Numerical ocean circulation modeling

Dale B. Haidvogel has been a leader in the development and application of alternative numerical ocean circulation model...

Author: Dale B. Haidvogel | Aike Beckmann

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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. 1 Environmental Impact of Land Use in Rural Regions P.E. R$etna, P. Groenendijk and J.G. Kroes Vol. 2

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 0 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) 1. 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.

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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

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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

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NCAR Community Ocean Model, NCOM), and other institutions. During the 1970’s, two competing approaches to vertical discretization and coordinate treatment made their way into ocean modeling. These alernatives were based respectively on vertical discretization in immiscible layers ( “layered” models) and on terrain-following vertical coordinates (“sigma” coordinate models). The former envisions the ocean as being made up of a set of non-mixing layers whose interface locations adjust in time as part of the dynamics; the latter assumes coordinate surfaces which are fixed in time, but follow the underlying topography (and are therefore not geopotential surfaces for non-flat bathymetry). In keeping with 1970’sstyle thinking on algorithms, both these model classes used (and continue to use) low-order finite difference schemes similar to those employed in the GFDL-based codes. Today, several examples of layered and sigma-coordinate models exist. The former category includes models designed and built at the Naval Research Lab (the Navy Layered Ocean Model, NLOM), the University of Miami (the Miami Isopycnic Coordinate Ocean Model, MICOM), GFDL (the Hallberg Isopycnic Model, HIM), the Max Planck Institute in Hamburg, FRG (the OPYC model), and others. In the latter class are POM (the Princeton Ocean Model), SCRUM (the S-Coordinate Rutgers University Model), and GHERM (the GeoHydrodynamics and Environmental Research Model), to name the most widely used in this class. More recently, OGCM’s have been constructed which make use of more advanced, and less traditional, algorithmic approaches. Most importantly, models have been developed based upon Galerkin finite element schemes e.g., the triangular finite element code QUODDY (Dartmouth University) and the spectral finite element code SEOM (Rutgers). These differ most fundamentally in the numerical algorithms used to solve the equations of motion, and their use of unstructured (as opposed to structured) horizontal grids. General Description of Contents

The goals of this volume are, first, to present a 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, to explore comparative model behavior with reference to a set of quantifiable and inexpensive test problems; and lastly, to

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demonstrate how these principles and issues arise in a particular basin-scale application. Our focus is the modeling of the basin-scale to global ocean circulation, including wind-driven and thermohaline phenomena, on spatial scales of the Rossby deformation radius and greater. Smaller-scale processes (mesoscale eddies and rings, sub-mesoscale vortices, convective mixing, and turbulence; coastal, surface and bottom boundary layers) are not explicitly reviewed. It is assumed from the outset that such small-scale processes must be parameterized for inclusion of their effects on the larger-scale motions. The related concepts of approximation and parameterization are central themes throughout our exposition. As we emphasize, the equations of motion conventionally applied to “solve for” the behavior of the ocean have been obtained via a complex (though systematic) series of dynamical approximations, physical parameterizations, and numerical assumptions. Any or all of these approximations and parameterizations may be consequential to the quality of the resulting oceanic simulation. It is therefore important for new practictioners of oceanic general circulation modeling to be aware of sources of solution sensitivity and potential trouble. We provide many examples of each. Chapter 1 offers a brief introduction to the derivation of the oceanic equations of motion (the hydrostatic primitive equations) and various oftenused approximate systems. Beginning with the traditional equations for conservation of mass, momentum, mechanical energy and heat, we show how these equations are modified within a rotating, spherical coordinate system. These continuous equations have many conservation properties; conservation of angular momentum, vorticity, energy and enstrophy are discussed. Various approximations are necessary to arrive at the accepted equations of oceanic motion. We review the arguments for the traditional, Boussinesq, and hydrostatic approximations, and the assumption of incompressibility, and how they relate to conservation properties such as energy and angular momentum. Lastly, additional approximations yield furthersimplified systems including the beta-plane, quasigeostrophic and shallow water equations. Chapter 2 discusses why we cannot solve the oceanic equations of motion directly. Instead, we must find approximate solutions using discrete numerical solution procedures. Two levels of discretization are involved the approximation of functions and the approximation of equations; we review a variety of approaches to each. Solutions of the discretized equations

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of motion can differ, sometimes dramatically, from the solutions of the original continuous equations. Sources of approximation error, with illustrative examples drawn from the one-dimensional heat and wave equations, are given. Alternative approaches to time differencing (e.g., explicit-in-time, implicit-in-time and semi-implicit) are also reviewed. Additional numerical considerations arise when seeking solutions in two or more spatial dimensions (Chapter 3). Among these are the occurrence of tighter time-stepping stability restrictions, the need for fast solution procedures for elliptic boundary value problems, and the possibility of horizontally staggered gridding of the dependent variables. The latter is of particular interest in that different choices for the horizontal lattice have direct effects on numerical approximation errors and discrete conservation properties. As an example of these effects, the propagation characteristics of a variety of wave phenomena (inertial-gravity, planetary waves) are examined on several traditional staggered grids, showing the types of numerical approximation errors that can occur. Four well-studied ocean models of differing algorithmic design are described in detail in Chapter 4. Among these are examples utilizing alternate vertical coordinates (geopotential, isopycnal, and topography-following), horizontal discretizations (unstaggered, staggered grids), methods of approximation (finite difference, finite element), and approximation order (low-order, high-order). The semi-discrete equations of motion are given for each model, as well as a brief summary of model-specific design features. Chapter 5 describes why the “complete” equations of motion derived in Chapter 1 are not really complete. Because of omitted, though potentially important, interactions between resolved and unresolved scales of motion (the “closure problem”), we must specify parameterizations for these unresolved phenomena. Processes for which alternative parameterizations have been devised include vertical mixing at the surface and bottom oceanic boundaries, lateral transport and mixing by subgridscale eddies and turbulence, convective overturning, and topographic form stress. The origin and form of these parameterizations are reviewed. Simple two-dimensional test problems are introduced in Chapter 6 to demonstrate the range of behaviors which can be obtained with the four models of Chapter 4 even under idealized circumstances. The processoriented problems address a range of processes relevant to the large-scale ocean circulation including wave propagation and interaction (equatorial Rossby soliton), wind forcing (western boundary currents), effects of strat-

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ification (adjustment of a vertical density front), and the combined effects of steep topography and stratification (downslope flow, alongslope flow). Substantial sensitivity to several numerical issues is demonstrated, including choice of vertical coordinate, subgridscale parameterization, and spatial discretization. Chapter 7 examines the current state of the art in non-eddy-resolving modeling of the North Atlantic Ocean. After a brief review of simulation strategies and validation measures, we describe three recent multiinstitutional programs which have sought t o model the North Atlantic and to understand numerical and model-related dependencies. Taken together, these programs provide further illustration of the controlling influences of the numerical approximations and physical parameterizations employed in the model formulation. Nonetheless, model validation against known observational measures shows that, with care, numerical simulation of the North Atlantic Basin can be made with a considerable degree of skill. Finally, Chapter 8 speculates briefly on promising directions for ocean circulation modeling, in particular the prospects for novel new spatial approximation treatments.

Acknowledgements

The early chapters in this book are an abbreviated version of lecture notes developed over the past 20 years for graduate-level courses in ocean dynamics and modeling. The first author thanks the Woods Hole Oceanographic Institution, the Naval Postgraduate School, the Johns Hopkins University and Rutgers University for their support of this instructional development. The test problems described in Chapter 6 have benefitted from the encouragement and support of Terri Paluszkiewicz and the Pacific Northwest National Laboratory. The authors also acknowledge the Institute of Marine and Coastal Sciences of Rutgers University and the Alfred-WegenerInstitute for logistical and financial support during the completion of this monograph. Discussions with, and helpful comments by, several colleagues have significantly improved this volume. We are particularly grateful for the insightful suggestions made by Claus Boning, Eric Chassignet and Joachim Dengg. Lastly, we note with thanks the many technical contributions of Kate Hedstrom, Hernan Arango and Mohamed Iskandarani.

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Acknowledgements

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Chapter 1 THE CONTINUOUS EQUATIONS 1.1 Conservation of Mass and Momentum . . . . . . . . . . . . . . 1.2 Conservation of Energy and Heat . . . . . . . . . . . . . . . . . 1.3 The Effects of Rotation . . . . . . . . . . . . . . . . . . . . . . 1.4 The Equations in Spherical Coordinates . . . . . . . . . . . . . 1.5 Properties of the Unapproximated Equations . . . . . . . . . . 1.5.1 Conservation of angular momentum . . . . . . . . . . . 1.5.2 Ertel’s theorem . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Conservation of mechanical energy . . . . . . . . . . . . 1.6 The Hydrostatic Primitive Equations . . . . . . . . . . . . . . . 1.6.1 The Boussinesq approximation . . . . . . . . . . . . . . 1.6.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . 1.6.3 The hydrostatic approximation . . . . . . . . . . . . . . 1.7 Initial and Kinematic Boundary Conditions . . . . . . . . . . . 1.8 Approximate Systems . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 The beta-plane (Cartesian) equations . . . . . . . . . . 1.8.2 Quasigeostrophy . . . . . . . . . . . . . . . . . . . . . . 1.8.3 The shallow water equations . . . . . . . . . . . . . . .

1 1 6 9 13 15 15 16 18 19 20 21 21 25 26

Chapter 2 THE 1D HEAT AND WAVE EQUATIONS 2.1 Approximation of Functions . . . . . . . . . . . . . . . . . . . . 2.1.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . .

37 38 38

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27 30 33

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2.1.2 Piecewise linear interpolation . . . . . . . . . . . . . . . 2.1.3 Fourier approximation . . . . . . . . . . . . . . . . . . . 2.1.4 Polynomial approximations . . . . . . . . . . . . . . . . Approximation of Equations . . . . . . . . . . . . . . . . . . . . 2.2.1 Galerkin approximation . . . . . . . . . . . . . . . . . . 2.2.2 Least-squares and collocation . . . . . . . . . . . . . . . 2.2.3 Finite difference method . . . . . . . . . . . . . . . . . . Example: The One-dimensional Heat Equation . . . . . . . . . Convergence, Consistency and Stability . . . . . . . . . . . . . Time Differencing . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The wave equation . . . . . . . . . . . . . . . . . . . . . 2.5.2 The friction equation . . . . . . . . . . . . . . . . . . . The Advection Equation . . . . . . . . . . . . . . . . . . . . . . Higher-order Schemes for the Advection Equation . . . . . . . Sources of Approximation Error . . . . . . . . . . . . . . . . . 2.8.1 Phase error / damping error . . . . . . . . . . . . . . . 2.8.2 Dispersion error and production of false extrema . . . . 2.8.3 Time-splitting “error” . . . . . . . . . . . . . . . . . . . 2.8.4 Boundary condition errors . . . . . . . . . . . . . . . . . 2.8.5 Aliasing error/nonlinear instability . . . . . . . . . . . . 2.8.6 Conservation properties . . . . . . . . . . . . . . . . . . Choice of Difference Scheme . . . . . . . . . . . . . . . . . . . . Multiple Wave Processes . . . . . . . . . . . . . . . . . . . . . . Semi-implicit Time Differencing . . . . . . . . . . . . . . . . . . Fractional Step Methods . . . . . . . . . . . . . . . . . . . . . .

39 40 45 47 47 48 48 51 55 58 60 66 67 71 73 73 79 79 80 80 84 86 87 89 90

Chapter 3 CONSIDERATIONS IN TWO DIMENSIONS 3.1 Wave Propagation on Horizontally Staggered Grids . . . . . . . 3.1.1 Inertia-gravity waves . . . . . . . . . . . . . . . . . . . . 3.1.2 Planetary (Rossby) waves . . . . . . . . . . . . . . . . . 3.1.3 External (barotropic) waves . . . . . . . . . . . . . . . . 3.1.4 Non-equidistant grids, non-uniform resolution . . . . . . 3.1.5 Advection and nonlinearities (aliasing) . . . . . . . . . . 3.2 Time-stepping in Multiple Dimensions . . . . . . . . . . . . . . 3.3 Semi-implicit Shallow Water Equations . . . . . . . . . . . . . 3.4 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conservation of Energy and Enstrophy . . . . . . . . . . . . . . 3.6 Advection Schemes . . . . . . . . . . . . . . . . . . . . . . . . .

93 93 95 100 106 108 108 109 111 112 115 118

2.2

2.3 2.4 2.5

2.6 2.7 2.8

2.9 2.10 2.11 2.12

Contents

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Chapter 4 THREE-DIMENSIONAL OCEAN MODELS 121 4.1 GFDL Modular Ocean Model (MOM) . . . . . . . . . . . . . . 123

4.2

4.3

4.4

4.5

4.1.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 4.1.2 System of equations . . . . . . . . . . . . . . . . . . . . 4.1.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 4.1.4 Spatial discretization, grids and topography . . . . . . . 4.1.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 4.1.6 Time-stepping . . . . . . . . . . . . . . . . . . . . . . . 4.1.7 Additional features . . . . . . . . . . . . . . . . . . . . . 4.1.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . S-coordinate models (SPEM/SCRUM) . . . . . . . . . . . . . . 4.2.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 4.2.2 System of equations . . . . . . . . . . . . . . . . . . . . 4.2.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 4.2.4 Spatial discretization, grids and topography . . . . . . . 4.2.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 4.2.6 Temporal Discretization . . . . . . . . . . . . . . . . . . 4.2.7 Additional features . . . . . . . . . . . . . . . . . . . . . 4.2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . Miami Isopycnic Model (MICOM) . . . . . . . . . . . . . . . . 4.3.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 4.3.2 System of equations . . . . . . . . . . . . . . . . . . . . 4.3.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 4.3.4 Spatial discretization, grids and topography . . . . . . . 4.3.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 4.3.6 Temporal discretization . . . . . . . . . . . . . . . . . . 4.3.7 Additional features . . . . . . . . . . . . . . . . . . . . . 4.3.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . Spectral Element Ocean Model (SEOM) . . . . . . . . . . . . . 4.4.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 4.4.2 System of equations . . . . . . . . . . . . . . . . . . . . 4.4.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 4.4.4 Spatial discretization, grids and topography . . . . . . . 4.4.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 4.4.6 Temporal discretization . . . . . . . . . . . . . . . . . . 4.4.7 Additional features . . . . . . . . . . . . . . . . . . . . . 4.4.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . Model Applications . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 125 125 128 129 130 131 133 133 133 136 136 140 142 142 144 145 145 145 147 148 149 150 150 151 152 152 152 153 154 157 159 161 162 162

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Chapter 5 SUBGRIDSCALE PARAMETERIZATION 5.1 The Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Overview of Subgridscale Closures . . . . . . . . . . . . . . . . 5.3 First Order Closures . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Constant eddy coefficients . . . . . . . . . . . . . . . . . 5.4 Higher Order Closures . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Local closure schemes . . . . . . . . . . . . . . . . . . . 5.4.2 Non-local closure schemes . . . . . . . . . . . . . . . . . 5.5 Lateral Mixing Schemes . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Highly scale-selective schemes . . . . . . . . . . . . . . . 5.5.2 Prescribed spatially varying eddy coefficients . . . . . . 5.5.3 Adaptive eddy coefficients . . . . . . . . . . . . . . . . . 5.5.4 Rotated mixing tensors . . . . . . . . . . . . . . . . . . 5.5.5 Topographic stress parameterization . . . . . . . . . . . 5.5.6 Thickness diffusion . . . . . . . . . . . . . . . . . . . . . 5.6 Vertical Mixing Schemes . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The vertical structure in the ocean . . . . . . . . . . . . 5.6.2 Surface Ekman layer . . . . . . . . . . . . . . . . . . . . 5.6.3 Stability dependent mixing . . . . . . . . . . . . . . . . 5.6.4 Richardson number dependent mixing . . . . . . . . . . 5.6.5 Bulk mixed layer models . . . . . . . . . . . . . . . . . . 5.6.6 Bottom boundary layer parameterization . . . . . . . . 5.6.7 Convection . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Comments on Implicit Mixing . . . . . . . . . . . . . . . . . . .

