev. 100,683-689. Bumdge, D. M. (1979). Some aspects of large scale numerical modelling of the atmosphere. In “Proceedings of the ECMWF Seminar on Dynamical Meteorology and Numerical Weather Prediction,” Vol. 2, pp. 1-78, ECMWF. Charney, J. G. (1947). The dynamics of long waves in a baroclinic westerly current. J. Meteorol. 4, 135-162. Charney, J. G., and DeVore, J.G. (1979). Multiple flow equilibria and blocking. J. Atmos. Sci. 36, 1205-1216. Charney, J. G., and Eliassen, A. (1949). A numerical method for predicting the perturbations of middle latitude westerlies. Tellus 1, 38-54. Cubasch, U. (1981). Preliminary assessment of long-range integrations done with the ECMWF global model. ECMWF Tech. Memo. No. 28, pp. 1-21. Eady, E. T. (1949). Long waves and cyclone waves. Tellus 1, 33-52. Gandin, L. S. (1963). “Objective Analysis of Meteorological Fields.” Gidrometeorol. Izda., Leningrad (Israel Program for Scientific Translations, Jerusalem, 1965). Geleyn, J.-F., and Hollingsworth, A. (1979). Economical analytical method for the computation of the interaction between scattering and line absorption of radiation. Contrib. Atmos. Phys. 52, 1-16. Geleyn, J.-F., Hense, A., and Preuss, H. J. (1982). A comparison of model generated radiation fields with satellite measurements. Beitr. Phys. Atmos. 55, 253-286. Girard, C., and Jarraud, M. (1982). Short and medium range forecast differences between a spectral and a grid-point model. An extensive quasi-operational comparison. ECMWF Tech. Rep. No. 32, pp. 1-178.

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Haseler, J. (1982). An investigation of the impact at middle and high latitudes of tropical forecast errors. ECMWF Tech. Rep. No. 31, pp. 1-42 Hollingsworth, A., Arpe, K., Tiedtke, M., Capaldo, M., and Savijarvi, H. (1980). The performance of a medium-range forecast model, in winter-impact of physical parametenzations. Mon. Weather Rev. 108, 1736-1773. Hollingsworth, A., Arpe, K., Lorenc, A. C., Tracton, M. S., Cats, G., Uppala, S., and Kallberg, P. (1985). The response of numerical weather prediction systems to FGGE level 11-b data. Part I. Analysis. Q. J. Meteorol. SOC. 111, 1-66. Hoskins, B. J., and Simmons, A. J. (1975). A multi-layer spectral model and the semiimplicit method. Q. J. R. Meteorol. SOC. 101, 637-655. Kessler, E. (1969). On the distribution and continuity of water substance in atmospheric circulations. Meteorol. Monogr. 10, 1-84. Krishnamurti, T. N., Ingles, K., Cocke, S., Kitade, T., and Pasch, R. (1983). Details of low latitude medium range weather prediction using a global spectral model. 11. Effects of orography and physical initialisation. Fl. State Uniu. Rep. 83-11, 1-206. Kuo, H.-L. (1974). Further studies of the parameterization of the influence of cumulus convection in large-scale flow. J . Atmos. Sci. 31, 1232-1240. Lange, A,, and Hellsten, E. (1983). “Results of the WMOiCAS NWP Data Study and Intercomparison Project for Forecasts for the Northern Hemisphere in 1981-82.” WMO Short- and Medium-range Weather Predict. Res., F’ubl. Ser. No. 2. World Meteorol. Organ., Geneva. Leith, C. E. (1978). Objective methods for weather prediction. Annu. Rev. Fluid Mech. 10, 107- 128. Lorenc, A. C. (1981). A global three-dimensional, multivariate statistical interpolation scheme. Mon. Weather Rev. 109, 701-721. Lorenz, E. (1982). Atmospheric predictability experiments with a large numerical model. Teilus 34, 505-513. Louis, J.-F. (1979). A parametric model for vertical eddy fluxes in the atmosphere. Boundary Layer Meteorol. 17, 187-202. Machenhauer, B. (1977). On the dynamics of gravity oscillations in a shallow-water model with application to normal mode initialisation. Beitr. Phys. Atmos. 50, 253-271. Manabe, S., Hahn, D. G., and Holloway, J. L. (1979). Climate simulation with GFDL spectral models of the atmosphere: Effect of spectral truncation. CARP Publ. Ser. 22, 41-94. Miyakoda, K., and Chao, J.-P. (1982). Essay on dynamical long-range forecasts of atmospheric circulation. J . Meteorol. SOC.Jpn. 60, 292-308. Miyakoda, K., and Sirutis, J. (1977). Comparative integrations of global models with various parameterized processes of subgrid-scale vertical transports: Description of the parameterizations. Beitr. Phys. Atmos. 50, 445-487. Miyakoda, K., Hembree, G. D., Strickler, R. F., and Shulman, I. (1972). Cumulative results of extended forecast experiments. I . Model performance for winter cases. M o n . Weather R e v . 100, 836-855. Miyakoda, K., Gordon, T., Caverly, R., Stern, W., Sirutis, J., and Bourke, W. (1983). Simulation of a blocking event in January 1977. Mon. Wecither Rru. 111, 846869. Namias, J. (1969). Seasonal interactions between the North Pacific Ocean and the atmosphere during the 1960’s. Mon. Weather Rev.97, 173-192. Pfeffer, R. L., ed. (1960). “Dynamics of Climate.” Pergamon, Oxford. Phillips, N. A. (1957). A coordinate system having some special advantages for numerical forecasting. J . Meteorol. 14, 184-185.

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Phillips, N. A. (1973). Principles of large scale numerical weather prediction. I n “Dynamical Meteorology” (P. Morel, ed.), pp. 1-95. Reidel Publ., Dordrecht, Netherlands. Rowntree, P. R. (1978). Numerical prediction and simulation of the tropical atmosphere. In “Meteorology over the Tropical Oceans,” p. 278. R. Meteorol. SOC.,London. Rowntree, P. R., and Bolton, J. A. (1978). Experiments with soil moisture anomalies over Europe. I n “The GARF’ Programme on Numerical Experimentation: Research Activities in Atmospheric and Ocean Modelling” (R. Asselin, ed.), Rep. No. 18, pp. 1-63. WMO/ICSU, Geneva. Sadourny, R. (1975). The dynamics of finite difference models of the shallow-water equations. J. Atmos. Sci. 32, 680-689. Shukla, J. (1984). Predictability of time averages. I n “Problems and Prospects in Long and Medium Range Weather Forecasting” (D. M. Burridge and E. Kallkn, eds.), pp. 109206. Springer-Verlag, Berlin and New York. Simmons, A. J., and Bengtsson, L. (1948). Atmospheric general circulation models, their design and use for climate studies. In “The Global Climate” (J. Houghton, ed.), pp. 3562. Cambridge Univ. Press, London and New York. Simmons, A. J., Wallace, J. M., and Branstator, G. W. (1983). Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci.40, 1363-1392. Smagorinsky, J. (1953). The dynamical influences of large scale heat sources and sinks on the quasi-stationary mean motions of the atmosphere. Q . J . R . Meteorol. SOC.79, 342366. Smagorinsky, J. (1969). Problems and promises of deterministic extended range forecasting. Bull. A m . Meteorol. SOC.50, 286-3 1 1 . Stem, W. (1982). Four-dimensional assimilation at GFDL using FGGE data. In “Proceedings of the 14th Stanstead Seminar on the Interaction Between Objective Analysis and Initialization” (D. Williamson, ed.), p. 127. Dept. of Meteorology, McGill University, Montreal. Sundqvist, H. (1981). On vertical interpolation and truncation in connexion with use of sigma system models. Atmosphere 14, 37-52. Tiedtke, M., Geleyn, J.-F., Hollingsworth, A., and Louis, J.-F. (1979). ECMWF modelparameterization of sub-grid scale processes. ECMWF Tech. Rep. No. 10, pp. 1-46. Volmer, J.-P., Deque, M., and Jarraud, M. (1983). Large-scale fluctuations in a long range integration of the ECMWF spectral model. Tellus 35, 173-178. Wallace, J. M., Tibaldi, S., and Simmons, A. J. (1983). Reduction of systematic forecast errors in the ECMWF model through the introduction of an envelope orography. Q.J. R . Meteorol. SOC.109,683-717. Wiin-Nielsen, A. (1979). Steady states and stability properties of a low-order barotropic system with forcing and dissipation. Tellus 31, 375-386. Winston, J. S., and Krueger, A. F. (1978). Diagnosis of the radiative heating over the Eastern Hemisphere in relation to the summer monsoon. Indian J . Meteorol., Hydrol. Geophys. 29(1/2), 259. Wiscombe, W. J., and Evans, J. W. (1977). Exponential sum fitting of radiative transmission functions. J . Comput. Phys. 24, 416-444.

EXTENDED RANGE FORECASTING K. MIYAKODA AND

J . SIRUTIS Geophysical Fluid Dynumics LaboraiorylNOAA Princeton University Princefon, New Jersey 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. An Evolution of 10-Day Forecast Performance . . . . . . 3. Examples of Monthly Forecasts. . . . . . . . . . . . . 4. A Projection of Seasonal Forecasts . . . . . . . . . . . 5. Postscript . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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I . INTRODUCTION The variability of the atmospheric circulation on a global scale appears to be subtle and complex, and yet a large portion of it may be predictable for quite a long period of time. How long, how much, and in what way this can be achieved is our interest. One approach to extended range forecasting is simply to integrate the differential equations of a general circulation model (GCM) in time, using a small time step, and to project the solutions to 10 days, 1 month, and one season ahead or beyond. We feel that this approach follows the basic spirit of the numerical weather prediction proposed by Richardson in the 1920s and demonstrated by Charney in the 1940s. The key aspect is to treat the relevant physics and particularly the advection terms in the hydrodynamic and thermodynamic equations as accurately as possible and thereby avoid the use of any statistical, mechanistic, or vastly simplified models in the basic framework. Accordingly, the time step for 30 the integration is not as large as 10 days of 1 month, but as small as 5 min, depending on the time-differencing scheme employed. At the same time, empirical relations or constants are limited to a minimum (ideally none) except for the purposes of computational efficiency. In this respect, the medium-range forecast, which covers about 10 days (see Bengtsson, Chapter 1) and the long-range forecast that covers the range beyond 2 weeks are not much different in most of the basic formulations. However, the degree of importance of each term in the equations changes, particularly in the areas of the hydrological cycle, some of the sub-grid-scale processes, and the lower boundary conditions. The last

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aspect eventually requires an air-sea-coupled model that expands the dimension of complexity to a considerable extent. One significant difference between the medium- and long-range forecasts is, however, that the stochasticity becomes increasingly dominant as the time range is extended. Any small uncertainty at the initial time grows due to the inherent dynamic characteristics, and in the monthly time range the uncertainty develops to a sizable amount. In this respect, it is desirable to modify the basic equations beforehand to the so-called stochastic-dynamic model, as demonstrated by Epstein (1969) for a low-order system of equations. If this approach were easily applicable to full-fledged GCM equations, there would have been a revolutionary change in the dynamic method of long-range forecasting. However, since no good idea has been proposed to make an efficient calculation for a stochastic-dynamic model, we continue to use the original GCM equations repeatedly, i.e., to use the “brute force” method. For this reason, it would not be inappropriate to start this essay with a discussion of the medium-range forecasts (Smagorinsky, 1969). In fact, it would be appropriate to make some nostalgic documentation about the “good old days” of Joe Smagorinsky’s laboratory, avoiding an excessive indulgence in a personal recollection of the past.

2. AN EVOLUTION OF ~O-DAY FORECAST PERFORMANCE In the 10-year period 1965-1975, a great deal of effort was devoted to the investigation of the feasibility of 2-week forecasts. Figures I and 2 are a display of forecast performance in chronological order. We take as an example day 10.5 of a Northern Hemisphere prognosis of 500-mb geopotential height for the case of March 1965. The reason why March 1965 was chosen is that a project for global forecasting was launched when one of the authors immigrated to the United States in 1965. Also at that time, the first daily world cloud maps made of numerous photomosaics from satellite TIROS IX became available. First, we started to send letters to data centers and operational centers to request meteorological data such as radiosonde and hemispheric maps or local weather maps. After that, it took three years to finish making a 4-day series of global maps and the gridded data for a number of parameters. R. H. Clarke from CSIRO, Australia (later the officer-in-charge of Commonwealth Numerical Meteorological Research Centre, Melbourne), was engaged in hand-map analysis full time for a

FIG.1. 500-mb geopotential heights at day 10.5 for 1 March 1965: (a) observed and (b) predicted by a hemispheric stereographic model (Hem N40; forecasting performed 1967); global models: (c) Kuri-grid model (Kun N48; 1968), (d) modified Kuri-grid model (Mod Kuri N48; 1973), and (e) latitude-longitude grid model (Lat.-long. N48; 1976). The contour interval is 60 rn.

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FIG. 1. (Continued)

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FIG. 1. (Continued)

year, and J. Sadler, University of Hawaii, worked on the analysis of the tropics, utilizing the satellite cloud imagery (Miyakoda et al., 1974). A hemispheric-stereographic GCM (Smagorinsky et al., 1965, 1967; Manabe et at., 1965) was already in use for the study of climate as well as prediction experiments. For example, two samples of 2-week forecasts with this model were presented at the first Global Atmospheric Research Programme (GARP) conference in Sweden in 1967. The grid resolution of the model was N40 and nine vertical levels, where the N40 denotes 40 gridpoints between a pole and the equator. Now the same model was applied to the initial condition of 1 March 1965. The forecast map of geopotential height is shown in Fig. 1 . Unfortunately, this particular result is not impressive. On the other hand, the global GCM with the Kurihara grid and nine vertical levels was just completed at that time (Kurihara and Holloway, 1967; Holloway and Manabe, 1971). However, for the purpose of a real data forecast, the N24 resolution (3.75' meridional distance) was somewhat poor, so we decided to construct the N48 resolution model. This task turned out to be nontrivial with the UNIVAC 1108 computer. An extraordinary amount of effort was expended to fit the gigantic model (for its time) in the computer; a data-segmentation technique was used. In order to calculate the terms of

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FIG.2. The same as in Fig. 1 , except using various sub-grid-scalephysics A, E, and F, with either 9 or 18 levels in the vertical: (a) observed, (b) predicted by F and L9 model (forecasting performed 1980), (c) A and L9 model (1980), (d) E and L9 model (1980), (e) A and L18 model (1978), and (f) E and L18 model (1977).

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FIG.2. (Continued)

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hydrodynamic equation, for example, only one grid box and six neighboring boxes were allowed in the central memory of the machine at one time. When the calculation was finished, this set of boxes was stored on the drum and the next set of boxes was called from the drum. It took 32 hr of machine time to execute a 1-day forecast calculation with the N48 Kuri-grid model. The Kuri-grid model was applied to the March 1965 initial condition, and the result was presented at the Tokyo international conference on numerical weather prediction in 1968. The map designated as N48 Kuri in Fig. 1 is the forecast result taken from the conference paper (Miyakoda et al., 1971). If one compares the N40 Hem and the N48 Kuri with the observations, a similarity to the observations may be more noticeable in the N48 Kuri than in the N40 Hem, though according to a report written at that time, “The global model’s performance was somewhat inferior to that of a hemispheric model due to excessive truncation error, and yet some advantage was recognized particularly in the tropics.” A noteworthy point is that in the Hovmoller (trough-ridge) diagram, “the global prediction is in better agreement with the observation than the hemispheric prediction”; the case was associated with the Pacific blocking event. Although the Kuri grid was designed to cover the whole globe homogeneously, forecast experiments indicated excessive, systematic truncation error in the polar region. This serious deficiency was found by Grimmer and Shaw (1967), Dey (1969), and Sankar-Rao and Umscheid (1969). They reported a spurious accumulation of mass over the polar regions and even the global spread of this adverse effect. To improve this point, Umscheid was requested to join the GFDL in 1970. He fixed the GCM in 4 months by increasing the number of grid points, particularly in the polar region. Theoretically, an ideal system would be a pure spherical grid. These grid systems are called the Modified Kuri (Mod Kuri) and the latitude-longitude (1at.-long.) grid, respectively. The number of grid points in a horizontal plane was increased from 5025 in the N40 Hem (for the Northern Hemisphere alone), to 9216 for the globe in the N48 Kuri, and to 10440 in the N48 Mod Kuri to 18240 in the N48 1at.-long. The results for the case of March 1965 are shown in (d) and (e) of Fig. 1 (taken from Umscheid and Bannon, 1977). It can now safely be stated that the prognoses resemble the observations. Improvements are particularly evident in the blocking ridges over Alaska and Europe and the cut-off low off California. For all these forecasts no physics was changed and the meridional resolution was kept at N48. Only the longitudinal space resolution was increased. By the way, the European Community was impressed by the results of

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the Mod Kuri model, and Hollingsworth from the ECMWF (European Centre for Medium Range Weather Forecasts, the director was then Win-Nielsen) visited the GFDL and adopted the model. The data set of 1 March 1965 was repeatedly utilized for tests in the ECMWF until they finally got fed up with this case. The subsequent 10-year period 1975-1985 is the second phase of the prediction experiment in which the major objective was the investigation of monthly forecasts. Figure 2 shows the evolution of the forecast performance in this phase though the prognoses in this figure are for day 10.5. The Mod Kuri model was used in all the maps of this figure, which was applied again to the case of March 1965. The variations in these experiments were due to the sub-grid-scale physics or the vertical resolution. In general, associated with the space discretization for the purpose of numerical integration, two things should be considered, i.e., the space truncation error and the sub-grid-scale physics. In the paper of Miyakoda and Sirutis (1977), a comparative study of three models was discussed, i.e., the A, E, and F physics models. The A model uses the so-called GFDL 1965 physics (Smagorinsky et al., 1965; Manabe et al., 1965), which is considered as the reference. The E model uses the turbulence closure theory of Mellor-Yamada (1974), which eliminates the dry-convective adjustment; it also employs the Monin-Obukhov boundary layer physics and incorporates soil heat conduction (Delsol et al., 1971). The F model uses the Arakawa-Schubert (1974) cumulus parameterization. The forecasts at day 10.5 of the A, E, and F physics with the L9 (9 vertical levels) model in Fig. 2 are taken from the same series of work [i.e., Miyakoda and Sirutis (1983)l. It appears that compared with the A model, the E and F model produce more intensified meandering (zonal asymmetry) of mid-latitude flow and better simulation of blocking ridges. The panels (e) and (f) of Fig. 2 are the forecasts by the N48L18 model with the A or E physics. The panels are laid out so as to make the comparison easy between the same physics model of different vertical resolutions. The L18 model appears to generate more intense troughs and therefore more transient eddies than the L9 model. The blocking ridges are better simulated and the zonal currents are concentrated in a narrower latitudinal band in the L18 than in the L9, though the results of the L18 model need further examination to confirm the validity. In summary, the model’s quality turned out to be essential for proving the feasibility of the medium-range forecast. The spatial resolution was most crucial, and the improvement by the sub-grid-scale physics is subtle and yet noticeable. The following remarks may be noteworthy.

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Retrospectively, standing at the year of 1968 in Fig. 1, it was extremely difficult to speculate what factors really contribute to an improvement of the forecast and to perceive whether the 10-day forecast was feasible in the first place. It was tempting to blame the quality of the initial condition, and, in fact, we once did. There were other factors for the candidates of forecast improvement: the real sea-surface temperature, the better initialization procedure, the better treatment of stratosphere, the consideration of stochasticity , a better orography , etc. The question was what priority should be given to them. It was critical to place these elements in the right order of importance. If the scientific priority was not properly assigned, a great deal of confusion could have occurred. OF MONTHLY FORECASTS 3. EXAMPLES

The monthly forecast experiment was started in 1975. The modified Kuri-grid model as well as the spectral model were tested, first for the case of March 1965 and then for the three January cases. The purpose of this preliminary study was to learn which model is suitable for the monthly integration. The case of January 1977, which was an extraordinarily cold winter over North America, was investigated extensively by using various models. The experiment (Miyakoda et al., 1983) revealed that the N48L9-E model performed well in simulating the blocking ridges, compared with the A physics model and with the spectral model of relatively low resolution (rhomboidal 30) of A and E physics. Based on this study, a decision was made in 1980 that the prediction model for the subsequent experiment would be the Mod Kuri N48L9-E model. The eight January cases taken for this experiment are the 1st January 1977, 1978, 1979, 1980, 1981, 1982, and 1983 and the 16th January 1979. The year 1979 is the FGGE (First GARP Global Experiment) year. The objective is to investigate whether the monthly forecasts are feasible and whether the stochasticity is a serious matter for the monthly time range. For the “brute force” stochastic system, three initial conditions based on the analysis of the GFDL, the NMC, and the ECMWF are used for each of the eight January cases. Before describing the results, the related auxiliary experiments will be mentioned. The A, E, and F sub-grid-scale physics are still being studied with respect to the eight January cases. In addition, the “envelope mountains” of the ECMWF are also being tested for the monthly time range. Wallace et al. (1983) have reported a substantial improvement for the medium-range forecast of these mountains. The ECMWF provided us

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with the ‘‘V2 standard deviation mountain.” We carried out an experiment on the impact of these mountains, where the envelope orography replaced the conventional orography in the F physics model. This model is hereafter referred to as the FM model. Examples of monthly forecasts are displayed in Fig. 3, in which the anomaly height maps for day 10-30 are shown, excluding the first 10 days that have the obvious skill. The anomaly 6z is the departure of geopotential height z at 500 mb from the January climatology, i.e., 6z = z - zc ,zc being the climatological value. The two samples are for 1 January 1977 and 1 January 1979 as the initial time, which were treated by the F model and the FM model, respectively, using the initial conditions based on the GFDL four-dimensional data assimilation (Ploshay el al., 1983a-d). The former is an excellent forecast (the correlation coefficient is 0.62), whereas the latter is a marginal one (the correlation coefficient is 0.28). In both cases, the forecasts show excessively low anomalies of geopotential heights compared with the observation, indicating that there is a systematic bias, which corresponds to the well-noted cooling tendency in the forecasts. Monthly forecasts are beyond the limit of predictability for the cyclone-scale weather, and yet long-range forecasts could become a reality if we restricted our concerns to the macroscale and slowly varying components of atmospheric circulation and weather. Indeed, there is a long history (Namias, 1968) of synopticians investigating a systematic behavior of “action centers” of pressure patterns, “teleconnections,” the “zonal index,” and the “southern oscillation.” These empirical findings on macroscale patterns are extremely valuable and indispensable for the GCM approach because the knowledge gives scope and guidance as to where an emphasis should be placed in the extended-range forecasts. Based on the groundwork of various researches in the past 30 years, Wallace and Gutzler (1981) (subsequently referred to as WG) have been successful in identifying and classifying teleconnection patterns that are geographically fixed. In general, there are several ways to represent mathematically the complex space and/or time variability of atmospheric fields, such as the spherical harmonic function, the Hough function, the EOF (empirical orthogonal function), and the WG teleconnection pattern. Among them, the last method is appealing to the monthly forecast studies, because first, the patterns at their extreme index directly correspond to the observed prominent anomaly patterns and therefore to the local weather events; second, in the forecasts, only the space (not time) variability is available because of the nature of the problem; and third, the

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teleconnection patterns have an approximate relation with the EOF (Wallace and Gutzler, 1981; Lau, 1981). Figure 4 illustrates the PNA (PacifidNorth American) patterns as an example. These patterns were derived from the 15-year series of 200-mb geopotential height maps for winter (Kinter, 1983). Figure 4b is the ensemble mean pattern for the case of positive index of PNA; 4a is that for negative index of PNA; 4c is the difference between them (the positive minus the negative index of PNA), and therefore the PNA anomaly pattern. The index will be explained later. In order to depict the characteristic features of a time-mean circulation pattern, the WG five-category teleconnection indices appear convenient and appropriate. Figure 4d is an example of the teleconnection indices for the 500-mb map of day 10-30 of March 1965. A set of five categories of indices characterizes a height field, i.e., WP (West Pacific), PNA, WA (West Atlantic), EA (East Atlantic), and EU (Eurasian). The index is calculated on the basis of a specific formula for the respective category by the use of height values. In the case of PNA, for example, height values at four points of the centers of maximum and minimum in the anomaly pattern (4c) are used. The value larger than 100 is hatched. Wallace and Gutzler (1981) noted that WP and WA represent the north-south dipoles over oceanic sectors, whereas, PNA, EA, and EU represent the wave trains oriented in the west-east direction over North America, Europe, and Eurasia (see also Fig. 8). From this set of teleconnection indices, the evolution of the 10-day mean, 500-mb-height fields are illustrated in Fig. 5, in which a comparison is made between the forecast and the observation. Figures 5a and 5b are the cases of 1 January 1977 and 1 January 1979, respectively. As seen in the former, the F model reproduced well the development of positive PNA and negative EA patterns after the middle of the period. In the latter, the FM model was detrimental in realizing the development of positive WA in the later stage, whereas in the E and F models, a positive WA did not grow but a negative WP became erroneously large. Figure 6 shows the performance comparison of the four models of sub-grid-scale physics, i.e., A, E, F, and FM, in terms of the correlation coefficients for 10-day mean height anomalies at 500-mb (Fig. 6a) and 1000-mb (Fig. 6b) levels between the observed and the forecasts, and also in terms of the root-mean-square errors of the same variables (Fig. 6c). The skill scores are calculated as the arithmetic mean of the individual scores for the four cases, i.e., March 1965, 1 January 1977, 1 January 1979, and 16 January 1979. It appears that the scores for the E model are

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in general better than for the A model and that the scores for the F model are better than those for the E model except for the rms error at 500 mb. It was discussed in Miyakoda and Sirutis (1983) that in the E model the blocking ridges are better simulated than in the A model and that in the F model the tropical precipitation is better simulated than in the E model. The effect of the FM model is peculiar. The forecasts with the FM model are best in all cases for the first 10 days, then for a yet unclear reason they deteriorate and then become better at the end of the month in terms of the correlation coefficient (three out of four cases have this tendency). On the other hand, the root-mean-square errors in the F model are always best (smallest) in all cases. It may be worthy to note that the transient eddies, defined as the departure from the monthly mean flow, are substantially reduced in the FM model. This agrees with the conclusion in a similar type by the study of U.K. Meteorological Office (Hills or Keeping). Figure 7 shows the root-mean-square-error distributions at 500 mb for the A, E, F, and FM models based on the four cases. The error is written

Az = z p r d

- Zobs

where Zprdis the predicted value and mean-square error is defined as

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is the observed value. The

where E is the ensemble mean, i is the index for the cases, n is 4 in our is the time mean, the average being taken over the last 20 case, and (E) days of the forecast. Looking at Fig. 7, one may note the following: (1) The error is largest in the A model and is smallest in the FM model. (2) Consistent with the ECMWF's findings, the errors in the FM model are reduced over the Aleutian and Atlantic region (in this case, Baffin Bay).

But new large errors are developed in FM model over Eastern Europe, Asia, and the North Atlantic, indicating that some mountains were unfavorably enhanced. Another point is that the maps of mean error, i.e., FIG.6. Correlation coefficients between the forecast and observed anomalies of (a) 500mb and (b) 1000-mb 10-day mean geopotential height for the Northern Hemisphere (9025"N); (c) root-mean-square errors of the 500-mb and 1000-mb 10-day mean height for the Northern Hemisphere (90-WNf.

FIG.7. The root-mean-square error of the 500-mb geopotential height for ensemble average of four cases and time mean over day 10-30 in monthly forecasts by the GCM with the A model (a), the F model (b), the E model (c), and the FM physics model (d). The contour interval is 40 m. The area of error between 160 and 200 rn is lightly shaded, and the area of error larger than 200 m is heavily shaded.

FIG.7. (Continued) 77

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E ( A Z ) (not shown here), are very similar to those of the root-mean-square error, i.e., d E ( A Z ) 2in Fig. 7; this implies that the systematic bias of the models influences the monthly (exactly day 10-30) mean error of individual forecast. At this point, it may be appropriate to discuss the rationale for using the 10-day filter in our study. As Oort and Taylor (1969) demonstrated, there is a substantial peak at day 2-7 in the frequency spectrum of the time series of wind speed near the surface and therefore presumably at 500-mb level as well. This peak corresponds to the traveling cyclones and anticyclones at middle and high latitudes. On the other hand, in the wave number-frequency spectrum of 500-mb geopotential height at 50”N, analyzed by Pratt (1977) and Straus and Shukla (1981), this peak is not found, but there is a hint of the existence of an energy source in the “medium-frequency, synoptic-scale waves, consisting of zonal wavenumber 5-10 with periods of day 2.8-6.9.” This apparent discrepancy is, however, easily reconciled: in Oort and Taylor (1969) the ordinate in the diagram is power multiplied by frequency instead of power. At any rate, it is generally accepted that the frequency range of day 2-7 includes the periodicity of baroclinically unstable waves. Blackmon (1976) and Blackmon et al. (1977) reported that the band-pass (2.5 - 6-day-period) and the low-pass (above-10-day-period) frequency fluctuations have distinctly different characteristics in structure, propagation, and evolution. For this reason, the low-pass filter of less than the frequency of (10 day)-’ appears adequate for the removal of probably unpredictable high-frequency components from the forecasts of monthly time range. It is interesting to note that the remaining part of the spectrum, i.e., stationary and low-frequency planetary and some synoptic-scale waves, constitute a major portion of the total variance (Blackmon et al., 1977). This means that our atmosphere has a “red” spectrum, which may make the long-range forecast more tractable. In our problem, the remaining part is the “signal” and the part we gave up, i.e., the high-frequency baroclinic waves, is the “noise.” The essential part for the long-range forecast is, therefore, the accurate prediction of the behavior of the signal. We have now come to discuss the results of the three-realization (integration) stochastic system. The stochasticity is the fluctuation of the signal and is therefore influenced by the feedback from eddy noise. The conclusions obtained so far are the following. For the monthly time scale, the separation of realizations from each other overall is not appreciable, supporting the result of Shukla (1981). It appears to take a considerable amount of time for the uncertainty (noise) to interact with the slowly varying component (signal) and to grow and spread. In some cases, however, the three realizations are quite separate from

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each other, for example, the case of January 1979, in which the separation is related to the simulation of blocking ridges. In fact, the standard deviation among the three realizations shows large values at high latitudes, 50 - 70"N, associated with the simulation of blocking ridge. Another point is that forecasts of 10-day, 20-day, or monthly mean geopotential height are better at 1000 mb than at 500 mb in terms of correlation coefficient of anomalies between the forecast and observation, the reason being as yet unclear (all eight cases have this feature). This feature is just opposite to the case of day-to-day forecasts on the medium range; i.e., the score of 1000 mb is substantially worse than that of 500 mb.

4. A PROJECTION OF SEASONAL FORECASTS Extrapolating from the past two decades, it may be reasonable to consider that the interest in the next decade will include the seasonal forecasts. The first question that naturally arises is whether the seasonal forecasts are really feasible, and then, if yes, how and to what time range they can be made. Gilman, a head of long-range forecasting at the Climate Analysis Center in Washington, D.C., (personal communication), told us that the upper limit may be four seasons. In dealing with the problem of the seasonal forecasts, perhaps we are now standing in the year corresponding to 1968 for the medium-range forecast described in Section 2. In order to embark on this venture, two directions of research appear to be envisaged; one is the basic research to unravel or investigate the mechanism of the variability of the slowly varying components, and the other is the modeling effort to simulate the process of the long-term evolution of the signal. The former may be divided into the study of internal dynamics of extratropics, external forcings (Lorenz, 1979), and tropical dynamics. The latter may be divided into the study of the improvement of model quality, stochastic prediction, and the air-sea-coupled model. The external forcings are important. Without these considerations, the seasonal forecasts are impossible. However, we will confine our discussion here to only the internal dynamics for the extratropics, because the internal dynamics has received less attention than the forcing in the past. Figure 8 illustrates the factors with relevance to the monthly and seasonal forecasts. In the low-pass variance field (filtered with period of 510 days) of the geopotential height at 300 mb, there are distinct centers of maximum variance in the Northern Hemisphere, i.e., the Aleutians, Iceland, and west of the Urals. These three centers, based on the data of

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FIG.8. The schematic illustration of the low-pass variance of geopotential height at 300mb and high-pass variance at 700 mb together with the jet stream at 300 mb, and the teleconnection indices of Wallace and Gutzler. The variances (standard deviation of height) at 300 rnb larger than 140 m are shown by contours, using a contour interval of 10 rn, whereas the variances at 700 mb larger than 30 m are shown by shading. The 300 mb height data are a 10-yr average and the 700-mb height data are a 9-yr average. (Courtesy of G-N Lau.)

Blackmon et al. (19771, are shown in Fig. 8, marked by the label 300 mb low frequency. Strictly speaking, the contours of the figure were not taken from the paper of Blackmon et al. (1977), but from the 10-year series data provided personally by G-N Lau (1981). These variances are the slowly varying components and are indeed the main targets of long-range forecasts. Closely attached to these centers, like shadows but located rather upstream of them, there are three elongated shades marked by the label 700-mb high frequency (Blackmon et al., 1977). They are the band-pass variance field (designed to retain fluctuations with periods between 2.5 and 6 days) of the geopotential height at 700 mb. These variances are associated with the “storm track” and therefore the baroclinic instability region. In this figure the five categories of Wallace-Gutzler’s teleconnection are also indicated at the respective geographical locations, together with the main jet streams. The questions of the internal dynamics are how these three low-pass variances are generated and what is the nature of their fluctuations. More specifically, whether they are initiated by the barotropic instability and subsequently modulated by the nonlinear interaction or

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whether they all partially produced by the interaction of baroclinic waves (Wallace, 1983). As has been speculated in terms of “zonal index,” the intensity of jet streams is a crucial factor for determining the large-scale flow pattern. One of the new aspects is, however, the meridional structure of the basic flow, which plays an important role together with the orographic effect in producing the realistic climatological flow fields (Hoskins and Karoly , 1981; Held, 1983; Nigam, 1983). The meridional profile of the zonal flow is also essential in generating the observed positive or negative PNA anomaly indices depending on the intensity of the zonal flow between 40 and 60”N (Kinter and Miyakoda, 1983). The barotropic instability associated with zonally asymmetric basic flow is another fundamental issue for understanding the slowly varying waves. This kind of instability is powerful enough to generate the observed magnitude of large-scale disturbances (Simmons et al., 1983). In fact, there is a speculation that the first barotropically unstable mode (the period is about 50 days, the e-folding growth rate is 7.8 days) may include some element of the PNA anomaly pattern. On the other hand, the concept of the baroclinic eddy forcing on the large-scale wave pattern is also plausible. Observational studies of Lau and Wallace (1979) and Holopainen et al. (1982) have shown that there is a considerable contribution from the nonlinear transfer of temperature and vorticity due to the transient eddies to the low-pass variance in the key regions. Fredericksen (1982, 1983) presented a similar theory based on the baroclinic process and a heuristic treatment of “random phase ensemble average.” The “E vector” that Hoskins et al. (1983) proposed is a useful diagnostic tool to evaluate the feedback of the eddies onto the mean flow in three-dimensional space. It is an extension of the Eliassen-Palm flux. Perhaps the most outstanding problem above all in the internal dynamics is the blocking phenomenon. It is intriguing (Blackmon er al., 1977) to note that the three centers of low-pass variance in Fig. 8 correspond closely to the locations that showed the highest percentage frequency of blocking ridges. There is a large body of literature on the subject of blocking. The paper of Charney and DeVore (1979) is most revolutionary in articulating the definition of such an ambiguous phenomenon in terms of mathematical formulation. According to this theory, blocking is one of the multiple equilibria. Whether the theory of multiple equilibria is true in the real world is controversial (Dole, 1983). Another argument is that even if the multiple equilibria exist, they may not be stable due to baroclinic instability (Reinhold and Pierrehumbert, 1982). In a paper on the realistic GCM simulation of the 1977 winter blocking

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(Miyakoda ef at.. 1983), it was postulated that the effect of the transient eddy forcings together with a proper meridional profile of the basic flow upstream of the ridge may constitute the necessary condition for the development and maintenance of the blocking ridge. A similar concept was discussed by Illari and Marshall (1983) and Hoskins ef al. (1983). 5.

POSTSCRIPT

Ten-day forecasts are now considered to be almost feasible so far as the mid-level circulation pattern is concerned, though a great deal of further work is still needed. Our current concerns have been the simulation of monthly range variability by extending the deterministic 10-day forecast approach. In general, the global circulation patterns are a consequence of internal atmospheric dynamics and external boundary forcings. The former is represented by teleconnection characteristic and baroclinicity. There is an increasing amount of evidence to indicate that internal dynamics are indeed essential for the simulation of variability on a monthly time scale. In order to achieve it, the quality of the GCM should be further improved, not only in reducing the systematic bias (climatic drift) but also in raising the capability of reproducing the stationary and quasi-stationary large-scale circulation and of generating sufficient magnitude of the transient eddies. Yet it is certain that seasonal forecasts, if possible, can only be realizable by taking into account the effect of external forcings associated with the anomalies of sea-surface temperature, soil moisture, and snowhce cover (Shukla, 1984). A most impressive example was the winter of 1982-1983; El Nifio and the Southern Oscillation exerted undisputable influence on the gross weather around the world. For this reason, the importance of tropical dynamics and air-sea coupling is already clear. The teleconnection process between the tropics and the extratropics was speculated by Bjerknes (1969) and was tested with a GCM by Rowntree (1972). The Southern Oscillation process perceived by Walker and Bliss was tested with a GCM by Julian and Chervin (1978). The old hypotheses have been promoted to established facts (Horel and Wallace, 1981). The tropical atmosphere and ocean are yet a mystery and therefore fascinating. The Rossby gravity waves and Kelvin waves are ubiquitous in the tropics (Matsuno, 1966; Philander, 1979). The dynamic balance in the extratropics maintained by the geostrophic constraint does not exist, but a different order prevails.

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Shukla (1981) hypothesized that “the tropical atmosphere is potentially more predictable compared to the mid-latitudes, because changes in the boundary conditions produce large changes in the tropics.” It may be challenging to explore whether this hypothesis is valid.

ACKNOWLEDGMENTS We thank Dr. Y. Kurihara, Dr. S. Manabe, and Mr. L. Umscheid for reviewing this essay. We also appreciate the help of Mr. P. Tunison, Mr. J. Connor, and Mrs. J . Pege with the preparation of this paper and Dr. N-G. Lau for providing some figures.

REFERENCES Arakawa, A., and Schubert, W. H. (1974). Interaction of cumulus cloud ensemble with the large-scale environment. Part I. J . Atmos. Sci. 31, 674-701. Bjerknes, J. (1969). Atmospheric teleconnections from the equatorial Pacific. Mon. Weather Rev. 97, 162-172. Blackmon, M. L. (1976). A climatological spectral study of the 500 mb geopotential height of the Northern Hemisphere. J. Atmos. Sci. 33, 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, 10401053. Charney, J. G., and DeVore, J. G. (1979). Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci. 36, 1205-1216. Delsol, F., Miyakoda, K., and Clarke, R. H. (1971). Parameterized processes in the surface boundary layer of an atmospheric circulation model. Q. J . R. Meteorol. Soc. 97, 181208. Dey, C. H. (1969). A note on global forecasting with the Kurihara grid. Mon. Weather Rev. 97, 597-601. Dole, R. M. (1983). Persistent anomalies of the extratropical Northern Hemisphere wintertime circulation. In “Large-Scale Dynamical Processes in the Atmosphere” (B. J . Hoskins and R. P. Pearce, eds.), pp. 95-109. Academic Press, New York. Epstein, E. (1969). Stochastic dynamic prediction. Tellus 21, 739-759. Fredericksen, J. S. (1982). A unified three-dimensional instability theory of the onset of blocking and cyclogenesis. J. Atmos. Sri. 39, 969-982. Fredericksen, 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. Grimmer, M . , and Shaw, D. B. (1967). Energy-preserving integrations of the primitive equations on the sphere. Q. J . R. Meteorol. Soc. 93, 337-349. Held, I. M. (1983). Stationary and quasi-stationary eddies in the extratropical troposphere: Theory. In “Large-Scale Dynamical Processes in the Atmosphere” (B. J. Hoskins and R. P. Pearce, eds.), pp. 127-168. Academic Press, New York. Holloway, J. L., and Manabe, S. (1971).Simulation of climate by a global general circulation model. I. Hydrological cycle and heat balance. Mon. Weather Reu. 99,335-370. Holopainen, E. O., Rontu, L., and Lau, N-C (1982). The effect of large-scale transient eddies on the time-mean flow in the atmosphere. J . Atmos. Sci. 39, 1972-1984. Horel, J. D., and Wallace, J. M. (1981). Planetary scale atmospheric phenomena associated with the Southern Oscillation. M o n . Wearher Rev. 109, 813-829.

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Hoskins, B. J., and Karoly, D. J. (1981). The steady linear response of a spherical atmosphere to thermal and orographic forcing. J . Atmos. Sci. 38, 1179-1 196. Hoskins, B. J., James, I. N., and White, G. H. (1983). The shape, propagation and meanflow interaction of large-scale weather systems. J . Atmos. Sci. 40, 1595-1612. Illan, L., and Marshall, J. C. (1983). On the interpretation of eddy fluxes during a blocking episode. J . Atmos. Sci. 40, 2232-2242. Julian, P. R., and Chervin, R. M. (1978). A study of the Southern Oscillation and Walker Circulation Phenomenon. J . Atmos. Sci. 106, 1433-1451. Kinter, J. L. (1983). Barotropic studies of stationary extratropical anomalies in the troposphere. Ph.D. Dissertation, GFD Program, Princeton University, Princeton, New Jersey. Kinter, J. L., and Miyakoda, K. (1983). A numerical study of planetary scale waves and blocking patterns using barotropic models. In “Research Activities in Atmospheric and Oceanic Modeling. Numerical Experimentation Programme, World Climate Research Programme” (I. D. Rutherford, ed.), Rep. No. 5 , pp. 3.4-3.7. Kurihara, Y.,and Holloway, J. L. (1967). Numerical integration of a nine-level global primitive equations model formulated by the box method. Mon. Weather Rev. 95,509530. Lau, N-C (1981). A diagnostic study of recurrent meteorological anomalies appearing in a 15-year simulation with a GFDL general circulation model. Mon. Weather Rev. 109, 2287-23 1 1. Lau, N-C, and Wallace, J. M. (1979). On the distribution of horizontal transports by transient eddies in the northern hemisphere wintertime circulation. J . Atmos. Sci. 36, 18441861. Lorenz, E. N. (1979). Forced and free variations of weather and climate. J. Atmos. Sci. 36, 1367-1376. Manabe, S., Smagorinsky, J., and Strickler, R. F. (1965). Simulated climatology of ageneral circulation model with a hydrologic cycle. Mon. Weather Rev. 93, 769-798. Matsuno, T. (1966). Quasi-geostrophic motions in the equatorial area. J . Metmrol. Soc. Jpn., 44, 25-43. Mellor, G. L., and Yamada, T. (1974). A hierarchy of turbulence closure models for planetary boundary layers, J. Atmos. Sci. 31, 1791-1806. Miyakoda, K., and Sirutis, J. (1977). Comparative integrations of global models with various parameterized processes of subgrid-scale vertical transports: Description of the parameterizations. Contrib. Atmos. Phys. 50, 445-487. Miyakoda, K., and Sirutis, J. (1983). Impact of subgrid-scale parametenzations on monthly forecasts. In “ECMWF Workshop on Convection in Large-Scale Models,” pp. 231277. Miyakoda, K., Moyer, R. W., Stambler, H., Clarke, R. H., and Strickler, R. F. (1971). A prediction experiment with a global model of the Kurihara-grid. J . Meteorof.SOC.Jpn. 49, Spec. Issue, 521-536. Miyakoda, K., Sadler, J. C., and Hembree, G. D. (1974). An experimental prediction of the tropical atmosphere for the case of March 1965. Mon. Weather Reu. 102, 571-591. Miyakoda, K., Gordon, T., Caverly, R., Stem, W., Sirutis, J., and Bourke, W. (1983). Simulation of a blocking event in January, 1977. Mon. Weather Rev. 111, 846-869. Namias, J. (1968). Long-range weather forecasting-history, current status and outlook. Bull. Am. Meteorol. SOC.49, 438-470. Nigam, S. (1983). On the structure and forcing of tropospheric stationary waves. Ph.D. Dissertation, GFD Program, Princeton University, Princeton, New Jersey. Oort, A. H., and Taylor, A. (1969). On the kinetic energy spectrum near the ground. M o n . Weather Rev. 97, 623-636.

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Philander, S. G. H. (1979). Variability of the tropical oceans. Dyn. A m o s . Oceans 3, 191208. Ploshay, J., White, R., and Miyakoda, K. (1983a). FGGE level 111-B daily global analyses. Part I. NOAA Data Rep. ERL-GFDL-1, 1-278. Ploshay, J., White, R., and Miyakoda, K. (1983b). FGGE level 111-B daily global analyses. Part 11. NOAA Data Rep. ERL-GFDL-1, 1-285. Ploshay, J., White, R., and Miyakoda, K. (1983~).FGGE level 111-B daily global analyses. Part 111. NOAA Data Rep. ERL-GFDL-1, 1-285. Ploshay, J., White, R., and Miyakoda, K. (1983d). FGGE level 111-B daily global analyses. Part IV. NOAA Data Rep. ERL-GFDL-1, 1-282. Pratt, R. W. (1977). Space-time kinetic energy spectra in mid-latitudes. J. A m o s . Sci. 34, 1054- 1057. Reinhold, B. B., and Pierrehumbert, R. T. (1982). Dynamics of weather regimes: quasistationary waves and blocking. Mon. Weather Reu. 110, 1105-1 145. Rowntree, P. R. (1972). The influence of tropical east Pacific Ocean temperature on the atmosphere. Q . J . R . Meteorol. Soc. 102, 607-625. Sankar-Rao, M., and Umscheid, L. (1969). Tests of the effect of grid resolution in a global prediction model. Mon. Weather RPU.97, 659-664. ShuMa, J. (1981). Dynamical predictability of monthly means. J. A m o s . Sci. 38,2547-2572. Shukla, J. (1984). Predictability of time averages. Part 11. The influence of the boundary forcings. In “Problems and Prospects in Long and Medium Range Weather Forecasting” (D. M. Bumdge and E. Kallen, eds.), pp. 155-206. Springer-Verlag, Berlin and New York. Simmons, A. J., Wallace, J. hl., and Branstater, G. W. (1983). Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci. 40,1363-1392. Srnagorinsky, J. (1969). Problems and promises of deterministic extended range forecasting. Bull. A m . Meteorol. SOC.50, 286-311. Smagorinsky, J., Manabe, S., and Holloway, J. L. (1965). Numerical results from a ninelevel general circulation model of the atmosphere. Mon. Weather Rev. 93, 727-768. Srnagorinsky, J., Strickler, R. F., Sangster, W. E., Manabe, S., Holloway, J . L., and Hembrie, G. D. (1967). Prediction experiments with a general circulation model. In “Proceedings of the IAMAPlWMO International Symposium on Dynamics of Large Scale Processes in the Atmosphere,” pp. 70-134. Moscow. Straus, D. M., and Shukla, J. (1981). Space-time spectral structure of a GLAS general circulation model and a comparison with observations. J. A m o s . Sci. 38, 902-917. Umscheid, L., and Bannon, P. R. (1977). A comparison of three global grids used in numerical prediction models. Mon. Weuther Reu. 105, 618-635. Wallace, J. M. (1983). On the structure and the evolution of low-frequency atmospheric variability. I n Proceedings of the WMOlCASlJSE Expert Study Meeting on Longrange Forecasting, Princeton, December, 1982, “LRF Publ. Ser. No. 1, pp. 50-57. World Meteorol. Organ., Geneva. Wallace, J. M., and Gutzler D. S. (1981). Teleconnections in the geopotential height field during the Northern Hemisphere winter. Mon. Weather Rev. 109, 784-812. Wallace, J . M., Tibaldi, S., and Simmons, A. J. (1983). Reduction of systematic forecast errors in the ECMWF model through the introduction of an envelope orography. Q . 1. R . Meteorol. SOC. 109, 683-717.

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PREDICTABILITY J. SHUKLA Center for Ocean-Land-Atmosphere Interactions Department of Meteorology Wniuersity of Maryland College Park, Maryland

Introduction . . . . . . . . . . . . . . . . . . . . Classical Predictability Studies . . . . . . . . . . . . . 2.1. Simple Models . . . . . . . . . . . . . . . . . 2.2. Observations (Analogs) . . . . . . . . . . . . . . 2.3. General Circulation Models . . . . . . . . . . . . 3. Predictability of Space-Time Averages. . . . . . . . . . 3.1. Dynamical Predictability . . . . . . . . . . . . . 3.2. Boundary-Forced Predictability. . . . . . . . . . . 3.3. Prospects for Dynamical Extended-Range Forecasting . . 4. Some Outstanding Problems . . . . . . . . . . . . . . 4.1. Mean (Climate Drift) and Transient Predictability. . . . 4.2. Observational Errors and Model Errors . . . . . . . 4.3. Predictability of Predictability . . . . . . . . . . . 5. Concluding Remarks. . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . 1.

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1. INTRODUCTION

The mathematical equations governing the dynamics of the atmospheric flows are nonlinear, and the observed structure of the atmosphere is characterized by horizontal and vertical gradients of wind, temperature, and moisture that permit hydrodynamical and thermodynamical instabilities to grow. These characteristics of atmospheric motion are the primary reason for an upper limit on the deterministic predictability of atmospheric flows. In addition, the equations and the physical parameterizations used for prediction are not exact, and they introduce a source of error in predictions made with a model. Even if the models were perfect, small uncertainties in the initial state can grow due to the inherent instability of the flow and nonlinear interactions among motions of different space and time scales. The quantitative upper limit for deterministic prediction, even for an exact model, is determined by the growth rates and equilibration of the most dominant instabilities. During the past three decades there have been several attempts to estimate the upper limit for deterministic prediction of the instantaneous state of the atmosphere, to be referred to as the weather. There are some 87 ADVANCES IN GEOPHYSICS, VOLUME

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conceptual difficulties in amving at an estimate of upper limit for deterministic predictability. For example, how does one decide that an upper limit has been reached? The most common answer to this question has been that if the differences between two predictions made with identical models but slightly different initial conditions become equal to the differences between two randomly chosen weather maps (for the same time of year), then the upper limit of prediction has been reached. Even if we accept this as a working definition for the upper limit, we have yet to define a way of measuring the differences. For example, should it be the root-mean-square difference between the two maps averaged over all the appropriately scaled variables or correlation coefficients or some other measure of space time variability? It should be pointed out that the aforesaid working definition of the upper limit of predictability implicitly assumes that we already have a good knowledge of the climatology of the atmosphere and therefore we do not consider a prediction to be of any value if it does not improve on our pre-existing knowledge of the climatology. The concept of usefulness is, therefore, implicit in this definition, although it does not follow that the prediction for a range of time equal to the upper limit as defined above has any practical utility. There have been some attempts to distinguish between the upper limits of theoretical and practical (useful) predictability. In some of the earlier works, predictability was defined by a single parameter, the growth rate for root-mean-square error over large areas (one hemisphere or the globe). For example, if the doubling time for the error in temperature is about 3 days and if the initial error is about 1"C, assuming a constant growth rate with time, the error will become about 8°C in 9 days. If 8°C is the root-mean-square error between randomly chosen weather maps, the upper limit will be considered to be 9 days. The difficulty in defining predictability in terms of growth rate arises mainly because the growth rate depends on the structure of the initial large-scale flow and the value of the maximum permissible error strongly depends on the latitude and season. The growth rate and equilibration also depend on the variable under consideration. In this chapter, we shall first review the earlier attempts to determine the limits of atmospheric predictability. We shall describe the results from simple models and turbulence theories in Section 2.1, the analog method in Section 2.2, and results from global general circulation models in Section 2.3. In Section 3 we shall review the work on predictability of space-time averages, and in Section 4 we shall review some of the outstanding problems of predictability. In Section 5 we shall present the concluding remarks.

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2. CLASSICAL PREDICTABILITY STUDIES

We shall use the words classical predictability studies to refer to those works during the past 30 years in which an attempt was made to arrive at a quantitative estimate of the growth rate of an initial error and to determine the limits of predictability. Such studies have used either simple models of atmospheric flow or turbulence models or complex models of the general circulation of the atmosphere with explicit treatments of mechanical and thermal forcings. Historical records of the atmospheric flows have also been examined to find naturally occurring analogs and the growth rate of their initial differences.

2.1. Simple Models The first such study reported in the literature is the one by Thompson (1957), who showed, using a simple barotropic model, that the initial errors tend to grow with time and that the atmospheric flow is not predictable beyond a week. He emphasized the effects of the scale of initial error on predictability and commented on the relative merits of expanding the observational network and improving the models for weather prediction. It was implicit in his work that the instability of the atmospheric flow is the main reason for limits on predictability. He also introduced the concept of the error between two randomly chosen maps as a convenient upper limit of the error beyond which the flow is competely unpredictable. He also showed that the zonally averaged flow is more predictable than the unaveraged flow. Simple predictability experiments were carried out, serendipitously , by Lorenz (1963) in connection with his work on numerical integration of a simplified nonlinear baroclinic model. The motivation for integrating the model and producing a long-time series was to test the ability of a linear statistical model to predict the behavior of a hydrodynamical flow using the data generated by the nonlinear governing equations. However, since the computing facility available to Lorenz at that time was far inferior to the present-day personal computers, from time to time he had to re-enter the solution printed by the machine to continue the integration further. Since the machine did the computations with an accuracy of six significant digits, but printed out only three digits, solutions were changed in the last three digits every time new values were punched in. Lorenz noticed that if calculations were repeated with such rounding off, the solutions began to diverge and that for longer integrations they became quite different.

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Soon thereafter, Lorenz (1965) wrote a comprehensive paper on the predictability of a 28-variable atmospheric model. He used a two-layer quasi-geostrophic model having a zonal flow with two north-south modes and perturbations with three east-west wavelengths each with two north-south modes in each layer. He showed that the doubling time for the initial errors strongly depended on the structure of the flow [a conclusion repeatedly confirmed by the general circulation model (GCM) experiments and the experience of operational weather prediction] and that for synoptic-scale observation errors the doubling time may range from a few days to a few weeks. On the average, the doubling time was about 4 days. He also noticed that although instantaneous flow patterns become completely unpredictable after a few days, some properties of the flow remain predictable much beyond that time. This will suggest some possibility for predicting space-time averages. Several investigators (Robinson, 1967; Lilly, 1969; Lorenz, 1969a; Leith, 1971; Leith and Kraichnan, 1972; Lorenz, 1984) have also used turbulence models to determine the predictability of an idealized hydrodynamical flow. These models do not include spherical geometry and the rotation of the Earth, nor do they include thermal and mechanical forcing functions and the physical processes of radiation and condensation. They do, however, provide useful insight into the error growth characteristics due simply to nonlinear interactions among the various scales of the fluid motion. These studies necessarily require a priori assumptions about the spectra of the kinetic energy of the atmosphere, and the results are quite sensitive to such assumptions. Robinson (1967) proposed the idea of a virtual viscosity that would dissipate eddies of all sizes, but with the time taken for dissipation of a particular scale (which is a measure of the predictability time for that scale) depending on the rate at which energy from that scale is transferred to the scales at which “true” dissipation takes place. He used a -3 power law for the large-scale atmospheric energy spectrum. Simply stated, Robinson’s concept of limited predictability is based on an assumption that eddies get “diffused” or “dissipated away” in a finite time and therefore there is no hope for predictions beyond a few days. This line of reasoning is not consistent with our intuition (based on observations of the atmosphere) that eddies do not get dissipated but are maintained by well-defined physical processes, and hence the problem of predictability is not the nonexistence of eddies, but rather their growth, movement, and decay, as well as their interactions with other scales of motions. Robinson’s concept of dissipative time scales seems more appropriate for determining the time steps for numerical integrations of atmospheric models rather than for determining the predictability of the atmosphere.

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Lorenz (1969a) (also using a -8 power law for the energy spectrum) calculated the time taken for each scale of the motion to be totally unpredictable, defined as the state at which error energy in a given scale becomes equal to the energy at that scale in the initial prescribed spectrum. He found that the interactions take place only among the adjacent scales; however, the error in the smaller scales gets saturated rather quickly. Even if the synoptic scales were free of any error initially, errors from the neighboring smaller scales produced errors in synoptic scales within a day or so. Lorenz further suggested that the predictability would be increased if the energy spectra had a -3 power law rather than -3. Leith (1971) and Leith and Kraichnan (1972) used improved turbulence closure approximations and showed that for the two-dimensional eddy kinetic energy spectrum similar to the one observed in the atmosphere, the doubling time for error was about 2 days. It is rather interesting that these estimates of error-doubling time are quite close to the estimates made by current state-of-the-art GCMs and also the estimates made by Lorenz (1969b) by using analogs in the past observations of the atmosphere. In his paper, Lorenz (1984) has shown that the presence of a moderately strong spectral gap in the mesoscale range of the assumed energy spectrum will increase the predictability by about 3 more days. Lorenz’s calculations suggest that the error level at day 1 without the spectral gap will be about the same as the error at day 4 in the presence of the spectral gap. 2.2. Observations (Analogs)

Dr. J. Namias once remarked that the analog method of weather forecasting is as old as the second weather chart. Before the advent of the statistical and dynamical models for weather prediction, the analog method was perhaps the most commonly used technique for weather forecasting. Even now analogs are commonly used to make extended-range predictions. Lorenz (1969b, 1973) proposed an ingenious method of studying classical atmospheric predictability using naturally occurring analogs from past records of atmospheric observations. He proposed that if it were possible to find two rather closely resembling atmospheric states, the rate with which the differences between the two states grow would give a measure of the classical predictability error growth. He used five years (1963-1967) of twice-daily height data over the Northern Hemisphere for the 200-, 500-, and 850-mb surfaces to carry out his search

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for good analogs. He could not find good analogs: the difference (rms error) between the two states corresponding to his best analog pair was 62% of the error between two randomly chosen states. Lorenz found that the doubling time for error between these “mediocre” analogs was about 8 days. Since the main objective was to find the growth rate of small errors, Lorenz extrapolated the growth rate for small errors from the knowledge of the growth rate for large errors. For this he proposed a quadratic hypothesis for error growth rate that gave a doubling time of 2.5 days for small errors. It is rather remarkable that this estimate of doubling time is very close to the estimates from the state-of-the-art global general circulation models. Considering the crudeness of the technique utilized, the quantitative exactness of this result should be considered as a combination of Lorenz’s brilliance and serendipity. As Lorenz pointed out, a cubic hypothesis would have given a doubling time of 5 days, but the data did not show as good a fit. Lorenz further suggested that the chances of obtaining really good analogs does not seem to be good even for larger data sets. However, it may still be worthwhile to process large data sets once they are available. Gutzler and Shukla (1984) have analyzed 15 years (1963-1977) of winter season daily, 500-mb height observations for the Northern Hemisphere to search for analogs. They restricted their search to 500 mb only because of the equivalent barotropic nature of a significant part of the atmospheric variability. They examined the analogs for the planetary waves and the synoptic waves separately and also for limited spatial domains. They also looked for analogs for 5-day-mean maps. They found that by considering the 15-year data, but only at 500 mb, the root-mean-square (rms) error between the best pair of analogs was only about 50% of the rms error between randomly chosen maps. This percentage error was reduced to 40% if rms error was calculated for the planetary waves only (zonal wave numbers 0-4), and further reduced to about 32% for limited regions over the North American and European sectors. The error doubling time was also reduced for reduced errors between the analog pairs, indicating thereby a faster growth for smaller errors. They also examined the accuracy of short-range predictions based on the best analogs, and in each case, with the exception of synoptic waves (wave numbers 5-36), such short-range predictions were inferior to the corresponding persistence forecasts. This was true even for the 5-day-mean circulation maps. Surprisingly, the rms error between the best pair of 5-day-mean analogs was close to 69% of the error between two randomly chosen 5-day-mean maps. In summary, the work of Gutzler and Shukla using natural analogs supports the last statement of Lorenz’s (1969b) paper, “Probably we can

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gain some additional insight into our problem by processing the largest sample of data which we can assemble, but we must not expect miracles.”

2.3. General Circulation Models Charney et al. (1966) were the first to apply general circulation models (GCMs) to the study of the classical predictability of model-simulated atmospheric circulations. They utilized the three GCMs described by Smagorinsky (1963), Mintz (1964), and Leith (1965) to examine the growth rate of initial sinusoidal and random temperature error fields. The results were highly model dependent; the error growth characteristics were quite different for each of the models. The Leith model showed a rapid decay of the initial error for the first 4 days, followed by an error increase to half of its initial value up to day 7. After day 10, the error began to level off. Thus, the expected exponential growth of the initial error was not manifested by the Leith model. The Mintz- Arakawa model showed a near-exponential growth of error after an initial drop for a few days. The doubling time of the error was estimated to be about 5 days. The Mintz-Arakawa results were considered to be the most realistic because this was the only model that exhibited strong aperiodic behavior during a long-term (about 300 days) integration of the model and also because the error growth characteristics were similar to what one could expect from theoretical considerations. The Smagorinsky model was integrated with initial random and sinusoidal temperature error fields of various amplitudes (0.02, 0.1, 0.5, and 2.0 K). For small initial error amplitudes, the error grew very slowly for the first 30 days, after which it showed a doubling time of 6 to 7 days. However, the actual flow patterns showed a primarily periodic behavior. It should be remarked that this large divergence among the results of various models is merely a reflection of the fact that none of the models were realistic. Predictability experiments with state-of-the-art models available today will show more convergent results. Later papers on classical predictability studies with GCMs were reported by Smagorinsky (1969), Jastrow and Halem (1970), and Williamson and Kasahara (1971). The model used by Smagorinsky was the improved version of the earlier model described by Miyakoda et af. (1969) and Manabe et al. (1965). The model used by Jastrow and Halem was the improved and modified version of the earlier Mintz-Arakawa model. Williamson and Kasahara (1971) used the model developed at the National Center for Atmospheric Research (NCAR). One of the

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important conclusions of the Jastrow-Halem and Williamson-Kasahara papers was that the growth rate of the initial error depended on the resolution of the model; error growth was slower for coarse resolution models. The most comprehensive study of the classical predictability at that time was reported by Smagorinsky (1969), presented as the Wexler memorial lecture at the 49th Annual Meeting of the American Meteorological Society. For the first time, he raised the question of predictability of different spectral modes. In fact, perhaps because this study was so comprehensive (and perhaps because of the reputation of the author), no major work on classical predictability was published for the next 12 years, although a large effort was devoted to actual weather prediction. Smagorinsky presented results of error growth for various initial error amplitudes using two different model resolutions. For an initial random error of about 0.25”C, the doubling time was about 2.5 days, but it took about 7 days for the error of 1°C to double to 2°C. Although the nature of the error growth with time was consistent with the theoretical concepts of hydrodynamical instabilities, the quantitative estimate of the growth rate of the error was quite different from the one suggested by Lorenz, which used simple models or observed analogs. Lorenz’s estimates for the doubling time of large-scale error fields (as expected to be for a typical observational network) was about 2 to 3 days, whereas Smagorinsky’s model results suggested a doubling time of about 5 to 7 days. As evidenced by a 800-word footnote in Smagorinsky’s paper, this difference in the result produced very useful and sharp discussion in the field. It is now generally agreed that Lorenz’s estimates were quite close to the present estimates using the current state-of-the-art GCMs. Recently there has been a renewed interest in classical predictability studies. Some examples of the recent works are found in papers by Shukla (1981a), Lorenz (1982), Baumhefner (1984), and Shukla (1984a). Error growth rates have now been examined separately for the tropics and mid-latitudes, for winter and summer, for different space scales, for different hemispheres, and for different initial conditions. In this section, we shall summarize some conclusions from these studies. In particular, we shall show the actual results from some of the integrations of the GLAS climate model. These results are, naturally, model dependent, and it is quite likely that the results from a different GCM with different treatments of numerics and physics will give different quantitative results. We believe, however, that our results on the relative predictability of winter and summer seasons, tropics and mid-latitudes, Northern and Southern Hemispheres, and large and small scales and for random and systematic initial errors will remain unchanged for any GCM that produces a reasonable simulation of climate and its variability.

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The model used for these predictability studies has been described by Shukla et al. (1981). It is a global primitive-equation model with nine levels in the vertical and a horizontal resolution of 4" latitude by 5" longitude. The model includes parameterizations of radiation, convection, and fluxes at the Earth-atmosphere interface. The observed annual cycle of sea-surface temperature, soil moisture, snow, and sea ice is prescribed at the model grid points. We shall present results of 30-day integrations using nine winter initial conditions and four summer initial conditions. The model was first integrated with the observed initial conditions of 1 January 1975 (control run), and then two additional integrations were carried out with random perturbations in the initial conditions. The rms error between the control run and the first perturbation run will be referred to as E l l , and the rms error between the control run and the second perturbation run will be referred to as E12. Similarly, observed initial conditions of 1 January 1977 were integrated along with three perturbations, and rms error between this control and three perturbations will be referred to as Ezl, E22, and E z ~respectively. , The rms error between a control run starting from the observed initial conditions of 1 January 1978 and one perturbation run will be referred to as For each perturbation run, the statistical properties of the random perturbation to the initial conditions were the same (a spatially Gaussian distribution with zero mean and standard deviation of 3 m s - I in u and u components at all the model grid points and at all the levels), but the actual grid-point values of the random perturbations were different for different cases. These integrations were earlier used by Shukla (198 la) to study the dynamical predictability of monthly means. Similar integrations were carried out for the summer season by using the observed initial conditions in the middle of June as control run and three perturbation runs. In the past, most attention was paid to the error growth rate (or doubling time) as the key predictability parameter. This is not a very useful parameter, partly because it varies greatly for different values of the error and partly because the ultimate limit of predictability is not only determined by the growth rate, but also by the saturation value of the error. This becomes an important consideration when we are examining the predictability of different seasons and different parts of the globe. We have, therefore, presented the results for error growth with time, as well as the ratio of error to the standard deviation of daily fluctuations. A larger error growth does not necessarily mean lesser predictability because it will also depend on the equilibration value of the error that depends on the magnitude of the day-to-day fluctuations. Figures 1-3 show the results for sea-level pressure, geopotential height at 500 mb, and wind at 300 mb, respectively. Each figure has four panels,

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which represent the rms error (a) and the ratio of error to standard deviation (b) for the winter season and similarly (c) and (d) for the summer season. The error in panel (a) represents the geometric mean error, the square root of average value of squares of Ell, E12,E21, E22,E Z 3 ,and E31; and the standard deviation (STD) used in panel (b) represents the square root of the sum of squares for all daily values of deviations for all integrations for that season. These figures show the dependence of predictability on latitude, season, and the weather variable in question. The main conclusions from the results of the classical predictability studies described in the preceding section and presented by several other investigators are summarized in the following subsections.

2.3.1. Predictability of the Tropics and Extratropics. Since the growth rate and equilibration mechanisms for the tropical and mid-latitude instabilities are quite different, it is desirable to examine their predictability separately. It is recognized that there is considerable interaction between the tropics and mid-latitudes. However, since the time scales of growth and equilibration of synoptic-scale tropical disturbances is much smaller than that of the tropical-extratropical interactions, we are justified in examining their predictability separately. In an earlier paper (Shukla, 1981b), the author has shown that the limit of deterministic predictability for the tropics is only 3 to 5 days compared to 2 to 3 weeks for the mid-latitudes. This is because the standard deviation of the day-to-day fluctuations (which is the saturation value of errors) is much smaller in the tropics and because the instabilities associated with the growth of the tropical disturbances are driven by moist convection, leading to larger growth rates than those of the dynamical instabilities of the mid-latitudes, which are driven by horizontal or vertical wind shear. Some of the earlier studies on predictability examined only the global or the hemispheric average rms error, and since the tropical errors are small in magnitude, results were dominated by the mid-latitude errors. Thus the results on tropical predictability were overlooked. Figure 4 from Shukla (1981b) shows the rms error averaged over 10" latitude belts centered at 6, 30, and 58"N for sea-level pressure. The equilibration value of the error is largest for 58"N and smallest for 6"N, reflecting the latitudinal variability of the daily standard deviation. It is also seen that the initial growth of error is the largest for the tropics. Thus, a combination of faster growth rate and smaller equilibration value makes the tropical regions of the globe the least predictable for day-to-day weather forecasting. This conclusion is not inconsistent with the current experiences in operational weather forecasting, where it has been found that the skill of tropical forecasts is not better than that of a persistence forecast, even at day 2 or 3.

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This result is further supported by Figs. 1-3, in which it is seen that for each variable and each season, although the rms error is quite small in the tropics, the ratio of rms error to standard deviation is more than 0.5 within 1 to 5 days, whereas it takes about 5 to 12 days to reach that value for the mid-latitudes. These conclusions are based on an assumption of an idealized, initial error field over the whole globe. In reality, the observational network over the tropics is worse than that over the mid-latitudes, and the prospects for deterministic prediction of day-to-day weather in the tropics appear to be rather dim. The prospects for predicting space-time averages, on the other hand, are good. It will be shown in a later section that partly because of smaller day-to-day variability in the tropics, and partly because of strong influence of boundary conditions, the space-time averages are more predictable in the tropics than in the mid-latitudes.

2.3.2. Predictability of the Northern and Southern Hemispheres. In an article by Louis Purret (1976) in the NOAA magazine Smagorinsky conjectured, “I would suspect that there is a little less predictability in the Southern Hemisphere than there is in the Northern Hemisphere.” Our results support this conjecture. The equilibration value for the error is

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higher in the Northern Hemisphere than in the Southern Hemisphere. The absence of large, stationary asymmetric boundary forcings in the Southern Hemisphere reduced the amplitude and variability of planetary scales, which in turn reduces the equilibration value of the errors. The possibility of enhancement of predictability due to the presence of stationary forcings was supported by another study, in which we examined the predictability of the idealized atmosphere of an ocean-covered Earth, and it was found to be smaller than the predictability of the atmosphere with mountains at the Earth's surface. 2.3.4. Predictability during the Winter and Summer Seasons. As shown in Figs. 1-3 and discussed earlier with Shukla (1984a), circulations during the Northern Hemisphere winter are more predictable than those during the summer. This provides a good illustration for the point that the error growth rate alone is not an adequate parameter to describe predictability. Although the error growth rate is higher during winter than in summer, the day-to-day variability during winter is also considerably larger than during summer, so that it takes longer for the initial error to reach its saturation value during the winter season. It should be pointed out that on the basis of the values of error growth rate alone, Charney et al. (1966) had erroneously concluded that circulations during the summer season might be more predictable than those during the winter season. Results of operational numerical weather prediction are not inconsistent with the conclusion arrived at by our predictability studies.

2.3.5. Predictability of Planetary and Synoptic Scales. Smagorinsky (1969) was the first to examine the predictability of different scales separately, and he correctly concluded that the larger scales are more predictable than the smaller scales. However, it was not clear whether a short sample of one case could resolve the predictability of various scales. We examined this question again (Shukla, 1981a) by using six pairs of control and predictability integrations, and the results are reproduced in Fig. 5 . It is seen that for the latitude belt 40-60"N for 500 mb, predictability of planetary scales (wave numbers 0-4) is more than 4 weeks compared to about 2 weeks for synoptic scales (wave numbers 5-12). For wave numbers 13-36, predictability was only a few days. It is interesting to note, however, that the initial growth rate for both the planetary and synoptic scales is nearly the same. The doubling time of initial small errors is about 2.5 days and once the error has reached a value of about 25 m, the doubling time is close to 3 days. The higher predictability of the planetary waves is due to higher values of their amplitude and variability. If the error growth for the synoptic scales were attributed to the fast-growing dynamical instabilities at those scales, it will be of interest to determine

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the relative importance of planetary scale instability itself and the influence of synoptic scales in making the planetary scales unpredictable.

