Atmospheric Modeling, Data Assimilation and Predictability Eugenia Kalnay Frontmatter More Information

Cambridge University Press 978-0-521-79629-3 - Atmospheric Modeling, Data Assimilation and Predictability Eugenia Kalnay Frontmatter More information

Atmospheric modeling, data assimilation and predictability

This comprehensive text and reference work on numerical weather prediction covers for the first time, not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a modern and clear description of numerical methods, and the determination of initial conditions using weather observations (an important new science known as data assimilation). Finally, this book provides a clear discussion of the problems of predictability and chaos in dynamical systems and how they can be applied to atmospheric and oceanic systems. This includes discussions of ensemble forecasting, El Niño events, and how various methods contribute to improved weather and climate prediction. In each of these areas the emphasis is on clear and intuitive explanations of all the fundamental concepts, followed by a complete and sound development of the theory and applications. Professors and students in meteorology, atmospheric science, oceanography, hydrology and environmental science will find much to interest them in this book which can also form the basis of one or more graduate-level courses. It will appeal to professionals modeling the atmosphere, weather and climate, and to researchers working on chaos, dynamical systems, ensemble forecasting and problems of predictability.

Eugenia Kalnay was awarded a PhD in Meteorology from the Massachusetts Institute of Technology in 1971 (Jule Charney, advisor). Following a position as Associate Professor in the same department, she became Chief of the Global Modeling and Simulation Branch at the NASA Goddard Space Flight Center (1983Ð7). From 1987 to 1997 she was Director of the Environmental Modeling Center (US National Weather Service) and in 1998 was awarded the Robert E. Lowry endowed chair at the University of Oklahoma. In 1999 she became the Chair of the Department of Meteorology at the University of Maryland. Professor Kalnay is a member of the US National Academy of Engineering and of the Academia Europaea, is the recipient of two gold medals from the US Department of Commerce and the NASA Medal for Exceptional Scientific Achievement, and has received the Jule Charney Award from the American Meteorological Society. The author of more than 100 peer reviewed papers on numerical weather prediction, data assimilation and predictability, Professor Kalnay is a key figure in this field and has pioneered many of the essential techniques.

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-79629-3 - Atmospheric Modeling, Data Assimilation and Predictability Eugenia Kalnay Frontmatter More information

Atmospheric modeling, data assimilation and predictability

Eugenia Kalnay University of Maryland

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Library of Congress Cataloguing in Publication Data Kalnay, Eugenia, 1942– Atmospheric modeling, data assimilation and predictability / Eugenia Kalnay. p. cm. Includes bibliographical references and index. isbn 0-521-79179-0 – isbn 0-521-79629-6 (pbk.) 1. Numerical weather forecasting. I. Title. qc996 .k35 2002 551.63´4 – dc21 2001052687

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I dedicate this book to the Grandmothers of Plaza de Mayo for their tremendous courage and leadership in defense of human rights and democracy

Contents

Foreword xi

Acknowledgements xv

List of abbreviations xvii

List of variables xxi

1 Historical overview of numerical weather prediction 1

1.1 Introduction 1 1.2 Early developments 4 1.3 Primitive equations, global and regional models, and nonhydrostatic models 10 1.4 Data assimilation: determination of the initial conditions for the computer forecasts 12 1.5 Operational NWP and the evolution of forecast skill 17 1.6 Nonhydrostatic mesoscale models 24 1.7 Weather predictability, ensemble forecasting, and seasonal to interannual prediction 25 1.8 The future 30

2 The continuous equations 32

2.1 Governing equations 32 2.2 Atmospheric equations of motion on spherical coordinates 36 2.3 Basic wave oscillations in the atmosphere 37 2.4 Filtering approximations 47 2.5 Shallow water equations, quasi-geostrophic ﬁltering, and ﬁltering of inertia-gravity waves 53 2.6 Primitive equations and vertical coordinates 60

vii

viii Contents

3 Numerical discretization of the equations of motion 68

3.1 Classiﬁcation of partial differential equations (PDEs) 68 3.2 Initial value problems: numerical solution 72 3.3 Space discretization methods 91 3.4 Boundary value problems 114 3.5 Lateral boundary conditions for regional models 120