163 164 167 170 170 173 175 175 176 177 178 181 182 183 186 189 189 190 192 193 193 196 198 200

Chapter 6 PROCESS-ORIENTED TEST PROBLEMS 6.1 Rossby Equatorial Soliton . . . . . . . . . . . . . . . . . . . . . 6.2 Effects of Grid Orientation on Western Boundary Currents . . 6.2.1 The free-slip solution . . . . . . . . . . . . . . . . . . . . 6.2.2 The no-slip solution . . . . . . . . . . . . . . . . . . . . 6.3 Gravitational Adjustment of a Density Front . . . . . . . . . . 6.4 Gravitational Adjustment Over a Slope . . . . . . . . . . . . . 6.5 Steady Along-slope Flow at a Shelf Break . . . . . . . . . . . . 6.6 Other Test Problems . . . . . . . . . . . . . . . . . . . . . . . .

203 204 208 213 216 221 227 234 240

Chapter 7 SIMULATION OF THE NORTH ATLANTIC 243 7.1 Model Configuration . . . . . . . . . . . . . . . . . . . . . . . . 243 7.1.1 Topography and coastline . . . . . . . . . . . . . . . . . 244

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7.3

7.4 7.5 7.6 7.7 7.8 7.9

7.1.2 Horizontal grid structure . . . . . . . . . . . . . . . . . 7.1.3 Initialization . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Spin-up . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenological Overview and Evaluation Measures . . . . . 7.2.1 Western boundary currents . . . . . . . . . . . . . . . . 7.2.2 Quasi-zonal cross-basin flows . . . . . . . . . . . . . . . 7.2.3 Eastern recirculation and ventilation . . . . . . . . . . . 7.2.4 Surface mixed layer . . . . . . . . . . . . . . . . . . . . 7.2.5 Outflows and Overflows . . . . . . . . . . . . . . . . . . 7.2.6 Meridional overturning and heat transport . . . . . . . 7.2.7 Water masses . . . . . . . . . . . . . . . . . . . . . . . . 7.2.8 Mesoscale eddy variability . . . . . . . . . . . . . . . . . 7.2.9 Sea surface height from a rigid lid model . . . . . . . . North Atlantic Modeling Projects . . . . . . . . . . . . . . . . . 7.3.1 CME . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 DYNAMO . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 DAMEE . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity to Surface Forcing . . . . . . . . . . . . . . . . . . . Sensitivity to Resolution . . . . . . . . . . . . . . . . . . . . . . Effects of Vertical Coordinates . . . . . . . . . . . . . . . . . . Effects of Artificial Boundaries . . . . . . . . . . . . . . . . . . Dependence on Subgridscale Parameterizations . . . . . . . . . Dependence on Advection Schemes . . . . . . . . . . . . . . . .

Chapter 8 Appendix A

THE FINAL FRONTIER

xix

244 245 246 248 248 250 252 252 253 253 254 255 256 257 259 259 260 261 262 263 267 276 277 281 283

Equations of Motion in Spherical Coordinates

287

Appendix B

Equation of State for Sea Water

289

Appendix C

List of Symbols

291

Bibliography

295

Index

312

Chapter 1

THE CONTINUOUS EQUATIONS

The equations which describe the oceanic general circulation are modified versions of the Navier-Stokes equations, long used in classical fluid mechanics. The essential differences are the inclusion of the effects of rotation, an important dynamical ingredient on the rotating Earth, and certain approximations appropriate for a thin layer of stratified fluid on a sphere. In addition, the ocean differs from other fluid media in the existence of multiple thermodynamic tracers (temperature and salinity), and a highly nonlinear equation of state. Nonetheless, much of the following derivation of the continuous equations follows that in other areas of computational fluid dynamics. For those interested in a more thorough treatment of the subject of geophysical fluid dynamics, the following condensed discussion may be supplemented with the excellent texts by Cushman-Roisin (1994) and Pedlosky (1987).

1.1

Conservation of Mass and Momentum

We begin by deriving the equations for conservation of mass and momentum in an inertial reference frame. These equations, suitably modified for the earth’s rotation and supplemented with equations governing the evolution of thermodynamic tracers, are the building blocks for the equations of oceanic motion. In the following, we adopt an Eulerian point of view in which time rates of change are considered at a fixed point (or volume) in space; an analogous derivation following fluid particles can also be performed (see, e.g., Milne-Thomson, 1968). Referring to Fig. 1.1, consider the changes in time between a system

THE CON TIN U 0 US EQ UATIONS

2

M

=v

t=O Fig. 1.1

t>O The control volume at t = 0 and a short time later.

having constant mass (designated M ) and a system of constant volume ( V ) . Let BV be the property of interest (mass, momentum or tracer) within the control volume,

where b is the amount of the property per unit mass, and p is the density of the fluid. Further, let B; and BXt represent the inventories of property B within the control volume at times 0 and At (some small time later). Since the volumes M and V coincide at time t , we have

At time At, we have

or, by subtraction of Eq. (l.l),

Bg

-By

= Bxt

-Br

+ B,V,, - BL .

3


The time rate of change over the interval At is, therefore,

B%

-BY

-B -

X ~- B,V

At which, in the limit of vanishing At, becomes At

BL, - B: At

+

dBM - dBV dt at

+-aBIut at

,

(1.2)

.

Now, the second term on the right-hand side of Eq. (1.2) may be written

=

at

lv

pb(t7. Z)ds

,

where fi is the unit normal to the bounding surface 6V. Since V is fixed in time,

aBV so that Eq. (1.2) becomes

dBM -dt

= d",

[IM

]

bpdV =

-(pb)dV :t

+

lv

pb(t7-Z)ds

.

Using the divergence theorem to replace the surface integral,

we obtain

=L

dBM

dt

[ T + V . ( p b . 3 ]dV .

(1.3)

Since these relations must hold for an arbitrary volume, we may take the limit dV + 0. In the fixed mass system (p6V = constant):

dBM

and Eq. (1.3) becomes d a -(b)phV = -(pb)GV dt at

+v

*

(pbV76V

T H E CONTIN V O US EQVA TIONS

4

or

a

d P z ( b ) = -(Pb) at

+ v . (Pbv') .

(1.4)

The statement of conservation of mass is obtained by setting b = 1:

a -at (p)+V.(pV')=O

.

(1.5)

This can be written in the alternate forms d -(p) v'. v p p v . v' = 0 at or

+

+

-dP+ p v . v ' = o ,

dt

where

-d p_ - ap

+v'.vp

dt dt is the total, or material, derivative. Conservation of momentum ( b = v') is obtained from Eq. (1.4) in a similar manner, with the result

d dv' = -(pq v . (pv"> = (forces acting on V) , dt dt where the final equality is a consequence of Newton's second law of motion. Here, v'v' is the dyadic product which is in component form (Pielke, 1984)

C

+

p-

= pv'(v-q+pv'.vv'+(v'*vp)v' . Using the mass conservation equation to simplify [;(P)+v'.vP+Pv.v' \

-! [E -vv',1 v'+p

-+v'

1

=O

the statement of conservation of momentum becomes dv' = (forces) .

pz C


5

The forces acting on the control volume include those which act throughout the body of the fluid (body forces) and those which act on the fluid surface (surface forces). Of the former, we consider at the moment only the gravitational force

fi = -

s,

(pgVz)dV ,

where -Vz = -i is the local vertical (downward) direction. Two important classes of surface forces exist. Pressure forces act normal to the surface (-pn'ds), so that +

Fp = where I is the unit tensor. Viscous forces

where T n' gives the traction on the face whose normal is n'. The stress tensor T is most easily understood in its equivalent matrix form:

where rij is the component of stress acting in the direction i on the plane whose normal lies in the direction j . For a Newtonian fluid: 2

+

T = - - p ( V . 8)I p[Vv'+ (Vv')T] , 3 where p is the molecular viscosity (see, e.g., Batchelor, 1967). However, for reasons that will be discussed in Chapter 5 , this form for the stress tensor is typically replaced with other viscous parameterizations. The forces acting on the fluid as a consequence of these viscous terms can be thought of as arising from the matrix vector product (see Fig. 1.2):

I;(=)!( Note that

rij

722

:;.):()I; Tzy

72%

is symmetric, ie., r i j = r j i .


6

z

/

TX%

P

X

/

/

Y

Fig. 1.2 Components of the viscous stress tensor.

After substitution of these external forces, the momentum equation assumes its traditional form: dv' p-=-Vp+V.T-pgVz dt

1.2

.

(1.7)

Conservation of Energy and Heat

The first law of thermodynamics states that the rate of change of energy in a system is equal to the net rate of addition of heat minus the net rate of work done by the system. Now, total energy is the sum of kinetic, potential and internal energies; that is,

7

Conservation of Energy and Heat

Here v 2 = v'. v' and e is the internal energy per unit mass. From Eq. (1.4), with b = gz el

f+ +

= w. If there is a heat flux

since

>

.

(1.37)

As a consequence of this partition, and the relative magnitudes of po and b, it is also appropriate to replace occurrences of p(x, y, z , t ) with the reference value po everywhere except when gravitational forces or spatial and temporal variations in density are essential (the Boussinesq approximation). For example, under the Boussinesq approximation, the horizontal pressure gradient terms are approximated as 1

-vp P

1

zz

Po

1 (1 + P I P O )

VP

The Hydrostatic Primitive Equations

N

21

1 - v p , Po

where an approximation error of O(ij/po),assumed to be small, has been made.

1.6.2

Incompmssibility

Expanding the statement of mass conservation, the use of Eq. (1.37) yields

p o ( V . i ? ) + i j ( V - d ) dij +-=O dt

.

Clearly, the second term in the above equation is smaller than the first and may be ignored. If the characteristic length and time scales of the perturbation density 6 are comparable to those of the velocity components, then term three is also smaller than the first term by a factor of ( i j / p o ) . This non-quantitative scaling suggests that a good approximation for the ocean would be

v*v'=o.

(1.38)

A more thorough derivation of the statement of incompressibility requires that both advective velocities and wave phase speeds be much less than the speed of sound in seawater, and that the density scale height ( p / l a p / a z J ) be much less than the fluid depth (Batchelor, 1967). These conditions are typically well satisfied for large-scale flow in the ocean, though not always in the atmosphere. 1.6.3

The hydrostatic approximation

Consider oceanic circulation in the upper 1,000 meters of the water column, for which the following typical magnitudes are appropriate:

--

(ax,a4J) p

-

a = 0.65 x lo7 m ar (u,v) U = 0.1 - 1.0 m/sec 20 UH/a at a/U p , = 1000 x 104N/m2 p p o = 1000 kg/m3 . O(1) H = 1000 m

-

r

--

-

THE CONTIN U O U S EQUATIONS

22

Table 1.1 Scaling of the vertical momentum equation. du,

2Ru cos q5

UzH

252u

dt

7~ ( I O -- ~ ~

O ( I O - ~-

o(10-~ -

pe PO H

9

o(10) o(10)

Accordingly, the terms in the vertical momentum equation in spherical coordinates have the representative magnitudes shown in Table 1.1. Thus, the primary balance for the assumed scales of motion is between the gravitational force and the vertical pressure gradient, the hydrostatic approximation: (1.39) A complete justification of the hydrostatic approximation requires a demonstration that perturbations to the mean hydrostatic state are themselves hydrostatic, and that scales of motion typical of mesoscale circulation features are also hydrostatic. See, for example, Holton (1992). With the hydrostatic and Boussinesq assumptions, the momentum equations in spherical coordinates are du

- -dt

dv -

-

dt 0

+

u v t a n 4 uw 2Rv sin 4 - 2Rw cos 4 1 aP r r porcos4bx u 2 t a n 4 YW 1 aP 2 R u s i n 4 - -- , r r P O T a4 _ _1 _aP_ _ SP Po a z Po .

'

Unfortunately, these approximated momentum equations have a problem: they correspond to the unphysical mechanical energy equation d (u2

+ v2) = -pgw 2

-8.Vp-

POW

-(u

r

2

+v2)

-2Rp,uwcos~ .

Extra terms have arisen in the mechanical energy budget because of the deletion of terms from the vertical momentum equation. These extra terms

The Hydrostatic Primitive Equations

23

correspond to spurious sources and sinks of mechanical energy. Hence we must reject the approximate momentum equations in their present form. To recover an appropriate energy conservation statement, we must make the further assumption that w = 0 in the u and v momentum equations (apart from Note that the removal of these extra terms is justified by the thinness of the ocean1, for which it is expected that

g).

w/(u,v)

-

HIL

the vector form of the inviscid, hydrostatic primitive equations becomes

dfi -+fix2 = -VP dt dP - -- - S P / P o

aZ

(1.41)

(1.42)

'Elimination of the extra Coriolis terms is often referred to as the traditional appmximation.

