atmospheric modeling, data assimilation and...

Atmospheric modeling, data assimilationand predictability

This comprehensive text and reference work on numerical weather predictioncovers for the first time, not only methods for numerical modeling, but also theimportant related areas of data assimilation and predictability.

It incorporates all aspects of environmental computer modeling including anhistorical overview of the subject, equations of motion and their approximations,a modern and clear description of numerical methods, and the determination ofinitial conditions using weather observations (an important new science knownas data assimilation). Finally, this book provides a clear discussion of theproblems of predictability and chaos in dynamical systems and how they can beapplied to atmospheric and oceanic systems. This includes discussions ofensemble forecasting, El Nino events, and how various methods contribute toimproved weather and climate prediction. In each of these areas the emphasis ison clear and intuitive explanations of all the fundamental concepts, followed bya 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 thisbook which can also form the basis of one or more graduate-level courses. It willappeal to professionals modeling the atmosphere, weather and climate, and toresearchers working on chaos, dynamical systems, ensemble forecasting andproblems of predictability.

Eugenia Kalnay was awarded a PhD in Meteorology from the MassachusettsInstitute of Technology in 1971 (Jule Charney, advisor). Following a position asAssociate Professor in the same department, she became Chief of the GlobalModeling and Simulation Branch at the NASA Goddard Space Flight Center(1983–7). From 1987 to 1997 she was Director of the Environmental ModelingCenter (US National Weather Service) and in 1998 was awarded the Robert E.Lowry endowed chair at the University of Oklahoma. In 1999 she became theChair of the Department of Meteorology at the University of Maryland.Professor Kalnay is a member of the US National Academy of Engineering andof the Academia Europaea, is the recipient of two gold medals from the USDepartment of Commerce and the NASA Medal for Exceptional ScientificAchievement, and has received the Jule Charney Award from the AmericanMeteorological Society. The author of more than 100 peer reviewed papers onnumerical weather prediction, data assimilation and predictability, ProfessorKalnay is a key figure in this field and has pioneered many of the essentialtechniques.

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Cambridge University Press978-0-521-79629-3 - Atmospheric Modeling, Data Assimilation and PredictabilityEugenia KalnayFrontmatterMore information

http://www.cambridge.org/9780521796293

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Atmosphericmodeling, dataassimilation andpredictability

Eugenia KalnayUniversity of Maryland






cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town,Singapore, São Paulo, Delhi, Mexico City

Cambridge University PressTh e Edinburgh Building, Cambridge cb2 8ru, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521796293

© Eugenia Kalnay 2003

Th is publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2003Th ird printing with corrections 20067th printing 2012

A catalogue record for this publication is available from the British Library

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

isbn 978-0-521-79179-3 Hardbackisbn 978-0-521-79629-3 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of fi rst printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.






I dedicate this book to the Grandmothers of Plaza de Mayo for their tremendous courage andleadership 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 nonhydrostaticmodels 10

1.4 Data assimilation: determination of the initial conditions for thecomputer 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 tointerannual 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 ofinertia-gravity waves 53

2.6 Primitive equations and vertical coordinates 60

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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 physicalprocesses 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 theirrelation to OI 168

5.6 Advanced data assimilation methods with evolving forecast errorcovariance 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, andLyapunov 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-latitudesand in the tropics 249

6.7 The role of the oceans and land in monthly, seasonal, andinterannual 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 stationweather forecasts 276

References 283

Index 328






Foreword

During the 50 years of numerical weather prediction the number of textbooks dealingwith the subject has been very small, the latest being the 1980 book by Haltiner andWilliams. As you will soon realize, the intervening years have seen impressive devel-opment and success. Eugenia Kalnay has contributed significantly to this expansion,and the meteorological community is fortunate that she has applied her knowledgeand insight to writing this book.

Eugenia was born in Argentina, where she had exceptionally good teachers. Shehad 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 ArgentineNational Weather Service! But a military coup took place in Argentina in 1966 whenEugenia 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 heran assistantship with Jule Charney at the Massachusetts Institute of Technology.She was the first female doctoral candidate in the Department and an outstandingstudent. In 1971, under Charney’s supervision, she finished an excellent thesis on thecirculation of Venus. She recalls that an important lesson she learned from Charneyat that time was that if her numerical results did not agree with accepted theory itmight be because the theory was wrong.