4 Introduction to the parameterization of subgrid-scale physical processes 127

4.1 Introduction 127 4.2 Subgrid-scale processes and Reynolds averaging 129 4.3 Overview of model parameterizations 132

5 Data assimilation 136

5.1 Introduction 136 5.2 Empirical analysis schemes 140 5.3 Introduction to least squares methods 142 5.4 Multivariate statistical data assimilation methods 149 5.5 3D-Var, the physical space analysis scheme (PSAS), and their relation to OI 168 5.6 Advanced data assimilation methods with evolving forecast error covariance 175 5.7 Dynamical and physical balance in the initial conditions 185 5.8 Quality control of observations 198

6 Atmospheric predictability and ensemble forecasting 205

6.1 Introduction to atmospheric predictability 205 6.2 Brief review of fundamental concepts about chaotic systems 208 6.3 Tangent linear model, adjoint model, singular vectors, and Lyapunov vectors 212 6.4 Ensemble forecasting: early studies 227 6.5 Operational ensemble forecasting methods 234 6.6 Growth rate errors and the limit of predictability in mid-latitudes and in the tropics 249 6.7 The role of the oceans and land in monthly, seasonal, and interannual predictability 254 6.8 Decadal variability and climate change 258

Contents ix

Appendix A The early history of NWP 261

Appendix B Coding and checking the tangent linear and the adjoint models 264

Appendix C Post-processing of numerical model output to obtain station weather forecasts 276

References 283

Index 328

Foreword

During the 50 years of numerical weather prediction the number of textbooks dealing with the subject has been very small, the latest being the 1980 book by Haltiner and Williams. As you will soon realize, the intervening years have seen impressive development and success. Eugenia Kalnay has contributed significantly to this expansion, and the meteorological community is fortunate that she has applied her knowledge and insight to writing this book. Eugenia was born in Argentina, where she had exceptionally good teachers. She had planned to study physics, but was introduced to meteorology by a stroke of fate; her mother simply entered her in a competition for a scholarship from the Argentine National Weather Service! But a military coup took place in Argentina in 1966 when Eugenia was a student, and the College of Sciences was invaded by military forces. Rolando Garcia, then Dean of the College of Sciences, was able to obtain for her an assistantship with Jule Charney at the Massachusetts Institute of Technology. She was the first female doctoral candidate in the Department and an outstanding student. In 1971, under Charney’s supervision, she finished an excellent thesis on the circulation of Venus. She recalls that an important lesson she learned from Charney at that time was that if her numerical results did not agree with accepted theory it might be because the theory was wrong. What has she written in this book? She covers many aspects of numerical weather prediction and related areas in considerable detail, on which her own experience enables her to write with relish and authority. The first chapter is an overview that introduces all the major concepts discussed later in the book. Chapter 2 is a pre- sentation of the standard equations used in atmospheric modeling, with a concise

xii Foreword

but complete discussion of filtering approximations. Chapter 3 is a roadmap to numerical methods providing a student without background in the subject with all the tools needed to develop a new model. Chapter 4 is an introduction to the parameterization of subgrid-scale physical processes, with references to specialized textbooks and papers. I found her explanations in Chapter 5 of data assimilation methods and in Chapter 6 on predictability and ensemble forecasting to be not only inclusive but thorough and well presented, with good attention to historical developments. These chapters, however, contain many definitions and equations. (I take this wealth as a healthy sign of the technical maturity of the subject.) This complex- ity may be daunting for many readers, but this has obviously been recognized by Eugenia. In response she has devised many simple graphical sketches that illustrate the important relations and definitions. An added bonus is the description in an appendix of the use of Model Output Statistics by the National Weather Service, its successes, and the rigid constraints that it imposes on the forecast model. She also includes in the appendices a simple adaptive regression scheme based on Kalman filtering and an introduction to the generation of linear tangent and adjoint model codes. Before leaving the National Centers for Environmental Prediction in 1998 as Director of the Environmental Modeling Center, Eugenia directed the Reanalysis Project, with Robert Kistler as Technical Manager. This work used a 1995 state-of- the-art analysis and forecast system to reanalyze and reforecast meteorological events from past years. The results for November 1950 were astonishing. On November 24 of that year an intense snowstorm developed over the Appalachians that had not been operationally predicted even 24 hours in advance. This striking event formed a test situation for the emerging art of numerical weather prediction in the years immediately following the first computations in 1950 on the ENIAC computer discussed in Chapter 1. In 1953, employing his baroclinic model, and with considerable “tuning” Jule Charney finally succeeded in making a 24-hour forecast starting on November 23 1950 of a cyclonic development, which, however, was still located some 400 kilometers northeast of the actual location of the storm. This “prediction” played a major role in justifying the creation of the Joint Numerical Weather Pre- diction Unit in 1955 (Chapter 1). By contrast, in the Reanalysis Project, this event was forecast extremely well, both in intensity and location Ð as much as three days in advance. (Earlier than this the associated vorticity center at 500 mbs had been located over the Pacific Ocean, even though at that time there was no satellite data!) This is a remarkable demonstration of the achievements of the numerical weather prediction community in the past decades, achievements that include many by our author. After leaving NCEP in 1998, Eugenia was appointed Lowry Chair in the School of Meteorology at the University of Oklahoma, where she started writing her book. She returned to Maryland in 1999 to chair the Department of Meteorology, where