THE CONTIN UO US EQUATIONS

24

aW

V*ii+= o (1.43) az -d p _ - 0 , (1.44) dt where IT = (u, v) is the horizontal velocity vector. The term f = 252 sin $J is often referred t o as the planetary vorticity. In component form, the hydrostatic primitive equations on the rotating earth take the following form: du uv tan $J - +fv---

dt

dv

TE -

-u2 tan $J

1

aP

T E COS $J

ax

-f"---j-&1 aP

dt TE dP - = -PS/Po 8.2 dP = o dt v.77 = 0 ,

(1.45)

where

(1.46) In most practical situations, the dependence of density on temperature and salinity must be explicitly represented. In such circumstances, the density transport Eq. (1.44) is replaced by the conservation statements for T and S, as well as an equation of state:

dT -

dt dS dt

= o

(1.47)

= o

( 1.48)

and P = p(T,S,P)

.

(1.49)

In principle, the latter is a complex nonlinear function of potential temperature, salinity and pressure ( p = p o p ) . Jackett and McDougall (1995) discuss the UNESCO equation of state and some of the issues associated with its implementation. In some settings, the equation of state may be linearized about some central values of temperature and salinity

Initial and Kinematic Boundary Conditions

25

+

[p = a(To- T ) p(S - So)];however, this is only appropriate over rather narrow ranges of T , S and p .

1.7 Initial and Kinematic Boundary Conditions The hydrostatic primitive equations are a coupled set of nonlinear partial differential equations in the seven dependent variables (u, u , w, T , S, p, p ) expressed as functions of three spatial coordinates and time. To fully specify their solution, a sufficient number of initial and boundary conditions to close the equations must be specified. In the absence of explicit dissipative terms, which we discuss below in Chapter 5, the inviscid equations require only kinematic boundary Conditions at the sea surface, bottom and sidewalls of the region of interest. These kinematic conditions require that fluid parcels on each surface remain on that surface. Two alternative treatments of the kinematic boundary condition at the sea surface are in common use in ocean modeling. The former treats the sea surface as a rigid plate corresponding to the fixed geopotential level z = 0. Differentiation with respect to time yields the upper kinematic boundary condition for a rigid l i d

w=o

(1.50)

z = o .

Note that this is an approximation to the true time-varying position of the free sea surface z = q ( x , y , t ) , for which the surface kinematic boundary condition is (1.51) The removal of the free sea surface via the approximate upper boundary condition (1.50) eliminates certain wave phenomena whose restoring force depends on the motion of the sea surface. The most notable of these are surface gravity waves, whose removal offers significant computational advantage. We return to this point below. At the sea bottom, z = -H(z,y), the appropriate kinematic boundary condition is

dH w=-u--u-

ax

dH

ay

z=-H(x,y)

.

(1.52)

26

THE CONTINUO US EQUATIONS

Lastly, assuming the sidewalls of our domain to be solid boundaries, we require (d*fi)=O

.

(1.53)

Alternate treatments are required if the lateral boundaries allow exchange of mass with neighboring regions. Instances in which this is necessary include mass influx due to riverine input and/or exchanges with adjacent marginal seas, and sub-global-scale applications in which open boundary conditions are introduced to represent the effects of the excluded portions of the global ocean. A discussion of the formulation of open boundary conditions is beyond the intent of this work; we refer the reader to Orlanski (1976), Chapman (1985) and Stevens (1990, 1991) for examples of their implementation. An example of the influences of open boundary conditions in a realistic basin-scale setting is given below in Chapter 7. Lastly, initial conditions [e.g., Z(z,y, z , t = O)] are required for all prognostic variables. For the hydrostatic primitive equations, the prognostic variables include u,w , T , S, and 77 (if there is a free sea surface). The remaining dependent variables may be obtained from the diagnostic equahydrostatic balance ( P ) ,and the equation of state tions for continuity (w), (PI *

1.8

Approximate Systems

The hydrostatic primitive equations (HPE) are thought to incorporate the minimum number of approximations consistent with long-term simulation of the meso-to-global-scale ocean circulation. Further levels of approximation t o the equations of motion are nonetheless possible. Each, though more restrictive in a dynamical sense, accords simplicity of solution and/or interpretation beyond that offered by the HPE. Three such approximate systems are briefly reviewed next: the Cartesian (beta-plane) system, the quasigeostrophic equations, and the shallow water equations. Each of these has been, and continues to be, used in large-scale ocean modeling for regional and/or dynamically idealized process modeling. As we discuss in Chapter 6, they also lend themselves to the construction of process-oriented test problems for numerical ocean models. Other sets of approximate systems, obtained by higher-order scaling or systematic filtering of wave processes, have been advanced in the liter-

Appmzimate Systems

27

ature, though none has yet achieved a wide acceptance for use in global ocem modeling. Among these latter systems are the large-scale geostrophic equations (LSG, Colin de Verdikre, 1988, 1989; Maier-Reimer et al., 1993), the balance equations (Lorenz, 1960; Gent and McWilliams, 1983a,1983b; McWilliams et al., 1990; Allen, 1991), and various filtered systems (Browning et al., 1990).

1.8.1

The beta-plane (Cartesian) equations

Beginning with the Boussinesq, adiabatic, hydrostatic primitive equations, and the representative scales

the equations (1.41)-( 1.45) become, in nondimensional form,

du -1 LIP Ro- - Rod(uv tan 4) - v sin 4 = -dt cos+ dX dv dP RoRo6(u2tan 4) + u sin 4 = -dt 84 &J dW --a u + - - d ( v t a n 4 ) + = o c o s + a x a+ dZ -db = o dt dP - = - b ,

+

aZ

where (1.54) (1.55) (1.56)


28

are the aspect ratio, the Rossby number and the scaled buoyancy, respectively. Now, introducing a local planar approximation, centered at some central latitude (4,):

4

=

+ o + G

x

=

sx,

the spherical metric terms are, t o leading order in 6 (6 relative to the wind direction on the northern (southern) hemisphere at the surface and decay exponentially downward. The "depth of frictional influence" (also called the "Ekman depth") is defined as the depth at which the wind influence is reduced to e-ll and for A, constant is (5.16) Viscosity values of A , = to m2s-l give reasonable mid-latitude values (10-50 meters) for the Ekman layer depth. Note that the Ekman

192

SUBGRIDSCALE PARAMETERIZATION

layer is not identical with the mixed layer because it relates solely to the depth of wind influence. Although of similar magnitude, the Ekman depth is not necessarily identical with the mixed layer depth as defined by the vertical variation in density. The choice of vertical viscosity is closely related to the vertical grid spacing near the boundaries. It is often assumed that it is sufficient to have at least one grid point within the boundary layer, whose thickness is in turn set by the vertical viscosity. So rather than choosing a vertical viscosity in accordance with an a priori estimate of Ekman layer thickness, the value of vertical viscosity is often chosen such that the Ekman depth is equal to the local vertical grid spacing. Fortunately, internal details of the boundary layer are of little importance for large-scale dynamics because the wind-driven ocean circulation is forced by the vertical velocity at the base of the Ekman layer, which is the result of the divergence of the vertically integrated Ekman transport in the Ekman layer. Therefore, it is the integrated properties of the Ekman layer, rather than specific details of their internal vertical structure, which matter, and these integral properties are reasonably well reproduced at low vertical resolution. The wind stress is

where pa is the density of air, CD the drag coefficient and W10 the wind speed 10 meters above the ocean. A typical value for CD is 0.003. Note that with sea ice present, the stress at the ice-ocean interface is ?I

= TilVi - VI(Vi

- v) ,

where the relative motion of water v and ice vi has to be considered. 5.6.3

Stability dependent mixing

In the vertical, stratification plays a major role in determining the strength of mixing, and a simple extension of the vertically constant mixing coefficient has been widely used. This parameterization scheme, originally suggested by Gargett (1984, 1986), proposes that vertical diffusivity be inversely proportional to the local Brunt-Vaisala frequency

Vertical Mixing Schemes

193

Under this assumption, mixing is largest in the deep ocean. For numerical reasons, the above form is usually modified by the introduction of a minimum value for vertical diffusivity. Also, an upper limit may need to be chosen to avoid problems with an overly severe vertical diffusive stability criterion. With these modifications, the scheme becomes

[See also Section 5.6.7 on convection.] 6.6.4

Richadson number dependent mixing

A vertical diffusion scheme originally developed for the tropical ocean is the Richardson-number-dependent mixing of Pacanowski and Philander (1981). Defining the local Richardson number

-Lee Po

Ri =

02

(2)2+(&2

'

a general form for the vertical viscous and diffusive coefficients is M -

AW - (1+ clRi)"

+A$

and

where the coefficients c1 and n are 5 and 2, respectively. The maximum vertical mixing coefficient is set to A$ = 10-2m2s-1, and the background viscosity and diffusivity are A: = 10-4m2s-' and ATb = 10-5m2s-1. Although originally intended for use in tropical oceans, the scheme has been used successfully in mid- and high-latitude oceans as well. In this adaptive and nonlinear parameterization, both the stratification and vertical current shear are taken into account. For small Ri, the coefficients are large, i e . , strong shear and weak stratification lead to enhanced vertical mixing. 5.6.5

Bulk mixed layer models

In recent years, the surface boundary layer has been the focus of a large number of modeling studies. The time-dependence of mixed layer variables


194

(depth, temperature, salinity and passive tracers) is dominated by processes on a wide range of time scales, and many different approaches have been used to simulate the observed diurnal and seasonal cycles with numerical models. See Large et al. (1994) for a review. Many of these methods have been validated against observational data from one or more locations (for instance the ocean weather ships), with varying degrees of success. Often, the surface mixed layer is portrayed as being vertically homogeneous or quasi-homogeneous, with a rapid variation in properties below. This has led to the development and use of so-called bulk mixed layer models, which predict the mixed layer depth and the (homogeneous) concentrations of thermodynamic and passive tracers. Most of the bulk models stem from the original work of Kraus and Turner (1967). They are all local, vertically integrated (zero-dimensional) models, based on the turbulent kinetic energy equation with various closure assumptions and simplifications. The equations for Kraus-Turner (KT) bulk mixed layer models are based on the vertically integrated turbulent kinetic energy (TKE) equation as given by Niiler and Kraus (1977)

where w e is the entrainment velocity at the base of the mixed layer, 6b the buoyancy jump across the interface between mixed layer and interior, u* = fiis the friction velocity at the sea surface for a given wind stress r , Bo is the surface buoyancy flux (positive upward), I is the radiative flux; and c1, c2, c3 are prescribed coefficients of proportionality. The KrausTurner model (5.18) describes diagnostically how the mixed layer depth changes due to the following processes:

0

0

the loss of TKE due to the entrainment of denser water from below the mixed layer, the gain of TKE due to turbulence generated by the shear of the mean flow at the base of the mixed layer, the gain in TKE from the wind work, the change of TKE due t o surface buoyancy fluxes, and the change in TKE due to solar heating within the water column by the penetrating component of solar radiation.

In this formulation, frictional losses of turbulent kinetic energy have to be taken into account by an appropriate choice of the coefficients s, m and

Vertical Mizing Schemes

195

n. In the Kraus-Turner formalism, dissipative effects are assumed to be proportional to the different production terms themselves, which requires the specification of several coefficients which are associated with the wind forcing, the buoyancy flux and the velocity shear terms. For the TKE generated by wind forcing, all models are based on the assumption that friction causes an exponential decrease of the available TKE with the depth of the mixed layer (Gaspar, 1988). Similarly, the kinetic energy generated by the buoyancy flux is also subjected to an exponentially decaying loss. Note that this approach might be problematic in case of poorly resolved surface boundary layers in the ocean model. Different approximations to the TKE equation have frequently been made. Often, the velocity shear term is neglected. According to Niiler and Kraus (1977) this is an acceptable simplification in situations where the mixed layer depth exceeds the penetration depth of inertial oscillations. Another common choice is to neglect the insulation term, which is equivalent to the assumption that all solar radiation is absorbed directly at the ocean surface, where it is immediately converted to heat. Occasionally,even the buoyancy flux is neglected. However, this does not mean that effects of buoyancy forcing are completely absent. Surface heating, for example, will still stabilize the water column by changing the stratification through the model tracer equation. Surface cooling can also result in convective instability, which is treated separately by convective adjustment (see below). The coupling of such a model to an ocean general circulation model is far from straightforward. Both physical and numerical issues arise, in particular having to do with the vertical resolution . The implementationspecific choices made will therefore quite often lead to slightly different behaviors of the mixed layer model. For example, although not necessarily intended to do so, the mixed layer model essentially enforces static stability near the surface. This in turn might interfere with the ocean model’s convection parameterization, depending on the order in which these steps are performed in the ocean model. Note also that while tracers are completely homogenized within the bulk mixed layer, momentum is usually not affected. A vertical velocity shear is still permitted. It is not clear whether this is inconsistent with the fundamental design of the model, or is actually dynamically realistic. For these reasons, the bulk mixed layer approach is perhaps best suited for inclusion in layered models (e.g., Section 4.3), while discrete level models are more easily coupled with a turbulence closure scheme.

196

5.6.6


Bottom boundary layer pammeterization

The interface between (moving) water and the solid earth creates a boundary layer structure similar to the planetary boundary layer of the atmosphere. Bottom boundary layers (BBLs) are regions of enhanced levels of turbulence, generated by mechanical shear and geothermal fluxes, although the contribution of the latter is small outside of certain geologically active zones and usually neglected. They contain an inverse Ekman spiral and, closer to the bottom, a logarithmic vertical profile of the flow component tangential to the boundary. In this layer the momentum, vorticity and energy supplied to the ocean by surface forcing are removed from the systernlo. The boundary conditions a t the sea bed are no normal flow,

and, to good approximation, no tangential flow: VIII-h

=0

.

Except in isolated locations ( e.g., hydrothermal vents), there is no normal flux of tracers across the water-earth interface. The appropriate boundary condition (e.g., for density) is dp

dh

da: (a,)

dp

+

ay

(s> dh

+

2

=0

'

The first terms are typically ignored, because topographic gradients are small. They may, however, be important a t steep topography. From the ocean modeler's perspective, the main effect of physical processes near the lower boundary of the ocean has long been considered to be a sink of momentum, vorticity and energy for the ocean circulation. However, the small-scale structure of the BBL may be complex, especially over sloping topography, and BBL dynamics is important for deep water spreading and hence for the thermohaline circulation in the ocean. Generally speaking, the BBL is usually several meters to tens of meters thick; thus, BBLs are rarely resolved in today's ocean models. As a consequence, the true (no-slip) boundary condition at the bottom is relaxed to a 'ONote, however, that in coarse resolution ocean models, a significant fraction of energy is dissipated at lateral boundaries.