2.3.6. Predictability of High- and Low-Resolution Models. Smagorinsky (1969) compared the error growth for models with two different resolutions. The low-resolution model had 20 grid points between the equator and pole, with a grid size of 640 km at the pole and 320 km at the equator. The high-resolution model had 40 grid points between the equator and pole. He did not find any significant difference in the doubling time

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for initial errors for high- and low-resolution models, although the persistence error for the high-resolution model was clearly larger than that for the low-resolution model. Subsequent papers by Jastrow and Halem (1970) and Williamson and Kasahara (1971) showed that the doubling time of the error decreased for higher spatial resolution of the model and that the doubling time for the synoptic-scale errors was closer to 3 days rather than 5 days as reported by Smagorinsky.

2.3.7. Predictability of “Balanced” and “Unbalanced” Initial States. Daley (1980) has shown that the error growth in the rotational component of the flow could be rather small if the initial error was only in the gravitational component. This might explain, at least partially, the faster growth rate of spatially coherent initial errors compared to purely random errors. In a GCM, through convection and other diabatic processes, errors in gravitational component will also soon feed back to the rotational components. Besides the question of dynamical balance between the mass and motion fields, there is also the question of consistency between the observed initial conditions used as input for a GCM integration and the boundary conditions of sea-surface temperature (SST), soil moisture, sea ice, snow, etc., used in the model. It is quite conceivable that this inconsistency can be an additional source for error growth. However, it can be argued that in classical predictability studies with GCMs, the errors due to imbalances in the initial and boundary conditions should not be too large because of the “perfect model” assumption in which time growth of a small initial perturbation is examined. We have attempted to address this question by repeating our predictability error growth calculations for an initial condition that was obtained after integrating a GCM for 30 days. Figure 6 shows the 500-mb rms error averaged for 40-60”N for six pairs of integrations during the winter season. The curves labeled E l l , E21,and E31 refer to the rms error between a control run that started from an observed initial condition and a perturbation run. In each case, the control run was extended up to 60 days. At the end of the 30 days, similar random perturbations were introduced and the rms error calculated for 15 days. The curves labeled E i I ,E ; I , and Eil in the lower part of Fig. 6 show the time growth of rms error for the three cases corresponding to E l l , and E31, respectively. It is quite clear that the growth rates of error for integrations starting from day 30 of the control run are smaller than those from the observed initial conditions. This result suggests that improvements in methods for analysis and initialization hold promise for improvements in short-range weather prediction. It is difficult to determine whether the above reduction in the error growth was due to a better balance between the mass and motion fields (i.e.,

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reductions in the gravitational component of the flow) or due to reductions in the inconsistency between the atmospheric flow and the underlying boundary conditions. 2.3.8. Dependence of Predictability on the Structure of Large-Scale Flow. Since the nature of error growth is determined by the nature of the dominant instabilities, which in turn depends on the dynamical structure of the flow fields, it is natural to expect that different initial conditions will show different predictability characteristics. Figure 6 shows the time series for the error field for six different pairs of control and predictability integrations. The random perturbation in the initial conditions had the same statistics (zero mean and standard deviation of 3 m s - I for u and u components) for all the cases. The curves labeled Ell,E21, and EN refer to three different initial conditions. It is clearly seen that the error growth rate depends on the initial conditions. It is therefore natural to expect that, for this reason alone, the skill of numerical weather prediction will

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not be the same on each day, although in operational forecasting the quality and quantity of input observations could also vary from one day to the other.

2.3.9. Dependence of Predictability on the Structure and Magnitude of the Initial Error. Figure 6 also shows the effects of structure of the initial error field on predictability. For example, the curves labeled E l l , Ez2,and Ez3not only had identical planetary- and synoptic-scale flow, but even the statistics of the random perturbation to the initial conditions were the same. The only differences were the actual grid-point values of the random error, and that was sufficient to produce differences in the rate of error growth. This again provides at least a partial explantation for the day-to-day changes in the skill of operational numerical weather prediction. In another set of experiments, we smoothed the initial random error over the oceans, corresponding to the assumption that oceanic errors are more systematic than those over land, which can be assumed to be mostly instrumental and therefore random. The magnitudes of the individual grid-point values were adjusted to keep the standard deviation of the error field the same as that for the globally random error. We found that the growth rate for spatially coherent initial error was larger than that for the random initial error. These results raise some interesting questions about the relative virtues of possible observing systems with uniform but large error over the whole globe compared to the existing system of relatively smaller error over the land and larger errors over the oceans. The dependence of error growth rate on the size of the error itself was well recognized from the earlier pioneering works of Lorenz (1969a) and Smagorinsky (1969). Lorenz had shown that very small errors confined to very small scales grow much faster than larger errors at larger scales. Smagorinsky had also shown that the smaller the error, the faster the growth rate. There is a smallest scale that can be resolved by a GCM, and GCM calculations can address the question of error growth only for scales larger than that. In earlier predictability studies by Shukla (1981b, 1984a), it was found that the doubling time for tropical errors was quite large. This was merely a manifestation of the fact that the error over the tropics had almost reached their saturation value, and there was no possibility for the error to double again. For example, an initial rms error of 1°C in temperature, of 3 mb in sea-level pressure (Jastrow and Halem, 1970), was already comparable to the saturation value of the error in the tropics. 2.3.10. Predictability of Different Variables. Figures 1-3 show the results for predictability of sea-level pressure, geopotential height, and

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wind field, respectively. Since these fields are dynamically coupled, it is reasonable to see small differences in predictability characteristics for different variables, especially in the mid-latitudes where the dynamical coupling is quite strong. However, in the tropics the mass and motion fields are not strongly coupled, and in relation to the mid-latitudes, the day-to-day changes in sea-level pressure and temperature are quite small compared to the wind field. Following the criteria of error growth and error equilibration, the wind field in the tropics seems to be a little more predictable than the pressure or temperature field. We had carried out similar predictability calculations for rainfall for the four summer runs reported by Charney and Shukla (1977) and Shukla (1981b), and we found that deterministic predictability for rainfall was even smaller than that for the circulation variables. Lorenz (1982) has examined the 10-day forecasts produced by the European Centre model for 100 consecutive days and has calculated error growth between model integrations starting from consecutive days. Assuming that the analyzed fields on two consecutive days do not differ greatly, the rms error between two integrations will give estimates of error growth similar to the ones obtained in the classical predictability studies. While the doubling time for the smallest observed error of 25 m was about 3.5 days, Lorenz estimated the doubling time for small errors to be about 2.5 days. He further introduced empirical methods for improving the forecast error, but the improvement in forecast was also accompanied with a decrease in the doubling time for the small errors. In this paper, Lorenz introduced the concept of lower and upper bounds on predictability. The lower bound on predictability refers to the minimum accuracy with which forecasts can be made, and the upper bound on predictability refers to the maximum error for forecasts at a given range. The performance of the current operational numerical weather prediction (NWP) models, therefore, gives an estimate of the lower bound of predictability (i.e., there is a possibility of doing better than that), while classical predictability studies give an estimate of the upper bound on predictability (i.e., we cannot do better than that). Improvements in NWP models and observing systems will lead to a larger lower bound and a smaller upper bound on predictability. Lorenz estimated the lower and upper bounds on predictability for the European Centre model and estimated that even without further improvements in 1-day forecast, 10-day forecasts as good as the present 7-day forecasts can be made. The range of predictability could be further extended by 2 more days by halving the 1-day forecast error.

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3. PREDICTABILITY OF SPACE-TIME AVERAGES It has generally been recognized that although the upper limit for prediction of instantaneous weather lies somewhere between 1 and 3 weeks, space-time averages of weather elements could be predicted for periods beyond this limit. During the last 5 to 10 years, a large body of observational and numerical modeling works have been reported that have collectively established a physical basis for dynamical prediction of monthly and seasonal averages. In the following sections, we shall present a brief review of the recent work and several remaining outstanding problems that need to be addressed. We shall first describe the mechanisms for the variability of monthly and seasonal averages and then examine the potential for their predictability. A convenient conceptual framework to describe the mechanisms of variability is afforded by the following two categories (Shukla, 1981a): (1) internal dynamics and (2) boundary forcings. ( I ) Internal dynamics: Even if there were no changes in the external forcing functions and even if the boundary conditions at the Earth’s surface were constant, there will be changes in day-to-day weather and in monthly and seasonal averages. These will occur due to the combined effects of dynamical instabilities and nonlinear interactions among different scales of motions, the interaction of fluctuating zonal winds with quasi-stationary mechanical and thermal forcings, etc. Monthly and seasonal averages can be made different by sampling different segments of this evolving nonperiodic flow. We shall further describe the predictability of internal dynamics in Subsection 3.1. ( 2 ) Boundary forcings: Slowly varying boundary forcings due to anomalies of sea-surface temperature, soil moisture, sea ice, snow, etc., can produce anomalous sources and sinks of heating and moisture that can influence the amplitudes and phases of planetary-scale waves, which, in turn, can change the location, intensity, and frequency of synopticscale disturbances. Since we shall confine our discussion only to the monthly and seasonal time scales, we shall not consider the external forcings due to fluctuations in solar or other extraterrestrial energy sources. There has been considerable interest in determining the relative importance of the internal dynamics and boundary forcings for the observed interannual variability of monthly or seasonal averages. Due to the strong coupling between the internal dynamics and boundary forcing mechanisms, it is not possible to determine their relative roles by

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analyzing observed data without making some drastic assumptions about the role of one or the other [see, for example, the paper and correspondence by Madden. 1976; Shukla, 1983a; Madden, 19831. One possible way to gain some insight into the problem is by idealized numerical experiments with GCMs, which can be integrated with and without boundary condition anomalies. Intercomparison between such integrations can suggest the possible role of the boundary conditions. Some attempts have been made in this direction (Charney and Shukla, 1977, 1981; Lau, 1981), but the conclusions remain questionable because they were based on comparison of numerical simulations with actual observations of the atmosphere, rather than on the comparison of two simulations (with and without the anomalous boundary forcing) from the same model. If model simulations without the changing boundary conditions can produce variances comparable to the observations, there is no justification to conclude that the boundary conditions are not important because internal dynamics can be overemphasized in such a hypothetical simulation. Numerical experiments with several GCMs have been carried out to determine the influence of boundary forcings due to regional anomalies of SST or soil moisture, etc., and they suggest an important role of boundary forcings for interannual variability of monthly and seasonal averages. 3.1. Dynamical Predictability

Since monthly and seasonal average anomalies are primarily determined by low-frequency, planetary-scale flow patterns, predictability of planetary waves will crucially determine the predictability of space-time averages. Predictability of planetary scales can be limited either by the instabilities at their own scale or by their interactions with highly unstable synoptic scales. If the growth and decay of the planetary waves were completely determined by their interactions with the synoptic scales, there will be no real hope for predictions beyond the limits of deterministic predictability. However, there is no evidence that that is the case for the atmospheric flows. A quantitative determination of the role of synoptic scales in the evolution and equilibration of planetary scales is quite essential to realize the potentials of dynamical predictability. Dynamical predictability of monthly means was investigated by the author using the GLAS climate model (Shukla, 1981a). The model was integrated for 60 days with three different observed initial conditions during three different years. These were supposed to represent large differences in the initial conditions. Six additional 60-day integrations

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were made after changing the observed initial conditions by superposition of random perturbations with root mean square of 3 m s-* in u and u components. These were supposed to represent small differences in the initial conditions. It was hypothesized that for a given averaging period, if the rms error among the time averages predicted from largely different (observed) initial conditions became comparable to the rms error among time averages predicted from small differences (random perturbations) in the initial conditions, the time averages would be considered to be unpredictable. It was found that the variances among the first 30-day means for largely different initial conditions were significantly different from the variances due to random perturbations, and it was concluded that the first 30-day means were dynamically predictable. It was also found that the next 30-day means (days 31-60) were not dynamically predictable. It has been pointed out by Dr. Y. Hayashi of the Geophysical Fluid Dynamics Laboratory (GFDL) (personal communications) that based on analysis of variance presented in our paper, it is not appropriate to declare the lack of predictability for second 30-day means, and the possibility remains that even the second 30-day means could be dynamically predictable. This was an idealized study, in the spirit of classical predictability studies for day-to-day weather prediction, and actual forecast experiments must be carried out to determine the predictability of monthly or seasonal averages. Miyakoda et al. (1983) have presented an example of a dynamical prediction for 30 days. This is an excellent illustration for potential of extended-range, dynamical predictability using advanced models for dynamics and physics. It is natural to expect that all initial conditions will not be equally predictable, but even one good example provides encouragement to pursue it further. There is some indication that the blocking situations have relatively greater predictability (Bengtsson, 1981). 3.2. Boundary-Forced Predictability

If the changes in the boundary conditions at the Earth’s surface were able to produce changes in the atmospheric circulation that were large and coherent enough to be distinguishable from the natural variability of the internal dynamics, boundary forcings would provide additional potential for predictability of space-time averages. Based on the correlations between the observed changes in boundary conditions, atmospheric circulation, and rainfall and also on GCM sensitivity studies with prescribed changes in the boundary conditions, it has been suggested that

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under favorable structures of the large-scale flow and appropriate locations of the boundary anomaly, significant and predictable changes in the atmospheric flow can indeed be produced. Changes in the boundary conditions produce local changes in the surface heat flux and moisture convergence, which in turn produce deep heat sources that can influence the remote, as well as the local, circulation. A summary of several such experiments carried out with the GLAS climate model has been presented in Shukla (1982, 1984b). Similar experiments have been, and are being, carried out at several other GCM groups around the world. However, to our knowledge not a single case of model integration has been reported in which observed global boundary conditions of all the slowly varying fields (SST, soil moisture, sea ice, snow, etc.) were used to integrate the observed initial conditions. We hope that the encouragement provided by the results from regional boundary anomalies will lead to study of predictability for global boundary conditions. We shall present here, as an illustration, one example of a sensitivity study with the GLAS climate model using the observed SST anomalies over tropical Pacific during the winter of 1982-1983 (Fennessy et al., 1985). In Fig. 7a, the observed SST anomaly during January 1983 was added to the climatological SST to integrate the model for 60 days. This integration is referred to as the “anomaly run,” and a similar integration with climatological SST is referred to as the “control run.” Such pairs of integrations were made for three different initial conditions. The difference between the anomaly and control runs averaged for three pairs for the period days 11-60 for precipitation (b) and the rainfall anomaly calculated from the observed outgoing long-wave radiation for 1982- 1983 winter (c) is also shown in Fig. 7. The outgoing long-wave radiation anomalies are changed to rainfall anomalies by using empirical relations developed by Arkin (1983). It is gratifying that the model calculations have been able to simulate rather well the location as well as intensity of rainfall anomaly. Similar results were obtained by several other modeling groups (Liege Colloquium on Hydrodynamics, May 1984) who used similar SST anomalies, although quantitative differences were found for different models with different parameterizations of boundary layer and moist convection. A comprehensive study of the role of tropical SST anomalies has been carried out by Lau and Oort (1985) in which they have examined a 15-year integration of the GFDL model, with the observed SST anomalies over the equatorial Pacific. The results are most remarkable, especially for the tropics. In the simulation without the SST anomalies (Lau, 1981), there was no evidence for the planetary-scale seesaw of surface pressure

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referred to as the Southern Oscillation, which is one of the most dominant modes of the tropical variability; whereas in the simulation with the SST anomalies, the Southern Oscillation is simulated remarkably well. The observed correlations between the tropical SST anomalies and mid-latitude circulation is also simulated remarkably well in the 15-year simulation with SST anomalies. These results, combined with some of Philander and Seigel (1985) and other ocean modeling groups on simulation of the oceanic circulation and SST that use the prescribed atmospheric wind stress forcing from the observations, suggest that the predictability of the coupled ocean-atmosphere system could be larger than the predictability of the atmosphere alone.

3.3. Prospects for Dynamical Extended-Range Forecasting A large body of observational, theoretical, and GCM results collectively suggests that there are good prospects for dynamical prediction of monthly and seasonal averages. Miyakoda el al., (1983) have already demonstrated the existence of extended-range predictability. Recent works by Miyakoda and his group at GFDL has shown that limits of predictability of time-averaged flow can be further extended by improving the dynamical model, especially the physical parameterizations of the model. The factors that have provided hope for dynamical extended-range forecasting (DERF) can be summarized as follows: (1) The planetary scales are more predictable than the synoptic scales. (2) Slowly varying boundary conditions at the Earth’s surface can produce significant and predictable changes in the time-averaged atmospheric circulation. (3) There exists a reasonable conceptual framework to understand the structure and evolution of atmospheric variability at medium- and longtime scales. (4) Global atmospheric GCMs are now able to simulate well the important features of the mean and transient components of atmospheric circulation. (5) Tropical ocean GCMs are also able to simulate the response of the prescribed atmospheric forcing reasonably well. (6) Advances in computer technology, space observations, communication, and data-processing techniques make it feasible to carry out a large number of integrations of atmospheric and oceanic GCMs by using real-time observations of global initial and boundary conditions.

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It is recognized that extended-range integrations with the atmospheric GCMs still show systematic errors (climate drift), and models must be continuously improved to reduce the climate drift. However, as previously suggested by the author (Shukla, 1983b), an appropriate mean climate drift can be subtracted from the predicted time averages to reduce the forecasting error. 4. SOMEOUTSTANDING PROBLEMS

Although it is possible to list a very large number of problems that need to be understood in the general area of predictability, we have chosen to comment only on the following three problems, which require further discussion. 4.1. Mean (Climate Drift) and Transient Predictability

Operational numerical weather prediction centers are constantly trying to improve their day-to-day forecasts by improving the parameterizations of model physics, increasing the resolution of the model, and improving the quality of data and data-analysis techniques. One of the important sources of error has been referred to as the “systematic error,” which is considered to arise mainly due to the “climate drift” of the model. The systematic error is generally defined as the average error for a large number of forecasts. Wallace et al. (1983) have shown that changes in the mountain heights (envelope orography) produced clear reductions in the systematic error. It is quite likely that systematic errors can be further reduced by changing the diabatic heating and dissipation mechanisms in the models. Climate drift for NWP models can be diagnosed by making long-term integrations of the forecast model with appropriate boundary conditions and comparing the simulated climate with observations. Systematic errors can also be reduced by statistical corrections to the forecasts (Faller and Lee, 1975; Lorenz, 1977). These methods have not been too popular with the operational NWP centers because they do not provide any physical insight, and although it might reduce the forecast error, it is not possible to understand either the cause of the error or reasons for improvement. The general assumption has been that improving the forecasts by changing the model adds to our understanding, whereas changing the forecast empirically does not. Systematic errors of NWP models have been examined mainly for flow parameters because it is possible to verify against observations. Similar

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analysis of systematic errors for rainfall, cloudiness, and vertical distributions of heating will be useful for determining the sources of systematic errors. Based on predictability studies for the tropics and a limited number of tropical forecasts, it was suggested (Shukla, 1981b) that systematic errors in tropical heat sources are so large and develop so fast that short- and medium-range forecasts could be improved by prescribing the large-scale tropical heat sources rather than calculating them by using model dynamics and physics. Some recent work at the European Centre (M. Tiedtke, personal communication) seems to support this conjecture. The extensive experience of operational numerical weather prediction and the predictability studies described in an earlier section suggest that some initial conditions are clearly more predictable than the others. Assuming that the quality and quantity of data do not change from one day to the other, and assuming that the model remains the same, the only factor that remains to be considered to explain the transient behavior of predictability is the dynamical structure of the initial state. An outstanding problem in weather forecasting is to identify the important features of the initial state that might reveal the predictability properties of the flow. For example, if it were true that in a large number of cases highly amplified presistent blocking ridges were predictable for longer periods [as has been suggested by Bengtsson (1981)], it will be possible to attach a higher confidence to a prediction that maintained the initial amplified blocking ridge. These considerations suggest a need for detailed synoptic study of predictability. We are not aware of any comprehensive study of predictability as a function of the synoptic structure of flow. For example, is it likely that location and intensity of the jet streams, intertropical convergent zones (ITCZ), or Walker cells could affect predictability? Although forecasts ultimately degrade either due to inadequacies of the models or the initial data, if there were significant relationships among the large-scale features of the initial flow and its associated predictability, such relationships can be exploited to improve the operational predictions. It should be noted that any statistical correction to the forecast based on past records will be helpful only in reducing the systematic errors and will not affect the errors that depend on the structure of the initial state.

4.2. Observational Errors and Model Errors It is rather interesting that the very first paper on predictability (Thompson, 1957) discussed here was motivated, at least in part, by the considerations of relative importance of good initial data and good models

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for weather forecasting. This question is equally, if not more, valid today as it was about three decades ago. During the past 10 years, there has been a large increase in the use of observations from satellites to define the initial state for NWP. However, major improvements in short- and medium-range forecasting appear to have come from better models. Arpe et al. (1985) have suggested that the observational errors are the dominant factor only for first 1 to 2 days of the forecast, after which the errors are dominated by model errors. This is a rather tricky question because model errors also contribute to the amplification of the initial observational errors, and we would have no way to know the time taken for the model-produced errors to become large if there were no observational errors to start with. Determination of relative roles of observational and model errors for short- and medium-range forecasting needs further work. 4.3. Predictability of Predictability

It is well established that there is an upper limit on predictability of weather. However, it is also evident that within that limit there can be changes in predictability that depend on the structure of the initial state. Earlier we discussed the possibility of identifying the important synoptic features of the flow that might provide some clue to the accuracy of the forecasts. Some formal procedures have also been suggested (Epstein, 1969; Leith, 1974; Hoffman and Kalnay, 1983) to make a quantitative determination of the reliability of the forecast. Leith (1974) suggested that instead of one forecast from a given initial condition, several (say about eight) forecasts from the same initial conditions can be prepared by integrating the model with slightly perturbed initial conditions. Divergence among these various forecasts will be a measure of the instability of the initial state and therefore a possible measure of the reliability of the forecast. Hoffman and Kalnay (1983) suggested that rather than perturbing the initial state (either randomly or systematically), forecasts from successively observed initial states can be combined, with suitable weighting functions, to produce a better forecast and to estimate the reliability of the forecasts. This suggestion eliminates the need for additional model integrations because such forecasts are produced routinely. If divergence among predictions from slightly different initial conditions were a good measure of the reliability of the predictions, these methods would provide not only a prediction (average of all integrations) of the flow, but also a prediction of the predictability of the flow. Although

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these methods have not yet been tried operationally, chances of their success will remain limited for forecast models with large systematic errors and climate drift. Persistence of atmospheric flows for several days may also reduce the advantages of using observed initial conditions during that period. For example, forecasts from several initial states within a 2- to 3-day period could be very similar, but each could be very different from the observations. An examination of the forecast errors for operational NWP models suggests a tendency for the persistence of correlation between forecasts from consecutive days. Since the decorrelation time for the atmospheric flows is about 5 days, this further suggests that predictability depends on circulation regime. This also suggests that errors of predictions from initial conditions in the immediate past can be a useful guide for predictability on a given day. It should be noted that the statistical correction techniques that use the error history for a large number of forecasts in the past will not be able to take into account the transient nature of predictability. For producing forecasts from a given day, the current operational NWP methods do not use any information from either the analyses or the forecasts during the last several days. The only exception is the use of a short-term forecast as a first guess for the analysis. Optimal interpolation techniques require the use of spatial structure functions, which are derived from past data over a much longer period than the decorrelation time of the atmosphere. Since weather forecasting is considered to be an initial value problem and since the prediction equations are highly nonlinear, there is no compelling reason to use the past history of forecast errors. However, considering the inadequacies of the models as well as of the observations, it should be possible to use information on the deficiencies of the forecasts from initial conditions in the recent past to improve short- and medium-range forecasts.

5. CONCLUDING REMARKS There is a complete agreement among scientists that the instantaneous weather is not predictable at infinite range. In fact, there is no serious challenge to the statement that the instantaneous weather is not predictable even beyond 2 to 3 weeks. As implied by the work of Lorenz (1982), it may be convenient to discuss the predictability at day 1 and at days beyond day 1 separately. Lorenz has shown that even if we could not improve predictions at day 1 , there is potential for improving predictions beyond day 1. There is little room for disagreement on this

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point. However, the question of predictability at day 1 needs more discussion. Is it possible to make significant reductions in the current 1-day forecast errors? Lorenz’s work implies that it is highly unlikely. The argument is as follows: There are, and there always will be, scales of motion unresolved by the NWP models, and even if there were no errors in the resolved scales, the errors of the unresolved scales would quickly make the resolved scales unpredictable. The underlying assumption is that by a reduction in the grid size of the model and by the introduction of more sophisticated and complex physical processes, the growth rates of errors will be increased, and therefore forecast errors at day 1 will remain nearly same as that for the current models. We are not quite sure about the validity of these conjectures because there is no evidence that the current 1-day forecast errors are mainly due to the unresolved scales. In fact, there is some evidence to the contrary, viz., that the current I-day forecast errors are also due to errors in the observations at the synoptic scales. It does not appear to be unreasonable to expect that by improving the current NWP models and the description of the initial state at the current resolution, 1-day forecast errors could be reduced without increasing the growth rate of the error. Estimates of the rates at which the unresolved scales influence the synoptic scales, and thereby the planetary scales, have been made only for simple models that do not have forcing and dissipation mechanisms, and there is no guarantee that these estimates will hold good for more realistic models of the atmosphere with well-defined forcing functions. The lower curves labeled E i I , E;g, and Eil in Fig. 6 suggest that the growth rate of the initial error can be reduced considerably by improving the initial conditions and reducing the inconsistency between the initial conditions and the boundary conditions. In the opinion of this author, we are not yet at a stage where the problems due to unresolved scales and the intrinsic instability of the flow are the primary factors contributing to the forecast error at day 1 or 2. It is not unlikely that the errors in defining the synoptic and planetary scale itself and in parameterizing the diabatic forcings at the synoptic and planetary scale are the primary reasons for the short-range operational NWP forecast errors. It is also of interest to note that the standard deviation of short-range forecast errors does not show any preferred areas of maxima in the storm track regions, which would have been expected from the classical predictability arguments of fast growth rates in those regions. Recent works on the predictability of space-time averages (Miyakoda, personal communication) indicate that the prospects for dynamical long-range forecasting of monthly and seasonal averages are quite good.

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However, this needs to be substantiated by a reasonably large number of actual forecasts. REFERENCES Arkin, P. A. (1983). Ph.D. Thesis, Univ. of Maryland, College Park. Arpe, K., Hollingsworth, A., Lorenc, A. C., Tracton, M. S., Uppala, S., and Kallberg, P. (1985). Q.J.R.Meteorol. Soc. 111, 67-101. Baumhefner, D. P. (1984). In “Predictability of Fluid Motions” (G. Holloway and B. J. West, eds.) pp. 169-180. American Institute of Physics, New York. Bengtsson, L. (1981). Tellus 33, 19-42. Charney, J. G., and Shukla, J. (1977). Symposium on Monsoon Dynamics. New Delhi, 1977. Charney, J. G., and Shukla, J. (1981). In “Monsoon Dynamics” (J. Lighthill and R. Pearce, eds.) pp. 99-109. Cambridge Univ. Press, London and New York. Charney, J. G., Fleagle, R. G., Riehl, H., Lally, V. E., and Wark, D. Q. (1966). Bull. Am. Meteorol. SOC.47, 200-220. Daley, R. (1980). Mon. Weather R e v . 108, 1719-1735. Epstein, E. S. (1969). Tellus 21, 739-759. Faller, A. J., and Lee, D. K. (1975). Mon. Weather Rev. 103. 845-855. Fennessy, M., Marx, L., and Shukla, J . (1985). Mon. Weather Rev. 113, 858-864. Gutzler, D. S., and Shukla, J. (1984). J . Atmos. Sci. 41, 177-189. Hoffman, R. N., and Kalnay, E. (1983). Tellus 35A, 100-118. Jastrow, R., and Halem, M. (1970). Bull. A m . Meteorol. Soc. 51, 490-513. Lau, N. C. (1981). Mon. Weather Rev. 109,2287-2311. Lau, N. C., and Oort, A. (1985). In “Coupled Atmosphere-Ocean Models,” Elsevier., Amsterdam (in press). Leith, C. E. (1965). In “Methods in Computation Physics,” (B. Alder and S. Fernbach, eds.) Vol. 4, pp. 1-28. Academic Press, New York. Leith, C. E. (1971). J. Atmos. Sci. 28, 148-161. Leith, C. E. (1974). Mon. Weather Rev. 102, 409-418. Leith, C.E., and Kraichnan, R. H. (1972). J . Atmos. Sci. 29, 1041-1058. Lilly, D.K. (1969). Phys. Fluids, Sirppl. ZI, 24-249. Lorenz, E. N. (1963). “Trans N.Y. Acad. Sci.,” 25, 409-432. Lorenz, E. N. (1965). Tellus 17, 321-333. Lorenz, E. N. (1969a). Tellus 21, 289-307. Lorenz, E. N. (1969b). J. Atmos. Sci. 26, 636-646. Lorenz, E. N. (1973). J . Appl. Meteorol. 12,543-546. Lorenz, E. N. (1977). Mon. Weather Rev. 105, 590-602. Lorenz, E. N. (1982). Tellus 34, 505-513. Lorenz, E. N. (1984). I n “Predictability of Fluid Motions” (G. Holloway and B. J. West, eds.) pp.133-139. American Institute of Physics, New York. Madden, R. A. (1976). Mon. Weather Rev. 104, 942-952. Madden, R. A. (1983). Mon. Weather Rev. 111, 586-589. Manabe, S.,Smagorinsky, J., and Strickler, R. J. (1965). Mon. Weather Rev. 93,769-798. Mintz, Y. (1964). WMO-IUGG Symp. Res. Deu. Aspects Long-Range Forecasting 66, 141155. Miyakoda, K., Smagorinsky, J., Strickler, R. F., and Hembree, G. D. (1969). Mon. Weafher Rev. 97, 1-76.

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Miyakoda, K., Gordon, C. T., Caverly, R., Stem, W. F., Sirutis, J., and Bourke, W. (1983). Mon. Weather Rev. 111, 846-869. Philander, S. G. H., and Seigel, A. D. (1985). In “Coupled Atmosphere-Ocean Models,” Elsevier, Amsterdam (in press). h r r e t t , L. (1976). NOAA Mag. 16-17. Robinson, G. D. (1967). Q . J . R . Meteorol. SOC.43, 409-418. Shukla, J. (1981a). J . Aimos. Sci. 38, 2547-2572. Shukla, J. (1981b). NASA Tech. Memo. 83829. Goddard Space Flight Center, Greenbelt, Maryland. Shukla, J. (1982). NASA Tech. Memo. 85092. Goddard Space Flight Center, Greenbelt, Maryland. Shukla, J. (1983a). Mon. Weather Rev. 111, 581-585. Shukla, J . (1983b). Proc. WMO-CASINSC Expert Study Conf. Long-Range Forecasting, Princeton, 1, 142-153. Shukla, J. (1984a). I n “Predictability of Fluid Motions” (G. Holloway and B. J. West, eds.) pp. 449-456. American Institute of Physics, New York. Shukla, J . (1984b). In “Problems and Prospects in Long and Medium Range Weather Forecasting” (D. M. Bumdge and E. Kallen, eds.) pp. 155-206. Springer-Verlag, New York. Shukla, J., Straus, D., Randall, D., Sud, Y., and Marx, L. (1981). NASA Tech. Memo. 83866. Goddard Space Flight Center, Greenbelt, Maryland. Smagorinsky, J. (1963). Mon. Weather Rev. 91, 99-164. Smagorinsky, J. (1969). Bull. A m . Meteorot. SOC.50, 286-311. Thompson, P. D. (1957). Tellus 9, 275-295. Wallace, J. M., Tibaldi, S., and Simmons, A. J. (1983). Q. J . R. Meteorol. SOC. 109, 683718. Williamson, D. L., and Kasahara, A. (1971).J . Atmos. Sci. 28, 1313-1324.

DATA ASSIMILATION W.

BOURKE,

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SEAMAN,

and K. PURI

Bureau of Meteorology Research Centre Melbourne. Australia

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Evolution of Assimilation and the FGGE . . . . . . . . . . . . . . . . . . . . . 3. Components of Four-Dimensional Assimilation Systems . . . . . . . . . . . . . . . 3.1. Observational Data Base . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Optimum Interpolation (01) . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Model Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Prediction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Characteristics of Some Current Assimilation Schemes . . . . . . . . . . . . . . . 4.1. Continuous Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Intermittent Assimilation . . . . . . . . . . . . . . . . . . . . . . . . . 5. RoleofFour-Dimensional Assimilation in ResearchandOperations . . . . . . . . . . 5.1. Research Implications of Four-Dimensional Assimilation . . . . . . . . . . . . . 5.2. Research on Four-Dimensional Assimilation Procedures . . . . . . . . . . . . . 5.3. Long-Term Operational lmplications of Four-Dimensional Assimilation . . . . . . . 6. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Numerical weather prediction (NWP) is a central aspect of activities in all major operational weather centers throughout the world. While the capacities differ, the principles are commonly shared; i.e., the weather prediction problem is defined as an initial value marching problem. The quality of NWP hinges crucially on the accuracy of specifying the initial conditions, on the appropriateness of boundary conditions, and on the ability to model mathematically the dynamics and physical processes of the evolving atmosphere. Advances in NWP in the past 15 years have been heavily influenced by the sophistication and capacity in mathematical computation and the burgeoning data base available from space-based observing systems. In particular, the two traditionally separated functions, objective numerical analysis and numerical model prediction, have been merged to yield what is now commonly described as four-dimensional data assimilation. In this procedure the numerical prediction model is employed to coordinate in time and space the irregularly distributed asynoptic data typically available from the international community of meteorological services. The spur to this development of data assimilation has been the space-based observing systems and in particular the temperature I23 ADVANCES I N GEOPHYSICS, V O L U M E

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soundings from the sun-synchronous, polar-orbiting satellites, although objective analysis schemes utilizing what is now termed intermittent assimilation were in use before satellite-based temperature soundings were available. The advent of this asynoptic observational data base, on the one hand, has defined a more complex objective analysis problem but at the same time has led to a much more comprehensive use of the prediction equations. The predictive component of the assimilation system provides an additional source of information, and an objective is to obtain a sequence of fields of dependent dynamic variables that is consistent with both the observations and physical laws of atmospheric flow. The observed variables defining large-scale atmospheric motion are the three-dimensional distribution of the horizontal components of the wind, the atmospheric pressure, the temperature, and the humidity. For large-scale flow the hydrostatic equation is particularly appropriate and vertical motion may accordingly be diagnosed. Furthermore, a reference level of pressure such as mean sea-level pressure, and the temperature field together with the hydrostatic relation and the geostrophic mass-wind relationship, is sufficient to define the three-dimensional distribution of pressure, temperature, and wind except perhaps in tropical latitudes in which the geostrophic relationship is inappropriate. Charney et al. (1969) in a pioneering paper on this subject suggested that the requisite model initial conditions (in the extratropics) of pressure, temperature, and wind in numerical prediction could be obtained in principle by assimilating only temperature data such as those available from satellites. The importance of this subject was recognized at an early stage by the Global Atmospheric Research Programme (GARP) Joint Organizing Committee (JOC), which sponsored the International Symposium on Four-Dimensional Data Assimilation in Princeton at the Geophysical Fluid Dynamics Laboratory (GFDL) in April 1971. Before proceeding, it is appropriate to elaborate a little on the Global Atmosphere Research Programme. A central objective of the program was to improve our understanding and explanation of atmospheric behavior, thereby leading to more comprehensive atmospheric models and more accurate prediction. A key arm of the GARP Joint Organizing Committee was the Working Group on Numerical Experimentation (WGNE) that had been established in 1968. The main aim of the WGNE was to identify a program of numerical experiments and to coordinate the studies of co-operating research groups. Foremost among the problem areas identified by the WGNE, and given continuing attention, was four-dimensional assimilation. In the present essay, the development of four-dimensional data

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assimilation through the decade of planning prior to the First CARP Global Experiment (FGGE) and its current status are reviewed. Reviews of data assimilation have already been presented by McPherson (1 975) and by Bengtsson (1975), and a more detailed account of research up until the mid-seventies will be found in these references. A monograph edited by Bengtsson et af. (1981) provides considerable technical detail on current data assimilation methods. 2. EVOLUTION OF ASSIMILATION AND

THE

FGGE

In the years immediately subsequent to the 1971 meeting in Princeton at the GFDL, there followed a significant expansion of research activity. Much of this research up until the mid-seventies was concerned with using model-simulated data in perturbed model reruns to analyze observational network requirements and assimilation procedures. The consensus of this time was that these “identical twin” experiments, while being very valuable, provided a too optimistic view of the possibilities of four-dimensional assimilation capabilities. Experiments based on real observational data approaching the FGGE global requirements became possible with the Nimbus-6 satellite launched in June 1975 that carried the most advanced infrared and microwave sounding instruments hitherto placed in orbit. This instrumentation was the precursor of the operational Tiros-N launched in October 1978 for the FGGE year. Consequently, from the mid-seventies onward it was possible to experiment with global temperature coverage such as anticipated as integral and key components both of the FGGE year and of the ensuing operational satellite program. The earlier generation of vertical temperature profile radiometer (VTPR) measurements from the NOAA-2/4 satellites had been available operationally since December 1972 and had, of course, been assessed in a number of studies in terms of impact on analyses and prediction as discussed in the review by Bengtsson (1975). However, the value of observing system simulation experiments (OSSEs) within the context of identical twin experiments in the development of four-dimensional assimilation was undoubted. The methodology of assimilation itself was established, and very important significant conclusions were obtained indicating the clear need for a reference level and the inadequacy of temperature measurements alone in defining the state of the tropical atmosphere. The inadequacy of temperature measurements alone in the tropics, as originally foreshadowed by Charney et al. (1969), was unanimously identified within the numerical experimentation conducted in the early seventies at the GFDL (Gordon et al., 1972), by

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Kasahara and Williamson (1972) at the National Centre for Atmospheric Research (NCAR), and by Jastrow and Halem (1973) at the Goddard Institute for Space Science (GISS). This facilitated the JOC recommendation that adequate coverage of the tropical zone by geostationary satellites to enable wind specification from cloud displacements would be a critical requirement for FGGE [World Meteorologic OrganizatiodInternational Council for Scientific Unions (WMO/ICSU), 1971bl.The need for vertical wind profile sounding between IO"N and 10"s supplementing the World Weather Watch (WWW) network and the cloud vector winds was also demonstrated in these studies (WMO/ICSU, 1974). Early planning for the FGGE also identified clearly the need for a reference level in the Southern Hemisphere (WMO/ICSU, 197la). This was initially to be achieved by a fleet of 450 constant-level balloons. A modest network of drifting buoys was identified as relevant to provide data in areas of persistent cloudiness in the latitudes 50-65"s. Subsequently, the constant-level balloon program did not eventuate, but an expanded program of drifting buoys throughout the Southern Hemisphere provided one of the major observational successes of the FGGE. The value of the early observational simulation studies has clear testimony even today; the operational polar-orbiting satellites, the modest operational deployment of buoys in the Southern Hemisphere in the early eighties, and the foreshadowed increase in buoys in the mid-eighties demonstrate this. Some experiments conducted in the first half of the seventies considered among other things the question of asynoptic versus synoptic insertion of temperature soundings (Kasahara, 1972). These experiments were a little inconclusive and the range of procedures used today still reflects some uncertainty. Further and more realistic OSSEs were undertaken by extracting simulated data from sophisticated models and then utilizing lower resolution and less comprehensive models for simulated assimilation and data base studies. Simple updating of nearest grid points was at this time being replaced by the more traditional objective analysis procedures, such as the successive correction method (SCM) and optimum interpolation (01). The nonidentical twin experiments provided some indication of the influence of an incomplete data base and of observational and forecast error and highlighted the requirement for detailed knowledge of the magnitude and structure of the expected observational errors. In particular there was good qualitative agreement between experiments at the NCAR (Williamson, 1975) and at the United Kingdom Meteorological Office (UKMO) by Lorenc (1975) in nonidentical twin OSSEs in defining the overall requirements of the

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assumed special observing systems; quantitatively, however, the 0 1 analysis step in the UKMO system provided superior model assimilation of data. Difficulty in drawing definitive conclusions with regard to data networks was clearly identified at this time and related to deficiencies in utilizing the information content of the observations. An important suggestion from the UKMO/OSSE studies of Lorenc (1975) was the overall improved performance over the Southern Hemisphere with a widely dispersed drifting-buoy network in comparison to a closer concentration of the buoys in regions of high variability and short longitudinal span. A second major study conference on four-dimensional data assimilation was again sponsored by the GARP/JOC and conducted in Paris late in 1975 (WMO/ICSU, 1976). Problems of particular concern in assimilation systems that were highlighted at this conference included (1) spurious excitation of gravity waves in the model atmosphere; (2) difficulties of combining observations, forecast, and climatology in the most effective and efficient manner; (3) verification procedures for assessing analyses; (4) the analysis and data problems of low latitudes; and ( 5 ) clarification of the relevant merits of the intermittent and the continuous approaches to assimilation.

Papers presented at the Paris meeting by Rutherford and by Schlatter foreshadowed a developing consensus for the 0 1 method of analysis. Rutherford presented details of a three-dimensional multivariate 01 scheme that was already operational in the Canadian Weather Service. The multivariate aspect provides mass-wind coupling through model prediction error covariances. This was a more sophisticated approach than the earlier useful demonstrations of inferring geostrophic corrections to model wind fields in the presence of assimilated mass data alone (Rutherford, 1973; Hayden, 1973). The multivariate approach was perceived to be more appropriate for intermittent data analysis than for continuous insertion where the mass-wind coupling would be achieved simply by model dynamics. Two factors associated with the general problem of assimilation, namely, the need for mass-wind coupling and the need to control excessive excitation of spurious gravitational oscillations, appeared to be handled more readily in the multivariate approach. At the meetings of the WMO Executive Committee Inter-Government Panel on FGGE in February 1976, two commitments were offered to produce FGGE level-IIIB data sets. These offers came from the European Centre for Medium Range Weather Forecasts (ECMWF) and

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the GFDL; the JOC expressed concern in April 1977 at the absence of further offers to produce level-IIIB data sets (WMO/ICSU, 1977). At this time it was also becoming apparent that the resources available for the FGGE would be less than expected. The formal observational requirements for the global experiment to be held in 1979 had been defined in the prior decade of planning and comprehensive observing systems simulation experiments. These formal requirements called for the observing systems in the extratropics of both hemispheres to provide at a lateral resolution of 500-km soundings of wind and temperature, surface pressure, humidity, and sea temperatures, with a vertical resolution of seven levels for the soundings (four troposphere, three stratosphere) and two degrees of freedom for humidity. In the equatorial tropics the formal requirements were essentially as in the extratropics but with more stringent requirements for wind data. The expense of the enhanced tropical wind observing system was such that it could only be implemented for two special observing periods. An example of the decrease in resources available for the FGGE was the omission of the constant-level balloon observing system in the Southern Hemisphere. Further OSSE studies undertaken at the UKMO (Bromley, 1978) identified a deterioration in analysis and prediction in the Southern Hemisphere upon omission of the constant-level balloons poleward of 30"sand dramatically demonstrated the need for at least one reference level in the Southern Hemisphere such as the surface-pressure observations from drifting ocean buoys. The studies by Bromley provided OSSE assessment of analysis and prediction error for three different FGGE data distributions: (1) the basic observing system corresponding to the normal meteorological observing network of WMO; (2) the basic system plus the ideal extensions via special observing systems; and (3) the basic system plus actual special observing systems then committed to FGGE.

Additional studies by Larsen et al. (1978) provided theoretical estimates of analysis error based on interpolation theory for these configurations of observations. The OSSE assimilation studies and analysis error estimates provided by Bromley (1978) and by Larsen et al. (1978) were such that the WMO Executive Committee Inter-Governmental Panel on FGGE was able to agree in early 1977 that a global-atmospheric experiment could be conducted. The observational data base had largely been identified as adequate, although the resources committed left some doubt regarding

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the adequacy of the tropical observing system where the lateral resolution of wind soundings would barely meet the now proposed requirement of 700 km. It is appropriate in an essay on data assimilation to record these details on the FGGE planning. In particular, the detailed assimilation studies led to a clear commitment in the late seventies to the Global Weather Experiment by the JOC and the WMO Executive Committee InterGovernmental Panel. During the period leading up to the FGGE it had also been possible to conduct assimilation experiments on subsets of the final observing systems. In the first of these, the GARP Atlantic Tropical Experiment (GATE) conducted in 1974 provided an early opportunity to evaluate four-dimensional assimilation in the tropics. The tropical domain presented at that time and even now an especially formidable task both from the analysis and prediction viewpoint; the tropical flow is relatively decoupled from the pressure field due to the weakness of the Coriolis force, and the complex convective physical processes remain difficult to parameterize on the scales typically available to global assimilation systems. Miyakoda et al. (1976, 1982) evaluated the assimilation methodology developed at the GFDL on the GATE data, initially in near real time for the entire 101 days and subsequently for 34 days with the complete final GATE data set. The 34-day GATE assimilation was the precursor to the system used for FGGE level-IIIB production at the GFDL (Stern and Ploshay, 1983). These analyses showed that the assimilation system could produce a reasonable and consistent picture for the easterly wave disturbances in the tropical Atlantic. However, the study highlighted the difficulty of maintaining dynamical consistency between the mass and wind fields and the dependence of the analysis scheme on the parameterization of physical processes included in the predictive component of the system. A second major experimental evaluation of observing systems was designed as a prototype of the FGGE. This study was conducted at the National Meteorological Center (NMC) in Washington. It is described in the report by Desmarais et al. (1978) and was referred to as the Data Systems Test (DST). The DST in August-September 1975 (DST-5) and February-March 1976 (DST-6) produced the most extensive global meteorological data base in existence at that time. The basic observing network of the WMO was augmented by the polar-orbiting satellite Nimbus-6 temperature and moisture soundings, cloud vector winds from geostationary satellites SMS- 1 and SMS-2, constant-level balloon winds and special aircraft data. The NMC assimilation system at this time consisted of the operational spectral Hough analysis scheme (Flattery,

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1970) and a global prediction model (Stackpole, 1976) with a 12-hr analysis-forecast cycle for DST-5 and 6-hr cycle for DST-6. Of particular concern in these studies was the inability to show significant impact on prediction from the use of Nimbus-6 temperature soundings in the Northern Hemisphere; the analyses in the Southern Hemisphere were assessed as improved due to the use of the satellite soundings, but the inability to define a reference level accurately was identified as a major limitation. The lack of positive impact of satellite soundings on 72-hr prediction in the Northern Hemisphere studies was accompanied by the recognition that the temperature retrievals were underestimating the spatial variance in the thermal structure of the atmosphere. These assimilation and forecast studies at the NMC were compared with similar experiments at the GFDL and the Goddard Laboratory for Atmospheric Science (GLAS), where positive impact of satellite soundings on prediction had been found. However, the NMC forecasts were identified to be superior to both the GFDL and the GLAS systems even in the absence of satellite soundings in the NMC system. It was thus apparent from these comparative studies that assimilation and data impact results were highly dependent on the inherent capabilities of the respective analysis and forecast systems (Tracton and McPherson, 1977). The NMC assessment of these experiments attributed the trivial impact of soundings in the NMC system in contrast to the beneficial impact in the GFDL and the GLAS systems, to the superior first guess of the NMC assimilation system relative to the GLAS, and to the superior analysis method at the NMC relative to the GFDL. An all-pervasive problem throughout these assimilation studies, and highlighted at the Paris study conference in 1975, had been that of the spurious excitation of inertia-gravity oscillations in the model atmosphere. Numerous algorithms to solve this problem had been suggested, but the one that now stands out as a decisive breakthrough was that of nonlinear normal mode initialization (NNMI). The NNMI procedures as now commonly used were developed by Machenhauer (1977); an equivalent algorithm was independently developed at the same time by Baer (1977). This breakthrough was not achieved in time to have a major impact on the OSSEs that contributed so decisively to the design of the FGGE observational network. However, the implementation of the dataassimilation systems at the two IIIB data producers for the entire FGGE year included the NNMI as a key component. The discussion to this point has been primarily concerned with the role of assimilation systems in the planning for the FGGE. The end product of GARP was not intended to be simply an ensemble of data sets, but a path to specific answers to physical problems arising in explaining the behavior

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of the atmosphere and to practical problems of predicting its future behavior. At the outset, however, it was necessary to quantify the information content of the projected data base from observing systems that could be implemented. It was similarly recognized that these observing systems reflected a reasonable extension of existing national plans. Five years after the FGGE, the global observational network, in fact, enjoys a substantial enlargement in capacity, reflecting operational deployment of some of the FGGE special observing systems. Consequently, it is now possible to conduct not simply OSSEs, but rather observing system experiments (OSEs) utilizing the FGGE data as well as current operational data, and much of the assimilation research in the recent years has been devoted to that end. The current maturity of assimilation systems is indicated by their widespread application in operational analysis and prediction. The following sections of this chapter discuss in detail the components of a number of current four-dimensional assimilation systems, the characteristics of several specific approaches, recent research, operational results in assimilation and prediction, and an assessment of future trends.

3. COMPONENTS OF FOUR-DIMENSIONAL ASSIMILATION SYSTEMS In describing the current approach to four-dimensional assimilation, it is appropriate to consider the following four components: the observational data base, the objective analysis, the initialization algorithms, and the prediction model. These subsystems are discussed in the following sections. Some detail is provided about analysis and initialization procedures; the prediction model component is only briefly discussed as it is covered elsewhere in this volume.

3.1. Observational Data Base The WWW observing system reached an unparalleled peak in 1979 during the FGGE. In addition to the Tiros-N and NOAA-5 polar-orbiting satellites, there were, for example, five geostationary satellites from which cloud vector winds could be derived, about 300 drifting buoys in the Southern Hemisphere, an enhanced tropical system including aircraft dropwinsondes, constant-level balloons, and special tropical observing ships. The data base available in the eighties is less comprehensive than that of the FGGE year, especially in the tropics and Southern Hemisphere.

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The data base typically used in global analysis and prediction at the present time consists of (1) surface reports from land stations, (2) surface reports from ships and drifting buoys, (3) manually prepared surface reports of sea-level pressure, (4) upper-air reports from land stations and ships, ( 5 ) aircraft reports, (6) cloud motion wind vectors, (7) upper-air reports of temperature from polar-orbiting satellites, and (8) upper-air humidity data derived from satellite imagery and synoptic cloud observations.

A nontrivial aspect of all objective analysis schemes is the preprocessing of the raw data. Data need to be recognized, decoded, regrouped, and reformated, and in addition the individual messages are condensed to optimize the information content. Preliminary validation of data includes checks on internal consistency, checks relative to climatology, and a hydrostatic check on reported height and temperature. A detailed discussion of these procedures is not appropriate in the present chapter. With the basic validated data assembled and sorted with respect to observation type and time of observation and in three dimensions, the comparison of the data with the contemporaneous estimate of the prediction model then constitutes the first step in the analysis procedure. It is common practice in intermittent assimilation to group the data within 6-hr time windows centered upon the synoptic times of 0600, 1200, 1800, and 0000 GMT. With increasingly higher horizontal resolution in numerical models, this time window may be reduced to minimize the asynoptic time error. Alternatively, the data may be utilized in continuous insertion, in which case the observing and insertion times are closely synchronized in the evolving prediction model atmosphere. Some current procedures assign slightly larger observation errors to data that are off-centered relative to the nominal observing time. 3.2. Optimum Interpolation (Or} An essential component of data assimilation is objective analysis. An excellent review of that subject has been presented by Gustavsson (1981). As foreshadowed in Section 2, some consensus has emerged with the widespread use of the 01 approach, and the present discussion will focus on this procedure. The basic idea of the 01 method is to utilize the statistical covariance

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properties of observational errors and background (“first guess”) field errors in order to minimize the mean square error of interpolation. Its earliest application in meteorology was by Eliassen (1954), but the approach was much further developed by Gandin (1963). Both these authors utilized a background field of climatology. The method was soon extended by Eddy (1967) to use a background field generated from the observations themselves and by Kruger ( 1969) to use a numerical prediction as a first guess. Rutherford (1973) introduced multivariate 01 that enabled the simultaneous use of mass and wind observations. With increasing computer power, fully three-dimensional multivariate schemes soon became feasible (Schlatter, 1975; Bergman, 1979; Lorenc, 1981). Probably the most appealing aspect of 01 is that it provides a logica framework within which to take the following factors into account, namely: (1) the spatial distribution of observations relative to one another and relative to grid points, (2) the error characteristics of different observing systems, (3) the information available from earlier data (by using a forecast background field and a forecast error covariance function), and (4) the quasi-geostrophic and hydrostatic relationships among variables.

Additional desirable features of 01 include in-built data checking and the availability of an interpolation error estimate. It should nevertheless be emphasized that 01 is optimal only to the extent that the presumed observational and forecast error covariances reflect the corresponding true covariances. In practice, however, 01 appears to be not unduly sensitive to small changes in the presumed covariance structure. In generalizing 01 analysis to be multivariate, the formalism requires in addition to autocovariances the introduction of cross covariances between, for example, different field types of mass and wind and the accompanying assumption that prediction errors in the mass and motion field behave in accordance with the geostrophic and hydrostatic relationships. Univariate correlations of geopotential or mass prediction errors are typically modeled in an analytic positive definite differentiable form that is horizontally isotropic. The most commonly used correlation function is the Gaussian exponential function p(r) = exp(-0.5S/b2), where r is the distance over which correlation is being considered and b is a fixed coefficient defining the horizontal scale of the parameter to be analyzed. The most common approach to obtain correlations for wind prediction errors is to apply the geostrophic wind equation

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to the isotropic correlation functions for geopotential. Similarly, cross correlations between wind and mass increments can also be derived analytically. In the tropical latitudes the geostrophic constraint is inappropriate and the multivariate schemes are gradually decoupled into univariate analyses as the equator is approached. 3.3. Model Initialization At some point in the four-dimensional assimilation cycle, it is necessary to commence model prediction from the most recent specification of the model atmosphere. There remains a serious problem. Primitive-equation models, unlike geostrophic models, admit higher-frequency gravity wave solutions that can have amplitudes substantially in excess of their counterpart in the real atmosphere. The low-frequency Rossby-mode component of the model is of prime interest, but this component can be obscured by gravitational mode oscillations occurring on a time scale similar to that of data insertion (e.g., 6 hr). These gravitational oscillations arise from imbalances between the mass and wind field. There have been a number of long-standing procedures for suppressing these spurious oscillations based on analysis of linearized forms of the equations of motion, on scale analysis, and on the fact that the large-scale atmosphere is essentially in geostrophic balance outside the tropics. These schemes usually have been formulated in pressure coordinates and have not been particularly effective or appropriate in primitive-equation models using the terrain following sigma coordinates. Implicit in the preceding remarks concerning high-frequency oscillations is the notion that the free normal modes of the discretized primitive equations can be readily evaluated. The identification of these free modes of oscillation of the primitive equation prediction model is accomplished by linearizing the equations around a simple basic state, which is commonly taken as an atmosphere at rest with temperature variation in the vertical only. The specification of a zero-wind mean state permits a simple decoupling of the three-dimensional eigenvalue problem into a series of two-dimensional eigenvalue problems. The vertical decoupling gives rise to a number of characteristic vertical modes, one corresponding to each discrete level in the model. The series of decoupled two-dimensional eigenvalue problems is equivalent to the Laplace tidal equations, where the mean free height is given by the eigenvalues or equivalent depths associated with each vertical mode. The horizontal eigenfunctions of the Laplace tidal equations are readily classified into two categories, namely, Rossby and gravity modes. For a given vertical structure, the frequency

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of the modes and the pairwise existence of eastward- and westwardpropagating gravity modes permits ready identification. It is thus possible, with the common vertical structure of the atmosphere assigned to both the mass and momentum fields, to project the model state variables, or their tendency, onto the horizontal structures identifiable as either Rossby or gravity normal modes. The direct use of normal modes of the actual models was suggested by Dickinson and Williamson (1972). They proposed that initial data be expanded in terms of model-normal modes and that the amplitudes of the unwanted modes be then set to zero. This scheme was especially effective in linearized models but failed to suppress the spurious noise in the requisite nonlinear calculation. More recently Machenhauer (1977) and Baer (1977) independently proposed a nonlinear normal mode scheme (NNMI). Machenhauer proposed that rather than setting the amplitude of the unwanted modes to zero, it is the tendency of these modes that should be set to zero. This turns out to be a superb solution, but it implies a nonlinear equation, the solution of which requires an iterative process. The scheme does not have guaranteed convergence but in practice with some restrictions does converge and does lead to a very well balanced state. The iterative scheme is usually performed only for the larger-scale vertical modes. For example, in a nine level model it is usual to initialize only the first three or four vertical modes. The linearized state is that of an atmosphere at rest and this is necessary for vertical decoupling. In performing the NNMI, there is some scope for variation in the linearization specification with aspects of the dynamics omitted from the linearization implicitly included in the nonlinear term; commonly the nonlinear iteration involves only the adiabatic component of the full prediction model. One effect of excluding diabatic effects from the initialization step is a serious depletion of the tropical divergent circulations. A number of solutions has been found to alleviate this problem. One method is to replace the NNMI by an incremental linear NMI in which gravity modes in the increments in the model state due to insertion of data are removed (Pun el al., 1982). Since this scheme does not directly affect the background model field, it is designed to preserve the circulation developed by the model during data assimilation. Two types of NNMI schemes have been proposed to retain the tropical circulation. The first is based on the result that the tropical divergent circulation maintained by convective processes influence mainly the low-frequency gravity modes in a model (Puri and Bourke 1982; Puri 1983a) and these low-frequency modes are excluded from initialization by using a frequency cutoff. This scheme effectively controls the high-frequency gravity wave noise and at the same time retains the tropical divergent circulation. The second type

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of NNMI has been proposed by Wergen (1982). In this scheme the diabatic heating is obtained by integrating the model prior to initialization for a few time steps and time averaging the diabatic forcing, which is then projected onto those low-frequency large-scale modes identified with convective heating and which can then be included in the nonlinear forcing during iteration. The theory of NNMI has proved to be a very valuable practical tool in data assimilation schemes and additionally a simple diagnostic analysis of the effectiveness of model memory of inserted data. An example of the data rejection that is a major problem in data assimilation can be seen in Fig. 7 of Bourke ef al. (1982), which shows that some mass information present in initial analyses is lost during NNMI. The concept of slow and fast data manifolds developed by Leith (1980) has provided a powerful method for analyzing such problems. Daley (1978), Daley and Puri (19801, and Tribbia (1982) have investigated the effectiveness of variational constraints within NNMI schemes by using the shallow-water equations. Puri (1983b) and Temperton (1984) have applied constrained schemes to multilevel assimilation systems and demonstrated that constrained NNMI can yield a reduction in the loss of information that occurs in unconstrained NNMI. The application of the NNMI algorithm has been restricted largely to use in the global and hemispheric domain and is particularly well suited to spectral models of the atmosphere. Recent developments have seen the application of NNMI schemes to limited-area prediction models (Briere, 1982; Bourke and McGregor, 1983), and this will no doubt facilitate the development of assimilation systems for regional application. 3.4. Prediction Models

The numerical prediction models used in research and operational four-dimensional assimilation typically have been developed for the purposes of medium-range weather prediction. These models are high-resolution, comprehensively parameterized deterministic models as sophisticated, if not more so, than general circulation climate models. The models typically span the global or hemispheric domain and utilize the primitive equations. The numerical formulation of the models in recent times has seen an increasing use of spectral models that commonly have from 10 to 15 levels and a horizontal resolution of from rhomboidal-30 to triangular-63 wave number truncation. The representation of model variables in the vertical is discrete, and normal finite difference algorithms are used with constraints to ensure conservation of energy, mass, and

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momentum. Accompanying the development of spectral models has been a growing consolidation in the alternative but longer-standing formulations of primitive equations in which finite differences are used in all three space dimensions. In particular, a staggered disposition of variables on latitude-longitude grids spanning the globe or hemisphere is now the most widely used method. The numerical details of both spectral and finite difference methods are reviewed in a comprehensive volume on global modeling of the atmosphere (Arakawa and Lamb, 1977; Bourke et al., 1977). Two features of prediction models that affect assimilation systems are the use of the terrain-following sigma coordinates and the use of temperature as the thermodynamic variable. In multivariate analysis a sigma coordinate system does not readily allow application of the geostrophic wind law, and temperature is less easily used than is geopotential. Accordingly, the analysis component of the assimilation cycle is frequently implemented in pressure coordinates. The model prediction is thus converted from sigma to pressure coordinates, corrected by available data, and the resultant increments or changes interpolated back to the model sigma domain. Univariate analysis in sigma coordinates is quite tractable and diagnostic increments to provide mass-wind coupling are also possible. However, in such schemes cross validation of the first guess and observations of different variables is not available, and variationally constrained blending is relatively complicated in other than pressure coordinates. The physical parameterizations of the prediction models routinely used in research and operational application of four-dimensional assimilation are quite comprehensive. These parameterizations may include (1) radiative forcing with climatological absorber amount specification (cloud amount may be diagnosed or quite commonly specified in terms of zonal mean values), (2) stability-dependent modeling of the constant flux layer of the boundary layer, (3) convective process parameterization yielding both large-scale condensation and small-scale penetrative convection, (4) hydrologic cycle modeling over land including prediction of surface and subsurface temperature and soil moisture content (sea-surface temperature is specified by monthly mean climatology), and (5) horizontal and vertical diffusive parameterizations.

Of additional concern is the question of the assimilation cycle requiring refinements in parameterization. A conspicuous omission from the preceding list is the inclusion of a diurnal forcing in the radiative heating

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calculations. Given that the time scale of assimilation cycles is of the order of 6 hr and that the diurnal changes in thermal fields within the atmosphere and at the surface over continents can be substantial on this time scale, the absence of such a diurnal component in the model may lead to biases in the background fields and to spurious rejection of observed data. Preliminary experiments with diurnal forcing in assimilation studies have, however, been very limited, and there is surprisingly little evidence to suggest that this is an important requirement. The time-step integration of the prediction model component of the assimilation cycle is typically effected by a semi-implicit algorithm in spectral models (Robert, 1969; Bourke, 1974) and increasingly by economical split-explicit schemes in finite difference models (Gadd, 1978). The computational overhead of implicit time differences in spectral models is only slight. Both of the time integration schemes allow the model evolution to proceed on time scales of significance in the meteorological context and not to be constrained unnecessarily by the small time steps associated with the time scale of the low-amplitude, high-frequency gravitational modes occurring in the models.

4. CHARACTERISTICS OF SOMECURRENT ASSIMILATION SCHEMES

As mentioned in Section 2, two approaches to insertion were under serious consideration in the mid-seventies, namely, continuous assimilation and intermittent assimilation. Originally, continuous assimilation implied insertion of data only at the nearest model time step (Bengtsson, 1975), but it is now understood to include continuous insertion in which the same data may be repeatedly inserted within a short time span. Also all current assimilation systems proceed only in the forward direction, with the implication that later data do not influence the analyses at earlier times, as would be possible in iterative schemes such as discussed by Talagrand ( 1981). At present both continuous and intermittent approaches are in use, reflecting in some sense the lack of consensus with regard to the most appropriate way to proceed. Continuous insertion of data is the basis of the assimilation systems used at the GFDL (Stern and Ploshay, 1983) and the UKMO (Bell, 1983); intermittent assimilation is the basis of the systems used at the ECMWF (Bengtsson et al., 1982), at the NMC (Kistler and Parrish, 1982), at the Japan Meteorological Agency (JMA) (Kanamitusu et al., 1983), at the Canadian Weather Service (Rutherford, 1975), and at the ANMRC (Bourke et al., 1982). The intermittent analysis schemes are usually multivariate in that the complete atmospheric

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representation of all dynamic variables is analyzed simultaneously, while the continuous schemes are more often univariate. These distinctions are not necessarily rigid, since intermittent univariate analysis is also commonly used. However, multivariate continuous insertion appears not to have been tested among the various research and operational groups, perhaps due to the substantial computational requirement. To illustrate the basic differences between the two approaches, the operational system at the ECMWF (Bengtsson et al., 1982) utilizing the intermittent multivariate approach will be considered, together with the research scheme at the GFDL (Stern and Ploshay, 1983) that utilizes the continuous insertion in a univariate framework. These two centers operated such schemes in the production of the FGGE analyses, as the official level-IIIB data producers. Variations within these two basic approaches, as developed by other groups, will be noted as appropriate.

4.1. Continuous Assimilation

The essential characteristic of this approach is the continuous insertion of data into the model at each time step. As used at the GFDL, these data are determined by three-dimensional univariate 01 in pressure coordinates. The univariate statistical interpolations to a regular latitude-longitude grid are performed at 2-hr intervals within each 12-hr span, on 19 mandatory pressure levels from 1000 to 0.4 mb, at the model grid points for u , u, T, q and p s . The six separate 01 analyses within the 12-hr span are provided with the most recent synoptic analysis for the first guess. The preprocessed 01 data are then inserted into the evolving model forecast with a weighting corresponding to their reliability. In the absence of data the model is unperturbed, while with highly reliable data the model solution would be essentially replaced by the insertion data at appropriate grid points. The continuous component of the GFDL assimilation involves the repeated insertion of the 0 1 analyzed data at each time step within the 2-hr time interval. To provide time continuity for the repeated insertion, the six individual 2-hr analyses are interpolated in time defining a time base of preprocessed data that is continuous for the 12-hr span. A schematic of the GFDL assimilation scheme shown in Fig. la illustrates these procedures. The NNMI is performed every 6 hr. The GFDL scheme initializes the first 7 of 18 vertical modes via four iterations of the NNMI. A frequency cutoff restricts the initialization to those free modes having a characteristic oscillation period of less than 6 hr. The prediction model component of the GFDL assimilation scheme, summarized in

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first guess application

prepared by 3-dimensional. u n i v a r i a t e , optimum i n t e r p o l a t i o n PSL.u.v.T.q on p r e s s u r e l e v e l s (no d a t a i n s e r t e d a t upper two m d e l levels) N40 gaussian, 1 9 p r e s s u r e l e v e l s 12-hr a s s i m i l a t i o n r e s u l t s every 1 2 h r (+ 1-hr d a t a window)

ASSIHILATION d a t a injected i n t o a global s p e c t r a l model, using weighted t i m e i n t e r polation uariabtee P*.T. 5 ( v o r t i c i t y ) , D(divergence), q on s i p a l e v e l s resolution R30L18 (rhomboidal t r u n c a t i o n a t 30 wavea. 1 8 a l e v e l s ) time integration semi-implicit @t-8%20 minutesf u n c t i o n of CFL c r i t e r i o n ) lateral diffueion I(D2 vertical mixing l e n g t h method ( t o 3 Irm depth), diffusion d r y convective adjustment (above l e v e l 18) boundary Zayer Monia-Obukhov process s p e c t r a l l y truncated topography developed by F e l s and Schwarrkopf radiation

method

*

* * *

*

(i) clouds - c l i m a t o l o g i c a l monthly mean for each l a t i t u d e (ii)application-diurnal variation; short- and long-wave r a d i a t i o n c a l c u l a t i o n every 2 h r

* *

sea-surface temperature land-surface temperature moisture

0019

-2 27

0052 - 3 x ) O W

-A

16

01%

-5

13

0223 - b

II

om

9

-7

0376 -8

16

0-

b2

-V

0542 -I0

10

0624 -11

1P

0703 --I2

30

O m -13

21

O W -14

14

0901 --150W. 0948 --16OA!

O W -17

017

o m -1aoo;

RAND monthly c l i m a t o l o g i c a l nornmls. y e t varying d a i l y

determined by s u r f a c e h e a t balance. using 3 s o i l l e v e l s t o model h e a t f l u x l a r g e - s c a l e condensation a t 80%humidity s a t u r a t i o n , cumulus p a r a m e t e r i z a t i o n by moist convective adjustment

INITIALIZATION

* method application

n o n l i n e a r normal mode, 7 v e r t i c a l modes (only modes with p e r i o d s s h o r t e r than 6 h r a d j u s t e d ) every 6 h r

ARCHIVE (m/s). T(.K). mixing r a t i o (g/g). g e o p o t e n t i a l h e l g h t (geop. meters). RH(%), v e r t i c a l v e l o c i t y ( m b / s ) , s u r f a c e wind s t r e s s T ~ ,T~ (N/m2)

variables

U.V

grid

1.875’ l a t i t u d e l l o n g i t u d e , 19 p r e s s u r e l e v e l s (9 Pressure l e v e l s for MR and RH)

-

,

DATA ASSIMILATION

141

Fig. Ib, is an 18-level global spectral model truncated at rhomboidal wave number 30. A detailed description of the model is given in Gordon and Stem (1982). Aspects of the GFDL scheme that are particularly distinctive include the extent of preprocessing and the insertion of data at each model time step. Continuous insertion is also utilized in the operational and research assimilation system developed at the UKMO. This system uses repeated interpolation and insertion of data at each time step of the forward-running forecast model. The time integration of the model is an economical split explicit scheme. The basic assimilation procedure was developed by Lorenc (1976) and used in the FGGE planning OSSEs discussed earlier. The UKMO scheme employs univariate 01 and the ongoing model forecast is adjusted by weighted corrections. The UKMO scheme derives the weights from the 6-hourly application of the 0 1 algorithm and keeps these constant for the 6-hr period. No explicit initialization procedures are used.

4.2. Intermittent Assimilation

The intermittent method of assimilation is perhaps the most widespread approach taken at present. Here the observations from a time span, typically within 3 hr of a nominal analysis time, are used to correct a 6-hr forecast made from the previous analysis. This approach ignores the asynoptic error associated with grouping the data into time blocks of 6 hr and relies entirely on the prediction model capability to coordinate the data in the time domain. A very comprehensive example of the intermittent assimilation is that utilized at the ECMWF. Here 01 employed at 6-hourly intervals, is applied in a three-dimensional multivariate form. The simultaneous analysis of geopotential and wind data is performed in pressure coordinates at 15 standard levels from 1000 to 10 mb. To the extent that the effective observational data base changes only slowly from grid point to grid point, the analyzed increments of geopotential and wind on pressure surfaces are locally close to nondivergent. Outside the tropical latitudes, the multivariate aspect of the scheme also retains an almost FIG. 1 . The GFDL assimilation system: (a) Schematic overview of the data-processing system during the FGGE. OPI denotes optimum interpolation, and the dashed arrows indicate the first guess used for OPI. Data are inserted into the model every time step as indicated by the full arrows, and a nonlinear normal mode initialization, denoted by N , is performed every 6 hr. (b) Model configuration. [From Stern and Ploshay (1983).]

