Dale B Haidvogel has been a leader in the development and application of alternative numerical ocean circulation models for nearly two decades Since receiving his PhD in Physical Oceanographyfrom the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution in 1976, his research activities have spanned the range from idealized studies of fundamental oceanic processes to the realistic modeling of coastal and marine environments He currently holds the position of Professor II in the Institute of Marine and Coastal Sciences at Rutgers, the State University of New Jersey Aike Beckmann received his PhD in oceanography from the Institute for Marine Research in Kiel, Germany, and has been working in the field of numerical ocean modeling since 1984 His research interests include both high-resolution process studies and large-scale simulations of ocean dynamics, with special emphasis on topographic effects He is currently a senior research scientist at the Alfred Wegener Institute for Polar and Marine Research in Bremerhaven, Germany, where he heads a group working on high-latitude ocean and ice dynamics SERIES ON ENVIRONMENTAL SCIENCE AND MANAGEMENT Series Editor: Professor J.N.B Bell Centre for Enwironrnenfal Technology, Imperial College Published Vol Environmental Impact of Land Use in Rural Regions P.E R$etna, P Groenendijk and J.G Kroes Vol Numerical Ocean Circulation Modeling D.B Haidvogel and A Beckmann Forthcoming Highlights in EnvironmentalResearch John Mason (ed.) NUMERICAL OCEAN CIRCULATION MODELING Dale B Haidvogel Rutgers University, USA Aike Beckmann Alfred Wegener Institute for Polar & Marine Research, Germany Imperial College Press Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co Re Ltd P Box 128 Farrer Road, Singapore 912805 LISA oflce: Suite lB, 1060Main Street, River Edge, NJ 07661 UK oflce: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-PublicatlonData Haidvogel, Dale B Numerical man circulation modeling I Dale B Haidvogel, Aike Beckmann p cm (Series on environmental science and management :vol 2) Includes bibliogcapbicalreferences and index ISBN 1-86094-114-1 (alk paper) Ocean circulation Mathematical mdoels I Beckmann.A (Aike) 11 Title 111 Series GC228.5.H35 1999 551.47'01'015118 dc21 99- 19666 CIP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library First published 1999 Reprinted 2000 Copyright Q 1999 by Imperial College Press All rights reserved This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers MA 01923 USA In this case permission to photocopy is not required from the publisher Printed in Singapore by Uto-Print To our daughters, Ilona and Annika Preface Until recently, algorithmic sophistication in and diversity among regional and basin-scale ocean circulation models were largely non-existent Despite significant strides being made in computational fluid dynamics in other fields, including the closely related field of numerical weather prediction, ocean circulation modeling, by and large, relied on a single class of models which originated in the late 1960’s Over the past decade, the situation has changed dramatically First, systematic development efforts have greatly increased the number of available models Secondly, enhanced interest in ocean dynamics and prediction on all scales, together with more ready access to high-end workstations and supercomputers, has guaranteed a rapidly growing international community of users As a result, the algorithmic richness of existing models, and the sophistication with which they have been applied, has increased significantly In such a rapidly evolving field, it would be foolhardy to attempt a definitive review of all models and their areas of application Our interest in composing this