Vertical Mizing Schemes

197

partial slip condition. The most simple parameterization considers a linear bottom stress

'?I-*

-.

= 76 = r b v b

,

which enters the prognostic equation for the bottom velocity through the vertical viscosity terms. Here, the bottom resistance coeficient r g has units of [ms-l]. A typical value is 0.0002 ms-'. An alternative is to assume that the bottom stress acts as a body force on the lowermost model layer (or level)

From dimensional considerations, it may be understood in this case that r; is an inverse damping time scale; time scales lying between several weeks to months are typically specified. Differences between these formulations arise if the vertical resolution varies with depth and/or location. Occasionally, a deflection angle is taken into account, ie., 7;

T :

= =

+ vbsincr,) (-ubsincu, + 'ubcos(Y,)

Tb ('LLbCOSCXv Tb

.

The veering angle a, is typically set to a value between 0" and 30". Note that for a 90" deflection, there is no energy loss. Higher-resolution studies, and those in shallow water, require a more realistic treatment of the bottom stress term than that given by a scaleindependent linear (Rayleigh) damping. Measurements have shown that the bottom stress is more closely approximated by a quadratic stress law which, in analogy to the surface stress in Eq. (5.17), can be written as

where v* is the so-called friction velocity. The quadratic bottom stress coefficient should, in principle, represent the effect of variable bottom roughness, but is usually set to a constant. Values for the non-dimensional coefficient of between 0.001 and 0.003 are often used. It should be noted that on certain staggered grids ("C","E") averaging is necessary to estimate the nonlinear bottom drag. Identical flow fields will therefore experience a different bottom stress on different grids.

198


Attempts have also been made to include the effects of wave motion in the BBL, especially in models that do not explicitly include tides. Assuming the bottom flow to be a linear superposition of the mean advection and a periodic (e.g., tidal) component v = V + IvT(eiWt

,

the temporal average over one (tidal) period leads to an approximated bottom friction term of T+ b = - c , v b d w

(5.19)

where VT is the m s tidal velocity. Note that Eq. (5.19) is not a parameterization for tidal stress (see Section 5.5.5). Its effect is to increase the drag coefficient for weak mean flows, where the wave generated turbulence cannot be assumed to be proportional to the macroscopic flow. It should be emphasized that the large-scale bottom slope does not explicitly enter these parameterizations. There are, however, important processes related to the BBL over a sloping bottom. One of these is down-slope flow of dense bottom water (see Section 6.4), the other rectified along-slope flow (Section 6.5). One-dimensional BBL models have been used by Davies (1987) and Keen and Glenn (1994)- Specialized two-dimensional models of the bottom boundary layer (akin to the surface boundary layer models) do exist and have been used to study the dynamics of bottom gravity plumes. For example, Jungclaus and Backhaus (1994) use a reduced gravity, vertically integrated primitive equation model in a two-layer system, where only the denser layer is active. Their model has a movable upper boundary and includes entrainment of ambient water from above. Only recently (e.g., Killworth and Edwards, 1999; Song and Chao, 1999) have such BBL models been fully coupled to an ocean general circulation model. 5.6.7

Convection

Static instability of the water column can occur as the result of surface thermohaline forcing especially in high latitudes. Direct heat loss to the atmosphere, mainly in the Greenland-Iceland-Norwegian Seas, the Labrador Sea and the Weddell and Ross Seas, but also in the Mediterranean and other marginal seas, is the main reason for unstable stratification. Brine released during the growth of sea ice can also destabilize the underlying fluid. In addition, static instability can arise outside the polar oceans by

Vertical Mazing Schemes

199

advective processes in upwelling regions along the ocean boundaries, or, on even smaller scales, in fronts. Such statically unstable situations do not persist for a long time; they are removed by vigorous and small-scale vertical motion, called “convective plumes”, which transport the denser water downward, leading to a vertical homogenization of the water column. The compensating upwelling motion does not occur localized but as a large-scale and weaker upward flow. The penetration depth of the convection depends on the initial stratification of the fluid ($reconditioning”), and the strength and time scale of the cooling/evaporation (Schott et al., 1994; Send and Marshall, 1995). For more details and an overview on recent findings the reader is referred to Send and K k e (1998). Non-hydrostatic models, in which the total time derivative of the vertical velocity is retained, have shown that highly nonlinear dynamics governs three-dimensional rotating convection in geophysical fluids (Jones and Marshall, 1993; Sander et al., 1995). The hydrostatic approximation eliminates such processes in primitive equation models; the HPE have no direct means of producing vertical accelerations in response to occurrences of heavy water over light. Thus static instability has to be removed by other means, since persistent unstable stratification in a numerical primitive equation model would cause internal wave growth, often leading to catastrophic model behavior. Two alternate parameterizations of convection processes are typically used in numerical primitive equation ocean models. One of these, referred to as “convectiveadjustment” eliminates static instability by an instantaneous vertical mixing of tracers in statically unstable water columns, until all occurrences of unstable stratification are removed. Unfortunately, static stability is not guaranteed for simple schemes that compare and homogenize adjacent grid boxes only, even if the water column is cycled several times (Ftahmstorf, 1993). A complete convection scheme has to take into account the mixing product and determines the maximum depth of convection in one sweep’l . The other strategy for the removal of static instability involves an increase of the vertical diffusion coefficient in order to diffuse away any un“Note that this parameterization for convection is not very efficient on high performance vector computers (as it requires a large number of conditional checks and repeated reevaluations of the nonlinear equation of state) and may lead to load imbalances on parallel machines.

200


stable stratification by increased vertical mixing. This strategy fits nicely within the framework of both low- and higher-order turbulence closure schemes. Advantages of this approach include improved vectorization, and a temporally and spatially more gradual transition from statically unstable to stable conditions which tends to excite many fewer internal waves than does instantaneous convective adjustment. A possible disadvantage is that static instability may persist over an unphysically long time. Typical values for the vertical diffusion coefficient in case of static instability are 1 to 100 m2s-l. These large values of vertical diffusivity may pose a severe constraint on the time-step, if the vertical diffusion term is treated explicitly. Therefore, this method is often applied in combination with an implicit vertical diffusion scheme (Section 2.11). See also Yin and Sarachik (1991). Both methods (convective adjustment and increased vertical mixing) are found to have similar qualitative behavior, although no rigorous test and systematic comparison has been performed. More recently, the so-called penetrative plume parameterization has been developed (Paluszkiewicz and Romea, 1997), which is based on an embedded parcel approach.

5.7

Comments on Implicit Mixing

In addition to physically motivated subgridscale parameterizations described in the previous subsections, numerical models may need some “smoothing” of the prognostic variables to avoid excessive build-up of gridscale noise and possible computational instability (see Chapter 2). This makes it generally difficult to discriminate between numerically and physically motivated mixing, especially because the former may exceed the latter under some circumstances. Also, most numerical schemes come with some amount of implicit smoothing. These implicit effects are hidden in the numerical algorithms: for example, the averaging operators needed for models with staggered grids have a net smoothing effect on the prognostic fields. The same is true for some time-stepping schemes. Simple comparisons of explicit smoothing coefficients of different models may thus be misleading. Finally, it should be noted that the hydrodynamic equations establish a close connection between lateral and vertical processes. Therefore, mixing in any direction will also have an effect on the structure of the fields in the other directions. For example, the lateral mixing of density is equiv-

Comments on Implicit Mizing

201

alent to vertical diffusion of vorticity, as can be seen from the linearized quasigeostrophic equations (Section 1.8.2):

a st"2$

a+ aw = o +P z -f = AV2p .

The foregoing equations may be combined to form a single equation for the quasigeostrophic potential vorticity [see Eq. (1.61)]:

where the operator on the right-hand side represents a vertical mixing of relative vorticity. It is also worth noting that mixing of momentum and tracers is not entirely independent due to the dynamical constraints of geostrophy. Increased mixing of momentum can replace some of the effects of tracer mixing (and vice versa).

Chapter 6

PROCESS-ORIENTED TEST PROBLEMS

The discussion in the preceding chapters raises the issue of which type of model to use in a given application. In the best of all possible worlds, it would not matter. That is, all models would produce the same qualitative physical behavior at fixed resolution, and each would approach the same (true) solution as temporal and spatial resolution were improved. Regrettably, the situation is not so fortunate. Numerical models of differing algorithmic formulation are often found to produce contradictory qualitative behavior, or to have questionable convergence properties. Nor is a single choice of model or numerical algorithm always superior to another. Depending on choice of application, and the standard of merit (cost, accuracy, robustness, most smooth solution, etc.), any model may “outperform” another. Given these realities, it is important that we understand, and that we be able to quantify, the behavior and properties of alternate models and methods. Ultimately, ocean circulation models such as those described in Chapter 4 must be intercompared in fully realistic settings. Examples of this approach are described below. Nonetheless, these realistic, threedimensional intercomparisons are difficult to conduct in a clean fashion (that is, under identical parametric circumstances for all models), are usually expensive to perform, and are difficult to analyze and to interpret. Another, more efficient, means of contrasting model behavior is to devise an inexpensive set of process-oriented test problems with which alternative numerical formulations can be assessed against stated standards of merit. This approach has proven useful in the atmospheric sciences (e.g., Williamson et al., 1992), but has not yet been adopted to any appreciable 203

204


extent in ocean modeling. We describe next several types of process-oriented test problems. For some of these problems analytic or semi-analytic solutions exist, which form the basis for model comparison. Other problems, though analytically intractable, are included to explore the range of model responses in important idealized limits for which known solutions do not exist. The majority of these test problems have been applied to the four models from Chapter 4. All model solutions have been obtained by the authors under identical parametric circumstances (i. e., the models are configured to solve the same initial boundary value problem), and are presumably indicative of model behavior in the general problem class under study. (These tests are also useful as a preliminary measure of model correctness, in the case of newly coded models.)

6.1

Rossby Equatorial Soliton

This test problem considers the propagation of a Rossby soliton on an equatorial @-plane,for which an asymptotic solution exists to the inviscid, nonlinear shallow water equations. In principle, the soliton should propagate westwards at fixed phase speed, without change of shape. Since the uniform propagation and shape preservation of the soliton are achieved through a delicate balance between linear wave dynamics and nonlinearity, this is a good context in which to look for erroneous wave dispersion and/or numerical damping. A schematic diagram of the domain and expected properties of the solution is shown in Fig. 6.1. Perturbation solutions to the Rossby soliton problem are available to both zeroeth (Boyd, 1980) and first order (Boyd, 1985). The geometry is a long equatorial basin, bounded on all four sides by rigid vertical walls and at the bottom by a flat, level surface. The equations governing the fluid motion are the inviscid shallow water equations on an equatorial @-plane (see Section 1.8.3). Following Boyd, we nondimensionalize with H = 40 cm, L = 295 km, T = 1.71 days and U = LIT = 1.981 m/s. In the resulting non-dimensional system, input parameter values are: lateral basin size (-24. 5 2 5 24.; -8.0 5 y 5 8.0), resting depth ( H = 1.) and beta (p = 1.). The asymptotic solution is constructed by adding the lowest and first

Roesby Equatorial Soliton

205

I

t>O

Fig. 6.1 Schematic diagram of the Rossby soliton test problem.

order solutions: 21

= u(o) + u(1)

2)

= $4

+#)

h = h(0) + h(') where u is the zonal velocity, v is the meridional velocity and h is the surface height anomaly. The superscripts refer to the order of the asymptoticseries. The zero-order solution is

while the first-order solution is given by

P R 0 CESS- ORIENTED TEST PROBLEMS

206

where

3

= -2Btanh(B()q

.

The only free parameter in the problem is the amplitude of the soliton B which should be kept smaller than 0.6 in order not to affect the accuracy of the asymptotic expansion. The value used below is B = 0.5. U ' , V', and h' are given by the infinite Hermite series,

with the coefficients un, vn and h, listed in Table 6.1. The Hermite polynomials can be computed with the recurrence formula (Abramowitz and Stegun, 1992): HO(5)

= 1, H l ( 5 ) = 25, H n ( 2 ) = ZZHn-1(2) - 2(n - l)Hn-Z(~),n2 2 .

Using the zeroeth order solution as initial conditions, we have computed the evolution of the Rossby soliton for a total of 40 time units. The solution for the lowest symmetric mode wave ( n = l) is used. During this interval, the soliton propagates westwards across several of its characteristic widths. (Comparable results are obtained for the more complete, first-order asymptotic solution.) Our interest in this test problem is to investigate spurious dispersion effects, and how they relate to the choice of horizontal resolution and the order of the approximation used in the numerical solution. To illustrate these effects, we have obtained numerical solutions from one of the secondorder, finite difference models (SCRUM) and from the higher-order, finite element model (SEOM). [Both models offer an easy option to solve only the depth-integrated (shallow water) equations.] The problem as posed is inviscid. Although most models will occasionally require finite values of viscosity for numerical reasons, both SCRUM and SEOM were successfully run on this non-turbulent problem with zero explicit viscosity.