142

W. BOURKE, R. SEAMAN, AND K. PURI

geostrophic relationship between analyzed geopotential and wind corrections. The incremental changes to the 6-hr forecast first-guess fields are interpolated from pressure to sigma coordinates and applied as corrections to the current model state in the sigma domain. A distinctive aspect of the ECMWF multivariate scheme is the definition of large overlapping analysis volumes from 660 to 1330 km square and one-third of the atmosphere deep. The multivariate 0 1 may utilize up to 200 separate observations in this three-dimensional volume to define increments of height and wind simultaneously, thus reflecting an appropriate mass-wind relationship. The final analyzed increments consist of weighted combinations of the increments from the overlapping volumes. The ECMWF prediction model is currently a 15-level global spectral model truncated at triangular wave number 63. The FGGE IIIB analyses were produced by utilizing a grid-point precursor to this model that effectively had a slightly lower resolution. A schematic of the ECMWF assimilation scheme and a summary of the prediction model are shown in Fig. 2a, b, respectively. At the completion of each 6-hourly 0 1 , the NNMI is applied, with two iterations and initialization of four vertical modes. Although not implemented in the IIIB-analysis system, an innovation is the inclusion of diabatic forcing in the initialization (Wergen, 1982). This is similar in effect to a frequency cutoff as used at the GFDL in that the model prediction and analysis of the divergent circulations in the tropics are better preserved. The United States operational global analysis and prediction system as developed at the NMC are very similar in broad strategy to that used at the ECMWF. The prediction model is a global spectral model, and the archived analysis is produced from a multivariate 01 applied at 6-hourly intervals, utilizing the NNMI. Other centers utilizing an intermittent assimilation scheme include the Canadian Weather Service, the JMA, and the ANMRC. The Canadian approach parallels that of the ECMWF scheme. The last two groups use spectral models in the prediction step, while 01 is applied as a two-dimensional multivariate scheme at the JMA and as a univariate scheme at the ANMRC. The discussion hitherto has neglected to mention the analysis and prediction of the moisture content of the atmosphere. While moisture is undoubtedly a key aspect of the thermodynamics of the atmosphere, Smagorinsky et al. (1970) discussed whether the moisture specification needs to be defined explicitly, since the forward-running model dynamics may be sufficient to generate an appropriate moisture specification. A

143

DATA ASSIMILATION

detailed discussion of the analysis of moisture in the context of the ECMWF scheme has been given by Tibaldi (1982). This scheme reflects the widespread use of a simple SCM analysis of moisture, as part of the normal assimilation cycle. The data base for moisture analysis is substantially inferior to that of the thermal field, although the polar-orbiting satellites do provide profiles of the precipitable water between several layers in the troposphere, in addition to temperature soundings. However, the quality of these data are such that they have not been used routinely in operational or research application. 5. ROLEOF FOUR-DIMENSIONAL ASSIMILATION IN RESEARCH A N D OPERATIONS

Four-dimensional assimilation has been developed during the past 15 years to the point that it is an essential component of numerical analysis and prediction systems in both research and operations. In research the most visible demonstration of this has been the production of twice-daily global analyses for the entire FGGE year by the ECMWF and the GFDL. In operations, the improving performance of medium-range global and hemispheric prediction at centers such as the ECMWF, the UKMO, and the NMC are clear evidence of the practical benefits of research in this area. (a1

18 GHT

00 GMT

Interpolation

06 GnT

1 2 CHT

Interpolation

In terpolatron

O + P

3-Dimensional Analysis Interpolation P * O

Initializatior

Forecast

I

* +6 * hr

I

3-Dimensional

*.

3-Dimensional

Interpolation

Interpolation

Interpolation

Initialization

Initialization

Initialization

etc

Forecast

FIG.2. The ECMWF assimilation system: (a) schematic overview of the data-processing system during the FGGE and (b) model configuration. [From Bengtsson er a / . (1982). From Birlletin ofrhe American Mereorologicd Sociefy, copyright 1982 by the American Meteorological Society.]

ANALYSIS $,u,v,q(for

PREDICTION pr300)

a

p(mb) 10 20 30 50 70

0.025 (al) 0.077 0.132 0.193 0.260 0.334 0.415 0.500 0.589 0.678 0.765 0.845 0.914 0.967

100 150 200 2 50 300 400

500 700 850 1000

T,u.v,q

North

o u Vertical and horizontal (latitude-longitude) grids and dispositions of variables in the’analysis (left) and prediction (right) coordinate systems.

ANALYSIS

Method

3-dimens ional multivariate

Independent variab Zes

A , cp, P # t

Dependent variub l e s Grid

Nonstaggered, standard pressure levels

First guess D a t a assimi~ationfrequency

6 , u.

" 8

(15-analysis levels, see above)

9

6-hr forecast ( complete prediction mode 1 )

6-hr (f3-hr w ndow)

INITIALIZATION

Method

N o n l i n e a r n o r m a l mode, 5 v e r t i c a l modes, n o n a d i a b a t i c

PREDICTION

-

P VI

Independent variables

x,

Dependent variables

T , u, v , q , P,

Grid

S t a g g e r e d i n t h e h o r i z o n t a l (Arakawa C - g r i d ) . Uniform h o r i z o n t a l ( r e g u l a r l a t / l o n g ) . Nonuniform v e r t i c a l s p a c i n g O f l e v e l s ( s e e above).

Finite difference scheme

Second o r d e r a c c u r a c y

Time integmtion

(At = 15 min) ( t i m e f i l t e r w = 0.05) 15 m4 s-l L i n e a r , f o u r t h o r d e r ( d i f f u s i o n c o e f f i c i e n t = 4.5.10 )

Horizontal diffusion

(0,

Q,

t

Leapfrog, semi-implicit

Earth surface

A l b e d o , r o u g h n e s s , s o i l m o i s t u r e , snow, and i c e s p e c i f i e d g e o g r a p h i c a l l y . A l b e d o , s o i l , m o i s t u r e and snow time d e p e n d e n t .

Orography

I n c l u d e d , m o d e r a t e l y smooth.

Physical parameterization

(i) (ii)

Boundary eddy f l u x e s d e p e n d e n t on r o u g h n e s s l e n g t h and l o c a l s t a b i l i t y (Monin-Obukov) F r e e - a t m o s p h e r e t u r b u l e n t f l u x e s d e p e n d e n t on m i x i n g l e n g t h and R i c h a r d s o n number

( i i i ) Kuo c o n v e c t i o n scheme

(iv) (v) (vi) (vii)

F u l l i n t e r a c t i o n between r a d i a t i o n and c l o u d s F u l l hydrological cycle Computed l a n d t e m p e r a t u r e , no d i u r n a l c y c l e Climatological sea-surface temperature

FIG.2. (Continued)

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W. BOURKE, R. SEAMAN, AND K. PURI

5.1. Research Implications of Four-Dimensional Assimilation It is now clear, after the FGGE and the production of the level-IIIB analyses, that the meteorological community has widely disseminated global analyses of very high quality. The assimilation systems that were developed for the FGGE, and a subset of the FGGE observing system has been maintained, enabling global analyses to be produced routinely at centers such as the ECMWF, the NMC, and the UKMO and more recently at the JMA. With these high-quality global analyses now available for a period of 5 years it is possible to address a range of questions fundamental to meteorological science. These include

(1) What are the limits to predictability, not only of day-to-day weather, but also of the aggregated atmospheric variables referred to as climate on the intra- and interseasonal time scales? (2) What are the factors governing intra- and interseasonal variations in weather? Of particular interest at the time of writing is the research program into the tropical oceans and global atmosphere, formally known as TOGA. This program is concerned with assessing the extent to which the time-dependent behavior of the tropical-ocean global-atmosphere system is predictable on the time scales of months and years. An essential component of TOGA is the ability to define the variations in the global atmospheric circulation, thermodynamics, and hydrological cycle. The 1982/1983, El Nifio episode has given particular emphasis to these studies. Indeed the TOGA program has drawn attention to the development of data-assimilation systems for the description of the temperatures, circulation, and pressure fields of the upper layers of the global ocean. An operational ocean-thermal-analysis forecast system has been developed by Clancy and Pollak (1983). The atmospheric data-assimilation system in these pioneering studies is providing the surface stress and the heat flux at the air-sea interface, and the concept of assimilation has thus already been expanded to the coupled atmosphere-ocean domain. As the atmospheric assimilation systems have improved, numerical weather prediction has been usefully extended in time scale. Medium-range weather prediction is now considered to be successful for time scales of the order of 1 week in the Northern Hemisphere mid-latitudes and to 4 days in the Southern Hemisphere. The limit to medium-range predictability has been addressed by Lorenz (1982). He

DATA ASSIMILATION

147

concludes that estimates of instantaneous weather patterns that are better than guesswork nearly 2 weeks in advance appear to be possible, and efforts to achieve this are clearly warranted. Such useful extensions in skill will no doubt depend on ability to enhance current assimilation systems as well as the prediction models themselves.

5.2. Research on Four-Dimensional Assimilation Procedures In the preceding discussion much has been said of the current four-dimensional assimilation procedures. The refinements to this approach are an ongoing activity at all the major research and operational numerical weather centers throughout the world. The various assimilation systems now available differ in detail in many respects from each other. Intercomparisons of data-assimilation schemes have been undertaken in a joint study by the ECMWF, the UKMO, and the NMC (Hollingsworth et al., 1985). These intercomparisons have concentrated on use of the FGGE data set. Certain differences in the quality of prediction to 3 days can be attributed to differences in the respective analyses and thereby to the assimilation systems. Even with the FGGE data base, it is evident that the current assimilation schemes do not always define the three-dimensional, global large-scale flow in a similar fashion. Although these differences no doubt reflect inevitable uncertainties in the initial state, it is also likely that the exploitation of the data base is less than optimum. Procedures for assimilating single-level data, such as surface pressure and cloud vector winds in particular, remain less than well defined. Indeed, differences in some of the rather ad hoc methods adopted in various approaches could only be expected to give rise to differing analyses. Many analysis differences in the intercomparison study cited above were also associated with differences in quality control and data selection. For example, the ECMWF system in some cases was assessed to be averaging inconsistent data, the NMC system to be rejecting some data, and the UKMO system to be accepting most data but sometimes in an unbalanced fashion. The absence of an explicit geostrophic constraint between mass and wind in the UKMO system at that time was identified as a contributory factor to poor forecasts in the Southern Hemisphere. A global energetics study of the FGGE analyses produced by the GFDL and the ECMWF for the two special observing periods has been conducted by Kung and Tanaka (1983). There are sharp contrasts in the energy transformations depicted in the analyses that are attributed to

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W. BOURKE, R. SEAMAN, AND K . PURI

specific differences in the four-dimensional assimilation procedures. The major differences are thought to be associated with (1) the differing NNMI and in particular the initialization of selected modes in the GFDL scheme and (2) the more geostrophic character of the multivariate analyses from the ECMWF in comparison to the univariate approach at the GFDL. In seeking to improve assimilation methods there are many areas of potential modifications. Given the widespread use of 01, it is necessary to specify expected prediction error covariances among all variables and positions that are to be analyzed. The assumed relationships in widespread use are unlikely to be optimum, and by using the large amounts of data accumulated from operational systems it is now possible to derive better models of auto- and cross-correlation functions for prediction error. Such studies are in progress at a number of centers, and although these results will be system dependent, it is anticipated that as assimilation procedures improve some consensus in modeling prediction error could be reached. Improvements in assimilation will also accompany improved realism in the predictive model component. The forecast models provide both the first-guess specification for 01 and often the fields for preliminary quality control of observations. The view of the study conference in Exeter (WMO/ICSU, 1982), as part of the numerical experimentation program on observing systems, was that considerable improvements can still be made in models using higher resolution, better numerical techniques, and improved parameterizations. The breakthrough in initialization now attributed to the NNMI is undoubted. However, the role of the NNMI in the tropics and in the vicinity of high orography is not fully understood. To some extent, the broad specification of the Hadley circulation can be retained by including some diabatic forcing in the initialization step (Wergen, 1982) or by simply bypassing the initialization of the low-frequency large-scale modes associated with the Hadley flow (Puri and Bourke, 1982; Puri, 1983a). Also the detailed specification of the diabatic heating at the initial forecast time is considered necessary if the divergent wind field over the tropics in particular is to be analyzed and predicted correctly. The ability both to analyze and to predict in the vicinity of high orography is clearly of relevance as evidenced by close attention given to specification of the numerically enhanced or “envelope orography” (Wallace et al., 1983). This approach has resulted in some improvement in prediction beyond 4 days, although it is also associated with some degradation at 1 day.

DATA ASSIMILATION

149

5.3. Long-Term Operational Implications of Four-Dimensional Assimilution

The immediate operational implications of current assimilation systems are evidenced by the quality of medium-range predictions routinely available. The longer-term implications arise from the ability to conduct ( I ) OSEs with the enhanced data base available from the FGGE year and (2) OSSEs with proposed future enhancements of observational systems. The JOC study conference in Exeter (WMO/ICSU, 1982) drew attention to the importance of research utilizing the FGGE data. At this meeting the particular concern was the coordination of numerical experimentation of relevance to the design of the future WWW system. The meeting considered a number of observing systems, including satellite temperature soundings, cloud vector winds, and drifting-buoy pressure observations and their relevance to maintaining accurate global analyses. The current range of assimilation systems enabled certain conclusions to be drawn about the observational requirements for global analysis and prediction. Among these conclusions were the following: (1) Atmospheric soundings from polar-orbiting satellites are an essential element of the global observing system, (2) cloud vector winds contribute significantly to tropical and Southern Hemisphere analysis although showing little impact on prediction, (3) drifting buoys in the Southern Hemisphere provide large positive impact on analysis and prediction, and (4) the impact of single-level observing systems is significantly dependent on the particular assimilation system used.

The ability to identify these factors in OSEs highlights the current value of existing assimilation systems. It is unlikely that the WWW network will approach the comprehensiveness of the FGGE year in the near future, and it is therefore crucial to carefully identify those components of key importance. There is now widespread activity in the numerical experimentation community to assess further the impact of particular observing systems in a closely controlled manner. For example, the same data base and broad strategies are to be utilized by a number of operational and research centers using data from the first special observing period of the FGGE. This intercomparison will highlight the response of analysis and prediction systems to degradation of the data base and should provide some further limited but quantitative measures of the impact of various observing systems. It should be noted that OSEs are inherently difficult

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W. BOURKE, R. SEAMAN, AND K. PURI

and expensive and that the results are not necessarily as decisive and quantitative as the designers of observing systems require. It is also clear that the utilization of the current observational data base could be enhanced. An obvious example is the potential for very-highresolution temperature soundings from the operational polar-orbiting satellites. These data are presently available at a horizontal resolution of from 250 to 500 km. With enhanced global processing, such as performed locally at a number of centers, the global data base for temperature soundings could approach a resolution of the order of 100 km. The numerical resolution of the assimilation and prediction systems under current development at the NMC, the GFDL, the ECMWF, and the UKMO could be expected to require that data base in the near future. In addition to improving the horizontal resolution of temperature soundings, the scope for improving the quality of the soundings themselves is also of considerable importance and is strongly coupled to present-day assimilation endeavors. In particular, recent reconsideration of direct inversion methods (Smith et al., 1984), in contrast to current operational statistical retrieval methods, affords the promise of more accurate soundings using first-guess temperature information from the assimilation systems. The present data base supporting global assimilation and prediction systems is almost totally dependent on space-based observing systems. Proposed enhancements of these satellite systems include substantially improved temperature soundings from an infrared interferometer (Smith, 1979, 1984) and the possibility of improved direct determination of wind fields. A particularly interesting system under consideration, known as WINDSAT, could provide high-quality wind measurements over the globe at horizontal resolutions of less than 100 km (Huffaker, 1978). The WINDSAT project is particularly exciting in view of the known behavior of current assimilation prediction systems. The wind measurements via a satellite-based infrared Doppler radar have been suggested as a long-term solution to the wind field specification necessary for improved prediction. The information content of wind measurements is relatively more valuable than mass measurements in the tropical domain and for smaller scales over the entire globe. However, the lead time of space-based sensors is long and even now the earliest practical demonstration of the WJNDSAT technique is proposed for a Space Shuttle flight in 1990. Accordingly, it is necessary to simulate the capabilities of such a system if indeed it is to materialize. The current assimilation systems and largescale prediction models afford the option to assess quantitatively such proposed new observing systems.

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151

6. CONCLUSION Four-dimensional assimilation has reached substantial maturity following the decade of research leading up to the FGGE and the more recent implementation of operational assimilation systems at centers throughout the world. The value of the assimilation OSSE studies in the seventies has been verified by the success of the current observing and assimilation systems in providing the meteorological community with comprehensive analyses spanning the global atmosphere for the first time. Thirteen years after the first conference o n four-dimensional assimilation in 1971 under the sponsorship of the JOC and hosted by Dr. Smagorinsky at the GFDL, it is clear that great progress in this field has been made. The capacity to develop global assimilation and prediction systems further and to design an optimum observing system for WWW in the decade ahead is a sign of great vitality in this important component of meteorological science.

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system for FGGE. I n “The GARP Programme on Numerical Experimentation,” Rep. No. 16, pp. 1-54. World Meteorol. Organ./Int. Counc. Sci. Unions, Geneva. Charney, J., Halem, M., and Jastrow, R. (1969). Use of incomplete historical data to infer the present state of the atmosphere. J. Atmos. Sci. 26, 1160-1163. Clancy, R. M., and Pollak, K. D. (1983). A real-time synoptic ocean thermal analysis/ forecast system. Prog. Oceanogr. 12, 383-424. Daley, R. (1978). Variational non-linear normal mode initialization. Tellus 30,201-218. Daley, R., and Pun, K. (1980). Four-dimensional data assimilation and the slow manifold. Mon. Weather Rev. 108, 85-99. Desmarais, A., Tracton, S., McPherson, R., and van Haaren, R. (1978). “The NMC Report on the Data Systems Test” (NASA Contract S-70252-A G ) . U.S. Dept. of Commerce, Natl. Oceanic Atmos. Admin., Natl. Weather Serv., Washington, D.C. Dickinson, R. E., and Williamson, D. L. (1972). Free oscillations of a discrete stratified fluid with application to numerical weather prediction. J. Atmos. Sci. 29, 623-640. Eddy, A. (1967). The statistical objective analysis of scalar data fields. J. Appl. Meteorol. 6, 597-609. Eliassen, A. (1954). “Provisional Report on Calculation of Spatial Covariance and Auto Correlation of the Pressure Field,” Rep. 5 . Inst. Weather Climate Res. Acad. Sci., Oslo. Flattery, T. (1970). “Spectral Models for Global Analysis and Forecasting,” Air Weather Serv. Tech. Rep. 242, pp. 42-54. U.S. Naval Academy, Annapolis, Maryland. Gadd, A. J. (1978). A split explicit integration scheme for numerical weather prediction. Q. J . R. Meteorol. SOC. 104, 569-582. Gandin, L. (1963). “Objective Analysis of Meteorological Fields.” Gidrometeorol. Izd., Leningrad (Israel Program for Scientific Translations, 1965). Gordon, C. T., and Stem, W. F. (1982). A description of the GFDL global spectral model. Mon. Weather Rev. 110, 625-644. Gordon, C. T., Umscheid, L., and Miyakoda, K. (1972). Simulation experiments for determining wind data requirements in the tropics. J. Atmos. Sci. 29, 1064-1075. Gustavsson, N. (1981). A review of methods for objective analysis. I n “Dynamic Meteorology: Data Assimilation Methods” (L. Bengtsson, M. Ghil, and E. Kallen, eds.), pp. 1776. Springer-Verlag, Berlin and New York. Hayden, C. (1973). Experiments in four-dimensional assimilation of Nimbus 4 SIRS data. J . Appl. Meteorol. U , 425-435. Hollingsworth, A.. Lorenc, A. C., Tracton, M. S., Arpe. K., Cats, G., Uppala, S., and Klllberg, P. (1985). The response of numerical weather prediction systems to FGGE 11-b data. Part I. Analyses. Q . J. R. Meteorol. Soc. 111, 1-66. Huffaker, R. M., ed. (1978). Feasibility study of satellite-borne radar global wind monitoring system. NOAA Tech. memo. ERL WPL-37. U.S. Dept. of Commerce, Washington, D.C. Jastrow, R., and Halem, M. (1973). Simulation studies and the design of the first CARP global experiments. Bull. Am. Meteorol. SOC.54, 13-21. Kanamitsu, M., Tada, K., Kudo, T., Sato, N., and Isa, S. (1983). Description of the JMA operational spectral model. J. Meteorol. Soc. Jpn. [Ill 61, 812-828. Kasahara, A. (1972). Simulation experiments for meteorological observing systems for GARP. Bull. Am. Mereorol. SOC. 53, 252-264. Kasahara, A., and Williamson, D. (1972). Evaluation of tropical wind and reference pressure measurements: Numerical experiments for observing systems. Tellus 24, 100-1 15. Kistler, R., and Parrish, D. (1982). Evolution of the NMC data assimilation system: September 1978-January 1982. Mon. Weather Rev. 110, 1335-1346.

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Kruger, H. B. (1969). General and special approaches to the problem of objective analysis of meteorological variables. Q. J. R . Meteorol. Soc. 95, 21-39. Kung, E. C., and Tanaka, H. (1983). Energetics analysis of the global circulation during the special observing periods of FGGE. J. Atmos. Sci. 40,2575-2592. Larsen, G., Little, C., Lorenc, A., and Rutherford, I. (1978). Analysis error calculations for the FGGE. In “The GARP Programme on Numerical Experimentation,” Rep. No. 16, pp. 55-1 17. World Meteorol. Organ./Int. Counc. Sci. Unions, Geneva. Leith, C. (1980). Nonlinear normal mode initialization and quasi-geostrophic theory. J . Atmos. Sci. 37,958-968. Lorenc, A. C. (1975). Results of observing systems simulation experiments for the First GARP Global Experiment. In “The GARP Programme on Numerical Experimentation,” Rep. No. 10, pp. 37-68. World Meteorol. Organ./Int. Counc. Sci. Unions, Geneva. Lorenc, A. C. (1976). Results of some experiments assimilating observations from a simulation of the FGGE observing system into a global circulation model. I n “The GARP Programme on Numerical Experimentation,” Rep. No. 1 I , pp. 358-374. World Meteorol. Organ./Int. Counc. Sci. Unions, Geneva. Lorenc, A. C. (1981). A global three-dimensional multivariate statistical interpolation scheme. Mon. Weather Rev. 109, 701-721. Lorenz, E. N. (1982). Atmospheric predictability experiments with a large numerical model. Tellus 34, 505-513. Machenhauer, B. (1977). On the dynamics of gravity oscillations in a shallow water model, with applications to normal mode initialization. Beitr. Phys. Atmos. 50, 253271. McPherson, R. D. (1975). Progress, problems, and prospects in meteorological data assimilation. Bull. A m . Meteorol. Soc. 56, 1154-1166. Miyakoda, K., Umscheid, L., Lee, D. H., Sirutis, J., Lusen, R., and Pratte, F. (1976). The near-real-time, global, four-dimensional analysis experiment during the GATE period. Part I. J . Atmos. Sci. 33, 561-591. Miyakoda, K., Sheldon, J., and Sirutis, J. (1982). Four-dimensional analysis experiment during the GATE period. Part 11. J. Atmos. Sci. 39, 486-506. Pun, K. (1983a). The relationship between convective adjustment, Hadley circulation and normal modes of the ANMRC spectral model. Mon. Weather Rev. 111, 23-33. Puri, K. (1983b). Some experiments in variational normal mode initialization in data assimilation. Mon. Weather Rev. 111, 1208-1218. Pun, K . , and Bourke, W. (1982). A scheme to retain the Hadley circulation during nonlinear normal mode initialization. M o n . Weather Rev. 110, 327-335. Puri, K . , Bourke, W., and Seaman, R . (1982). Incremental linear normal mode initialization in four-dimensional data assimilation. Mon. Weather Rev. 110,~1773-1785. Robert, A. J . (1969). The integration of a special model of the atmosphere by the implicit method. Proc. WMOIIUGG Symp. Numer. Weather Predicr. 1968, pp. VII-9VII-24. Rutherford, I. (1973). Experiments in the updating of P. E. forecasts with real wind and geopotential data. Prepr., Conf. Probability Stat. A f m o s . Sci., 3rd, 1973, pp. 198-201. Rutherford, 1. D. (1976). An operational three-dimensional multivariate statistical objective analysis scheme. In “The GARP Programme on Numerical Experimentation,” Rep. No. 1 1 , pp. 98-121. World Meteorol. Organ./Int. Counc. Sci. Unions, Geneva. Schlatter, T. W. (1975). Some experiments with a multivariate statistical objective analysis scheme. Mon. Weather Rev. 103, 246-257.

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Smagorinsky, J., Miyakoda, K., and Strickler, R. F. (1970). The relative importance of variables in initial conditions for dynamical weather prediction. Tellus 22, 141-157. Smith, W. L. (1979). The use of radiance measurements for sounding the atmosphere. J. Atmos. Sci. 36, 566-575. Smith, W. L. (1984). Passive remote sounding from meteorological satellites. Ext. Abstr. Ausiralas. Conf. Phys. Remoie Sens. Aimos. Ocean, 1st 1984, pp. 36-39. Smith, W. L., Woolf, H. M., Hayden, C. M., Schreiner, A. J., and Le Marshall, J. F. (1984). The physical retrieval T O W Export Package. In “The Technical Proceedings of the First International TOVS Study Conference” (W. P. Menzel. ed.). CIMSS University of Wisconsin, U.S. Stackpole, J. D. (1976). The National Meteorological Center eight-layer global forecast model. Prepr., Conf. Weather Forecast. Anal. 6ih, 1976, pp. 112-116. Stem, W. F., and Ploshay, J. J. (1983). An assessment of GFDL’s continuous data assimilation system used for processing FGGE data. Prepr., Conf. Numer. Weather Predict., 6ih, 1983, pp. 90-95. Talagrand, 0. (1981). On the mathematics of data assimilation. Tellus 33, 321-339. Temperton, C. (1984). Variational normal mode initialization for a multilevel model. M o n . Wearher Rev. 112, 2303-2316. Tibaldi, S. (1982). The ECMWF humidity analysis and its general impact on global forecasts and on the forecast in the Mediterranean area in particular. Riv. Meteorol. Aeronaur. 42, 309-328. Tracton, M. S . , and McPherson, R. D. (1977). On the impact of radiometric sounding data upon operational numerical weather prediction at NMC. Bufl. A m . Meteorol. Soc. 58, 120 1-1209. Tnbbia, J. J. (1982). On variational normal mode initialization. Mon. Weather Rev. 110, 455-470. Wallace, J. M., Tibaldi, S., and Simmonds, A. J. (1983). Reduction of systematic forecast errors in the ECMWF model through the introduction of an envelope orography. Q. J. R . Meteorol. SOC. 109, 683-717. Wergen, W. (1982). Incorporation of diabatic effects in non-linear normal mode initialization. In “The GARP Programme on Numerical Experimentation,” Rep. No. 3, pp. 2.82.10. GARPIWCRP, World Meteorol. Organ./lnt. Counc. Sci. Unions, Geneva. Williamson, D. L. (1975). Observing systems simulation experiments for the First GARP Global Experiment. In “The GARP Programme on Numerical Experimentation,” Rep. No. 10, pp. 97-124. World Meteorol. Organ./Int. Counc. Sci. Unions, Geneva. World Meteorological Organization/lnternational Council of Scientific Unions (WMO/ ICSU) (1971a). “Report of the Fifth Session of the Joint Organizing Committee for GARP, Bombay, 1971,” sect. 5.3.1, p. 15. WMO/ICSU, Geneva. World Meteorological Organization/International Council of Scientific Unions (WMO/ ICSU) (1971b). “Report of the Sixth Session of the Joint Organizing Committee for CARP, Toronto, 1971,” sect. 5.3.2, pp. 17-18. WMO/LCSU, Geneva. World Meteorological Organization/lnternationalCouncil of Scientific Unions (WMO/ ICSU) (1974). “Report of the Eighth Session of the Joint Organizing Committee for CARP, London, 1973,” sect. 4.2.4, pp. 9-10. WMO/ICSU, Geneva. World Meteorological OrganizationlInternational Council of Scientific Unions (WMO/ ICSU) (1976). Proceedings of the JOC Study Group Conference on Four-Dimensional Data Assimilation, Paris, 1975. In “The GARP Programme on Numerical Experimentation,” Rep. No. 11, pp. 1-5. WMO/ICSU, Geneva. World Meteorological Organization/lnternational Council of Scientific Unions (WMOI

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ICSU) (1977). “Report of the Thirteenth Session of the Joint Organizing Committee for GARP, Stockholm, 1977,” sect. 3.1.3, p. 3 . WMOlICSU, Geneva. World Meteorological Organizationllnternational Council of Scientific Unions (WMOl ICSU) (1982). JSC study conference on observing systems experiments, Exeter. 1982. I n “The GARP Programme on Numerical Experimentation,” Rep. N o . 4, pp. 1-23. GARPIWCRP, WMOJICSU, Geneva.

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Part II

MESOSCALE DYNAMICS

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PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS RICHARD A. ANTHES YING-HWA Kuo DAVIDP. BAUMHEFNER RONALD M. ERRICO W. BETTGE THOMAS National Center for Atmospheric' Reseurch* Boulder, Colorado I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. I . Pessimistic Outlook for Mesoscale Predictabihty Based on Atmospheric Spectra and

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1.2. Counterarguments and More Optimistic Points of View . . . . . . . . . . . . . 2. Classic Predictability Experiments and Their Relationship to Mesoscale Predictability . . 3. Preliminary Predictability Study with a Mesoscale Model . . . . . . . . . . . . . 3.1. Observed Atmospheric Evolution 10-1 I April 1979 . . . . . . . . . . . . . . 3.2. Summary of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Introduction of Perturbation on Initial Conditions. . . . . . . . . . . . . . . 3.4. Results of Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 4. Discussion and Comparison with a Predictability Study Using a Global Model . . . . . 5. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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It is well known that nonlinear numerical models of the global atmospheric circulation that are integrated forward in time from slightly different initial conditions will produce solutions that diverge with time, until the differences (in a root-mean-square sense) become as large as the differences expected between two randomly chosen simulations from a large ensemble of cases. In models of synoptic-scale atmospheric motions (those with horizontal scales greater than 2000 km), the time at which the variance of a pair of solutions with initially small differences (often considered to be representative of errors in the observations) reaches the error variance associated with two randomly chosen atmospheric states is considered to be the limit of atmospheric predictability. The estimation of the limits to predictability of synoptic-scale motion has attracted the attention of many fluid dynamicists and atmospheric scientists over the past

* The National Center for Atmospheric Research is sponsored by the National Science Foundation. 159 ADVANCES IN GEOPHYSICS. VOLUME

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two decades. The present estimations of the limit to predictability of large-scale motion range from about 10 days (Baumhefner, 1984) to more than a month (Shukla, 1981a). While progress in improving synoptic-scale predictions using global numerical models has been steady over the past 25 years, corresponding progress in the prediction of important mesoscale* phenomena, such as convective precipitation systems, has been slow. However, recent advances in scientific understanding and in technology (remote-sensing systems from the ground and space and supercomputers) have made it possible to envisage significant improvements in predicting mesoscale atmospheric events [University Corporation for Atmospheric Research (UCAR), 19831. This possibility raises the important questions of what are the inherent limits to mesoscale predictability and what physical factors determine these limits. Two properties of the atmosphere are considered important in limiting atmospheric Predictability. First is the existence of instabilities (e.g., baroclinic, inertial, and convective instabilities) that cause neighboring trajectories in phase space to diverge. A second property is the nonlinear interactions between different components of the wave spectrum. These interactions depend on the initial distribution of energy in the different wave numbers and on the number of waves the model can resolve. Uncertainties and errors in the resolvable-scale waves and errors introduced by the neglect of unresolvable scales grow with time and spread throughout the spectrum, eventually contaminating all wavelengths and destroying the forecast. The study of the growth and spread of errors through wave-number space in homogeneous turbulence models has been an important scientific method in estimating the limits to predictability of large-scale atmospheric flows (Thompson, 1957; Robinson, 1967, 1971; Lorenz, 1969a; Leith, 1971; Leith and Kraichnan, 1972). Results from these studies have been used to draw pessimistic conclusions concerning mesoscale predictability (Tennekes, 1978). Although counterarguments will be presented later that suggest more positive conclusions, it is worthwhile to consider first the arguments based on observed atmospheric spectra and turbulence theories.

* In this paper, the scale definitions proposed by Orlanski (1975) are adopted. The mesoscale refers to horizontal scales of motion ranging from 2 to 2000 km. It is subdivided into three scales, the meso-y scale (2-20 km), the meso-p scale (20-200 km), and the meso-a scale (200-2000 km). The emphasis of this paper is on the rneso-a scale.

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1 .I. Pessimistic Outlook for Mesoscale Predictability Bused on Atmospheric Spectra and Turbulence Theory

Figure 1 shows the horizontal spectra of the wind from several studies based on data from the middle and upper troposphere (Lilly and Petersen, 1983). These spectra, which confirm spectra found in earlier studies [e.g., Leith (1971)], indicate a relatively flat region for wave numbers 1-10 (corresponding to scales of motion of roughly 4000-40,000 km). In the range of wave numbers 10 to about 20 (roughly the wavelengths 2000-4000 km), the energy decreases in proportion to k-3, where k is the horizontal wave number. In this well-known region of the -3 power law, the atmosphere behaves like a two-dimensional fluid with energy tending to be trapped in low wave numbers while enstrophy cascades to high wave numbers (Leith, 1971). In models of fluids with spectra that follow the -3 power law, the slow transfer of energy from smaller to larger scales implies that errors introduced in the shortest scales resolvable by the model and by neglect of unresolvable scales will be slow to contaminate the larger scales, indicating extended predictability for these scales of motion. WAVELENGTH A, km 104

103

102

10

IO-~ I o-’ WAVE NUMBER K, m-’, K = E T / A

FIG.1 . Horizontal spectra of the horizontal wind in the middle and upper troposphere. The solid line curves were obtained from analysis of aircraft flight data, the dotted line curve from Doppler radar, and the others principally from sonde data. [-, Nastrom and Gage Lilly and Petersen (1983); ----, Vinnichenko (1970); ...., Balsley and Carter (1982); -, (1982); -.-, Chen and Wiin-Nielsen (1978).j [From Lilly and Petersen (1983).]

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Figure 1 and other studies indicate that, for small scales of motion, the spectrum shows a slower decrease of energy with decreasing scale, with a slope approximately proportional to k V 3 . At very small scales (microscale, 1-1000 m), the kP5l3 slope has been well established; this region corresponds to the three-dimensional (3-D) inertial subrange. Thus, the mesoscale represents a transition from the k-3 power law associated with synoptic scales of motion to a k-513 power law associated with microscale three-dimensional turbulence. In a 3-D turbulent fluid with a -3 power law, energy moves toward both higher and lower wave numbers, at a rate greater than the transfers in a fluid with a -3 power law. In a simulation of a 3-D turbulent regime with a -8 power law and initially no error for wave numbers k < k, and total error for wave numbers k > k, ,Leith and Kraichnan (1972) found that the total error first decreased as information propagated from lower to higher wave numbers. Later, as the initially sharp division between regions with and without errors broadened, errors in the small scale moved toward larger scales and the total error increased. Leith and Kraichnan’s (1972) calculations indicated that any scale of motion would be contaminated by errors within a time that is approximately 10 times the lifetime of a local eddy. The more rapid transfer of energy from small to large scales in three-dimensional turbulence compared to two-dimensional turbulence indicates less inherent predictability for atmospheric systems that behave as 3-D turbulence. Since the mesoscale spans scales of motion ranging from the synoptic scale, which behaves as 2-D turbulence, to the microscale, which behaves as 3-D turbulence, one would expect that mesoscale atmospheric systems, especially at the smaller scales (meso-y, 2-20 km) would have considerably less inherent predictability than synoptic-scale systems. This is the essence of the argument presented by Tennekes (1978). To summarize, the pessimistic conclusion from turbulence studies and the observed atmospheric spectrum is that inevitable errors or initial uncertainties in the small scales of motion will propagate toward larger scales and will reach the mesoscale sooner than the synoptic scales, rendering the former less predictable. Reducing the uncertainty in the smaller scales by increasing the observational density would be very costly and, because of the relatively rapid rate of energy transfer on this scale of the atmosphere, which has a -$ spectrum, would increase the predictability of mesoscale motions only marginally.

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1.2. Counterarguments and More Optimistic Points of View

Several counterarguments can be made to suggest that prediction of some important mesoscale phenomena is not as hopeless as the preceding conclusions indicate. First, the observed atmospheric spectra represent a statistical description of atmospheric structure involving averages over space and time. However, many mesoscale events are highly intermittent. Moreover, the atmospheric structures when they occur are highly organized and may not be as random as the -5 inertial range of 3-D turbulence. Certain mesoscale circulations may have peculiar dynamic structures that resist the cascade of energy to larger and smaller scales. For example, tropical cyclones, which are meso-a-scale phenomena, sometimes persist for several weeks. Severe rotating thunderstorms in a sheared environment seem to have lifetimes and predictability much longer than would be expected from turbulence theories. Lilly (1984) suggests that the reason for this enhanced predictability is that these storms are characterized by large values of helicity, the dot product of vorticity and velocity. Three-dimensional turbulence simulations (AndrC and LeSieur, 1977) indicate that flows that possess high values of helicity resist turbulent decay. According to Lilly (1984), this conclusion helps explain the apparently successful simulation over a period of several hours of a real thunderstorm with a coarse-resolution, poorly initialized, three-dimensional model (Wilhelmson and Klemp, 198 1). A second important factor that influences the behavior of many atmospheric mesoscale phenomena that has not been considered in the predictability studies using turbulence models is the effect of boundary forcing. Surface inhomogeneities including elevation and surface characteristics (albedo, heat capacity, and moisture availability) generate many mesoscale phenomena (such as mountain waves, sea breezes, and convection) and modulate their behavior. Known surface inhomogeneities, if incorporated properly in numerical models, are likely to increase the predictability of motions they force. Anthes (1984) classified the development of mesoscale weather systems into two types: (1) those resulting from forcing by surface inhomogeneities and (2) those resulting from internal modifications of large-scale flow patterns. Land-sea breezes, mountain-valley breezes, mountain waves, heat island circulations, coastal fronts, and dryline and moist convection are often generated by the first mechanisms. Fronts and jetstream phenomena, generated by shearing and deformation associated with large-scale flows, belong to the second class.

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The preceding classification is based on the generation of mesoscale phenomena. Lilly (1984) considers the genesis, evolution, and decay of mesoscale systems in his classification of mesoscale flows into four categories: (1) nearly homogeneous turbulent flow with well-defined variance spectra, (2) frontal and jetlike near discontinuities, (3) flows produced in response to small-scale topographic forcing, and (4) large-amplitude instabilities such as convective storms. Lilly’s first category seems to be the one to which the turbulence studies of the preceding section are relevant. For the other three, as indicated here and by Anthes (1984), Lilly (1984), and UCAR (1983), there is evidence of greater predictability than would be implied by the energy cascade arguments. An optimistic hypothesis is that most of the significant mesoscale atmospheric phenomena belong to one of the last three categories, in which case there is hope for skillful predictions using deterministic methods, provided that the synoptic-scale motions are predicted correctly.

2. CLASSICPREDICTABILITY EXPERIMENTS AND THEIRRELATIONSHIP TO MESOSCALE PREDICTABILITY In studies pertaining to the predictability of synoptic-scale motions, predictability has referred to the growth of small errors in the initial conditions (Charney et al., 1966; Smagorinsky, 1969; Lorenz, 1969~; Williamson, 1973). The theoretical upper limit of predictability is considered to be the time required for the initial small error to grow to the expected error variance E between two randomly chosen states. This latter value is related to the value of the temporal variance u2of the time series of a given variable x by

E = 2u2 (2.1) Three methods have been utilized to estimate the rate of growth of errors and the associated theoretical limits on predictability. The first, and the one used in this study, is the divergence of pairs of solutions of a numerical model that have initial conditions very close to each other (Charney et af., 1966; Smagorinsky, 1969; Jastrow and Halem, 1970; KaSahara, 1972). A second method, summarized in the preceding section, calculates the growth of errors in homogeneous turbulence models. In a third method, Lorenz (1969b) examined the rate of divergence of pairs of near analogs in the real atmosphere. Because atmospheric variabilities u2as well as the growth rate of small errors varies with geographic regions, the theoretical limit to predictability, as given by the classic definition, also varies with location. For exam-

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ple, because u is less in low latitudes than in middle latitudes, the limit to deterministic predictability is shorter in the tropics, provided that the error growth rates are the same (Shukla, 1981b). Even in fixed geographic regions, some weather types, such as those involving mesoscale convection, may be more sensitive to small errors than are other types, and hence predictability vanes with synoptic patterns. When global models are used in classic predictability studies, the likelihood of possible errors in the overall estimate of synoptic-scale predictability is minimized (though not eliminated) by consideration of the entire Earth, which is likely to have a full complement of representative weather patterns over the period of the predictability experiment. That is, erroneous estimates of predictability introduced by weather types unrepresentative of the climatological data base used to derive u2 are likely to be small. However, if classic predictability methods are used to study mesoscale weather phenomena on regional domains, the problems of sampling errors and case dependency of results become much larger, Regional domains (of typical size 5000 x 5000 km) often contain only one or two energetically active systems during a relatively short period (0-48 hr). The different systems almost certainly respond quite differently to initial errors, to the physical forcing at the Earth’s surface, and to the forcing by the physics internal to the model (such as latent heating). In wintertime-developing frontal systems with strong dynamical forcing, for example, initial errors may grow at a different rate than in summertime convective systems with weak dynamic forcing. If the preceding examples are valid, it is clear that care must be taken in making predictability estimates on the mesoscale. Not only do the estimates of natural variability vary greatly among different mesoscale regimes, the growth of initial errors and errors introduced by the physical parameterization also is likely to behave quite differently from one regime to another. It should be apparent from the earlier discussion that considerable quantitative work needs to be done on estimating the predictability of different mesoscale systems. The next section utilizes a threedimensional, primitive-equation model to estimate the predictability of meso-a-scale motions in a regime characterized by organized moist convection.

3. PRELIMINARY PREDICTABILITY STUDYWITH A MESOSCALE MODEL Compared to the amount of effort dedicated to the study of the predictability of large-scale atmospheric motions, very little attention has

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been directed toward quantitative studies of the predictability of mesoscale weather systems. Considerable effort has been put into the development and testing of mesoscale models, however, as reviewed by Anthes (1983). The success of these models in predicting and simulating many mesoscale atmospheric phenomena such as fronts, dry lines, polar lows, orographic waves, mesoscale convective complexes, and mesoscale precipitation features embedded within cyclones indicates that these models have reached a level of maturity that makes quantitative estimates of predictability possible. The use of mesoscale or regional models for predictability studies is not a straightforward extension of the methods used in applying global models to the predictability of large-scale motions. In addition to the sampling problems and the different role of physics discussed earlier, the presence of lateral boundary conditions (LBC) in mesoscale models introduces an added complexity. Not only do errors in the initial data and errors in model physics limit predictability, but errors introduced by the LBC, which are not present in global models, contribute additional errors to the predictions (Baumhefner and Perkey, 1982). Conversely, accurate LBC supply information to the mesoscale models that may improve the predictability by limiting error growth. This section makes an estimate of the growth of errors in a limited-area, mesoscale model for a single case study involving mesoscale convective systems. A series of 48- and 72-hr simulations beginning at 0000 GMT 10 April 1979 are made to investigate the growth of errors in a mesoscale model resulting from variations in initial conditions and lateral boundary conditions. During this period in time, special mesoscale data were collected over the central portion of the United States as part of the Severe Environmental Storm and Mesoscale Experiment (SESAME). The results of the simulations are presented following a description of the observed atmospheric behavior during this period and a summary of the model.

3.1. Observed Atmospheric Evolution 10-11 April 1979 During the 48-hr period beginning 0000 GMT 10 April 1979, organized convection developed over Oklahoma and Texas. This convection spawned a variety of severe weather, including heavy rain, large hail, and at least 12 tornadoes (Alberty et al., 1980). These storms killed 56 people, injured 1916, and caused damage of several hundred million dollars (Moore and Fuelberg, 1981). The evolution of the large-scale features and their interaction with the

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mesoscale have been discussed in several papers (Carlson et al., 1980, 1983; Moore and Fuelberg, 1981; Anthes et al., 1982); therefore, only a brief overview of the evolution of the synoptic-scale motions over the modeling domain is presented here. The lower troposphere during the 72-hr period was characterized by two large cyclones, one moving slowly eastward from the Rockies and the other moving northeastward along the Atlantic coast (Fig. 2). Ahead of the western low, a strong pressure gradient was associated with low-level southerly flow with maximum winds at 850 mb in excess of 25 m s-' (not shown). This flow carried extremely moist air (mixing ratios in excess of 20 g kg-') from the Gulf of Mexico northward into the southern plains, supplying the moisture for the intense convection. At 500 mb, a deep trough was located over the Rockies during the period (Fig. 3). Jet streaks propagated around this trough system during the 72-hr period. The vertical circulations induced by these propagating wind maxima have been linked to the development of the low-level southerly jet and possibly the outbreak of the intense convection over Texas and Oklahoma that began around 2300 GMT 10 April (Moore and Fuelberg, 1981). 3.2. Summary of Model

The model is based on the one described by Anthes and Warner (1978). The vertical coordinate is u = ( p - pt)/(ps- p , ) , where p is pressure, p s surface pressure, and pt the constant pressure at the top of the model (100 mb).Thenumberofalevelsis 16(0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.78, 0.84, 0.89, 0.93, 0.96, 0.98, 0.998, 1.01, which gives 15 layers of unequal thickness at which the temperature, moisture, and wind variables are defined. The horizontal grid contains 46 points in the north-south direction and 61 points in the east-west direction; the grid size is either 80 or 160 km in these simulations. The parameterization of surface and planetary boundary layer (PBL) processes is described by Zhang and Anthes (1982). In this scheme vertical fluxes of heat, moisture, and momentum are calculated explicitly between layers. Under stable conditions, these turbulent fluxes are represented by a local Richardson number. In contrast, under unstable conditions (free convection) the vertical mixing occurs through mixing between convective elements originating at the surface and environmental air in the PBL. The ground temperature and surface sensible and latent heat fluxes are predicted from a surface energy budget. Short- and long-wave radiation is considered in the surface energy

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FIG.2. Observed sea-level pressure over large domain at (a) OOOO GMT 10 April 1979, (b) oo00 GMT 12 April 1979, and (c) OOOO GMT 13 April 1979. Contour interval is 4 rnb.

PREDICTABILITY OF M ESOSCALE ATMOSPHERIC MOTIONS

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(C)

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w

lmo

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loo0 90° 80° FIG.2. (Continued)

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budget but not in the free atmosphere. These radiative fluxes depend upon the model-simulated cloud cover in a parameterization developed by Benjamin (1983). The cumulus parameterization and treatment of nonconvective precipitation follow methods developed by Kuo (1965, 1974) and Anthes (1977). In the convective parameterization, the total latent heat release is proportional to the vertically integrated moisture convergence ; the vertical distribution of the convective heating is specified from a constant profile, as in Anthes et al. (1983). A summary of the constants in the model for these simulations is given in Table I. The initialization procedure involves an interpolation of the National Meteorological Center's (NMC) operational global analysis to the Lambert-conformal grid of the model. This interpolated analysis is used as a first guess in an analysis scheme based on successive scans and the use of radiosonde data at standard and significant levels. Following this enhancement, the vertically integrated mean divergence is removed. Two methods of implementing the LBC are used in the simulation. In most of the experiments, we use the porous-sponge method developed by Perkey and Kreitzberg (1976). In this method, the large-scale tendencies of a prognostic variable a, obtained from either a large-scale model or from observations, are blended in the rows or columns of grid points near

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N

4 0'

30'

20’

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IOO'

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W 120' 110' 100' 90' 80’ 70’ 60’ FIG.3. Observed 500-mb heights (solid lines, contour interval 60 m) and temperatures (dashed lines, contour interval 8°C) at (a) 0000 GMT 10 April 1979, (b) 0000 GMT 12 April 1979, and (c) OOOO GMT 13 April 1979.

I71

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150' 140' 130"

50°40"300 20"

(C)

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FIG.3. (Conrinued)

the boundary with the model-generated tendencies. A second method varies from the first in that the temperature, pressure, and specific humidity are specified from a large-scale model or observations at the lateral boundaries throughout the simulation. This second method is similar to that of Anthes and Warner (1978). Both methods of treating the lateral boundaries generate considerable numerical noise in places; however, the porous-sponge condition generates somewhat less noise than the flux-dependent boundary conditions. The noise produced at the TABLEI. CONSTANTS AND PARAMETERS IN MODELSIMULATIONS Value

Parameter Horizontal array size Number of layers in vertical Constant pressure at top of model p t Grid size A s Time step A t Roughness length over land Albedo over land Moisture availability Soil conductivity Soil heat capacity per unit volume

46

X

61

I5 100 mb 80 or 160 km 160 or 320 s 0.1 m 0.2 0.7 10-3 kJ m-' s-l K lo00 kJ m - ) K I

I

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RlCHARD A. ANTHES ET AL.

boundary by either scheme does not appear to significantly affect the solutions more than several grid points near the boundary. 3.3. Introduction of Perturbation on Initial Conditions

In classic predictability experiments using large-scale models, the growth of differences between two simulations with small variations in the initial condition is considered to be a measure of the inherent uncertainty in numerical forecasts. While any perturbation to the initial conditions will ultimately lead to large differences in simulations on global scales, the growth of differences occurs mainly from differences in the slow meteorological modes of the model rather than from differences in the gravity wave modes, because gravity modes have little effect on the forecasts (Daley, 1981). In the relatively short periods (typically 48 hr) of mesoscale forecasts for which gravity modes are not important, randomly perturbed initial conditions might lead to misleading error-growth rates if most of the initial “error” or uncertainty is projected onto gravity modes rather than slower modes. Maximum growth of differences between two simulations is expected to occur when most of the initial differences is in the slow modes. In these simulations, we introduce initial perturbations in the wind and temperature fields in a consistent way to help ensure that significant differences are present in the slow modes at the initial time. These perturbations include significant energy in all scales resolved by the model, and hence the damping of the total initial perturbation by the model’s frictional parameterization should be unimportant. Fried et al. (1979) showed that a mesoscale model retained the general shape and amplitude of a small-scale perturbation in the initial vorticity field over a 12-hr simulation. We follow their approach here, but instead of introducing a single perturbation designed to represent a real atmospheric circulation feature, we introduce many random perturbations to simulate uncertainties in the initial conditions resulting from observational and analysis errors. The perturbation introduced at the nth point (x, , y , ,p , ) = x, takes the form

c,,

5‘

= t,F(p)G(x,y)

(3.1)

where is the maximum perturbation vorticity introduced in the vicinity of x, and

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

x = (xu -

Y =

xn),

(yij

173

- Yn)

where i,j are the indices of model grid points. Each nth perturbation in vorticity is an ellipsoid with a horizontal scale specified by r,, a vertical scale specified by Ap,, and an eccentricity specified by E, . These parameters are determined for each point x, in the model domain at which perturbations are concentrated by a random selection of parameters in the range

15”l 5 100 km

5

r,, 5 500 km

100 mb

5

Ap,

0.8

5 E, 5

5

400 mb

(3.4)

1.2

The range Z of the amplitude of vorticity perturbations is 0.2 X s-I in s-I in Exp. 4. In Exps. 2 and 4 (Table 11),the points Exp. 2 and 1.0 x x, correspond to every sixth point at all levels in the model domain, which gives a separation of 5As (800 km in large-scale experiment, 400 km in small-scale experiment). An example of the perturbations introduced by this method is shown in Fig. 9 later in the chapter. A property of the perturbations described by (3.2)-(3.4) is that, as the spacing between the x, approaches the grid size As, n approaches the number of grid points, and r, and Ap, approach zero, the specification becomes equal to adding random perturbations of vorticity at every grid point. Following the calculation of { ’ ( x , y , p ) for the model domain, consistent temperature perturbations are calculated from the balance equation as follows. First, the perturbation streamfunction I$’ is calculated from the 5’ field according to (3.5) with Q = 0 on the boundaries. A perturbation geopotential +’ is then calculated from the balance equation V2*’

= {’

W’ = fW- 2m2[(+;y)2- Gx$~yl + pJ?I + Y+;

(3.6)

wherefis the Coriolis parameter, m is the map scale factor, p = dfldy and y = dflax. Here x and y are the horizontal axes in the Lambert-conformal map projection. The perturbation temperatures are then calculated from +‘ by using the hydrostatic equation.

TABLE11. Experiment number

As

S U M M A R Y OF SIMULATIONS

Initial time

Duration of simulation (hr)

160

OOZ

72

Observed

No

Control simulation over large domain

2

160

10 April 0oz 10 April

72

Same as Exp. 1

Yes

3

80

OOZ

72

From Exp. 1

No

47

Same as Exp. 3

Yes

OOZ 10 April OOZ 10 April

72

From Exp. 2

No

12

Same as Exp. 3

No

OOZ

48

Observed

No

48

Persistence (steady state)

No

48

From Exp. 1

Yes

48

Same as Exp. 3

Yes

Little or no growth of differences from perturbed initial condition Control simulation over small domain; linear growth of differences from Exp. 1 Little or no growth of differences up to 47 hr, when noise near boundary causes termination of simulation Slow growth of differences to 48 hr and then no further growth to 72 hr Flux-dependent LBC; linear growth of differences to 48 hr, little change after 48 hr Generally linear growth of differences to 72 hr Linear growth of differences, greatest differences of all simulations at 48 hr Initialized from heavy smoothing of data at 24 hr from Exp. 3; no growth of differences during following 48 hr Random perturbations of amplitude 0.2”C and 0.3 m s-I added to initial temperature and wind fields of Exp. 3

1

(km)

Lateral boundary condition

Initial perturbations

10 April 4

80

OOZ

10 April 5

80

6

80

7

80

8

80

9

80

10

80

10 April OOZ 10 April 00z I 1 April OOZ

10 April

Remarks

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

175

3.4.Results of Numerical Simulations In this section, the results of nine numerical simulations of the SESAME-I case are discussed. These experiments, summarized in Table 11, are designed to contribute to an understanding of the predictability of the atmosphere on the meso-a and synoptic scales over a limited area. The two major aspects of predictability investigated here are the effect of uncertainties in the initial conditions and the effect of LBC on the simulations. The model physics is identical in all the simulations. The first two simulations (Table 11) utilize a 160-km grid that covers the domain shown in Fig. 2. Experiment 1 may be considered the control simulation on the large scale; it is used for comparison with Exp. 2, which differs only in perturbed initial conditions and for providing LBC for several of the high-resolution simulations on the smaller-scale domain. Experiment 3 utilizes an 80-km grid on the small domain shown in Fig. 4. It may be considered the control simulation on the fine mesh. Experiment 4 differs from Exp. 3 only in perturbed initial conditions: A comparison of the time rate of change of differences between Exps. 1 and 2 or Exps. 4 and 3 represents an analog to the classic predictability studies of global models. Experiment 5 is identical to the control Exp. 3 except that the LBC are derived from Exp. 2 (perturbed initial conditions on large scale) rather

w 1100 100" 90" 80" 70" FIG.4 . Simulations at 48 hr (0000 GMT 12 April 1979, designated 79041200) on small domain (Exp. 3) of sea-level pressure (contour interval 4 mb).

176

RICHARD A. ANTHES ET A L .

than Exp. 1. A comparison of the growth of differences between Exps. 5 and 3 with that of Exps. 4 and 3 gives an estimate of the relative source of forecast error resulting from small uncertainties in LBC versus those resulting from uncertainties in initial conditions. Experiment 6 utilizes the same data for the LBC as Exp. 3, but the flux-dependent numerical method of treating the lateral boundaries is used rather than the porous-sponge technique. A comparison of the two pairs of simulation (Exps. 6 and 3 versus Exps. 5 and 3) gives an estimate of the uncertainties introduced by the method of incorporating lateral boundary data versus those introduced by variations in the data itself. In another simulation investigating the source of errors introduced by the LBC, Exp. 7 is run with observed data from the 12-hr analyses, interpolated in time. Observed data have been used in this way in previous limited-area simulations with this model (Anthes et al., 1982, 1983). Experiment 8 is a final simulation with different lateral boundary data; it utilizes the data at the initial time (0000 GMT 10 April) for the entire forecast (steady-state boundary conditions). Experiment 9 considers the effect of a heavy horizontal smoothing of model data at 24 hr of Exp. 3 and a subsequent extension of the forecast to 72 hr. A comparison of the last 48 hr of Exps. 9 and 3 gives an estimate of the impact of mesoscale structure in the initial data on a 48-hr forecast. In comparing pairs of experiments, rms differences in several variables are calculated over all grid points on the interior portion of the domain. This interior grid consists of 38 points in the north-south direction and 53 points in the east-west direction. All layers are included in the calculation, with data in each layer weighted by the mass of that layer. 3.4.1. Control Simulations on Large and Small Domains. Before discussing the effect of variations in initial and LBC on the simulations, we present a brief discussion of the 48- and 72-hr control simulations on the large and small domains (Exps. 1 and 3). The simulation of Exp. 3 utilizes time-dependent LBC from Exp. 1 (a one-way nested grid calculation). The small-scale forecast of sea-level pressure at 48 hr (0000 GMT 12 April) shown in Fig. 4 is in general agreement with the observations at that time (Fig. 2b). The cyclone over the Great Plains has a minimum pressure of 990 mb compared to the observed value of 984 mb. The location of the center is simulated over central Colorado rather than central Kansas, an error of about 400 km. The anticyclone over James Bay and the cyclone off the coast of the Canadian maritimes are simulated in the correct locations with nearly the correct intensity. The large-scale simulation of sea-level pressure resembles the small-scale simulation over the United States (not shown).

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

177

The 48-hr simulations of the 500-mb flow on both the coarse- and finemesh domains are also in general agreement with the observations. The closed low over the western United States is located southeast of the observed position and has a minimum height about 50 m higher than observed. The cyclone over eastern Canada is simulated well, but the cyclone in northern Canada, near the boundary of the domain, is not well simulated, possibly because of its nearness to the northern boundary. The 72-hr simulations of sea-level pressure on both the large domain (Fig. 5a) and small domain (Fig. 6a) show considerable errors when compared to the observations (Fig. 2c). The cyclone over the Great Plains is located too far to the south and the strong cyclone approaching the Pacific coast along the Canadian border is not simulated. At 500 mb, the simulations look more similar to the observations (compare Figs. 5b, 6b, and 3c). The major error in the large-scale simulation (Fig. 5b) is the underdeveloped cyclone near the Pacific northwest coast, while the major error in the fine-mesh simulation is the underdeveloped ridge in the eastern part of the United States and the Great Lakes region. Figure 7 shows the temporal variation of the root-mean-square (rms) differences of the u and v components, the specific humidity, and the temperature between Exps. 1 and 3 . The differences are computed over the interior portion of the smaller domain (38 x 53 grid) of Exp. 3, with the coarse-resolution model data interpolated to the finer-mesh grid points when necessary. These differences can be caused by a combination of slightly different initial conditions, different LBC, different truncation errors, different behavior of the model physics in response to a factor of 2 difference in grid length and to the fact that the fine-mesh simulation can resolve slightly smaller scales of motion. The growth of differences in these simulations, which differ in many ways, serves as a useful benchmark for later comparisons when there are fewer differences in the simulations. The growth of differences in Exps. 1 and 3 in all of the variables is approximately linear over the 72-hr period. The rms velocity component errors increase from about 0.3 m s - I at i = 0 to about 2.0 m s-' at 72 hr, an increase of about 0.55 m s-I per day (Table 111). The rms difference in specific humidity increases from about 0.07 g kg-' at t = 0 to 0.3 g kg-I at 72 hr, an increase of about 0.1 g kg-' per day. The temperature difference increases from 0.14 to 0.6 K, an increase of about 0.15 K per day. The differences in accumulated rainfall also increase linearly with time, from 0 at t = 0 to 0.27 cm at 72 hr (not shown). 3.4.2. BehaviorofPerturbations in Initial Conditions. In Exps. 2 and 4, random perturbations to the wind and temperature and moisture fields are

178

RICHARD A. ANTHES ET A t

N 40’

30’

20’

10"

w

1200

w

160'

11Oo

tooo

150" 140' 130"

90-

80’

10’ 5 0 0 4 0 0 3 ~200

60’

(b)

W 120' 110' 100" 90" 80O 70° 60’ FIG.5. Simulations at 72 hr (0000 GMT 13 April) on large domain (Exp. 1) of (a) sea-level pressure (contour interval 4 mb) and (b) 500-mb heights (solid lines, contour interval 60 rn) and temperature (dashed line, contour interval 8°C).

PREDICTABILITYOF MESOSCALE ATMOSPHERICMOTIONS

179

1200

w

1100

w

110"

100'

90"

80"

70"

goo

800

70°

FIG.6. Simulation at 72 hr (0000GMT 13 April) on small domain (Exp. 3 ) of (a) sea-level pressure (contour interval 4 mb) and (b) SW-mb height (solid lines, contour interval 60 m) and temperature (dashed line, contour interval FC).

180

RICHARD A. ANTHES ET A L .

50

~

4.0

-

~

,

~

,

~

~

(a)

E

Y

-

> 7

2.0

l

-

-

-

-

-

-

-

h

3.0

~

-

-

c

-

0

t

l

l

l

l

"

l

0.5

1

1

I

1

1

1

1

- (b) - 0.3 0.4

A

I

t

l 1

l 1

r ,

l

1

l

1

l

1

1

I .o

I

-

- 0.8 - 0.6

-

[

TABLE111. SUMMARY OF ROOT-MEAN-SQUARE DIFFERENCES BETWEEN PAIRSOF SIMULATIONS Root-mean-square differences 24

v

Experiments

Time

(m s-I)

(m s - l )

4 (g kg-I)

1 and 3 1 and 2 3 and 4

72 hr 72 hr 47 hr 72 hr 72 hr 48 hr 48 hr 48 hr 48 hr

1.8 0.1

2.1 0.1 0.4 0.1 0.8 2.1 2.4 0.2 0.3

0.30 0.04 0.09 0.03 0.17 0.24 0.27 0.05 0.15

3 and 5 3 and 6 3 and 7 3 and 8 3 and 9 3 and 10

0.5

0.2 0.9

1.7 3.3 0.2 0.2

T (K)

p ” (mb)

Growth rates

Accumulated precipitation (cm)

0.60

1.30

0.27

0.05 0.19 0.05

0.09 0.40 0.05 0.19 1.00 2.00 0.20

0.03

0.23 0.73 1.07 0.06 0.12

0.02

0.11 0.03

Wind (rn S K I day-’) 0.55 0.01

-0.01 0.05

0.09

0.28

0.16 0.26 0.04 0.02

0.95

1.43 0.00 0.05

Specific T PS humidity (g kg-’ day-’) (“C/day) (mb day-’) 0.08 0.01

0.01 0.01 0.06 0.12 0.14 0.01 0.05

0.15 0.00 -0.03 0.02 0.08 0.37 0.54 0.00 -0.05

0.27 0.00 0.00 0.02 -0.04 0.50

1 .oo -0.15 -0.02

182

RICHARD A. ANTHES E T A L .

added to the initial conditions of Exps. 1 and 2, with 2 in (3.4) equal to 0.2 X s-l in Exp. 2 and 1.0 x s-l in Exp. 4. The magnitudes of these perturbations (in an rms sense) for Exp. 2 are 0.1 m s-l for the wind components, 0.05"C for temperature, and 0.02 g kg-' for the specific humidity. The temporal behavior of the rms differences between the two pairs of simulations are examined over the 72 hr of the simulations as in classic predictability experiments. Figure 8 shows the temporal behavior of the rms differences between Exps. 1 and 2 of the wind components, temperature, and specific humidity. In sharp contrast to similar experiments using global models, the differences grow slowly, if at all, over the 72-hr period. The rms u- and u-component differences both increase from 0.09 to 0.12 m s-l. The difference in specific humidity increases from 0.02 to 0.04 g kg-', Yhile the temperature difference shows no growth. The surface pressure difference (not shown) also remains nearly constant over the period. The difference in accumulated rainfall over the domain increases in time from 0 at t = 0 to 0.03 cm at 72 hr. These small increases are much less than those between Exps. 1 and 3. Because of the little or no growth of differences between the pair of large-scale simulations, the amplitude of the initial perturbation was increased by a factor of 5 for the higher-resolution simulations on the small domain. The perturbation vorticity , stream function, and temperature at 700 mb are illustrated in Fig. 9. Maximum temperature perturbations of more than 2°C and maximum wind perturbations of over 6 m s-l are present at this level. Figure 10 shows the temporal variation of the rms differences out to 47 hr. Experiment 4 was terminated at this point at which numerical noise at one of the grid points near the lateral boundary exceeded a prescribed value. In spite of the shorter time period, the results appear conclusiveeven the larger perturbations are not growing. The differences in wind components and specific humidity remain constant, while the differences in temperature actually decrease slightly over the period. The rainfall difference increases from 0 to 0.10 cm after 24 hr, then remains nearly constant out to 47 hr (not shown). The similar behavior of both the large-scale and small-scale simulations-little or only very slight growth of differences in initial conditions on time periods of 48 to 72 hr-differs greatly from the nearly exponential growth of initial differences in global models." For this particular synoptic situation, the model is not sensitive to these perturbations in the initial

* As shown in a later section, a global model does show growth of small differences in this region over the 3-day period.

183

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

0.5

0.4

0.3

E

v

> 7

0.2

0.I

0

I-

t (hr) FIG.8. Temporal variation of rms differences between Exps. 1 and 2: (a) u and u components and (b) specific humidity q and temperature T .

I84

RICHARD A. ANTHES E T A L .

w

100" 90" 80" 70" FIG. 9. Perturbation field introduced into initial conditions of Exp. 4: (a) vorticity (contour interval 1 X s-'), (b) stream function (contour interval 5 x lo5 m2 s-') and isotachs (dashed lines, contour interval 2 m s - ' ) , and (c) temperatures (contour interval 0.5"C). lIO0

I85

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS (C)

N 50°

40" I200

3oo

conditions. The small-scale features that develop in the latter stages of the simulations develop in a similar way regardless of the variations in initial conditions. While it is obviously dangerous to generalize on the basis of one case study, these results suggest that the large-scale flow, together with forcing from the surface due to topographic effects and land-water differences, are more important in determining the mesoscale features 2472 hr into the forecast than are variations in the initial conditions. This result may be interpreted to imply less of a need for meso-a-scale observations in initializing regional models of this type for forecasts over this time range.

3.4.3. Response of Simulation to Variations in Lateral Boundary Conditions. One interpretation of the lack of significant growth of differences in initial conditions of the two pairs of limited-area simulations discussed in the preceding section is that the LBC exert a major control on the simulations. This effect is a fundamental difference from global model experiments. Miyakoda and Rosati (1977) found limited-area model simulations to be sensitive to the LBC. They concluded that the major errors came from inaccurate data supplied at the boundaries and that errors introduced by the LBC could be a major source of forecast error over the limited-area model domain in time periods as short as a day. Orlanski (1983) found that a 48-hr simulation initialized at 1200 GMT 9 April 1979

186

RICHARD A. ANTHES ET A L .

I.o

0.8

-

0.6

I

u)

E v

> 0.4

3

0.2

0 0.5

I .o

0.4

0.8

i 0.3 m

0.6

n

s

-I n

m

0

I

Y

0.2

0.4

0.2

01 0

I

I

10

I

I

20

I

I

30

1

I

40

1

0 50

t (hr) FIG. 10. Temporal variation of rms differences between Exps. 3 and 4: (a) u and u components and (b) specific humidity q and temperature T .

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

187

was more sensitive to variations in the LBC than to small differences in initial conditions. The LBC from a global forecast played a dominant role in determining the position and strength of the synoptic, as well as the mesoscale features in the limited-area solution. Baumhefner and Perkey (1982) showed that errors generated at the lateral boundary of a limited-area model with horizontal resolution 2.5" (latitude and longitude) propagated into the interior of the domain at speeds of 20 to 30" of longitude per day. In agreement with Miyakoda and Rosati's (1977) results, these errors were associated more with inaccurate data supplied at the lateral boundaries than with the method of specification. In the case studied by Baumhefner and Perkey, the errors introduced by the LBC were much less than the total forecast error. To examine the effect of the LBC on the predictability of this case, four simulations are run with different boundary conditions. Experiment 5 is the same as Exp. 3 except that it obtains boundary data from the largescale Exp. 2 rather than Exp. 1. We showed in the preceding section that Exp. 2 differed only slightly from Exp. 1; i.e., there was little growth of the initial differences. Thus, a comparison of Exps. 5 and 3 tests the sensitivity of the 80-km simulation to small variations in the LBC. The fine-mesh simulation is not sensitive to variations in the LBC of this order of magnitude (Table 111). The simulated fields are nearly identical in Exps. 5 and 3 and the rms differences level off at 48 hr to about 0.2 m s-I for the wind components, 0.03 g kg-' for the specific humidity, and 0.05"C for the temperature (not shown). The rms difference in accumulated rainfall is 0.03 cm. In the second simulation investigating the sensitivity of the simulation to the LBC, Exp. 6 utilizes the same data on the boundaries as did the control simulation (Exp. 3). However, the numerical method of treating the LBC differs; in Exp. 6 the flux-dependent method described earlier was used instead of the porous-sponge method used in the other simulations. Figure 11 shows the temporal behavior of the rms differences. After a nearly linear growth during the first 48 hr the differences gradually level off and remain constant during the last 24 hr of the simulation. The rms differences after 72 hr are small (Table HI), and the simulated fields over the verification region are quite similar. This comparison indicates that the numerical technique of applying the LBC has relatively little impact on the simulations. In the third simulation with varying LBC, Exp. 7 is run with observed data supplied to the boundaries rather than model data. The observed data are obtained by interpolating the analyses between the 12-hr observation times. These boundary data may be more accurate than the model-generated data, especially near land, but are probably not as

188

RICHARD A. ANTHES ET A L .

2.5

2.0

-

A

I

a

1.5

E

Y

w . I

3

1.0

0.5

0 I .o

0.8

0.6 A

Y

Y

0.4

t

0.2

c

I4

0 0

I-

10

20

30

40

50

60

70

0 80 and u

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

189

compatible numerically with the solutions on the interior of the fine-mesh domain. As shown by the increase of rms differences (Fig. 12), the use of observed rather than model data on the LBC has a relatively large impact on the simulation. The differences at 48 hr are as great or greater than the differences at 72 hr in the other pairs of experiments (Table HI).There are also noticeable differences in the simulated fields. For example, the sealevel pressure field at 48 hr of Exp. 7 (Fig. 13) indicates that the center of the Great Plains cyclone is located over the Texas-Oklahoma Panhandle rather than western Colorado (Fig. 4). The sea-level isobar pattern is closer to the observed (Fig. 2b). Other fields (winds and temperature) at other levels also show noticeable differences. In the final experiment investigating the effect of the LBC on the simulation, Exp. 8 is run with steady-state boundary data equal to those at the initial time. We expect these data to be less accurate than those of all the previous experiments and therefore that the differences in simulations after 72 hr should be large. The growth of the rms differences (Fig. 14) confirms this expectation. Not only are the differences after 48 hr greater than those of any of the other pairs (Table HI),the growth rates are generally more than double those of the previous pairs. Inspection of most of the simulated fields (the sea-level pressure map is shown in Fig. 15) shows large differences from those of Exp. 3, and Exp. 8 is clearly an inferior simulation. In summary, small differences in the lateral boundary conditions, either in the data or in the numerical method of implementation, have a relatively minor effect on the simulations of this case. However, large variations in the boundary data (observed versus modeled and steady-state versus time-dependent) produce relatively large differences in the simulations, and those differences are greater than those arising from perturbations in the initial conditions. 3.4.4. Effect of Removing Small-Scale Information at 24 hr of Control Simulation. An important practical problem in the design of future observing systems is the extent to which mesoscale atmospheric structure must be observed and initialized in numerical models to obtain accurate mesoscale forecasts. It is well known that mesoscale models develop mesoscale features during forecasts or simulations, even when initialized with only large-scale conditions (Miyakoda and Rosati, 1977; Orlanski and Ross, 1977; Ross and Orlanski, 1978, 1982; Anthes et al., 1982). Such simulations can be used to estimate the impact of initial mesoscale structure on the subsequent simulation or forecast by reducing the mesoscale information at some point in the simulation through smoothing or filtering.