volume is more modest yet, we feel, more important In particular, we seek to review the fundamentals upon which the practice of ocean circulation modeling is based, to discuss and to contrast the implementation and design of four models which span the range of current algorithms, and finally to explore and compare the limitations of each model class with reference to both realistic modeling of basin-scale oceanic circulation and simple two-dimensional idealized test problems The latter are particularly timely With the expanded variety and accessibility of today’s ocean models, it is now natural to ask which model might be best for a given application Unfortunately, no systematic comvii viii Preface parison among available large-scale ocean circulation models has ever been conducted Replicated simulations in realistic basin-scale settings are one means of providing comparative information Nonetheless, they are expensive and difficult to control and to quantify The alternative - the development of a set of relatively inexpensive, process-oriented test problems on which model behavior can be assessed relative to known and quantifiable standards of merit - represents an important and complementary way of gaining experience on model performance and behavior Although we direct this book primarily towards students of the marine sciences and others who wish to get started in numerical ocean circulation modeling, the central themes (derivation of the equations of motion, parameterization of subgridscale processes, approximate solution procedures, and quantitative model evaluation) are common to other disciplines such as meteorology and computational fluid dynamics The level of presentation has been chosen to be accessible to any reader with a graduate-level appreciation of applied mathematics and the physical sciences Ocean Models Today There are, at present, within the field of ocean general circulation modeling four classes of numerical models which have achieved a significant level of community management and involvement, including shared community development, regular user interaction, and ready availability of software and documentation via the World Wide Web These four classes are loosely characterized by their respective approaches to spatial discretization and vertical coordinate treatment The development of the first oceanic general circulation model (OGCM) is typically credited to Kirk Bryan at the Geophysical Fluid Dynamics Laboratory (GFDL) in the late 1960’s Following then-common practices, the GFDL model was originally designed to utilize a geopotential (z-based) vertical coordinate, and to discretize the resulting equations of motion using low-order finite differences Beginning in the mid-l970’s, significant evolution in this model class began to occur based on the efforts of Mike Cox (GFDL) and Bert Semtner (now at the Naval Postgraduate School) At present, variations on this first OGCM are in place at Harvard University (the Harvard Ocean Prediction System, HOPS), GFDL (the Modular Ocean Model, MOM), the Los Alamos National Laboratory (the Parallel Ocean Program, POP), the National Center for Atmospheric Research (the 306 Bibliogrophy Lynch, D.R., J.T.C Ip, C.E Naimie, and 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forecast errors in the ECMWF model through the introduction of an envelope orography Quart J Roy Meteor SOC.,109, 683-717 W a g , D.