Oceanic Circulation Models: Combining Data and Dynamics NATO ASI Series Advanced Science Institutes Series A Series presenting the f8SU/tS of activ;ties sponsored by the NATO Science Committso, which aims at th8 disS8mination of advanced scientifIC and tschnofogicaf know/edge, with a Vi8W to strengthsning links betwesn scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation S Physics London and New York C MathematIcal Kluwer Academic Publishers and Physical Sciences Dordrechl, Boston and london D Behavioural and SocIal Sciences E Applied Sciences F Computer and Systems Sciences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, london, H Cell Biology Paris and Tokyo Series C: Mathematical and Physical Sciences -Vol. 284 Oceanic Circulation Models: Combining Data and Dynamics edited by David L. T. Anderson Department of Atmospheric, Oceanic and Planetary Physics, Oxford University, Oxford, U.K. and JOrgen Willebrand Inslilut fur Meereskunde, Universitat Kiel, Kiel, ERG. Kluwer Academic Publishers Oordrecht I Boston / London Published in cooperation with NATO Scientific Affairs Division Proceedings of the NATO Advanced Study Institute on Modelling the Ocean General Circulation and Geochemical Tracer Transport Les Houches, France February 15-26, 1988 Library of Congress Cataloging In Publication Data NATO Advanced Study Institute on 'ModellIng the Ocean General CIrculatIOn and Geochemical Tracer Transport' (1988 Les Houches, Haute-Savoie, France) Ocean circulation models combining data and dynamics ! edited by David L.T, Anderson and Jurgen Hillebrand. p, cm. -- (NATO ASI series, SerIes C, Mathematical and physical sciences; vol. 284) "Proceedings of the NATO Advanced Study Institute on 'Modelling the Ocean General Circulation and GeochemIcal Tracer Transport' held In Les Houches, France, February 15-26, 1988." Inc I udes index. ISSrJ 0-7923-0394-6 1. Ocean circulatlon--Cornpute~ simulatlcn--C~ngresses. 2, Cherrlcal oceanography--Computer slmulatlon--Congresses, I. Anderson, '0. L. T. <David L. T, ) II. Hillebrand, J, (Jurgen), 1941- III. TItle, IV: Series, NATO ASI series,. Series C, MathematIcal and physical sCIences; no. 284. GC228.5.N38 1988 551.47' 01--dc20 89-15518 ISBN-I 3: 978-94-010-6946-5 e-ISBN-I 3: 978-94-009-1013-3 DOl: 10,1007/978-94-009-1013-3 Published by Kluwer Academic Publishers, P.O. Box 17,3300 M Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed In the U.S,A, and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U,S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P,O, Box 322, 3300 AH Dordrecht, The Netherlands, Printed on acid free paper All Rights Reserved © 1989 by Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo copying, recording or by any information storage and retrieval system, without written permission from the copyright owner. This book contains thc proceedings of a NATO Advanced Research Workshop held within the programme of activities of the NATO Special Programme on Global Transport Mechanisms in Ihe Goo-Sciences running from 1983 to 1988 as part of the activities of the NATO Science Committee. Other books previously published as a result of the activities of the Special Programme are: BUAT-MENARD. P. (Ed.) - Th~ Role of Air-Sea Exchange in G~och~mical Cycling (CIS5) 1986 CAZENAVE, A. (Ed.) - Earth Rotation: Solved ond Unsolw:d Problems (CIS7) 1986 WILLEBRAND. J. and ANDERSON, D.L. T. (Eds..) - l.clrge-Scole Transport Processes in OctarlS and AlI1IQspher~ (C I 90) 1986 MCOUS, C. and NlCOUS, G. (Eds.) - I"ev~rsible Phenom~M and Dynamical Systems Analysis In Geosciences (CI92) 1986 PARSONS. I. (Ed.) - Origins ofI gn~ous l.clyering (CI96) 1987 LOPER, E. (Ed.) -StruClure and DYMmics of Partially Solidified Systems (E125) 1987 VAUGHAN, R. A. (Ed.) - Remote Sensing Applications in Meteorology and Clinullology (C201) 1987 BERGER. w. H. and LABEYRIE, L. D. (Eds.) - Abrupt CUmaric Change _ Evidence and Implications (C216) 1987 VISCONTI, G. and GARCIA, R. (E.ds.) - TrarlSport Processes in the Middle Almospher~ (C213) 1987 SIMMERS, I. (Ed.) -Estimalion of Natural Recharge of Groundwater (C222) 1987 HELGESON. H. C. (Ed.) - Chemical TrarlSporl in M~lasomatlc Proctsses (018) 1987 CUSTODIO, E., GURGU!. A. and LOBO FERREIRA. J. P. (Eds.) _ Groundwater Flow and Qua/ity Modelling (024) 1987 ISAKSEN, I. S. A. (Ed.) - Tropospheric Ozon~ (C227) 1988 SCHLESINGER, M.E. (Ed.) - Physically-Based Modelling and Simulation of Climate and Climatic Change 2 vols. (C243) 1988 UNSWORTH. M. H. and FOWLER. D. (Eds.) - Acid Deposition al fligh Elevation Siles (C252) 1988 KISSEL, C. and LAY. C. (Ed5.) - Paleomagnetic Rotalions and Continental Deformation (C254) 1988 HART, S. R. and GULEN, L. (Eds.) -CrUSIIMamle Recycling at Subduction Zones (C258) 1989 GREGERSEN, S. and BASHAM, P. (Eds.) - Earthquakes at North-Allantic Passive Margins: Neotectonics and Poslglacial Rebound (C266) 1989 MOREL-SEYTOUX, H. J. (Ed.) -UnSalurated Flow in Ilydrologic Modeling (C275) 1989 BRIDGWATER. D. (Ed.) - Fluid Movements - Element Transport and the Composition of the CrUSI (08\) 1989 ANDERSON. D.L.T. and WILLEBRAND. J. (Eds.) - Ocean Circulation Models: Combining Data and Dynamics (C284) 1989 CONTENTS Preface xvii List of Participants xix Tracer Inverse ProbleIIlS Carl Wunsch 1. Introduction 1 1.1 The General Problem ..........•...•......•..•...•..•...•..•..•........ 1 1.2 On Determinateness .................................................. 13 2. Interpolation and Map Making 15 2.1 Interpolation ........................•......•......•.........•........ 15 2.2 The Gauss·Markov Theorem .......................................... 18 2.3 Determining a Mean Value ............................................ 25 2.4 A Priori Information ..............•...•..•.•....•......•..•..•........ 27 3. Simple Estimation 28 3.1 Elementary Least.Squares ............................................. 28 3.2 Underdetermined Systems ............................................ 38 3.3 Errors in the Observations ............................................ 43 3.4 Resolution ........................•...•..•..•...•..•...•..•..•........ 44 3.5 Rowand Column Scaling ...........•......•..•...•..•.........•...... 47 3.6 Generalized Inverses .............................•...•..•..•...•...... 48 3.7 Other Estimation Procedures ......................................... 49 3.8 Inverse Methods and Inverse Problems ................................ 54 4. Using Steady Tracers 55 4.1 The Background .......................................•..•..•........ 55 4.2 Where Steady Tracer Inverse Methods are Going .......•..•........... 56 4.3 Eclectic Modelling .....................................•..•........... 57 4.4 Remaining Steady Tracer Issues ....................................... 58 5. Time Dependent Problems 59 VIIi 5.1 Observational Realities and Boundary Controls ........................ 59 5.2 Modifying the Model .......................... _. ........•............ 66 Appendix. Some Notes on the History of Inverse Methods in Ocean Circulation Problems ......................................................... 69 A Geometrical Interpretation of Inverse Problems Dirk Olbers 1. The Overdetermined Case 79 2. The Underdetermined Case 81 3. The Singular Value Decomposition (SVD) 87 3.1 The Overdetermined-Underconstrained Case ..............•..•........ 90 3.2 Again the Undcrdetermined Case ..................•...•.....•........ 91 3.3 The Tapered Cut-off Solution ........... , ...... , ...................... 92 Determining Difl'usivities from Hydrographic Data by Inverse Methods with Applications to the Circumpolar Current Dirk Olbers and Manfred Wenzel 1. Introduction to ill-posed and Noisy Problems 95 1.1 A Singular Example ...........................•......•..•............ 96 1.2 Some Noisy Examples ................................................ 98 2. The Physical Model 101 2.1 "'Equations of Motion and the Level-of-no-motion Problem ............ 102 2.2 Parametrization of Mixing ........................................... 104 3. The Inverse Model 109 3.1 The .8-Spiral Method ................................................ 110 3.2 Mass Conservation ........................................•.......... 112 4. Examples: Circulation and mixing in the Southern Ocean 114 4.1 A Singular and Some Well-posed Problems ........................... 115 4.2 The Diabatic Model ...............................•..•..•........... 