190

RICHARD A. ANTHES ET AL.

5.0

4.0

-

I

cn

3.0

E w

>

;2.0

I.o

0 0.5

I .o

0.4

0.8

- 0.3

0.6

m s

V 0

Y

0,

w

I-

= 0.2

0.4

0.I

0.2

0

0

I

I

10

1

I

20

I

I

30

1

I

40

I

10

50

t (hr) FIG. 12. Temporal variation of rms differences between Exps. 3 and 7: (a) u and u components and (b) specific humidity q and temperature T.

191

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS N

50"

40"

30"

w 110" 100" 90" 80" 70" FIG. 13. Simulation at 48 hr (0000 GMT 12 April) from Exp. 7 of sea-level pressure (contour interval, 4 mb).

A comparison of the subsequent simulation beginning with smoothed data with the original simulation provides some insight into the importance of mesoscale features in the initial conditions. In a set of four simulations that were similar to the approach described above, Orlanski and Ross (1983) investigated the sensitivity of dry and moist simulations of a cold front to smoothed initial conditions of temperature and moisture. Simulations at 24 hr of a control experiment were reinitialized after 24 hr with conditions consisting of exact velocities, but with temperature and moisture data modified by smoothing either in the horizontal or in the vertical. In the horizontal smoothing procedure, the 60-km control resolution was filtered to a resolution of 540 km. In the vertical smoothing procedure, the 1-krn vertical resolution was filtered to 3-km resolution in the troposphere. In the subsequent 24-hr dry simulations, the initial differences of the temperatures were reduced from a range of about 5.1 to 1.YC, while the differences in vector wind increased from zero to less than 2 m ss' after 24 hr. The differences in the vertically smoothed dry experiment were even less and also remained small with time. Thus, these simulations showed no growth of initial perturbations. In the moist simulations, the effects of the heavy horizontal smoothing persisted in time over the region of intense mesoscale convection, but the

192 5.0*

I

- (a) 4.0

-

1

I

I

I

I

I

I

I

,

-

t (hr) FIG. 14. Temporal variation of rms differences between Exps. 3 and 8: (a) u and u components and (b) specific humidity q and temperature T .

PREDICTABILITY OF MESOSCALE ATMOSPHERIC MOTIONS

193 N

50"

40"

I200

30"

w 110" 100" 90" 80" 70" FIG. 15. Simulation of 48 hr (0000 GMT 12 April) from Exp. 8 of sea-level pressure (contour interval 4 mb).

differences did not grow during the 24 hr. However, the intensity of the frontal precipitation was weakened considerably by the heavy horizontal smoothing of the water vapor field. Orlanski and Ross (1983) concluded from their experiments that a dense (in the horizontal) network of thermodynamic soundings with poor vertical resolution is preferable to a sparse network with good vertical resolution. In Exp. 9 of Table 11, we study the impact of a horizontal smoothing of all model fields at 24 hr of Exp. 3 on the subsequent 48-hr simulation. The smoothing consists of 50 applications of a nine-point smoothing operator. With each pass of the operator, a variable at a grid point is replaced by the mean consisting of itself plus the variables of the nearest eight points. The effect of the smoothing operation is to reduce horizontal gradients of temperature and specific humidity and to reduce the amplitude of the jet streams. However, no features of scale greater than 2Ax (160 km) are completely eliminated. Figure 16 shows the rms differences between the smoothed Exp. 9 and the control Exp. 3. The initial differences of about 0.2 m s-' in the wind components remain nearly constant during the 48 hr. The specific humidity differences show a slight growth, but the temperature differences also show no growth. These results are consistent with those of Orlanski and Ross (1983) and indicate that the simulations are not sensitive to small

194

RICHARD A. ANTHES ET A L .

0.5

1

- (a) 0.4

I

I

I

1

I

I

I

I

-

-

0

0.3 -

Y)

-

E

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0.1

-

0 0.10

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I

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1

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I

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(b)

0.08

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- 0.10

-

-

- 0.08

-

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variations in the magnitude of mesoscale gradients of temperature, wind, and moisture. 4. DISCUSSION AND COMPARISON WITH A PREDICTABILITY STUDY USINGA GLOBALMODEL

The predictability experiments with the limited-area model described in the preceding section showed a different behavior from that of global models in that two simulations with small variations in initial conditions showed little or no growth in the differences. Two hypotheses can be advanced to explain this absence of growth out to 72 hr. First, the same LBC may be preventing different evolutions of the flow on the interior of the domain by providing identical large-scale information to the periphery of the pairs of simulations. If the large-scale flow, together with the forcing of the Earth's surface through orography and energy fluxes, is controlling the evolution of the mesoscale features as suggested by Anthes (1984), then one would expect little sensitivity of mesoscale forecasts to variations in initial conditions. In contrast, if large regions of instabilities to small-scale perturbations existed in the initial large-scale fields, one would expect a much greater sensitivity to variations in the initial conditions. A second hypothesis concerning the lack of growth of the initial difference is that the synoptic weather type over the limited area was, by chance, more stable to initial perturbations than typical global circulations, which always contain some regions that are sensitive to initial perturbations. In order to test this hypothesis, we perform a predictability experiment with a global forecast model initialized at the same time as the regional model (0000 GMT 10 April 1979) and compare the growth rates of differences from this model to those of the limited-area model. The growth rate is calculated over the Northern Hemisphere and over a portion of North America corresponding approximately to the 80-km limited-area model domain (25-53"N, 68-1 12"W). The global model used in this predictability study is the NCAR Community Climate Model (designated CCMOB). Williamson (1983) provides a complete documentation of the model. The version of the spectral model used in the predictability study here utilizes a rhomboidal truncation, with N = rn + M, where m is the Fourier wave number, M the largest Fourier wave number, and N(m) the highest degree of the associated Legendre polynomial. Here M = 30, so that the resolution is designated as rhomboidal-30 (R-30). In the control global simulation, the CCMOB is initialized at 0000 GMT

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10 April 1979 with the European Centre for Medium Range Weather Forecasts (ECMWF) operational analysis us; a nonlinear normal mode method (Errico, 1983). A second forecast is identical to the first, except that the initial conditions are perturbed by the addition of random temperature differences of maximum amplitude 03°C. The two forecasts are integrated for 10 days in a classic predictability experiment. Figure 17 shows the growth of rms differences between the pair of global forecasts over the Northern Hemisphere, as well as, the limited area over North America. The rms differences over both domains show a similar, nearly exponential growth after the first day. This growth is similar to that from two simulations with an R-15 version of the CCM

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using simulated January data as initial conditions (dotted circles in Fig. 17) (Baumhefner, 1984). The clear result displayed in Fig. 17 is that the particular case chosen for the predictability studies using the limited-area model does not exhibit anomalous behavior in a global model, even when differences are considered only over the domain of the limited-area model. Thus, the lack of error growth in the limited-area model simulations may be attributed to the effect of the lateral boundary conditions. Figure 18 compares the temporal rate of change of rms 500-mb geo-potential height differences from the global model experiments with those from several of the limited-area simulations. An additional 80-km simulation, initialized with random temperature and wind perturbations I

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superimposed on the control simulation (Exp. 10, Table 11) is shown in Fig. 18. The lack of error growth in the limited-area model experiments, even though the initial differences are generally greater, is apparent. The comparison of error-growth rates in the limited-area model simulations with those from the global simulations suggests that the lack of error growth in the former is caused by the imposition of identical (or similar) large-scale information on the lateral boundaries. If this result applies generally to limited-area, meso-a-scale simulations, the important implication is that specification of accurate large-scale boundary data is more important in the forecast range from approximately 24 to 72 hr than are mesoscale details in the initial conditions. These results also support the hypothesis that meso-a-scale motions may have predictability on these time scales, provided that accurate large-scale initial conditions and lateral boundary data and realistic physics are part of the regional modeling system.

5. SUMMARY A N D CONCLUSIONS This chapter has discussed a number of aspects of the predictability of mesoscale atmospheric circulations including a comparison with previous studies of the predictability of synoptic-scale motions. The first section reviews some results from turbulence theory that suggest that the predictability of mesoscale motions should be considerably less than that for synoptic-scale motions. A counterargument is then presented that gives some reasons for optimism that some mesoscale systems may have greater predictability than that suggested by turbulence theory. Coherent structures in fluids may have spectra quite different from those of the “average” atmosphere and resist turbulent decay. Physical forcing at the Earth’s surface, such as by mountains, may contribute to extended predictability. Finally, mesoscale phenomena such as fronts are known to evolve from purely large-scale initial conditions in models that have adequate resolution. The second section reviews classic, large-scale predictability studies, in which the growth of small differences in initial conditions between a pair of otherwise identical simulations is used to estimate the upper limit of atmospheric predictability. Current estimates of this limit to the predictability of synoptic-scale motions range from about 10 to 30 days. Application of the classic method of estimating predictability to mesoscale domains is not straightforward, however, because of the increased possibility of studying unrepresentative cases, differences in

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predictability in different geographic regions, possibly greater relative importance of physical forcing at the Earth’s surface in high-resolution models, and finally, the importance of the effect of lateral boundary conditions on mesoscale-model forecasts. In the third section, some preliminary numerical simulations are made with a meso-a-scale model (horizontal resolution 80 or 160 km) to investigate some of the aspects influencing mesoscale predictability. In contrast to the previous large-scale studies, these simulations show little or no growth of differences in initial conditions over the time period 0 to 72 hr. Several types of initial differences are examined, ranging from random small-scale perturbations to perturbations on the scale of the regional domain; the lack of growth in all of these experiments is notable. In contrast, the simulations are more sensitive to large variations in lateral boundary conditions, which suggests that the correct specification of time-dependent, synoptic-scale data on the lateral boundaries is more important than details in the mesoscale initial conditions, at least for the one case studied here. The last section discusses results from a global forecast model initialized at the same time as the mesoscale forecasts. The error-growth rate in the global model pair of simulations was consistent with that of previous large-scale studies, even over the limited-area domain of the mesoscale model. This result suggests that the growth of errors over this region depends on interactions with scales of motion larger than that of the mesoscale domain. The imposition of the same or only slightly varying lateral boundary conditions apparently prevented the growth of initial differences in the limited-area forecasts. The most important practical result suggested by these experiments is that meso-a-scale models depend critically on accurate specification of the large-scale atmospheric variables at the lateral boundaries. For these simulations, minor differences in initial conditions on the mesoscale had no significant impact on the forecasts out to 72 hr. Considerable further study is needed to verify and extend these results to other synoptic situations and with models with alternative physics, methods of supplying lateral boundary conditions, and variations in the initial perturbations. In addition, rms differences are only a gross measure of the differences in simulations; examination of other measures of difference and a decomposition of the differences by scale would be useful. Finally, these results are not likely to be applicable to meso-p- or meso-y-scale models when a relatively large fraction of these models’ domain may exhibit various instabilities (such as convective instability). Under such conditions, we expect the evolution of the simulation to be considerably more sensitive to the initial conditions.

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ACKNOWLEDGMENTS This work was partially supported by the U S . Environmental Protection Agency under Interagency Agreement DW 930144-01-0, but has not been subject to EPA review procedures. Roger Daley, Akira Kasahara, and Ray Pierrehumbert provided thorough reviews and helpful comments. Ann Modahl provided expert typing and editorial assistance.

REFERENCES Alberty, R. L., Burgess, D. W., and Fujita, T. T. (1980). Severe weather events of 10 April 1979. Bull. Am. Meteorol. SOC.61, 1033-1034. Andre, J. C., and LeSieur, M. (1977). Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J . Fluid Mech. 81, 187-207. Anthes, R. A. (1977). A cumulus parameterization scheme utilizing a one-dimensional cloud model. Mon. Weather Reu. 105, 270-286. Anthes, R. A. (1983). Regional models of the atmosphere in middle latitudes. Mon. Weather Rev. 111, 1306-1335. Anthes, R. A. (1984). Predictability of mesoscale meteorological phenomena. In “Predictability of Fluid Motions” (G. Holloway and B. J. West, eds.), pp. 247-270. Am. Inst. Phys., New York. Anthes, R. A., and Warner, T. T. (1978). Development of hydrodynamic models suitable for air pollution and other mesometeorological studies. Mon. Weather Rev. 106, 10451078. Anthes, R. A., Kuo, Y.-H., Benjamin, S. G., and Li, Y.-F. (1982). The evolution of the mesoscale environment of severe local storms: Preliminary modeling results. Mon. Weather Rev. 110, 1187-1213. Anthes, R. A., Kuo, Y.-H., and Gyakum, J. R. (1983). Numerical simulations of a case of explosive marine cyclogenesis. Mon. Weather Reu. 111, 1174-1 188. Balsley, B. B., and Carter, D. A. (1982). The spectrum of atmospheric velocity fluctuations at 8 km and 86 km.Geophys. Res. Lett. 9, 465-468. Baumhefner, D. P. (1984). The relationship between present large-scale forecast skill and new estimates of predictability error growth. I n “Predictability of Fluid Motions” (G. Holloway and B. J. West, eds.), pp. 169-180. Am. Inst. Phys., New York. Baumhefner, D. P., and Perkey, D. J. (1982). Evaluation of lateral boundary errors in a limited-domain model. Tellus 34,409-428. Benjamin, S. G. (1983). Some effects of surface heating and topography on the regional severe storm environment. Ph.D. Thesis, Pennsylvania State University, University Park. Carlson, T. N., Anthes, R. A., Schwartz, M., Benjamin, S. G., and Baldwin, D. G. (1980). Analysis and prediction of severe storms environment. Bull. Am. Metenrol. SOC.61, 1018- 1032. Carlson, T. N., Benjamin, S. G., and Forbes, G. S. (1983). Elevated mixed layers in the regional severe storm environment: Conceptual model and case studies. M o n . Weather Rev. 111, 1453-1473. Chamey, J. G., Fleagle, R. G., Lally, V. E., Riehl, H., and Wark, D. Q. (1966). The feasibility of a global observation and analysis experiment. Bull. Am. Meteorol. SOC.47, 200-220.

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Chen, T.-C., and Wiin-Nielsen, A. (1978). Non-linear cascades of atmospheric energy enstrophy. Tellus 30, 313-322. Daley, R. (1981). Predictability experiments with a baroclinic model. Almos.-Oceun 19, 77-89. Emco, R. M. (1983). “A Guide to Transform Software for Nonlinear Normal-Mode Initialization of the NCAR Community Forecast Model,” NCAR Tech. Rep. NCAR/TN217+IA. Natl. Cent. Atmos. Res., Boulder, Colorado. Fried, G . D., Cahir, J. J., and Anthes, R. A. (1979). Effect of subjectively enhancing the initial vorticity field in a short-range numerical forecast. J . A p p l . Meteorol. 18, 11881204. Jastrow, R., and Halem, M. (1970). Simulation studies related to GARP. Bull. Am. Meteorol. Sac. 51,490-513. Kasahara, A. (1972). Simulation experiments for meteorological observing systems for GARP. Bull. A m . Meteorol. Soc. 53, 252-264. Kuo, H. L. (1965). On formation and intensification of tropical cyclones through latent heat release by cumulus convection. J . Atmos. Sci. 22, 40-63. Kuo, H. L. (1974). Further studies of the parameterization of the influence of cumulus convection on large-scale flow. J . Afmos. Sci. 31, 1232-1240. Leith, C. E. (1971). Atmospheric predictability and two-dimensional turbulence. J . Armos. Sci. 28, 145-161. Leith, C. E., and Kraichnan, R. H. (1972). Predictability of turbulent flows. J . Atmos. Sci. 29, 1041-1058. Lilly, D. K. (1984). Some facets of the predictability problem for atmospheric mesoscales. In “Predictability of Fluid Motions” ( G . Holloway and B. J. West, eds.), pp. 287-294. Inst. Phys., New York. Lilly, D. K., and Petersen, E. L. (1983). Aircraft measurements of atmospheric kinetic energy spectra. Tellus 35A, 319-382. Lorenz, E. N. (1969a). The predictability of a flow which possesses many scales of motion. Tellus 21, 289-307. Lorenz, E. N. (196913). Atmospheric predictability as revealed by naturally occurring analogues. J . Atmos. Sci. 26, 636-646. Lorenz, E. N. (1969~).Three approaches to atmospheric predictability. Bull. Am. Meteorol. SOC. 50,345-349. Miyakoda, K . , and Rosati, A. (1977). One-way nested grid models: The interface conditions and the numerical accuracy. Man. Weather Reu. 105, 1092-1107. Moore, J. T., and Fuelberg, H. E. (1981). A synoptic analysis of the first AVE-SESAME 1979 period. Bull. Am. Meteorol. Sac. 62, 1577-1590. Nastrom, G . D., and Gage, K . S. (1982). A first look at wave number spectra from GASP data. Tellus 35A, 383-388. Orlanski, I. (1975). A rational subdivision of scales for atmospheric processes. Bull. Am. Meteorol. Soc. 56, 527-530. Orlanski, I. (1983). The influence of nesting in limiting mesoscale predictability. Paper presented at IAMAP General Assembly, Hamburg, August, 1983. Orlanski, I., and Ross, B. B. (1977). The circulation associated with a cold front. Part I . Dry case. J . Atmos. Sci. 34, 1619-1633. Orlanski, I., and Ross, B. B . (1983). Simulations of mesoscale phenomena using smoothed thermodynamic initial conditions (unpublished manuscript). Perkey, D. J . , and Kreitzberg, C. W. (1976). A time-dependent lateral boundary scheme for limited-area primitive equation models. Man. Weather Reu. 104, 744-755.

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Robinson, G. D. (1967). Some current projects for global meteorological observation and experiment. Q. J . R. Meteorol. SOC.93, 409-418. Robinson, G. D. (1971). The predictability of a dissipative flow. Q. J . R. Meteorol. SOC.97, 300-312. Ross, B. B., and Orlanski, I. (1978). The circulation associated with a cold front. Part 11. Moist case. J. A m o s . Sci. 35, 445-465. Ross, B. B., and Orlanski, I. (1982). The evolution of an observed cold front. Part I. Numerical simulation. J . Atmos. S c i . 39, 296-327. Shukla, J. (1981a). Dynamical predictability of monthly means. J. A m o s . S c i . 38, 25472572. Shukla, J. (1981b). “Predictability of the Tropical Atmosphere,” NASA Tech. Memo 83829. Natl. Aeron. Space Admin., Washington, D.C. (Available from National Technical Information Service, Springfield, Virginia.) Smagorinsky, J. (1969). Problems and promises of deterministic extended range forecasting. Bull. Am. Meteorol. SOC. 50, 286-31 1. Tennekes, H. (1978). Turbulent flow in two- and three-dimensions. Bull. Am. Meteorol. SOC. 59,22-28. Thompson, P. D. (1957). Uncertainty of initial state as a factor in the predictability of largescale atmospheric flow patterns. Tellus 9, 275-295. University Corporation for Atmospheric Research (UCAR) (1983). “The National STORM Program-Scientific and Technological Bases and Major Objectives.” Report submitted by the University Corporation for Atmospheric Research, Boulder, Colorado, to the National Oceanic and Atmospheric Administration in fulfillment of Contract NA81RAC00123. [Edited by R. A. Anthes, National Center for Atmospheric Research, Boulder, Colorado, which is sponsored by the National Science Foundation, January 1983.1 Vinnichenko, N. K. (1970). The kinetic energy spectrum in the free atmosphere-1 second to 5 years. Tellus 22, 158-166. Wilhelmson, R. B., and Klemp, J. B. (1981). A three-dimensional numerical simulation of splitting severe storms on 3 April 1964. J . Atmos. Sci. 38, 1581-1600. Williamson, D. L. (1973). The effect of forecast error accumulation on four-dimensional data assimilation. 1.Atmas. S c i . 30, 537-543. Williamson, D. L. (1983). “Description of the NCAR Community Climate Model (CCMOB),” NCAR Tech. Note NCAR/TN-210+STR. Natl. Cent. Atmos. Res., Boulder, Colorado. Zhang, D., and Anthes, R. A. (1982). A high-resolution model of the planetary boundary layer sensitivity tests and comparisons with SESAME-79 data. J . Appl. Meteorol. 21, 1594- 1609.

THERMAL AND OROGRAPHIC MESOSCALE ATMOSPHERIC SYSTEMS-AN ESSAY ROGERA. PIELKE Department of Atmospheric Science Colorudo State University Fort Collins. Colorado 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 2. Summary of Major Research Accomplishments . . . . . . . . . . . . . . . . . . 204 2.1. Thermal Mesoscale Systems . . . . . . . . . . . . . . . . . . . . . . . 20s 2.2. Orographic Mesoscale Models . . . . . . . . . . . . . . . . . . . . . . 212 3. Research Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4. Eventual Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 19

1. INTRODUCTION There are two major reasons to investigate any scientific question. First there is an interest to evaluate relationships among variables that are constrained by a preassumed set of conditions. In mathematical terminology these constraints are referred to as axioms. Using these constraints, sets of relationships are postulated and theorems established that satisfy these axioms. No relationship to the real world is necessary, of course, in setting up these axioms, and only our imagination limits the form of the constraints. When we apply this approach to physics, however, the assumed constraints are based on the deduced behavior of actual physical systems. In their application to the representation of tropospheric mesoscale systems, these constraints include the ideal gas law; first law of thermodynamics; the second law of motion; and conservation relations for mass of air, for water substance, and for atmospheric pollutants. Pielke (1984, Chapter 2) and other similar texts describe the use of these constraints to derive the physical equations that are applied to describe mesoscale atmospheric flow. Once these physical relations as applied to mesoscale systems are established, an investigator can use them in order to determine the interrelation among the dependent variables that appear in the mathematical statement of the physical relations. Assumptions can be made and the equations simplified based on analyses of the order of magnitude of different terms in the equations. On the mesoscale, these 203 ADVANCES IN GEOPHYSICS, VOLUME

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assumptions can include the hydrostatic approximation to the vertical pressure distribution and the neglect of compressibility in the continuity of mass of air equation. The tools used to quantitatively evaluate the interrelation among the dependent variables can involve the linearization of the equations and their resultant solution by exact mathematical techniques, their approximation by a discrete representation and then evaluation on a computer, or their interpretation using water-tank or wind-tunnel modeling. These approaches, discussed in depth in Pielke (1984), are the means by which the mesoscale meteorologist investigates the interrelation of physical quantities based on the assumed constraints of the laws of physics. In contrast to pure mathematical studies, these relationships must relate to actual physical flows since they are based on fundamental physical concepts. The second reason to investigate a scientific question is that the interrelation among dependent variables can be used to assess the impact of a deliberate or inadvertent change of one variable on another. These relationships can also be used to predict the future state of these dependent variables, given their current distribution with respect to one another. By basing the relation of one variable in terms of the others, consistent with fundamental physical constraints, there is some confidence that the assessments and forecasts are realistic. Two of the reasons that complete confidence is not possible are that the modeling tools used to determine the interrelations are not perfect and our ability to specify the entire initial fields is incomplete. Thermal and orographic flows represent two types of mesoscale systems in which both theoretical and practical studies have been made. Thermal flows, as defined here, represent circulations that result from differential heating along the ground surface and the resultant upward mixing of this differential heating by turbulence or its transmission upward by differential net radiative flux divergence. Orographic mesoscale systems occur as larger-scale flow is perturbed by irregular terrain as air advects toward topographic obstacles. The theoretical and practical reasons for studying these two types of systems are discussed in the next section. An in-depth discussion of thermal and orographic mesoscale flows is presented in Pielke (1984, Chapter 13).

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and verify these models; however, physical understanding and practical assessments and forecasts are only possible by using some type of model, whether it be quantitative or qualitative. The development of quantitative modeling tools to understand and simulate thermal and orographic flows has developed on somewhat different tracks. An understanding of this division is essential in order to appreciate some of the difficulties and achievements in obtaining better knowledge of the two types of mesoscale flows.

2.1. Thermal Mesoscale Systems Thermal mesoscale systems can be cataloged into land and sea breezes, mountain-valley flows, lake-effect storms, and urban circulations (Pielke, 1982). Of these four features, land and sea breezes have been studied most extensively by using both observational and modeling tools. Thermally induced mesoscale circulations are strongly influenced by nonlinear advection and turbulent diffusion, as discussed, for example, by Martin and Pielke (1983), although idealized linear studies have been fruitful in developing an initial understanding of this type of mesoscale system. From this author’s perspective, the most valuable analytic studies occurred in the 1940s with the work of Defant (1950) and Haurwitz (1947). Haurwitz demonstrated, using the circulation theorem, the relationship between differential heating and land and sea breezes. The fundamental concept of a “thermal” mesoscale system is based on that type of analysis. Defant’s work was summarized effectively in “Compendium of Meteorology” (Defant, 1951), in which a qualitative description of both the land and sea breeze and mountain-valley flows was provided. Of perhaps even greater utility, however, was his earlier work (Defant, 1950), which described in detail the mathematical formulation of a land- and sea-breeze model that although suffering from simplifying assumptions (e.g., the water cooled/warmed to the same degree and at the same rate as the land warmed/cooled), nevertheless captured the essence of the land-seabreeze phenomena. Martin and Pielke (1983) found his model to be an effective tool to examine the adequacy of the hydrostatic assumption. Without adequate computer power, however, and the development of accurate and stable computation schemes, the incorporation of nonlinear effects into these analytic models could not be accomplished. The 1950% therefore, were characterized by a refinement of the linear models in order to remove some of the grosser assumptions applied by Defant (1950).

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During this time period, however, the use of computers to solve the equations of motion on the synoptic scale expanded [e.g., as reviewed in Namias, (1983)l. It was, therefore, only a matter of time before the computer would be used to simulate smaller-scale atmospheric systems. In this author’s opinion, the first effective use of a computer to simulate a thermal-mesoscale system was Estoque’s (1961, 1962) pioneering two-dimensional sea-breeze model. With the incorporation of a finite difference approximation to the nonlinear advection terms, Estoque was able to show the concentration of the low-level convergence by nonlinear terms, as well as the interaction of the large-scale prevailing flow with the sea-breeze-induced advection pattern (e.g., see Fig. 1). Estoque’s work represented the premier model for the 1960s with a number of investigators not only performing theoretical idealized studies using his model, but actually applying the model to specific geographic areas [e.g., Moroz (1967)l. Beginning in the late 1960s, however, two major components of Estoque’s model were questioned-his use of forward-upstream differencing and of a vertically differentiated form of the continuity of mass of air equation. Molenkarnp (1968) pointed out the large implicit diffusion of such upstream differencing, which he claimed was of the same order as the physical diffusion of heat and momentum by turbulence. Since the vertical diffusion of heat, in particular, is so important in the generation and evolution of the sea breeze, the conclusion drawn by the professional community was that upstream differencing was an inappropriate computation tool to use in all mesoscale models. The inadequacy of this solution technique is generally recognized today, although it was shown in Mahrer and Pielke (1977) that for sea-breeze simulations the vertical diffusion of heat and momentum is, in fact, much larger than the corresponding computation diffusion so that upstream differencing can be accurately used for such problems. The second major criticism of Estoque’s model involved its rnathematical formulation. As shown in the classic paper by Neurnann and Mahrer (1971), the use by Estoque (1961, 1962) of a vertically differentiated form of the continuity of mass of air equation in order to obtain an additional boundary condition can introduce a fictitious source or sink of mass. Therefore, the physical realism of the results from such a model would be suspect. Nonetheless, despite these shortcomings, Estoque’s work was and is generally recognized as being pioneering. His model represented a major technological as well as scientific advance. An innovative alternative approach to vertical cross-sectional models

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such as that developed by Estoque, introduced during the mid-l960s, was a horizontal cross-sectional boundary-layer model in which a single layer was used to represent the vertical direction. Pioneered by Lavoie (1972), such mixed-layer models applied to specific geographic areas represented an original attempt to include spatial variations in geography (in the first Lavoie study, the area was in the vicinity of Lake Erie) and yet still retain the computation economy of a two-dimensional model. Lavoie’s model was later applied to other regions [e.g., Oahu, Lavoie (1974); the Black Hills of South Dakota, Chang (1970); Alberta, Raddatz and Khandekar (1979)l and even today rates as an effective and efficient modeling tool for boundary-layer flow that is perturbed by terrain in the presence of substantial synoptic flow and strong overlying thermodynamic stratification. In the late 1960s, computer power continued to expand. It was, therefore, only a matter of time before three-dimensional sea-breeze simulations would be attempted. McPherson (1968) made the first such attempt, to the author’s knowledge. In his study, the influence of an embayment (which was to qualitatively represent Galveston Bay, Texas) on the sea breeze was studied. Estoque’s model was used as the basis for McPherson’s model. At the time of McPherson’s integrations, computer power was only marginally able to handle his simulation using a grid mesh of 70 X 15 X 20, so that only one production run was possible using the (at the time) state-of-the-art CDC 6600 computer. Approximately 6 hr of central processing time was required in order to run his model for 24 hr on that computer system. Because of a magnetic tape failure only 18 hr were actually integrated. Nevertheless, despite these shortcomings (as well as his use of a computationally poor approximation to the advection terms), his work demonstrated that technology had evolved to the state at which three-dimensional simulations were possible. In 1971 William Cotton proposed to the Experimental Meteorology Lab director, Joanne Simpson, that a three-dimensional sea-breeze model should be developed to examine the influence of sea-breeze-induced convergence zones on cumulus convection in the Florida Area Cumulus Experiment (FACE) target area. It was at this point that the author of this chapter was employed, which culminated after 18 months in the model reported in Pielke (1974). The approach used in that study differed somewhat from Estoque’s model (although the author was motivated by Estoque’s papers) in that similarity theory was used in order to represent the vertical turbulent diffusion of heat, moisture, and momentum. In addition, the actual coastline configuration of the south Florida peninsula was digitized and the turbulent fluxes fractionally weighted as to the proportion of land and water in a grid area.

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FIG.I . Sea breeze with (a) zero synoptic flow at I100 local time, (b) zero synoptic flow at 1700 local time, (c) 5-m-s-I synoptic offshore flow at 1100 local time, and (d) 5-m-s-' synoptic offshore flow at 1700 local time. Vectors give the landward and vertical circulation,

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full lines the temperature change from OXOO. Dashed lines are velocity components (meters per second) into the figure. [From Estoque ( 1962). From Jortrnctl of Atmospheric- Science. copyright 1962 by the American Meteorological Society.]

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The results reported in Pielke (1974) were encouraging not only because it demonstrated that three-dimensional mesoscale model simulations had come of age, but also because even dry thermally forced mesoscale systems exert a strong controlling influence on deep cumulonimbus systems not only in their initiation state, but throughout much of their life history. The model runs reported in that paper, however, still required substantial central processing and turn-around time (i.e., time between submission and return of a computer job). The author could only submit 1 hr of simulation time per day (with t h e results written on tape) so that a 10-hr model run required 10 days to complete on the National Meteorological Center CDC 6600. The size of the model grid (33 x 36 X 7) was constrained so that it fit into the central processing unit without any need to spool data into and out of the central core. During the 1970s and continuing into the early 1980s, the use of three-dimensional model simulations of thermally induced mesoscale systems expanded. Examples include Carpenter (1979), Warner et al. (1978), Mahrer and Pielke (1976), Hsu (1979), Kikuchi et al. (1981), and McNider (1981). McNider’s (1981) work, for instance, represented the first successful three-dimensional model simulation of both drainage winds and the resultant deep out-valley flow due to the cold air pooling in an open valley. His model integration reproduced quantitatively what Defant (1951, p. 665) had heuristically illustrated for flow within an open valley under zero synoptic flow. A more thorough review of thermally forced mesoscale models is presented in Pielke (1984). As the major computing centers obtained later generation computers (e.g., the CDC 7500 and later the CRAY at the National Center for Atmospheric Research: the CDC 205 at Colorado State University, at the Geophysical Fluid Dynamics Lab, at the United States National Meteorological Center, and at the British Meteorological Office), our ability to run three-dimensional mesoscale codes has increased. At present, computer capabilities exist to run these three-dimensional models in real time for mesoscale-sized areas. The parameterizations required to use these tools are felt to have been developed to a level of realism in which accurate simulations are possible for most types of thermally induced mesoscale systems (exceptions are discussed in Section 3). Also, the theoretical basis for this type of flow developed by using analytic and idealized two-dimensional numerical mesoscale models appears to be well understood. Unfortunately, operational weather services have continued to

THERMAL A N D OROGRAPHIC MESOSCALE SYSTEMS

21 1

concentrate resources on synoptic- and hemispheric-scale forecast models despite a series of demonstrations that skill with these larger-scale models has plateaued [American Meteorological Society (AMS), 1979, 19831. Several European countries, however, such as the United Kingdom and France, are currently performing prototype testing of operational mesoscale forecast models. In the author’s opinion, part of the reason for the emphasis on larger-scale models is political because the perceived responsibilities of a country such as the United States is to provide a nationwide model. Therefore, resources for operational model development have concentrated in specific national centers such as the National Meteorological Center (NMC). One rationale for this approach is that in a federal form of government such as exists in the United States, states should support the implementation of mesoscale models since these tools are specific to small-scale, often substate-sized, regions. Since states, traditionally, however, have not been involved in weather forecasting, mesoscale models have not been developed or implemented under state jurisdiction. Perhaps in the future, state offices of climatology (which were originally a federal service) could be used as a focal point for the development of state forecast centers that use mesoscale models in parallel to the federal government’s application of synotic- and larger-scale models. In contrast to traditional weather forecasting, however, one would suspect that government agencies that are tasked to minimize and warn of environmental atmospheric health hazards would take full advantage of our skill at simulating thermally induced mesoscale systems. After all, such effects as the recirculation of pollutants by meteorological flow systems would be expected (and have been shown) to have a major impact on the concentrations of atmospheric pollution. In the United States, agencies that are mandated to provide environmental atmospheric poilution assessments include the Department of Energy and the Environmental Protection Agency. Unfortunately, these and other related agencies assess air-quality impacts in complex terrain by using highly idealized approaches that are valid only over flat, homogeneous terrain in the absence of a substantial mesoscale or smaller perturbation to the flow (American Meteorological Society, 1981). The argument currently given to retain these tools is that they are consistent. Never mind that they are often not accurate! A paper by this author (Pielke et al., 1983) attempts to illustrate one situation in which the current assessment approach (e.g., using constant wind direction with height and no plume splitting) would clearly be in error.

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2.2. Orographic Mesoscale Models In contrast to thermally forced mesoscale models, the early pioneering work on the effect of terrain on prevailing large-scale flow was generally performed by fluid dynamists rather than meteorologists, as was the case with much of the study of thermally forced mesoscale systems. This difference in background, unfortunately, has generated something of a split between fluid dynamists and meteorologists currently working in the numerical simulation of orographic flows and has, in the author’s opinion, limited the application of current modeling tools to real-world assessments. The contributions of the fluid dynamists to the fundamental physical understanding of this type of mesoscale system, however, have been substantial. The approach of these investigators, applying their studies of flow in a stratified fluid to the atmosphere, has been to model flow over orographic barriers as a wave phenomena. The early analytic studies of Queney (1947, 1948) and Scorer (1949) and later that of Eliassen and Palm (1960), for instance, brought into focus the appropriate form and direction of group wave energy propagation. Long’s (1954) analytic model is recognized as a brilliant study that has been extended to provide exact solutions [e.g., Lilly and Klemp (1979) and Raymond (1972)l to a specific subset of flow over terrain obstacles of finite amplitude. Baines (1977) discusses limitations of Long’s solution, however, for actual stratified flow over an obstacle. An effective summary of these studies as viewed by a fluid dynamist is given in Smith (1979). The first successful nonlinear numerical simulation of atmospheric mesoscale orographic flow appears to be that of the doctoral thesis of Hovermale (1965). Regretably , Hovermale’s two-dimensional modeling study was never submitted for publication in the peer-reviewed literature. In the 1970s, however, several other two-dimensional numerical model simulation studies did appear in the literature, including that of Mahrer and Pielke (1975), Klemp and Lilly (1978), and Peltier and Clark (1979). At about the same time, a linear model of orographic flow as reported in Klemp and Lilly (1975) was used to model the actual flow over a specific orographic barrier (the Colorado Rockies in an east-west cross section through Boulder, Colorado) on a specific day in which a damaging downslope wind storm occurred. The ability of even a linear model to produce credible solutions of actual flows emphasizes (in contrast to thermally forced mesoscale flows) the apparent frequent dominance of the pressure gradient force [an approximately linear effect, see Pielke (1981, p. 300)] as opposed to nonlinear advection in controlling the structure of orographic flow.

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Klemp and Lilly concluded from their linear study that it is the partial downward reflection of upward-propagating wave energy from the tropopause that causes severe orographically induced wind storms. A controversy, however, has developed regarding the importance of this mechanism to cause the extremely large wind speeds observed occasionally to the lee of mountain barriers. Peltier and Clark (1979) and Clark and Peltier (1977) using a nonlinear numerical model argued that it is the downward reflection of wave energy from breaking waves at any level in the atmosphere that causes these strong winds. An exchange of correspondences in 1980 (Klemp and Lilly, 1980; Peltier and Clark, 19801 illustrates this controversy. At the same time that the dynamists (and a few meteorologists) were developing prognostic numerical orographic models, a number of investigators introduced what can be called kinematic or diagnostic orographic flow models. By greatly simplifying the equations of motion, relatively economical simulations of airflow around and over terrain obstacles could be produced. These models, however, suffered from either a need to input a substantial number of observations and/or a requirement to specify a priori the height at which the disturbance in the atmospheric flow due to the orography disappears. Although as shown by the more complete numerical models, simulation results are often very sensitive to the specification of the top boundary condition, these kinematic models appear to be useful tools when

(1) a strong inhibition to upward motion occurs in the lower troposphere (i.e., strong thermodynamic stratification above the top of the planetary boundary layer); (2) a substantial fraction of the terrain lies above that altitude, and (3) differential heating and cooling within the atmosphere or along the terrain are small. The airflow under this situation moves more or less parallel to the ground around the topographic obstacle. Examples of kinematic models include that of Dickerson (1978), Fosberg ef (if. (1976), Collier (1977), Danard (1977), and Sherman (1978). Also, over the long term, such models appear able to provide realistic estimates of the climatological conditions, for instance, as shown by Rhea (1977) for mean monthly wintertime orographic precipitation in western Colorado. The limitations on these simple models, however, have not been adequately determined, although, for example, one would expect Rhea’s model to be in error during convective precipitation events or when air flows over or around more isolated peaks as opposed to the ridges found in western Colorado. In addition, the application of the more complete

2 14

ROGER A . PIELKE

prognostic models has not been performed for specific geographic locations except in a few specialized cases [e.g., Clark and Gall (1982) and Seaman (1982) for Elk Mountain, Wyoming; Mahrer and Pielke (1976) for Barbados]. Part of the reason for this lack of application is the same as for thermally driven mesoscale systems; i.e., there is no effective organization, at least in the United States, to routinely utilize such models for either weather forecasting or environmental assessments. An additional complication, however, in this author's opinion. is the insistence of a number of investigators that most of the basic physical understanding of wave dynamics and boundary layer structure in complex terrain must be underlo5

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stood before applications are made. This approach is shortsighted, however, and is contrary to the philosophy used in synoptic and hemispheric modeling when tools such as the equivalent barotropic model were applied to represent atmospheric flow even though it clearly had deficiencies in representing the actual atmosphere. This limited tool, however, had a major impact on improving weather forecasting [e.g., see Namias (1983)l. A third problem with applying orographic flow models to actual geographic regions is the difficulty in adequately specifying the terrain forcing. As a necessary condition for accurate simulations of airflow over rough terrain, the dominant topographic variations must be represented in the model. For some mountain areas, such as for the Blue Ridge Mountains of Virginia (see Fig. 2a), the terrain is smooth enough at the shorter scales such that grid intervals can be defined that will provide the computational resolution of most of the variability in orography. In other locations, however, such as in western Colorado (see Fig. 2b), there is much more terrain variance on the small scales, which, unless the smaller scales can be adequately parameterized, makes impossible an accurate model simulation using reasonable computer resources (Young and

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ROGER A. PIELKE

Pielke, 1983). In addition, as pointed out by Walmsley et uI. (19821, the influence of these terrain variations on atmospheric flow is weighted by the wave number, so that the small-scale features have a disproportionately large effect.* A fourth problem involves the difficulty of obtaining sufficient spatial density in observations, particularly at upper levels, with which to obtain a reasonably accurate initial field of the dependent variables. Even when the mesoscale system is strongly forced by terrain variability, a realistic specification of the state of the atmosphere for the initialization of a model integration is essential for an accurate model simulation. In irregular terrain, this initial state of the atmosphere can seldom, if ever, be characterized adequately by using observations with only limited spatial resolution. Nevertheless, despite these difficulties, resources should be expended to test these models in actual geographic regions in order determine their limitations as well as their skills. Such an approach will better delineate the needed research that must be performed in order to improve the skill of the models. It will also, however, permit these tools to be used for real-world assessments without waiting for all the questions regarding orographic flow to be answered. 3.

RESEARCH

AREAS

It is this author’s opinion that the major problems associated with improved simulations of orographic and thermally forced mesoscale flows are, in general, practical, not theoretical. The basic mathematical relations that describe these flows, for instance, are well understood [e.g., see Pielke (1984)l. It is in the discretization and numerical solution of these equations, in the pararneterizations of subgrid scale and of source/sink terms, and in the specification of initial and boundary conditions, however, that improvements are needed. Only when chemical effects become important does there appear to be a need for substantial new theoretical insight into the basic equations.

* A similar difficulty, of course, arises if the spatial distributionof differential heating is on a small scale. Spectra of this forcing of thermally driven mesoscale systems, however, have not been explored yet, however, since wave dynamics are not as important with those systems as with orographic Rows. With thermal mesoscale flows, vertical and horizontal turbulent diffusion occur, which then generate a pressure gradient force. The spatial scale of the forcing is therefore smoothed somewhat by this diffusion of heat. With orographic Row the degree of smoothing of small-scale terrain forcing by wave-wave interactions (and by turbulent diffusion) is not known.

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TABLEI. EXAMPLES OF ENGINEERING QUESTIONS 1. What level of sophistication is required in the parameterization of the boundary layer

2. 3.

4. 5.

6.

in mesoscale models? Is a refined version of first-order closure sufficient as suggested in Pielke (1984, Chapter 7), or is higher-order closure required? How detailed a treatment of long- and short-wave radiative flux divergence is required in order to represent realistically this type of forcing in mesoscale models? Are more accurate solutions of mesoscale flows possible at less expense using recently discussed techniques such as finite element or Lagrangian schemes? Have microcomputers advanced to the stage at which three-dimensional model simulations can be performed o n those systems? Can the linear and nonlinear components of a model be evaluated separately such that more economical and much more accurate results are achieved, as suggested in Weidman and Pielke (1983)? Can satellite sensing systems be improved sufficiently to provide measurements of sufficient spatial resolution to permit accurate initializations of mesoscale models in complex terrain?

Listed in Table I is a brief compilation of a number of engineering tests that must be performed in order to improve mesoscale models. In Table I1 are listed examples of applied research questions that can be addressed by using these tools. Neither list is all-inclusive; however, it illustrates the type of research emphasis that this author feels is important. It also illustrates that as larger and faster computers become available, the satisfactory answer to these questions will be easier to achieve.* * Maximum computer power at present is represented by systems such as the CYBER 205 and the NCAR CRAY IA. These systems have o n the order of I to 2 million words of central core capable of a practical calculation rate of 10 to 140 megaflops (million floating point operations per second). By 1986, plans currently suggest practical calculation rates of 320 to 2000 megaflops may be possible with a central core of over 250 million words.

TABLE 11. EXAMPLES OF APPLIED RESEARCH OUESTIONS 1. How significant are terrain variations on the subgrid scale on mesoscale flows? If they

are significant, what is an effective and accurate parameterization? 2. When and under what circumstances does the hydrostatic assumption fail to represent accurately the pressure distribution in atmospheric flows? 3. When and under what circumstances does the wind deviate substantially from gradient wind balance above the planetary boundary layer as a function of the scale of an atmospheric feature? 4. What influence does pollution and its effect on the atmospheric radiation budget play on the development and evolution of mesoscale systems? 5 . What are the physical mechanisms that determine whether air flows up and over a heated or cooled terrain obstacle as opposed to moving around it or being blocked?

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4. EVENTUAL GOALS Eventually, thermally forced and orographic mesoscale models will be used in real time for weather forecasting and environmental assessments. With the development of 32-bit minicomputers much of this practical work can soon be performed cheaply and efficiently at many locations. Mesoscale modeling, with its need to provide simulations at many geographic areas, is clearly closely linked to the minicomputer revolution. At the same time that a need exists to proliferate the capability to integrate mesoscale models, however, there is also a requirement to nuture several operational/research/academic centers that focus research in mesoscale atmospheric modeling. These centers would be similar in concept to the Geophysical Fluid Dynamics Laboratory at Princeton; the Environmental Research Laboratory of the National Oceanic and Atmospheric Administration (NOAA), the National Center for Atmospheric Science, Colorado State University and the University of C O ~ O rado in Colorado; and the National Severe Storms Laboratory and the University of Oklahoma in Oklahoma. The additional requirements, however, are that an operational component to interact with the research group would be included, and the operational, research, and academic units would be colocated in the same building. The closest thing to this model in meteorology of which I am aware was the presence of the National Hurricane Center, the National Hurricane Research Laboratory, the Experimental Meteorology Laboratory, and the Department of Atmospheric Science in the same building at the University of Miami in the early 1970s. (Unfortunately, what was a very effective environment for scientific-operational interactions was subsequently broken up later in the 1970s by a series of unfortunate administrative decisions.) The Prototype Office of Forecasting program during the summer of 1982 represents an example in which a short-term blending of operational and research individuals to predict weather along the Colorado Front Range was recently performed. This type of approach is also being proposed for the STORM Central program (Zipser, 1984).*

* Thermal mesoscale systems may be involved in the evolution of areas of cumulonimbus convection, which is the focus of the STORM Central program. Cotton (1983), referring to Banta (1982) and others in his research group, has suggested that deep cumulus cloud activity over the Rocky Mountains, initiated at least in part by daytime upslope flow, may coherantly interact in order to initiate these mesoscale convective systems. These mesoscale features are associated in the central United States with tornadic and severe thunderstorms early in their life cycle and Rash floods later in their evolution [a summary of these systems for 1982 is given in Rodgers ef al. (198311.

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This meshing of capabilities with its real-time exposure to mesoscale weather is vital to the most cost-effective, rapid improvement in our accurate simulation and prediction of weather on this scale. The presence of the academic community with its resource of students assures continuity and fresh ideas in the continued advancement of applying modeling tools t o thermally forced and orographic flows.

ACKNOWLEDGMENT The author wishes to thank I. Orlanski for the opportunity to contribute to this volume. Support to prepare this contribution was provided by the National Science Foundation, Division of Atmospheric Sciences/Meteorology under grant #ATM-8304042, and by the National Park Service grant #NA81RAH00001. The typing and preparation of the manuscript were very competently performed by S. Rumley. W. R. Cotton and Y . Mahrer are thanked for recommending revisions and additions to the draft version of this chapter.

REFERENCES American Meteorological Society (AMS) (1979). Statement on Weather Forecasting. Bull. A m . Meteorol. Soc. 64, 1453. American Meteorological Society (AMS) (1981). “Air Quality Modeling and the Clean Air Act Recommendation to EPA on Dispersion Modeling for Regulatory Application.” PB83-106237. AMS, Boston. American Meteorological Society (AMS) (1983). Policy statement of the American Meteorological Society on weather forecasting. Bull. A m . Meteorol. Soc. 64, 1385-1387. Baines, P. G . (1977). Upstream influence and Long’s model in stratified flows. J . Fluid Mech. 82, 147-159. Banta, R . M. (1982). An observational and numerical study of mountain boundary-layer flow. Ph.D. Dissertation, Dept. of Atmospheric Science, Colorado State University, Fort Collins. Carpenter, K. (1979). An experimental forecast using a nonhydrostatic mesoscale model. Q . J . R . Meteorol. Soc. 105, 629-655. Chang, C . (1970). A mesoscale numerical model of airflow over the Black Hills. Master of Science Thesis, South Dakota School of Mines and Technology, Rapid City. Clark, T. L., and Gall, R. (1982). Three-dimensional numerical model simulations of airflow over mountainous terrain: A comparison with observations. M o n . Weather Rev. 110, 766-791. Clark, T. L., and Peltier, W. R. (1977). On the evolution and stability of finite-amplitude mountain waves. J . Atmos. Sci. 34, 1715-1730. Collier, C. G. (1977). The effect of model grid length and orographic rainfall efficiency on computed surface rainfall. Q . J . R . Meteorol. SOC. 103, 247-253.

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Cotton, W. R. (1983). Up-scale development of moist convective systems. “Lecture for Course on Mesoscale Meteorology, Pinnarpsbaden, Sweden, June, 1983 (available from Dept. of Atmospheric Science. Colorado State University, Fort Collins, Colorado, 80523). Danard, M. (1977). A simple model for mesoscale effects of topography on surface winds. Mon. Weather Rev. 105, 572-581. Defant, F. (1950). Theorie der land- und seewind. Arch. Mereorol.. Geophvs. Bioklirnatol.. Ser. A 2, 404-425. Defant, F. (1951). Local winds. In “Compendium of Meteorology” (T. F. Malone, ed.), pp. 655-672. Am. Meteorol. SOC.,Boston, Massachusetts. Dickerson, M. H. (1978). MASCON-A mass-consistent atmospheric flux model for regions with complex terrain. J. Appl. Meteorol. 17, 241-253. Eliassen, A., and Palm, E . (1960). On the transfer of energy in stationary mountain waves. Geophys. N o r v . 22, 1-23. Estoque, M. A. (1961). A theoretical investigation ofthe sea breeze. Q . J. R . Meteorol. Soc. 87, 136-146. Estoque, M. A. (1962). The sea breeze as a function of prevailing synoptic situation. J . Armos. Sci. 19, 244-250. Fosberg, M. A., Marlatt, W. E., and Krupnak, L. (1976). Estimating airflow patterns over complex terrain. U.S.For. Serv., Rocky M t . For. Range Exp. Stn., Res. Pap. RM-162, 1-16. Haurwitz, B. (1947). Comments on the sea breeze circulation. J . Meteorol. 4, 1-8. Hovermale, J. B. (1965). A nonlinear treatment of the problem of airflow over mountains. Ph.D. Thesis, Pennsylvania State University, University Park. Hsu, H.-M. (1979). Numerical simulations of mesoscale precipitation systems. Ph.D. Dissertation, Dept. of Atmospheric Oceanic Science. University of Michigan, Ann Arbor. Kikuchi, Y.,Arakawa, S., Kimura, F., Shirasaki, K., and Nagano, Y. (1981). Numerical study on the effects of mountains on the land and sea breeze circulation in the Kanto district. J . Meteorol. SOC.Jpn. 59, 723-738. Klemp, J. B., and Lilly, D. K . (1975). The dynamics of wave-induced downslope winds. J . Atmos. Sci. 32, 320-339. Klemp, J. B., and Lilly, D. K. (1978). Numerical simulation of hydrostatic mountain waves. J . Atmos. Sci. 32, 78-107. Klemp, J. B . , and Lilly, D. K. (1980). Mountain waves and momentum flux. GARP Publ. Ser. 30, 116-141. Lavoie, R.L. (1972). A mesoscale numerical model of lake-effect storms. J. Atmos. Sci. 29, 1025- 1040. Lavoie, R. L.(1974). A numerical model of trade wind weather over Oahu. M o n . Wearher Rev. 102,630-637. Lilly, D. K . , and Klemp, J. B. (1979). The effects of terrain shape on nonlinear hydrostatic mountain waves. J . Fluid M e c k . 95, 241-261. Long, R. R. (1954). “Some Aspects of the Flow of Stratified Fluids. 11. Experiments with a Two-Fluid System,” Tech. Rep. No. 4. Johns Hopkins University, Baltimore, Maryland. McNider, R. T. (1981). Investigation of the impact of topographic circulations o n the transport and dispersion of air pollutants. Ph.D. Dissertation, Dept. of Environmental Sciences, University of Virginia, Charlottesville. McPherson, R. D. (1968). “A Three-dimension Numerical Study of the Texas Coast Sea Breeze,” Rep. No. 15. Atmospheric Science Group, College of Engineering, University of Texas. Austin.

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Mahrer, Y., and Pielke, R. A. (1975). The numerical study of the airflow over mountains using the University of Virginia mesoscale model. J. Atmos. Sci. 32, 2144-2155. Mahrer, Y.,and Pielke, R. A. (1976). The numerical simulation of the airflow over Barbados. Mon Weuther Rev. 104, 1392-1402. Mahrer, Y., and Pielke, R. A. (1977). A numerical study of the airflow over irregular terrain. Beitr. Phys. Atmos. 50, 98-113. Martin, C. L., and Pielke, R. A. (1983). The adequacy of the hydrostatic assumption in seabreeze modeling over flat terrain. J. Atmos. Sci. 40, 1472-1481. Molenkamp, C. R. (1968). Accuracy of finite difference methods applied to the advection equation. J. Appl. Meteorol. 7, 160-167. Moroz, W.J. (1967). A lake breeze on the eastern shore of Lake Michigan: Observations and model. J . Atmos. Sci. 24, 337-355. Namias, J. (1983). The history of polar front and air mass concepts in the United States-an eyewitness account. Bull. A m . Meteorol. SOC.64, 734-755. Neumann, J., and Mahrer, Y. (1971). A theoretical study of the land and sea breeze circulations. J . Atmos. Sci. 28, 532-542. Peltier, W. R., and Clark, T. L. (1979). The evolution and stability of finite-amplitude mountain waves. Part 11. Surface wave drag and severe downslope windstorms. J . A m o s . Sci. 36, 1498-1529. Peltier, W. R., and Clark, T. L. (1980). Reply. J. Atmos. Sci. 37, 2122-2125. Pielke, R. A. (1974). A three-dimensional numerical model of the sea breezes over south Florida. Mon. Weather Rev. 102, 115-134. Pielke, R. A. (1981). Mesoscale dynamic modeling. Adv. Geophys. 23, 186-344. Pielke, R. A. (1982). The role of mesoscale numerical models in very-short-range forecasting. In “Nowcasting: A New Approach to Observing and Forecasting the Weather” (K. A. Browning, ed.), pp. 207-221. Academic Press, New York. Pielke. R. A. (1984). “Numerical’ Meteorological Modelling,“ Academic Press, New York. Pielke, R. A., and Kennedy, E. (1980). “Mesoscale Terrain Features,” Rep. No. UVAENVSCI-MESO-1980-1. Available from R. A. Pielke, Dept. Atmos. Sci., Colorado State Univ., Ft. Collins, 80523, Pielke, R. A., McNider, R. T., Segal, M., and Mahrer, Y. (1983). The use of a mesoscale numerical model for evaluations of pollutant transport and diffusion in coastal regions and over irregular terrain. Bull. Am. Meteorol. SOC. 64, 243-249. Queney, P. (1947). “Theory of Perturbations in Stratified Currents with Applications t o Air Flow over Mountain Barriers,” Misc. Rep. No. 23. Dept. of Meteorology, University of Chicago, Univ. of Chicago Press, Chicago, Illinois. Queney, P. (1948). The problem of air flow over mountains: A summary of theoretical studies. Bull. A m . Meteorol. Sac. 29, 16-26. Raddatz, R. L., and Khandekar, M. L. (1979). Upslope enhanced extreme rainfall events over the Canadian western plains: A mesoscale numerical simulation. Mon. Weather Rev. 107, 650-661. Raymond, D. J. (1972). Calculation of airflow over an arbitrary ridge including diabatic heating and cooling. J. Atmos. Sci. 29, 837-843. Rhea, 0. J. (1977). Orographic precipitation model for hydrometeorological use. Ph.D. Dissertation, Dept. of Atmospheric Science, Colorado State Univ., Fort Collins. Rodgers, D. M., Howard, K. W., and Johnston, E. C. (1983). Mesoscale convective complexes over the United States during 1982. M o n . Weuther Rev. 111, 2363-2369. Scorer, R. S. (1949). Theory of waves in the lee of mountains. Q. J . R. Meteorol. SOC.75, 41-56.

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Seaman, N. L. (1982). “A Numerical Simulation of Three-dimensional Mesoscale Flows over Mountainous Terrain,” Rep. No. AS 135 (NSF Grant No. ATM-77-17540). Dept. of Atmospheric Science, University of Wyoming, Laramie. Sherman, C. E. (1978). A mass-consistent model for wind fields over complex terrain. J. Appl. Meteorol. 17, 312-319. Smith, R. B. (1979). The influence of mountains on the atmosphere. Adv. Geophys. 21, 87230. Walmsley, J. L., Salmon, J. R., and Taylor, P. A. (1982). On the application of a model of boundary-layer flow over low hills to real terrain. Boundary Layer Mereorol. 23, 17-46. Warner, T. J., Anthes, R. A., and McNab, A. L. (1978). Numerical simulations with a threedimensional mesoscale model. Mon. Weather Rev. 106, 1079-1099. Weidman, S. T., and Pielke, R. A. (1983). A more accurate method for the numerical solution of nonlinear partial differential equations. J. Compur. Phys. 49, 342-348. Young, G. S., and Pielke, R. A. (1983). Application of terrain height variance spectra to mesoscale modeling. J. Atmos. Sci. 40, 2555-2560. Zipser, E. (1984). Storm-Central phase, preliminary program design. Prepared by the National Center for Atmospheric Research in consultation with, and on behalf of, the Interagency Team for STORM-Central, Boulder, Colorado, January, 1984.

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS I. ORLANSKI, B. Ross, L. POLINSKY, A N D R. SHAGINAW Geophysical Fluid Dynamics LahoratotylNOAA Princeton University Princeton, New Jersey

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2. Baroclinic Waves and Fronts . . . . . . . . . . . . . . . . 2.1. Quasi-Geostrophic Effects. . . . . . . . . . . . . . . . 2.2. Semigeostrophic Effects . . . . . . . . . . . . . . . . 2.3. Ageostrophic Effects. . . . . . . . . . . . . . . . . . 3. Mature Front. . . . . . . . . . . . . . . . . . . . . . . 3.1, The Significance of Frontal Collapse. . . . . . . . . . . . 3.2. Dynamic Balance in a Mature Front. . . . . . . . . . . . 4. What Observed Features Can Be Explained by Theory? . . . . . 5. What Other Processes Are Important in Frontogenesis? . . . . . 5.1. Structure of the Cold Front . . . . . . . . . . . . . . . 5.2. Frontogenetical Terms . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . .

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1 . INTRODUCTION Because so much has been written about atmospheric fronts in extended reviews and textbooks (Pettersen, 1956; Palmen and Newton, 1969; Pedlosky, 1979; Hoskins, 1982), this paper will not attempt to provide a thorough review of the subject, but rather will endeavor to describe breakthroughs in the development of our understanding of fronts* and to indicate some of the questions yet to be answered satisfactorily. It would be fair to state that the pioneering work of Bjerknes and his collaborators at the Bergen School in the early twenties is to the study of atmospheric fronts what the contemporary work of N . Bohr and his associates in Denmark is to our understanding of atomic structure. The observation by Bjerknes (1919) of convergence lines at boundaries between air masses has had profound implications for theoretical and applied meteorology. Prior to this verification, the possibility that air masses in the atmosphere were separated by surfaces of discontinuity could only be inferred from incomplete observational data. It is quite remarkable, therefore, that Margules (1906) was able to develop a rather detailed theory of atmospheric discontinuities 13 years

* Although this review will deal with the general subject of atmospheric fronts. primary emphasis will be given to the structure and dynamics of surface cold fronts. 223 A D V A N C E S I N GEOPHYSICS, VOLUME

28B

Copyright b 1985 by Academic Press. Inc. All right\ of reproduction in any form reserved.

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1. ORLANSKI, B . ROSS, L. POLINSKY, AND R . SHAGLNAW

prior to the observational description of Bjerknes. Not only did Margules formulate the theoretical basis for the dynamics of these fronts, but he also recognized the important concept that “the potential energy depends on the horizontal temperature distribution. It appears to be the main source of storm energy and be converted directly into wind [kinetic] energy .” But Margules’s primary contribution is his well-known formula relating the equilibrium slope a of the interface between two air masses ( 1 and 2) having different density p and horizontal wind speed U along the front: a = (2~/g)[(pzUz- p1U1)4pz - p1)l

where R is the angular velocity of the rotating frame of reference and g the gravitational acceleration. This formula was the primary theoretical concept used by the Bergen School to define their observed fronts. The observations of Bjernkes and his collaborators, using a fairly dense surface network and sparse aerological data, provided the basis for their concept of the evolution of surface cold waves and suggested a link between surface fronts and cyclones. Their results were summarized by Bjerknes (1919) in the classic paper “On the Structure of Moving Cyclones.” Eighteen years later, J. Bjerknes (1937) identified the upper-air wave and pointed out its role in cyclogenesis. This discovery led, in turn, to the simultaneous discovery of baroclinic instability by Charney (1947) and Eady (1949). As Charney (1975, p. 1 1 ) comments, At a time when our knowledge of the upper atmosphere was still gained largely by indirect inference from surface observations and from a few upper air ascents, [Bjerknes] accurately described the sequence of events linking the formation of the surface cyclone with the upper wave. . . Throughout this exciting period of discovery, the more complete picture of the atmosphere provided by new and improved sounding networks offered a continual challenge to the developing theory. In particular, Charney (1975) was puzzled by the fact that whereas a clear correlation had been established between long upper-air waves and primary surface waves, no correspondence had been found between the secondary frontal waves (with wavelength -1000-2000 km) studied by Bjerknes and Solberg (1921), Solberg (1928), Kotschin (1932), Eliasen (1960), and Orlanski (1968) and the long upper-air waves (with wavelength -3000-6000 km) described by baroclinic instability theory. Whereas the two-layer numerical experiment of Phillips (1956) demonstrated a link

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

225

between baroclinic and surface waves, no similar link has been found between baroclinic waves and waves associated with frontal instabilities.

2. BAROCLINIC WAVESA N D FRONTS The primary breakthrough in our understanding of the relationship between surface frontogenesis and the evolution of planetary waves resulted from the primitive-equation numerical simulation described by Phillips (1956). In this numerical solution, growing finite-amplitude baroclinic waves produced regions of more intense temperature gradients suggestive of frontal zones. One could have inferred from this that the deformation field associated with such waves could provide the forcing necessary to produce frontogenesis. The proof of this hypothesis was provided several years later by Williams (1967) in a two-dimensional numerical simulation of the evolution of an Eady wave. Williams showed that the constantly growing, unstable Eady wave produced very intense, frontlike temperature gradients near the surface. After 5 days of model integration, these gradients were so intense as to make the numerical representation, with 50-km horizontal resolution, unacceptably inaccurate. During the early 1970s, Hoskins and Bretherton (1972) presented an analytic model of frontogenesis that demonstrated how the Eady wave can produce a temperature discontinuity in a finite period of time. In a series of outstanding papers, Hoskins and his collaborators (Hoskins and Heckley, 198 1 ; Hoskins and West, 1979) explained the frontogenetical processes that can occur during the evolution of baroclinic waves. In particular, comparison of the theoretical model results with Williams’s numerical solution showed that Hoskins’s equation system with the semigeostrophic approximation produced most of the features that Williams’s primitive-equation system provided, but with the advantage that an analytic solution could be found for the frontogenesis problem. Furthermore, the Hoskins-Bretherton model generated temperature discontinuities in a finite time without the deficiency of numerical breakdown that Williams encountered. Hoskins (1982) has reviewed theoretical work on frontogenesis and has summarized recent advances in our understanding of frontogenetical processes in idealized flows. Let us now briefly summarize some of the basic features of frontogenesis using the quasi-geostrophic and semigeostrophic approximations. The processes by which the thermal wind balance along the front is maintained can be better seen from the

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equations of the vertical wind shear and horizontal potential temperature gradient, shown here in the X-Z plane:

I

I

4

--(gel) dt

=

-gV&

1 *

v0 - W W ,

I1

11 - gv,

*

v0,

-

gV,\

vO

where

The roman numerals above each term indicate the level of approximation assumed. The subscripts g and a indicate geostrophic and ageostrophic components, respectively. Hoskins (1982) uses the quasi-geostrophic (terms I) and semigeostrophic (terms I and 11) forms of (2.1) and (2.2) to illustrate how the cross-stream circulation in a very simplified frontal model acts to preserve the thermal wind balance f V,, = g0,. In the present discussion, we will show how this balance is maintained for the more realistic jet configuration used by Orlanski and Ross (1977) (Fig. 1) in an integration of the full nongeostrophic equations, starting from idealized geostrophic initial conditions.

2 . 1 . Quasi-Geostrophic Effects

-

-

The terms -gVgx V0 from (2.1) and -fV, VV, from (2.2) each represent the quasi-geostrophic forcing function Q , in which the dominant In the frontal configuration associated with Fig. 1 , the term is fUgZVgX. along-front wind V consists of a midtropospheric jet centered above the maximum surface temperature gradient. Therefore Q1 will be initially antisymmetric about this jet (the shear Ug2is a positive constant) and positive (negative) to the left (right) of the jet maximum. The cross-stream circulation cells, indicated by dashed lines in Fig. 1, are roughly antisymmetric at the beginning of the integration, a result that is explainable by this symmetry of Ql and the use of the quasi-geostrophic approximation (terms I). In this approximation, only the first two terms are retained from the right-hand side of (2.1) and (2.2). If Q , is positive (as

ADVANCES I N THE THEORY OF ATMOSPHERIC FRONTS 15

227

(a) 100

10

km

mb 500

5

0

0

500

km

lo00

(bl

10

km

mb

5

500

lo00

0 0

500

km

lo00

FIG.1. Plots of potential temperature 0 (solid contours) and perturbation stream function +’ (dashed contours) (a) 3.00 hr and (b) 10.87 hr after a two-dimensional frontal model was initialized with idealized conditions. Locations of maximum and minimum are indicated by X and N, respectively. [After Orlanski and Ross (1977). From Journul of the Atmospheric Sciences, copyright 1977 by the American Meteorological Society.]

to the left of the front in Fig. I ) , then (2. I ) predicts that gt), will increase (since its forcing term is +IQ,l), while (2.2) indicates that f V g 7 will decrease (since its forcing term is -1QlI). To counteract this disruption of the thermal wind balance go., = fVg,, an ageostrophic cross-stream circulation develops whereby the third term in (2. l), - N 2 w , , becomes negative (to the left of the front) to offset the increase of Q , in (2.1). Likewise, the third term in (2.2), -f2Uaz, becomes more positive to balance Q Iin (2.2). If cross-front geostrophic balance is assumed to be

228

1. ORLANSKI, B. ROSS, L. POLINSKY. A N D R . SHAGINAW

maintained, then the quasi-geostrophic terms on the right-hand sides of (2.1) and (2.2) define the diagnostic equation for the cross-stream circulation [as studied by Sawyer (1952, 1956) and Eliassen (1959, 1962)].

2.2. Semigeostrophic Effects As pointed out by Hoskins (1982), the use of the more complete semigeostrophic approximation (terms I and 11) produces more realistic asymmetries in the frontal structure. The equation of the vertical component of the relative vorticity 5 at the ground z = 0 is

I

I

11

(d,ldt)( = -fD - { D - V,

I1 *

Vg

(2.3)

For example, under the semigeostrophic assumption, the second term on the right-hand side of (2.3), -50, is included in the equation system. As a result, cyclonic vorticity intensifies in convergence zones and weakens in divergence zones, effects that the quasi-geostrophic system cannot produce. Similarly, the semigeostrophic term -gU,,O, [the last term of (2. l)] enhances temperature gradients in convergence zones and weakens them in divergence zones, Such effects help to produce the asymmetry in frontal features, such as the cross-stream circulation, which is evident in the solution of Orlanski and Ross after 1 1 hr of integration (Fig. lb). Figure l b also shows relative vorticity and horizontal convergence to be maximum at the surface, as one would expect. The presence of persistent vortex stretching near the surface in the semigeostrophic equations produces an unbounded growth of vorticity until the sernigeostrophic approximation is no longer valid (Hoskins, 1982). The semigeostrophic forms of (2.1)-(2.3) do not provide any ageostrophic mechanism by which to prevent this precipitous growth. Certainly the inclusion of eddy diffusion and surface friction can provide a limiting mechanism (Hoskins and Bretherton, 1972).

2.3. Ageostrophic Effects Another possible limiting mechanism, caused by terms neglected in the semigeostrophic approximation, has been proposed by Orlanski and Ross (1984). In the quasi-geostrophic and semigeostrophic approximations, the balancefc = V’p replaces the tendency equation for the horizontal divergence D. However, if we consider the full equation, then

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

111

I

(d,ldt)D

= f <-

I

229

111

V2p - (U.r)2

(2.4)

The approximations discussed in the preceding subsection consider the divergence tendency equation (2.4) to consist of only the first two terms on the right-hand side, namely, f <= V 2 p .If the rapidly growing vorticity predicted by the semigeostrophic frontogenesis model causesf< to exceed V 2 p , then (2.4) with the tendency term included predicts a decrease in convergence (D becomes more positive), thereby reducing the vortex stretching and helping to limit the unbounded vorticity growth. In summary, while frontogenetical models are able to explain the rapid frontal intensification, it is still questionable what factors maintain fronts in an approximate steady state for several days, as often is observed in nature.

3. MATUREFRONT

Historically, fronts have been viewed as surfaces of discontinuity that separate air masses of different densities. Even in more recent times, the concept of a front as a discontinuity continues to be important, as indicated by the prevalent belief that frontogenetical solutions can only be successful if a discontinuity develops in a finite period of time. In retrospect, Williams’s (1967) solution of frontogenesis is seen to have been more successful than was believed at the time of its publication. In 1967 the failure of the frontal solution to produce a discontinuity because of limitations of the finite-difference technique was viewed as a major shortcoming, even though very intense gradients developed in the unstable Eady wave.

3.1. The Significance of Frontal Collapse Why have researchers considered it necessary for fronts to collapse to a discontinuity in a finite time? First of all, it is important that the time scale of the generation of sharp frontal gradients be shorter than the time scale of the baroclinic waves that drive them. In addition, intense gradients imply frontal scales that are much smaller than planetary scales. Accordingly, the time when the solution becomes discontinuous at the surface gives a well-defined, albeit artificial, limit to the frontogenetical solution.