-P., 1984 Mutual intrusion of a gravity current and density front formation J Phys Oceanogr., 14, 1191-1199 Wajsowicz, R.C., 1986: Free planetary waves in finite-difference numerical models J Phys Oceanogr., 16,773-789 Webb, D.J., 1995: The vertical advection of momentum in Bryan-Cox-Semtner ocean general circulation models J Phys Oceanogr., 25, 3186-3195 Winton, M., R Hallberg, and A Gnanadesikan, 1998: Simulation of densitydriven frictional downslope flow in z-coordinate ocean models J Phys Oceanogr., 28, 2163-2174 Wilkin, J.L., and K.S Hedstrom, 1998: Gridpak Users Guide On: http://marine.rutgers.edu/po/gridpak.html Wilkin, J.L., J.V Mansbridge, and K.S.Hedstrom, 1995: An application of the capacitance matrix method to accommodate masked land areas and island circulations in a primitive equation ocean model Int J Num Meth Fluids, 20, 649-662 Williamson, D.L., J.B Drake, J.J Hack, R Jacob, and P.N Swartztrauber, 1992: A standard test set for the numerical approximations to the shallow water equations in spherical geometry J Comput Phys., 102, 211-224 Wunsch, C., D.B Haidvogel, M Iskandarani, and R.Hughes, 1998: Dynamics of the long-period tides Prog Oceanogr., 40, 81-108 Yin, F.L.,and E.S.Sarachik, 1994: An efficient convective adjustment scheme for ocean general circulation models J Phys Oceanogr., 24, 1425-1430 Zalezak, S.T., 1979: h l l y multi-dimensional flux-corrected transport algorithms for fluid J Comput Phys., 31,335-362 Index CPP, 131, 144 f77,131, 144, 151 angular momentum, 15 approximate difference equation, 54 approximation h-p, 152 Boussinesq, 20 centered, 49 CTCS, 68 error, 73, 109 finite difference, 122 finite element, 122 FTCS, 58, 68 FTUS, 69 Galerkin, 47, 52, 56 hydrostatic, 21 least squares, 40 one-sided, 49 piecewise linear, 39 planar, 28 polynomial, 45 thin-shell, 23 TOUS, 70 traditional, 23 Arakawa Jacobian, 117 artificial viscosity, 78 aspect ratio, 28 Asselin filter, 150, 178 asynchronous integration, 131 Austausch coefficient, 171 “B” grid, 125, 151 “ C grid, 132, 140, 145, 148 acceleration centripetal, 12 Coriolis, 12 accuracy, 118 adaptive grid, 145 ADE, 55 adiabatic fluid, 18 adjustment geostrophic, 95, 100 gravitational, 95, 221, 227 advection scheme centered, 67 ELAD, 119 FCT, 119 flux-corrected transport, 119 FTUS, 69 locally adaptive, 119 semi-Lagrangian, 118 TOUS, 70 TOUW, 119 TVD, 119 upstream, 69 upwind, 69 advection schemes, 221 amplification factor, 55 balance equations, 27 312 Index baroclinic leakage, 126 barotropic h i d , 33 barotropic-baroclinic coupling, 154 basin modes, 211 betccplane, 26, 27, 101 biharmonic operator, 178 body force, bolus velocity, 187 bottom boundary layer, 190, 196 bottom layer flow,271 bottom resistance coefficient, 197 bottom torque, 139 boundary condition free slip, 140, 210, 213 free-slip, 213 kinematic, 25 no slip, 126, 140, 158, 197, 210, 216, 271 open, 26 boundary conditions, 25 boundary layer, 138, 175 bottom, 132, 176, 182, 190, 196, 198, 253, 271 lateral, 177,211 surface, 190 boundary layer model, 176 Boussinesq approximation, 20 box-concept, 125 breadt h-first-search algorithm , 161 brine release, 198 Brunt-VaisQa, 175 Brunt-VaisQa frequency, 192 buffer zone, 259 bulk mixed layer, 150 bulk mixed layer model, 169 bulk mixed layer models, 193 buoyancy, 28 cabbeling, 151, 275 capacitance matrix, 115, 136 cell Reynolds number, 181 centered approximation, 49 centripetal acceleration, 12 313 force, 12 potential, 12 CFL criterion, 70 Chebyshev polynomials, 138, 139 climatology, 245 closure k - E , 175 k - 1, 175 first order, 170 higher order, 173 local, 175 non-local, 175 second order, 174 closure problem, 164,167 CME, 259 coastally trapped waves, 183 cold water sphere, 190 collocation method, 40, 48 collocation point, 155 computational mode, 62,80 conjugate gradient method, 125 conservation of angular momentum, 15 energy, 6, 115 enstrophy, 18, 115 heat, mass, 1, mechanical energy, 18 momentum, 1, conservation properties, 84 consistency, 55 control volume, convection, 199 convection layer, 190 convective