207

Rossby Equatorial Soliton

Table 6.1 Hermite series coefficients for the Rossby soliton test problem.

n

0 1 2 3 4

5 6 7 8 9 10 11 12 13 14

15 16 17 18 19 20 21 22 23 24 25 26

Table 6.1 Un

1.789276 0 0.1164146 0 -0.3266961e-3 0 -0.1274022e-2 0 0.4762876e-4 0 -0.1120652e-5 0 0.1996333e-7 0 -0.2891698e-9 0 0.3543594e-11 0 -0.3770130e-13 0 0.3547600e-15 0 -0.29941 13e-17 0 0.2291658e-19 0 -0.1 178252e-21

Vn

0 0 0 -0.6697824e-1 0 -0.2266569e-2 0 0.9228703e-4 0 -0.1954691e-5 0 0.2925271e-7 0 -0.3332983e-9 0 0.2916586e-11 0 -0.1824357e-13 0 0.4920950e-16 0 0.6302640e-18 0 -0.1289167e-19 0 0.1471189e-21 0

rln

-3.071430 0 -0.3508384e-1 0 -0.1861060e-1 0 -0.2496364e-3 0 0.1639537e-4 0 -0.4410177e-6 0 0.8354759e-11 0 -0.1254222e-9 0 0.1573519e-11 0 -0.1702300e-13 0 0.162 1976e-15 0 -0.1382304e-17 0 0.1066277e-19 0 -0.1178252e-21

208


A standard resolution of 97 by 33 grid points (AS = Ay = 0.5) was chosen for both models. (In SEOM, this was obtained using a uniform mesh of 12 by 4 ninth-order elements.) A conservative value of 0.02 was used for the time-step in all cases. Figures 6.2a-d, produced using SEOM, show the typical behavior of the soliton. Shortly after initiation, the Rossby soliton sheds an eastward-propagating equatorial Kelvin wave as it adjusts from its initial state and begins to propagate westward. During this initial adjustment, the soliton loses approximately 7% of its initial amplitude. Note that the initial state of the model is inexact both because of finite numerical resolution and because the analytical solution is itself approximate. Both SEOM and SCRUM undergo roughly equivalent initial adjustments, leading us to conclude that the initial imbalance is primarily due to the asymptotic nature of the analytical solution. Following the initial adjustment, the Rossby soliton propagates westward with reduced loss of amplitude. By 40 time units, the SEOM result (Fig. 6.2d) shows only a further loss of 2%; a weak trail of dispersing Kelvin waves can be seen immediately behind the soliton. The lower-order methods used in SCRUM yield somewhat greater dispersion and amplitude loss. At the standard resolution, SCRUM loses a total of approximately 13% of its initial surface signature by time 40 (Fig. 6.2e). These dispersive effects increase roughly quadratically when the resolution is degraded by a factor of two (Fig. 6.2f). Interestingly, the phase speeds at which the solitons move westward appear to be largely unaffected by choices of resolution and approximation method, showing soliton propagation to be a robust feature independent of solution technique.

6.2

Effects of Grid Orientation on Western Boundary Currents

The wind-driven, vertically integrated mass transport in the ocean is one of the major attributes of the large-scale ocean circulation. Sverdrup (1947) first recognized the dominant balance between latitudinal dependence of the Coriolis parameter and the curl of the wind stress which sets the interior pattern of the wind-driven gyres. Shortly thereafter, linear viscous theories were proposed t o explain the occurrence of regions of enhanced currents on the western margins of the oceans (western boundary currents) and the dissipative removal of vorticity input to the oceans by the atmo-

Effwts of Grid Orientation on Western Boundary Cumnts

209

0

0

0

0

0

I 0

--

0

--

0

0

Fig. 6.2 Rossby soliton solutions: ( a d ) Surface displacement for the Flossby soliton problem obtained using SEOM at t=0,8,24,40 time units. The corresponding maximum values are 16.7, 15.6, 15.5, and 15.3 non-dimensional units. (e) SCRUM result at 40 time units, maximum value 14.5. (f) SCRUM result at half the horizontal resolution, maximum value 11.5.


210

sphere. In Stommel’s (1948) model, the westward intensification and the existence of strong western boundary currents are associated with the retarding effects of linear bottom stress. Munk (1950) extended the linear theory by including lateral viscous terms of harmonic form. The basis for these prior investigations is the barotropic (vertically averaged) vorticity equation

-V2$ d

at

+ J($, 0”) + p-a$ ax =

ii.

V x 7‘+ AhV41C, ,

(6.1)

where .JI is the horizontal mass transport streamfunction, and the horizontal velocity components are given by

a$ --

= I

v

=

dY a$ ax .

The steady, inviscid and linearized version of the barotropic vorticity equation (a balance between the third and fourth terms in Eq. (6.1), that is, vorticity input due to the wind and north/south motion of water parcels) is called the Svev-drup balance. In primitive equation models, the dynamically equivalent system are the shallow water equations (SWE, see Section 1.8.3)

dhu -+at

dhuu

dhvu - fhv ax + ay dhv dhuv dhvv -+-+at ax ay + f h u

d q dhu -+-+at ax

dhv

ay

=

-gh-

a77 + rZ + AhV2hU

dX

=

-gh-

87)

dY

+ + AhV2hV T ’

= 0 .

The SWE are used here. This test problem investigates the accuracy of different numerical realizations of this linear homogeneous wind-driven flow in a square basin with sinusoidal wind forcing. The dependence of the solution on increasingly fine grid spacing is quantified for different horizontal grid arrangements (“B” and “C”, as well as unstructured grids), for rotated grids relative t o the model domain, and for both free-slip (no stress) and no-slip boundary conditions. Of particular interest are the consequences of grid orientations which lie at an angle to the western boundary (and hence necessitate a “staircase” representation of the boundary).

Eflects of Grid Orientation on Western Boundary Currents

211

The dimensions of the model domain are taken to be L, = L, = 1000 km and H = 5 km. Although much smaller than actual ocean basin dimensions, this choice is sufficient to illustrate the effects of boundary orientation and convergence in this test problem. A purely zonal wind stress with cosine (single gyre) meridional structure T

= -T~cos(~F~/L,)

is applied, and all advection terms are explicitly set to zero. This basin is placed on a mid-latitude @-plane. Model parameters are

Dissipation of vorticity is accomplished by a harmonic lateral viscosity with a coefficient Ah = 540 rn2s-l, giving a boundary layer width LM,,,,~ = (Ah/@)lI3 of 30 km. The configuration of, and solution to, this problem are illustrated in Fig. 6.3. Convergence is tested for three different horizontal resolutions (50, 25 and 12.5 km; all on the order of the boundary layer width), for three different orientations of the numerical grid relative to the zonal direction (0", 17" and 45") and for two different lateral boundary conditions (free-slip and no-slip). (Note that the boundary condition enters the problem only via lateral viscous terms.) The rotated basin results illustrate the effect of step-like representation of a curved coastline in ocean models that use masking to represent irregular lateral boundaries. The integration of these models to steady state is less straightforward than it may seem from the relatively simple configuration. This is because the spin-up phase is dominated by large-amplitude eigen-oscillations of flat bottom ocean basins, known as basin modes, with periods of

where m and n are integers describing the horizontal mode numbers. The gravest mode in a rigid lid model (m= 1, n = 0, RD = 00) has a period of

SJV3'780tTd J S 3 J C73JN31210-SS33021d

ZIZ

213

Effects of Grid Orientation o n Western Boundary Cumnts

= 22.85 days. For the prescribed amount of viscosity, the damping time is about 60 years.' scale of these basin-scale oscillations Tdiss = Several strategies can be used to speed-up the convergence process to the steady state. The obvious method is to initialize the model with the analytical solution, if known. This method, however, is not very successful if the numerical solution differs globally from the analytical initial field. Another option is to reduce the amplitude of the basin modes by increasing the forcing slowly over the first few years. Although this may be helpful in some cases, it does not always lead to shorter integration periods. The most economical integration here was obtained when the model was started with bottom friction as an additional energy sink; during the course of the integration the coefficient is slowly reduced to zero. This led to typical integration periods of 15 to 35 years, depending on resolution and boundary condition.

2'10

6.2.1

The free-slip solution

McCalpin (1995)derives an analytical expression for the volume transport streamfunction to the Stommel and Munk problems. The 540 m 2 c 1 viscosity solution for free-slip boundary conditions is

= [1.194501.106

-1.191364.108 - e(x15) +1.994869. e(x2z) -3.137464 - lo3 ' cos(X3z) . e(-A42) +1.768437. lo3 . sin(A3z) . e ( - x r z ) ] sin(7r 10-6y) m3s-' +

with A1

A2

= 2.630050* 10-gm-' = 3.352986 10-5m-1

-

(6.3)

'Although we do not pursue it here, these basin modes represent a good inviscid numerical test problem in their own right. Initializing a linearized model from the known modal solution, spurious damping and dispersion can be monitored.

.!

, ,

I

,

Effects of Grid Orientation on Western Boundary Currents A3

= 2.869657 - lO-‘m-’

Ad

= 1.676624-10-5m-1

215

.

The free-slip case has a maximum of I,/$’“ = 19.14 Sv. Table 6.2 summarizes the results. Of the finite difference models, the “C” grid performs generally better than the “B” grid, notably for the unrotated case. The 12.5 km resolution experiment reproduces the maximum to within 0.3% (“C”grid) and 5% (“B” grid). For increasing rotation angle, the solution deviates further from the analytical solution; the error for 45” rotation can be up to 15%, even for the highest resolution of 12.5 km. Both finite difference grids are very similar at 45”. The ms-error depends approximately quadratically on the resolution. We also note that pointwise extrema (like the maximum streamfunction here) are not guaranteed to converge uniformly for the marginal resolution we have used. For a boundary layer thickness of 30 km, all three resolutions are too coarse, but they represent common choices for basin-scale numerical models (see also Chapter 7). The inferior accuracy of the “B” grid can be explained by the placement of the velocity points near the boundary: in case of the “B” grid the closest non-zero velocity point is one grid distance from the boundary, while the nearest point for tangential velocities is only one half gridpoint on the “C” grid. The spatial structure of the streamfunction difference for the free-slip case is shown in Fig. 6.4. The finite difference solutions are plotted for the highest of our resolutions (12.5 km), and all show a O(lO)%maximum error except the unrotated “C” grid run. In general, the rotated free-slip runs experience an increasingly large lateral stress at the boundaries. This is confirmed by Fig. 6.5, where the dependence of the maximum streamfunction on the rotation angle is plotted. Both finite difference grids show a decrease of the maximum transport with rotation of the grid; the “B” grid is systematically in error due to the smaller effective domain size. Large effects are found even for small rotation angles: a few steps in the coastline can change the dynamical balance of the boundary current entirely. Table 6.2 shows that coarser models are not affected as much by the rotation of the grid. In contrast (Table 6.2), the spectral finite element solution shows fasterthan-algebraic convergence (e.g., a factor of ten improvement with a doubling of resolution) and little or no dependence on the degree of rotation between the inner and edge elements. An 11th order, 25 km average resolu-

PROCESS-ORZENTED TEST PROBLEMS

216 1.10

I

I

I

5

10

I

1

I

I

I

I

I

1.05 1 .oo

0.95

0.90 0.85 0.80

0.75

0.70

0

15

20

25

30

35

40

45

Fig. 6.5 Maximum transport (relative to the unrotated value) for the free-slip experiments at 12.5 km resolution as a function of grid rotation. Both the “B”-grid (open circles) and the “C”-grid (asterisks) model results are fitted to an exponential with 8.75 degrees e-folding scale. The FE model results (crosses) are nearly independent of the rotation angle.

tion grid is significantly better than both finite difference grids with double the number of horizontal gridpoints. 6.2.2

The no-slip solution

Unfortunately, the analytical solution with no-slip boundaries is more difficult to obtain. [The separation of variables method employed by McCalpin (1995) to determine the free-slip solution does not apply to the no-slip problem.] Instead, we have chosen a high-resolution model simulation as the reference solution2 . The streamfunction pattern exhibits the well-known westward intensification with a narrow recirculation cell. The maximum 2Briefly, the SEOM model on a uniform, 11-th order elemental grid of 61 by 61 points is used to obtain the reference solution. Given the polynomial approximation used in SEOM, this reference solution can be interpolated to any interior point to provide reference data for other models. A stand-alone computer program has been written t o compute the required error measures.

Effects of Grid Orientation on Western Boundary Currents

217

Table 6.2 Results from the horizontal convergence experiments with free-slip boundary is the ratio of computed to conditions. The relative transport amplitude analytic solutions at the location of the maximum computed value.

I

---grid

angle [“I

21

“FE”

45

F’ree-sliD

# of points

41 41 41

&ms

0.8978

0.9999 1.0005 1.0001

[SV] comments

0.8550 0.5591 0.4675 2.2511 1.5125 1.1441 2.8184 1.4061 0.8837 0.8943 0.3434 0.1049 1.0709 0.8016 0.8950 1.2406 1.2724 0.9777 0.4569 0.1358 0.0636 0.0066 0.0021 0.0307 0.0023 0.0588 0.0028

6th order 11th order 6th order 9th order 11th order 9th order 11th order 9th order 11th order

218


transport is $,”” = 15.95 Sverdrups, and the maximum boundary current velocity 5.2 cm s-l. The solution is symmetric about a zonal line in the center of the basin. Table 6.3 summarizes the no-slip results. Generally, the conclusions on the overall performance of “B” and “C” grids for free-slip boundary conditions also hold here: the “C” grid slightly over-estimates the amplitude of the streamfunction, while the “B” grid under-estimates it. The “C” grid results are also generally closer to the true solution. It is, however, interesting t o note that there is almost no dependence on the rotation angle; the no-slip solutions are less seriously degraded by a step-like coastline. The spectral finite element model results are again significantly better than any of the finite difference solutions. With the exception of low-order spectral expansion runs with only 6 polynomials, the results are at least an order of magnitude more accurate than finite difference runs with twice the number of degrees of freedom. Figure 6.6 shows the error patterns for several SEOM runs. The under-resolved experiment exhibits large error and a strong north-south asymmetry. All other runs are extremely accurate. The error patterns, however, do reflect the elemental structure, especially for the rotated grids (see Fig. 6.6e and f). A systematically reduced boundary current transport is expected from all basin-scale simulations that use masking on a regular and quasi-regular grid. Considering the large effects on the linear solutions, the nonlinear dynamics must also be greatly affected by the step-like representation of the coastline. This has been investigated recently by Adcroft and Marshall (1998). It is unclear whether the model behavior can be improved by more accurate formulations of the boundary conditions (see, e.g., Verron and Blayo, 1996).

Effmts of Grid Orientation on Western Boundary Currents

219

Table 6.3 Results from the horizontal convergence experiments with no-slip boundary conditions. The relative transport amplitude ($Jtmaz/$JNS)is the ratio of computed to analytic solutions at the location of the maximum computed value.