230

I . ORLANSKI. B. ROSS, L. POLINSKY. A N D R . SHAGINAW

Hoskins and Bretherton (1972), Pedlosky (1979), and Blumen (1981) give a simple argument to explain the development of the discontinuity in the semigeostrophic system. From conservation of potential vorticity and the assumption of semigeostrophy (even for time scales shorter than a pendulum day, 2+i~/f), it was shown that surfaces of constant V have constant slopes in the x-z plane and that these slopes do not vary in time. They therefore conclude that the ageostrophic cross-front wind U, should be only a function of the along-front wind V at the surface; i.e., CJ, = F ( V ) . Then the surface convergence U,, should be proportional to vorticity V , since UaX= F V ( V ) V , . In this case the term -50 in the vorticity equation (2.3) will be the positive quantity -t2FV assuming F v < 0; therefore vorticity growth will be unbounded. Theoretical studies of other nonlinear systems (Witham, 1974; Boyd, 1980) have demonstrated that nondispersive waves can develop discontinuities in finite times. What factors determine the conditions under which geophysical fluids will develop similar discontinuities? In its simplest form, the shallow-water equation system can be shown to produce wave breaking due to differential advection of the wave crest and trough (Boyd, 1980). A simplified set of equations, relevant to the surface front, can be obtained from the inviscid momentum equations on a flat surface (so that w = 0): (dldt)U, - fV,

0

(3.1)

(dVldt) + fU,= 0

(3.2)

=

and

Solutions to (3.1) and (3.2) can be determined for two different limits. First, if we assume V , = 0 (i.e., V = V , ) , then (3.1) reduces to (dU,ldt) = (aU,lat)

+

U,(aU,ldx)

=

0

This is the equation studied by numerous authors, e.g., Witham (1974); it produces a discontinuity in a finite time. In the other extreme, we may assume that V = V , so that V is completely nongeostrophic. A complete steady-state solution can then be found to this nonlinear system, namely, U * ( X ,t)]”*+ C arcsin

~

U(X, t ) UO

V

where

=

t(X

- ct)f

(3.3a)

T [ U $- U2(x,t)]’” (3.3b)

ADVANCES I N THE THEORY OF ATMOSPHERIC FRONTS

U’

23 I

V'

-2000

-1000

0

1000

2000

X' (km)

X FIG.2. (a) Distribution of across-front velocity U' and along-front velocity V' at a height of 0.5 km after 5 days of integration of a primitive equation, 2-D frontal solution. [After Williams (1967). From Journal of lhe Atmospheric Sciences, copyright 1967 by the American Meteorological Society.] (b) Distribution of quantities U and V from equation (3.3) described in the text. Primed quantities are dimensional, while unprimed variables are nondimensional.

The tendency d l d t has been replaced by C(dldx), with C a constant propagation speed.* Equation (3.3) (plotted in Fig. 2b) shows a remarkable similarity to the fields at z = 500 m (Fig. 2a) from Williams's (1967) solution. Since this solution represents a nonlinear steady-state result like a hydrodynamic bore propagating with constant phase speed C and does not exhibit collapsing characteristics, one would expect that Williams's frontal solution would have gone to this steady solution, rather than to the limit of frontal collapse, if numerical inaccuracies were not present. Some solutions by Houghton (1969) of the shallow-water equations with rotation are also very relevant to this discussion. These equations were found to exhibit a collapsing character if inertial waves are suppressed; i.e., f = 0. Inclusion of the dispersive ageostrophic effects

* If C >>

U,, the solution reduces to a linear inertial wave.

232

1. ORLANSKI, B . ROSS, L. POLINSKY, A N D R. SHAGINAW

when rotation is present leads to a more balanced solution such as (3.3). One may then conclude from the preceding simple models that the occurrence of discontinuities in a finite time is largely the result of approximations applied to the primitive equations. These approximations apparently can eliminate dispersive effects such as inertial waves that tend to prevent frontal collapse.

3.2. Dynamic Balance in a Mature Front The possible frontolytical effect of ageostrophic processes is an important element in our understanding of the dynamics of mature, quasi-steady fronts. Such ageostrophic mechanisms provide an alternative to the widely held view that discontinuities can only be prevented through the inclusion of diffusive effects (Williams, 1974). As explained in Section 2, the full divergence tendency equation (2.4) shows that as vorticity increases due to vortex stretching in areas of low-level convergence, negative feedback produces divergence tendency, thereby reducing convergence and vortex stretching. This effect thus produces a phase shift between the vorticity and divergence maxima and reduces the need for viscous dissipation to damp vorticity growth (Orlanski and Ross, 1984). When considering the dynamic balances in a mature front, it is important to consider not only the dynamic response of the front to an imposed large-scale deformation field, but also the intensity of the synoptic-scale forcing itself. For example, although the two-dimensional front models treated numerically by Williams (1967) and Orlanski and Ross (1977) appear to have many similar features, Williams’s solution experiences continuous frontogenetical growth, whereas the front in Orlanski and Ross’s model evolves into a mature, quasi-steady state (Fig. lb) after an initial period of frontogenetical adjustment. These dramatic differences are explainable by differences in the configurations used to initialize each model. The Eady wave used in Williams’s initial conditions is unstable, even in its finite-amplitude state. The stability criterion for this Eady wave is dependent on the horizontal wavelength in that Eady waves are neutrally stable when their wavelength is less than roughly 2000 km, as is the case in the solution of Orlanski and Ross (1977). Accordingly, Williams’s synoptic-scale disturbance is constantly intensifying, so one would expect the front embedded in this field to exhibit a similar growth. On the other hand, the fact that the Orlanski-Ross front remains effectively steady in a neutral wave demonstrates that finite-amplitude, slowly growing baroclinic waves can support frontal regions in a quasi-

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

233

steady configuration. Many of the long-lived fronts observed in the real atmosphere may fall into this category. Standing issues that still remain to be resolved regarding mature fronts are (1) what dynamic balance between frontogenetical and frontolytical processes occurs in realistic long-lived fronts, and (2) what determines the characteristic scale of such fronts? We will attempt to answer the first later in this chapter. Regarding the latter question, it is well known that the aspect ratio between the depth and width of a surface front is of order N l f , where N is the Brunt-Vaisala frequency. Stated in a slightly different way, the frontal width is the order of the Rossby radius of deformation, while the height is determined by the depth of penetration of the front (Orlanski and Polinsky, 1983). Figure 3 shows a comparison among the analyzed vertical structures of three different observed fronts. The moderately intense cold front analyzed by Ogura and Portis (1982) (Fig. 3a) has a horizontal scale near the surface on the order of 200 km. Sanders’s (1955) front (Fig. 3b) is considerably more intense, with horizontal scales of 50 km or less. Finally, the time-height cross section of potential temperature shown in Fig. 3c is taken from microwave radiometer measurements described by Decker (1984) during the passage of a cold front aligned perpendicular to, and east of, the Rocky Mountains. The frontlike structure shown exhibits two horizontal scales, the wider one passing in a time of order 50 min and the narrower one in about 4 min. The fronts shown in Figs. 3a and 3b are more classic in their structure and horizontal scales, while the “front” in Fig. 3c is similar to the “Southerly Buster” described by Baines (1980) as a Kelvin jet flow that forms ahead of cold fronts in the presence of orography. These three fronts show the wide range of scales exhibited by fronts observed in nature.

4. WHATOBSERVED FEATURES CAN BE EXPLAINED BY THEORY? The extensive observational studies of the Bergen School that confirmed the existence of surface fronts did not, of course, reveal the structure of the cross-stream circulation associated with frontal surfaces. In fact, this circulation was inferred from quasi-geostrophic theoretical considerations by Sawyer (1952, 1956) and Eliassen (1959, 1962) rather than from the sparse upper-air data existing at that time. This theory indicates that whenever the horizontal temperature gradient increases in

0

200

400

600

800

lOOOkrn

10000

200

400

s

4000-

z

5 3000I

2000-

1611r

800

304

5000 -

2 -i

600

302

1000 km

------

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

235

the front as a result of differential advection in a horizontal deformation field, a thermodynamically direct vertical circulation must develop in order to produce a corresponding increase in the vertical wind shear. It is through this vertical circulation that the mesoscale and synoptic-scale features of the frontal system interact. The horizontal scale of the frontal waves discussed in Section 1 is an order of magnitude larger than the mesoscale features of the cross-stream frontal circulation. It is only recently that the use of dense observing networks has enabled scientists to determine simultaneously the threedimensional structure of the wind and temperature fields and thereby to obtain more evidence of the processes that act to produce, maintain, and ultimately destroy the sharp temperature gradients within frontal surfaces. Nevertheless, since the atmospheric front is one of the most prevalent features of extratropical weather maps, it is surprising that only a few attempts have been made to examine in detail these frontogenetical and frontolytical processes in actual observed fronts. An early effort to do this was made by Sanders (1955) in a case study of an intense surface cold front. His careful analysis of data from a rather dense observing network indicated that air that enters the front from the warm air zone near the surface experiences frontogenetical intensification one to two orders of magnitude larger than typical in the free atmosphere. As this air moves up the isentropes within the frontal zone, frontolytical effects act to weaken temperature and wind gradients, thereby producing the typical frontal structure in which gradients are maximum close to the surface. Because of the apparently realistic behavior of the Hoskins-Bretherton (referred to here as HB) frontogenesis model, Blumen (1980) has made a one-to-one comparison of the predictions of this idealized model and Sanders’s analysis. He found qualitative agreement regarding details of the horizontal wind and temperature fields. Major discrepancies were evident, however, in the details of the vertical frontal circulation itself. In particular, the vertical velocity, which was most intense at the middle levels of the model, was less intense and broader in scale than the narrow rising jet that Sanders’s analysis showed to occur above the zone of maximum cyclonic vorticity within the surface front. As Blumen has pointed out, this narrow jet produces the most significant frontogenetical effects in the observed front by vertically tilting isentropic surfaces and isopleths of the horizonal long-front wind. An explanation of the discrepancies between model and observations was demonstrated by Blumen to be due to the absence of a boundary layer and turbulent mixing processes within the model. More recently, observations from the dense upper-air network of the 1979 SESAME/AVE field experiment have permitted Ogura and Portis

236

1. ORLANSKI. B. ROSS, L. POLINSKY, AND R. SHAGINAW

(1982) to produce a more coherent picture of observed mesoscale features of a cold front. This study analyzes a cold front that passed through the SESAME network on 25-26 April 1979. Although severe storms formed along the surface front during this period, the Ogura-Portis analysis of frontogenetical and frontolytical effects in the cross-stream circulation is limited to adiabatic processes. This analysis indicates a direct cross-stream vertical circulation, with moist air ascending above the surface front but with the upgliding flow within the frontal zone encountering a secondary circulation at midlevels. Also the horizontal temperature gradient and vertical vorticity are maximum near the ground, as predicted by the HB model and observed by Sanders (1955). The fact that the horizontal temperature gradient is smaller in the warm, rather than the cold, sector is also stressed as agreeing with the HB model prediction as calculated by Blumen (1980). The theoretical model fails to predict other important observed features. Specifically, the observed horizontal convergence and cyclonic vorticity are the same order of magnitude and are concentrated in zones of similar widths (approximately 300 km). Also, as pointed out by Blumen’s comparison with Sanders’s analysis, the vertical ascending motion analyzed by Ogura and Portis is located at low levels within the front, rather than the middle levels as the HB model predicted. A close comparison of the adiabatic frontogenetical functions was done only with Sanders’s results. Both sets of results are qualitatively similar although the magnitudes of Sanders’s terms are an order of magnitude larger because of his more intense front. Only the general structure of the vertical circulation was compared with the HB results. In summary, both Blumen’s analysis of Sanders’s results and Ogura and Portis’s analysis show qualitative agreement in coarser observed frontal features with the HB model results but fail to show agreement in important details such as the structure of the vertical motion field and the effect of frontogenetical and frontolytical forcing. Even the inclusion of Ekman pumping in the planetary boundary layer (Blumen, 1980) was unable to produce a realistic vertical jet as is observed. Such disagreement is not surprising in view of the simplifying assumptions of sernigeostrophy and two-dimensionality in the model and the absence of realistic processes such as moist convection and turbulent mixing. The important question to be addressed is which of these missing elements is needed to make the theoretical simulation of the cold front more realistic? This question points out the serious gap that remains in our understanding of the processes that govern frontal dynamics.

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

237

5 . WHATOTHERPROCESSES AREIMPORTANT I N FRONTOGENESIS? In attempting to include all of the processes that are important in frontogenesis, one realizes that it is no longer possible to employ a simple idealized frontal model. Some of these processes that are regarded as important are the effect of moisture on the frontal environment, the interaction of convective systems with the frontal circulation, and the influence of the planetary boundary layer. These phenomena are notoriously difficult to include as simple parameterizations (Rao, 1966; Orlanski and Ross, 1977; Ross and Orlanski, 1978; Blumen, 1980; Keyser and Anthes, 1982). In addition, the most widely used assumption of all, the approximation of two-dimensionality , may also be questioned. Today, with the availability of high-speed computers, we now have the capability to produce model simulations of observed fronts with most of these important processes included. Recent simulations of the evolution of a cold front in a moist environment (Ross and Orlanski, 1982; Orlanski and Ross, 1984) have produced realistic mesoscale features such as the presence of dual updrafts similar to those analyzed in observed squall lines. Unfortunately, in the case study described by Ross and Orlanski, observations could not be analyzed in sufficient detail to warrant an intercomparison with the cross-stream circulation in the model solution. In fact, the literature contains very few detailed comparisons of mesoscale model results with observations. Recent observational experiments using dense observing networks [e.g., Severe Environment Storm and Mesoscale Experiment (SESAME)] now can provide a more complete picture of the mesoscale structure of fronts. The existence of these observational data sets and the availability of three-dimensional mesoscale numerical models offers us the opportunity for a detailed comparison of modeled fronts (including frontogenetical processes) with their observed counterparts. Therefore, we will present in this section a comparison of a new mesoscale numerical simulation with the analysis by Ogura and Portis (1982) of the SESAME cold front of 25-26 April 1979.*

5.1. Structure of the Cold Front The formulation of the model used here has been described in detail by Ross and Orlanski (1982). The specific form of the model, including recent * Anthes et al. (1982) have also provided a numerical simulation of this case; however, they did not attempt to compare their results with Ogura and Portis’s results.

238

1. ORLANSKI, B . ROSS, L. POLINSKY. AND R . SHAGINAW

modifications, is described by Orlanski and Polinsky (1984). The model is initialized at 0000 GMT 25 April 1979from the GFDL/FGGE (First GARP Global Experiment) global analysis over a domain covering the eastern two-thirds of the United States. Boundary data, only used locally where inflow occurs, are obtained by the same initialization procedure, using the GFDL/FGGE analysis at synoptic times (0000 GMT and 1200 GMT). Due to space limitations, we will confine our discussion to those features of the observed front described by Ogura and Portis (1982).* Our primary intent is to make a detailed one-to-one comparison of model results with their analysis. The broad features of the observed front, such as frontal position, geopotential height, and wind field structure, are well simulated by the model. In fact, a close comparison of the mesoscale solution with either the GFDL/FGGE or the National Meteorological Center (NMC) operational analysis would not be appropriate due to the great disparity between model and observing network resolutions. On the other hand, the OP analysis of the dense SESAME network observations provides a more useful test of the validity of the current mesoscale simulation. As in OP, we will confine our discussion to the time 0200 GMT 26 April 1979, which is 26 hr into the model integration. Apparently, this time was chosen by Ogura and Portis because it corresponded to the beginning of severe convection along the cold front. The vertical component of the surface relative vorticity in the simulation [Fig. 4a (OP8)] shows the same frontal alignment as that observed. A cellular structure appearing on the analyzed vorticity field is far less pronounced in the simulation. General agreement is found between strong temperature gradients, vorticity [Fig. 4a (OP8)], and surface convergence [Fig. 4b (OP9)]. An important finding of OP was that the analyzed surface maxima of convergence and vorticity were of roughly the same magnitude. This was a major discrepancy between OP’s analysis and idealized theoretical models, which predicted much weaker convergence compared to vorticity. Although the present model simulation shows good agreement with OP in the magnitude of the convergence, the simulated vorticity maximum is twice as large. The absence of precise mesoscale observations regarding the position of the front and its associated convection requires that some aspects of the model results be compared with detailed satellite photographs. To a first approximation, the location and column-integrated content of liquid water in the model solution can be related to corresponding satellite imagery. Figure 5 (OP20) provides such a comparison. Allowing for

* This paper will also be referred to here as OP. Their figures will be designated by OP followed by their figure number.

, -

B

FIG.4. (a) Horizontal distribution of vertical vorticity (units of s - ’ ) at height z = 0 m from frontal simulation at 0200 GMT 26 April. Region where surface is above 500 m height is not contoured. (b) Distribution of horizontal divergence (units of 1O-j s) at height z = 0 m. Line segment A-B indicates cross section used in later figures.

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

24 I

differences in mapping between satellite and model results, one can see good agreement in position for the two major cloud zones, labeled C and D. Cloud system C is associated with the frontal system under discussion here. Locations of intense precipitation zones (not shown here or in OP) show good agreement for the two storm systems C and D. The structure of the frontal circulation is shown in Fig. 6 (OP17, OP21) in a vertical cross section perpendicular to the frontal system along the line AB of Fig. 4b. The position of this line is similar to but several hundred kilometers north of that chosen by OP. Striking similarities and some differences are evident between this structure and OP’s results. The upper-level jet maximum, analyzed by OP to be 40 m s-I, is only 30 m s-l

1

1

N

-500

0

x

500

(km)

FIG.6. Vertical cross section (line A-B of Fig. 4b) of along-front velocity V (solid contours) and potential temperature (dashed contours) from simulation at 0200 GMT 26 April. Vectors denote cross-front velocities. Stippled region indicates V > 20 m s-I. Cross-hatching indicates V < 0 m s-I.

242

1. ORLANSKI. B . ROSS. L. POLINSKY. A N D R . SHAGINAW

here. (The GFDL/FGGE analysis shows only 27 m s-l.) Nevertheless, the region in which winds exceed 20 m s-I (stippled zone in Fig. 6) is similar in extent to that of OP. The similarities between the two fields for the lowest 5 km are remarkable. Predominant features are a northerly surface jet (at 950 mb or 500 m) to the left of the strong temperature gradients and a southerly low-level jet ahead of the front (at 850 mb or 1500 m). The southerly jet ahead of the front has been frequently observed and was speculated by Keyser and Anthes (1982) to be the result of planetary boundary layer effects. We would suggest, however, that the presence of moist convection could also play an important role, since a similar solution produced without moisture for the present case did not show this low-level jet. The circulation within the plane of the cross section, indicated in Fig. 6 by vectors, includes the removal of a IO-m-s-’ rightward translation, corresponding to the assumed translation of the cold front (as in OP21). Similarities with observations are apparent, including inflow of warm air into the front at low levels, upgliding motion on an incline steeper than the isentropes, and sinking motion in the cold air to the rear of the front. A primary difference is the simulated weak downward motion ahead of the upper-level jet, which does not appear in the OP analysis. This sinking motion appears to be associated with the low-pressure system to the southeast of this domain (cloud system D in Fig. 5). Conceivably, the OP analysis of vertical motion was unable to capture this feature. Distinguishing features of the frontal zone as a separate entity from the larger baroclinic system are clearly evident in the vorticity, the vertical velocity, and the magnitude of the potential temperature gradient as shown in the vertical cross sections in Fig. 7. The vertical component of vorticity [Fig. 7a (OP18)] has a similar structure to observations. Two maxima of cyclonic vorticity are identifiable, one due to the upper-level jet and the other associated with the surface front. The magnitude of the calculated frontal vorticity is double its analyzed value, as mentioned in the discussion of Fig. 4a. Although such differences might be considered to be acceptable, we believe that such disparities may reflect the coarseness of the observing network used by OP. The vertical velocity field [Fig. 7b (OP19)I shows dual updraft maxima similar to the observed maxima in OP’s figure at 650 and 950 mb. Although direct quantitative comparison between simulated and observed fields of upward motion is difficult because of differences in analyzed quantities (velocity w in z coordinate versus velocity w in p coordinate), a rough comparison indicates the model upward motion to be twice as intense as its analyzed equivalent. The sinking motion to the rear of the

ADVANCES I N THE THEORY O F ATMOSPHERIC FRONTS

243

x (km) X (km) FIG.7. Comparison of fields in cross section A-B (Fig. 4b) for (a) vertical vorticity m s - l ) , (c )water vapor (solid contours, gm kg-') and tempers-l), (b) vertical velocity K ature (dashed contours, "C), and (d) horizontal potential temperature gradient lV&l m-I). Stippled areas indicate positive fields in (a) and (b) and values greater than 24 x K m-I in (d). Heavy dashed line indicates lines of maximum vorticity. front agrees with observations. Ahead of the front, the simulation shows sinking motion of less than 2 cm S K I ,whereas the analysis shows slightly larger intensity and more variability. Note that the position of the frontal zone, defined as the line of maximum cyclonic vorticity and indicated in Fig. 7 by a heavy dashed line, shows a vertical penetration (to 5 km) similar to that determined from observations (see OP18 and neglect that portion of the vorticity

244

1. ORLANSKI. B . ROSS, L. POLINSKY, A N D R. SHAGINAW

maximum line that is associated with the upper-level jet). The position of the strong upward motion occurs to the right of this vorticity line (Fig. 7b). Finally, the cross section of water vapor mixing ratio and temperature [Fig. 7c (OP16)l shows reasonable agreement with observations. The region of largest horizontal potential temperature gradient IvH81, shown by stippling in Fig. 7d, overlaps the line of maximum vorticity. A point we will stress below in comparing dry and moist solutions is that the slope of the vorticity line as well as the region of large IvH81 are much steeper than the potential temperature contours, as seen in Fig. 7d. This can also be seen in the vectors above the surface front in Fig. 6. An analysis of this solution indicates that fluid parcels in the moist environment will tend to follow lines of constant equivalent potential temperature in the same manner as they follow constant-8 lines in the dry case. This conclusion that the moist environment can directly change the circulation of the front is an important result.

5.2. Frontogenetical Terms To this point in the discussion, emphasis has been given to the extent to which the 26-hr simulation results agree with the detailed observational analysis of OP. However, one of the goals of this study will be to utilize the completeness of this numerical solution to provide insight into the role of the various frontogenetical and frontolytical forcing mechanisms in the maintenance of the front. To this end, the analysis of those terms that enhance or weaken potential temperature gradients may provide us with clues as to which processes are most important. Following Miller (1948), Sanders (1955), and others, one may derive a prognostic equation that describes the time variation of the magnitude of the horizontal gradient of potential temperature IvH81. A frontogenetical function may then be defined as (d/dt)lVHBl,the time rate of change of \VH8/following a fluid parcel. This is similar to the leftmost term of (2.1). The basic equation for the conservation of potential temperature may be differentiated in the horizontal to obtain an expression for this frontogenetical function:

dt

pHel = CONVERGENCE + DEFORMATION + TILTING

+ DIFFUSION + DIABATIC

where CONVERGENCE

-

1 21vHe)

-(8: +

OW, + VJ

(5.1)

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

TILTING

e

- -[O,W,

245

+ 8,~,]

IvHel

DIFFUSION

1 -[VH IvHO1

and DIABATIC

1

-[VH~

OH]

lvHOl

The quantity F is the diffusion term in the original prognostic equation for 0, and H is the heating due to condensation/evaporation. Both Sanders (1955) and OP have attempted to evaluate the first three (adiabatic) terms on the right-hand side of (5.1) for observed fronts. Also, Blumen (1980) has evaluated these adiabatic terms for the idealized Hoskins-Bretherton model of a dry front. We feel that it is important to clarify the role of the other terms, namely, the diabatic and the diffusive terms, in the context of our observed front simulation.* To make the effects of moisture more apparent, we will also compare terms in the complete moist solution with those from a solution in which moisture is excluded. Figure 8 shows a comparison between the dry and moist solutions of the convergence, deformation, and tilting terms, t which are the only contributors to the frontogenetical function in the term-by-term analyses of the three papers mentioned earlier. The moist terms in Fig. 8 should be compared to those shown by OP (OP25) for the same time. In order to provide an uncluttered view of the structure in both the moist and dry cases, we use different contour intervals (as indicated in the upper-right-hand corner of each frame) in each case. An inspection of Fig. 8 reveals that corresponding terms for the dry and moist cases have the same signs at low levels in each case. Specifically, convergence and deformation are both frontogenetical, while tilting is frontolytical near the surface. At these levels, the magnitudes of the moist terms are three to four times larger than the corresponding dry terms. The similarity between terms ends above 1.5 km in conjunction with the

* All terms shown here were averaged over 30 min (I5 time steps) of model integration. t Palmen and Newton (1969, p. 261) have also noted the potential importance of diabatic effects that could not be analyzed in Sanders’s (1955) study. (The authors thank Dan Keyser for pointing out this reference to them.)

DRY I

CONVERGENCE ,

,

,

,

CNT-2

8-

MOIST

, ~

76-

N 4

1~

,-2, -500

,

,

/b

,

0

,

,

/

--

/ 11 , ~

500

-500

0

500

X (kmi

X (km)

DEFORMATION

-E Y N

-500

0

500

X (km) TILTING 87-

6

T5

Y N

A

3 2 1

-500

0

X (km) X (km) FIG.8. Comparison of adiabatic terms from dry and moist simulations at 0200 GMT in vertical cross section A-B (Fig. 4b). Contour interval for each frame is shown in upper right corner in units of K m-I s-l. Cross-hatching and stippling indicate terms less than - 12 X K m-I s-l and more than 12 X K rn-’ s-I, respectively.

ADVANCES I N THE THEORY O F ATMOSPHERIC FRONTS

247

separation of the dry and moist vorticity maximum lines (dashed lines, defined as in Fig. 7). In fact, the shallower slope of the vorticity lines in the dry, compared to moist, solutions is another indication of the tendency of the fluid parcels to follow isentropes in the dry case and isolines of equivalent potential temperature in the moist case. Certainly, deeper penetration of the front is a dominant feature of the moist solution. Comparison of the moist terms in Fig. 8 with those of OP (OP25) show very good agreement with regard to sign, vertical penetration, and general structure. The increased intensity of the simulated terms in comparison with the analyzed terms is consistent with the apparent trend for modeled gradients to be more intense than analyzed ones. The results clearly indicate the tilting term to be the largest by far of the three moist adiabatic terms above the surface. As shown by OP, it is strongly negative (frontolytical) at a position along and slightly to the left of the vorticity maximum line. Figure 7b shows this to be the case because w, is maximum here (while 8, > 0). Since tilting dominates the other two adiabatic terms in this region, the frontogenetical function as the sum of these three terms alone in OP is quite negative, implying that strong frontolytical processes dominate here. Previous model simulations (Orlanski and ROSS, 1984) indicate that in moist convection the vertical advection of potential temperature (adiabatic cooling) is roughly compensated by latent heat release due to condensation. From that result, one would expect that in the presence of moist convection the tilting term (being derived from the horizontal gradient of this vertical advection) would be largely compensated by the corresponding diabatic term in (5.1). In other words, if the tilting term produces a frontolytical effect, the diabatic term will produce an opposing frontogenetical effect. The diabatic term, as calculated from the model solution (Fig. 9a), shows a similar structure to that of the tilting term in Fig. 8. As expected, large frontogenetical effects occur ahead of the vorticity maximum line. The sum of the tilting and diabatic terms is shown in Fig. 9b. Strong frontolytical effects occur on the cold side of the front, while weak frontogenesis is evident in the middle levels on t h e warm side ahead of the front. Diffusion (Fig. 9c) produces frontolysis which reinforces that shown in Fig. 9b on the cold side of the front. Finally, with these additional terms now available, we are able to address the question of what terms contribute to frontogenesis and frontolysis. The substantial derivative of IVHOl, shown in Fig. 9d, is the true frontogenetical function.* A strong frontolytical region exists on the

* This term is computed independent of the other terms. The sum of all terms produces a K m-’s-I. residue of only 16 x

248

I. ORLANSKI, B . ROSS, L. POLINSKY, A N D R . SHAGINAW

8 7

6

6

-E

T5 Y N

5

Y

4

N 4

3

3

2

2

1

1

-500

0

-500

500

0

500

N

-500

0

500

X (km) Xikd FIG.9. Comparison of nonadiabatic terms and substantial derivative term in units of K m-I s - I with contour interval of 12 x lo-"' K m-I s-l. Cross-hatching and stippling K m-I s I and more than 12 x lo-'" K m-I s-l, indicate terms less than -12 x respectively. (a) Diabatic, (b) diffusion, (c) diabatic and tilting, and (d) substantial derivative.

cold side of the front at middle levels, with tilting and diffusion being the largest contributors to the frontolysis. Regarding the intensity of this frontolytical effect, however, one should recognize that its large magnitude is due, in part, to the fact that IVH81 is generally small in this region (Fig. 7d). Two regions of frontogenesis appear on the warm side of the vorticity line, one near the surface and the other at around 5 km. The mechanisms for surface frontogenesis are convergence and deformation. The frontogenesis that occurs above 2 km seems to be largely due to the excess of the diabatic term over the tilting term (Fig. 9b).

ADVANCES IN THE THEORY OF ATMOSPHERIC FRONTS

249

These results are consistent with the differences evident between the moist and dry solutions above the low levels. They also seem to be consistent with our intuition that a moist frontal solution will have a deeper structure than a dry one. The accepted view has been that this deep circulation is associated with the moist convection ahead of the front and that the frontal circulation itself is not significantly altered by convection. In fact, the simulation indicates that the entire frontal circulation changes in a moist environment. In a dry environment, the main circulation of the front follows isentropic surfaces, whereas in a moist (saturated) environment, parcels follow moist pseudoadiabats (i.e., surfaces of constant equivalent potential temperature), thereby modifying the entire cross-stream circulation. The steeper slope of the frontal structure in the moist case is the result of the weaker static stability of the moist environment. The present discussion can be extended to attempt to answer the question posed by Charney, mentioned at the beginning of this paper, as to how surface frontal waves are connected to upper-level baroclinic waves. We know how finite-amplitude waves produce frontogenesis and that the resulting surface fronts can be unstable (Eliasen, 1960; Orlanski, 1968). Now we can envision that, in a moist atmosphere, after surface frontal waves have developed due to secondary instabilities, they are able to communicate with the upper levels of the troposphere through the moist diabatic processes. This communication is achieved through the mechanism whereby short unstable frontal waves modulate areas of moist convection, thereby modifying the moist environment in the upper levels. We have shown above that the frontal circulation is not only modified directly by convective systems, but also feels the influence of the less stable moist environment. The current level of our understanding of fronts provides us with considerable confidence in our understanding of the mechanisms governing the generation and maintenance of atmospheric fronts. On the other hand, there is a whole range of unanswered questions regarding secondary effects of fronts: How do fronts interact with orography and how are they associated with lee cyclogenesis? What conditions determine the type of moist convection, such as the many different kinds of rainbands, that fronts produce? How do fronts affect mesoscale convection in general? What role do fronts play in the generation of mesoscale convective complexes, comma clouds, and coastal cyclogenesis? A review of the progress made in these areas in recent years is beyond the scope of the present discussion. There is, however, considerable evidence that many mesoscale convective systems develop in association with a preexisting frontal system (Orlanski and Polinsky,

260

YOSHIO KURIHARA

enough to adequately resolve the hurricane vortex and, if feasible, its inner structures. However, to cover an entire integration domain with a fine mesh is not practical. Under these circumstances, we developed a movable nested mesh system in which the inner meshes of fine resolutions are telescopically nested. The positions of these inner meshes are relocated as the hurricane moves, so that the vortex center is always found near the centers of the inner meshes. The above system can easily be reduced to a uniform grid configuration. In making the time integration of the nested mesh model, different time steps are used for the meshes with different grid spacings. Listed in Table I1 are specifications of some of the grid systems that have been used in our model. A single-mesh system was used in the simulation of tropical storm genesis, a triply nested mesh model in the hurricane landfall experiment, and a quadruply nested mesh model in the study on the hurricane eye. When performing the numerical simulation experiments in a research mode, we usually specify rather simple idealized initial conditions. This enables us, at least mathematically, to distinguish between the basic state and the perturbation fields at the initial time. Such a distinction is more or less ambiguous when observed data are used for the model initialization. To carry out the numerical integration of the model in a research capacity, we have treated the lateral side of the computational domain as either open or closed boundary depending on whether a basic flow field was present or not. In the following section, some of the results are presented that we obtained in numerous experiments conducted at the GFDL.

TABLE11. SPECIFICATIONS OF SOME GRIDSYSTEMSa ~

Grid spacing Grid system

East-west

North-south

Domain size (grid number)

Time step (s)

Uniform mesh Triply nested mesh Mesh 1 Mesh 2 Mesh 3 Quadruply nested mesh Mesh 1 Mesh 2 Mesh 3 Mesh 4

8 (“long.)

8 (“lat.)

40 x 40

90

1 (“long.)

1 (“lat.)

t (“long.)

4 (“lat.) 4 (“lat.)

45x37 33 x 33 22 x 22

I50

f (“long.)

90 cosine (km) 30 cosine 4 (km) 10 cosine (km) 5 cosine 6 (km)

90 (km) 30 (km) 10 (km) 5 (km)

41x41 36x36 36x36 32x32

126 42 14 7

+

50 25

The grid spacing, domain size, and time step used for the time integration of the models are listed. b stands for latitude.

NUMERICAL MODELING OF TROPICAL CYCLONES

26 1

3. NUMERICAL SIMULATION OF TROPICAL CYCLONES 3.1. Tropical Storm Genesis The evolution of tropical storms from easterly waves has been studied with the GFDL hurricane model. We found that the same initial wave disturbance can be transformed into either a developing storm or a nondeveloping system, depending on both the environmental and the surface conditions. Figure 3 shows the 4-day sequence in a case of tropical storm development that took place in a simulation model. It shows that as the minimum sea-surface pressure decreases, the pattern of flow at the model level 8 (-930 m) changes from that of a wave trough to a distinct vortex. The analysis results of the experiments performed under many different conditions indicate that the evolution of a tropical storm from an easterly wave is strongly influenced by the following factors, roughly in the order of importance: (1) the moisture content in the planetary boundary layer, (2) the static stability of the air above the low-level disturbance, (3) the vertical profile of the basic flow, and (4) the horizontal profile of the basic flow. The condition (1) seems to be a crucial factor; the maintenance of sufficient moisture in the planetary boundary layer of an easterly wave is an almost necessary condition for the wave to be transformed into a tropical depression. In our numerical models for the tropics, a dry vortex with tropical storm intensity has never developed. It is reasonable to assume that the amount of water vapor in the boundary layer tends to be large over the ocean with high surface temperature. Observations indicate that the sea-surface temperature in the areas in which most of the tropical depressions develop is higher than 26-27°C. The threshold sea-surface temperature for storm development in our models was found to correspond well with the above value. In any case, to simulate the tropical storm genesis, the boundary layer of the model has to be adequately moist. The model integration starts with an idealized easterly wave that is superposed on an arbitrarily specified basic state. During the early stage of the integration, a broad region of low-level convergence is generaliy found at the trough region of the wave where the relative vorticity is positive. The moist convection occurs in the same area, sometimes with extension to the east of the trough. As the liberated latent heat causes warming of the air in middle troposphere above the surface trough of the easterly wave, a circulation is induced due to the solenoidal effect. This circulation consists of low-level conver-

t=O

t=24

I

t=48 t=72 I

1008

1006

rnb 1004

1002 I

I

I

I

48 24 72 96 hr FIG. 3. Time variation of minimum surface pressure (millibar) in a simulation experiment of a tropical storm genesis. Streamlines and isotachs (meters per second) at level 8 (-930 m) of the model at 0,24,48,72, and 96 hr are also shown. [Based on Kurihara and Tuleya (1981). From Monthly Weather Reuiew, copyright 1981 by the American Meteorological Society.]

NUMERICAL MODELING OF TROPICAL CYCLONES

263

gence, upward motion in the trough region, upper-level divergence, and subsidence of the air in the outer area. While the low-level air converges toward the wave trough region, it picks up moisture from the ocean during its travel from the surrounding area, enabling the moist convection in the trough region to continue. The low-level convergence also causes vertical stretching of the absolute vorticity. Thus the relative vorticity at the trough tends to increase. These processes continue when the warming of the air is sustained above the area of surface convergence. In this case, the warming of the middle to upper levels and the surface convergence together make a positive feedback mechanism linked by the moist convection. An important feature of such a mechanism is that the upper-level warming and the surface vorticity increase proceed concurrently to enable the released latent energy to be effectively preserved within the developing vortex system. Since the upper-level warming or the formation of a warm core implies a decrease of surface pressure, this feature suggests that a gradient wind balance exists between the wind and pressure fields at the low levels. When the model integration is extended, the trough of the easterly wave is progressively transformed into a tropical depression and then to a tropical storm. The preceding scenario of tropical storm genesis breaks down if the warm core is not formed above the low-level disturbance. Factors ( 2 ) and ( 3 ) (i.e., the static stability and the vertical shear of the basic flow) are both related to the formation of a warm core at the early stage of model integration. The upward motion takes place in the region of boundary layer convergence, and it acts to lower the middle-level temperature in stably stratified air. This tendency counteracts the effect of heating due to the moist convection. The more stable the air is, the weaker the net warming is for the same amount of condensation of water vapor. The net effect can even be negative. Thus, the static stability is a parameter that significantly controls the formation of the warm core. In addition, the static stability at low levels affects the vertical mixing of the momentum, heat, and moisture in the boundary layer, as well as their exchange at the surface. The static stability is determined partly through the radiative transfer process. It may be of interest to note that in the numerical experiment without the effect of radiation, the rate of decrease of the minimum surface pressure was reduced about 50% as compared with the experiment having that effect. As mentioned before, if the warm air remains above the low-level disturbance, then the solenoidal field effectively works to intensify the disturbance. This condition requires that the upper-level wind U , which advects the warm air and the propagation velocity c of the low-level disturbance, is about the same. Since the intrinsic phase velocity of the

264

YOSHIO KURIHARA

easterly wave is westward, i.e., the wave moves to the west without the basic flow, the magnitude of the quantity U - c becomes small when the basic flow has an easterly vertical shear; i.e., the easterly component increases with height. A series of experiments were performed in which the vertical shear of the basic flow was varied for each experiment while the same wave was specified at the initial time. The maximum surface vorticity, maximum surface wind, and central surface pressure of the disturbance after 96-hr integration in each experiment are listed in Table 111, which also includes the vertical shear, measured by U (150 mb)-U (850 mb), as well as a measure of the vertical coupling U (-335 mb)-c. It indicates that a moderate easterly shear is most favorable for the development of tropical storms in our model. Figure 4 shows the east-west vertical cross sections, passing through the disturbance center, of the wind field relative to the moving system at 48 hr for two cases of the experiment. The top part of the figure is for the experiment with westerly shear of 15 m s-' and the bottom part for that with easterly shear of - 15 m s-*. At 48 hr, the difference of surface pressure patterns between the two cases was not dramatic. However, a large difference is evident in the structure in the free atmosphere. In the easterly shear case, the deep radial-vertical circulation was already organized and accompanied with a warm core above the disturbance. In contrast, the warm core and the associated circulation did not evolve in the westerly shear case. It is clear that the upper-level warming is an important condition in the storm genesis. The horizontal shear of the basic flow [factor (4) in the earlier list] also affects the evolution of the disturbances. The vorticity associated with the horizontal shear modifies the effect of vorticity stretching. Our numerical results indicate that although the horizontal shear of basic flow is not necessarily required for the tropical storm genesis, the low-level cyclonic TABLE111. DISTURBANCES EVOLVED AFTER 9 6 - INTEGRATION ~ ~ UNDERTHE DIFFERENT VERTICAL SHEARS OF THE BASIC WIND.",^ Easterly shear

Westerly shear

U (150 mb) - U (850 mb) at the initial time (m s-I)

-22.5

-15.0

-7.5

0

7.5

15.0

Max. surface vorticity s-I) Max. surface wind (m s - * ) Min. surface pressure (mb) U (-335 mb) - c (m s-I)

170 16.8 1000.8 -6.7

365 22.5 993.5

317 22.2 997.1 0.5

175 16.2 1002.1 3.2

155 13.8 1003.5 5.4

63 10.0 1006.9 3.1

-3.0

From Kurihara and Tuleya (1982a). At the initial time, the maximum surface vorticity, maximum surface wind, and minimum surface pressure were 43x s - &9, m s-', and 1008.4 mb, respectively, in all cases. a

U

0.5

1 .o U

0.5

1 .o (b) west 5' center e a s t 5' FIG.4. Longitude-height distribution, through the disturbance center, of the wind relative to the moving system, at 48 hr, for the nondeveloped disturbance in a westerly shear case (a)

and the developed disturbance in an easterly shear case (b). Arrows show the zonal and vertical components, and isopleths indicate distribution of the meridional component (meters per second). The pressure normalized by the surface value is taken along the ordinate. I = 48 hr. [From Kurihara and 'I'uleya (1982a).I

266

YOSHIO KUKIHARA

shear and, to a lesser degree, the upper-level anticyclonic shear are conducive for tropical storm genesis. A schematic diagram, Fig. 5 , summarizes the factors involved in the genesis of a tropical storm within the easterly wave that were investigated by our model. The existence of sufficient moisture in the planetary boundary layer over a large area is always necessary for the tropical storm genesis. Other factors determine whether the environmental state is favorable or not for genesis. To formulate a simple criterion for the development of each individual disturbance based on these factors seems to be quite difficult since the sensitivity of the storm genesis to one factor can be considerably modulated by another. 3.2. Intensijication of Tropical Storms

A tropical storm may develop further into a strong vortex or hurricane. During the rapid intensification period, a positive feedback mechanism proceeds, which accelerates the radial-vertical circulation, strengthens FAVORABLE CONDITION

DETRIMENTAL CONDITION

upper levels

middle levels

planetary boundary layer ocean

I

high SST L

I

I I

l o w SST

reduces e v a p o r a t i o n

FIG.5 . Conditions that can influence the genesis of a tropical storm in an easterly wave.

I I

NUMERICAL MODELING OF TROPICAL CYCLONES

267

moist convection and builds up a distinct warm moist core. In the upward motion branch of the circulation, the conditional instability is neutralized and the static stability becomes moist neutral. It should be noted that since the intensification of a vortex is done through a feedback mechanism, a small change in a certain part of the chain of events can cause a significant change in the vortex intensity. For example, when the evaporation from the ocean in the small inner area of the vortex was suppressed in one experiment, a drastic weakening of the hurricane took place in less than a day. The water-vapor-budget analysis showed that the amount of evaporation in this area before being suppressed was negligible compared with the large, low-level, horizontal flux convergence of water vapor into the vortex. However, such large flux convergence was possible in the presence of the small evaporation. When the evaporation was suppressed, the warm core weakened slightly, which induced the weakening of the radial-vertical circulation and the associated moisture convergence in the boundary layer; thus further weakening of the warm core and hence of entire vortex system ensued. The sharp drop in the minimum surface pressure during an intensification period is in accordance with the formation of an intense warm core. The analysis of the mass budget indicates that the net loss of mass from the central part of the vortex was caused from a small difference between two big quantities, i.e., the large outflux of the air at the upper levels and the large boundary-layer influx. The strong inflow at low levels, upward motion in the inner region, and outflow at upper levels all contribute to the vorticity budget through the horizontal and vertical advection, stretching, and twisting of the vorticity . The net result is to cause contraction and intensification of the vortex. When the vorticity near the center increases to a large value, the role of horizontal mixing in the vorticity budget seems to become significant. At the middle levels in the troposphere, the mean azimuthal component of the wind in the vortex maintains gradient wind balance with the large pressure gradient. In the boundary layer, the balance is modified by the frictional effect as well as the inertia effect, i.e., the advection of the radial momentum. At the upper levels, the balance of forces is not simple because of a more complicated flow pattern.

3.3. Structure of Hurricanes The gross features of the intense, compact vortices simulated in the hurricane models were similar with each other, although the intensity and size differed from one vortex to another. Besides the grid resolution of the

268

YOSHIO KURIHARA \ 200 -

600 -

I

I

I

I

1

I

0 20 40 60 80 km FIG. 6. Height-radius distribution of mean azimuthal wind (meters per second) of a numerically simulated tropical cyclone. Dash-dotted line shows the ridge lines in the distribution. [From Kurihara and Bender (1982).]

model, the differences in the initial state and surface-boundary condition of the model apparently caused the structural differences among the mature vortices. Presented in Figs. 6 through 9 are some of the results obtained from the experiment using the quadruply nested mesh model shown in Table 11. In

0

20

I

40

I

I

60

I

1

80

I

km

FIG.7. Height-radius distribution of mean stream function ( lo7 m2 mb s ) of a numerically simulated tropical cyclone, indicating the mean radial-vertical circulation. The vertical velocity is zero along the dotted lines. The eye is indicated by shading. [From Kurihara and Bender (1982).]

"C 8. Vertical profiles of temperature at the points A (in the eye), B (in the eye wall, 20 km west of A), and C (in the environment, 75 km west of A) in a numerically simulated tropical cyclone. Relative humidities (in percent) are indicated by numbers along the curves. t = 46.55 hr. [From Kurihara and Bender (1982)l. FIG.

0.0 1

20

40 west

c

20

40 km east

FIG.9. West-east vertical cross section, through the eye, for the radial-vertical wind (arrow) and azimuthal wind (contour, meters per second) at one instant in the numerical simulation of a tropical cyclone. Areas with sinking motion are shaded. The pressure normalized by the surface value is taken along the ordinate. [From Kurihara and Bender ( 1982)l.

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YOSHlO KURIHARA

this case, a weak circular vortex was specified in a calm environment at the initial time. When the time integration of the model proceeded, an intense vortex developed with a compact eye or relatively warm and dry air column surrounded by an area of strong convection usually called the eye wall. The mean azimuthal wind and the mean radial-vertical circulation in the above vortex are shown in Figs. 6 and 7, respectively. Figure 6 indicates that the vertical wind shear is very large in the boundary layer. It is clearly shown in Fig. 7 that two distinct circulation systems exist: one within about a IO-km radius and a larger system surrounding it. The branch of sinking motion in the former system corresponds to the eye. The upward motion branches of the two systems constitute the eye wall. The large area of upward motion in the eye wall is connected to the upperlevel outflow of the outer circulation system. Vertical profiles of the temperature and the relative humidity within the eye and eye wall and in the outer environment are shown, respectively, in Fig. 8. The air within the eye is quite warm and dry, the temperature deviation from the surrounding environment being 14.5"C at 300 mb. Within the eye wall, the air is saturated above the boundary layer and the sounding curve coincides with a moist adiabatic ascent. In the outer region, the atmosphere is relatively dry and stable for dry adiabatic ascent. The hurricane vortex includes asymmetric features in addition to the axisymmetric structure just shown, and the former structure plays an important role for the maintenance of the latter. An asymmetric part of the wind, i.e., the deviation from the azimuthal mean, may be called the eddy motion. To illustrate the asymmetry of the wind field, the distribution of wind on a west-east vertical cross section passing through the vortex center is presented in Fig. 9. We clearly see that with the addition of the eddy contribution to the mean structure, an air parcel within the eye typically does not sink all the way along the stream line as shown in Fig. 7. Rather, it enters into the eye from the eye wall, where it tends to sink a short distance, and is then absorbed into the eye wall. Other analysis results indicate the presence of other important asymmetric features within the eye wall, including cells of strong convection that rotate cyclonically at a much smaller speed than the azimuthal wind within the eye wall. It was found that the mean circulation and the eddies played compensating roles in the maintenance of the simulated hurricane eye. The mean sinking motion within the eye caused the warming and drying effect. This tendency was counterbalanced in our model mainly by the effects of horizontal eddy mixing, i.e., the transport of heat from a relatively warm eye region to the eye wall and the moisture flux from the eye wall into the eye. Near the bottom of the eye and the eye wall, the eddy motion plays a role in balancing the budget of the angular momentum.

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3.4. Spiral Bands and Comrna Vortices Spiral bands and a comma-shaped band were also simulated in our numerical models. Figure 10 shows the distribution of the vertical motion at -3.3-km level obtained from an experiment in which a strong vortex and accompanying spiral bands developed in the absence of the mean basic flow. In this case, spiral bands with a width of about 100 km behaved like outward-propagating internal gravity waves, with the spiral pattern in the vertical motion field extending to almost 1000 km from the vortex center. Within the band, a nodal point was generally found at the middle

FIG. 10. Distribution of w (vertical p velocity, dpldt, in lo-) mb s - ' ) at level 6 (-3300 rn) at one instant in the model integration. Area of negative w is shaded. Center of spiral bands are indicated by dotted lines. [From Tuleya and Kurihara (1975). From Joouvnal of the Atmospheric Sciences, copyright 197s by the American Meteorological Society.]

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level in the vertical profiles of vertical motion and temperature perturbation. It seems that the spiral bands in our model were excited near the outer edge of the eye wall where a strong radial shear existed in the mean azimuthal wind. According to a simple analytical study, the most-preferred azimuthal wave number of the bands was two. Once the bands propagate away from the eye wall, they lose their energy source and decay through dissipation. Thus, the bands seem to develop as a result of local instability of the mean azimuthal flow. The contribution of the spiral bands to the energy and angular-momentum budgets in the outer region of a vortex were quite small compared with that of the mean radial-vertical circulation. An interesting feature observed in some of our hurricane simulation experiments is the dependency of the vortex structure on the direction of the mean basic flow. Figure 11 shows two vorticity fields near the top of the boundary layer. The first developed in a uniform easterly flow of - 5 m s - I and the other evolved from the same initial disturbance but in a uniform westerly flow of +5 m s-l. The pattern in the former case has no strong asymmetry, while that in the latter case appears stronger and exhibits a comma-shaped tail attached to the vortex. We performed several experiments to investigate the formation mechanism of this comma vortex. It was found that the horizontal advection of the absolute vorticity relative to the moving weak disturbance tends to produce a pair of vorticity tendencies, i.e., a negative tendency mainly due to the planetary vorticity advection to the east of the disturbance and a positive one located north of it. The appearance of such a tendency field is dependent on the

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FIG.1 1 . Distribution of level-9 (-440 m) vorticity (in S-I) for the experiments with uniform easterly flow of -5 m s-I (a) and with uniform westerly flow of +5 m s-' (b), respectively. Shaded areas indicate values greater than 5 x lo-' s-I, I = 96 hr. [From Tuleya and Kurihara (1981). From Monthly Weather Reuiew, copyright 1981 by the American Meteorological Society.I

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structure and intensity of the disturbance. If the pair of vorticity tendencies occurs, it induces confluence of the mean westerly flow into the middle of the pair, from the southwest to the northeast. On the other hand, the same tendency forces the mean easterly flow to diverge around the pair. Thus, in the case of the mean westerly flow, a broad zone of relatively strong wind is formed to the southeast of a vortex. Apparently, the enhanced air-sea interaction beneath the above zone as well as the moist convection within it make a contribution to the transformation of the zone into a narrow band, in which the vorticity intensifies. In this manner, a comma-shaped tail of the vortex evolves.

3.5. Landfall of Hurricanes When hurricanes or tropical storms make landfall, their surface wind decreases and the central surface pressure increases as they move inland. Shown in Fig. 12 is the distribution of the maximum surface wind during the passage of the storm for an experiment in which the hurricane made landfall onto a flat land. It shows an abrupt decrease of the maximum surface wind a t the shoreline, with a rapid decrease inland of the area of hurricane force winds. This suggests a rapid decay of the vortex after

27" N

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FIG. 12. Horizontal distribution of maximum low-level (-68-m) wind during the tropical storm passage for the experiment with surface temperature fixed at 298 K over the land and at 302 K over the ocean. Roughness length over the land was set to 25 cm. Shading indicates hurricane force winds (>33 m s-I). Land is to the west of the indicated coastline, and the dots indicate the fine mesh resolution of -17 krn at 2S"N. [From Tuleya et a / . (1984). From Monthly Weather Review, copyright 1984 by the American Meteorological Society.]

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landfall. The analysis results of the landfall simulation experiments clearly indicate that the decay of hurricanes shortly after the landfall is primarily due to the suppression of the latent energy supply over the land rather than the frictional dissipation of kinetic energy. Upon landfall, the low-level wind speed, in particular the tangential component of the wind, is reduced in response to the increase of surface roughness. However, the increased surface friction causes an increase in the inflow component despite the decrease in the total wind speed. Such a feature in another case of hurricane landfall is shown in Fig. 13. Provided that the moisture in the boundary layer is abundant, this could result in an increased influx of moisture, enabling the storm to deepen after the landfall. This was confirmed in an experiment in which the evaporation was not suppressed over the land and the same surface temperature was assumed over the ocean and the land. One of the factors to reduce the evaporation over the land is the relatively low land-surface temperature. In an experiment run with such a condition, the cool land surface not only lowered the saturation mixing ratio of the water vapor at the surface, but also reduced the conditional instability at the low levels and altered the boundary-layer inflow. Thus, the precipitation rate over the land in this experiment was decreased as compared with the case of higher land-surface temperature. Another experiment indicated that the response of the surface wind to a

1

110 km radius

:

S-

FIG.13. Vertical profiles of tangential and radial wind in the planetary boundary layer of a storm before and after the landfall, i.e., over ocean and over land. Profiles are those of average wind at a radius of 110 km from the storm center. [From Tuleya and Kurihara (1978). From Journal o f t h e Atmospheric Sciences. copyright 1978 by the American Meteorological Society.]

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nonhomogeneous surface roughness distribution was rather quick. In this experiment, the surface wind of the storm exhibited an observable decrease near the location of large surface roughness. Except for the boundary layer, the structural change of the model hurricanes after moving onto the flat land was gradual. When the hurricane was over the ocean, the kinetic energy of the storm was maintained by the balance among the three budget components, i.e., the kinetic energy generation, the smaller amount of dissipation, and the export of kinetic energy by the upper-level outflow. Several hours after the landfall, the kinetic energy generation decreased and became comparable with the dissipation, while the export at the upper levels did not change very much. Thus, the total budget resulted in a gradual decrease of the kinetic energy of the entire storm system. The decay rate of the hurricanes after the landfall can be affected in the presence of mountains. When a mountain range, about 500 km wide, 1 km high, and 2000 km long, was placed along the shoreline, the decay rate of the storm was considerably enhanced as it transversed the mountain. When the hurricane approached the mountain, intense precipitation was induced in the area in which the strong wind was up along the mountain slope, reducing the moisture content of the hurricane vortex. Also, the moisture convergence in the boundary layer was reduced because the air in the elevated boundary layer was gradually drier, and moreover the wind near the mountain ridge acted to remove moisture from the storm region at a large rate. Consequently, when the vortex moved to the lee side of the mountain range, the supply of latent energy to the vortex was greatly reduced, causing the enhanced decay. In this experiment, the mean basic flow field was also modified by the mountain; this change affected the storm track. In the above case, the storm slightly intensified before its landfall, apparently because of the mountain effect. Indeed, the influence of mountains can reach a tropical cyclone in various ways even when the storm is still several hundred kilometers upstream of mountains. Such an influence depends not only the length scale, height, and shape of mountains, but also on the position, strength, and movement of a tropical cyclone and the condition of environmental flows. In an experiment incorporating a mountainous island similar to Taiwan, the dry air originating at the surface of high mountains was advected into the center part of a storm to cause an abrupt weakening of a tropical cyclone over the ocean. In this case, the storm moved along a curved track around the mountain block with an accelerated speed. In the other experiment with a stronger basic flow, the surface vortex landed, stalled at the windward side of the mountain, and weakened while a secondary vortex formed at the leeside. The latter one

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moved away when it was connected to the middle-level circulation that transversed the mountain. 4.

S O M E CHALLENGING

ISSUESIN

THE

FUTURE

A number of subjects need to be considered in the future studies to meet our research goals, i.e., (1) to improve our understanding of the basic mechanisms involved in the life cycle of tropical cyclones and (2) to investigate the capability of the numerical models in the prediction of tropical disturbances. Although many problems may still be studied with the existing hurricane models, effort will be continued in the future to upgrade the performance standard of the models. Of course, as progress is made in the numerical modeling studies with these objectives, the advancement in the area of observation, data processing, and analysis must occur simultaneously.

4.1. Improvement of Numerical Models

In the improvement of the parameterization scheme, the effect of cloud dynamics, including the cumulus friction and microphysics of clouds (e.g., the formation of snow and graupel, their melting, and the effect of the resulting downdraft), have to be carefully evaluated and incorporated, if necessary. How to cope with the nonhydrostatic character of the moist convection is a problem we cannot pass over. One of the physical factors that is missing in the current GFDL models is the response of the ocean to the passage of a tropical cyclone. Since the behavior of the storm is greatly influenced by the ocean surface condition, the inclusion of the ocean layers in the model will be given special attention. Another factor to be considered is the diurnal variation of the radiation. Development of a better numerical integration scheme is also an important part of the model improvement. In the case of a limited area model, the conditions at an open lateral boundary of the domain have always to be watched with caution.

4.2. Basic Study

So far only limited cases of tropical storm genesis, i.e., the transformation of the trough region of an easterly wave into a tropical storm, have

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been investigated with the numerical simulation models. Studies should be extended to include the formation of a tropical storm in the intertropical convergence zone as well as from cloud clusters. The structure and behavior of a tropical cyclone at its mature stage are influenced by the conditions of the environment, the boundary conditions at the ocean or land surface, and the convective activity within it. We have to study how the environmental flow, which appears as asymmetric when viewed from the vortex, affects the symmetric structure of the storm. Also investigation is needed on the transient feature of the storm structure. Various factors contribute to the movement of tropical cyclones. A vortex in a calm environment slowly moves by the effect of planetary vorticity advection and possibly in response to nonhomogeneous surface conditions. When it is embedded in a large-scale flow or begins to interact with other vortices, its movement can be quite erratic. The numerical study on this subject is of practical importance in light of current major concern in the prediction of tropical cyclones. While some of the intense vortices decay over the ocean or after making the landfall, others move to the middle latitudes and become extratropical systems. The transition process in the latter case is one of the research topics in the future.

4.3. Prediction of Tropical Cyclones

The practical limit of the numerical prediction of tropical cyclones depends on (1) the degree of accuracy to which the numerical models can simulate the physical processes pertaining to the time change of the storms, (2) the accuracy with which the initial conditions are represented in the model, (3) the speed with which the time integration is performed, and (4) other factors. It seems that the prediction of the intensity of tropical storms will be probabilistic. This is because the evolution of a vortex is strongly related to the effects of moist convection, and the treatment of moist convection in numerical models inevitably involves hypotheses or statistical considerations on the cloud physics and dynamics. On the other hand, the prediction of movement of the well-developed vortices, i.e., tropical storms, may be made with a higher level of certainty as compared with the forecasting of intensity. A key problem is whether a strong vortex can be resolved and maintained reasonably well in numerical models. The problem just mentioned points to the importance of the develop-

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ment of good analysis-initialization systems. It is desirable that the initialized fields include components that are primarily related to the diabatic effects. Otherwise, the initial disturbances in the model will tend to weaken in the early period of the time integration. It is reasonable that the model initialization be made with the use of the prognostic system to be used subsequently. Therefore, to construct numerical models of high quality is essential for attaining to a good model initialization.

APPENDIX.GFDL HURRICANE MODELS Structure of the GFDL hurricane models, either the original framework or later modifications, has been explained in the following papers. Listed below are the specific papers in which a scheme for treating a particular physical or mathematical element in the currently working models is described. 1 . Governing Equations (a) The system of equations: The system on a constant f plane is given in Kurihara and Tuleya (1974) (Section 2a); a more general system in the spherical coordinates is written in Kurihara and Bender (1980) (Section 2a, Appendix C ) . (b) Surface exchange: See Kurihara and Tuleya (1974) (Section 3b); land-surface conditions are explained in Tuleya and Kurihara (1978) (Section 2b) and Kurihara and Bender (1983) (sections 2a and 2b). (c) Moist convection: A method of parameterization is described in Kurihara (1973), also outlined in Kurihara and Tuleya (1974) (section 3c); important modifications are made in Kurihara and Bender (1980) (Appendix C). (d) Vertical mixing: See Kurihara and Bender (1980) (Appendix A). (e) Horizontal mixing: See Kurihara and Tuleya (1974) (Section 3a). ( f ) Radiation: See Kurihara and Tuleya (1981) (Section 2). (g) Orography: Treatment of orography is explained in Bender et al. (1985). 2. Numerical Schemes of Model Integration (a) Vertical resolution: All models use the resolution specified in Kurihara and Tuleya (1974) (Section 2b). (b) Horizontal resolution: Grid systems being used are a uniform

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mesh in Kurihara and Tuleya (1981) (Section 2); a movable triply nested mesh in Kurihara and Bender (1980) (Section 2d); a movable quadruply nested mesh in Kurihara and Bender (1982) (Section 2). (c) Movable nested mesh: The design is explained in Kurihara et al. (1979); also see Kurihara and Bender (1980) (Sections 2g and 2h). (d) Finite differencing: At present, the scheme in Kurihara el al. (1979) (Section 2d) is used; also see Kurihara and Bender (1980) (Sections 2f and 2g). (e) Lateral boundary conditions: See Kurihara and Bender (1980) (Section 2c) for the open condition; another scheme was proposed in Kurihara and Bender (1983); a revised scheme is being tested. As to the noise control, see Kurihara and Bender (1980) (Section 2i and Appendix C). Also see Kurihara and Tuleya (1981) (Section 2) for a constraint imposed on the model. ( f ) Problems related to the initialization: Application of the reverse balance equation is explained in Kurihara and Bender (1980) (Section 3 and Appendix B). Concerning the planetary boundary layer initialization, see Kurihara and Tuleya (1978) and Kurihara and Bender (1979); a different approach is given in Kurihara and Bender (1983) (Appendix B). (8) Time integration: The iteration scheme is described in Kurihara and Tripoli (1976), an important note on the numerical values of the weights is given in Kurihara and Bender (1980) (Section 2e) and Kurihara and Tuleya (1981) (Section 2).

ACKNOWLEDGMENTS It is my great pleasure to acknowledge the constant encouragement and stimulus for research I have received from Joseph Smagorinsky since I first saw him in Tokyo in 1960.1 feel fortunate for having been able to study the many problems relating to tropical cyclones in the excellent research environment that he created and provided for us. Above all, 1 appreciate the trust he has placed in the GFDL Hurricane Dynamics Project; it has been vital to the growth and productivity of the project. I am grateful to my colleagues who have participated in the project for sharing both difficult times and enjoyable moments: M e w s . R. E . Tuleya and M. A. Bender, the GFDL; Mr. G . J. Tripoli, Colorado State University; Mr. A. A . Gaeta, U S . Navy; and Mr. M. Kawase, the GFD Program, Princeton University. In addition, a large number of people have helped the hurricane research at the GFDL through administrative support, scientific advice, interest and criticism, execution of computer jobs, preparation of manuscripts, and so on. I sincerely thank all of them. In preparing the present manuscript, I owe thanks to Mr. M. A. Bender for review, Ms. J. Kennedy, Mr. P. Tunison and his group, and Mr. J . Conner for all their assistance.

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REFERENCES Anthes, R. A. (1982). Tropical cyclones, their evolution. structure, and effects. Meieorol. Monoxr. 41, 1-206 [ISBN 0-933876-54-81, Bender, M. A., and Kurihara, Y. (1983). The energy budgets for the eye and eye wall of a numerically simulated tropical cyclone. J . Meteorol. SOC. J p n . 61, 239-243. Bender, M. A,, Tuleya, R. E., and Kurihara, Y . (1985). Numerical study of the effect of a mountain range on a landfalling tropical cyclone. Mon. Weurhrr Reu. 113, 567-582. Gray, W. M. (1981). Recent advances in tropical cyclone research from rawinsonde composite analysis. In “WMO Programme on Research in Tropical Meteorology,” pp. 1-407. World Meteorol. Organ., Geneva. Kurihara, Y . (1973). A scheme of moist convective adjustment. hfon. Weather Reu. 101, 547-553. Kurihara, Y. (1975). Budget analysis of a tropical cyclone simulated in an axisymmetric numerical model. J . A t m o s . Sci. 32, 25-59. Kurihara, Y. (1976). On the development of spiral bands in a tropical cyclone. J . Atmos. Sci. 33, 940-958. Kurihara, Y . , and Bender, M. A. (1979). Supplementary note on “A scheme of dynamic initialization of the boundary layer in the primitive equation model.” M o n . Weather Reu. 107, 1219-1221. Kurihara, Y . , and Bender, M. A. (1980). Use of a movable nested-mesh model for tracking a small vortex. Mon. Weather R e v . 108, 1792-1809. Kurihara, Y . , and Bender, M. A. (1982). Structure and analysis of the eye of a numerically simulated tropical cyclone. J . Meteoro/. Soc. Jpn. 60, 381-395. Kurihara, Y., and Bender, M. A. (1983). A numerical scheme to treat the open lateral boundary of a limited area model. M o n . Weather Reu. 111, 445-454. Kurihara, Y . , and Kawase, M . (1985). On the transformation of a tropical easterly wave into a tropical depression: A simple numerical study. J . Atmos. Sci. 42, 68-77. Kurihara, Y . , and Tripoli, G. J. (1976). An iterative time integration scheme designed to preserve a low-frequency wave. M o n . Weather R e v . 104, 761-764. Kurihara, Y . , and Tuleya, R. E. (1974). Structure of a tropical cyclone developed in a threedimensional numerical simulation model. J . A m o s . Sci. 31, 893-919. Kurihara, Y., and Tuleya, R. E. (1978). A scheme of dynamic initialization of the boundary layer in a primitive equation model. Mon. Weather R e v . 106, 114-123. Kurihara, Y . , and Tuleya, R. E. (1981). A numerical simulation study on the genesis of a tropical storm. Mon. Weather R e v . 109, 1629-1653. Kurihara, Y., and Tuleya, R. E. (1982a). Influence of environmental conditions on the genesis of a tropical storm. In “Topics in Atmospheric and Oceanographic SciencesIntense Atmospheric Vortices” (L. Bengtsson and J . Lighthill. eds.), pp. 71-79. Springer-Verlag, Berlin and New York. Kurihara, Y., and Tuleya, R. E. (1982b). On a mechanism of the genesis of tropical storms. Proc. R e g . Sci. Cotf. Trop. Mrteurol., 1982. pp. 17-18. Kurihara, Y . , Tripoli, G. J., and Bender, M. A. (1979). Design of a movable nested-mesh primitive equation model. M o n . Weather Reu. 107, 239-249. Mellor, G. L., and Yamada, T. (1974). A hierarchy of turbulence closure models for planetary boundary layers. J . A t m o s . Sci. 31, 1791-1806. Ooyama, K. V . (1982). Conceptual evolution of the theory and modeling of the tropical cyclone. J . Meteorol. Soc. J p n . 60, 369-380. Sasamori, T. (1970). A numerical study of atmospheric and soil boundary layers. J . A m o s . Sci. 27, 1122-1137.

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Smagorinsky, J. (1963). General circulation experiments with the primitive equations. Part I. The basic experiment. M o n . Weather Reu. 91, 99-164. Tuleya, R . E., and Kurihara, Y . (1975). The energy and angular momentum budgets of a three-dimensional tropical cyclone model. J . Atmos. Sci. 32, 287-301. Tuleya, R. E., and Kurihara, Y. (1978). A numerical simulation of the landfall of tropical cyclones. J . Atmos. Sci. 35, 242-257. Tuleya, R. E . , and Kurihara, Y . (1981). A numerical study on the effects of environmental flow on tropical storm genesis. Mon. Wearher Rev. 109, 2487-2506. Tuleya, R. E., and Kurihara, Y. (1982). A note on the sea surface temperature sensitivity of a numerical model of tropical storm genesis. M o n . Weather Reu. 110, 2063-2069. Tuleya, R. E., and Kurihara. Y. (1984). The formation of comma vortices in a tropical numerical simulation model. Mon. Weather Reu. 112, 491-502. Tuleya, R. E . , Bender, M. A., and Kurihara, Y. (1984). A simulation study of the landfall of tropical cyclones using a movable nested-mesh model. Mon. Weather Reu. 112, 124136.

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NUMERICAL WEATHER PREDICTION IN LOW LATITUDES T. N . KRISHNAMURTI Depurtment of Meteorology Floridu State Uniuersiry Tulluhussee. Florida

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 1 . 1 Scope of Simple Models Based on Conservation Laws . . . . . . . . . . . . . . 285 2. Initialization: Dynamic, Normal Mode. and Physical . . . . . 2.1 Physical Initialization . . . . . . . . . . . . . . . . . 2.2 Humidity Analysis . . . . . . . . . . . . . . . . . . 3. Parameterization of Physical Processes . . . . . . . . . . . 3 . 1 . Parameterization of Cumulus Convection . . . . . . . . 3.2. Radiative Parameterization . . . . . . . . . . . . . . 4. Medium-Range Prediction of Monsoon Disturbances . . . . . 5. On the Prediction of the Quasi-Stationary Component . . . . . 6. Scope of Future Research. . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . .

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293 302 306 . . . . . 306 . . . . 313 . . . . . 3 I4 . . . . . 323 330 . . . . . . . . 331

1. INTRODUCTION

In this chapter on prediction we shall address the scope of real-data numerical weather prediction efforts with regional and global models over the tropics. The regional prediction covers models based on simple conservation laws (absolute and potential vorticity) and more general primitive-equation models with detailed physical parameterization. The global model is a multilevel, high-resolution spectral model with detailed physics and initialization. Much progress has been made in recent years in the development of numerical weather prediction models for the low latitudes. The World Weather Watch in low latitudes has been enhanced by the inclusion of a variety of surface- and satellite-based observations. The availability of faster computers has enabled a larger volume of tests on analysis, initialization, sensitivity to physical parameterization, and statistical evaluation of numerical weather prediction. It has been possible in recent years to define initialization schemes such that divergent motions retain important information on the Hadley- and Walker-type vertical circulations. Measures of precipitation rates at the initial time have been constructed from a mix of satellite radiances and rain-gauge data sets. These, in turn, have been used to restructure the humidity analysis such that the implied rain by a cumulus parameterization scheme is close to the “observed measures.” Over rain-free areas a humidity reanalysis, above 283 A D V A N C E S IN GEOPHYSICS, V O L U M E

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Copyright Q 1985 by Academic Press. lnc All rights of reproduction In any form reserved.

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the boundary layer, is structured to provide an advective radiative balance that appears to improve the evolution of the divergent wind. Specific studies have focused on investigations of westward passage of African waves, tests of cumulus parameterization schemes utilizing GATE observations, formation and motion of tropical depressions, and prediction of time-averaged motion field (i.e., the stationary components). Historically, the single-level models were first tested by many scientists over the tropics in the 1960s. The aim of these efforts was to assess the performance (or skill) of single-level models against climatology and persistence. It was then apparent that single-level models simply could not perform better than climatology or persistence. Lack of observations in the tropics placed a heavy emphasis on climatology for the first-guess field in the tropical objective analysis of the initial as well as the verification states. Thus, for these obvious reasons, numerical prognosis could not outperform climatology or persistence when root-mean-square errors were assessed over large domains. The present experience with GATE, the FGGE, and MONEX (a list of acronyms is presented in Table I) is gratifying in that substantial skill over both persistence and climatology has been demonstrated over certain regions of the tropics. This is largely attributable to a better density of observations from a mix of surface- and space-based platforms. Furthermore, the modeling aspects, with respect to the handling of conservation laws, boundary conditions, testing against linear stability theory and the care exercised in the handling of linear and nonlinear computational instability, have improved considerably since 1 960. Charney’s (1963) scale analysis over the tropics essentially stressed the importance of barotropic dynamics and the nondivergent character for the large-scale flows. The exceptions were regions, such as the ITCZ, in which deep convective processes were present. That picture was revised TABLEI. LISTOF ACRONYMS CARP MONEX GATE ECMWF TIROS-N INSAT GWE FGGE IIIb NMI ITCZ NWP

Global Atmospheric Research Program Monsoon Experiment GARP Atlantic Tropical Experiment European Centre for Medium-Range Weather Forecast Operational U.S. polar-orbiting satellites during 1979 Geostationary satellite launched by India in 1983 Global Weather Experiment First GARP Global Experiment Analyzed data at grid points produced during FGGE Normal mode initialization Intertropical convergence zone Numerical weather prediction

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285

by Holton (1969) upon consideration of the gravitational character of the tropical motions and the treatment of a realistic scale height of the atmosphere. The study of Krishnamurti (197 1) identified planetary-scale divergent circulations over the entire tropics. This led to the thinking that even over nonconvective regions in which large-scale descent is ever present, an essential balance between the adiabatic warming and radiative cooling is present (Holton, 1972). Thus, one is led to the conclusion that singlelevel models based on simple conservation laws (of absolute or potential vorticity) can only have a very limited skill in actual numerical weather prediction. These simple laws are only applicable near a level of nondivergence at which the barotropic character may be more relevant. Although that level is close to 600 mb in the tropics (over most regions), it is not the most appropriate level from the point of view of identifying tropical easterly type waves. Many such waves have their largest amplitude between 850 and 700 mb. It is the future motion of such waves that is a problem of much practical relevance. Thus, most tests with single-level models are carried out at these two levels. The choice of 850 mb has at times been dictated by the availability of a large number of cloud motion vectors. That choice was not very prudent in some series of numerical applications, since the day-to-day evolution of the motion field was also influenced by the strong divergent wind. That was especially true over the region of the winter monsoon in which the skill of the single-level models was found to be marginal even at the end of 24 hr.