adjustment, 150, 199 convective plumes, 199 convergence, 44, 51, 55, 109, 152,215, 239 coordinates Cartesian, 26, 101, 111, 134, 187 curvilinear, 133, 148 generalized, 14 generalized adaptive vertical, 286 generalized vertical, 121 314 geopotential, 123, 271 horizontally adaptive, 286 inertial, isopycnic, 145, 275 level, 123 non-Cartesian, 13 s, 275 sigma, 133, 134 spherical, 13, 22, 134, 287 terrain-following, 133 vertical, 121, 123, 125, 133, 136, 145, 148 z, 123 Coriolis acceleration, 12 parameter, 29, 95 term, 131, 159 terms, 59 coupled modeling, 283 critical level, 108 curvilinear coordinates, 133, 148 DAMEE, 259, 261 data assimilation, 261 deformation, 181 Denmark Strait, 253 diagnostics, 131, 144 diapycnal mixing, 145, 275 diffusive velocities, 179 diffusivity molecular, 163 of heat, of salt, discretization horizontal, 152 spatial, 71, 125, 155 temporal, 58, 159 vertical, 136, 148 dispersion relation, 101 domain of accuracy, 97 DWBC, 249 DWBC, deep western boundary current, 185 dynamic equilibrium, 245 Index dynamic pressure, 23 DYNAMO, 259, 260 eastern recirculation, 252 ECMWF, 247, 261, 262 eddy coefficient, 171 eddy form stress, 185 eddy variability, 250 eddy-induced transport velocity, 187 eigenvalue problem, 32 Ekman depth, 191 layer, 190 number, 163 spiral, 196 elliptic equation, 112, 125, 254 elliptic solver ADI, 114 CG, 125 conjugate gradient, 114 direct, 113 Gauss-Seidel, 114 iterative, 114 multi-grid, 114 SOR, 114 energy, 6, 116 conservation, 6, 23, 117 EKE, 256 EPE, 256 internal, kinetic, 6, 84, 166, 256 mechanical, 18 MKE, 256 MPE, 256 potential, 6, 256 transformation, 256 turbulent kinetic, 167, 194 enstrophy, 18, 116 enstrophy conservation, 18 entrainment, 198, 254 entropy, 8, 184 envelope method, 244 epineutral surface, 176, 182 equation Index advection, 67, 110 barotropic vorticity, 115,210 Burger’s, 84 elliptic, 112, 254 friction, 59, 66 heat, 109 kinetic energy, 166 Reynolds averaged, 166 semi-discrete, 111, 128, 140, 149 TKE, 167, 194 wave, 60,109 equation of state, 8, 24, 124, 136, 147, 151, 289 error aliasing, 80 amplitude, 60 approximation, 109 damping, 73 dispersion, 79 phase, 60,73, 79 pressure gradient, 138, 234 timesplitting, 79 truncation, 138 Ertel’s Theorem, 16 factorization, 90 filtered equations, 27 filtering, 26, 83, 169 Asselin, 150, 178 Fourier, 130 Shapiro, 142, 178 spectral, 161 finite difference method, 48 FLAME, 278 Florida Current, 250 form stress, 183 forward-backward scheme, 147 Fourier coefficients, 42 Fourier filtering, 130 Fourier series, 40 free sea surface, 25, 106, 124, 142 explicit, 125, 147 implicit, 125 friction equation, 59 315 FTUS, 69 Galerkin approximation, 47, 56 method, 152, 157 generalized vertical coordinate, 121 geometrical flexibility, 152 geostrophic adjustment, 100 geostrophy, 33 GFDL, 123 Gibbs phenomenon, 44, 46 global climate system, 283 gravitational adjustment, 95, 221, 227 gravity, 13 grid “A”, 95,98 “B”, 95, 99, 104, 125, 127 “C”, 95, 100, 105, 140, 148 “D”, 95 “E”, 95 adaptive, 145 Arakawa, 93 block-structured, 115 curvlinear, 143 finite difference, 95 isotropic, 244 masking, 115, 211 Mercator, 244 non-staggered, 140 orthogonal curvlinear, 143 rotated, 127 staggered, 93, 197 unstructured, 115, 152,285 grid generation, 143 grid index, 39 group velocity, 95, 101 Gulf Stream, 250 separation, 251, 263 system, 249 harmonic operator, 173 heat conservation, heat equation, 51 Hovmoller diagram, 230 316 HPE, 19, 23, 24, 121 hydrographic data, 245 hydrostatic approximation, 22 correction, 138 inconsistency, 238 hydrostatic primitive equations, 19 IfM Kiel, 259 implicit mixing, 200 incompressibility, 18, 21 initial conditions, 25 initialization, 245 instability baroclinic, 186, 257, 263 barotropic, 186, 257 