[“I -grid

“B”

angle

# of points

0

17

-‘IB”

“C”

45

0

17

45

0

17 45

No-slip

21 41 81 21 41 81 21 41 81

21 41 81 21 41 81 21 41 81 21 41 81 41 21 41 41 41 41 41

$rm*

0.9672 0.9720 0.7663 0.8763 0.9282 0.5793 0.8166 0.9477 1.0607 1.0118 1.0023 1.0402 0.9698 0.9917 1.0113 0.9844 1.0056 0.8588 1.0510 0.9998 1.0002 1.0077 1.0003 0.9996 0.9998 1.0007 1.0000

[Svl

0.8527 0.4104 0.4465 1.6017 0.8744 0.6739 2.2106 1.2974 0.5729 0.4790 0.1390 0.0374 0.7398 0.4739 0.2303 0.6995 0.4186 0.0860 0.9245 0.1990 0.0125 0.0040 0.0550 0.0018 0.0152 0.0050 0.0412 0.0055

comments

6th order 6th order 6th order 9th order 11th order 11th order 9th order 11th order 9th order 11th order


220

+

c

Fig. 6.6 Streamfunction difference plots for the western boundary current test problem for SEOM with no-slip boundary conditions; (a-c) 6th order truncation and 4x4, 8 x 8 and 16x16 elements, respectiveIy; (d-f) 11th order truncation and 4x4 elements and Oo, 1 7 O and 45O rotation, respectively. Contour intervals are: (a) 1 Sv.; (b) 0.16 Sv.; (c) 0.015 Sv.; (d) 0.0016Sv.; (e) 0.006 Sv.; and (f) 0.006 Sv.

Gravitational Adjustment of a Density l h n t t>O

t=O

‘

22 1

h

9 V

Fig. 6.7 Schematic diagram of the gravitational adjustment test problem (p2

6.3

> pl).

Gravitational Adjustment of a Density Front

The uniform advection of a passive scalar, initially distributed in some specified shape, is a traditional test of behavior for advection algorithms (see Rood, 1987). A related, but more demanding, problem involves the gravitational adjustment of a two-density-layer system, initially separated by a vertical wall (Wang, 1984). The system is illustrated in Fig. 6.7. At time zero, the vertical wall dividing the two immiscible fluids is removed; thereafter, the fluid layers adjust at the internal gravity wave phase speed to form a stably stratified, two-layer system. During and after the adjustment, sharp density fronts divide the two layers both horizontally and vertically. Density fronts of this type are often observed in estuaries; Geyer and Farmer (1989) discuss this in the context of F’raser estuary. This problem would presumably be best handled with an isopycnal model (such as MICOM), which automatically respects the layered structure of the solution. The large majority of ocean circulation models, however, use non-isopycnal coordinates, and therefore must carefully consider how to deal with sharp density fronts. As discussed in Section 2.8, loworder schemes are typically either diffusive or dispersive in nature. In the former instance, positive-definiteness of tracer fields can be preserved, but often at the expense of excessive smoothing of sharp transitions. The firstorder upwind scheme is the traditional example of this trade-off. Dispersive schemes (of which centered differencing is an example) are much less dif€usive; however, they are not guaranteed to be either monotonic or positive definite. (That is, they can produce artificial tracer extrema and “wig-

222

PR 0 CESS-0R IENTED TEST PROBLEMS

gles".) In order to maintain smoothness, large values of explicit diffusivity are therefore often necessary. The consequences of these effects are illustrated in Table 6.3 and in Figs. 6.8 to 6.10. The latter show the resulting density structure after a simulated time of 10 hours for three different models (MICOM, SCRUM and SEOM) and several advection schemes. The model domain is 64 km long and 20 meters deep, and spatial resolution is fixed at 0.5 km in the horizontal and 1 meter in the vertical. The initial density contrast across the vertical front is 5 kg m-3. Table 6.3 summarizes the properties of the various solutions at hour 10. The forms of the discrete advective operator [centered, upwind and Smolarkiewicz] are noted in the table, as are the values of explicit (harmonic) viscosity and diffusivity used. As tests of model performance, Table 6.3 also tabulates the resulting minimum and maximum density values, and the estimated normalized adjustment phase speed. (The latter should equal 0.0, 5.0 and 1.0, respectively.) Since it uses density as the independent vertical coordinate, the MICOM results are characterized by exact preservation of the minimum and maximum values of the density field, an obvious virtue in this problem. Nonetheless, the phase speed exhibited by the adjusting front is extremely sensitive to the applied subgridscale parameterization. The most surprising and unusual aspect is that for small values of X the phase speed of the front is actually overestimated. Also, higher vertical resolution does deteriorate the results, by increasing the noise and changing the phase speed (Fig. 6.8). A centered-in-space treatment of advection produces unacceptably noisy evolution of the density front unless it is accompanied by elevated levels of lateral smoothing (Fig. 6.9a,b,e). Reasonable reproduction of the adjustment speed of the front is obtained for a viscosity value of 50 m2s-'; however, spurious tracer extrema remain large even at this value of viscosity. The upwind scheme (Fig. 6.9d) produces somewhat less smooth, more slowly propagating fronts; however, the maximum and minimum values of tracer concentration are precisely preserved. Lastly, the more elaborate iterative scheme of Smolarkiewicz and Grabowski (1990) produces reasonable adjustment speeds at reduced viscosity values (10 m2s-'); however, severe oscillations (though no spurious extrema) are produced behind the advancing front.

223

Gmvitational Adjustment of a Density h n t

Table 6.4 Minima, maxima and phase velocity relative to the analytical solution for the gravitational adjustment experiment after 10 hours of integration. The phase velocities are determined by computing the horizontal distance between the surface and bottom location of the 2.5 o-units density contour, divided by the time since the release of the front. X is the constant used for the deformation dependent viscosity.

model

MOM MOM MOM MICOM MICOM MICOM MICOM MICOM MICOM SCRUM SCRUM SCRUM SCRUM SCRUM SCRUM SEOM SEOM

advection scheme / resolution centered centered upstream 2-layer 2-layer 2-layer 2-layer 6-layer 6-layer centered centered Smol. upstream centered Smol. high-order high-order

Table 6.4 diffusivity / viscosity 50150 20120

-/5/X=0.03 5/X=0.06 5/X=0.12 50/X=0.00 5/X=0.03 50/X=0.00 100/100 50/50 -/50 -/50 lop0 -/lo 100/100 50150

-2.090 -5.542 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.156 -1.678 -0.013 -0.001 -5.169 -0.017 -0.00044 -0.540

phase speed 7.090 10.543 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.138 6.603 5.000 5.000 9.744 5.000 5.000 5.601

=

0.962 0.991 0.907 1.192 1.037 0.799 0.757 0.953 0.715 0.913 0.927 0.928 0.906 0.955 0.896 0.912 0.932

224

PROCESS- 0RIENTED TEST PROBLEMS

4.5

3.5

2.5

1.5

0.5

Fig. 6.8 MICOM results for the gravitational adjustment test problem. The density front after 10 hours of integration: (a) 2-layer, Ah = 5 m2s-',X=0.03; (b) %layer, Ah = 5 m2s-',X=0.06; (c) 2-layer, Ah = 5 m2s-',A=O.12; (d) 2-layer, Ah = 5 m2s-',X=0.00; (e) 6-layer, Ah = 5 m 2 s - ' , k 0 . 0 3 ; and (f) 6-layer, Ah = 5 m 2 s - ' , k 0 . 0 0 .

Gmuitational Adjustment of a Density Front

225

Fig. 6.9 SCRUM results for the gravitational adjustment test problem: (a) centered advection, A, = Ah = 100 m2s-'; (b) centered advection, A , = Ah = 50 m2s-'; (c) upstream advection of tracers, A, = 50 m2s-1, Ah = 0; (d) Smolarkiewicz advection of tracers, A, = 50 rn2s-', Ah = 0; (e) centered advection, Av = Ah = 10 rn2K1;and (f) Smolarkiewica advection of tracers, A , = 10 m2s-l, Ah = 0.

226

P R 0 CESS- ORIEN T E D T E S T PROBLEMS

Fig. 6.10 Comparison of results from SEOM and SCRUM for the gravitational adjustment test problem: (a) SEOM, A , = Ah = 100 m2s-'; (b) SCRUM, A , = Ah = 100 m 2 K 1 ; (c) SEOM, A , = Ah = 50 m's-'; and (d) SCRUM, A , = Ah = 50 m's-'.

Gravitational Adjustment Over a Slope

227

Finally, the high-order finite element solutions (Fig. 6.10) are quite similar in propagation speed and smoothness to the results obtained with centered differences using SCRUM. Both models require sufficient explicit viscosity and diffusivity (in the range of 50 to 100 mks units) to produce smoothly well behaved solutions. Interestingly, and (to us) unexpectedly, the high-order method produces smaller spurious extrema than does the low-order model with centered differencing. Although this seems consistent with the overall higher accuracy of the spectral element formalism, spectral-based schemes are known to be particularly susceptible to over- and under-shooting. This test shows, however, that once a level of “adequate” resolution is reached, the properties of the high-order solution (including over- and under-shooting) are better than those from low-order centered schemes.

6.4

Gravitational Adjustment Over a Slope

The combined effects of steep bathymetry and strong stratification pose a particular challenge for the models reviewed in Chapter 4. In particular, each choice of vertical coordinate must deal with a potentially troublesome numerical approximation issue. For z-coordinate models, the source of potential trouble lies in the discretization of the topography which, without special treatment, is approximated as a series of discontinuous steps. Terrain-followingcoordinates, by their very nature, do not suffer this problem; however, they introduce the possibility of spurious pressure gradient effects, as mentioned in Section 4.2. Lastly, models formulated in isopycnic coordinates need to deal with the numerical issues associated with vanishing layer thicknesses where and when isopycnic layers intersect topographic features. Notwithstanding the inherent numerical difficulties in their simulation, a variety of oceanic processes combining the effects of stratification and topography are important on both regional and basin-wide scales. Examples include physical processes occurring at seamounts [e.g., Taylor caps (Chapman and Haidvogel, 1992), lee waves (Chapman and Haidvogel, 1993), and the resonant generation of seamount-trapped waves (Haidvogel et al., 1993; Beckmann and Haidvogel, 1997)], wind- and buoyancy-driven circulation on continental shelves [e.g., outflows (Jiang and Garwood, 1996), shelfbreak fronts, and upwelling], and downslope flow accompanying deep water

PROCESS- ORIEN TED TEST PROBLEMS

228

110

Fig. 6.11

t>O

Schematic diagram of the downslope flow test problem (pz

> pl).

production by marginal seas (Price and Baringer, 1994). The latter are particularly consequential for the water mass properties of the global ocean in general, and the North Atlantic in particular. For instance, outflow of dense water across the Denmark Strait, and subsequent downslope flow and mixing, are a main ingredient in the production of North Atlantic Deep Water (Dickson and Brown, 1994). Unfortunately, it is well known that the representation of these processes in existing coarseresolution large-scale ocean circulation models has not been a great success. Whether this is a reflection of inherent numerical limitations of the numerical models themselves, or of the subgridscale closures being used, is not known. To explore the numerical issues involved with downslope flow, we have developed a simple two-dimensional (z-2) test problem that investigates the flow of dense water down a steep topographic slope in the absence of rotation. A schematic diagram of the test problem is shown in Fig. 6.11. With the single (important!) exception of the underlying bathymetry, the downslope flow test problem is identical in concept to the gravitational adjustment test. At the initial time, a vertical density front is situated near a steep topographic slope. Upon release, the dense water will slump, and form a narrow ribbon of dense water which moves rapidly down the slope. In the absence of rotation and an ambient stratification in the deep basin, the downslope flow will reach the bottom, where it should eventually form a stagnant abyssal layer. (Variations on this test problem involving the addition of rotation and ambient stratification are more dynamically complex. Although we do not report on this here, in these more elaborate

Gravitational Adjustment Over

Q

Slope

229

configurations the production of geostrophically balanced shelf break fronts and internal solitary gravity waves can be seen.) As far as we know, there is no analytical solution to this test problem. Even though it is therefore difficult to quantify the results, it is revealing to seek solutions to this problem for several reasons. First, because of the steep topography involved, it is expected that models having differing vertical coordinate treatments will produce differing answers indicative of their respective idiosyncrasies. Second, this is not an easy test problem on which to produce a stable numerical solution; it is therefore a good context in which to assess model robustness. Lastly, the role of the choice of subgridscale closure on the character of the descending plume is not clear. An important issue is the extent to which the downslope adjustment process will be sensitive to lateral smoothing closures. The model domain is two-dimensional, with a horizontal dimension of 0 2 x 5 200 km. The topography used is the steep tanh-profile:

where Hmin = 200 m Hm,, = 4000 m L, = 10 km zo = 100 km

.

The resulting water depth increases from a minimum of 200 meters to an abyssal depth of 4000 meters in a distance of about 20 km. The maximum topographic slope is just under 20 percent. At the initial time, a vertical front with a five sigma-unit density contrast is located at x = 60 km, and the velocity field is at rest. Initialized in this way, each of the four models from Chapter 4 has been advanced in time for 10 hours, a time sufficient in most model runs for the plume to reach the base of the topographic slope3. Horizontal resolution 3Do a quick back-of-the-envelope calculation. You will find that the average rate of horizontal movement is approximately 3 m/sec! In actuality, instantaneous horizontal speeds were much greater in some cases, and pointwise vertical velocities also exceeded 1 m/sec. At these speeds, it is likely that non-hydrostaticeffects would become important.

230

PROCESS- ORIENTED TEST PRO E L EMS

was prescribed as 1 km in all model runs. (In the case of SEOM, 40 sixthorder elements were used.) Figure 6.12 shows a time history of the downslope flow at intervals of 2.5 hours obtained using SEOM. For this simulation, 21 points (5 fifth-order elements) were used in the vertical. The elemental partition in the vertical was chosen using a simple analytical stretching to put greatest resolution at the bottom. In order to obtain stable simulations, it was found empirically that SEOM needed an along-topography diffusivity coefficient of 1000 rn2/see. This may be interpreted as the amount of harmonic diffusivity necessary to produce a smoothly resolved plume as it descends and steepens under the influence of nonlinear advection. The results in Figure 6.12 were obtained with a geopotentially oriented viscosity of 1000 m2/sec. With these subgridscale assumptions, SEOM shows a rapidly descending plume of dense water which nearly reaches the base of the topographic slope by hour ten. A more easily understood manner of presenting the results from the four models is in the form of a Hovmoller diagram in which the value of density along the bottom (0 5 x 5 200 km) is plotted a t successive times (time increasing upwards). Examples of the resulting Hovmoller diagrams are shown in Fig. 6.13 for MOM and MICOM. The value of this manner of presentation is that the rate of downslope propagation, and the strength of the density contrast at the nose of the plume, are immediately and clearly identifiable. Figure 6.13 displays the phase diagrams for MOM and MICOM. The zcoordinate results show a rather slow down-slope advancement of the front, with almost constant phase speed. During this time, the density contrast is reduced to 20% for centered and 10%for upstream advection. The isopycnic model produces a faster descent, and no dilution of the densest water mass. Also it produces a plume head that is thicker than the tail. Multi-layer configurations of MICOM produce even faster downslope movement, but are also quite noisy. Lastly, we show an astonishing sensitivity of these results to subgridscale parameterization. Figure 6.14 presents four Hovmoller diagrams summarizing results obtained from the SEOM and SCRUM models. The first diagram is taken from the SEOM simulation whose time history was shown above (Fig. 6.12). With along-terrain mixing of density, but geopotentially oriented viscosity, the SEOM results are similar to those from MOM

Gravitational Adjuetment Over a Slope

231

rnox = 5.0, rnln = 0.0

rnax = 5.4, rnln = 0.007

rnax = 5.1, mln = -0.01

max = 5.1, rnln = -0.08

. . .