1.1. Scope of Simple ModeIs Based on Conservation Laws With the availability of a network of data, during GATE (in 1974), from west Africa and the eastern Atlantic Ocean, extensive experimentation began that appeared most promising. The Global Weather Experiment in 1979 provided a unique opportunity to extend such studies over most of the tropics. These numerical experiments revealed large regional differences in the behavior of the models based on simple conservation laws. Extensive integrations have been carried out with two regional limitedarea models with a horizontal resolution on the order of 100-km mesh. The first of these is the well-known barotropic model in which the initial stream function is obtained from an analysis of the tropical wind field. In the construction of such a model, care must be exercised to assure that the model is capable of resolving barotropic instability. For that purpose, a channel model was first constructed that had a cyclic boundary condition in the zonal direction and a constant stream function along the meridional boundaries. The finite differencing scheme includes the usual sec-

286

T. N. KRISHNAMURTI

ond-order Arakawa's (1966) Jacobian that conserves the quadratic invariants. Model tests were first carried out with zonal flows (with horizontal shear) that satisfy the necessary condition for the existence of barotropic instability and that exhibit finite growth rates (in accordance with the linear dynamics) for certain synoptic scales (=a few thousand kilometers) in the zonal direction. Perturbations of about the same scale were tested and shown to grow in the nonlinear model. The construction of a model, in which horizontal shear supplies energy to perturbation thereby increasing curvature vorticity of the perturbation, is not a simple matter. Scale selection is predicted only when the model utilizes a constant stream function zonally along the meridional boundaries (consistent with the vanishing of the amplitude function according to the linear theory). Furthermore, the periodic boundary conditions need to be applied on a domain whose zonal scale is equal to that of the wavelength under investigation. Once such preliminary tests are made, one takes some liberty with the extension of the model domain for the real data NWP problem. In fact, after these initial tests we have even relaxed the requirement of a constant stream function along x, by specifying a stream function that varies along x but is independent of time; this is shown to improve the prediction over the tropics since it provides a more realistic steady forcing at its subtropical boundaries. However, this specification does suffer from the fact that on occasions the tropical wave energy growth (or decay) is directly a consequence of the imposed time-invariant boundary forcing. Figure 1 illustrates an example of a forecast made during GATE with this prescription. In this sample, the skill of prediction was very high-up to 96 hr. In this figure, the observed flow field is shown in (b) while (c) shows the corresponding predicted field. At 96 hr, the track of the wave (observed and predicted) is shown in (a). This regional model has been examined in many hundred experiments over each of the following regions: Caribbean (summer), eastern Atlantic and West Africa (summer), Indian monsoon regions (summer), winter monsoon region (Indonesia, Malaysia, Australia, Indochina during northern winter), and western Pacific Ocean (northern summer). Useful skill of the single-level barotropic model was found over West Africa and the Indian subcontinent during the northern summer months, while the lowest skill, measured by vector root-mean-square (rms) error,

FIG.1. (a) Track of observed and barotropically predicted easterly wave during GATE, June 3-7, 1974, 122. (b) Observed streamlines and isotachs, 700 mb. on September 7, 1974, 122. (c) The 96-hr predicted streamlines and isotachs, 700 mb, valid on September 7, 1974, 122.

(a) 25 N 2ON. 15N.

ION

*

5 N 0 ’ I

5 s

35

4QW

30

25

20

I5

I0

5 W

0

5E

I 0

I5

2OW

(b) 25 N 20 N I5 N ION 5N

ON 5 - s-

40W35

30

25

30

25

20

15

10

15

10

5W

0

5E

I 0

15

20E

OE

5E

I0

I5

20E

(C)

25 N

20N

15N ION

5 N

ON 5 s

4 0 W 35

20

5W

288

T. N . KRISHNAMURTI

was noted over Southeast Asia and the South China Sea during the northern winter months. Table I1 illustrates the rms errors of the rotational wind over three regions at the end of 24, 48, 72, and 96 hr. As may be noted, the performance of the single-level model over the winter monsoon region is poor. Over the African and Indian summer monsoon domain the skill over persistence at 96 hr is remarkable. The second model based on the conservation of potential vorticity is the familiar shallow-water system. This is a primitive equation model in which the three equations momentum (along x and y ) and the mass continuity equation describe its basic dynamics. This is an extremely useful low-order model for tropical numerical weather prediction. The finite difference versions of the model, whose performance is described later, requires proper handling of time differencing and the nonlinear advective terms. The use of semi-implicit time differencing schemes has become rather routine in its application over middle latitudes as well as over tropical beta planes. Although its use has been shown to damp highfrequency gravitational modes, it does not seem to affect the amplitude of mixed Rossby gravity waves and Kelvin-type waves that are central to the tropical wave dynamics. In the model, where most of the following tests were made, a semi-Lagrangian advection scheme proposed by Krishnamurti (1962) and Mathur (1970) was used. The domain invariants (for a closed system) are the area-averaged mean height of the free surface, the total energy, the mean potential vorticity, and the mean-square potential vorticity. The current design of this model (Krishnamurti et al., 1980a) shows that these domain invariants can be preserved to within 1% of their initial value during the 96-hr forecast. The inclusion of bottom topography (or smoothed mountains) in the shallow-water system of equations is relatively straightforward and the basic invariants are still preserved. The assignment of a mean height for the free surface is usually based on a trial-and-error approach. The intensity of divergence can be controlled somewhat by the choice of this parameter. For the prediction of tropical wave disturbances a choice of 2 km for the mean height of the free surface seemed appropriate. We have also noted that the maximum height of mountains is limited to 1 km for stable integrations, thus for a mesh size of 100 km the slope of the terrain is limited to Figures 2 and 3 show some examples of 24- and 48-hr forecasts made with this model for the prediction of African waves. Part (a) in these illustrations shows the observed flow field at 700 mb (heavy lines show stream lines; thin lines, isotachs; very heavy lines show the trough line of the African waves). Part (b) shows the results of a forecast made with the barotropic model, while (c) shows the forecasts made with the single-level primitive-equation model applied to the 700-mb level. The phase speed of

TABLE11. ROOT-MEAN-SQUARE ERRORSTATISTICS

FOR

48 hr

24 hr

VECTOR WIND 96 hr

72 hr

Region"

BARO

IL-PE

PERSIST

BARO

IL-PE

PERSIST

BARO

IL-PE

PERSIST

BARO

IL-PE

PERSIST

African monsoon Asian monsoon (summer) I Asian monsoon (summer) I1 Asian monsoon (winter) I Asian monsoon (winter) I1

4.82

4.97

5.47

6.15

6.02

6.58

6.87

6.31

6.49

7.09

6.44

6.66

5.6

4.3

5.04

5.5

4.9

5.8

6.7

5.5

6.5

6.9

6.0

7.0

4.4

4.5

4.7

5.1

5.1

5.3

5.9

6.0

6.1

6.2

6.7

7.3

4.9

5.3

4.9

6.0

6.4

5.9

7.0

7.4

6.4

7.7

8.1

6.6

6.6

6.2

6.4

8.9

8.2

8.4

9.8

9.4

9.2

10.7

9.8

9.6

Asian monsoon region I: 30"E-90"E, 3WS-40"N (summer), 67.5"E-I5WE, 18.7YS-33.75"N (winter); Asian monsoon region 11: 90"E-I50"E, 30"s40"s (summer), 150"E-127.5"W. 18.75"S-33.75"N (winter).

NUMERICAL WEATHER PKEDIC’IION IN LOW LATITUDES

29 I

the westward passage of African waves is handled reasonably well by both models up to 48 and 96 hr, respectively. We have noted from a large sample of such forecasts with the GATE and FGGE/WAMEX observations that such quality forecasts of African waves are indeed possible. Table 11 also includes the vector root-mean-square error statistics for the two models. The single-level primitive equation model with smoothed orography performs better than persistence and climatology for periods on the order of 96 hr. As stated earlier, the performance over several other regions of the tropics is poorer. 2. INITIALIZATION: DYNAMIC, NORMAL MODE,A N D PHYSICAL In the tropical modeling effort, initialization appears to be necessary because one wants to retain gravitational divergent modes such as the Hadley-Walker as well as storm-scale divergent circulations. Classic static initialization procedures that make use of quasi-geostrophic type omega equations underestimate divergent motions. Dynamic initialization has been used extensively in our modeling effort, that entails a forwardbackward integration of the dynamical equations (Miyakoda and Moyer, 1968). No physics is included within this process. Although this process converges to a slowly varying atmospheric state, the absence of diabatic and dissipative processes results in an incorrect equilibrium with errors in the amplitude of divergent motions. Ascent (or descent) results in adiabatic cooling (or warming) that would otherwise be balanced by diabatic heating (or cooling) over t h e tropics. After forward-backward integrations (which require some 18 hr of l-hr forward and l-hr backward integration), the diabatic and dissipative processes are switched on as the actual forward prediction is carried out. A readjustment of the balance between the adiabatic and diabatic processes occurs during the first 24 hr of integration. This readjustment is not catastrophic and the ensuing fields in the tropics do seem quite realistic. The major advantage of the dynamical initialization seems to be its ability to provide a reasonable windpressure adjustment that is usually not far from the gradient wind or nonlinear balance laws. The fields that encounter the largest changes during the dynamical initialization are the surface pressure, the divergent wind, and the geopotential heights. The rotational wind and the temperature fields exhibit very small changes. FIG. 2. (a) Observed streamlines and isotachs of the 700-mb wind field, September 7, 1974, 122. (b) Predicted 24-hr barotropic forecast valid at 122 on September 7. 1974. Ic) Predicted 24-hr, single-level, primitive-equation forecast valid at 122 on September 7, 1974.

FIG.

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

293

After several years of experimentation with the normal mode initialization (NMI) (Daley, 1981; Puri and Bourke, 1982; Kitade, 1983; Errico, 1984), it is now becoming apparent that a convergence of the Machenhaur (1977) method of NMI can be implemented for a few vertical modes of the atmosphere. Recent contributions on this problem show that diabatic and nonlinear processes can also be included with this initialization to resolve the divergent circulations of the low latitudes. Kitade, working with Florida State University’s global spectral model, showed that an underrelaxation procedure for the solution of the Machenhaur equation for the amplitude of Rossby and gravitational modes produces a slow convergence in all of the 10 FGGE cases he investigated. This holds much promise for the inclusion of diabatic heating in the initialization problem. Kitade demonstrated that the inclusion of the NMI improves the prognosis of the 24hr rainfall over the tropics. Some of the major advantages of this elegant method are, however, lost if the initial analysis of the humidity is poor. This usually results in the incorrect placing of regions of moisture convergence and errors in diabatic forcing and divergent wind that remain in spite of a careful NMI. In recent years we have carried out a large number of observational studies on the scales and temporal variability of divergent circulations. On the basis of the results of these studies, we have noted a useful alternative for the initialization of the global model. The rotational wind at the full resolution of the global model is evaluated from observations and retained initially, while only the five long waves of the divergent wind (based on the analyzed wind field) are retained. That prescription of the initial divergent wind retains the essential features of the planetary-scale Hadley and east-west circulations. However, it does not assure a remedy for the inconsistencies within the humidity analysis. We have carried out numerous global forecasts with this method. It does suffer from an eventual contamination of the divergent wind after 2 to 3 days of the forecast. The prediction of the rotational wind, nevertheless, is quite promising up to 6 or 7 days. Figure 4 illustrates an example of the observed and predicted velocity potential (Fig. 4a,b) at the end of 2 days and of the stream function (Fig. 4c,d) at the end of 6 days.

2.1. Physical Initialization Based on the results of a number of numerical prediction experiments, we have confirmed that the differential heating between land and ocean is an important and critical factor in the investigation of phenomenon such as for the onset of monsoons over the Indian subcontinent. The preonset

294

T. N . KRISHNAMURTI

3OE

30E

60E

60E

120E

90E

I20E

150E

I50E

FIG.4. (a) Observed velocity potential at 850 mb on July 3, 1979, 122, units los m s - l . (b) Predicted velocity potential at 850 mb (at hour 48) valid on July 3, 1979, 122. units 10’ m2s-I. (c) Observed stream functions at 850 mb on July 7, 1979, 122. units los m2 s-I. (d) Predicted stream functions at 850 mb (at hour 144) valid on July 7 , 1979, 122, units lo5 m2 s-I.

period during the month of May shows a rather persistent flow field in the monsoon region. At low levels, the circulation exhibits anticyclonic excursions over the Arabian Sea, flowing essentially parallel to the west coast of India from the north. Over the Indian subcontinent, the major feature is a shallow heat low over northern India. Our findings, to be described briefly later, may be stated as follows: “A seemingly stable climatological flow appears to exist day after day over the monsoon region. However, this flow is, in fact, quite unstable to the configuration of large-scale differential heating.” As the heat sources commence, a rapid north-westward movement toward the southeastern edge of the Tibetan Plateau occurs, an interesting configuration of the large-scale divergent

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

295

40N

30N

ON

30S 30E

60E

90E

I20E

150E

W E

I20E

150E

( d)

40N 30 N

ON

30S 3OE

60E FIG.

4. (Continued)

circulation. A favorable configuration for a rapid exchange of energy from the divergent to the rotational kinetic energy develops. Strong low-level monsoonal circulations evolve, with the attendant onset of monsoon rains. That appeared to be the scenario during the year of the Global Experiment, 1979. In order to test this observational sequence, a series of short-range numerical prediction experiments was initiated. The experiments differed from each other in the definition of the initial heat sources. The differential heating between the Arabian Sea and the southeastern edge of the Tibetan Plateau is described by a net cooling over the ocean (dominated by radiative forcing) and a strong net heating over the foothills of the Himalayas (dominated by convective forcing). This strong net heat-

296

T. N . KRlSHNAMURTl

ing occurs over regions of organized cumulus convection where a large net supply of moisture is available. In order to provide such a forcing in the different initial states for the proposed experiments, we extracted the divergent wind and humidity field from three different epochs in the monsoon evolution: (1) springtime, (2) preonset, and (3) postonset. The rotational wind, pressure, and temperature fields were kept identically the same for all three experiments. Since the divergent wind and the humidity fields were different in each case, the application of a cumulus parameterization scheme that depended on moisture convergence gave rise to different measures of latent heating initially. The rotational wind, in each experiment, described the preonset circulations. Figure 5 describes the initial state at 850 mb in these experiments (Krishnamurti and Ramanathan, 1982). The 96-hr forecasts of the low-level flow field at 850 mb for the three respective experiments are shown in Fig. 6a-c. The strong monsoon onset response in Fig. 6c, when a more northerly heat source was deployed initially, is clearly evident here. These experiments were more phenomenological in their design. However, they demonstrate a strong sensitivity of the onset to the humidity analysis as well as to the specification of the initial divergent wind. Other aspects of this study relate to diagnostic investigations of the transfer of energy from the divergent to the rotational wind. These are described by energy exchange functions (Krishnamurti and Ramanathan, 1982). Here the energy equations are cast into a system of three equations, i.e., the rotational kinetic energy equation, the divergent kinetic energy equation, and the available potential energy equation. When these equations are expressed as integrals over a closed mass of the atmosphere, they conserve the total energy (rotational plus divergent plus available potential) in the absence of heat sources, sinks, and dissipative processes. A number of major inferences on the workings of a differentially heated system during the onset of monsoons can be made with this system of equations:

Here K+, Kx, and APE denote the aforementioned energy quantities, respectively. The energy exchange functions are enclosed within braces, where a positive sign for an exchange function denotes an exchange of energy from the first member to the second. In the context of the onset, one first notes that during this period K$ increases with time; a negative definite dissipation 09 requires that energy must be transferred from the

298

T. N . KRISHNAMURTI

I J J Y

205

jgT Y

305 .~ JOE

S-U_u2dd-n2--J’4--4

-JyAdd_y-4/92JJY24+wJ

-J-J~-Jdd-.f.&J J2J-J- \ \ \ 4-J-J-J-VOE

50E

60E

70E

EOE

90E

IOOE

I IOE

IZOE

130E

IltOE

FIG.6. The 96 forecasts of the wind fielc, at 850 mb for the three respective experiments. The full wind barbs denote 5 m s-’ while the half barbs denote 2.5 m s - I . The response in the near equational region in the three experiments: (a) the response for springtime moisture convergence, (b) the response for the preonset moisture convergence, and (c) the response for the postonset moisture convergence.

divergent to the rotations motions; i.e., ( K x . KJI)is positive. The strong evolution of divergent circulation during this period (as was noted from observations) implies that K X increases with time, with a negative definite dissipation Dx;and from the aforementioned requirement on the energy exchange from the divergent to the rotational motions, we draw the next major inference, which is that divergent motions must receive energy from the available potential energy. That process happens to be the wellknown covariance among the vertical velocity and the temperature field. An analogous argument on the third equation requires that a net generation of available potential energy must take place in a system in which the rotational and divergent motions are amplifying. Figure 7 shows the energy exchanges in the three respective experiments. It is of interest to note that when a favorable configuration of the initial heating is selected

150E

300

T. N . KRISHNAMURTI

-30 -20 -40

-

1

1

I

1

I

I

I

1

I

I

0

12

24

36

40

60

12

04

96

TIME (HOURS) FIG.7. Energy exchange from the divergent to the rotational component (Kx. K+) for the three respective numerical experiments. The domain of integration and energetics is the same as the map domain of Fig. 6.

(corresponding to Fig. 6c), a large energy exchange, as stated in the aforementioned scenario, follows in the numerical experiment. These results demonstrate a large sensitivity of the prediction in low latitudes to the initial analysis of the humidity field. That is an area of major research under the area of physical initialization. While examining these same processes in real-data forecasts, especially with a global model, we have confirmed that an accurate humidity analysis in low latitudes was essential for a definition of the heat sources and sinks. Even the presently available FGGE IIIb data analysis of the humidity field suffers from major inconsistencies with respect to regions of cloud cover as shown by satellite radiance data sets. Short-range prediction experiments frequently show a rapid deterioration of the divergent wind not only over regions of cloud cover, but also in relatively clear areas. That strongly suggested

NUMERICAL WEATHER PREDICTION I N LOW LATITUDES

30 I

Preinitiolization I l l b Data Sets

I*

Sotellite and Rain Gauge

7 Rainfall ( R ) Analysis Modify Humidity Field q Consistent With Advective-Radiative Balonce

.

Modify Divergent Wind Consistent With Kuo’s Scheme

Dynornic lnitializotion

Modify Humidity Field q Consistent With Kua‘s Scheme

Modify Humidity Field q Consistent With Advective-Radiotive Bolonce

Spectrol Prediction

FIG.8. A schematic flow chart of the physical-dynamical initialization carried out within the global model. For a more detailed description of this chart, see Krishnamurti et a / . (1984).

that moisture supply was not being properly defined over convective areas and the radiative forcing was not being calculated accurately in rainfree areas in which the errors in the vertical distribution of humidity are large. In this situation, the models do not provide a reasonable radiative cooling for the cloud-topped (nonprecipitating) mixed layers. Figure 8 presents an outline of a physical initialization procedure that we have been experimenting within the global model. The physical initialization procedure is structured around a dynamical initialization and essentially provides a more reasonable humidity analysis. Krishnamurti et al. (1984) have discussed in detail the procedures involved in this method. Essentially, it consists of the following components. (1) Analysis of the “observed rain” from a mix of rain-gauge and satellite radiance information. This entails determination of a statistical multiple repression among rain-gauge data, satellite infrared radiance,

302

T. N. KRISHNAMURTI

and its time rate of change at a collection of collocated rain-gauge sites. The regression coefficients are next used to determine a first-guess field based on the daily values of the radiances and their time rate of change. The next step is an objective analysis of the FGGE IIc rain gauge data (some 3000 to 5000 observations per day) over the global tropics with the aforementioned first-guess field. (2) The humidity analysis is restructured in the rain areas (as determined above) to a cumulus parameterization scheme-in this case the Kuo scheme with a moistening parameter b = 0. Thus the humidity reanalysis at all vertical levels is minimized to provide an initial computed rainfall rate close to the observed rainfall rate. (3) Over rain-free areas, a proposal for a reanalysis of the humidity field has been made that seeks an advective-radiative balance. The radiative parameterization (described in the next section) is based on a emissivity-absorptivity method. A large sensitivity to the calculated radiances results from moisture distribution where it encounters changes in the cloud specifications. A reanalysis of the humidity can render the atmosphere cloudy from a cloud-free situation. The cloud specification is based on threshold values of relative humidity in a vertical layer. The fractional areas of low, middle, or high clouds can be altered from a reanalysis of the humidity field. That results in a change in the net radiative heating at the Earth’s surface and in the vertical column. Given adequate wind observations from the composite observing systems (WWW, cloud winds, commercial aircraft), the premise here is that the divergent winds defined from these observations over the tropics are superior to those obtained from any indirect methods (Oort and Peixot, 1983, p. 481). Although the advective-radiative balance is applied rigidly at a level just above the planetary boundary layer, this still requires a modification of the entire vertical profile of humidity. The profile is slowly altered in a sequence of experiments at each point such that the final radiative cooling at the reference level balances (closely) the advective temperature change (Krishnamurti et al., 1984). Thus the physical initialization aims toward reasonable rainfall rates in the rainy areas and an advective-radiative balance elsewhere. Figure 9a,b illustrates an example of the observed rain and that obtained from a reanalysis of the humidity field. 2.2. Humidity Analysis

Here we shall describe the initial humidity data. The present data sets come from (1) conventional surface and upper-air World Weather Watch

NUMERICAL WEATHER PREDICTlON IN LOW LATITUDES

303

I

eoE ~ O E IOOE IIOE I Z O E I ~ O E MOE .II)OE FIG.9. (a) Observed rainfall rate obtained from a mix of satellite and rain-gauge observations (mm/day) for July I , 1979. (b) Initialized rainfall rate obtained at the end of the physical-dynamical initialization (midday) for July 1, 1979. ~ O E

~ O E ~ O E ~ O E TOE

304

T. N . KRISHNAMURTI

Network, (2) layer-averaged specific humidity from remotely sensed observations of polar-orbiting satellites, and (3) humidity structure functions over satellite-inferred regions of deep convective cloud cover based on statistical compositing procedures. The conventional data have been analyzed by many global analysis groups such as the ECMWF and the GFDL. This is of reasonable high quality over land areas but is of marginal usefulness over the vast data-void oceanic areas. The satellite surroundings provide a very useful data set for the analysis of the humidity field. The Tiros-N operational vertical sounder (TOVS) has been described by Smith et al. (1979). In this series of satellites three channels centered around 8.3, 7.3, and 6.3 km provide layer-averaged precipitable water between the Earth’s surface and 300 mb. Cadet (1983) has described the use of this data set over the monsoon region. The initial method of retrieval entails a statistical approach in which the eigenvectors of a coefficient matrix are evaluated. The satellite data sets are collocated onto the positions of the radiosonde observation via a simple bilinear interpolation in space. The coefficients determined from the covariance matrix are updated over periods of about 6 days. This method is only applied to regions that are cloud free. Gruber and Watkins (1979) have addressed the deficiencies of this method over disturbed regions. Cadet also notes that in extreme situations in which the atmosphere is very dry or very moist, this method does not always match the radiosonde observations at each and every sounding. Cadet attributes these discrepancies to the arrival of extreme dry or moist air over regions in which a week-old data set does not provide reasonable retrieval coefficients. In some instances we have noted that the 1000-mb specific humidity can be smaller than the layeraveraged specific humidity (obtained from satellite retrieval) between 1000 and 700 mb. This appears to be inconsistent with respect to the neighboring radiosonde data and the depth of the mixed layer. Figure 10 illustrates a sample of observations from Tiros-N passes over the global tropics during a single day. These data are 29 hr of the synoptic time that is 12 GMT in this instance. Both 03- and 15-hr local time data sets are used here-they describe the equatorial crossing time for the polar-orbiting satellite. Over cloudy areas in which the equivalent black-body temperatures of the outgoing long-wave radiation are lower than a threshold value -240 K, one expects deep convective clouds and cirrus anvils to be present. Such areas are easy to map from polar-orbiting or geostationary satellite data sets. In a pilot study, Cadet (1983) collocates such satellite information onto the sites of the WWW soundings. Using data sets for 1979, he determined composited vertical distribution of relative humidity over

306

T. N . KRISHNAMURTI

such deep convective areas. In our analysis of the humidity field, we have identified such deep convective areas from satellite data sets and extracted the aforementioned structure functions of relative humidity. These were then used as a part of the data sets for the objective analysis of the humidity field.

3.

PARAMETERIZATION OF PHYSICAL PROCESSES

Large-Scale Condensation. If an ascent of air is encountered in a stable saturated (defined here as 80%) environment, then the excess water vapor beyond saturation is made to fall out as rain and the atmosphere is heated accordingly in each layer. Air-Sea Interaction. In the regional model, the surface transfers of momentum and sensible and latent heat are handled via the usual bulk aerodynamic formulas. The bulk transfer coefficients are based on GATE studies (Businger and Seguin, 1977). The drag coefficient for the definition of the surface stress is dependent on wind speed but not on stability. In the global model, the surface transfers are based on similarity theory (Businger et al., 1971) and are stability dependent. Planetary Boundary Layer. A simple K theory for the vertical diffusion of the surface fluxes in the planetary boundary layer is used. The coefficients of vertical transfer are determined via simple mixing length concepts.

3.1. Parameterization of Cumulus Convection The current version of our global spectral model utilizes a variant of the Kuo scheme that is structured to GATE observations (Krishnamurti et al., 1980b, 1983a). The first of these studies dealt with the observations over the hexagonal ship array of GATE seeking a relationship between the observed rain (as measured by radar and rain gauge) and the net largescale moisture convergence. As was first noted by Thompson et al. (1979), a very close relationship between these two quantities is indeed present. That is reflected in a simple version of Kuo’s scheme in which the rainfall estimates are parameterized as the net available supply of moisture. Results of these calculations, shown in Fig. 11, exhibit a very close correspondence between the calculated (solid line) and the observed (dashed line) measures. It should be noted that Lord (1982) demonstrated a similar success in the specification of rainfall rates from an application

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

307

FIG. 11. Calculated and observed rain from the use of the Kuo scheme (the moistening parameter b = 0) during the last phase of GATE. These are the so-called semiprognostic calculation described in Krishnamurti et al. (1983a).

of the Arakawa-Schubert theory (1974). In a prognostic model, that prescription of available moisture supply leaves no room for the moistening of a vertical column by cumulus convection. The situation was usually remedied by the choice of a strong vertical diffusion of moisture that was accomplished by a large value of the vertical exchange coefficient. A limitation of this was that the diffusive process acted equally strong in nonconvective areas, resulting in an overall increase of humidity above the planetary boundary layer nearly everywhere. To overcome these difficulties, the Kuo scheme was posed as a two-parameter problem. A moistening parameter b and a mesoscale moisture convergence parameter q were determined from a statistical regression approach utilizing the GATE data sets. Following Krishnamurti et al. (1983a), we shall denote the large-scale supply of moisture for cumulus convection by the relation

In addition to this large-scale supply, it is assumed that there exists a mesoscale supply proportional to ZL . Thus we express the net supply Z by the relation

z = Z L ( 1 + q)

(3.2)

where -q is an undetermined mesoscale convergence parameter. As in Kuo (1974), we introduced a moistening parameter b that is defined by the

308

T. N. KRISHNAMURTI

relation

M

=

+ q)b,

ZL(I

R = ZL(I

+ q)(l - b)

(3.3)

where M is the part of the net supply that goes into moistening and R denotes the rainfall rate. Following Kanamitsu (1973, we may write a maximum supply required to produce a grid-scale cloud by the expression

Here AT denotes a cloud time scale. As noted by Kanamitsu (1975) and Krishnamurti et al. (1983a), the last term in (3.4) permits a smooth transition from convective to stable large-scale condensation heating when it is encountered. This maximum supply is further divided into the two respective parts,

Q

+ QO the respective proportions of moistening and rain, i.e., =

(3.5)

Qq

and Thus, once the two unknowns of the problem, -q and b, are known, uq and ae are determined. The prediction equations take the form

ae

a0 dP

-+V*VVB+w-=aa, dt

d0

rsiO1 “ap

(3.8)

4s, - T4

(3.9)

-

+

and

a

4 at

+ v .vq = U

These are, respectively, the thermodynamic and the moisture equations. For the sake of the present discussion, we have only considered the convective parameterizations, other sources and sinks of the problem are, of course, added on to the right-hand side of (3.8) and (3.9). The multiple-regression approach based on GATE observations consisted of regressing the quantities M/ZLand R/ZLagainst a large number of large-scale variables. That was done utilizing the special GATE ship array data sets. Screening regression (stepwise linear regression) showed that

NUMERICAL WEATHER PREDICTIONI N LOW LATITUDES

309

the most promising candidates for regression were the vertically integrated vertical velocity w and the relative vorticity 5 at 700 mb (where the amplitude of GATE waves, i.e., the African waves, were the strongest). Thus we have the additional relations

MIIL = a l l + blw

+ CI = a25 + bzw + c2

(3.10)

RIIL

(3.11)

The best-fit values of the constants a t , b1, c1, u 2 , b 2 ,and c2 are described in Krishnamurti et al. (1983a). In the course of numerical weather prediction, the predicted values of 5 and w are used to determine MIIL and RIIL from (3.10) and (3.1 1); they in turn determine b and q from a solution of (3.3). Finally, the magnitudes of a4 and ae, provided by (3.6) and (3.7), close this system of parameterization. Discussions on the performance of this scheme in semiprognostic and prognostic applications were presented in Krishnamurti et al. (1983a, 1984). The scheme provided reasonable measures of heating and rainfall; however, it has been found to be somewhat deficient in describing the vertical distribution of moistening; excessive moistening in the planetary boundary layer seems to be related to an absence of a downdraft mechanism in deep convection. The results of calculations of semiprognostic estimates of rainfall rate during the third phase of GATE were compared with observed estimates. These are shown in Fig. 11. The correspondence of semiprognostic estimates to observed ones is quite reasonable. Similar rigorous tests of cumulus parameterization schemes in the prognostic context are usually not possible due to a lack of the observed measures of heating, moistening, and rainfall rates. Recently, I have carried out some experiments with the assistance of my colleagues Dr. Richard Pasch and Mr. Simon Low-Nam. In these tests we selected African waves that arrived over the GATE ship array some 48 hr after the initial state from West Africa. The obvious advantage in the selection of these cases is that one can compare predicted values (at around 48 hr) of the heating, moistening, and rainfall rates with the observed counterparts. The observed motion field at hours 24,48, and 72 (during September 4-6, 1974) are shown in Fig. 12a-c. This illustrates the westward passage of an easterly wave at 850 mb during GATE. The results of actual numerical weather prediction [with a regional multilevel grid-point model described in Krishnamurti et al. (1979)] of this wave is shown in Fig. 13a-c. The model carries the easterly wave westward with a reasonable phase speed over the GATE ship array during these 72 hr. These are the 850-mb flow fields at hours 24, 48, and 72. The observed and predicted rain over the GATE ship array around 12ON-17"W is shown in Fig. 14. The 12-hourly

70W

60W

50W

40W =30W

2OW

IOW

-

OE

IOE

70W 6 0 W 5oW 4 0 W 30W 20W IOW OE IOE FIG. 12. The motion field based on observations at 700 mb on September 4(a), 5(b), and 6 ( c ) , 1974 (122) over the GATE domain. The forecasts described in this section started on September 3, 1974 (122). 3 10

4OW

30W

20W

low

OE

1 0E

20E

40W 30W 20w low OE IOE 20 E FIG. 13. Results of a 72 numerical weather prediction with a regional multilevel grid-point model, corresponding, respectively, to the map times of the 700-mb flow fields shown in Fig. 12 for September 4(a), 5(b), and 6 ( c ) . 1974 (122). 311

312

T. N. KRISHNAMURTI I5

-

13 -

14

12

-

II

-

10

m

-

w

9-

a a

8-

F 7-

6-

54-

3-

O

122 18 00 06 3RD SEPT

12

ia

00

0s

12

ia

00

os

12

ia

00 122 ?TI4 S€PT

00

TIME FIG. 14. Histograms of the observed and predicted rain at 12"N, 17"W obtainr:d from radar-rain -gauge-based observations and the regional multilevel primitive-equation model.

totals for the same 96-hr forecast are shaded. The predicted rain at this location is somewhat underestimated. The discrepancy in part is attributed to the resolution of the numerical prediction model (11 levels, 100km mesh). The observed rain is based on the calibration of radar reflectivity that integrates the rain over a much smaller resolution. This test of the cumulus parameterization appears satisfactory, although one must note that the cumulus parameterization via the regression approach was developed from GATE data sets; thus a prognostic test with the same data sets is not entirely independent. The predicted and observed vertical profiles of the heating and moistening profiles are shown in Fig. 15. These vertical profiles are for the period of heaviest rain on September 5, 1974. The correspondence of the calculated profile Ql (apparent heat source) to the observed is in reasonable agreement, while that of the Q2 (the apparent moisture sink) is poor. We believe that further work is necessary in this area of Parameterization.

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

313

3.2. Radiative Parameterization The present version of our multilevel grid-point and global spectral models includes a fairly sophisticated radiative parameterization based on Chang (1979). The various elements of this parameterization include (1) emissivity method for calculating long-wave radiation, (2) absorptivity method for calculating short-wave radiation, (3) energy balance at the Earth’s surface, (4) variable zenith angle of the sun, (5) definition of clouds based on relative humidity thresholds and cloud feedback process, (6) prescribed surface albedo, and (7) ground wetness defined as a function of surface relative humidity and surface albedo. The obvious question is how important is a specification of the radiative process for short- to medium-range numerical weather prediction in low latitudes. The downward branches of the Hadley-Walker circulations occupy a much larger area of the tropics and subtropics compared to the ascending branches along the ITCZ. The essential balance in these downward regions is one between radiative cooling and adiabatic warming. Cooling rates on the order of 1 to 3°C per day are obtained from simple order of magnitude considerations. An absence of radiative cooling results in rather large accumulative errors in medium-range numerical weather prediction. The divergent wind exhibits a major deterioration in time. On more regional and shorter time scales, the inclusion of the diurnal cycle of radiative heating and cloud feedback processes have been shown to affect the amplitude of eddy motions over the GATE region on the time scale of 3 to 4 days (Krishnamurti et al., 1979; Slingo, 1980). Figure 16 shows the time evolution of meridional kinetic energy in the prediction of an African wave (September 4-7, 1979). Results of three different experiments from the study of Krishnamurti et al. (1979) are shown here. These show, respectively, the results for (1) no heating, (2) complete heating with a fixed zenith angle of the sun, and (3) diurnally varying solar radiation, cloud feedback, and complete heating. It appears from these studies carried out at Florida State University and the U.K. Meteorological Office that a strong interaction between the diurnal and African waves occurs with a transfer of energy to the latter. These results are still quite preliminary since the precise mechanism for such energy exchanges have not been clarified. In the experiment with no heating, the differential heating between West Africa and over the ocean to its south is absent. This is shown to result in a collapse of the West African monsoon in a matter of 3 to 4 days. The role of heating is evidently quite important in supplying energy to the monsoon (basic flows) and to the waves via instabilities and interactions.

314

T. N. KRISHNAMURTI 100-

-m

200

-

300

-

400-

z

Y

5003

c

cn

n

-

!A 600-

LL L

L

-

700 -

800

-

Go/ DAY FIG.15. (a) The observed and the predicted vertical profiles of the apparent heat source ("C/day) between hours 36 and 48. (b) The analogous vertical distribution for the apparent moisture sink. Solid line indicates observed values, broken line predicted.

Careful analysis of the role of radiative heating on the maintenance of heat lows that appear in proximity to the African and Asian monsoon is required. Such sensitivity studies in the prediction context deserve to be addressed. 4. MEDIUM-RANGE PREDICTION OF MONSOON DISTURBANCES Two studies on tropical cyclogenesis, carried out with the global model, will be presented here. These are real-data forecasts on the medium-range time frame. The data sets for these experiments were extracted from the FGGE and MONEX during June and July 1979. In both instances, the FGGE IIIb data analysis produced by the ECMWF were used as a firstguess field and additional MONEX data sets were incorporated via a simple successive correction method. The ECMWF analysis scheme is

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES 100

3 15

-

2cQ-

300 -

-

-r$ m

400-

500-

3 v)

v)

$

600.

a 700 -

800

-

900. \

-4 -2

( b) 0

2

4

6

8

10

12

14

Co / DAY FIG.15. (Continued)

described in Lorenc (1981) and the FSU analysis is described in Krishnamurti et af. (1983b). The following is based on recent studies of Krishnamurti et al. (1983b, 1984). The onset of monsoon rains commenced over central India around June 18. The circulations on June 11 were typical of the preonset period, as in Fig. 17. This shows the streamlines and isotachs over the Indian region. The ensuing week was characterized by a buildup of strong low-level westerlies over the Arabian Sea and the formation of a tropical storm (named the onset vortex) over the eastern Arabian Sea. This storm eventually moved northward and finally northwestward toward the Arabian coast prior to its dissipation (Krishnamurti et af., 1981). Numerical prediction of the onset during 1979 raised at least three challenging problems, namely, the prediction of the buildup of westerlies, the formation and motion of the onset vortex, and the commencement of rains. A large number of prediction experiments were carried out to assess the impact of data, physics, resolution, and the definition of orography. The

316

I 0

T. N. KRISHNAMURTI

I

I

I

I

I

15

30

45

60

75

I

1

9 0 9 6

Forecast t i m e in hours

FIG. 16. The results based on numerical weather prediction (96-hr) during GATE (Krishnamurti et af., 1979). Here the time evolution of eddy kinetic energy over the GATE domain of Fig. 13 is shown for three different radiative heating parameterizations.

studies clearly showed that the dense MONEX observations were a critical addition to the FGGE data sets. Figure 18 illustrates a sample printout of the data at 850 mb; this includes observations from a variety of surfaceand space-based platforms. The critical data sets are the high-resolution cloud winds from geostationary satellites and the soundings from dropwindsonde research aircraft and research ships. Our results show that the prediction of the onset, with the global model, were vastly superior with these data sets. We have not examined the details of the onset with respect to its sensitivity to various parameterizations of the planetary boundary layer and the radiative processes. However, we have examined the sensitivity of the monsoon onset to various versions of the cumulus parameterization discussed in Krishnamurti et al. (1983a). Such tests were also carried out by the European Centre for Medium-Range Weather Forecasts, the U.K. Weather Service, and French Weather Service.

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

317

FIG. 17. Streamlines and isotachs (m s - ' ) over the MONEX domain on June 11, 1979, 122. This is the initial state over this region in a global medium-range prediction.

FIG.18. Typical data distribution during MONEX and streamline isotach (m s-I) analysis at 850 mb. July 7, 1979, 122.

318

T. N. KRISHNAMURTI

These studies show that the classical Kuo scheme underestimates heating, and the consequent evolution of the monsoon onset is very slow; even after 7 days none of the aforementioned salient features are described by the model. A version of the Kuo scheme in which the moistening parameter b is set equal to zero and the available supply of large-scale moisture is used to provide heating is a superior scheme. Although the onset of strong monsoon westerlies and the commencement of monsoon rains are reasonably predicted by this method, it fails to simulate a reasonable structure of the onset vortex or its track. Resolution experiments were carried out with respect to both horizontal and vertical resolution of the spectral model. When 29 waves (rhomboidal) and 5 vertical levels were used, a forecast of the onset was found to be quite poor. When 29 waves (rhomboidal) and I 1 vertical levels were used, the forecasts showed a marked improvement in simulating the onset of monsoon westerlies in the lower troposphere. Further experiments were continued with 42-wave triangular truncation. That version of the model produced the best results when an enhanced orography was included. This was the envelope orography proposed by Wallace et al. (1983). Experiments with and without the envelope orography showed that the mountain chains around the Arabian Sea and the Himalayas had an important role in the evolution of the monsoon circulations. Around the Arabian Sea the principal mountain ranges are the Western Ghats along western India, the Madagascar Mountains, the East African Highlands, and the Ethiopian Mountains. The envelope orography is steeper compared to the normal orography. It adds almost a kilometer to the heights of each of these principal mountain chains. The original tabulation of mountains comes from a U.S. Navy tape of the orography on a 10-min resolution. The transform grid for a 42-wave triangular truncation has a resolution of around 200 km. The Gaussian grid elementary squares of the transform grid contain about 144 high-resolution orography grid points. A mean height h and a standard deviation u of the high-resolution grid data are evaluated for each of the elementary Gaussian grid squares. The envelope orography used here is defined by the relation h=h+2u Figure 19a-c illustrates the observed and predicted motion field at 850 mb on day 6 of the prediction. The observed (Fig. 19a) field illustrates the strong monsoonal flows and onset vortex occupying most of the northern Arabian Sea. The prediction with the regular orography (Fig. 19b) is not as impressive as that carried out with the envelope orography (Fig. 19c). The major defect in the prediction of the onset with the regular orography was in the path of the onset vortex. Although this storm formed at the

3 19

NUMERICAL W E A r H E K PRE1)IC'I'ION I N LOW LATITUDES

TABLE111. RMS ERROR OF VECTOR WIND,30"N-3OoS, 850 MB (METERSPER SECOND)

Mode 1Ier

1

2

3

4

5

6

7

ECMWF 1 ECMWF2 FSU NMC RPN Persistence

3.67 3.69 4.4 3.5 4.3 4.2

5.01 4.98 5.5 4.7 5.7 5.7

5.91 5.87 5.7 5.8 6.7 7.0

6.36 6.27 6.1 6.5 7.1 6.6

6.50 6.38 6.5 6.6 7.6 6.7

6.99 6.76 6.8 6.9 8.7 7.4

7.25 6.78 6.7 6.9 8.9 6.8

correct time and place, it first moved eastward into India prior to an eventual westward motion toward the northern Arabian Sea. This eastward motion was entirely absent when the envelope orography was deployed. Thus it appears that the offshore meridional motion of such storms is strongly controlled by a steeper orography. The prediction of the track with the envelope orography was nearly accurate to about 7 days. Other aspects of this study (vertical structure, mechanisms of onset) are discussed in Krishnamurti et al. (1983b). A detailed intercomparison of forecasts for this same storm was carried out by about seven modeling groups. The details of these intercomparisons are presented by Temperton et al. (1983). In these studies the focus was on the aforementioned features of the monsoon and on the error statistics of the respective models. Tables I11 and IV show the root-meansquare wind errors at 850 and 200 mb over the global tropics for these intercomparisons. There are some marked differences in the performance of the different models in the tropics. Temperton et al. (1983) have alluded these differences largely to resolution and cumulus parameterization schemes deployed within each model.

TABLEIV. RMS ERROROF VECTORWIND, 30"N-305, 200 M B (METERSPER

SECOND)

Day ~

Modeller ECMWF 1 ECMWF 2 FSU NMC RPN Persistence

~

1

2

3

4

5

6

7

6.62 6.66 7.6 4.7 7.5 8.3

9.78 9.73 10.6 9.3 10.8 10.9

11.39 11.37 13.8 11.5 14.0 13.2

12.46 12.35 15.8 14.3 16.1 15.1

13.59 13.32 17.7 13.3 18.0 16.0

14.75 14.40 18.7 14.4 20.3 16.7

15.75 15.24 19.6 15.6 22.9 16.1

320 L(

T.N. KRISHNAMURTI

ON

30N

ON

305 3

'iON

30k

Oh

30s 3

FIG.19. Observed and predicted 144-hr wind field over the MONEX domain on June 17, 1979, 122. (a) Streamlines and isotachs based on observations. (b) Predicted fields with regular orography. (c) Predicted field with envelope orography (speed m s-l).

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

FIG.

32 1

19. (Continued)

The second major study was on the formation of a monsoon depression that formed over the northern Bay of Bengal around July 4-5, 1979. A number of experiments, all starting on July 1, 1979, were carried out for 10 days with the global model. As before, the best skill in predicting cyclogenesis was noted with a higher-resolution version of the model (42 waves triangular and 11 levels). Major improvements occurred when a reanalysis of the humidity was based on the proposed physical initialization described earlier. The initial state for this case was characterized by zonal westerlies in the lower troposphere over the northern Bay of Bengal. Figure 20 illustrates the flow field at 850 mb on July 1 , 1979, 122. Results of numerical weather prediction at day 6 of integration (July 7, 1979, 122) are shown in Fig. 21a,b. Figure 18 is the observed field for July 7, 1979, 122. The two numerical prediction experiments respectively denote results with and without the proposed physical initialization. The inclusion of physical initialization improves the initial humidity analysis and also provides an improvement of diabatic heating and the initial rainfall rates. Both experiments succeed in the formation of the depression, although the intensity of landfall around day 6 of the forecast is better described by the experiment with the physical initialization. The track of the depression is due westward and is handled quite well by the global model.

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

323

We have described the formation of two tropical depressions on the medium-range time scale. Are these just two isolated examples of success or is there a message here? The results here are based on a gradual evolution of models and the data base. It is our contention that the combination of the FGGE/MONEX did provide an unprecedented data set that was not available over most other regions of the tropics. This data set enabled us to define somewhat better initial rainy regions, initial diabatic forcing, and initial divergent wind (Krishnamurti et al., 1984). The divergent wind errors, especially on the large planetary scale, were smaller when the physical initialization was invoked. We have also investigated the energy transformations over a local domain, emphasizing the role of the horizontal shear on monsoonal low-level flow in the initial stage and thereafter the importance of cumulus convection that aided the transfer of eddy available potential to the eddy kinetic energy. Since the formation and motion of a monsoon depression is a rather important problem in the Indian subcontinent, it is necessary that further studies on this problem be continued with several other cases. The FGGE/MONEX data sets shown in Fig. 18 was an exceptional situation. Only three well-defined monsoon depressions formed during that summer. Thus only limited studies with a larger sample of storms defined by adequate initial observations are possible at this stage. The recent geostationary satellite INSAT appears very promising for providing high-resolution cloud winds. These data sets along with a collection of commercial ship and aircraft observations can provide a useful data base for such future studies. With respect to the question whether models need be global or whether these studies can be carried out with regional models, it seems that the planetary-scale aspects are quite important for mediumrange prediction and the global model does seem to be far superior, especially due to the importance of a long fetch of the cross-equatorial flows.

5. ON THE PREDICTION OF THE QUASI-STATIONARY COMPONENT Joseph Smagorinsky addressed the problem of predicting the quasistationary component in 1953. He stressed the importance of zonally asymmetric heat sources (arising from the land-ocean configuration) in determining the middle-tropospheric quasi-stationary components. In the ensuing 30 years much progress has been made in the actual prediction of those quasi-stationary components, although an understanding of the mechanisms for its maintenance remains an unsolved problem. In a large number of prediction experiments with the high-resolution global model (42 waves triangular, 11 levels), full physics, and envelope

324

325

t

N 0

I

$

-

Z

0

2

0 0

Z

t

0

h

v

cn

326

z 0

m

w

a

m

0

Ln

(D

0

ffl

0 a7

cn

FIG.22. (a) 10-day mean 20-mb flow field (June 11-21, 1979, 12z) based on observations. (b) 10-day mean 200-mb flow field (June 11-21, 1979, 122) predicted by the global model.

CD

z 0

Z 0

cn

328

Z 0

m

(3

w

Ln 0

m

cn 0

co

cn 0 m

( b) 90N

60N

30N

ON

305 W

N \o

605

905

0

90U

6OW

30W

OE

FIG.23. (a) 7-day mean 200-mb flow (December 10-16, 1978, 122) based on observations. (b) 7-day mean 200-mb Row field ( December 10-16 1978, 122) predicted by the global model.

330

T. N . KRISHNAMURTI

orography, we noted a considerable skill in the prediction of the timeaveraged flows (up to 7- or 10-day time range). All of these were based in the FGGE/MONEX data set. As stated earlier, it is not possible to determine the elements (data sets, model resolution, initialization, or physical parameterization) that were critical for this success. This requires a large numerical experimentation program in which the sensitivity of the timeaveraged state to a number of such elements needs to be assessed. Such studies are currently being carried out at Florida State University. Figures 22 and 23 illustrate two such results of medium-range predictions at 200 mb. Here (a) illustrates the time-averaged flow field and (b) illustrates the counterpart based on a prediction from the global model. In these instances, we have also noted that the model has a larger skill in the prediction of the rotational wind as compared to the divergent wind. The comparison, illustrated in Figs. 22 and 23, shows that most large-scale time-averaged features are handled extremely well by the global model. A reasonable prediction of a 10-day averaged state simply implies a good still in the daily prediction (up to 5 or 6 days) that was apparent in many of the forecasts.

6. SCOPE OF FUTURE RESEARCH

Given adequate observations from surface- and space-based platforms, the scope for future progress in the numerical weather prediction in low latitudes appears quite promising. Major improvements in the analysis of motion, thermal, mass, and humidity fields will be forthcoming. Here we foresee a phenomenological thrust where the detailed three-dimensional definition of the initial state on many scales will be based on statistical structure functions as well as dynamical principles. Definition of tropical convective areas (based on satellite observations) wil be closely integrated within the analysis schemes. The initialization scheme will retain the gravitational Rossby and mixed mode that are relevant to the description of the divergent motion on many scales. Improved resolution will contain adequate information to predict the initial formation of tropical waves, tropical depressions, and tropical storms. Strategies for nested grids (stationary or movable grids) will be necessary for handling the problems on the formation and motion of hurricanes (or typhoons) starting from a depression or a tropical storm stage. Concurrently, major refinements in the parameterization of the physical processes is expected. Vertical column models of physical parameterization require a detailed synthesis among boundary layer, cumulus convection, and radiative pro-

NUMERICAL WEATHER PREDICTION IN LOW LATITUDES

33 1

cesses. Definitions of clouds, cloud feedback process, diurnal changes, and orographic influences will continue to be refined. The future observing systems will include radiance information from several operational geostationary satellites around the globe. The enhanced WWW and observations from commercial ships and aircraft will provide a major improvement of tropical data base. The large data gaps over tropical oceans and the Southern Hemisphere will rely on remote sensing that will provide measures of sea-surface temperature, Earth radiation budget, rainfall rates, soundings of temperature and humidity cloud winds, cloud cover, and oceanic wind stress (scatterrometer). Thus the opportunities for improved numerical weather prediction and related theoretical developments appears very promising at this stage.

ACKNOWLEDGMENT It was in 1958 that I first had an occasion to hear Professor Joseph Smagorinsky present, in Chicago, an excellent lecture on his multilevel primitive-equation model. He was already quite well known for his work on the role of land-ocean asymmetry in the investigations of the quasi-stationary components. Over the years, many of our students of dynamic and synoptic meteorology have benefited from his contributions and perspectives. The Global Weather Experiment owes its success to Joe in many ways. Having worked closely with him on the Monsoon Experiment, I, like most others. have seen many unique and admirable scientific-administrative qualities in his ways of doing things. The present study also, to me, reflects much of his influence over the years. Our research reported here was supported by the NOAA grant NA 82AA-D-00004 and the NSF grant ATM-8304809 and ATM-7819363-04. We are indebted to Dr. Rex Fleming, Dr. Joseph Huang, Dr. Jay Fein, and Dr. Pamela Stephens for their support. The computations reported here were carried out at the National Center for Atmospheric Research that is sponsored by the National Science Foundation.

REFERENCES Arakawa, A. (1966). Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. J . Cornput. P h y ~l, . 119-143. Arakawa, A., and Schubert, W. H. (1974). Interaction of a cumulus cloud ensemble with the large scale environment. Part I. J . Atrnos. Sc,i. 31, 674-701. Businger, J. A,, and Seguin, W. (1977). Transport across the air-sea interface; sea-air surface fluxes of latent and sensible heat and momentum. In “Proceedings of the GATE Workshop,” pp. 441-453. Natl. Cent. Atmos. Res., Boulder, Colorado. Businger, J. A., Wyngaard, J . C., Izurni, Y . , and Bradley, E. F. (1971).Flux profile relationship in the atmospheric surface layer. J . Atrnos. Sci. 28, 181-189.

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Cadet, D. (1983). Mean fields of precipitable water over the Indian Ocean during the 1979 summer monsoon from the TIROS-N sounding system. In “Satellite Hydrology” (M. Deutich, D. R. Wiesnat, and A. Rango, eds.), 5th Annu. Symp. Am. Water Resour. ASSOC.,pp. 115-123. Chang, C. B. (1979). “On the Influence of Solar Radiation and Diurnal Variation of Surface Temperatures on African Disturbances,” Rep. No. 79-3, pp. 1-157. Dept. of Meteorology, Florida State University, Tallahassee. Charney, J. G. (1963). A note on large-scale motions in the tropics. J. Atmos. Sci. u),607609. Daley, R. (1981). Normal mode initialization. Reu. Geophys. Space Phys. 19, 450-468. Emco, R. M. (1984). The dynamical balance of a general circulation model. M o n . Weather Rev. (to be published). Gruber, A., and Watkins, C. D. (1979). Preliminary evaluation of initial atmospheric moisture from the TIROS-N sounding system. In “Satellite Hydrology” (M. Deutich, D. R. Wiesnet, and A. Rango, eds.), 5th Annu. Symp. Am. Water Resour. Assoc., pp. 115123. Holton, J. R. (1969). A note on the scale analysis of tropical motions. J. Atmos. Sci. 26,770111. Holton, J. R. (1972). “An Introduction to Dynamic Meteorology.” Academic Press, New York. Kanamitsu, M. (1975). “On Numerical Prediction over a Global Tropical Belt,” Rep. No. 75-1, pp. 1-282. Dept. of Meteorology, Florida State University, Tallahassee. Kitade, T. (1983). Nonlinear normal mode initialization with physics. Mon. Weather Reu. 111, 2194-2213. Krishnamurti, T. N. (1962). Numerical intergration of primitive equations by a quasi-lagragian advective scheme. J . Appl. Meteorol. 1, 503-521. Krishnamurti, T. N. (1971). Tropical east-west circulations during the norther summer. J . Atmos. Sci. 28, 1342-1347. Krishnamurti, T. N., and Ramanathan, Y.(1982). Sensitivity of monsoon onset of differential heating. J . Atmos. Sci., 39, 1290-1306. Krishnamurti, T. N., Pan, H., Chang, C. B., Polshay, J., and Oodally, W. (1979). Numerical weather prediction for GATE. Q. J . R. Mereorol. Soc. 105, 979-1010. Krishnamurti, T. N., Pasch, R. J., and Ardanuy, P. (1980a). Prediction of African waves and specification of squall lines. Tellus 32, 215-231. Krishnamurti, T. N., Ramanathan, Y ., Pan, H . , Pasch, R., and Molinari, J. (1980b). Cumulus parameterization and rainfall rates. I. Mon. Weather Rev. 111, 815-828. Krishnamurti, T. N., Ardanuy, P. A., Ramanathan, Y., and Pasch, R. (1981). On the onset vortex of the summer monsoon. M o n . Wearher Reu. 109, 344-363. Krishnamurti, T. N., Low-Nam, S., and Pasch, R. (1983a). Cumulus parameterization and rainfall rates. 11. M o n . Weather Reu. 111, 815-828. Krishnamurti, T. N., Pasch, R., Pau, H., Chu, S., and Ingles, K. (1983b). Details of lowlatitude numerical weather prediction using a global spectral model. I. J. Meteorol. Sac. Jpn. 61, 188-207. Krishnamurti, T. N., Ingles, K., Cocke, S., Pasch, R., and Kitade, T. (1984). Details of lowlatitude medium-range numerical weather prediction using a global spectral model. 11. J . Meteorol. SOC. Jpn. (to be published). Kuo, H. L. (1974). Further studies of the parameterization of the influence of cumulus convection on large-scale flow. J . A m o s . Sci., 31, 1232-1240. Lord, S . J. (1982). Interactions of a cumulus cloud ensemble with the large-scale environment. 111. Semiprognostic test of the Arakawa-Schubert theory. J . Atmos. Sci. 39,88103.

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Lorenc, A. (1981). A global, three-dimensional, multivariate statistical interpolation scheme. Mon. Weuther Rev. 109, 701-721. Machenhauer, B. (1977). On the dynamics of gravity oscillations in a shallow water model, with application to normal mode initialization. Beitr. Phys. Artnos. 50, 253-271. Mathur, M. B . (1970). A note on an improved quasi-Langragian advective scheme for primitive equations. M o n . Wenfher Reu. 98, 214-219. Miyakoda, K . , and Moyer, R. W. (1968). A method of initialization for dynamical weather forecasting. Tellus 20, 113-128. Oort, A. H., and Peixot, J. P. (1983). Global angular momentum and energy balance requirements from observations. In “Theory of Climate” (B. Saltzman, ed.), pp. 355-490. Academic Press, New York. Pun, K., and Bourke, W. (1982). A scheme to retain the Hadley circulation during nonlinear normal mode initialization. Man. Weuther Reu. 110, 327-335. Slingo, J . M. (1980). A cloud parameterization scheme derived from GATE data for use in a numerical model. Q . J . R . Meteorol. S O C . 106, 747-770. Smagorinsky, J. (1953). The dynamical influence of large-scale heat sources and sinks o n the quasi-stationary mean motions of the atmosphere. Q . J . R. Metearol. Soc. 79,342-366. Smith, W. L., Woolf, H. M., Hayden. C. M., Wark, D. Q . , and McMillin, L. M. (1979). The TIROS-N operational vertical sounder. Bull. A m . Mereorol. Soc. 60, 1177-1187. Temperton, C., Krishnamurti. T. N., Pasch, R., and Kitade, T. (1983). “WGNE Forecast Comparison Experiments,” Rep. No. 6, pp. 1-104. WCRP, World Climate Research Program, World Meteorol. Organ., Geneva. Thompson, R. M., Jr., Payne, S . W.. Recker, E. E., and Reed, R. J. (1979). Structure and properties of synoptic-scale wave disturbances in the intertropical convergence zone of the eastern Atlantic. J . Atmos. Sci. 36, 53-72. Wallace, J. M . , Ribaldi, S . , and Simmons, A . J . (1983). Reduction of systematic forecast errors in the ECMWF model through the introduction of envelope orography. Q . J . R . Meteorol. Soc. 109, 683-718.

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Part IV

TURBULENCE AND CONVECTION

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SUB-GRID-SCALETURBULENCE MODELING J. W. DEARDORFF Department of Atmospheric Sciences Oregon State University Coruallis, Oregon 1.

2. 3. 4. 5.

Introduction: The Need for Grid-Scale Reynolds Averaging . . . . . . . . . . . . The Effect of Grid-Volume Reynolds Averaging . . . . . . . . . . . . . . . . . The Sub-Grid-Scale Eddy Coefficient . . . . . . . . . . . . . . . . . . . . . . Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . FutureOutlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

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1. INTRODUCTION: THENEEDFOR GRID-SCALE

REYNOLDS AVERAGING Sub-grid-scale turbulence modeling involves the direct numerical integration of the governing equations of motion, continuity, and (usually) thermodynamics within a domain of size sufficient to capture or resolve most of the kinetic energy of the system being modeled. The domain is subdivided into a network of grid points at which the physical variables are centered or into an equivalent array of Fourier spectral components, and future values of the variables are predicted numerically step by step in time. The grid scale is the distance between these grid points in any of the three coordinate directions. However, scales of motion smaller than two grid intervals in length can never be resolved by the numerical model, although they, of course, exist within the system simulated. Since the governing equations contain partial derivatives of the dependent variables, and since the derivatives within the actual system modeled almost always contain most of their intensity on scales much smaller than the grid scale, the governing equations must first be averaged or filtered, at least conceptually, to remove these high wave numbers and high-frequency fluctuations. Only then can the (filtered) derivatives be approximated by differences in values separated by one or two grid intervals along the appropriate spatial coordinates. This averaging process is, therefore, not performed over entire slices of the physical domain, nor over an ensemble of realizations, as in ensemble mean turbulence modeling. Instead, the averaging, named after Reynolds (1895), is applied rather locally over limited volumes of space of the order of several Ax * Ay Az, where Ax, Ay, Az are the grid intervals of the 337 ADVANCES IN GEOPHYSICS. VOLUME

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numerical model. (This small-scale Reynolds averaging can also be considered to extend over time, by an interval comparable to At, the model time-step increment. However, this usually does not seem necessary since numerical stability requirements demand that A t be sufficiently small that calculated changes in time proceed smoothly if spatial gridscale averaging has been performed.) In this way, the motions on scales greater than twice the respective grid intervals can be resolved explicitly, provided that the effect of the sub-grid-scale motions has been satisfactorily represented.

2. THEEFFECTOF GRID-VOLUME REYNOLDS AVERAGING The sub-grid-scale effects show up within the Reynolds averaged equations as extra terms called Reynolds fluxes, or Reynolds stresses in the case of the equations of motion. Only nonlinear terms generate these Reynolds fluxes. Interestingly, the form of the Reynolds-averaged equations depends little if at all on the scale of filtering employed or on the grid interval of the model. Hence, the equations as averaged after the method of Reynolds (1895) often appear in the literature without any mention of the averaging scale that has been conceptually employed and sometimes without any indication that averaging has been performed. It is usually left to the reader to deduce whether an averaging scale extending on the order of a grid interval on each side of each model grid point is involved along the three spatial coordinates. The grid-scale averaging can be considered to be of various types (e.g., top-hat, Gaussian, damped cosine) and to extend over a volume (surrounding each grid point or surrounding all other points, too) whose size depends on the whim of the investigator. How does the size of the averaging volume affect the sub-grid-scale Reynolds fluxes that emerge from the model calculations? Clearly, the greater the Reynolds-averaging scale is considered to be, the greater should be the fraction of the total flux calculated by the model that resides on the subgrid scale and the smoother should be the explicitly resolved velocity field calculated by the model. If the sub-grid-scale Reynolds flux is approximated by a sub-grid-scale eddy coefficient multiplying the appropriate velocity or scalar gradient within the model, the effect of utilizing an increased Reynolds-averaging volume is thus implemented by utilizing an increased sub-grid-scale eddy coefficient. Also, the greater the Reynolds-averaging scale is considered to be, for a given model grid network, the more closely Reynolds (1895) averaging assumptions will be satisfied locally. Only if these are closely satisfied

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will the total turbulent flux equal the sum of the resolvable-scale flux and what is usually considered the sub-grid-scale flux. For a given type of sub-grid-scale averaging, one expects that there exists an optimum Reynolds-averaging volume in relation to the grid scales, for which the net turbulence statistics calculated by the model (sum of resolvable and subgrid scales) agree most closely with available turbulence statistics of the system being simulated. In particular, the model-output spectral statistics for wavelengths of four to two grid intervals may be examined to see whether they agree well with the statistics for the actual system after the latter have been filtered by using the same filter as employed on the governing equations. (This information may be known, for example, if the model grid interval lies within the inertial subrange of the actual turbulence being simulated.) If the model uses too little or no Reynolds averaging, this fact will quickly make itself known within a three-dimensional numerical integration at large Reynolds number as a piling up of turbulent intensity on a scale of two-grid intervals. But due to the great computing expense and research time necessary to perform “large-eddy” turbulence modeling numerically, the question of optimal Reynolds-averaging filter has not received enough study. An excellent discussion on sub-grid-scale modeling for large-eddy simulations is that of Wyngaard (1982). In that discussion, Wyngaard mathematically points out the full meaning of the topic and recent directions that large-eddy models have taken in relating the scales of motion too small to resolve back to those that are explicitly calculated by the model. Therefore the present discussion is brief and omits mathematical aspects. The expense to be paid for large-eddy simulations yields two great advantages over ensemble-mean methods. First, since the energy-containing range of the turbulence is directly resolved, the role of coherent structures can be examined, and many detailed statistics on resolvable scales can be explored [e.g., see Peskin (1974) and Moin and Kim (1982)l. Second, the turbulence-closure assumptions that must be made for the sub-grid-scale Reynolds fluxes are less crucial to the model results than are those that must be made within ensemble-mean turbulence models (which are therefore much cheaper to run on the computer). In particular, once grossly satisfactory results are obtained in a problem by using a model with a particular grid network, improved results are almost guaranteed to occur with an increase in computer resolution, provided that the sub-grid-scale eddy coefficient depends on grid interval in a manner to be mentioned. It may be noted that in most instances large-eddy simulations are carried out for the purpose of examining some aspects of the statistics of the

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turbulznce, and not for the purpose of predicting the detailed time-dependent, irregular shapes of these eddies as they evolve from particular initial conditions. For this reason, the initial conditions utilized are largely irrelevant, with up to 500 to 1000 time steps needed to reach a stage at which the large eddies are in statistical equilibrium with the calculated mean field gradients. Increases in computer speed and capacity are thus always welcomed in large-eddy modeling since the increased resolution that may be etiployed poses no difficulty in acquiring the arbitrary initial conditions. 3. THESUB-GRID-SCALE EDDYCOEFFICIENT

To Wyngaard’s (1982) discussion, one need add only a historic note: that sub-grid-scale modeling originated in the paper of Smagorinsky (1963). In that key paper, Smagorinsky established the form of the subgrid-scale eddy coefficient K that still receives much use today. That is, K is proportional to the magnitude of the resolvable-scale velocity deformation, multiplied by the square of the representative grid interval. Smagorinsky (1963) applied this eddy diffusion to the quasi-two-dimensional large-scale atmospheric general circulation, utilizing horizontal velocity components in calculating the velocity deformation. It was up to Lilly (1967) to show that for three-dimensional turbulent motions Smagorinsky’s formulation was exactly what was needed to cascade resolvable-scale turbulence energy, on scales lying within the inertial subrange, to the sub-grid scale at which the dissipation of turbulence energy occurs. Lilly (1967) also provided the first realistic estimate of the coefficient of proportionality in the sub-grid-scale eddy-coefficient formulation. A more recent estimate has been made by Yoshizawa (1982). It is perhaps ironic that Smagorinsky’s (1963) sub-grid-scale formulation has found more use in three-dimensional large-eddy turbulence modeling than in the originally intended global-circulation applications. Since the 1960s, the need for horizontal eddy diffusion in global circulation models has fallen sharply as improved finite-difference methods [e.g., Arakawa (1966) and Phillips (1959)I were developed that eliminated numerical instabilities. At the same time, it became realized that in quasigeostrophic or two-dimensional flow there is only a slight cascade of kinetic energy to scales much smaller than those at which energy is fed in (Kraichnan, 1967; Lilly, 1971), in great contrast to three-dimensional turbulent flow. There has been some study on what the most appropriate horizontal eddy-diffusion formulation may be for two-dimensional flow (Leith, 1968). However, because of the presently perceived relative unim-

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portance of horizontal diffusion on synoptic scales, some global circulation models presently do not use any explicit horizontal sub-grid-scale eddy diffusion (Gates and Schlesinger, 1977).

4. RECENTDEVELOPMENTS The exclusive use of a sub-grid-scale eddy viscosity has been found by Clark et al. (1977) and more extensively by Bardina et al. (1983) to yield local Reynolds stresses that are very poorly correlated with the actual ones, as determined from calculations on a grid mesh of up to (1 28)3. Only the net dissipation provided by the eddy-viscosity model and the associated net transfer of energy from resolvable to sub-grid scales can be made correct. Bardina et al. (1983) were able to develop a much improved subgrid-scale Reynolds stress formulation that makes explicit use of the smallest resolvable eddies. The improvement is especially marked for shear-flow turbulence. Interestingly, the new formulation, the "scalesimilarity model," still includes the Smagorinsky sub-grid-scale eddy viscosity K to provide the dissipative property needed. Their work, along with that of McMillan and Ferziger (1979), also goes far in explaining a puzzling finding of Deardorff (1971) that the K proportionality constant needs to be reduced by a factor of at least two in flows dominated by mean shear. That result is now explained as a stabilizing rotational effect of the mean shear that vanishes only for irrotational flow. In the work of Bardina et al. (1983), an accurate defiltering method is also developed for representation of the sub-grid-scale turbulence energy itself, which must be added to the energy of the explicitly calculated, filtered motions to obtain the net turbulence energy. By using a (32)3grid, excellent comparisons were obtained against measurements of homogeneous-shear turbulence and turbulence in rotational flow. 5 . FUTURE OUTLOOK

Because the calculating power of large computers is continuing to advance steadily, the outlook is promising that the cost of performing largeeddy simulations will continue to decline significantly, while the realism of the model results will continue to improve. Among the outstanding problems still to be solved, however, are (1) what is the most appropriate filter for use in grid-scale Reynolds averaging, (2) just how the filter and averaging volume influence the magnitude of the sub-grid-scale eddy coefficient, (3) how K should be affected by use of grid intervals that may

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differ in the three-coordinate directions [see Deardorff (1971)], (4) how the K formulation may differ for grid volumes adjacent to a bounding surface (Schumann, 1975), (5) how much larger can the K for scalar subgrid-scale diffusion be than that for momentum, and (6) what are the conditions under which the K-type sub-grid-scale closure or the scalesimilarity model may need to be replaced by more complicated secondorder sub-grid-scale modeling [see Deardorff (1973)] in representation of the locally varying sub-grid-scale Reynolds stresses.

REFERENCES Arakawa, A. (1966). Computational design for long-term numerical integration of the equations of fluid motion. Two-dimensional incompressible flow. Part I. J . Cornput. Phys. l, 119-143. Bardina, J., Ferziger, J. H., and Reynolds, W. C. (1983). “Improved Turbulence Models Based on Large Eddy Simulation of Homogeneous, Incompressible, Turbulent Flows,” Rep. TF-19. Thermosci. Div., Dept. of Mechanical Engineering, Stanford University, Stanford, California. Clark, R. A., Ferziger, J. H., and Reynolds, W. C. (1977). Evaluation of subgrid scale turbulence models using a fully simulated turbulent flow. J. Fluid Mech. 91, 1-16. Deardorff, J. W. (1971). On the magnitude of the subgrid scale eddy coefficient. J. Cornput. Phys. 7, 120-133. Deardorff, J. W. (1973). The use of subgrid transport equations in a three-dimensional model of atmospheric turbulence. J . Fluid Eng. 95, 429-438. Gates, W. L., and Schlesinger, M. E. (1977). Numerical simulation of the January and July global climate with a two-level atmospheric model. J. Afmos. Sci. 34, 36-76. Kraichnan, R. (1967). Inertial ranges in two-dimensional turbulence. Phys. FtUids 10, 14171423. Leith, C. E. (1968). Two-dimensional eddy viscosity coefficients. In Proc. WMO-ZUGG Symp. Numer. Weather Predict. Japan Meteorological Agency, Tokyo. Lilly, D. K. (1967). The representation of small-scale turbulence in numerical simulation experiments. Proc. ZBM Sci. Comput. Symp. Enuiron. Sci., IBM Form No. 320-1951, pp. 195-202. Lilly, D. K. (1971). Numerical simulation of developing and decaying two-dimensional turbulence. J . Fluid Mech. 45, 393-415. McMillan, 0. J., and Ferziger, J. H. (1979). Direct testing of subgrid-scale models. AZAA J . 17, 1340-1346. Moin, P., and Kim, J . (1982). Numerical investigation of turbulent channel flow. J . Fhid Mech. 118, 341-377. Peskin, R. L. (1974). Numerical simulation of Lagrangian turbulent quantities in two- and three-dimensions. Adv. Geophys. MA, 141-163. Phillips, N. A. (1959). An example of nonlinear computational instability. In “The Atmosphere and the Sea in Motion” (B. Bolin, ed.), pp. 501-504. Rockefeller Inst. Press in association with Oxford Univ. Press, New York. Reynolds, 0. (1895). On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos. Tmns. R. SOC.London, Ser. A 186, 123-164.