What has she written in this book? She covers many aspects of numerical weatherprediction and related areas in considerable detail, on which her own experienceenables her to write with relish and authority. The first chapter is an overview thatintroduces 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

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xii Foreword

but complete discussion of filtering approximations. Chapter 3 is a roadmap to nu-merical methods providing a student without background in the subject with allthe tools needed to develop a new model. Chapter 4 is an introduction to the pa-rameterization of subgrid-scale physical processes, with references to specializedtextbooks and papers. I found her explanations in Chapter 5 of data assimilationmethods and in Chapter 6 on predictability and ensemble forecasting to be not onlyinclusive but thorough and well presented, with good attention to historical devel-opments. These chapters, however, contain many definitions and equations. (I takethis 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 byEugenia. In response she has devised many simple graphical sketches that illustratethe important relations and definitions. An added bonus is the description in an ap-pendix of the use of Model Output Statistics by the National Weather Service, itssuccesses, and the rigid constraints that it imposes on the forecast model. She alsoincludes in the appendices a simple adaptive regression scheme based on Kalmanfiltering and an introduction to the generation of linear tangent and adjoint modelcodes.

Before leaving the National Centers for Environmental Prediction in 1998 asDirector of the Environmental Modeling Center, Eugenia directed the ReanalysisProject, with Robert Kistler as Technical Manager. This work used a 1995 state-of-the-art analysis and forecast system to reanalyze and reforecast meteorological eventsfrom past years. The results for November 1950 were astonishing. On November 24of that year an intense snowstorm developed over the Appalachians that had notbeen operationally predicted even 24 hours in advance. This striking event formeda test situation for the emerging art of numerical weather prediction in the yearsimmediately following the first computations in 1950 on the ENIAC computer dis-cussed in Chapter 1. In 1953, employing his baroclinic model, and with considerable“tuning” Jule Charney finally succeeded in making a 24-hour forecast starting onNovember 23 1950 of a cyclonic development, which, however, was still locatedsome 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 eventwas forecast extremely well, both in intensity and location – as much as three daysin advance. (Earlier than this the associated vorticity center at 500 mbs had beenlocated 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 weatherprediction community in the past decades, achievements that include many by ourauthor.

After leaving NCEP in 1998, Eugenia was appointed Lowry Chair in the Schoolof 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 toensemble forecasting and data assimilation. We look forward to future contributionsby Professor Kalnay.

Norman Phillips






Acknowledgements

I drafted about two thirds of this book while teaching the subject for the first timeat Oklahoma University, during the fall of 1998. OU provided me with a supportiveenvironment that made it possible to write the first draft. I made major revisions andfinished the book while teaching the course again in the fall in 1999 through 2001 atthe University of Maryland. The students that took the course at UM and OU gaveme essential feedback, and helped me find many (hopefully most) of the errors in thedrafts.

In addition, several people helped to substantially revise one or more of themanuscript 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 weatherprediction, especially in Europe, reproduced in an appendix. Alfredo Ruiz Barradasreviewed Chapter 2. Will Sawyer reviewed and made major suggestions for improve-ments for Chapter 3. Hua-lu Pan influenced Chapter 4. Jim Geiger reviewed Chapter 5and pointed out sections that were obscure. Jim Purser also reviewed this chapter andnot only made very helpful suggestions but also provided an elegant demonstration ofthe equivalence of the 3D-Var and OI formulations. Discussions with Peter Lyster onthis 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 outputpost-processing. Shu-Chih Yang and Matteo Corazza carefully reviewed the completebook, including the appendices, and suggested many clarifications and corrections.

I am grateful to Malaquias Pena, who wrote the abbreviation list and helpedwith many figures and corrected references. Dick Wobus created the beautiful 6-day

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xvi Acknowledgements

ensemble forecast figure shown on the cover. Seon Ki Park provided the linear tangentand adjoint code in Appendix B. The help and guidance of Matt Lloyd and SusanFrancis of Cambridge University Press, the editing of the text by Maureen Storey,and the kind foreword by Norm Phillips are also very gratefully acknowledged.

I began to learn numerical weather prediction (NWP) in the late 1960s from pro-fessors at the University of Buenos Aires, especially Rolando Garcia and RubenNorscini, and from the inspiring book of P. D. Thompson. At MIT, my thesis ad-visor, Jule Charney, and the lectures of Norm Phillips and Ed Lorenz, influencedme more than I can describe. The NWP class notes of Akio Arakawa at UCLA andthe NCAR text on numerical methods by John Gary helped me teach the subject atMIT. Over the last 30 years I have continued learning from numerous colleaguesat 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 importantexperience came from a decade I spent as Director of the Environmental ModelingCenter at the National Centers for Environmental Prediction, where my extremelydedicated colleagues and I learned together how to best transition from research ideasto operational improvements.