0

im

2030

im

200

Fig. 6.12 Downslope flow at 2.5 hour intervals obtained with the SEOM model. The maximum and minimum density anomaly values are indicated.

PROCESS- 0RIENTED T E S T P R 0BL EMS

232

10 0

8 7

s6 a 5

5

- 4 3 2 1

(4

0 0

20

40

60

100 120 140 160 160 200

80

x

(W

0

20

40

60

80

100 120 140 160 180

0

x (km)

Fig. 6.13 Hovmoller diagrams for MOM and MICOM in the down-slope flow problem: (a) MOM, centered-in-space advection, 500 m2s-l explicit geopotential diffusivity and viscosity; (b) MOM, upstream advection for both momentum and tracers; (c) MICOM, 2 layer; (d) MICOM, 6-layer. For both MICOM runs a diffusivity of 200 m 2 K 1 was used in combination with a X value of 0.2. For MICOM, the diagnostic quantity shown is the thickness of the densest layer. The dotted line indicates the front (defined as exceeding 1 meter in thickness).

Gravitational Adjustment Over o Slope

"0

20

40

60

80

233

100 I20 140 160 180 200

Fig. 6.14 Hovmoller diagrams for the downslope flow problem obtained with SCRUM and SEOM: (a) SEOM with along-terrain mixing of tracers (1000 m2/sec) and geopotential mixing of momentum (1000 m2/sec); (b) SEOM with along-terrain mixing of tracers (1000 rn2/sec) and along-terrain mixing of momentum (1000 m2/sec); (c) SCRUM with along-sigma mixing of momentum and tracers (1000 m2/sec) and centered differencing of advection; and (d) SCRUM with along-sigma mixing of momentum and tracers (1000 m2/sec) and Smolarkiewicz advection of density.

234


and MICOM. In particular, the advancing plume descends to about the base of the topographic slope (Fig. 6.14a), albeit retaining by that time a more substantial density contrast than MOM. If, however, the subgridscale viscous operator is aligned with the along-terrain direction (thereby smoothing normal to the nose of the descending plume) the speed of descent is enhanced by nearly a factor of two (Fig. 6.14b). Simple estimates of the possible range of influence of the along-terrain viscous operator [ie., ( v / ( t= l O h ~ s ) ) l /=~ 6 km]indicates that the enhancement in descent speed cannot be a simple matter of “nose diffusion”. Apparently, the along-terrain viscosity interacts synergistically with the momentum cycle to elevate the speed of the descending plume. At present, we have no explanation for this effect. Lest this behavior be attributed to model peculiarity, Fig. 6.14c,d show the outcome if SCRUM is run in an identical configuration (along-sigma viscosity and diffusivity). Although there are differences in the SCRUM results depending on the form of advection operator chosen, the basic propagation pattern is the same as with SEOM; after approximately seven hours, the nose of the descending plume has collided with the downslope wall of the computational domain (z= 200 km).(One difference is that the SCRUM model survives this collision, and continues to hour 10. The SEOM simulation fails shortly after the advancing plume “splashes into” the downstream boundary.) We note in closing that if SCRUM is configured with rotated (geopotential) viscosity, the speed of plume descent is slowed to approximately match that of the other three models.

6.5

Steady Along-slope Flow at a Shelf Break

The phenomenology of interest in this test problem is steady geostrophic along-slope flow adjacent to steep and/or tall bathymetric features. Timemean circulations of this type are known to arise from tidal rectification along the continental shelf and at subsea ridges, banks and seamounts. We are again interested in the consequences of differing numerical resolution and vertical coordinate systems, in particular, any spurious effects which may be associated with sigma-coordinate pressure gradient errors. Figure 6.15 shows a schematic diagram of the test problem configuration. We envision a two-dimensional (2,z ) shelf/slope topography under uni-


235

Fig. 6.15 Schematic diagram of the along-slope flow test problem: alongslope velocity (u) and density perturbation ( p ' ) .

form rotation

(fo =

The total density field is

1x

where the resting stratification

with

.

H p = 1OOOm Fixed parameter values are po g

= 1000. a-units = 9.81 m/s2

,

yielding a first Rossby deformation radius of 28.5 km in the deep fluid. We assume a density perturbation that has its maximum amplitude at the bottom and decays exponentially upward into the fluid with an efolding scale of H p . Also, it is largest in shallow water (H,in) and decays exponentially with increasing depth: p'(z, z ) =

[e-(z+h(Z))/Hp

I

e1-h(Z)/Hmin

.

PROCESS- ORIEN T E D TEST PROBLEMS

236

Using the hydrostatic relationship, the dynamic pressure by vertical integration becomes

The corresponding three-dimensional velocity field is then inferred from geostrophy and continuity: +,I)

=

zI(z,z)

=

w(y,z)

=

1 dP

f o 8Y

=o

+--f1o da Px

-lo

(E+$)dz=O

.

For convenience, we adopt the same topography as in Section 6 . 4 , here interpreted as as a steep continental shelf, positioned at the center of the model domain: h(x) = Hmin

1 + -(Hmaz 2

- Hmin)(l+ tanh((~ -z o ) / L ) )

where

Hmin = 2 0 0 m HmQz = 4 0 0 0 m

L , = 10 k m xo = 100 km

The along-slope velocity then becomes zI(x,z)

=

-

[

(Hp

+ Hmin)(Hmaz - H

'

2HminLsfopo

The vertical scale depth of the density perturbation is set to

Hp=50m

.

7


237

For a positive density anomaly (Ap' > 0 ) ,the analytic solution corresponds to a bottom-intensified dome of cold water sitting atop the shelf, and to a geostrophically balanced anti-cyclonic current. A density anomaly of O(O.1) a-unit produces a maximum current of O(10) m / s . Using the analytical solution for p' as initial conditions for the density field, the test proceeds as follows. The initial pressure field is integrated numerically by the model to obtain an initial distribution of pressure. In turn, this pressure field is used to compute a geostrophically balanced velocity on the model grid. Note that these pressure and velocity fields, having been numerically computed, will be in error to a greater or lesser degree depending on the model and the spatial resolution chosen. Initialized in this manner, and with no forcing or dissipation, a properly coded numerical model should show no subsequent evolution of the alongslope flow or its accompanying density field. (Nonlinear effects are trivial given the two-dimensionality of the problem.) This result, in itself, is not particularly interesting, though it is an easy test of correctness of coding. The numerically derived velocity field can, however, be compared with the analytic solution for the along-slope velocity component to assess errors arising from the discrete resolution, the particular representation of topography used, associated pressure gradient errors, and any other modelspecific algorithmic details related to the determination of pressure and geostrophic velocities. The measures of error we investigate here are, first, the maximum error in along-slope velocity divided by the maximum analytic velocity value (a normalized error in pointwise velocity) and, second, the rms value of the velocity error divided by the m s of the analytic velocity field (a normalized bulk error measure). These error statistics have been determined for several of the models from Chapter 4 as a function of horizontal and vertical resolution, as described next. Note that no explicit time-stepping of the model (only model initialization) is required to produce the necessary output; hence, this is a particularly inexpensive test to perform. For terrain-following coordinate models, an issue of particular interest is the occurrence of pressure gradient errors, their relationship to horizontal and vertical mesh refinement, and the circumstances under which convergent results (vanishing errors) are obtained. Figures 6.16 to 6.18 show the n n s error results for SPEM4 The r-value (see Section 4.2.4) is 0.13, 0.07 4The evaluation of the maximum error reveals a very similar picture.

238

PROCESS- ORIENTED TEST PROBLEMS 5 0 111

0 O'J

one 10

0 07

0 Ob 0 05

n 0.1

20

0 03

0 02

u UI

.-

n In 1 000

40016

1000

2000

4000

Fig. 6.16 Relative error for the along-slope flow problem in SPEM, as a function of horizontal grid distance (z-axis, in meters) and number of vertical levels (y-axis), illustrating the pressure gradient error for a positive density anomaly: (a) in a pure a-coordinate configuration; (b) in a stretched s-coordinate configuration. The background density field has been subtracted in these results.

and 0.03 for the three horizontal resolutions, respectively. (The topography has a 20% slope.) In the first two cases, the background density stratification is subtracted. Still, there is an error of a few percent, except for the very high horizontal resolution cases. The non-convergence of the numerical solution in the coarser horizontal resolution cases is due to hydrostatic inconsistency (JanjiC, 1977; Mesinger, 1982; Beckmann and Haidvogel, 1993), which excludes further improvements for increasing vertical resolution. This is also true for the stretched vertical coordinate (Fig. 6.16b). If the background stratification is included, the errors can be much larger (up to 80% for a density perturbation of 0.1); in that case, however, they decrease linearly with increasing vertical resolution (Fig. 6.17). The asymmetry for positive and negative density anomalies is illustrated for a density perturbation of 1.0 (Figs. 6.18a,b). It is caused by the pressure gradient error arising from the background stratification which adds an erroneous prograde ( i e . , positive) flow to the numerical solution. For the geopotential model, the error as a function of horizontal and vertical resolution is shown in Fig. 6.19. Two aspects seem noteworthy: one is that much finer grid spacing is necessary to obtain similar error levels as in the previous s-coordinate model; the other is the non-uniform reduction of the error, which is found to have a minimum at about A z l A y = 2.5.10-3.


239

5 I00 0 90

0 80

10

o 70 0 60 0 50

o 40

ICI

0 30 0 23 0 10 11

.A

1000

2000

4000

Fig. 6.17 Same a8 in Fig. 6.16 but for a positive density anomaly of 0.1 o-units including the background stratification.

5

5 0 10

0 10

n 09

0 09

o nn

Id

n wi

u un

18

0 07

n on

0 00

0 05

o ns u 04

o nd

;0

‘-Ll

0 on

1000

2000

4000

n 00

1000

2000

4000

Fig. 6.18 Same as in Fig. 6.16 but for a (a) positive , and (b) negative density anomaly of 1.0 n-units including the background stratification.

Both results are of course due to the poor treatment of the bottom boundary layer by the step-like representation of the topography. The problem is aggravated because the solution depends on the gradient of the topography, Also, the lateral boundary which the model sees as either zero or condition for pressure at the steps ($$ = 0) is not a good approximation to the true solution. This strong sensitivity to the ratio of the horizontal and vertical grid spacing has also been found in other convergence studies (see, e.g., Haidvogel and Beckmann, 1998). In particular, the convergence rate can be

2.

240

PROCESS- ORIENTED TEST PROBLEMS

Fig. 6.19 Relative error for the along-slope flow problem in MOM as a function of horizontal (x-axis) and vertical (y-axis) resolution.

very slow, even if both horizontal and vertical grid spacing are reduced simultaneously. Lastly, we note that the expected result that the higher-order finite element model shows uniformly lower error levels than either the 0- or Icoordinate models (Fig. 6.20). Both r m s and maximum errors decrease approximately exponentially with increasing horizontal resolution. Similar behavior is found for a case in which the tanh-folding scale of the topography is halved, resulting in a 20% maximum slope, though overall error levels increase by between one and two orders of magnitude. 6.6

Other Test Problems

The preceding test problems illustrate the general remark that considerable differences among numerical algorithms may arise even in rather simple dynamical settings. The sources of model inaccuracy and contrasting behavior are many; however, several issues predominate. Of these, the three that seem most evident in these test problems are the issues of horizontal spatial approximation (staggered versus unstaggered, low-order versus high-order, boundary-conforming versus staircase), vertical coordinate treatment (geopotential, terrain-following, isopycnal), and the advection of tracers (degree of monotonicity, order of approximation). The underlying effects, often not small, of the chosen viscous/diffusive subgridscale closures are also emphasized, particularly in front-producing problems, and

Other Test Problems

241

1 0-2

/

10-3

20% SLOPE /:~ERRoR~

10.'

10-5

104

10-'

0

I

2

3

4

5

Fig. 6.20 Maximum and m s errors for the along-slope flow problem in SEOM as a function of horizontal grid distance (x-axis, in kilometers) for a positive density anomaly of 0.1. A spectral truncation of six (horizontal) and five (vertical) was used in all cases. The lines correspond to results obtained with an unstretched (equidistant) placement of elemental boundaries in the vertical. Errors are slightly reduced if the elemental grid is stretched towards the bottom (dots).

more generally in limits of strong stratification and steep bathymetry. We note in closing this chapter that related conclusions have been reached in other studies in which systematic model/algorithm comparisons have been conducted. Simplified studies of form stress over a wind-driven coastal canyon demonstrate considerable inter-model variation in the structure and magnitude of residual currents, and caution against the use of "staircase" representations of continental shelf topography (Haidvogel and Beckmann, 1998). Hecht et al. (1995) discuss the potential advantages of upwind-weighted advection schemes for ocean tracer transport in a sheared

242


western boundary current. Lastly, process-oriented simulations of a selfadvected dense water plume over sloping topography have been used in vaxious configurations (e.g., the “dam break” problem introduced by Jungclaus and Backhaus, 1994 and Jiang and Garwood, 1995) to evaluate numerical issues like the implementation of topography, resolution and subgridscale mixing (see, cg., Beckmann and Doscher, 1997; Killworth and Edwards, 1999).

Chapter 7

SIMULATION OF THE NORTH ATLANTIC

Ocean models like those described in Chapter 4 are used for a wide variety of applications, ranging from idealized process studies to dynamically inclusive simulations of the circulation in regional and basin-to-global-scale oceanic domains. In addition to these purely oceanographic applications, ocean models are also part of climate system studies (see, e.g., Trenberth, 1992 for an introduction) and form the basis for coupled physical-biogeochemical studies in the marine environment (Hofmann and Lascara, 1998). As an important example, and representative of many ocean modeling studies, this chapter will focus on the simulation of the wind-driven and thermohaline circulation in the North Atlantic Ocean.