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Schumann, U. (1975). Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J . Cornput. Phys. 18, 376-404. Smagorinsky, J. S. (1963). General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev. 91,99-164. Wyngaard, J. C. (1982). Boundary-layer modeling. In “Atmospheric Turbulence and Air Pollution Modelling” (F. T. M. Nieuwstadt and D. van Dop, eds.), pp. 69-106. Reidel Publ., Boston, Massachusetts. Yoshizawa, A. (1982). A statistically-derived subgrid model for the large-eddy simulation of turbulence. Phys. Fluids 25, 1532-1538.

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ENSEMBLE AVERAGE, TURBULENCE CLOSURE GEORGEL. MELLOR Geophysical Fluid Dynamics Program Princeton University Princeton, New Jersey

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 2. The Turbulence Macroscale and Turbulence Closure . . . . . . . . . . . . . . . . 347 3. Averaging Distance for Measurements in the Atmosphere and Oceans and for Numerical Models.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 4. Numerical Modeling Applications and Horizontal Diffusion. . . . . . . . . . . . . . 355 5. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 357 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. INTRODUCTION

A bit of personal history is appropriate to this paper. In 1966 I was involved in models and experiments on neutral, rotationless, turbulent boundary layers; I had never heard of the Geophysical Fluid Dynamics Laboratory (GFDL) or Joseph Smagorinsky. Shortly thereafter all of this changed; the GFDL and Smagorinsky moved to Princeton, and, not coincidentally, I developed a keen interest in turbulent boundary layers in which the Coriolis parameter f and the square of the Brunt-Vaisala frequency N2 are nonzero. On the other hand, it was useful to me to have initially confined my attention to “simple” turbulent flows corresponding to the limiting cases, f - . 0 and N -+ 0. It is here that comparatively well documented, stationary flows had been created in the laboratory yielding unambiguous turbulence data. Just a few examples are decaying grid turbulence, turbulent wake and jet flows, turbulent boundary layer flows, pipe flows, channel flows, flows with heat transfer (where, nevertheless, Nz is small compared to velocity gradients and the flows are nearly neutral), and flows over curved walls that tend to be stable or unstable for convex or concave walls, respectively. In 1966 we were just learning that simple, but properly scaled, eddyviscosity hypotheses-requiring only two or three empirical, nondimensional constants-together with the numerical solution of the mean boundary-layer equations of motion, produced an algorithm with a surprisingly wide predictive range. For example, boundary layers, forced by a range of mainstream pressure gradients, were simulated very well (Mel345 ADVANCES IN GEOPHYSICS, VOLUME

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lor, 1967; Mellor and Gibson, 1966; Mellor and Herring, 1968, Part I; Cebecci and Smith, 1968) as they were by other schemes that bore some resemblence to present-day, second-moment closure (Bradshaw and Femss, 1968; Mellor and Herring, 1968, Part 11).Furthermore, there was no doubt that solving differential equations for mean properties was an improvement over earlier integral methods that, although they did and still serve a useful purpose, were much more heavily laden with empiricism and more limited in predictive range. Notice that the computer was and is very much involved in this process. Its continuing development is steadily reducing the need for that portion of empiricism based on the need to solve. At Princeton in 1960, we nunlerically integrated our first turbulent boundary-layer equation-laboriously since Fortran was not yet available-on an IBM 615; today, the same problem is easily accornmodated on most microcomputers. The ergodic idea of an ensemble average being the equal of a time average in stationary flow and also the equal of a space average in a homogeneous flow is easy to accept in the laboratory where flows that were both stationary and homogeneous in at least one space coordinate were commonly created. Thus one could chop up a time record of, say, velocity into pieces longer than a macrotime scale and claim that the pieces represented independent experiments or members of an ensemble. Although harder to do, one could, in principal, measure the same flow in the homogeneous (usually cross-flow) direction and again chop up the record into pieces of length larger than the macrolength scale to again obtain statistically independent members of an ensemble. Thus, in these flows, one is easily convinced that temporal, spatial, and ensemble averages are equivalent. Proceeding further, the basis for second-moment, turbulence closure is the hope that turbulence in one mean flow environment is sufficiently similar to turbulence in another mean flow environment so that thirdmoment, ensemble averages can, to some useful degree of approximation, be universally related to second-moment averages. Thus, turbulence closure is a means of interpolating and, it is hoped, extrapolating a limited data set to a wider range of problems. The process has been singularly enhanced by the second-moment, closure hypotheses of Rotta (1951) and Kolmogorov (1941), which have been implemented and elaborated by a number of researchers (Donaldson and Rosenbaum, 1968; Hanjalic and Launder, 1972; Lewellen and Teske, 1973; Lumley and Khajeh-Nouri, 1974). Our own interest was heightened by the realization that the effects of density stratification as measured in an atmospheric field experiment (Businger et al., 1971) could be replicated fairly well by a straightforward extension of Rotta’s hypothesis wherein the requisite nondimensional

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constants were obtained exclusively from laboratory data for neutral flows (Mellor, 1973; Mellor and Yamada, 1982). (Mellor and Yamada contain many additional references .) This was really a surprising, extrapolatory result, which we reproduce here in Fig. 1; it encouraged further development and applications. The same model also simulated the stabilizing or destabilizing effects of boundary flows on convex or concave walls (Mellor, 1975). Geophysical flows are generally not stationary. Nevertheless, as the converse of the measurement process, ensemble-average closure schemes should supply information applicable to horizontal averages. This, of course, fits into the business of numerical simulation since, generally, computer resource limitations require that the horizontal grid element be very large (in fact, much too large, as will be discussed in Section 4) compared to turbulence scales. Furthermore, again related to insufficient computer resources, the hydrostatic approximation is usually invoked, removing the possibility of explicitly calculating small-scale turbulence. On the other hand, the companion article in this volume by Deardorff discusses the explicit numerical calculation of small-scale turbulence where the hydrostatic assumption is not invoked and where the horizontal grid scale must be on the order of the vertical grid scale, which, in principal, should be small compared to the turbulence macroscale. “Sub-grid-scale” closure is required to account for scales of turbulence between the inertial subrange and the Kolmogorov dissipation scale. These are calculations proffered in the spirit of basic research to provide information for ensemble-mean closure hypotheses. Consider, briefly, the very much smaller molecular scales. For example, in the kinetic theory of gases, the mean-free path is small compared to the problem scale (e.g., boundary-layer thickness) and “closure” is greatly facilitated. Derived constitutive relationships are local relationships, and constitutive coefficients are properties of the fluid. Conversely, the major difficulty with turbulence closure is that turbulence macroscales are not very small compared to the problem scale. 2. THETURBULENCE MACROSCALE AND TURBULENCE CLOSURE In this chapter the word turbulence will be reserved for small-scale, three-dimensional motions that are random but whose ensemble average properties are approximately deterministic in the sense that they are related, either through laboratory measurements or closure models, to the mean velocity and density gradients. We also restrict attention primarily to one-point turbulence closure, although reference will be made to the

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5 FIG.1. Near-surface atmospheric boundary-layer data from Businger et al. (1971) compared with the turbulence closure model (Mellor, 1973) in the form of Monin-Obukhov variables. (a) and (b) &-, are proportional to the vertical velocity and temperature gradients, respectively, and 6 is proportional to distance from the ground; the variables are made nondimensional by using surface stress, heat flux, the gravity constant, and the coefficient of thermal expansion. These results were the first indication that second-moment closure extrapolated into the range (N21 >0 [see also Lewellen and Teske (197311.

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two-point correlation equations and two-point closure. In this chapter the term mean variable refers to the ensemble average of a variable or the first moment of that variable’s probability density, whereas the word couariance or variance will refer to the second moment of the joint probability density of two variables relative to their means or the probability density of a single variable relative to its mean, respectively. The turbulence integral macroscale A is defined as the integral of the two-point, longitudinal correlation function, integrated with respect to the distance separating the two points. In the outer 80% or so of a boundary layer adjacent to a solid surface, the macroscale is approximately one-half the boundary-layer thickness. We have obtained this number from the measurements of Schubauer and Klebanoff (1951); about the same value is also obtained from their measurements of the lateral correlation when converted by using the isotropic relationship. Thus, the integral macroscale is not small-comparedto the boundary-layer thickness, a fact that has provoked criticism that local closure relationships are simply not “valid.” However, in spite of this, local closure works well-surprisingly well. The evidence is all over journal papers. In fact, even the first moment closure of the 1960s demonstrated detailed, predictive skill (Kline e f al., 1968), although second-moment closure has greater predictive range for a single set of empirical closure constants. The question is not: Is local closure valid? The question is: Why does it work so well? A partial answer, in the case of turbulent flows near solid surfaces, is that a large part of the adjustment of mean properties from its mainstream value to the wall value occurs where the turbulent scales are small. Most models require that length scales be proportional to the distance to the wall, and this generally will reproduce the reliable, but not totally universal, law of the wall. The second-moment equation for velocity, the Reynolds stress transport equation is

+ ua u-+ g,Zp + giujp kuj

ax,+

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GEORGE L. MELLOR

Uppercase letters denote ensemble mean flow variables, lowercase letters are turbulence variables, and overbars denote their ensemble averages. The first and second terms on the left are the mean material derivative of uiuj and the corresponding Coriolis term. The remaining terms on the left are the shear and buoyancy production terms. The terms on the right require closure modeling. Other equations involving temperature and humidity for the atmosphere or temperature and salinity for the oceans are needed, together with appropriate equations of state to account for gravity and density stratification. However, in this chapter, I wish to concentrate on Eq. (2. l), referring to my papers and those of others already cited for consistent closure extensions for the other equations. The two major closure hypotheses for Eq. (2.1) are due to Kolmogorov (1941) and Rotta (1951). Rotta provided closure for the important, pressure-velocity correlation term (second term on the right-hand side of Eq. (2.2). This term plays a role analogous to the Boltzmann integral in the kinetic theory of gases (Mellor and Herring, 1973). It can be related to third and lower moments of velocity. Rotta's hypothesis can be interpreted as follows: The most general, local, linear closure relation is (2.2a) One can add nonlinear terms to (2.2a) (Hanjalic and Launder, 1972; Lumley and Khajeh-Nourri, 1974) at the expense of having to add more empirical constants. One can also add a term proportional to the buoyancy flux giwjp but, apparently, the associated empirical constants are approximately null (Mellor, 1973). The next step is to approximate the fourth-order tensors as isotropic tensors proportional to aii 8km and its permutations. Considering the continuity equation and symmetry in i andj, one obtains

where II is a macrolength scale and CI a constant. A deficiency in (2.2b) is that, in application to near-wall boundary-layer flow, the equation will yield 2(cross-flow variance) = w’i (variance normal to the wall), whereas data indicate that 0.5 < w2/$ < 0.8. Monin (1965) [see also Mellor and Herring (1973)l observed that this deficiency can be removed by including nonisotropic terms like A;Aj 8 k m together with the isotropic terms in the Cijkmcoefficients in Eq. (2.2a); here, A; is a unit vector normal to the wall. Additional empirical coefficients that must be functions of distance from the wall are required. Therefore, there exists a tradeoff between adding

ENSEMBLE AVERAGE, TURBULENCE CLOSURE

35 1

further empiricism and some increase of simulation accuracy of the turbulent energy components. This probably would not result in a significantly improved accuracy for mean properties. By contracting (2.2a), the pressure-velocity correlation term disap- of - the term is to pears through continuity. This means that one function redistribute energy among the energy components u2, u2, and 2;thus Rotta called it the “energy distribution term.” It will also be seen that the right side of (2.2b) serves the same function. Kolmogorov hypothesized that small-scale turbulence is isotropic even if the larger scales are not. The result is that (2.3a) where the turbulent energy dissipation E = u(aui/8xk)2.Further, Kolmogorov proposed that for a large Reynolds number, E is independent of viscosity u. Let E = vu&ft, where ud and q d = (u3/&)l14are the dissipation velocity and length scales. If u varies for a problem with fixed macrovelocity and length scales, then the dissipation velocity and length scales adjust so that (aui/8xk)2 = u$/qft is simply inversely proportional to u. Thus, since E is independent of viscosity E =

q3/Al

(2.3b)

where q = ( ~ 7 and ) A, ~are ’~ macrovelocity and length scales.* The rationale for (2.3b) is clearer in two-point correlation space. We refer only to the contracted and shorthand form of the correlation equation for the turbulence velocity components at x and x + r, which is F(x, r)

=

ask a2R -_ - u ark

(2.4a)

ar:

Sk = uk(x)ui(x)ui(x+ r ) - uk(x)ui(x)ui(x - r)

(2.4b)

R = ui(x)ui(x + r)

(2.4~)

All terms are dependent on time in addition to x and r. The full equation set may be found in Hinze (1975, p. 235), where it may be seen that F(x,r) contains tendency, advection, diffusion, and production terms for the correlation function R(x,r). For the simplest case of decaying homoge-

* For a boundary layer we find that A , is about 3 times the integral macroscale A, which as previously mentioned is about 0.5 times the boundary-layer thickness. In relatively low Reynolds numbers, grid-produced, decaying turbulence we find that A, is about 4 to 5 times the integral macroscale (Dickey and Mellor, 1980); thus, demonstrating some consistency of one type of flow with another seemingly disparate flow.

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GEORGE L. MELLOR

neous turbulence F = aR(x,r)/at. The second term in Eq. (2.4a) is the gradient of the triple correlation, whereas the third term is the two-point dissipation function. At r = 0, the single-point, turbulence energy equation is recovered, the second term vanishes and third term is the dissipation. [The resulting equation is identical to the contraction of Eq. (2.1)]. Figure 2 shows the behavior of the terms in eq. (2.4) for small r. All of the terms or their derivatives, when scaled on q d and u d , are universal (Dickey and Mellor, 1979) in the range 0 < r < C q d (as is the energy spectrum for a large wave number) (Gibson and Schwartz, 1965), where it may be shown that C is proportional to (A ~ / v ) ~ ' ~ . For 0 < r 5 Cqd, F = const = --E. Outside of the dissipation range where, say, the separation distance r > 20 q d , one finds that the two-point dissipation function, third term in Eq. (2.4a), is negligible; the sum of all the terms in the first term is balanced by the triple correlation gradient, the second term. The latter is governed by larger-scale dynamics. The role of the two-point dissipation function is to fill in the void established by the triple-correlation gradient function as it necessarily decreases to zero as r + 0. (There is some analogy here with the behavior of the viscous stress near a smooth wall that fills in the void as the Reynolds stress necessarily limits to zero as the wall is approached.) Thus, (2.3b) is really a model for the second term evaluated outside of the dissipation

rhd FIG.2. The two-point turbulent kinetic energy equation F(x,r) includes tendency, advection, production, and diffusion terms. r is the separation distance and qdis the Kolmogorov dissipation length scale. The departure 1 - F > 0 is for a finite Reynolds number, and I - F 0 as the Reynolds number approaches infinity.

-

ENSEMBLE AVERAGE. TURBULENCE CLOSURE

353

range and simply states that Skdecreases from a value of order q3when T)d << r << A to zero when r = order (A). Thus, v(a2R/a& = (aSk/ark)20Td q3/A. Aside from Eqs. (2.2) and (2.3), one must add hypotheses for the diffusion terms in Eq. (2.1). They are not as important as the terms we have discussed here, but they are not insignificant in many applications, and they introduce additional macroscales. A major assumption in most models is that the various macroscales are proportional to each other and the proportionality coefficients are the same for a usefully wide range of turbulent flows. The success of the models indicates that the assumption is approximately valid, although it is possible in the laboratory to create flows where the error in this assumption is manifest (Newman et al., 1981). Since Eq. (2.1) is a single-point equation, it does not contain lengthscale information. One must appeal to the two-point equations or, equivalently, the spectral equations for this information. Here we restrict attention to the two-point equations. Models by Hanjalic and Launder (1972), Lumley and Khajeh-Nourri (1974), and their successor models use a length-scale equation that may be obtained by twice differentiating Eq. (2.4) with respect to the separation distance and then letting the separation distance limit go to zero. After multiplication by the viscosity, this becomes a dissipation transport equation. There is a problem here because in Eq. (2.4) the second and third terms change rapidly in the range 0 < r < 20 q d , whereas the first term does not; their derivatives at r = 0 are very large compared to the derivatives of the first term. An analysis taking into account relevant velocity scales as well as length scales indicates that the second derivatives of the larger than the second derisecond and third terms are of order (Rq/v)1’2 vative of the first term at r = 0. The strategy invoked by Lumley and Khajeh-Nourri is to equate the sum of d2 (second term)/dd and a2(third term)/a& evaluated at r = 0 to a model closure term of lower order, i.e., to the same order as a2F/ad. We believe that this strategy is fundamentally wrong. First of all, what is needed is a macroscale, whereas the details of (2.4) around r = 0 clearly scales on dissipation scales. Furthermore, since (2.4) is universal for small r, there is no new information to be obtained other than that provided in Fig. 2. The case is clear when one attempts to solve (2.4) as a function of r and t (Domaradzki and Mellor, 1984). One must then model Sk(r,t) as a closure function of R(r,t). Since (2.4) is universal for small r, one need not be bothered with modeling viscous effects directly; one need only solve the problem for r > 20qd and then match the viscosity-indepen-

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GEORGE L. MELLOR

dent solutions to the universal function* involving R(r) (just as the viscosity-independent velocity profiles may be matched to the law of the wall when solving the mean turbulent boundary-layer problem). Conversely, the aforementioned dissipation transport equation and closure model have no apparent rational continuation into r > 0. Rotta’s strategy and the one we have adopted is to use the integral of (2.4). After averaging over the sphere r = const, Eq. (2.3) is integrated over r. The result is an equation for q21, where 1 is proportional to the integral macroscale. t (Analogous to the role of the von Karman, momentum integral equation that was used to provide boundary-layer thickness information in the integral theories from the 1940s to the 1960s. Now, when one solves differential equations for the mean velocity, this information is part of the solution.) The resultant equation is quite empirical and more suspect than, say, Rotta’s or Kolmogorov’s hypothesis. Nevertheless, it is a macro-length-scale equation that recognizes that eddy structures are advected and that is made to reproduce properties of law of the wall and homogeneous turbulence decay. Other applications of the complete model indicate that the macro-length-scale equation supplies length scales that yield observable boundary-layer profile scales. However, studies do indicate that mean flow properties are not overly sensitive to moderate variations in the macro-length scale. 3. AVERAGING DISTANCE FOR MEASUREMENTS IN THE ATMOSPHERE AND OCEANS AND FOR NUMERICAL MODELS

We next identify a minimum averaging distance to obtain ensemblemean statistics from data. The normalized root-mean-square error for where ~, u, is the variance, A the mean velocity U is (u,/U)(~A/L,) integral macroscale, and L, a horizontal measurement distance (Bendat and Piersol, 1971). Assuming Gaussian probability distributions, the normalized root-mean-square error for any of the variances (second moments) is estimated as (A/L,)1’2. If we accept an error of lo%, the measurement distance L, should be at lest lOOA. A corresponding averaging * It is the structure function [u(x + r) - ~($1’

= 2 ( 2 - R ) nondimensionalized on q d and that is universal for large Reynolds number and small r . On this point, we also note that Kolmogorov hypothesized that the structure function is isotropic for small r, from which (2.3a) follows. t Since all macroscales are assumed to be proportional, the length-scale equation could be rescaled so it is the equation for q2A, q2hl,or the product of q2 and any macroscale. For purely historical reasons, we have chosen to scale the equation so that 1 - K Z as z+-0, where K is the von Karman constant. vd

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355

time is 100Rl U ,, where U,is a translation velocity relative to the turbulence field being observed. We use again the estimate that the integral macroscale is about one-half of the boundary-layer thickness. Thus, for an atmospheric boundary layer, typically 1000 m thick, one needs a 50-km horizontal averaging distance; for a translation velocity of, say, 10 m s-l, an averaging time of about an hour and a half is needed. In the ocean surface layer, typically 50 m deep, one needs a horizontal averaging distance of 2 km and for a translation speed of, say 0.2 m s-I, an averaging time of about 3 hr is needed. What about numerical model grid requirements? Assuming that the horizontal grid elements Ax and Ay are equal, for numerical values associated with each areal cell to represent ensemble average values, the grid elements should be greater than about IOA or 5 km for the atmosphere and 250 m for the ocean. There is little danger that these conditions will be challenged for some years, since due to limited computer resources, models generally exceed these grid element lengths by one, two, or more orders of magnitude. 4. NUMERICAL MODELINGAPPLICATIONS AND HORIZONTAL DIFFUSION

Our turbulence closure model has been used by a number of atmosphere and ocean modelers, and some of these are referenced by Mellor and Yamada (1982). My own experience is in using the closure model embedded in a three-dimensional ocean model (Blumberg and Mellor, 1984). A problem immediately encountered in such large-scale models is the need to include horizontal diffusion in order to eliminate grid-scale noise, and this is on a very much more empirical basis now than is secondmoment turbulence closure. The choices for horizontal diffusivities are constant (dimensional!) values, the horizontal eddy viscosity of Smagorinsky (1969), and other forms such as the biharmonic operator (Holland, 1978). The effective values for the diffusivities are large. There are a couple of issues here that may not be clearly understood. First of all, second-moment closure does supply horizontal Reynolds stresses and fluxes, and there are process modeling applications requiring small-grid elements where these may be needed (e.g., cumulus convection or diffusion of a passive scalar close to its source). However, in applications in which the horizontal scale of mean motion is large compared to the vertical scale, scaling analysis leading to the boundary-layer approximation (which includes, incidentally, the hydrostatic approxima-

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GEORGE L. MELLOR

tion) shows that these horizontal turbulence stresses and fluxes are unimportant compared to the vertical stresses and fluxes. Thus, the horizontal diffusion required by most numerical models presumably represents unresolved, large-scale, horizontal advective processes. Typically, the diffusivities are many orders of magnitude larger than the values generated by the turbulence closure model. On the other hand, there is some small grid size for which this horizontal diffusion can be eliminated. In a recent modeling application to New York Harbor, the horizontal grid elements were 0.5 km (Oey et al., 1985). No horizontal diffusion terms were required and it is believed that horizontal dispersion due to resolved advective processes followed by vertical mixing, the so-called Taylor diffusion process (Taylor, 1954), was explicitly calculated. In open ocean calculations, the natural horizontal scale is the Rossby radius of deformation, a number around 20 to 40 km; one can speculate that grid sizes of 5 km or less may be required before horizontal diffusion terms can be eliminated completely. 5 . CONCLUDING REMARKS

Second-moment (or second-order), turbulence closure modeling is now more than 30 years old. However, a general burgeoning of model development and application and extension to geophysical flows began about 10 years ago. The turbulence models contain nondimensional empirical constants that are fixed by reference to a small subset of the available laboratory data. The models seem to do well in simulating a wide range of laboratory data outside of that subset. Direct comparison of geophysical data with model simulation is difficult since, oftentimes, simplifying assumptions such as horizontal homogeneity are required. Nevertheless, the available evidence does not indicate obvious error in the turbulence models per se. There are difficult extensions to be made involving phase change and chemical reactions. Otherwise, second-moment modeling has entered a stage of maturity at which, in our opinion, innovations with major geophysical consequences are unlikely; i.e., the closure hypotheses by Rotta and Kolmogorov may be modified but, in essence, will continue to provide the basis for closure. The past few years have seen the turbulence submodels incorporated into large, three-dimensional, time-dependent models of the atmosphere, oceans, and estuarine systems. In the last application we have noted that the horizontal resolution can be made fine enough so that horizontal diffusivities may be set to zero. There are then no adjustable parameters

ENSEMBLE AVERAGE, TURBULENCE CLOSURE

357

related to mixing in such a model. However, for most applications, the horizontal grid elements are large and require large horizontal diffusivities. Presumably, these diffusivities account for unresolved horizontal, sub-grid-scale advective processes, but a quantitative connection with such processes has not been established as it has in the case of secondmoment, turbulence closure models. Note that one could also write second-moment equations for these advection processes, in which case, one would need closure relations analogous to those in Eqs. (2.2) and (2.3).

REFERENCES Bendat, J. S., and Piersol, A. G. (1971). “Random Data: Analysis and Measurement Procedures.” Wiley, New York. Bradshaw, P., and Ferriss, D. H. (1968). Derivation of a shear-stress transport equation from the turbulent energy equation. Proc. AFOSR-IFP Stanford Conf. 1968, Vol. I , pp. 264-169. Blumberg, A. F., and Mellor, G . L. (1984). A description of a three-dimensional coastal ocean circulation model. In “Three-Dimensional Shelf Models” (N. Heeps, Ed.), Coastal Estuarine Sci. Vol. 5 . Am. Geophys. Union, New York (in press). Businger, J. A., Wyngaard, J. C., Izumi, Y., and Bradley, E. F. (1971). Flux profile relationships in the atmosphere and surface layer. J. Atmos. Sci. 28, 181-189. Cebeci, T., and Smith, A. M. 0. (1968). A finite difference solution of the incompressible turbulent boundary layer equations by an eddy viscosity concept. Proc. AFOSR-IFP Stanford Conf. 1968, Vol. I , pp. 346-355. Dickey, T. D., and Mellor, G. L. (1979). The Kolmogoroff r2’3law. Phys. FIuids 22(6), 10291032. Dickey, T. D., and Mellor, G. L. (1980). Decaying turbulence in neutral and stratified fluids. J. Fluid Mech. 99, 13-31. Domaradzki, J. A., and Mellor, G. L. (1984). A simple turbulence closure hypothesis for the tiiple-velocity correlation functions in homogeneous isotropic turbulence. J. Fluid Mech. 140,45-61. Donaldson, C. D., and Rosenbaum, H. (1968). “Calculation of the Turbulent Shear Flows through Closure of the Reynolds Equations by Invariant Modeling,” ARAP Rep. 127. Aeronaut. Res. Assoc. Princeton, Princeton, New Jersey. Gibson, C. H., and Schwartz, W. H. (1965). The universal spectra of turbulent velocity and scalar fields. J. Fluid Mech. 16, 365-384. Hanjalic, K., and Launder, B. E. (1972). Fully-developed assymmetric flow in a plane channel. J. Fluid Mech. 52, 689. Hinze, J. 0. (1975). “Turbulence.” McGraw-Hill, New York. Holland, W. (1978). The role of mesoscale eddies in the general circulation of the oceannumerical experiments using a wind-driven quasi-geostrophic model. J. Phys. Oceanogr. 8, 363-392. Kline, S. J., Morkovin, M. V., Sovran, G . , and Cockrell, D. J., eds. (1968). Proc. AFOSRIFP Stanford Conf. 1968. Koimogorov, A. N. (1941). The local structure of turbulence in incompressible viscous fluid

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for very large Reynolds number (in Russian). Dokl. Akad. Nauk SSSR 30, 301; S. K. Friedlander and L. Topper, “Turbulence.” Wiley Interscience, New York, 1961 (Engl. transl.). Lewellen, W. S., and Teske, M. E. (1973). Prediction of the Monin-Obukhov similarity functions from an invariant model of turbulence. J . Atmos. Sci. 30, 1340-1345. Lumley, J. S., and Khajeh-Nourri, B. (1974). Modeling homogeneous deformation of turbulence. A d v . Geophys. 18A, 162-192. Mellor, G. L. (1967). Turbulent boundary layers with arbitrary pressure gradients and divergent or convergent cross flows. AIAA J . 5 , 1570. Mellor, G. L. (1973). Analytic prediction of the properties of stratified planetary surface layers. J . A m o s . Sci. 30, 1061-1069. Mellor, G. L., and Gibson, D. M. (1966). Equilibrium turbulent boundary layers. J. Fluid Mech. 24, 255. Mellor, G. L. (1975). A comparative study of curved flow and density stratified flow. J . Atmos. Sci. 32, 1278-1282. Mellor, G. L., and Herring, H. J. (1968). Two methods of calculating turbulent boundary layers based on numerical solution of the equations of motion. I. Mean velocity field method. 11. Mean turbulent field method. Proc. AFOSR-IFP Standard Cony. 1968, Vol. 1 , pp. 275-299. Mellor, G. L., and Herring, H. J. (1973). A survey of the mean turbulent field closure models. AZAA J . 11, 590-599. Mellor, G. L., and Yamada, T. (1982). Development of a turbulence closure model for geophysic fluid problem. Rev. Geophys. Space Phys. 20, 851-875. Monin, A. S. (1965). On the symmetry properties of turbulence in the surface layer of air. Izv. Atmos. Oceanic Phys. 1, 45-54. Newman, G. R., Launder, B. E., and Lumley, J. L. (1981). Modelling the behaviour of homogeneous scalar turbulence. J. Fluid Mech. 111,217-232. Oey, L.-Y., Mellor, G. L., and Hires, R. I. (1985). A three-dimensional simulation of the Hudson-Raritan Estuary. Part I. Description of the model and model simulations. (Submitted for publication.) Rotta, J. C. (1951). Statistische Theorie nichthomogener Turbulenz. 1 . Z. Phys. 129, 547572. Schubauer, G. B., and Klebanoff, P. S. (1951). Investigation of separation of the turbulent boundary layer. Natl. Advis. Comm. Aeronaut., Rep. 1030. Smagorinsky, J. (1969). General circulation experiments with the primitive equations. I. The basic experiment. J. Meteorol. 14, 184-185. Taylor, G. I. (1954). The dispersion of matter in turbulent flow through a pipe. Proc. R . Soc. London, Ser. A 223,446-468.

THE PLANETARY BOUNDARY LAYER H. A. PANOFSKY* Department of Meteorology Pennsylvania State Universih, University Park, Pennsylvania

. . . . . . . , . , . . . . . . . . . , , . . . 1.4. The PBL in Strong Winds. . . . . . . . . . . . . . 1.5. Complex Terrain . . . . . . . . . . . . . . . . . The Equations in the PBL. . . . . . . . . . . . . . . . The Surface Layer . . . . . . . . . . . . . . . . . . 3.1. Profiles and Fluxes over Homogeneous Terrain , . , . . 3.2. Variances . . . . . . . . . . . . . . . . . . . . 3.3. Spectra and Cospectra . . . . . . . . . . , . , . . First- and Second-Order Closures. . . . . . . . . . , . . 4.1. First-Order Closure . . . . . . . . . . . . , . . . 4.2. Large-Eddy Exchange . . . . . . . . . . . , . . . 4.3. Second-Order Closure . . . . . . . . . . , , . . . Boundary-Layer Models . . . . . . . . . . . , . . . . 5.1. The Surface Layer over Complex Terrain. . . . . . . . 5.2. Modeling the Whole PBL . . . . . . . . . . . . . . Boundary-Layer Parameterization . . . . . . . , , , . . References. . . . . . . . . . . . . . . . . . . . . .

1. General Characteristics . . . . . . . . . . . 1.l. Definitions and Importance . . . . . . . 1.2. The Daytime Boundary Layer . . . . . . 1.3. The PBL at Night . . . . . . . . . . .

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359 359 360 361 363 363 364 361 361 310 312 311 377 319 380 380 380 381 383 383

1 . GENERAL CHARACTERISTICS 1 . l . Definitions and importance

The planetary boundary layer (PBL) is that part of the atmosphere that is affected by the characteristics of the ground by rapid vertical exchange processes, mostly by turbulence. Therefore, its characteristics vary strongly from day to night and from land to sea. The thickness of the PBL varies from 1 to 2 km on strongly convective days with strong winds over land to 100 m or less at night with clear skies and weak winds. Over the ocean, the thickness is typically less than 1 km and shows little diurnal variation. The lowest 10% of the PBL is called the “surface layer” in which conditions are relatively simple. Its importance lies in the fact that most human activity in the daytime and generally in strong winds takes place in the surface layer. * Emeritus. Present address: Scripps Institute of Oceanography, University of California at San Diego, La Jolla, California 92093. 359 ADVANCES I N GEOPHYSICS. VOLUME

28B

Copyright 0 1985 by Academic Presb. Inc. All nghts of reproduction in any form reserved.

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Because of the importance of turbulence in the PBL, special procedures have to be adopted in this region in models of the atmosphere on all scales.

1.2. The Daytime Boundary Layer There are two kinds of turbulence: mechanical and convective. Mechanical turbulence is strong with strong winds near the ground, and convection is strong when the atmosphere is strongly heated from below. Therefore, on sunny, light-wind days over land, convection dominates through the whole PBL. With moderate winds, mechanical turbulence becomes important near the ground. But the production of mechanical turbulence decreases with height more rapidly than that of convective turbulence; thus even with moderate winds, convection dominates at the top of the PBL and determines its thickness over land. Only in very strong winds are the characteristics of the whole PBL determined by the behavior of mechanical turbulence-both day and night over land and water. In general, convection is important over water only in regions of cold air over warm water. The thickness of the PBL in the daytime can be defined in two ways: Either it is the height h, which there is considerable turbulence; or it is zi , the height of the lowest inversion, that is, that part of the atmosphere affected by surface heating. Variable zi is usually a little smaller than h because the lowest part of the inversion is usually turbulent, both due to overshooting from below and because the wind shear in the inversion tends to be strong. In practice, zi and h are often used interchangeably. There are several ways for estimating h and zi: (1) By acoustic sounder: This instrument measures reflections from heterogeneous blobs of refractive index (essentially temperature) that cease to exist above h. Unfortunately, the range of most acoustic sounders is less than 1 km. (2) By constructing an adiabat from the surface to a morning sounding: The intersection is zi . This method fails after the time of maximum temperature because the temperature decreases and zi continues to rise slowly. (3) From the conditions of heat budget and continuity: The original equation was quite simple, allowing only for heating from below. It showed that zi is proportional to where H is the surface heat flux and time and t is counted from sunrise. Exact estimates of H are quite dimcult.

m,

Approximate estimates from relatively simple observations (e.g., insolation and cloudiness) have been suggested by de Bruin and Holtslag (1982)

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and others. These methods work well [for a more complete discussion, see Tennekes and Driedonks (1981)l. A useful parameter in the description of the daytime boundary layers is the Monin-Obukhov length -L, which is proportional to T ~ (7/ is~ the surface stress) and inversely proportional to heat flux. It is essentially independent of height, and the rationale for its definition will be given later. For z > -L, convection dominates and mechanical turbulence can be neglected. The principal length scale is zi . The velocity scale is w , = [ g H z , / ~ ~ p T where ] ” ~ , g is the acceleration of gravity, H the surface heat flux, cp the specific heat at constant pressure, p the density, and T the temperature. Typically, - L is 100 m on a clear windy day and 5 m on a clear day with weak winds. The region above z = - L is often called the “mixed” layer. Because of these basic scales, simple expressions exist in the mixed layer for the various characteristics of turbulence. For example, the standard deviation of vertical velocity is given by u ~ / w *=

+(z/zi)

(1.1)

and the spectral density of vertical velocity S , ( k l ) by klS(k1 ) / w Z

=

F(z/zi klL) 7

(1.2)

where kl is the wave number in the x direction, usually taken in the direction of the mean wind, and (b and F a r e universal functions that have been studied [e.g., Kaimal et al. (1976) and H#jstrup (1982)l. Also, in the mixed layer, wind direction, wind spread, and potential temperatures are constant with height. For z < zil10, we enter the surface layer, where different scaling laws apply. We will later discuss the characteristics of the surface layer in some detail. Sometimes the surface layer and mixed layer overlap, so that the laws of both regions are valid. In such a region, particularly simple conditions prevail, which are referred to as the conditions of “free convection.” For example, u, is proportional to z1I3and the standard deviation of temperature uTto z - I / ~ . 1.3. The PBL at Night

At night over land, maximum cooling occurs at the surface so that turbulence is damped by the stratification. Only mechanical turbulence can exist that, as we have seen, is driven by the wind. Hence, with light winds, turbulence is weak and vertical transfer of momentum, heat, and moisture is slow. The depth of the mixed layer is small, typically of the

362

H. A. PANOFSKY

order of 100 m, depending on wind speed. Turbulence decreases rapidly upward. At the top of the turbulent layer and above, heat transfer by infrared radiation becomes relatively important and extends surface cooling above the turbulent (mixed) layer. Therefore the PBL (the layer modified by surface changes) can be much thicker than the mixing depth (the turbulent layer). Typically, the PBL extends to the top of the surface inversion, but the mixing depth is much smaller. This difference has led to considerable confusion in the definition of the thickness of the nocturnal boundary layer. There are no successful formulas for estimating the mixing depth at night, partly because of the difficulty in interpreting the observations. Another difficulty is due to the slowness of the transfer processes, so that the PBL is not in equilibrium with the surface processes. A formula frequently used for mixing depth is h

=

a

m

f

(1.3)

where a is a constant (thought to be near 0.4), the friction velocity u* is T the surface stress, f the Coriolis parameter, and L the MoninObukhov length mentioned earlier. It is positive at night and grows rapidly with increasing wind speed. Since u* also varies rapidly with wind speed, (1.3) shows that h is very sensitive to wind speed (as would be expected). The mixing depth also is inversely proportional to
a,

dhldt

=

(ho - h ) / T

(1.4)

where ho is some equilibrium value of hl , such as that given by (1.3), and T a response time of the nocturnal atmosphere. However, there is no consensus about whether (1.3) or (1.4) is preferable. Further complications in the nocturnal boundary layer consist in the appearance of thin turbulent layers, separated from the main turbulent region by laminar layers. These thin layers are driven by local shear. Superimposed on the turbulence are presumably irregular gravity waves with periods of order 15 min. Typical spectra show considerable energy at periods of this order, separated from the high-frequency turbulence by a spectral gap. It is possible that these gravity waves cause the local shears responsible for the thin turbulent layers. Occasionally, gravity waves with regular periods are seen, driven by wind shears at higher levels [see, e.g., Finnigan and Einaudi (1981)l.

THE PLANETARY BOUNDARY LAYER

363

On some nights, the PBL undergoes abrupt changes due to local processes. There is a tendency for the wind above 100 m to accelerate after sunset when the mixing between upper and near-surface air has ceased. During some nights when the surface cooling is not excessive, the wind shear becomes sufficiently large to cause mechanical turbulence through a thick layer, producing increasing temperature, winds, and turbulence at the surface. This destroys the shear, and further cooling at the surface reestablishes the less turbulent conditions that had prevailed previously. Such “turbulent interludes” can be important because they can bring pollution in upper layers down to the surface. Altogether, conditions in the nocturnal boundary layer are complex, and somewhat controversial. Considerable clarification is needed for some of its features. 1.4. The PBL in Strong Winds

With very strong winds, only mechanical turbulence is important. In that case, turbulent energy is proportional to the square of the wind speed at a given level and increases with increasing terrain roughness. The thickness of the PBL is given by h

=

0.2ur/f

and u* can be estimated from where V is the wind speed at level z (typically 5-15 m) and zo the roughness length, which is a measure of surface roughness and will be discussed in more detail in Section 5 . Briefly, it varies from 0.01 cm over ice to several meters over forest or cities. Equations (1.5) and (1.6) then show that the boundary-layer thickness on windy days is proportional to the wind speed. The von Karman constant k is usually taken as 0.4, although estimates vary from 0.35 to 0.43. J. A. Businger (personal communication, 1984) is carrying out a careful field study to reduce this range of uncertainty.

1.5. Complex Terrain The preceding sections described some of the properties over flat, relatively uniform terrain. Clearly, hills, mountains, and coasts will distort and complicate features of the PBL. We will mention here just one practically important feature: if a nocturnal boundary layer, characterized by a

364

H . A. PANOFSKY

surface inversion (aTlaz > 0) passes over warmer and sometimes rougher terrain, a strongly mixed layer develops near the ground, with isothermal or even lapse conditions (aT/dz < 0). The depth of this depends on the roughness of the new surface and on heat input. It varies from 100 m or so over small cities to 300 m over New York. This depth is critical for nighttime pollution concentration over cities. Other special types of boundary layers over complex terrain were under active study in 1984. It is a large field of active research and will be discussed later.

2. THEEQUATIONS IN THE PBL

For many atmospheric problems, it is sufficient to consider the seven variables pressure p , density p, temperature T, and (specific) humidity q ; and the three velocity components u, u, and w , where w is usually the vertical component and u and u the horizontal components. Component u is taken either as west-east or along the surface wind, a convention we shall adopt here. For special problems, particularly those involving air pollution, many additional variables may be required to describe the concentrations of trace constituents-sometimes more than 100. We will not consider such problems. The behavior of the seven variables is governed by the seven equations: the equation of state, the first law of thermodynamics, the two equations of continuity for air and moisture, and the three components of the equation of motion (Newton’s second law). In the absence of strong horizontal terrain variations or fronts, horizontal derivatives can usually be neglected except in the expressions for pressure gradient forces. Molecular transfer terms are small compared to turbulent transfer terms (except in the lowest millimeter). Several of the basic equations are of the form daldt

=

b

(2.1)

where a is the concentration of a property per unit mass (e.g., momentum or enthalpy) and b the sum of sources and sinks. We then introduce a = Z + a’ and b = b + b’, where Zand 6 represent the large-scale components to be modeled; the primes suggest deviations from the large scale, which are only described statistically. In order to proceed further, the barred and primed quantities have to obey the Rey= = 0; primed and barred quantities are nolds postulates, e.g., uncorrelated; and a‘ = Ci. In addition, a must be differentiable. In general, it is not easy to satisfy all these conditions. Fortunately, there appears to be a spectral “gap” near a period of 1 hr. Hence 1-hr averages (or better

a

365

THEPLANETARYBOUNDARYLAYER

I-hr regression lines) seem to be reasonably satisfactory for ii and 6. These are essentially equivalent to data shown on smoothed mesoscale or synoptic-scale maps. Another simplification is the Boussinesq approximation; p’ is negligible compared to p , unless multiplied by gravity g. Thus, 7 p. This approximation is valid as long as the fluid velocities are small compared to the speed of sound. After other boundary layer approximations are made, the equations can be written

-

Here the coordinates x, y, and z are defined in the same directions as M , u, and w. Variable p is pressure, t is time, and S E and S, represent sources and sinks for enthalpy and humidity, respectively. The terms containing small-scalemotions have been given special symbols and names: -pu’w‘ and - ~ u w are denoted by 7, and 7? and are called the Reynolds stress components in the x and y directions, respeclively; - cppw’T’ is the vertical flux of enthalpy H, also called heat flux; and pw’q’ is denoted by E and represents the vertical flux of moisture. It is clear that Eqs. (2.2)-(2.8) constitute 7 equations with I 1 unknowns, where T,, 7?,H, and E are the “new unknowns” influencing the “old” variables, p, p , T, 4, U,V , and M,. In order to close the system, new variables have to be related to the old variables and the coordinates. This is the closure problem, which will be discussed in Section 4. One other equation is needed to understand the nature of small-scale motion: the eddy-energy equation. This equation is derived by multiplying the three unaveraged equations of motion by u’, u’, and w’, respectively, and adding them. The result can be written [subject to the same assumptions as Eqs. (2.2)-(2.8)] --

dddt

= -uT(aii/az) - u’w’(au/az)

+ ( ~ w ’ T ‘ / T ) M+ D + P

-

E

(2.9)

Here e is the eddy energy ( u ’ ~ + uf2 + d 2 ) / 2 ;M is a moisture correction

366

H. A. PANOFSKY

that is usually near one over land but can be important in moist air, particularly over tropical areas; D is the change of energy due to vertical divergence or convergence of the eddy flux; P represents energy change due to the work of small-scale motion against pressure forces; and E represents dissipation of eddy energy into heat and is proportional to molecular viscosity and the square of wind shear components. Since the viscosity of air is small, only small eddies with large shear contribute to this dissipation. The first two terms on the right represent production of mechanical turbulence. It is almost always positive. The same terms with opposite sign occur in the large-scale energy equation, so that they describe the degradation of large-scale into small-scale energy. The next term describes the production of convective energy. It tends to be positive in the day and negative at night over land. Variables P and D , at least in the lower PBL, have a tendency to cancel. When the production terms are large, small-scale energy reaches equilibrium quickly, and dddt can be neglected. Thus, in the lower PBL, mechanical plus convective production approximately balance dissipation. But the production terms produce relatively large “eddies,” whereas E can only dissipate small eddies. Hence there is a continuous transformation of large into small eddies. High in the PBL, production terms are small. There the balance is between convergence of flux (imported from below) and dissipation. The flux Richardson number Rf is defined by the ratio of negative convective production to mechanical production as (2.10) It is negative in the daytime over land; the larger its magnitude, the more important is convection. It is usually positive at night, when stratification is stable and tends to damp out mechanical turbulence. Condition Rf = 0 is typical for strong winds when mechanical turbulence is dominant. For Rf > 0.25, vertical turbulence disappears entirely. Under these conditions, gravity waves appear. When Rf = 0.25, it is called the “critical” Rf. Over the ocean, the sign of Rf is determined by the sign of the air-sea temperature differences. Qualitatively, Rf measures the relative importance of heat convection and mechanical turbulence. Quantitatively, it must be simulated in wind tunnels when atmospheric flow is to be modeled. In practice, Rf is often replaced by Ri,the gradient Richardson number or just Richardson number. The measurement of Ri does not require the measurement of small-scale quantities as Rf does. it is defined by

THE PLANETARY BOUNDARY LAYER

367 (2.11)

Here Yd is the dry adiabatic lapse rate and Ri related to Rf by a factor, usually of order 1. However, on strongly convective days, RflRi can grow to infinity and even change sign at heights of order 50 m. Under such conditions, the use of Ri should be avoided. As we shall see, Ri is particularly useful in characterizing different types of vertical dispersion. A priori, the variation of neither Ri nor Rf with height is known. For this reason, another quantity with qualitatively similar properties as Ri and Rf has often replaced their use. This is zlL, where L is the MoninObukhov length, defined in terms of surface fluxes and therefore independent of height. Empirically, Ri is related to zlL by approximately zlL

=

Ri

(Ri < 0, unstable)

zlL

=

Ri/(I - 5Ri)

(Ri > 0, stable)

(2.12)

Qualitatively, z / L also describes the relative importance of convective and mechanical turbulence; e.g., if L is negative and z >> -L, convective turbulence dominates. At height z << ILI, mechanical turbulence is important; at height z >> 1L1, it is not. In distinction to Ri and Rf, however, there is no critical value of zlL when vertical turbulence disappears. It just gets weak for large z/L. 3. THESURFACE LAYER

Over homogeneous terrain, the fluxes decrease approximately linearly with height. Therefore, they vary by a fraction of order 10% in the lowest 10% of the PBL. This variation can usually be neglected and the fluxes can be treated as independent of height in the lowest 10% of the PBL. This region is called the “surface layer” in which most of us live and work most of the time. Over homogeneous terrain, characteristics of profiles and turbulence statistics are simple and well understood. But homogeneous terrain is rare, and extension to complex terrain is difficult except in cases of simple types of terrain change, e.g., sudden change of roughness or in case of flow over hills.

3.1. Profiles and Fluxes over Homogeneous Terrain One of the great simplifications in the surface layer is that the wind direction is independent of height. Therefore we can align x and Ewith the

368

H. A. PANOFSKY

wind direction and disregard 6. We will also drop overbars in this section for convenience. In the important case of very strong (destructive) winds (Ri z/L 0, mechanical turbulence only), the wind profile is given by

- -

u

=

(u,/k) In(z/zo)

(3.1)

[see also Eq. (1.6)]. In principle, u is the mean wind component in the x direction; in practice, it is often replaced by the mean wind speed V. The roughness length of zo at a given site can be determined accurately from measurement of the wind profile; with some practice, qualitative estimates can be made by inspection of the site. The u, is again defined by G ,where T is now the surface stress, parallel to u. Equation (3.1) has been derived in different ways. As written, the ground, where u = 0, is defined by z = ZO.The equation is valid in the surface layer, but only if z > 20 zo . If the ground is covered by objects of height h (e.g., trees or houses), z must be replaced by z - d, where the displacement length d is of order 0.75h. In what follows, we will assume that d << z. Equation (3.1) implies that the ground is sufficiently rough that even at the surface, molecular friction is negligible; instead, momentum is transferred into the ground by differential pressure on roughness elements, that is, by “form drag.” Over very smooth terrain, e.g., water with very light winds, Eq. (3.1) has to be modified to include molecular viscosity. Equation (3.1) can be solved for T = put and shows that the surface stress is proportional to the square of the wind speed. But it also increases with increasing zo, which itself increases with wind speed over the sea. The exact relation is still controversial, because zo really depends on the state of the sea that is not uniquely a function of wind speed. Still, the relation is often used to estimate surface stress from wind observations and an assumed value of ZO. When stratification becomes important, Monin-Obukhov similarity generally works well in the surface layer. It postulates that if all variables are nondimensionalized in terms of L and u, ,universal relations develop that can be observed and, to some extent, derived in second-order closure models (see Section 4). Provided that zo << (L(,usually a good assumption, the expression for the wind profile can be generalized to u = u,/k[ln z/zo - +,n(~/L)l

(3.2)

Here JI, is a presumably universal function. For L > 0, it is usually taken as -4.7 zlL. However, if zlL > 1, turbulence is so weak that winds at different levels become disconnected and no profile law fits well. For

369

THE PLANETARY BOUNDARY LAYER

+

L < 0 , is usually represented by a complicated expression that was first suggested by Paulson (1970). It has also been tabulated [e.g., by Panofsky and Dutton (1984)l. Equation (3.2) is quite successful for vertical extrapolation of wind, given L and zo. There are simple ways to estimate L [see, e.g., Golder (1972)l. The solution of (3.2) for r = pul shows that, for a given wind and roughness, the surface stress is greater in unstable air ( L < 0) than in stable air ( L > 0). Figure I shows the change of wind from a representative clear day ( L < 0) to a clear night ( L > 0). Note that typically, in the surface layer, the wind is strongest in the middle of the day. But by extrapolating beyond the top of the diagram, it is clear that at some height (about 100 m) the diurnal variation disappears. Above that height, the daytime wind remains constant through the whole PBL. But, at night, the winds increase to a speed larger than geostrophic, forming a jet at several hundred meters, usually after midnight. Temperature and moisture profiles resemble Eq. (3.2). Instead of temperature, potential temperature 0 leads to simpler equations. In the PBL, it is sufficiently accurate to define 8 by T + Y d Z . The surface layer profile equations are then 8 - O0 = TJk[ln z/zoT - +h(z/L)I

(3.3)

q - qo = QJklln Z / Z O ~- Jr,(z/L)I

(3.4)

Here O0 and qo are the potential temperatures and specific humidity at z = ZOT and z = z O q r respectively; T, and Q, are defined by -H/c,pu, and -Elpu, . It is clear that the slopes of the profiles are proportional to H / u ,

I

I

I

I

I

1

I

2

3

4

5

6

wind speed, m/s

FIG.1. Diurnal variation of wind speed in the surface layer (schematic). [From Panofsky and Dutton (1984). “Atmospheric Turbulence.” Copyright 1984. Reprinted by permission of John Wiley and Sons, Ltd.]

370

H . A. PANOFSKY

and Elu, ,so that measured profiles can be used to estimate E and H. Here zOTand zo4 present problems and depend on surface properties in a complex manner. For more details, see Liu et al. (1979) and Blackadar and Jersey (1980). If temperature or moisture observations are available at several levels, then H and E can be found without running into difficulties with surface assumptions, since differences of the variables at two levels are independent of zOTor zoq. Over the ocean, however, only surface temperature is often given and q, T, and u are at one level. Therefore ZOT and zoq must usually be estimated for flux estimates over the sea. There are, of course, other methods for estimating the important surface fluxes-each with its special difficulties. If high-frequency records are available (frequencies > 1 Hz), the fluxes can be found from their definitions. Second, budgets of heat, momentum, and moisture can be used to find surface fluxes; and finally, spectra of wind, temperature, and moisture at very high frequencies lead to flux estimates. This will be explained later.

3.2. Variances

Variances of vertical and horizontal angles are important in the estimation of atmospheric diffusion, variances of wind speed (or the longitudinal wind component u) affect the stability of structures, and variances of scalars are related to the scattering of electromagnetic and acoustic waves. In the surface layer the standard deviation u of the velocity components should follow the Monin-Obukhov scaling and be of the form d u , = +(.?/I,)

(3.5)

where 4 represents universal functions, different for each velocity component. Actually, only the vertical component obeys an equation of the form of (3.5). But height does not affect the horizontal components; yet decreasing 1IL increases their fluctuations significantly. The strength of the horizontal variations of wind speed depends, instead of height, on the thickness of the PBL, zi,which also limits total eddy size. Thus the three equations are cr,,/u*= +,(Zi/L)

(3.6)

u,/u* = $*(Zi/L)

(3.7)

THE PLANETARY BOUNDARY LAYER

u,,/u*

=

+3(z/L)

371 (3 3)

where +I ,+ 2 , and +3 are fairly well observed in unstable air ( L < 0) [see Panofsky et al. (1977)l. Also +3 has been determined by second-order closure models [see Mellor (1973)l. Empirical expressions have been suggested for the three +’s that can be found, e.g., in the paper by Panofsky et aE. (1977). They all have the property that for large instability they depend on their arguments to the 4 power. In this limit, u* cancels. Then / ~ , u,~ and u, (in free convection) u,,,is proportional to ( g H M ~ l c , p T ) ’ and are proportional to ( g H M ~ ~ l c , p T )The ” ~ .latter expression turns out to be 0.58w, (see Section 1 for definition of w * ) . Note that for all three components, the standard deviation are of the form au, in strong winds (zlL + 0), where the a’s are constants. Thus the u’s are independent of height in the surface layer. The constants over homogeneous terrain are 2.5, 1.9, and 1.25 for u , u , and w, respectively. At higher levels and in unstable air, the standard deviations of the velocity components obey mixed layer scaling. The u’s of the horizontal components remain approximately constant with height, but those of the vertical component decrease toward the top of the PBL. [For details see Hgjstrup (1982).] For L > 0 (stable air) the properties of the u’s are not well understood. Both the 0’s and u, become small so that the ratios are poorly determined from measurements. Second-order closure (Mellor, 1973) suggests that the ratios of the sigmas to u* increase slowly with increasing zlL. In any case, with pure turbulence a,,,/u, u,/u, and u,Ju all become small. In practice, however, only u,,Iu dies out with increasing stability or increasing height. Horizontal fluctuations of low frequency may increase again, but probably not due to turbulence. As mentioned before, there is some evidence that gravity waves become important. Thus, the velocity fluctuations at a few 100 m recorrelated with surface pressure variations [see Zhou Leyi and Panofsky (1983)]. No significant differences between the normalized standard deviations of moisture and temperature have been observed in the surface layer. Both obey Monin-Obukhov scaling. For L < 0 Wesley et af. (1970) suggest that

u~-/T. = 2(1

-

18z/L)-”*

(3.9)

For L > 0 the scatter is large, and no expression can be fitted. Above the surface layer in convective conditions, a T / T ,decreases to a minimum in the central PBL and reaches values near those at the surface near z = z i . This is because warm air is mixed in from above.

372

H. A. PANOFSKY

3.3. Spectra and Cospectra

Most spectra of atmospheric turbulences are based on time series, over periods of the order of half an hour to an hour, usually with trends removed. Spectra have important advantages overcorrelation functions (which give the same information): estimates at different frequencies are independent of each other, and reactions of structures to atmospheric turbulence can be computed with relative simplicity if the response is linear. It is common to plot fS,(f) as function of In f, where S,(f) is the spectral density of a parameter a at frequency f. In this plot, the area between two frequencies gives the variance contributed by this frequency interval. Here fS,(f> represents variance per unit logarithmic frequency interval. Taylor’s “frozen wave” hypothesis is often assumed to be valid. In that case, local variation as a function of time represents upstream variation as the function of distance passing the observed with velocity V . In that case,fS,(f) = klS,(kl), where k l is the wave number in the mean wind direction. Taylor’s hypothesis has been checked by observations; it is quite good for high frequencies. At low frequencies, the turbulence is often transported more rapidly than by the local velocity. Spectra in crosswind directions have been measured from airplanes and rarely, from many anemometers along lines not parallel to the wind. Figure 2 shows, schematically, the spectra of horizontal and vertical

longitudinoI component

-

.c Y

v)

1

I

0.01

0.I

1

I f,hz FIG.2. Comparison of spectra of vertical and horizontal wind components in the atmospheric boundary layer (schematic). [From Panofsky and Dutton (1984). “Atmospheric Turbulence.” Copyright 1984. Reprinted by permission of John Wiley and Sons, Ltd.]

THE PLANETARY BOUNDARY LAYER

373

wind components in the surface layer. Note that the quasi-two-dimensional shape of the atmosphere suppresses the low-frequency vertical velocity components but not the horizontal components. The major portion of the spectra lies in the energy-generating region. Energy is transmitted from there through the “inertial subrange” to the dissipation range at wave numbers of order 10 cycles/cm, where the turbulence is dissipated at rate E. Conditions in the inertial range are particularly simple and important for various atmospheric problems. The dissipation range is not usually measured in the atmosphere, since the instruments generally used to measure atmospheric turbulence are too slow to respond to it. The inertial subrange is isotropic (its statistics are independent of the orientation of the coordinate system). Next, S,(kl) depends only on the rate of dissipation E and kl . Dimensional analysis then requires that

fSa(fl)= klS,(kl)

= U E ~ ‘ ~ ~ ; ~ ‘ ~

(3.10)

if a is a velocity component. The constant a is about 0.15 for the u component and 0.20 for the u and w components, iff is in cycles/time and kl in cycles/length. Equation (3.10) is valid for wave numbers in the range of about I/z< k l < 10 cycles,’ cm. The exact range is different for u, u, and w and depends on stability and other variables. The corresponding frequency range follows from the relationship f = klV (Taylor’s hypothesis). The importance of Eq. (3.10) is twofold: first, many structures have free periods in the frequencies of the inertial range, and accurate estimates of spectral density are needed there; and second, observations of the spectra in the inertial range furnish estimates of E that through the energy Eq.(2.9) can be connected with stress. In the surface layer, in equilibrium, Eq. (3.10) can be simplified to E =

(u?kz)+,(z/L)

+,

(3.11)

+,

Here = 1 in purely mechanical turbulence and describes the effect of convection on dissipation. In very strong winds, when the response of structures is most important, E is then given by [after substitution of Eq. ( 2 4 1 E

=

k2V-‘/z ln3(z/zo)

(3.12)

This, with Eq. (3.10), gives spectral estimates in the inertial range in strong winds. Conversely, if E has been determined from spectral measurements by use of Eq. (3.10), Eq. (3.11) yields u , , that is, the surface stress. This technique is easiest in strong winds or over the sea, where (be is usually

374

H . A. PANOFSKY

near 1. But, more generally, +E can be estimated if temperature information is available. The variance is the area under the spectrum. Since uwsatisfies MoninObukhov scaling, the w spectrum in the energetic range would also be expected to. Observations over uniform terrain indeed show that it does. Thus, the spectral density is described by klS,.(kl

)/US

=fS,,(f) =

F ( k l L , d L ) = G(klz,z/L)

(3.13)

Here F and G are universal functions. Empirical fits to data over good terrain have been given [e.g., by Kaimal et al. (1972)l. These are generally interpolation formulas between high frequencies, where Eq. (3.10) is valid, and low frequencies, where F and G are proportional to kl . If one considers the universal function G, it is clear that in mechanical turbulence, the spectrum depends only on klz = z / h , where h is the horizontal wavelength in the x direction. Hence, for example, the wavelength of the maximum klS(kl) for w is proportional to height. Measurements show that in strong winds the maximum wavelength is at about twice the height. With increasing convection, the wavelength at maximum energy becomes larger. [For details, see, e.g., Panofsky and Dutton (1984).] Since uuand (T, depend on zi/L rather than z / L , so must their spectra. But at high frequencies, they obey Eq. (3. lo), which is a function of z / L , as seen by Eq. (3.11). Therefore, as Kaimal (1978) was the first to point out, spectra of u and u for L < 0 (unstable) must have two components RIIIII

I I I IIIIII

I I

iiiiin

I I I illin

I I I 111111

7j

I I I imr

_i

lo-'

0.002 0.010

0.030

104

"SL

10-I

loo

lo1 n.a I

fq

lo2

lo3

lo4

V

FIG.3. The variation with height of the spectrum of the longitudinal wind component in a convective boundary layer. [After H$jstrup (198 1). Published with permission of editor, Boundary Layer Meteorology].

375

THE PLANETARY BOUNDARY LAYER

that obey different scaling. He designed spectra by combining two spectral forms of different types of linear interpolation on a log-log plot. Hgjstrup (1981) designed quite realistic spectra by requiring that klS(kllu2)

FL(zik1 zi/L) + FH(zk1)

=

(3.14)

9

Here FH contributes relatively little to the total energy, so that the effect of height on the variance is negligible. Since FH shifts to a decreasing kl with increasing height but FL does not, the two sections approach each other. At low levels, the spectra have double peaks (see Fig. 3). For numerical approximations to FH and FL for u and u, we refer to Hgjstrup (1981, 1982) or Panofsky and Dutton (1984). One characteristic of the horizontal spectra is that a very tiny addition of convection causes a large increase of spectral energy at low frequencies (Fig. 4). This is why earlier papers suggested different neutral limits when approached form stable and unstable sides. The sudden increase of energy with little convection was also noted in the earlier experiments by Deardorff (1970). In stable, windy air, the spectra of u and u obey Monin-Obukhov similarity [see Kaimal et af. (1972) for numerical forms]. However, as was noted earlier, with very light winds, spectra pick up strong low-frequency energy that is not due to turbulence but probably to gravity waves. Spectra of scalars also have their inertial range at high frequencies. These are particularly important because inhomogeneities of refractive index with these small wavelengths scatter electromagnetic waves.

=i

-

“I=.

v)

0.1

0.01

I

I

I

0.0001 0.001

I I 0.01 0.1 n =- f r

I

I

I

10

U

FIG.4. The variation with stability of the spectrum of the lateral wind component in a convective boundary layer. [After Hfijstrup (1981). Published with permission of editor, Boundary Layer Meteorology.]

376

H. A. PANOFSKY

The spectral densities of a scalar p are given by

k I S( k I )

==

0.23E

~

“ 3 k; ~ 2’3

(3.15)

Here x represents the rate of destruction of turbulent fluctuations of p by molecular action. If production and destruction of fluctuations are in balance, x can be related to other variables: - -

x = -w’P’

ap/az

(3.16)

For some types of waves, the refractive index depends mostly on temperature, with a small correction for moisture so that p’ is proportional to T . In that case, temperature measurements can be used to predict the back scatter of waves by use of Eqs. (3.15) and (3.16), combined with the scattering theory proposed by Tatarskii (1971). Conversely, measured intensity of the back scatter leads to estimates of w’T’, or heat flux. At longer wavelengths, scalar spectra obey Monin-Obukhov scaling. For quantitative formulas, we again refer to Kaimal et al. (1972) for temperature and Schmitt er af. (1979) for moisture. It is probably fair to say that spectra of wind components over uniform terrain in the surface and mixed layers are now well understood, as are the general features of spectra over complex terrain. However, serious controversies remain concerning scalar spectra, particularly the similarity of moisture and temperature spectra. Kaimal et al. (1972) also show that cospectra between w and T and w and u obey Monin-Obukhov scaling. These are important for measurement of w’Tl and w’u’, because they show the range of frequencies responsible for producing these fluxes. Further, if the instruments have insufficient frequency response, the cospectra allow the measurements to be corrected. Cospectra Co also have inertial ranges. Wyngaard and Cot6 (1972) have shown that f Co(f) falls off as ki4I3, much faster than the spectra. Thus there is little correlation between w and T o r w and u at high frequencies, consistent with the condition of isotropy. There now exists a considerable amount of information on two-point statistics in the atmosphere. It is easiest to describe these in terms of coherence and phase delay. Coherence acts like the square of a correlation coefficient, varying from one for perfect correlation to zero for no correlation. Unlike simple correlation, however, the coherence Coh is a function of frequency; usually coherence is best for large eddies (low frequencies). The phase delay also depends on frequency, usually being approximately proportional to it. A good engineering fit is provided by the Davenport hypothesis (Davenport, 1961): that the coherence between two like wind components sepa-

377

THE PLANETARY BOUNDARY LAYER

rated by Ax; (i = 1,2,3 for the three cardinal directions) is given by Coh(f,Ax;) = exp( -aif Ax;lV)

(3.17)

By Taylor’s hypothesis, fAxilV represents the ratio of separation to wave length. The decay constant aiis relatively small for unstable air and for very stable air in the presence of gravity waves. In the important case of strong winds, aiis proportional to Axilz; here the eddies near the ground are squashed and less coherent than above. For detailed formulas, see Bowen et al. (1983). In general, a; is much smaller for separation in the x direction than in the y and z directions. In fact, in strong winds, the variation of coherence with Ay is the same as that with Az, just as with Ay replacing Az. Phase delay in the vertical is due to wind shear; gusts arrive first at the higher levels. The slope of the eddies is of order one, which determines the phase differences at different levels. There is no phase delay between the observations at two points along a horizontal line at right angles to the wind. In the wind direction, the phase change corresponds to the signal velocity that at high frequencies equals the wind speed (Taylor’s hypothesis). For low frequencies, the signal velocity is typically faster than the local velocity because the “eddies” travel with the faster velocities at greater heights. For a more complete discussion of two-point statistics, see Panofsky and Dutton (1984). 4. FIRST-AND SECOND-ORDER CLOSURES In order to deal with the properties of the surface layer over complex terrain, or the whole PBL, we have to consider methods of closing the 7 equations with its 11 unknowns, Eqs. (2.2)-(2.8). The simplest method is the molecular analogy, also called first-order closure. Second-order closure is much more complex but can be used in situations in which firstorder closure fails. In the special case of exchange by large eddies close to the ground, where first-order closure is also inadequate, a relatively simple technique, which we shall call “large-eddy exchange,” works well. 4.1. First-Order Closure

In analogy to molecular transfer, it is assumed that the turbulent transfer flux of a property Kin the i direction, PUT, is down the gradient of iiin the xi direction, and its magnitude is proportional to the gradient of ii in that direction. Here a is the concentration of A per unit mass. Thus

378

H. A. PANOFSKY ~

pu;ua'= -Kaip(dii/dxi)

(4.1)

This equation defines K a i , the transfer or exchange or Austausch coefficient for property A in the xi direction. The magnitude of K indicates how large a flux of A in the xi can be accomplished by a given gradient. It is thus a measure of the strength of the mixing process. Since the symbol K is used quite universally, first-order closure is also called K theory. Also KOihas dimensions of length times velocity. It can be considered to be proportional to the product of eddy size (or mixing length) and the magnitude of eddy velocity component in the i direction. In this way Kaiis similar to the corresponding molecular transfer coefficients, which are proportional to mean free path and molecular velocities. Except in the lowest millimeter, K coefficients are very much larger than molecular coefficients so that the latter can be neglected. But whereas molecular coefficients depend only on temperature, K coefficients vary with all kinds of variables: height, roughness, wind speed, and stability. They can also depend on the distribution of sources and sinks of the variables being transported. To make matters more complex, the K s are sometimes considered as tensors, so that the transport in the i direction can depend on gradients in the j direction. Fortunately, this model is not used in the PBL. Since K theory implies that the eddy size is smaller than the radius of curvature of the profile of the property being transported, the flux is determined by the local gradient. Actually, with large eddies, the flux may be determined by differences of the transported property over large distances. For example, on a hot day there may be no local gradient of potential temperature at a height of 50 m. But w'T' is positive because upward motions bring in very hot air from near the surface. In the absence of strong horizontal gradients (the usual case for u, u, 8, and 4) in the PBL, we are only interested in vertical transfer. In that case, we put

'

These equations close the system (2.2)-(2.8), once the K's are given. It is always assumed that the K,'s for transfer of x momentum and y momentum are equal to each other; K , is called the eddy viscosity, Kh the eddy conductivity, and Kq the eddy diffusivity.

THE PLANETARY BOUNDARY LAYER

379

The properties of the K s in the surface layer over homogeneous terrain can be derived from the definitions and the profile laws of Section 3. For example,

where +,n = kzlu, d d d z is a function of zlL, which can be obtained from Eqs. (4.1) and (4.2). Similar expressions are valid for K h and K , , obtainable from (4.4) and (4.5). Above the surface layer, there is no sure guide for modeling the behavior of the K’s. In the simpler models, the K’s are determined with the use of one of the forms of (4.6) up to the top of the surface layer and then continued by an interpolation formula from there to zero at the top of the PBL. Sometimes forms are chosen above the surface layer, which are similar to (4.6). For example, a representative of eddy size kz is replaced by a quantity 1 that reaches a maximum in the middle of the PBL and goes to zero at the top or reaches a finite limit there. Since the wind begins to turn above the surface layer, aiddz is replaced by d/(dZIaz)* + (at7/az)2. Different forms have been used for (bm, where the argument is changed from z / L to Ri. Wipperman (1973) in his discussion lists over 30 distributions of the K’s. Fortunately, their behavior near the top of t h e PBL is not important because gradients are small there. At night, there often exists a region above the mixed layer, where there is negligible turbulence, but heat is still exchanged by infrared radiation. This situation is sometimes handled by introducing a fictitious Kh, larger than K,,, . This technique is not very precise because only radiative transfer in nearly opaque spectral regions can be described by K theory.