Finally, I would like to express my gratitude for the tremendous support, patienceand encouragement that my husband, Malise Dick, my son, Jorge Rivas, and mysisters Patricia and Susana Kalnay have given me, and for the love for education thatmy parents instilled in me.






Abbreviations

3D-Var Three-dimensional variational analysis4DDA Four-dimensional data assimilation4D-Var Four-dimensional variational analysisAC Anomaly correlationADI Alternating direction implicitAMIP Atmospheric Models Intercomparison Project

(frequently refers to long model runs in which theobserved SST is used instead of climatology)

AO Arctic oscillationAP Arrival point in semi-Lagrangian schemesARPS Advanced Regional Prediction SystemAVN NCEP’s aviation (global) spectral modelCAPS Center for Analysis and Prediction of StormsCFL Courant–Friedrichs–LewyCOAMPS US Navy’s coupled ocean/atmosphere mesoscale

prediction systemCONUS Continental USACPC Climate Prediction Center (NCEP)CSI Critical success index (same as threat score)DP Departure point in semi-Lagrangian schemesDWD German Weather ServiceECMWF European Centre for Medium-Range Weather ForecastsEDAS Eta data assimilation systemENIAC Electronic numerical integrator and computer

xvii






xviii Abbreviations

ENSO El Nino–Southern OscillationFASTEX Fronts and Storm Track ExperimentFDE Finite difference equationFDR Frequency dispersion relationshipFFSL Flux-form-semi-Lagrangian schemeGLE Global Lyapunov exponentsGPS Global positioning systemhPa hecto Pascals (also known as millibars)HPC Hydrometeorological Prediction Center (NCEP)JMA Japan Meteorological AgencyJNWPU Joint Numerical Weather Prediction UnitLFM Limited fine meshLLV Local Lyapunov vectorsMCC Mesoscale compressible community (model)MeteoFrance National Meteorological Service for FranceMJO Madden and Julian oscillationMM5 Penn State/NCAR mesoscale model, version 5MOS Model output statisticsNAO North Atlantic oscillationNASA National Aeronautics and Space AdministrationNCAR National Center for Atmospheric ResearchNCEP National Centers for Environmental Prediction

(US National Weather Service)NCI Nonlinear computational instabilityNGM Nested grid modelNLNMI Nonlinear normal mode initializationNMC National Meteorological CenterNOAA National Oceanic and Atmospheric AdministrationNORPEX North Pacific ExperimentNWP Numerical weather predictionNWS National Weather ServiceOI Optimal interpolationPDE Partial differential equationPDO Pacific decadal oscillationPIRCS Project to Intercompare Regional Climate SystemsPQPF Probabilistic quantitative precipitation forecastPSAS Physical space analysis schemePVE Potential vorticity equationRAFS Regional analysis and forecasting systemRAOB Rawinsonde observationRDAS Regional data assimilation systemRSM NCEP’s Regional Spectral Model






Abbreviations xix

RUC NCEP’s rapid update cycleSAC Standardized anomaly correctionSCM Successive correction methodSOR Successive overrelaxationSST Sea surface temperatureSWE Shallow water equationsTOGA Tropical ocean, global atmosphereTOVS TIROS-N operational vertical sounderTS Threat scoreUKMO United Kingdom Meteorological OfficeUTC Universal time or Greenwich time, e.g. 1200 UTC.

Frequently abbreviated as 1200ZWMO World Meteorological Organization






Variables

a radius of the EarthA analysis error covariance matrixB background error covariance matrixC covariance matrixCp, Cv specific heat at constant pressure, constant volumed innovation or observational increments vectorD fluid depthE( ) expected valuef Coriolis parameterg gravitational constantH linear observation operator matrixH observational operator, scale height of the atmosphereI identity matrixJ cost functionJM maximum number of grid points jK Kalman gain matrixL(t0, t) resolvent or propagator of TLMM TLM matrixN Brunt–Vaısala frequencyP projection matrixp pressure, probability, distribution functionq mixing ratio of water vapor and dry air massQ forecast model error covariancer position vector

xxi






xxii Variables

R observations error covariance matrixR root mean square error, gas constantRd Rossby radius of deformationR0 Rossby numberRE relative errorT temperatureTS threat scoreu,v eastward and northward wind componentsW weight matrixW vertical wind component, optimal weightx,y horizontal coordinatesδi j Kronecker deltaεa analysis errorεb background errorη absolute vorticity� geopotential heightϕ geopotential, latitudeλ longitudeλi global Lyapunov exponentρi j element i,j of the correlation matrix Cσ standard deviationσ 2 varianceψ streamfunctionω vertical velocity in pressure coordinates, spectral frequencyζ relative vorticity






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