nonlinear, 80 numerical, 54 static, 198 interpolating polynomials, 49 inviscid fluid, 18 isotropic grid, 244 Jacobian, 155 Arakawa, 117 Jacobian operator, 116 JEBAR, 35, 139, 248 Kelvin waves, 208 land masking, 115, 136 large-eddy simulation, 176 large-scale geostrophic, 27 Lax-Richtmyer equivalence theorem, 55 LBE, 112 least squares approximation, 40, 48 Legendre polynomials, 45, 155 LES, 176 linear balance equations, 112 load-balancing, 161 Loop Current, 250 LSG,27 Indez masking, 115 mass conservation, 1, massively parallel computers, 125 mean value theorem, 39 Mediterranean Water, 253 Mercator grid, 244 meridional heat transport, 255 overturning, 254, 269 mesoscale eddies, 256 metric coefficients, 14 terms, 28 minmax polynomial, 45 mixed layer, 150 mixed layer models, 193 mixing artificial, 78 rotated operator, 182 mixing length hypothesis, 171 mixing scheme adaptive, 169, 181 convective adjustment, 199 diapycnal, 145 eddy-mean flow, 186 eddy-topography, 183 epineutral, 142 Gent-McWilliams, 186 GM90, 277 harmonic, 181 harmonic horizontal, 278 higher order, 177 isopycnal, 278 isopycnic, 187 KPP, 169, 176 MY, 169 non-local, 176 PP, 169, 193 PWP, 169 Richardson number dependent, 193 Smagorinsky, 169, 181 stability dependent, 192 thickness diffusion, 186, 278 Indez turbulent closure, 176 vertical, 189, 200 mode baroclinic, 125 barotropic, 125 computational, 62,80 external, 125 internal, 125 physical, 62 model finite differences, 284 finite elements, 284 forcing, 246 horizontal grid structure, 244 initialization, 245 intercomparison, 260 multi-scale, 284 topography, 244 models non-hydrostatic, 199 modular code, 123, 133 molecular friction, 163 molecular viscosity, MOM, 123 momentum conservation, 1, momentum forcing, 246 monotonicity, 118, 119,169 Montgomery potential, 146 mortar elements, 161,286 multi-grid algorithm, 136 multi-scale modeling, 284 NCAR, 259 NCEP, 262 near-bottom circulation, 269 nesting two-way, 161, 285 NetCDF, 131, 144 Newton’s second law, Newtonian fluid, NHPE, 15, 162 non-hydrostatic primitive equations, 15 nonlinear instability, 80 317 nudging, 276 numerical instability, 54 Nyquist frequency, 64 objective analysis, 144 ocean circulation model DieCAST, 93, 132 DieCast, 262 DJM, 144 GFDLM, 259, 260 GHERM, 144 HAMSOM, 132 MICOM, 122, 145, 260, 262 MOM, 122, 123, 260 NLOM, 262 OPA, 132 OPYC, 151 POCM, 123 POM, 144, 262 POP, 123 QUODDY, 162 SCRUM, 122, 133, 262 SEOM, 93, 122, 152 SOMS, 93,132 SPEM, 122, 133, 260 ocean circulation models SEOM, 262 OGCM, 162 open boundaries, 130, 142, 276 open boundary conditions, 276 outflow, 253 overflow, 253 overturning meridional, 254, 269 overturning streamfunction, 254 parallel computers, 152 parallelization, 161 partial differential equation, 38 PDE, 38 penetrative plumes, 200 perturbation, 165 phase velocity, 95 physical mode, 62 318 physical-biological models, 283 physical-geological models, 283 piecewise linear approximation, 39 planetary boundary layer, 196 polar singularities, 127, 152 poleward undercurrent, 185 polynomial approximation, 45 Chebyshev, 45, 138 interpolating, 49 Legendre, 45, 155 minmax, 45 positive definiteness, 169 potential centripetal, 12 gravitational, 12 Montgomery, 146 pre-processing, 114 preconditioning, 199 pressure baroclinic, 35 dynamic, 23 force, gradient, 35, 123, 134, 139, 145 pressure gradient error, 138, 234 QG, 26, 30, 201 QGPVE, 32, 106, 112, 201 quasigeostrophy, 26, 30, 201 radiation condition, 130 Rayleigh friction, 197 rectification, 184 reference density profile, 139 reference frame inertial, rotating, reference level, 151 resolution eddy resolving, 162 eddy-resolving, 265 non-eddy permitting, 162 non-eddy resolving, 162 restoring, 247 Index restoring terms, 276 Reynolds