4.2. Large-Eddy Exchange

Large-eddy exchange is a technique to determine local changes of temperature or other variables in the low PBL when the magnitude of the variable in question is large at the surface and considerably smaller up to a height zt. Eddies are of various sizes, but all reach down to the surface. Thus, the flux is not proportional to the local gradient. Blackadar (1979) divided the layer he calls the “mixed” layer ( z 5 zt) into a number of equally thick thinner layers. In each layer a fraction m of surface air and local air is exchanged. Then m is given by Ei w(z), where E i s a constant determined from energy conservation and w(z>is a weighting function that decreases linearly with height from 1 at the surface to 0 at z = zt. Under

380

H. A . PANOFSKY

some conditions, a mixing layer is added above z1 to allow for overshooting. Blackadar (1979) finds that this technique is fast and computationally more stable than K theory and gives realistic results. 4.3. Second-Order Closure

For second-order closure, the system (2.2H2.8) is closed by obtaining additional equations for changes of second-order quantities such as WIT’ = H / c , p . To derive this equation, we multiply the thermal equation T‘,andthenadd of w’ and the equation for dwldt by-and average them. Analogous equations are derived of u t 2 ,v *, w f 2 u, ’ w ’ , u’T’,and TI2and, if needed, similar equations involving moisture. All these equations contain terms on the right that are, in effect, new unknowns, e.g., terms containing third-order properties such as w 7 . Different techniques are used by different scientists to close the system. In almost all cases, pressure terms are modeled by assuming that an anisotropic system will tend to restore isotropy. Third-order terms are put proportional to gradients of second-order terms. Mellor and Yamada (1982) have given a useful review of the derivation and techniques of closing the second-order closure equation. They go from a complete system (their level 4) to level 2 by an analysis of the relative magnitude of the different terms. Level 2, now quite commonly used, neglects total derivatives and diffusion terms. Blackadar (1979) has modified the set of level 2 equations and has obtained some useful results. For example, he is able to reproduce theoretically the surface layer dependence of u,,/u* and +m on z / L in the unstable surface very well. There have been many other attempts with varying degrees of success to use second-order closure to duplicate the observed universal functions in the surface layer. Most of these are given in the paper by Mellor and Yamada (1982). Other applications will be discussed later. A companion article in this volume by Mellor (Chapter 11) further discusses the present state of second-order closure research. 5 . BOUNDARY-LAYER MODELS

5.1. The Surface Layer over Complex Terrain Modeling of turbulence characteristics over complex terrain in the surface layer by first- or second-order closure has been quite successful for

THE PLANETARY BOUNDARY LAYER

38 1

particular special cases: the effect of a sudden change of surface roughness [see, e.g., Rao et al. (1974)l flow over general hills (Taylor et al., 1983), flow in and over canopies (Yamada, 1982), and nocturnal drainage flow (Yamada, 1981). Note that the papers quoted give prior studies of these topics. In order to understand the behavior of flow over complex terrain qualitatively, we begin with the changes of atmospheric flow following a change of surface roughness. Observations of spectra (and simple theory) suggest that high-frequency turbulence is modified more rapidly than is low-frequency turbulence [see Panofsky et al. (1982)l. Thus, surface stress and uwclose to the surface, adapt rapidly to the changing surface conditions, and are essentially in equilibrium with the local surface. In contrast, u,,and uu,with their low-frequency energy, “remember” their past history and thus represent the roughness of a wider area. Following a sudden change of roughness, an interface develops, the slope of which in the surface layer is of order &. The air above this interface has not yet realized the surface change, and the air below it has been modified. Wind profiles usually show a kink at this interface. Now, if one considers a tower, e.g., 100 m downward of the line of roughness change, one finds that u, changes with height, even below the interface. This is because at some distance above the surface, the air hit the interface later and was modified for a shorter period than surface air. Similarly, the local stress and effective roughness length change with height. It has been observed at least at four towers (Brookhaven, Boulder, Cabauw, and Cape Canaveral) that the effective roughness below 10 m is smaller than that above. In each case, with the roughness length below 10 m, z represents the roughness of the local ground cover; zo and (T,,, above 10 m represent the effect of roughness upstream, including form drag, if there is rolling terrain. If zo is needed for large-scale modeling, the values used should not only include surface cover roughness, but also form drag, and should be based on measurements high on towers or on measurements of variances on low-flying airplanes.

5.2. Modeling the Whole PBL In general, modeling of boundary layers over various types of increasingly complex terrain is a very active area of research in 1984. In models of the whole PBL, conditions at the top of the PEL, e.g., temperature and wind, are assumed to be given. There are equilibrium models and timedependent models. In the case of equilibrium models, the interest is in reproducing mean wind and temperature structures and in relating surface

382

H . A. PANOFSKY

fluxes to free-air conditions. In the case of time-dependent models, we are interested in predicting the development of conditions in the PBL over one or more days, given initial conditions and compared with observations. The first equilibrium model was the constant K, model, leading to the Ekman spiral. This is unsatisfactory for the atmosphere because K,,, actually becomes very small near the surface. However, two-layer K , models have been quite successful, beginning with the model by Taylor (1915). He added a lower layer in which the wind shear was parallel to the surface stress. There have been many two-layer and more complex K models since that time, generally reproducing the observed wind distribution well. More important, they reproduce the relation between geostrophic wind and surface stress and, when extended, between heat flux or moisture flux and the conditions above and below the PBL. Assuming that the general forms of the profiles are near z >> zo and z << zi and assuming that there exists a matching region in which both are satisfied, one obtains the two relations (no vertical shear of the geostrophic wind): kGlu, sin a

=

-B

kGlu, cos a

=

In

(5.1) df20 -

A

(5.2)

where G is the geostrophic wind and A and B are parameters independent of height but varying with stability. Angle a is between surface wind and isobars. The relations between A, B, and stability parameters have been found from observations, usually with considerable scatter. [For a recent discussion and references to earlier work see Brown (1982).] Once A and B are known, Eqs. (5.1) and (5.2) can be regarded as two equations in the two unknowns, u,lG and a,as a function of stability and Glfzo, the so-called surface Rossby number. Similar relations exist for (6Top-OO)/T*and Glfza. Thus, given G, zo, A , and B, the surface stress vector can be found. There is a multitude of time-dependent PBL models. For second-order closure models, see Mellor and Yamada (1982). The K models are described by Blackadar (1979). He also includes the large-eddy hypothesis for daytime mixing. Models are usually tested on observations at O’Neill [see Lettau and Davidson (1957)] or on the Wangara observations (Clarke et at., 1971). Agreement is usually good in the major features for several days, but details are often not reproduced well.

THE PLANETARY BOUNDARY LAYER

383

6. BOUNDARY-LAYER PARAMETERIZATION In summary, modeling of the PBL under convective conditions is quite satisfactory. Nocturnal boundary layers, particularly in light winds, still present difficulties. In experiments with atmospheric phenomena on the large scale and mesoscale, conditions in the boundary layer have to be treated specially because of the importance of turbulence, so that the surface fluxes of momentum, heat, and moisture are estimated correctly. There are two different techniques used with low-resolution and highresolution models. Here we mean by low-resolution that there is only one level to represent the whole boundary layer, so that, e.g., in Eq. (2.21, h / a z is approximated by -T&, where T~ is now the surface stress and the stress at the top of PBL is taken as zero. In the case of low resolution, the magnitude and direction of surface stress are estimated from Eqs. (5.1) and (5.2), with similar equations used to estimated surface heat and moisture flux over water, where the surface temperature and humidity are known. Over land, the same equations are needed to determine surface temperature and moisture after estimations the fluxes from heat and moisture budgets at the surface. In simpler models, the thickness of the PBL h is taken as a constant. In more sophisticated models, the behavior of h is predicted along with that of the other variables. [For details, see Deardorff (1972).] For the high-resolution models, the fluxes at various levels have to be included. [For a summary, see Blackadar (1979).] Most modelers use K theory with K given by one of the many techniques described before. An especially simple equation is usually added to represent the surface layer. Only rarely has second-order been combined with general models because of the large amount of additional computation. When second-order closure is used, it is in the simplified forms of the Mellor-Yamada level 2 or 24 [see Mellor and Yamada ( 1982)l. Still, second-order closure requires considerably more computer time than first-order closure does. Not all modelers agree whether the additional cost is worth the increased accuracy of second-order closure. REFERENCES Blackadar, A. K. (1979). High-resolution models of the planetary boundary layer. Adv. Environ. Sci. Eng. 1, 50-85. Blackadar, A. K., and Jersey, G. K . (1980). Efficient modeling of evapotranspiration. In “Beprints of the Fifth Conference on Atmospheric Turbulence, Diffusion and Air Quality, Atlanta,” p. 163-164. Am. Meteorol. Soc., Boston, Massachusetts.

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H. A. PANOFSKY

Bowen, A. J., Flay, R. G. J., and Panofsky, H. A. (1983). Vertical coherence and phase delay between wind components in strong winds below 20 m. Boundary Layer Meteorol. 26, 313-324. Brown, R. A. (1982). On two-layer models and the similarity functions for the PBL. Boundary Layer Meteorol. 24, 451-463. Clarke, R. W., Dyer, A., Brook, R. R., Reid, D. G., and Troup, A. J. (1971). “The Wangara Experiment: Boundary Layer Data,” Tech. Pap. 19. Div. Meteorol. Phys., CSIRO, Melbourne, Australia. Davenport, A. G. (1961). The spectrum of horizontal gustiness near the ground in high winds. Q. J . R . Meteorol. SOC.87, 194-211. Deardorff, J. W. (1970). Preliminary results from numerical integrations of the unstable planetary boundary layer. J . Atmos. Sci. 27, 1209-121 1 . Deardorff, J. W. (1972). Parameterization of the planetary boundary layer for use in general circulation models. Mon. Weather Reu. 100, 93-106. de Bruin, H. A. R., and Holtslag, A. A. M. (1982). A simple parameterization of the surface fluxes of sensible and latent heat during daytime compared with the Penman-Monteith concept. J . Appl. Meteorol. 21, 1610-1621. Finnigan, J. J., and Einaudi, F. (1981). The interaction between an internal gravity wave and the planetary boundary layer. Q.J . R . Meteorol. SOC. 107, 793-832. Colder, D. (1972). Relations among stability parameters in the surface layer. Boundary Layer Meteorol. 3, 47-58. HGjstrup, J. (1981). A simple model for the adjustment of velocity spectra in unstable conditions downstream of an abrupt change in roughness and heat flux. Boundary Layer Meteorol. 21, 341-356. HGjjstrup, J. (1982). Velocity spectra in the unstable boundary layer. J . Atmos. Sci. 39, 2239-2248. Kaimal, J. C. (1978). Horizontal velocity spectra in an unstable surface layer. 1.Atmos. Sci. 35, 18-24. Kaimal, J. C., Wyngaard, J . C., Izumi, Y.,and Cot& 0. R. (1972). Spectral characteristics of surface layer turbulence. Q. J . R . Meteorol. Soc. 98, 563-589. Kaimal, J. C., Wyngaard, J. C., Haugen, D. A., Cote, 0. R., Izumi, Y., Coughey, S. J., and Readings, C. J. (1976). Turbulence structure in the convective boundary layer. J . Almos. Sci. 33, 2152-2169. Lettau, H., and Davidson, B., eds. (1957). “Exploring the Atmosphere’s First Mile.” Pergamon, Oxford. Liu, W. T., Katsaros, K. B., and Businger, J. A. (1979). Bulk parameterization of air-sea exchanges of heat and water vapor including the molecular constraints at the interface. J . Atmos. Sci. 36, 1722-1735. Mellor, G. L. (1973). Analytic prediction of the properties of stratified planetary surface layers. J . Atmos. Sci. 30, 1061-1069. Mellor, G. L., and Yamada, T. (1982). Development of a turbulence closure model for geophysical fluid problems. Reu. Geophys. Space Phys. 20,851-875. Panofsky, H. A., and Dutton, J. A. (1984). “Atmospheric Turbulence: Models and Methods for Engineering Applications.” Wiley (Interscience), New York. Panofsky, H. A., Tennekes, H., Lenschow, D., and Wyngaard, J. C. (1977). The characteristics of turbulent velocity components in the surface layer under convective conditions. Boundary Layer. Meteorol. 11, 355-361. Panofsky, H . A., Larko, D., Lipschutz, R., Stone, C., Bradley, E. F., Bowen, A. J., and H@jstrup,J. (1982). Spectra of velocity components over complex terrain. Q. J . R . Meteorol. SOC.108, 215-230.

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Paulson, C. A. (1970). The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer. J . Appl. Mefeorol. 9, 857-861. Rao, K. S., Wyngaard, J. C . , and Cote, 0. R. (1974). The structure of a two-dimensional internal boundary layer over a sudden change of surface roughness. J . Atmos. Sci. 31, 738-746. Schmitt, K.F.,Friehe, C. A., and Gibson, C. H. (1979). Structure of marine surface layer turbulence. J . Atmos. Sci. 36, 602-618. Tatarskii, V. I. (1971). “The Effects of the Turbulent Atmosphere on Wave Propagation” (translated from the Russian by J . W. Strohbehn). Israel Program for Scientific Translations, Jerusalem. Taylor, G. I. (1915). Eddy motion in the atmosphere. Philos. Trans. R. Soc. London. Ser. A 215, 1-26. Taylor, P. A., Walmsley, J. L., and Salmon, J. R. (1983). A simple model of neutrally stratified boundary-layer flow over real terrain incorporating wave number dependent scaling. Boundary Layer Meteorol. 26, 169-189. Tennekes, H., and Driedonks, A. G. M. (1981). Basic entrainment equations for the atmospheric boundary layer. Boundary Layer Meteorol. 20, 515-529. Wesley, M. L., Thurtell, G. W., and Tanner, C. B. (1970). Eddy correlation measurements of sensible heat flux near the Earth’s surface. J . Appl. Meteorol. 9, 45-50. Wipperman, F. (1973). “The Planetary Boundary Layer of the Atmosphere.” Deutscher Wetterdienst, Offenbach, West Germany. Wyngaard, J. C., and CotC, 0. R. (1972). Cospectral similarity in the atmospheric surface layer. Q. J . R . Meteorol. Sac. 98, 590-603. Yamada, T. (1981). A numerical simulation of nocturnal drainage flow. J . Mefeorol. Soc. Jpn. 59, 108-122. Yamada, T. (1982). A numerical model study of turbulent airflow in and above a forest canopy. J . Meteorol. Soc. Jpn. 60, 439-445. Zhou, L.,and Panofsky, H . A. (1983). Wind fluctuations in stable air at the Boulder tower. Boundary Layer Meteorol. 25, 353-362.

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MODELING STUDIES OF CONVECTION YOSHIOGURA Department of Atmospheric Sciences Uniuersity of Illinois Urbana, Illinois 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2. Benard-Rayleigh Convection . . . . . . . . . . . . . . . . . 3. Complexity of Convection in the Atmosphere . . . . . . . . . . 4. Shallow Moist Convection . . . . . . . . . . . . . . . . . . 5. Deep Moist Convection. . . . . . . . . . . . . . . . . . . . 5.1. Life Cycle of Air-Mass Thunderstorm Cells . . . . . . . . . 5.2. Mesoscale Convective Systems. . . . . . . . . . . . . . . 5.3. Simulations of Long-Lived Squall Lines . . . . . . . . . . 5.4. Life Cycle of Mesoscale Convective Systems . . . . . . . . 6 . Feedback Effects of Cumulus Clouds on Larger-Scale Environments . 7. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . .

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

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

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

387 388 391 392 397 397 397 399 406 408 413 416

1. INTRODUCTION

About 20 years ago, this author published a review paper on modeling studies of atmospheric convection (Ogura, 1963a). It started with classifying atmospheric circulations depending on their horizontal scales and subsequently focused its review on those circulations that have comparable horizontal and vertical scales, namely, nonhydrostatic circulations. It was followed by a brief description of the anelastic system derived by Ogura and Charney (1962) and OguLa and Phillips (1962). This system excludes acoustic waves and provides a mathematical framework convenient for use in numerical studies of nonhydrostatic circulation. I then went on to review articles published by that time in the area of modeling studies of convection. There were not many to review, however. Advances made in this area were quite limited compared to those achieved in modeling studies and simulations of synoptic- and globalscale circulations. Recall that numerical *eather prediction using the quasi-geostrophic equations became operational long before that time. The first realistic simulation of the global circulation with the quasi-geostrophic system was made 7 years earlier (Phillips, 1956). Smagorinski (1963) had already presented the results of his general circulation experiments with the primitive equations. Therefore, the modeling study of convection was born quite late indeed. I concluded that review paper by expressing my conviction that the modeling study would prove to be 387 ADVANCES I N GEOPHYSICS, VOLUME

28B

Copyright 0 1985 by Academic Presb. Inc. All rights of reproduction in any form reserved.

388

YOSHI OGURA

extremely useful in increasing our understanding of the highly nonlinear fluid mechanics of atmospheric convection. This conviction was based partially on my own experience in the energy cascade problems in twoand three-dimensional turbulent flows (Ogura, 1962, 1963b). The situation has changed dramatically during the past 20 years. A wealth of observational data for atmospheric convection studies has been accumulated from measurements by various means, including radars, instrumented aircrafts, satellites, and rawinsondes in special field observations. This has made it possible for modelers to design their models properly, verify their predictions against observations, and improve their models, as it had been possible for modelers of synoptic and global scale circulations for a long time. In turn, the model results have aided us in understanding the underlying physics operating in complex atmospheric convection for which observations alone could not provide all dynamically consistent data. Interest among meteorologists in this area of research has grown to such a degree that during the past several years, it has become one of the three disciplines that has demanded most of the CRAY-1 computer time at the National Center for Atmospheric Research (the other two being numerical weather prediction and climate studies). The objective of this chapter is to put the current status of mathematical modeling studies of convection in historical perspective, assess future prospects, and pinpoint immediate problems, as requested by the editor of this book. The editor also suggested that this chapter present the author’s view of the topic rather than a comprehensive account of everybody’s work. This chapter follows that suggestion since I found it nearly impossible, and perhaps even undesirable, to try to present a comprehensive account of previous research for several reasons. The space available for presentation is limited while research under the heading of convection is wide in scope. Furthermore, several excellent review papers have been published on some specific topics; these will be referred to in the text. References in this chapter are, therefore, not intended to be exhaustive. 2. B~NARD-RAYLEIGH CONVECTION We begin this chapter with the classic BCnar3-Rayleigh convection that occurs in a fluid confined between two horizontal plates with the Rayleigh number Ra exceeding a certain critical value. The purpose here is to show that vastly different flow patterns can de1;elop in a fluid even in this, possibly the simplest of fluid situations. At the onset of convection, the fluid motion takes the familiar cellular pattern. The aspect ratio hlh (X is horizontal wavelength, h layer depth)

389

MODELING STUDIES OF CONVECTION

for cellular patterns is approximately 3. When Ra exceeds the critical value slightly, the flow pattern changes and two-dimensional rolls are the preferred mode of convection (Fig. 1). More important, the amplitudes of the roll vortices become finite and linear theories cannot adequately describe the behavior of convection. This is evident from laboratory studies that show that the spectrum of roll vortices contains a single sharp peak at a certain wavelength, whereas linear theory predicts a relatively broad band of exponentially growing disturbances [e.g., Busse and Whitehead (1971)l. A number of nonlinear stability analyses have been done, with the conclusion that two-dimensional rolls are indeed stable solutions only with respect to perturbations of infinitesimal amplitude for a certain range of Ra and Prandtl number Pr [e.g., Busse (1967)l. However, one question has not been completely answered concerning which wavelength of two-dimensional rolls is preferred for given values of Ra and Pr. In this regard, Ogura (1971) performed numerical experiments for two-dimensional rolls at mildly supercritical conditions. Unlike the

.. ..

10'.

U U

pv

U U

Turbulent flow 0

loe'

*

0 2

.

Time dependent three-dimensional flow

I

-

(3

w

J

105-

> n

a

0 0

0

u 5

0

. 103

lo-*

P

t

-F

I

I

lo-'

0

Steady two-dimensional flow

30

I

No motion 10

I 102

103

I 10'

PAANDTL NO.

FIG. 1. Regime diagram for Bknard-Rayleigh convection. The 0 represents steady cellular flow; 0 time-dependent cellular flow; 0 cells and transient bubbles (transitional); H transient bubbles; large-scale flow with tilted plumes; and * heat flux transitions. [From Krishnamurti and Howard (1981).]

-

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YOSHI OGURA

previous numerical experiments [e.g., Saltzman (1962)], he considered the large horizontal extent of the fluid so that many modes could be accommodated. His result shows that the scale selection depends on the initial conditions, implying that the motion is history dependent. However, his numerical experiments were done only for strictly two-dimensional rolls. From their laboratory experiment, Willis et al. (1972) noted that all mechanisms that can change the wavelength of rolls imply the existence of structure that is three-dimensional. Numerical experiments for three-dimensional Btnard convection were done by Somerville and Lipps (1971), Somerville and Gal-Chen (1979), and others. However, because of the computer limitations, no numerical experiments have been performed for three-dimensional convection with a large horizontal extent of the fluid. At this point, this author cannot help referring to a remarkable work of Lorenz (1963) that was done by using minicomputer rather than a supercomputer. Lorenz considered the fluid situation identical to the two-dimensional Benard-Rayleigh convection. Instead of integrating the full equations governing this fluid system as an initial value problem, he expanded the variables into an infinite number of modes [see Saltzman (1962)l and then set all but three of the modes to be identically zero. The chaotic behavior of the fluid motion represented by the numerical solutions had a great impact on our way of thinking of not only finite amplitude thermal convection, but also turbulent flows and climate changes. The Lorenz equations have been extensively discussed since then not only by atmospheric scientists, but also by hydrodynamists and applied mathematicians, as evidenced in a recent book by Sparrow (1982). Returning to Benard-Rayleigh convection, flow patterns take different forms as the Rayleigh number increases. This was extensively investigated in laboratory experiments by Krishnamurti (1970) and Krishnamurti and Howard (1981). Figure 1 summarizes their observations of flows found for various ranges of Rayleigh and Pandtl numbers. It includes the five transition curves. The flow is steady two-dimensional in the range between curves I and 11, as discussed before. It is steady three-dimensional cellular between curves I1 and 111, and time-dependent (quasiperiodic), three-dimensional cellular between curves I11 and V. In all these regimes (that is, to the lower right of curve V), the flow is cellular in the sense that a fluid parcel initially at some horizontal location is always confined to the vicinity of that location, horizontal excursion being limited by cell boundaries that have horizontal separation comparable to the layer depth. On the other hand, to the upper left of curve V, cellular flow disappears and is replaced by a random array of transient plumes. The new finding in Krishnamurti and Howard’s experiment (1981) is that with

MODELING STUDIES OF CONVECTION

39 1

a further increase of the Rayleigh number, large-scale turbulent flow develops in which these plumes drift in one direction near the bottom and in the opposite direction near the top of the layer. With the onset of this large-scale flow, the horizontal scale of motion has increased from that comparable to the layer depth to one comparable to the entire chamber width. The question of whether or not this large-scale flow has any counterpart in meteorological or geophysical motions has yet to be explored. 3. COMPLEXITY OF CONVECTION IN THE ATMOSPHERE In the preceding section, we showed that the fluid motion can be extremely complex even in a simple flow situation in which the fluid is confined between two horizontal plates and heated uniformly from below. It would then not be surprising that convection in the real atmosphere takes vastly different forms. This is manifested in the development of different types of convective clouds, as will be described in this chapter. Along with the vertical shear of the ambient wind (see Section 5), phase changes of water in the atmosphere may be one of the major factors that cause complexity in the evolution and structure of clouds. Whenever phase changes of water occur in some parts of the convective region, a local positive or negative buoyant force begins. Further, a stratiform cloud deck represents a radiative heat sink. For many years, physical processes involved in the initiation and growth of liquid and ice particles in clouds have been the major topic in cloud physics. However, it was only in the 1960s that meteorologists began to realize that there are strong interactions between cloud physics and dynamics. The water loading and evaporation from liquid and ice particles provide a negative buoyant force to convective clouds, thus exerting significant effects on their evolution (Section 5.1). In turn, cloud microphysical processes that determine the mixing ratios of liquid and ice particles as well as the size distributions of particles are controlled by humidity, temperature, and air motion, which are continuously changing with time and space during the evolution of clouds. Thus any successful deep cloud models must incorporate not only the sub-grid-scale turbulence, but also cloud microphysical processes. This requirement increases enormously the demand for computer time. The typical radii of condensation nuclei, cloud droplets, raindrops, and hail are of the order of and m, respectively. On the other hand, the typical horizontal size of the thunderstorm is lo4 m. Therefore, a numerical study of a thunderstorm has to cover length scales ranging from to lo4 m. However, there are gaps between the typical scales of cloud micro-

-

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physical processes and cloud motions we wish to describe. Consequently, microphysical processes are “parameterized” in many cloud models in which liquid and ice particles are represented by only several variables. In a typical three-dimensional deep cloud model, the model domain is represented by 7.4 X 104 grid points (48 x 48 X 32) with 1-km horizontal and 500-m vertical grid intervals, and only two categories of liquid drops are considered (Section 5.1). An integration of a model of this size with 6-s time step requires about 0.7 hr of central processor time on the CRAY-I computer for a I-hr integration (Klemp and Wilhelmson, 1978). An example of another line of research is the work of Takahashi (1979) in which the size distribution of liquid drops was represented by 59 categories while the third dimension in space was sacrificed. The need for a computer that runs one or two orders of magnitude faster than CRAY-1 is obvious. Other processes that make atmospheric convection complex include strong mutual interactions among the convective elements themselves (Section 5.3) and between these elements and the environment in which they are embedded (Section 6). Further, flows in the troposphere are turbulent virtually always and everywhere. This makes it difficult to apply directly the results obtained for laminar BCnard-Rayleigh convection described in Section 2 to atmospheric convection (Section 4). 4. SHALLOW MOISTCONVECTION

Shallow convection represents convective overturning with the vertical extent much smaller than the scale height of the atmosphere (-10 km). Conversely, the vertical extent of deep convection is comparable with the atmospheric scale height. Shallow convection in the atmosphere occurs generally when the lower troposphere is capped by an inversion layer. With the advent of the meteorological satellite program in the early 1960s, it became apparent to meteorologists for the first time that the cloud manifestation of BCnard-Rayleigh cellular convection frequently occurred over vast expanses of the Earth. Very soon, however, meteorologists also realized that some shallow clouds, although they looked similar to BCnard convection, could not be explained adequately by a simple application of Rayleigh theory. As an example, consider convection that develops over the Japan Sea during periods of cold air outbreaks. Convection takes the form of regular horizontal cloud streets that are aligned close to the direction of the wind. These cloud streets persist across the Japan Sea and frequently extend to the East China Sea, where a roll convection regime changes to a threedimensional cellular convection regime, in a seemingly similar fashion

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shown in Fig. 1. Similar cloud streets and three-dimensional cellular convection also develop south of the ice edge in the Bering Sea. This is shown schematically in Fig. 2a (Walter, 1980). Individual three-dimensional cells typically have diameters of 30 to 60 km. They form in the planetary boundary layer with depth of 1 to 2 km, generally capped with a welldefined inversion layer. These cells are often referred to as mesoscale cellular convection (MCC). Hubert (1966) identified two kinds of cell structure in MCC, namely, the open cells with cloud-free centers and cloud walls, and closed cells with cloudy centers and clear walls. Cloudiness is about 40% in regions of open MCC and about 90% in regions of closed MCC (Agee and Lomax, 1978).

I

ICE PACK

FLOW OF COLD AIR

ROLLS

OPEN CELLS

CELLS

--

FRONT

. , '/h-15 A/h-27 A- 3 0 k m A-32km h 2.25km h 1.25km

P-"

A/h-2.8 TRANSITION ~ - 3 k m REGION h - 1.1km A/h -6-7

-

d

-

FIG.2. (a) Schematic representation of how the convective regimes change with distance from the ice edge. (b) North-south cross section showing hlh for different distances from the ice edge. [From Walter (1980). From Monthly Weather Reuiew, copyright 1980 by the American Meteorological Society. I

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Thus we may define an open (closed) convective pattern as one in which most of the air is descending (ascending) (Helfand and Kalnay, 1983). Mesoscale cellular convection has attracted the interest of meteorologists partly because it forms over the vast expanses of the oceans (Agee et al., 1973). Thus (particularly with regard to the closed-cell structure) it constitutes an important component of the radiative-convective processes in the Earth's climate (Arakawa, 1975). Further, MCC itself poses the following interesting problems. The first problem is the aspect ratio of the cloud streets and MCC. Recall that the aspect ratio is defined as the ratio of the roll-cloud wavelength or the diameter of MCC (A) to the depth of the convection ( h , height of the cloud top or the inversion base). Figure 2b (Walter, 1980) shows that both h and hlh increase with the distance from the coastline or the ice edge along the rolls. A more recent study of Miura (unpublished paper) for cloud streets over the Japan Sea also shows a similar relation between hlh and h, with the values of hlh ranging from about 4 to 20. The fact that Xlh increases with the layer depth was found more than a half-century ago by Terada (1928) in his laboratory studies, which used a thin layer of alcohol, of horizontal roll vortices. Convection was driven by the evaporative cooling at the upper surface and the mean flow was generated by inclining the glass container. The values of Xlh he found were 3.2 for h = 0.05 m, increasing to 5.20 for h = 0.35 m. Two basic mechanisms were proposed to explain the occurrence of roll vortices. The first, inflection point instability, requires an inflection point in the velocity profile in the plane normal to the roll axis. This mechanism has been studied by Brown (1972), Faller ( 1963), Faller and Kaylor ( 1966), and Lilly (1966). Typical values of Xih for roll vortices with the maximum growth rate resulting from these studies are near 2 to 3 for stable, neutral, and slightly unstable conditions. Roll vortices are typically aligned about 20" from the direction of the gradient wind. The second mechanism for explaining roll vortices is that of buoyancy combined with wind shear. Here the rolls produced by buoyancy are aligned parallel to the direction of the vertical shear (Asai, 1970a,b, 1972) or shear gradient (J. Kuettner, 1959; P. Kuettner, 1971). Values of Xlh in these cases range from 2.5 to 2.8, similar to the values in the classic BCnard-Rayleigh convection. Thus none of these mechanisms adequately explains large values of hlh. Several hypotheses have been proposed to explain Xlh as large as 20 to 30. However, none of them seems to be convincing. A most popular one is that proposed first by Priestley (1962). It attributes large Alh to an anisotropy of the eddy diffusion coefficients. Linear stability analysis by Mitchell and Agee (1977) for MCC and that by Sun (1978) for cloud streets

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in the tropics adopted this hypothesis. While the anisotropy of the eddy diffusion coefficients is valid, the obvious weakness of this argument is the lack of basis on which the proper magnitude of the horizontal diffusion coefficient is determined. Another interesting problem related to MCC is determination of the mechanism(s) governing open cells and closed cells. Laboratory observations show that open cells are preferred in gases in which viscosity increases with temperature and hence decreases with height, whereas closed cells are preferred in liquids where viscosity decreases with temperature and hence increases with height. Tippelskirch’s (1956) experiments with molten sulfur indicated that the direction of the flow in his convection cells changed at 153°C;the viscosity of sulphur decreases with temperature below 153°C and increases with temperature above 153°C. Based on these observations, Hubert (1966) and Agee and Chen (1973) suggested that vertical variation of the eddy viscosity may determine the open or closed cells. However, it has not been verified by observations that the eddy diffusion coefficient indeed decreases (increases) with height in the main body of the planetary boundary layer with open(closed-) cell situations. Krishnamurti (1968a,b) studied the dependence of cell configuration on the time rate of change of the mean temperature of the fluid. She found that open cells occurred when the upper and lower boundaries were warming uniformly in time and that the convection pattern was closed when the boundaries were cooling uniformly in time. In a later series of papers, Krishnamurti (1974a-c) investigated yet another possible mechanism. She found that closed cells were stable in a uniformly rising fluid while open cells preferred a uniformly sinking fluid. Somerville and GalChen (1979) performed numerical simulations of three-dimensional BCnard convection with mean vertical motion. Their result shows that the presence of mean vertical motion induces a qualitative and striking change in the preference mode of Benard convection. However, Sheu and Agee (1977) observed that both open and closed cells occurred under the conditions in which the synoptic-scale subsidence prevailed. Helfand and Kalnay (1983) proposed another mechanism. Unlike the ordinary Benard convection, their convection is driven by horizontally uniform but vertically varying internal heating and cooling. Figure 3 shows the result from their numerical experiments. The horizontal extent of their model fluid was taken to be three times larger than the depth of the fluid layer. Consequently, three cells developed in case (a) in which the internal heating profile was symmetric with respect to the central plane of the fluid (i.e.,the upper half of the fluid was cooled and the lower half was heated at the same rate). Further, the width of the upward motion was the

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W 100

-

FIG.3. The horizontal profiles of dimensionless vertical velocity at the central plane of the fluid in the steady state for two different dimensionless internal heating profiles (a) and (b) shown in the inset. [From HeIfand and Kalnay (1983). From Journal ofAtmospheric Science, copyright 1983 by the American Meteorological Society.]

same as that of the downward motion in each cell in this case. However, as soon as asymmetry of the internal heating profile was introduced in this model, the flow suddenly changed and settled down into a single cell, as exemplified in case (b). Thus asymmetry of the internal heating profile increases the overall aspect ratio Xlh of the convective cell. Moreover, the relative width of the upward motion was much narrower than that of the downward motion when heating was concentrated near the bottom of the fluid layer. Conversely, cooling concentrated near the top of the fluid layer resulted in strong descending currents that were compensated by weak ascents over most of the horizontal area (i.e., closed cells). This mechanism is appealing since closed cells are observed to form when stratus cloud decks are broken up into stratocumulus clouds, and this process is undoubtedly driven by radiative cooling at the top of the cloud decks. However, the literature does not tell us whether all closed cells form in this way. Data gathered during the Air Mass Transformation Experiment (AMTEX) indicated that there were 14 open cell cases and 20 closed cell cases in the special observation network located over the warm Kuroshio current (Sheu and Agee, 1977). Thus there is a need to examine further the transition nature of MCC, both observationally and theoretically.

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5. DEEPMOISTCONVECTION 5.1. Life Cycle of Air-Mass Thunderstorm Cells The modern study of thunderstorms began with the U.S. government’s Thunderstorm Project (Byers and Braham, 1949), which investigated the common summertime thunderstorms of Florida and Ohio. These thunderstorms, which are now called “air-mass” thunderstorms, form almost daily during summer over much of the United States. They occur in widespread convectively unstable air masses characterized by low-level warm humid air and little vertical wind shear. The internal structure of an individual air-mass thunderstorm was found to be characterized by a more or less random pattern of “cells” that were the storm’s basic convective elements. Each cell within a thunderstorm progressed through a characteristic life cycle. Some early simulations of moist convection in the beginning of the 1960s discussed the evolution of thunderstorm cells [see Ogura (1963a)l. Since then, the life cycle of a single cell within a thunderstorm has been simulated quantitatively in numerous cloud models with different degrees of sophistication of cloud physics parameterizations. Figure 4 shows an example of the model results that illustrate in a fairly realistic manner the three stages (developing, mature, and dissipating) in the life cycle of a thunderstorm cell in light of radar observations (Houze and Hobbs, 1982). This modeling study clearly shows that a model thunderstorm cell progresses through a life cycle only when precipitating particles are included in a model. The water loading and evaporation from precipitating particles provide a negative buoyant force and are responsible for dissipation of a cell. Thus, in order to simulate a warm rain cloud, it is necessary to consider at least two categories of water drops: small cloud droplets and raindrops.

5.2. Mesoscale Convective Systems In the tropics and mid-latitudes, convective deep clouds are frequently organized to form mesoscale convective systems (MCSs). Here MCSs are broadly defined as all precipitation systems that have horizontal scales of 10 to 500 km and include significant deep convection during some part of their lifetime. In mid-latitudes, MCSs include rainbands, squall lines, mesoscale convective complexes, multicell storms, and supercell storms. Briefly stated, the multicell storm is similar to the air-mass thunder-

-

10

-Ec- 8 Y

16

P w

I 4 2 0

0

TIME(M1NUTES 1

TIMEIMINUTES)

12

r(b) 10

-

lot

5i Y

8-

5 6Y

Y

'2-

d 0:

Ib

2bTIME(M1NUTES) 30 tb sfo

1

Qo

10

o:

"[ (

10

lot

II 10

20

I

I

I

50 30 40 TIMEIMINUTES)

1

60

J

70

f

20

30 40 50 TIME(MINUTES)

60

1

oohdr++nkTk---hTIME(MINUTES1

FIG.4. Time-height cross sections of (a) vertical velocity (meters per second), (b) excess temperature (degrees Celsius), (c) liquid and solid water content (grams per kilogram), (d) content of cloud droplets (grams per kilogram), (e) content of raindrops (grams per kilogram), and (f) content of ice crystals (grams per kilogram) for a thunderstorm cell simulated by a one-dimensional cloud model. [From Ogura and Takahashi (1971). From Monthly Weather Reuiew, copyright 1971 by the American Meteorological Society.]

70

MODELING STUDIES OF CONVECTION

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storm already described, except that the cells form and move through the storm in a systematic and organized, rather than random, fashion (Section 5.3). The supercell storm takes on the character of a single giant quasisteady thunderstorm cell. However, both multicell and supercell storms differ from air-mass thunderstorms in that they occur in environments of strong vertical wind shear [see Wiesman and Klemp (1982) for further discussion]. Therefore their internal structure is significantly different from the structures of air-mass thunderstorms. In the tropics, precipitation is almost wholly convective in origin. Deep convective clouds are frequently organized to form large cloud clusters that contain rain areas covering up to approximately 5 x lo4 km2 in area. Two types of cloud clusters are recognized. Squall clusters are associated with tropical squall lines and characterized by their explosive growth, their distinct and generally convex shaped leading edge, and their rapid propagation. Nonsquall clusters do not form as dramatically or propagate as rapidly as the squall clusters. However, they are important because they are so numerous. The squall clusters are relatively rare. The reader is referred to the review articles of Houze (1981), Houze and Betts (1981), and Houze and Hobbs (1982) for more detailed descriptions of MCSs in the mid-latitudes and the tropics. 5.3. Simulations of Long-Lived Squall Lines

As described in Section 5.1, the life span of individual thunderstorms that form in the weak vertical shear environment is about 30 to 60 min. On the other hand, some MCSs are known to live much longer than 1 hr. Examples of extremely long-lived MCSs include an isolated cloud cluster that traveled from the eastern foot of the Tibetan Plateau to east of Japan (Ninomiya et al., 1981) and MCSs that formed in the mountains of central Colorado and traveled across the United States (Cotton et al., 1983; Wetzel et al., 1983). The process responsible for the shortness of the life span of air-mass thunderstorm cells is that the downdraft that develops inside the cell due to water loading cuts off the supporting upward motion. In contrast, the vertically sheared flow effectively dislocates the precipitating particles from the updraft area. This makes it possible for some MCSs to live longer or even to attain a quasi-steady state. With this recognition, some cloud modelers focused their effort on investigating the effect of the vertical wind shear on the structure and evolution of the thunderstorm. The early work in this research area included Orville (1968), Takeda (1971), Schlessinger (1973), Hane (1973), and Wilhelmson (1974). All of them except Wilhelmson used two-dimen-

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discussed here. The prime symbol in the preceding equations denotes deviations from area-averaged quantities. Values Ql and Q2 represent the apparent sources of heat and moisture, respectively, and can be measured either by observations or by grid values in a large or regional, scaleprediction model. Thus, Eqs. (6.lb) and (6.2b) represent the cloud heating and drying effects, respectively. They consist of three terms: vertical transport of sensible heat and moisture by cumulus convection, condensation of water vapor that occurs inside cumulus clouds, and evaporation from liquid and ice particles. Figure 10 shows the result from the cumulus ensemble model applied to

I

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TOTAL CLOUD EFFECT

I

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)

\

/ 1

1, 1.5 2.0 2.5 0.5 1.0 HEATING RATE (K hr 1 FIG.10. Heating rates by condensation c, evaporation e, and vertical transport of sensible heat by clouds F from a two-dimensional cumulus ensemble model. The sum of the three terms represents the total cloud effect (dashed line). The cloud-heating effect is estimated from the large-scale heat budget during the period from 06 through 12 GMT 12 August 1974 and is denoted by Ql - Q,(solid line). [From Tao (I983).] -1.5

-1.0

4.5

0

-'

MODELING STUDIES OF CONVECTION

41 1

the situation in the tropical rainband event for the period 6-12 GMT August 12, 1974 (Tao, 1983). Since the large-scale forcing is predominantly due to upward motion in the tropics, the observed w field shown in Fig. 9 was introduced into the model as the large-scale forcing. As noted earlier, we interpreted this w field at middle and upper levels as a consequence of convection, rather than a large-scale forcing to cause convection. This did not matter in Tao’s research since his interest was in statistical properties of clouds that were in a quasi-equilibrium state with the given large-scale forcing. Clouds were generated in the model by introducing small-amplitude perturbations at low levels. The model was run for 6 hr of physical time. The closeness of agreement between the time-averaged heating rate at each level of model clouds and the rate of cooling by large-scale motion estimated from the large-scale heat budget indicates that model clouds were indeed in a quasi-equilibrium state with the large-scale forcing. Figure 10 also shows that the heating rate by the vertical flux of sensible heat is one order of magnitude smaller than that by condensation at all altitudes, except, of course, in the subcloud layer. On the other hand, the maximum cooling rate by evaporation is comparable with the condensation heating rate in magnitude. Information of statistical properties of clouds gained from the cumulus ensemble model facilitated close examination of some cumulus parameterization schemes. Mr. C. Y. J. Kao and the author are currently examining the Arakawa-Schubert scheme (Arakawa and Schubert, 1974). This scheme will be noted as the AS scheme. Lord (1978, 1982) has extensively tested the AS scheme on the basis of a semiprognostic method (i.e., one time-step integration) by using the GATE phase 111 data. His conclusions were that the AS scheme predicts precipitation amounts very well and that the vertical profiles of cloud heating and drying effects agree fairly well with those estimated from the large-scale heat and moisture budgets when averaged over the entire phase 111. We have applied the AS scheme to the tropical rainband event described earlier with an algorithm different from Lord’s. Three-hourly rawinsonde observations were available for this event. As an example, Fig. 11 shows the cloud-heating rate predicted from the AS scheme by means of the one time-step integration in the developing stage of the rainband. Good agreement between the prediction and the observation based on the large-scale heat budget is evident. The cloud-heating profiles and moistening profiles at other observation times (diagrams are not shown) indicate equally good agreement. The AS scheme considers many cumulus clouds with different entrainment constants, and the cloud properties of each class of clouds are specified by one-dimensional, steady-state, entraining cloud model. Thus, the AS scheme permits the prediction of not only the net cloud-heating

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-0.5 0 0.5 1.0 1.5 2.0 HEATING RATE (K hr-’) FIG.11. Same as Fig. 10, but calculated by using the Arakawa-Schubert cumulus parameterization scheme for the situation at 09 GMT 12 August 1974.

-1.5

-1.0

effect [i.e., the sum of the three terms in Eq. (6.lb)], but also each term of Eq. (6.lb) individually. To satisfy our curiousity, this was computed and the result is shown in Fig. 11. A comparison of Fig. 1 1 with the result from the cumulus ensemble model (Fig. 10) shows that the AS scheme substantially underestimates the cooling rate by evaporation. This is not surprising since, in the AS scheme, all raindrops fall out from the system as soon as they form. Evaporation occurs from cloud droplets in the layer of detrainment that occurs only at the cloud top level. Interestingly, the AS scheme underestimates the rate of condensation also, with the result that the net heating rate is just about right. We have also simulated the evolution of the rainband under consideration. The model we used for this simulation was two-dimensional and

MODELING STUDIES OF CONVECTION

413

hydrostatic. The horizontal grid arrangement was similar to that of Yamasaki (1975). It was comprised of a grid 300 km in length with a 20-km interval (which we designated as the rainband area) and was flanked at each side by stretched grid areas to represent the environment. The AS scheme was incorporated into the model. In the simulation, there was no motion initially and convection was forced to occur by imposing lifting over the rainband area. The lifting, however, was confined to the layer below 600 mb, with the maximum value of 1.5 pb s-' at 850 mb, so as to correspond to the o field observed at 00 GMT (in Fig. 9). Figure 12 shows the air flow and the temperature anomaly from the initial condition at physical time of 7.5 hr. Since no rotational flow developed in this experiment, the heat released at the center of the rainband spreads laterally with the gravity waves, followed by the development of a mesoscale circulation. The AS scheme was originally designed to predict cloud effects on large-scale circulations whose time scale is much larger than that of individual convection cells so that the quasi-equilibrium hypothesis is applicable. Obviously, the small time scale of the rainband under investigation makes an application of the AS scheme less justifiable. Nevertheless, the AS scheme appears to be able to simulate the developing stage of the rainband qualitatively.

7. CONCLUDING REMARKS In this chapter we reviewed the progress of the modeling study of some aspects of atmospheric convection achieved during the past 20 years. The progress has been impressive and, more important, appears to have accelerated in recent years. This acceleration may be attributed to several factors. An increased availability of large, fast computers to modelers is one. The rapid accumulation of observational data from several field programs is another. An increased confidence among modelers in the capability of their models to actually simulate at least some aspect of the seemingly complex atmospheric convection is another. The last point may require a little elaboration. Unlike synoptic-scale motions, the moist atmospheric convection processes, with which most cloud modelers are dealing, have a characteristic that, when the atmosphere is saturated, small perturbations are unstable and, worse, perturbations of infinitesimally small wavelength have the maximum growth rate if the viscosity effect is ignored [e.g., Kuo (1961)l. This implies that if anything goes wrong in numerical calculations, the model soon produces irregular patterns. Consequently, it has been, and still is, an important

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105

210 315 420 525 HORIZONTAL DISTANCE (km)

630

735

10.8 14.41

w

I

3l 0

FIG.12. Temperature anomaly from the initial state in units of 0. I degrees Celsius (a) and the velocity field at 7.5 hr in units of meters per second (b), from a two-dimensional rainband model with the Arakawa-Schubert scheme. The model is slab symmetric and only the right half is shown.

part of research to sort out physically meaningful patterns from spurious ones. Related to this characteristic of moist convection, there is a problem that this chapter did not address, i.e.,the extreme sensitivity of the model result, in some problems, to environmental conditions or to the external

MODELING STUDIES OF CONVECTION

415

model parameters. For example, Chen (1980) made an extensive “sensitivity test” in his study of successive generation of new cells in the Raymer storms. He found that the model result is quite sensitive to slight changes in the moisture content and the vertical wind shear in the lower troposphere. In some cases no new cells formed at all, and in other cases five new cells formed in succession, as described in Section 5.4. We believe this sensitivity reflects reality: otherwise there would not be the wide variety of cloud echo patterns that one observes often within the detection range of a single radar. Ooyama (1982) noted that no deterministic prediction is possible for motions that have a horizontal scale comparable to the tropopause height on the basis of the three-dimensional energy cascade process to smaller scales. For practical purposes, cloud modelers will eventually have to deal with the probabilistic nature of storm formation in terms of the sensitivity of storms to, and the variability of, environmental conditions. Somehow cloud modelers have been able to keep spurious motion under control and have succeeded in simulating convective systems in a realistic manner, as described in this chapter. By virtue of the modeling study, model results provide complete, dynamically consistent data sets within the limitation of the specific model employed. These data sets have helped us enormously in interpreting observed data and in gaining insight into complex convective systems. Cloud modelers will continue to face great challenges. On top of the unsolved problems discussed in this chapter, there are phenomena such as downbursts, dry air entrainment from cloud tops, tornado formation, and clouds merging, just to name a few. We need a theoretical understanding of these phenomena as well as more observational documentation. To deal with such fine structures of storms, computers faster and larger than those currently available are required, since doubling the number of grid points in a three-dimensional domain implies a computer time 16 times longer. At the other side of the spatial spectrum of motions, the structure and evolution of storms are known to be modulated by the larger-scale environment. The requirement of taking a larger-model domain also demands greater computational power.

ACKNOWLEDGMENTS The author wishes to thank Su-Tzai Soong for his valuable comments and Robert Fovell for improving the manuscript. Karen Garrelts and Norene McGhiey typed the manuscript, and John Brother drafted the diagrams. The author is supported by the National Science Foundation under the grants ATM82-11786 and ATM82-10130. Funds for the former grant are jointly provided by the NSF and the National Oceanic and Atmospheric Administration.

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INDEX A Acoustic sounder, 360 Adiabatic effects in frontogenesis, 245-247 tropical numerical weather prediction, 29 1 Africa, weather forecasting, 43, 286, 288289, 291, 313-314 Air-mass thunderstorms, 397-399 Air Mass Transformation Experiment (AMTEX), 396 Albedo, solar radiation, 6 AMTEX, see Air Mass Transformation Experiment ANMRC data assimilation studies, 138, 142 Anomaly height maps, 66 Anticyclones, 6 “A” physics in weather forecasting model, 64-65, 67.15-76 Asia, weather forecasting, 27, 288-289, 3 14 Atlantic Ocean, weather forecasting, 43 Atmospheric flows characteristics of, 87 mesoscale, see Mesoscale atmospheric systems predictability, 87-122 balanced and unbalanced initial states, 106-107 boundary forcings, 110-1 15 classical studies, 89-109 different variables, 108-109 dynamical long-range forecasting, 1 15116 dynamical predictability, 1 1 1- 1 12 eddies, dissipation of, 90 error-doubling time, 90-95, 109 general circulation models, 93- 109 high- and low-resolution models, 105106 initial error, 108 internal dynamics, 110 lower bounds, 109 Northern and Southern Hemispheres, 103- 104

observational errors and model errors, 117-1 18 outstanding problems, 116-119 planetary and synoptic scale, 104-105 predictability of predictability, 118-1 19 simple models, 89-91 space-time averages, 90, 110-1 16 structure of large-scale flow, 107-108 systematic error, 116 transient predictability, 117 tropics and extratropics, 102-103 turbulence models, 90 upper limit, 87-89, 109 winter and summer seasons, 104 Atmospheric fronts, 223-252 ageostrophic effects, 228-232 baroclinic waves and fronts, 225-229 cold fronts, see Cold fronts collapse, 229-232 cross-stream circulation, 233, 235-236 dynamic balance, 232-233 frontogenetical terms, 244-250 mature front, 229-233 new mesoscale numerical simulation, 237-250 observed features and theory of, 233-236 quasi-geostrophic effects, 226-228, 233 semigeostrophic effects, 228, 230

B Balloon program, data assimilation, 126, 128, 131 Baroclinic effects, 224 and atmospheric fronts, 225-229 transient weather systems, 6 weather forecasting, 16, 43 seasonal forecasting, 80-82 Barotropic effects seasonal weather forecasting, 80-81 tropics, 284-288 BBnard-Rayleigh convection, 388-392, 395 Bering Sea, atmospheric convection over, 393

423

424

INDEX

Bjerknes, J., 223-224 Blocking phenomenon, in seasonal weather forecasting, 81-82 Blue Ridge Mountains, orographic mesoscale flow models, 214-215 Boundary forcing, mesoscale phenomena, predictability influenced by, 163 Boundary layer, see Planetary boundary layer C Canada, weather forecasting, 43 Canadian Weather Service, 138, 142 CCMOB, see Community Climate Model Central America, weather forecasting, 50 Circulation, 6; see also General Circulation Models Cloud feedback, tropical numerical weather prediction, 313 Clouds, 6 convection, modeling, studies of, 391394, 399,406-415 nonsquall clusters, 407 squall line, 403-405 medium-range weather forecasting, 13 microphysical processes, 391-392 Coherence, in surface layer, 376-377 Cold fronts, 233-236 structure of, 237-244 Colorado, orographiic mesoscale flow models, 214-215 Community Climate Model (CCMOB), 195 Convection, see also Moist convection modeling studies, 387-421 air-mass thunderstorm cells, 397-398 Btnard-Rayleigh convection, 388-391 closed cells, 395-396 complexity of convection, 391-392 cumulus clouds, feedback effect on larger-scale environments, 408-41 3 deep moist convection, 397-408 long-lived squall lines, 399-406 mesoscale systems, 166-195, 397-399, 406-408 open cells, 395-396 shallow moist convection, 392-396 planetary boundary layer, 360-361, 367, 371, 373 tropics, 284 Cooling, atmospheric, 6

weather forecasts, 42 tropical numerical prediction, 313 Cumulus clouds convection, in tropical numerical weather prediction, 306-312, 323 feedback effect on larger-scale environment, 408-413 Cross-stream frontal circulation, 233, 235236 Cyclones, see also Tropical cyclones surface fronts and, 224

D Data assimilation, in numerical weather prediction, 123-155 characteristics of current assimilation schemes, 138- 143 continuous assimilation, 138- 14I evolution of assimilation, 125-131 four-dimensional systems, 131-138 model initialization, 134-136 observational data base, 131-132 optimum interpolation, 132-134, 139, 141-142, 148 prediction models, 136-138 research and operations, role in, 143151

intermittent assimilation, 138-139, 141143 medium-range weather forecasting, 16-24 Data Systems Tests (DSTs), 129-130 Diabatic effects in frontogenesis, 245, 247-249 tropical numerical weather prediction, 291, 321,323 Diffusion, in frontogenesis, 245, 248 DLRF, see Dynamical long-range forecasting Drifting buoy program, data assimilation, 126, 128, 131, 149 Dry front, 245-247, 249 DSTs, see Data Systems Tests Dynamic initialization, tropical numerical weather prediction, 291 Dynamical long-range forecasting (DLRF), 115-1 16

E Eady wave, 225, 232 East Atlantic (EA) patterns, in monthly

INDEX

425

weather forecasting, 67 Extratropics internal dynamics of, 79 Easterlies tropical storm genesis from, 261, 263-266 predictability, 102-103 Eye, humcane, 269-270 weather forecasting, 43 tropical numerical prediction, 309 Eye wall, humcane, 269-270, 212 ECMWF, see European Centre for Medium Range Forecasts F Eddies FACE, see Florida Area Cumulus Experidissipation, 90 ment humcane, 270 First GARP Global Experiment (FGGE), 5, planetary boundary layer, 377-380 15-16, 19, 21, 35, 49-50 in sub-grid-scale turbulence modeling, atmospheric fronts, 238, 242 339-342 data assimilation, 125-131, 147, 149 transient, 81-82 monthly weather forecasts, 65 Eddy conductivity, 378 tropical numerical weather prediction, Eddy diffusivity, 378, 394-395 284, 293, 300, 302, 314, 316, 323, 330 Eddy-energy equation, 365-366 Florida Area Cumulus Experiment (FACE), Eddy viscosity, 341, 345, 355, 378, 395 207 Embayment, sea breeze influenced by, FM physics weather forecasting model, 66207 67, 15-76 Energy, Department of (U.S.), 211 F physics in weather forecasting model, 64Envelope orography, 43, 65-66, 148 67, 75-76 tropical numerical weather prediction, Free convection, 361, 371 318-3 19 Fronts, see Atmospheric fronts Enviromental Protection Agency (U . S . ) , Frozen-wave hypothesis, 372 21 1 E physics in weather forecasting model, 64G 65, 67, 75-16 Eurasian (EU) patterns, in monthly weather GARP, see Global Atmosphere Research forecasting, 67 Program Europe, weather forecasting, 28, 43 GARP Atlantic Tropical Experiment European Centre for Medium Range (GATE), 129 Weather Forecasts (ECMWF), 4-5, convection studies, 401-402, 406-407, 50-51 41 1 convection research, 14 tropical numerical weather prediction, data-assimilation system, 16-24, 127, 284, 286, 306-313, 316 138-139, 141-143, 146-148, 150 General Circulation Models (GCMs), 51, 82 error growth, 41, 43 dynamical long-range forecasting, 115mesoscale atmospheric motions, 196 116 models, 7, 11, 15, 24-26, 43-47 extended-range forecasting, 55-56, 59, 63 modified-Kuri model, 64 macroscale patterns, 66 monthly weather forecasts, 65 numerical experiments with, 111 Northern Hemisphere forecasts, 27-34, predictability studies, 93-109 36-40 Geophysical Fluid Dynamics Laboratory tropical numerical weather prediction, (GFDL) 304, 314, 316 atmospheric fronts, 238, 242 Extended-range weather forecasting, see data-assimilation studies, 125, 128, 130, Weather forecasting, extended-range 138-141, 143, 147-148, 150-151 External forcings, in seasonal weather forehurricane research, 256, 258-260, 278casting, 79, 82 219

426

INDEX

weather forecasting extended-range, 63-65 medium-range, 4, 51 tropical numerical prediction, 304 Geopotential height, 66-70, 74, 76, 78-80 predictability, 98, 105, 107-108 tropical numerical weather prediction, 29 1 Geostationary satellites, 129, 131 GFDL, see Geophysical Fluid Dynamics Laboratory GISS, see Goddard Institute for Space Science GLAS, see Goddard Laboratory for Atmospheric Science Global Atmosphere Research Program (CARP), 4 data-assimilation studies, 124, 130-131 Joint Organizing Committee (JOC), 124, 126-129, 149 two-week weather forecast, 59 Working Group on Numerical Experimentation (WGNE), 124 Global Weather Experiment (GWE), 16, 129 tropics, 285, 295 Goddard Institute for Space Science (GISS), 126 Goddard Laboratory for Atmospheric Science (GLAS), 130 climate model, 113 Gravity waves, 134-135 planetary boundary layer, 362, 366, 371, 377 squall lines, 405 weather forecasting, 17, 20 GWE, see Global Weather Experiment

H Hadley circulation data assimilation, 148 weather forecasting, 50 Hadley-Walker circulation, 313 Heating, atmospheric, 6 tropical numerical weather prediction, 313-314, 318, 321 Humidity analysis, in tropical numerical weather prediction, 283, 296, 300-306, 321 weather forecasting, 15-16

Humcanes, 255 decay of, 274-275 GFDL models, 258-260, 278-279 landfall, 273-276 numerical models, 256-260 structure, 267-270 I Identical twin experiments, 125 India medium-range monsoon predictions, 3 15, 319, 323 tropical numerical weather prediction, 286, 288, 293-302 Inertia-gravity oscillations, in model atmosphere, 130 INSAT satellite, 323 Instabilities, atmospheric predictability limited by, 160 Internal dynamics, in seasonal weather forecasting, 79-82 Intertropical convergence zone (ITCZ), 284 International Symposium on Four-Dimensional Data Assimilation, 124 Inversion layer, and shallow convection, 392-393 ITCZ, see Intertropical convergence zone

J Japan Meteorological Agency (JMA), 138, 142, 146 Japan Sea, atmospheric convection over, 392, 394 JMA, see Japan Meteorological Agency Joint Organizing Committee (JOC), see Global Atmosphere Research Program, Joint Organizing Committee

K Kolmogorov’s turbulence closure hypothesis, 350-351, 356 Kuri-grid weather forecasting model, 57, 59-60.63 L Land breezes, 205 Land-surface characteristics, weather forecasting, 15

427

INDEX Large-scale atmospheric flows mesoscale phenomena affected by, 163, 166, 169, 185, 199 observed variables, 124 predictability, 107-108, 160 tropics, 284-285 Latent heat release, and atmospheric heating, 6 Lateral boundary conditions (LBC), in mesoscale models 166, 169, 175-177, 185-189, 195, 199 Latitude-longitude grid weather forecasting model, 57, 60, 63 LBC, see Lateral boundary conditions Long-range weather forecasting, see Weather forecasting, extended-range forecasting M MCC, see Mesoscale cellular convection Medium-range weather forecasting, see Weather forecasting, medium-range Mesoscale atmospheric systems atmospheric fronts, 223-252 classification of mesoscale flows, 164 convention systems, 397-399,406-408 predictability of atmospheric motions, 159-202 atmospheric spectra, 161-162 classic predictability experiments, 164165, 198 global model, 195-199 optimistic points of view, 163-164 pessimistic outlook, 161- 162 preliminary study with model, 165- 195, 199 control simulations on large and small domains, 176-181 lateral boundary conditions, 166, 169, 175-177, 185-189, 195, 199 numerical simulations, results of, 175- 195 observed atmospheric evolution, 166- 167 perturbation in initial conditions, 172-173, 175, 177, 182-185, 195-196, 199 small-scale information, effect of removal of, 189-195 summary of model, 167-172

turbulence theory, 161-163, 198 thermal and orographic systems, 203-222 goals of mesoscale modeling, 218-219 research areas, 216-217 Mesoscale cellular convection (MCC), 393396 Mid-latitudes convection studies, 397, 406 energy transfer, 6 weather forecasting, 42 Mixed layer, planetary boundary layer, 361362, 379 Modified Kuri-grid weather forecasting model, 57, 60, 63-65 Moist convection deep moist convection, 397-408 air-mass thunderstorm cells, 397-398 long-lived squall lines, 399-406 mesoscale systems, 397-399, 406-408 shallow, 392-396 tropical cyclones, 258-259, 261, 263, 273, 277 weather forecasting, middle-range, 14 Moisture in frontogenesis, 245-247, 249 planetary boundary layer, 369-371, 376 tropical cyclones, 259, 261, 263, 266, 274 tropical numerical weather prediction, 306-309, 312, 318 Monsoon Experiment (MONEX), 284, 314, 316, 322-323, 330 Monsoons, 285,288-289,293-296, 313-314 medium-range prediction, 3 14-323 Mountains mesoscale orographic flows, 204, 212-219 tropical cyclones, effect on, 275-276 weather forecasting, 43 monthly, 65-66, 75 tropical numerical predictions, 288, 318-3 19 Mountain-valley atmospheric Rows, 205, 210 Multivariate statistical interpolation, 127, 139

N National Centre for Atmospheric Research (NCAR), 126 Community Climate Model, 195 convection studies, 388

428

INDEX

National Meteorological Center (NMC) data-assimilation studies, 129-130, 138, 142-143, 146-147, 150 mesoscale models, 21 1 monthly weather forecasts, 65 NCAR, see National Centre for Atmospheric Research Nimbus-6 satellite, 125, 129-130 NMC, see National Meteorological Center NMI, see Normal mode initialization NNMI, see Nonlinear normal mode initialization NOAA satellites, 125, 131 Nonidentical twin experiments, 126 Nonlinear normal mode initialization (NNMI), 130, 135-136, 139, 142, 148 Normal mode initialization (NMI), tropical numerical weather prediction, 293 North America mesoscale predictability, 196-197 weather forecasting, 27 Northern Hemisphere energy transfer, 6 mesoscale predictability, 196 predictability, 103-104 weather forecasting, 20, 26-40, 42, 50 extended-range forecasting, 74 medium-range prediction, 146 seasonal forecasting, 79 Numerical weather prediction (NWP) data assimilation, 123-155 extended-range forecasting, 55-85 improvement in, 3 low latitudes, 283-333 medium-range forecasting, 3-54 predictability, 87-122 Numerical weather prediction (NWP) models, 109, 119-120 systematic errors, 116 NWP, see Numerical weather prediction 0 Observing system experiments (OSEs), 131, 149-150 Observing system simulation experiments (OSSES), 125-128, 149, 151 Ocean-atmosphere system, predictability, 115 Oceans breezes, 205-209

mesoscale cellular convection over, 394 sea-level pressure mesoscale models, 176-179 predictability, 96, 103, 108-109 surface temperature, weather forecasting, 15, 21 Oklahoma, squall lines, 400-401 Orography, see Mountains OSEs, see Observing system experiments OSSEs, see Observing system simulation experiments

P PacificlNorth American (PNA) patterns, in weather forecasting, 67, 70, 81 Pacific Ocean boundary-forced predictability, 113 weather forecasting, 43 PBL, see Planetary boundary layer Phase delay, in surface layer, 376-377 Planetary boundary layer, 359-385 complex terrain, 363-361, 380-381 daytime, 360-361, 366, 369 definitions, 359 ensemble average turbulence closure, 345, 347-351, 354-355 equations, 364-367 equilibrium models, 381-382 first-order closure, 377-379 flows, 207 flux calculation of, 13 general characteristics, 359-364 high resolution model, 383 large-eddy exchange, 379-380 low-resolution model, 383 middle-range weather forecasting, 13-14 mixed layer, 361-362, 379 mixing depth, 362 models, 380-382 night, 361-364, 366, 369, 379 parameterization, 383 second-order closure, 371, 380, 383 in strong winds, 363 surface layer, 359, 361, 367-377 thickness, 359-360, 362-363, 370 time-dependent models, 382 tropical cyclones, 259, 261, 263, 266-267, 274 tropical numerical weather prediction, 306, 309

INDEX Planetary scale predictability, 104-105 PNA, see Pacific/North American patterns Polar-orbiting satellites, 129, 131, 149-150 tropical numerical weather prediction, 304 Polar regions cooling, 6 extended-range weather forecasting, 63 Pollution planetary boundary layer, 363-364 recirculation of pollutants, 21 1 Precipitation measurement of rate of, in low latitudes, 283 nonconvective, 14 weather forecasting, 15 Predictability of atmospheric flows, see Atmospheric flows, predictability Pressure, atmospheric, weather forecasting, 43 Prototype Office of Forecasting, 218

R Radiative parameterization, 13, 313-3 14 Rainband, evolution of, 406-407, 412-414 Rainfall predictability, 109 tropical numerical weather prediction, 293, 301-303, 306-309, 312, 321 Raymer storm, 405, 415 Relative humidity, hurricane, 269-270 Reynolds averaging, in sub-grid-scale turbulence modeling, 337-341 Reynolds fluxes (Reynolds stresses), 338, 349, 365 Rossby wave, 134-135 weather forecasting, 17 Rotta’s turbulence closure hypothesis, 350351, 354, 356

S Scandinavia, weather forecasting, 43 SCM, see Successive correction method Seas, see Oceans Seasonal weather forecasts, 79-82 Severe Environmental Storm and Mesoscale Experiment (SESAME), 166, 175, 235-236 atmospheric fronts, 237-238 Shallow-water system, 288

429

SI, see Statistical interpolation Siberia, weather forecasting, 43 Similarity theory, 13, 306 Snow, weather forecasting, 15, 21, 24 Snowmelt, weather forecasting, 15 Soil moisture, weather forecasting, 15, 21, 24 Solar radiation, 6 South China Sea, numerical weather prediction, 288 Southern Hemisphere predictability, 103-104 weather forecasting, 20, 26, 30, 35,42, 50 medium-rdnge prediction. 146 Southern Oscillation, I15 Space-based weather observing systems, 123-126, 129-131, 149-150 atmospheric fronts, 240-241 convection, 392 tropical numerical weather prediction, 304, 316, 323 Space-time averages, predictability, 110116 Space truncation error, in weather forecasting models, 64 Squall lines, long-lived, 399-406 SST anomalies, 113-115 Statistical interpolation (SI), 126-127 STORM Central program, 218 Stratosphere, weather forecasting, 29, 4243 Sub-grid-scale physics, in weather forecasting models, 64-65 Successive correction method (SCM), 126, 143 Summer predictability, 96, 98, 100, 104 weather forecasting, 29 Surface fronts, and cyclones, 224 Surface inhomogeneities, mesoscale phenomena, influence on, 163 Surface layer, planetary boundary layer, 359, 361, 367-377 complex terrain, 380-381 homogeneous terrain, 367-370 spectra and cospectra, 372-377 variances, 370-371 Synoptic-scale atmospheric motions, 159160, 175, 198 predictability, 104-105

430

INDEX

T Teleconnection patterns, in monthly weather forecasting, 66-67, 70-72, 80, 82 Temperature, see also Thermal atmospheric flows data-assimilation research, 125-126, 129130 planetary boundary layer, 369-371, 376 tropical cyclones, 259,261,263,269,272274 weather forecasting, 42-44 Thermal atmospheric flows, mesoscale systems, 204-211, 216-219 Thunderstorms, 397-400 Tip, typhoon, 256 Tiros-N satellite, 125, 131 tropical numerical weather prediction, 304-305 Tiros-N operational vertical soui.der (TOVS), 304 TOGA research program, 146 Tropical cyclones development of, 255 incidence, 255 numerical modeling of, 255-281 GFDL humcane models, 258-260, 278-279 hurricanes, 256-260 improvement of models, 276 intensification of tropical storms, 266267 landfall of hurricanes, 273-276 medium-range monsoon prediction, 3 14-323 prediction of cyclones, 277-278 spiral bands and comma vortices, 271273 structure of hurricanes, 267-270 tropical storm genesis, 261-266 Tropical storms, 255 genesis, 261-266, 276-277 intensification, 266-267 landfall, 273 Tropics boundary-forced predictability, 113-1 15 convection studies, 406 data-assimilation studies, 129 extended-range weather forecasting, 8283

heating, 6 mesoscale convection systems, 397, 399 mesoscale predictability, 165 numerical weather prediction in, 283-333 air-sea interaction, 306 cumulus convection, 306-312 dynamic initialization, 291 future research, 330-331 humidity analysis, 302-306 initialization, 291-306 large-scale condensation, 306 medium-range monsoon prediction, 314-323 normal mode initialization (NMI), 293 Parameterization of physical processes, 306-314 physical initialization, 293-302 planetary boundary layer, 306 quasi-stationary component, 323-330 radiative parameterization, 3 13-314 simple models based on conservation laws, 283-291 predictability, 102- 103, 109 systematic errors in heat sources, 117 squall lines, 402 weather forecasting, 16,20-21,26,30,35, 43, 49-50 Tropopause, weather forecasting, 43 Troposphere convection studies, 392, 407-408, 415 weather forecasting, 28-30, 42 Turbulence, 6 complex terrain, 380-381 convective, 360-361, 367, 371, 373 daytime, 360-361 hurricane models, 259 mechanical, 360-361, 363, 366-367, 373 mesoscale predictability and, 161-163, I98 night, 361-363 planetary boundary layer, 359 spectra, 372 sub-grid-scale modeling, 337-343 eddy coefficient, 340-341 effect of grid-volume Reynolds averaging, 338-340 future outlook, 341-342 need for grid-scale Reynolds averaging, 337-338 recent developments, 341

INDEX

vertical velocity, 361 Turbulence closure, ensemble average, 345358 averaging distance for measurements, 354-355 horizontal diffusion, 355-357 modeling applications, 355-357 turbulence macroscale, 347-354 Typhoons, 255

U United Kingdom Meteorological Office (UKMO), 126-128, 138, 141, 143, 146-147, 150 United States mesoscale models, national, 21 1 weather forecasting, 50 Univariate statistical interpolation, 139, 141 Upper-air waves, and cyclones, 224 V Vertical temperature profile radiometer (VTPR) measurements, 125 Vorticity, tropical cyclones, 255, 263-264, 267-268, 270-273, 215, 277 VTPR measurements, see Vertical temperature profile radiometer measurements

W WA patterns, see West Atlantic patterns Wave phenomena orographic meoscale flow models, 212213 spectrum component interactions, atmospheric predictability limited by, 160- 162 WCRP, see World Climate Research Programme Weather forecasting, see also Numerical weather prediction analog method, 91-93 data assimilation, 16-17, 123-155 extended-range, 55-85 dynamical long-range forecasting, 1 15117 monthly forecast, 65-79, 110 seasonal forecast, 79-82, 110 medium-range, 3-54 basic equations, 7-10 clouds, 13

43 1

data-assimilation systems, 16- 17 error patterns, 20, 23, 41-43 four-dimensional data assimilation, 16 moist convection, 14 monsoons, 314-323 nonconvective parameterization, 14 Northern Hemisphere, 27-40 numerical formulation, 10-12 numerical prediction models, 136, 146 observations, use and importance of, 15-24 operational applications and results, 24-35 physical and mathematical basis, 5-7 planetary boundary layer, 13-14 problems and prospects, 35, 40-50 radiation, 13 Southern Hemisphere, 30, 35 surface values, 15 ten-day forecast, 56-65, 82 tropics, 30, 35 predictability, 87-122 short-range forecasting, 3 West Atlantic (WA) patterns, in monthly weather forecasting, 67 Westerlies, weather forecasting, 42 West Pacific (WP) patterns, in monthly weather forecasting, 67 WGNE, see Global Atmosphere Research Program, Working Group on Numerical Experimentation Winds horizontal spectra, 161 long-lived squall lines, 402 planetary boundary layer, 360-363, 382 surface layer, 361-377 predictability, 100, 109 tropical cyclones, 255, 263-270, 273-275 tropical numerical weather prediction, 284-285,288-289,291,293,296-301, 313, 316, 319-323, 330 WINDSAT system, 150 Wind shear, 362-363, 377 convection studies, 394, 402, 405, 415 thunderstorm evolution, 399 Winter predictability, 96, 98, 100, 104 weather forecasting, 29, 43, 47, 50 WMO, see World Meteorologic Organization

432

INDEX

Working Group on Numerical Experimentation (WGNE), see Global Atmosphere Research Program, Working Group on Numerical Experimentation World Climate Research Programme (WCRP), 5 World Meteorologic Organization (WMO), 126-129, 148-149

World Weather Watch (WWW) in low latitudes, 149, 283, 302, 304 WP patterns, see West Pacific patterns WWW, see World Weather Watch Z

Zonal flow, 81 Zonal index, 80