Foreword xiii

she continues to do research on a range of topics, including applications of chaos to ensemble forecasting and data assimilation. We look forward to future contributions by Professor Kalnay.

Norman Phillips

Acknowledgements

I drafted about two thirds of this book while teaching the subject for the first time at Oklahoma University, during the fall of 1998. OU provided me with a supportive environment that made it possible to write the first draft. I made major revisions and finished the book while teaching the course again in the fall in 1999 through 2001 at the University of Maryland. The students that took the course at UM and OU gave me essential feedback, and helped me find many (hopefully most) of the errors in the drafts. In addition, several people helped to substantially revise one or more of the manuscript chapters, and their suggestions and corrections have been invaluable. Norm Phillips read an early draft of Chapter 1 and made important historical com- ments. Anders Persson wrote the notes on the early history of numerical weather prediction, especially in Europe, reproduced in an appendix. Alfredo Ruiz Barradas reviewed Chapter 2. Will Sawyer reviewed and made major suggestions for improvements for Chapter 3. Hua-lu Pan influenced Chapter 4. Jim Geiger reviewed Chapter 5 and pointed out sections that were obscure. Jim Purser also reviewed this chapter and not only made very helpful suggestions but also provided an elegant demonstration of the equivalence of the 3D-Var and OI formulations. Discussions with Peter Lyster on this chapter were also very helpful. D. J. Patil suggested many improvements to Chap- ter 6, and Bill Martin pointed out the story by Ray Bradbury concerning the “butter- fly effect”. Joaquim Ballabrera substantially improved the appendix on model output post-processing. Shu-Chih Yangand Matteo Corazza carefully reviewed the complete book, including the appendices, and suggested many clarifications and corrections. Iamgrateful to Malaquias Peña, who wrote the abbreviation list and helped with many figures and corrected references. Dick Wobus created the beautiful 6-day

xvi Acknowledgements

ensemble forecast ﬁgure shown on the cover. Seon Ki Park provided the linear tangent and adjoint code in Appendix B. The help and guidance of Matt Lloyd and Susan Francis of Cambridge University Press, the editing of the text by Maureen Storey, and the kind foreword by Norm Phillips are also very gratefully acknowledged. Ibegan to learn numerical weather prediction (NWP) in the late 1960s from professors at the University of Buenos Aires, especially Rolando Garcia and Ruben Norscini, and from the inspiring book of P. D. Thompson. At MIT, my thesis advisor, Jule Charney, and the lectures of Norm Phillips and Ed Lorenz, inﬂuenced me more than I can describe. The NWP class notes of Akio Arakawa at UCLA and the NCAR text on numerical methods by John Gary helped me teach the subject at MIT. Over the last 30 years I have continued learning from numerous colleagues at other institutions where I had the privilege of working. They include the Univer- sity of Montevideo, MIT, NASA/GSFC, OU, and UM. However, my most important experience came from a decade I spent as Director of the Environmental Modeling Center at the National Centers for Environmental Prediction, where my extremely dedicated colleagues and I learned together how to best transition from research ideas to operational improvements. Finally, I would like to express my gratitude for the tremendous support, patience and encouragement that my husband, Malise Dick, my son, Jorge Rivas, and my sisters Patricia and Susana Kalnay have given me, and for the love for education that my parents instilled in me.