averaging, 164 number, 164, 169 stress, 170 stress tensor, 166 Richardson number, 169, 193 rigid lid, 25, 106, 124, 125, 141, 258 Rossby number, 28, 30 radius, 32, 96, 185 soliton, 204 rotated mixing tensors, 182 rotation, scalability, 152 scaling, 19, 26 scheme 2-step, 59 3-step, 60 higher order, 70, 71, 173, 175, 177 higher-order , 169 sea surface height, 257 seamrface height variability, 257 semi-discrete equations, 52 semi-implicit scheme, 131 semi-Lagrangian schemes, 118 shallow water equations, 26, 33 Shapiro filter, 142, 178 shaved grid cells, 132 SOR, 125 spatial filters, 130 spectral approach, 139 spectral element method, 152 spectral model, 144 spherical coordinate system, 13 spherical coordinates, 13, 124 spherical harmonics, 152 split-explicit scheme, 160 sponge layer, 276 SSH, 257 stability, 55, 61, 109 stability analysis, 55, 60 staggered grids, 93 Indez staircase topography, 132 static instability, 199 statistical mechanics, 184 statistical methods, 250 statistically steady state, 245 steplike topography, 239 Straits of Gibraltar, 254 streamfunction, 32, 125, 136 stress, 35 stress tensor, stretched grid, 180 Sturm-Liouville problem, 32 subduction, 252, 269 surface epineutral, 176, 182 forces, stress, 246 surface boundary layer, 190 surface gravity waves, 25 surface mixed layer, 169, 190,253 surface pressure, 258 Sverdrup balance, 210 regime, 248 SWE, 26, 33, 154, 210 Taylor series, 38, 48 tensor rotation, 169 terrain following coordinates, 133 test problems, 203, 240 thermal conductivity, thermal wind, 33, 275 thermobaricity, 147,151, 275 thermocline ventilation, 269 thermodynamics first law, thermohaline circulation, 248, 254 thickness diffusion, 186, 189 thin shell model, 132 tidal modeling, 162 tidal stress, 198 time splitting, 125, 150 time-step, 54 time-stepping scheme 319 Adams-Bashforth, 59 Crank-Nicholson, 159 Euler backward, 59, 61, 66 Euler forward, 59,61,66 explicit, 59 implicit, 59, 159 leapfrog, 59, 66 leapfrog-trapezoidal, 64, 142 semi-implicit, 89, 131,160 split explicit, 90 split-explicit, 136, 142 third-order Adams-Bashforth, 159 trapezoidal, 59, 61,66 topographic control, 275 stress, 183 topography smoothing, 244 total variance diminishing scheme, 119 triangular finite elements, 162 truncation, 38 truncation error, 50 turbulent closure, 174 turbulent kinetic energy, 167 two-way nesting, 161 unstructured grids, 152 upstream advection, 181 user’s manual, 123, 133, 145 variational formulation, 157 velocity bolus, 187 eddy-induced transport, 187 group, 95, 101 phase, 95 ventilation, 252 vertical modes, 32 vertical stretching, 137 viscosity artificial, 78 kinematic, 163 molecular, 320 volumetric water mass census, 255 Von Neumann method, 55 vorticity planetary, 24 potential, 17, 271 relative, 17 warm water sphere, 190 water mass, 255 wave equation, 59, 60 wavenumber, 96 waves baroclinic, 101 barotropic, 101, 106 external, 95, 106 inertia-gravity, 95 inertial, 131 internal, 95 planetary, 100 planetary (Rossby), 95 Rossby, 100 WBC, 250 western boundary current, 208, 250 westward intensification, 100, 248 WOCE, 250 zero mass layers, 150 zonal flow bands, 252 Index ... and J.G Kroes Vol Numerical Ocean Circulation Modeling D.B Haidvogel and A Beckmann Forthcoming Highlights in EnvironmentalResearch John Mason (ed.) NUMERICAL OCEAN CIRCULATION MODELING Dale B... concise review of the fundamentals upon which numerical ocean circulation modeling is based; second, to give extended descriptions of the range of ocean circulation models currently in use; third,... applied mathematics and the physical sciences Ocean Models Today There are, at present, within the field of ocean general circulation modeling four classes of numerical models which have achieved a