122 Ocean Acoustic Tomography: a Primer Robert A. Knox 1. Introduction 141 2. Elementary Hydrodynamics and Acoustics 143 3. Ocean Sound Speed Distribution 146 4. Rays and Modes 147 5. Capsule Description of the Tomographic Method 157 6. A Simple Ray Example 160 7. Notes on Hardware Limitations and Observational Errors 162 8. Inversions 165 9. Some Results and Future Plans 168 10. Conclusions 180 11. Literature 183 The Circulation in the Western North Atlantic Determined by a Nonlinear Inverse Method. BerIe Mercier 1. Introduction 189 2. A Nonlinear Inverse Formalism 190 3. The Circulation in the Western North Atlantic 192 3.1 Motivation ....................................•..•...•..•...•.•.... ' 192 3.2 The Data Base ..........•....•..........•.....•..•...•..•..•....•... 193 3.3 The Dynamical Model ............................................... 194 3.4 A Priori Assumptions ................................................ 196 3.5 The Results .................•...•..•...•..•...•..•..••..•..•........ 197 Altimeter Data Assimilation into Ocean Circulation Models - Some Preliminary Results William R. Holland 1. Introduction 203 2. A Dynamic Initialization Scheme 204 3. Time and Space Dependence of the Observations 215 4. Geosat Assimilation in the Agulhas Retroflection Region 221 5. Discussion 230 Assimilation of Data into Ocean Models D. J. Webb 1. Introduction 233 2. Theory 234 2.1 The Kalman Filter .................................................. 235 2.2 Non-Linear Systems ...........•..•......•......•.........•..•....... 239 2.3 Alternative Approaches .............................................. 240 2.4 Adjoint System .................•...•..•......•.........•..•......... 241 3. Applications 244 3.1 Kalman Filters ...................•..•......•......•.........•....... 244 3.2 Objective Analysis ................................................... 244 3.3 Oceanographic Applications .......................................... 246 3.4 Projection Schemes .................•..•...•..•••.•..•..•..•......... 247 3.5 Adjoint Schemes ...................•..•......•......•................ 249 4. Conclusions 249 Driving of Non-linear Time-dependent Ocean Models by Observation of Transient Tracers - a Problem of Constrained Optimisation Jens Schroter 1. Introduction 257 2. A Control Problem 260 3. An Elegant and Efficient Way to Calculate the Gradient of the Cost Function 262 4. The Ocean Model and its Adjoint 265 5. Results 268 6. Model Parameters as Control Variables 276 7. Sensitivity 280 7.1 Error Covariance and Resolution ..................................... 280 ,; 7.2 Observational Analysis .........•..• _. .•..•.. _• ..•.. _• . _ •..•..•...... 282 8. Conclusions 283 Assimilation of XBT Data Using a Variational Technique J. Sheinbaum and D.L.T. Anderson 1. Introduction 287 2. Implementation of Variational Assimilation Using Lagrange Multipliers 289 3. Results 293 The Role of Real·Time Four-Dimensional Data Assimilation in the Quality Control, Interpretation, and Synthesis of Climate Data A. Hollingsworth 1. Introduction 304 2. The Importance of Accurate Data Assimilation for NWP 305 3. The Analysis Module 308 3.1 The 0/1 Algorithm ........... , ...•......•............... ' •. '• ...... 311 3.2 The 0/1 Filter .............................................•........ 313 3.3 The 0/1 Interpolator ............................................... 314 3.4 The Relationship Between the Filter and Interpolator .........•...... 315 3.5 General Comments ........................................•........ 315 4. Non-linear Normal Mode Initialization 316 5. The Forecast Model 318 6. Quality Control and Data Monitoring 319 6.1 Aireps ........... , ...................................... , •. , •....... 319 6,2 Radiosonde Monitoring ................... , . .................. , ..... 322 6.3 Remotely Sensed Wind Data ........................................ 323 6.4 Scatterometer Winds ....................................... , . , , ..... 327 6,5 Temperature Soundings from Satellites , ... , ................ , ........ 330 7. The Value and Limitations of Global NWP Datasets for Climate Studies 334 7.1 Advantages ..... "." .. , ............................................ 334 7.2 Limitations ....... ", .............................................. , 335