Abbreviations

3D-Var Three-dimensional variational analysis 4DDA Four-dimensional data assimilation 4D-Var Four-dimensional variational analysis AC Anomaly correlation ADI Alternating direction implicit AMIP Atmospheric Models Intercomparison Project (frequently refers to long model runs in which the observed SST is used instead of climatology) AO Arctic oscillation AP Arrival point in semi-Lagrangian schemes ARPS Advanced Regional Prediction System AV N NCEP’s aviation (global) spectral model CAPS Center for Analysis and Prediction of Storms CFL CourantÐFriedrichsÐLewy COAMPS US Navy’s coupled ocean/atmosphere mesoscale prediction system CONUS Continental USA CPC Climate Prediction Center (NCEP) CSI Critical success index (same as threat score) DP Departure point in semi-Lagrangian schemes DWD German Weather Service ECMWF European Centre for Medium-Range Weather Forecasts EDAS Eta data assimilation system ENIAC Electronic numerical integrator and computer

xvii

xviii Abbreviations

ENSO El NiñoÐSouthern Oscillation FASTEX Fronts and Storm Track Experiment FDE Finite difference equation FDR Frequency dispersion relationship FFSL Flux-form-semi-Lagrangian scheme GLE Global Lyapunov exponents GPS Global positioning system hPa hecto Pascals (also known as millibars) HPC Hydrometeorological Prediction Center (NCEP) JMA Japan Meteorological Agency JNWPU Joint Numerical Weather Prediction Unit LFM Limited fine mesh LLV Local Lyapunov vectors MCC Mesoscale compressible community (model) MeteoFrance National Meteorological Service for France MJO Madden and Julian oscillation MM5 Penn State/NCAR mesoscale model, version 5 MOS Model output statistics NAO North Atlantic oscillation NASA National Aeronautics and Space Administration NCAR National Center for Atmospheric Research NCEP National Centers for Environmental Prediction (US National Weather Service) NCI Nonlinear computational instability NGM Nested grid model NLNMI Nonlinear normal mode initialization NMC National Meteorological Center NOAA National Oceanic and Atmospheric Administration NORPEX North Pacific Experiment NWP Numerical weather prediction NWS National Weather Service OI Optimal interpolation PDE Partial differential equation PDO Pacific decadal oscillation PIRCS Project to Intercompare Regional Climate Systems PQPF Probabilistic quantitative precipitation forecast PSAS Physical space analysis scheme PVE Potential vorticity equation RAFS Regional analysis and forecasting system RAOB Rawinsonde observation RDAS Regional data assimilation system RSM NCEP’s Regional Spectral Model

Abbreviations xix

RUC NCEP’s rapid update cycle SAC Standardized anomaly correction SCM Successive correction method SOR Successive overrelaxation SST Sea surface temperature SWE Shallow water equations TOGA Tropical ocean, global atmosphere TOVS TIROS-N operational vertical sounder TS Threat score UKMO United Kingdom Meteorological Ofﬁce UTC Universal time or Greenwich time, e.g. 1200 UTC. Frequently abbreviated as 1200Z WMO World Meteorological Organization

Variables

a radius of the Earth A analysis error covariance matrix B background error covariance matrix C covariance matrix

Cp,Cv speciﬁc heat at constant pressure, constant volume d innovation or observational increments vector D ﬂuid depth E() expected value f Coriolis parameter g gravitational constant H linear observation operator matrix H observational operator, scale height of the atmosphere I identity matrix J cost function JM maximum number of grid points j K Kalman gain matrix

L(t0, t) resolvent or propagator of TLM M TLM matrix N BruntÐVa¬õs¬al¬a frequency P projection matrix p pressure, probability, distribution function q mixing ratio of water vapor and dry air mass Q forecast model error covariance r position vector

xxi

xxii Variables

R observations error covariance matrix R root mean square error, gas constant

Rd Rossby radius of deformation R0 Rossby number RE relative error T temperature TS threat score u,v eastward and northward wind components W weight matrix W vertical wind component, optimal weight x,y horizontal coordinates

δij Kronecker delta εa analysis error εb background error η absolute vorticity geopotential height ϕ geopotential, latitude λ longitude

λi global Lyapunov exponent ρij element i,j of the correlation matrix C σ standard deviation σ 2 variance ψ streamfunction ω vertical velocity in pressure coordinates, spectral frequency ζ relative vorticity