7.1 Model Configuration Numerical simulations of the large-scale ocean circulation strive to represent all potentially important processes with a degree of realism sufficient to reproduce the “important” observed phenomena. (There is of course the inevitable issue of which phenomenological attributes of the oceanic circulation are important. We focus below on the traditional measures of North Atlantic circulation, namely those few for which generally accepted quantitative estimates are available.) In addition to the choice of numerical model, the configuration of such a simulation requires the identification and preparation of model-specific datasets and decisions in several related areas.

243

244

7.1.1

SIMULATION O F T H E N O R T H A T L A N T I C

Topography and coastline

Topography and coastline data are typically obtained from one of the available gridded global datasets of ocean depth - e.g., the 5' (1112") resolution, ETOPOS database (NGDC, 1988). While certainly not correct in all details, these are regarded as generally sufficient for horizontal grids of comparable or lower resolution. An improved 2' (1/30") resolution topographic dataset based on satellite gravimetric measurements has recently been made available between 72"s and 72"N (GTOP030; Smith and Sandwell, 1997). A depth value has to be assigned to each model grid point. Different strategies are possible, including using the nearest neighbor, averaging all data points that fall within a given grid box, or using the envelope method t o determine the large-scale topography. In the latter approach, the computed area-mean water depth over a grid cell is reduced by the local (grid-cell) standard deviation of topography in order to account for subgridscale topographic variability (Wallace et al., 1983). Whichever of these three methods is used, additional "by hand" modifications are often found to be necessary to remove grid-scale variability and/or to adjust the width and depth of critical passages and sills (e.g., the Florida Straits). For geopotential coordinate models, single point peaks and depressions have to be eliminated (see Pacanowski, 1996); (T- and s-coordinate models require further smoothing of the bathymetry to reduce the pressure gradient error (see Section 4.2).

7.1.2

Horizontal grid structure

In large-scale ocean modeling with finite difference grids, a spherical coordinate system seems most appropriate. Straightforward latitude-longitude grids with a globally constant increment in degrees ( e . g . CME) have been used extensively, but suffer from the disadvantage of meridionally varying zonal grid spacing in kilometers. This distortion of the grid has been found undesirable. As an alternative, the use of an isotropic grid' has become a frequently employed strategy ( e . g . ,Semtner and Chervin, 1992; DYNAMO group, 1997). Here, the meridional grid spacing decreases gradually away from the equator such that all horizontal grid boxes are squares. It varies 'This arrangement of grid points is also called a Mercator grid because the resolution is equidistant in a Mercator projection.

Model Configuration

245

according to

where X ~ E Q is the zonal grid spacing in degrees at the equator and 4 the geographical latitude. The latitude locations for tracer points on such a grid can be computed analytically from 180

K

4 ( j ) = -arcsin[tanh(AXIEQ--(j 7T 180

- jEQ)]

,

where the index j references the j-th center of a grid box from the equator. This choice adds resolution to the higher latitudes, reduces the distortion of grid cells and does not require a reduced time-step. One should, however, be aware of the greatly increased memory requirements for models that extend beyond 60"N. Note that curvilinear and unstructured grids (e.g., Sections 4.2 and 4.4) offer additional flexibility in coastline definition and horizontal variation in resolution,

7.1.3 Initialization Assuming that we are interested in a proper representation of the oceanic thermohaline circulation, model initialization from either a homogeneous or a resting (level) isopycnal state is only affordable for very coarse resolution configurations because the diffusive processes in the deep ocean take thousands of years to reach an equilibrium. However, a dynamic equilibrium for the wind-driven and the fast modes of the thermohaline circulation can be reached within 10 years of integration with non-eddy resolving configurations and within 20-25 years for simulations with self-generated internal mesoscale variability. [In the latter case, the equilibrium is of course not truly steady ( i . e . , invariant in time) but rather a statistically steady state, wherein the running mean of total energy of the system has approached some constant or slowly varying level.] In these decadal-length experiments, the initialization of the model in principle requires hydrographic data (potential temperature and salinity) close to a realistic instantaneous state of the ocean. Unfortunately, no synoptic three-dimensional hydrographic data set is available at the required range of spatial scales. Therefore, initialization must be performed using a climatological dataset - i e . , one that has been compiled from multi-year observations and smoothed over some characteristic horizontal distance.

SIMULATION OF T H E N O R T H A T L A N T I C

246

Such atlases (monthly mean values with non-eddy resolving resolution and 35 standard depth levels) of the hydrography exist for the World Ocean beginning with Levitus (1982). More recent products with corrected and additional observations (1 degree; Levitus and Boyer, 1994; Levitus et al., 1994) and with higher horizontal resolution (1/4 degree; Boyer and Levitus, 1997) are also available. In general, the temperature data are considered to be more reliable than the salinity data. Often, the number of independent salinity measurements is sufficient only for seasonal or annual means. The initial state of a model simulation can also be taken from previous model integrations. In this way, the initial fields may be more in balance with the dynamical equations. This may be problematic, however, if multiple steady states of the circulation exist and the new model run is not free to move away from the prior solution (see, e.g., Beckmann et al., 1994a). Also, if the prior simulation has been performed in a sufficiently different parameter regime, spin-up of the new solution may be no less costly than if climatological initial conditions had been used. Finally, one should be aware that, due to the averaging procedures involved, there may be a mis-match between hydrographic data coverage and model topography in certain areas. Gridded climatological data sets may not contain enough details of the topography to be directly usable for highresolution applications. Then, extrapolation of the hydrographic fields is required, which might lead to inaccuracies. 7.1.4

Forcing

Gridded observational datasets are required as forcing fields at the ocean surface for such quantities as heat and fresh water fluxes and wind stress. Widely used wind products are Hellerman and Rosenstein (1983), Isemer and Hasse (1987), Da Silva et al. (1994), or Trenberth et al. (1990). The momentum forcing (surface wind stress) is typically computed from

where Ja is the atmospheric wind at the sea surface. The associated input of momentum to the ocean can be formulated in two different ways. The former is as a surface stress via the surface boundary condition

av'

AM-=? v

az

atz=O

Model Configurntion

247

on the vertical viscous term

The latter method assumes a body force over a finite depth (typically a vertical grid box)

-aa_-- .7‘ dt

AZ

If monthly mean wind stress data are specified, linear interpolation is used to avoid abrupt jumps in the forcing fields, which would cause the ocean to respond with strong inertial waves. Any interpolation, however, modifies the monthly mean value supplied to the ocean. Correction of this systematic modification of the forcing data is possible (e.g., Killworth, 1996). Unfortunately, this correction leads to a change in phase of the forcing fields; ie., certain phenomena occur earlier or later than in the original data. Ultimately, higher frequency (daily or hourly) winds may have to be used to avoid these effects. The thermohaline forcing of an ocean model is less straightforward. The specification of surface fluxes alone may lead to an undesirable drift of the model fields (Barnier, 1998) as it neglects the feed-back to the atmosphere entirely. Therefore, basin-scale models are often forced (exclusively or additionally) by Newtonian “restoring” to monthly mean climatological surface fields. A widely used scheme specifies the heat flux as the sum of a surface heat flux and a restoring term

where Tappare the so-called “apparent atmospheric temperature”. This equation can be combined to a single restoring term:

Q = (T*- T m r j a c e )

+

*

The data for T* = Tapp Ql/Q2 are often based on Han’s (1984) analysis. A conceptual disadvantage of forcing fields based on observations is that they are likely to be inconsistent, in the sense that largely unrelated datasets with different spatial and temporal resolution were used for these products. This can be avoided if forcing fields from atmospheric models (like analyses of ECMWF simulations; e.g., Barnier et al., 1995) are used.

SIMULATION OF T H E NORTH ATLANTIC

248

These data are also available at much higher temporal resolution, e.g., daily rather than monthly mean winds. 7.1.5

Spin-up

After the above issues have been addressed, and appropriate datasets chosen, first test integrations can be performed. This is the time for “plausibility checks” and initial “tuning” of the model, ie., the adjustment of parameters (treatment of topography, subgridscale closures, resolution) to optimize solutions. This first assessment of model behavior is mainly phenomenological (e.g., path of major currents, seasonal cycle in the upper ocean), supplemented by some general “rules of thumb” for important transports (Gulf Stream, Deep Western Boundary Current) and typical velocities and/or eddy energy levels. (Examples of these measures are introduced below.) One has to keep in mind, however, that model response during spin-up goes through several distinct phases, in accordance with the time scales of important physical processes. The sequence of adjustment is roughly as follows. Immediately after model initialization, the solution is dominated by the JEBAR effect and inertial adjustment t o the initial density field. (The latter is often accompanied by considerable high-frequency motions in the form of inertial and internal waves.) After weeks to months, the barotropic wind-generated response is set up (Sverdrup interior, westward intensification), followed by the analogous baroclinic response after 3 to 5 years. The thermohaline circulation in the main thermocline takes about a decade to be established, while the deep ocean is in equilibrium only after many centuries.

7.2 Phenomenological Overview and Evaluation Measures Simulations of the North Atlantic attempt to represent the most important phenomena and aspects of the wind-driven and thermohaline circulation and water mass distribution. Knowledge of the time-mean structure of the circulation and tracer fields, and their variability on both seasonal and shorter timescales are used to assess the performance of these models. The most widely used phenomenological attributes of the North Atlantic typically used for validation are discussed below. These are the path

Phenomenological Overview and Evaluation Measures

249

and strength of the Gulf Stream System (including the Florida Current), the basin-wide quasi-zonal circulation, the eastern recirculation, the Deep Western Boundary Current (DWBC), the seasonal cycles of mixed layer dynamics and water mass formation, and the three-dimensional patterns of eddy variability. A schematic overview of these features is shown in Fig. 7.1. The huge amount of data produced by basin-scale numerical simula-

.-

_. -

__ 1i

Deep Western Boundary Current Antarctic Bottom Water Meridional Heat Flux Mediterranean Salt Tongue

0 ovettow

--

-p* m0

Overturning Circulation Subtropical Gyre and Gulf Stream Subpolar Gyre Heat Exchange with Atmosphere Rings I eddies

Fig. 7.1 Schematic diagram of some important North Atlantic phenomena.

250

SIMULATION OF T H E N O R T H ATLANTIC

tions, even at relatively coarse resolution, requires specialized analysis procedures. Several analysis approaches are now standard. Instantaneous snapshots are a suitable way t o monitor the model during run-time, and a first quality check can be based on these fields. In addition, they can be used for movie sequences to show the temporal evolution of the prognostic fields. Typical are horizontal maps near the surface, or vertical sections. During model spin-up, tabulation of bulk measures such as instantaneous values of basin-integrated kinetic energy or enstrophy are a convenient means of initially monitoring the approach of the model simulation to a n eventual steady state. Once a steady state has been reached, time-mean fields represent the next higher level of diagnostic evaluation. It is common procedure t o compute seasonal, annual or even longer-term means for comparison with known climatologies. Deviations from the time-mean may be computed to produce estimates of rms eddy variability. Time series at single points (e.g., the Florida Straits transport) or along repeatedly visited sections [e.g., the (World Ocean Circulation Experiment) WOCE sections] form the basis for a direct comparison to observations. Regional balances can be calculated t o determine dynamical regimes (e.g., the subtropical gyre, subpolar gyre, equatorial dynamics, the Gulf Stream). Finally, statistical approaches can be applied to compute spectra, and correlation length scales both in time and space (e.g., Stammer and Boning, 1992).

7.2.1

Western boundary currents

The large-scale wind forcing in combination with the variation of the Coriolis parameter with latitude lead to a westward intensification of the midlatitude ocean circulation. In the North Atlantic, these dynamical influences are manifest in an inter-related system of boundary currents which includes the North Brazil Current, the Loop Current in the Gulf of Mexico, the Antilles Current, the Florida Current, the Gulf Stream (all northward flowing) and the Labrador Current (flowing in a southward direction). A first measure of the performance of a numerical model is the timemean transport and its seasonal variation of this boundary current system, along with the velocity structure and the time-mean path of the flow. One of the few places where direct transport measurements are available is the Florida Straits. The annual mean transport at 25.5"N is found t o be about 32 Sverdrups; its seasonal cycle has an amplitude of 4 Sverdrups. Other observational data for validation of the circulation in the subtropical North

Phenomenologicnl Overview and Evaluation Measures

251

Atlantic also exist (see Schott and Molinari, 1996) Farther north, one of the most prominent and robust features of the North Atlantic circulation is the Gulf Stream and its continuation as the North Atlantic Current. The Gulf Stream carries water from the Gulf of Mexico northward along the North American coast, until it leaves the western boundary at Cape Hatteras. From there, it continues as a free meandering jet. To the north, a cyclonic circulation is observed, the Northern Recirculation Cell. As described in more detail below, numerical models have (until recently) shown poor agreement with specific observed features of the Gulf Stream System - e.g., its separation at Cape Hatteras, its maximum transport, and the levels of mesoscale eddy variability associated with Gulf Stream meander, ring and eddy formation. Over the years, many hypotheses have been advanced to explain this general failure of numerical models. Proposed explanations have included deficiencies in large-scale wind or local thermohaline forcing, and insufficient resolution (which reduces the inertia and vorticity of the Stream, and poorly represents local topographic effects). Other numerical issues known to play a role in model realism also include the formulation of lateral boundary conditions and the vertical model coordinate. For an extensive overview on the various theories relating to Gulf Stream separation see Dengg et al. (1996). The confluence area of Gulf Stream and Labrador Current at around 50"N is another critical point. The relative strengths of both boundary current systems need to be represented accurately to obtain the observed flow pattern in the so-call Northwest Corner (see, e.g., Kase and Krauss, 1996). Below the wind-driven flow of the upper 1000 meters, a continuous band of southward flowing dense water forms the Deep Western Boundary Current (DWBC). It is part of a continuous cyclonic along-boundary flow all around the North Atlantic basin and observed all along the western boundary of the Atlantic. One of the main sources is the Denmark Strait overflow, which entrains surrounding North Atlantic water and forms a bottomintensified southward flow at 1200-1500 meters depth in the Irminger Sea. This DWBC gradually descends to 2000-2500 meters at the equator. The ability of a numerical model to represent this flow is essential for the largescale thermohaline circulation and the meridional heat transport.

252

7.2.2

SIMULATION OF THE NORTH A T L A N T I C

Quasi-zonal cross-basin flows

There are several predominantly zonal flows in the North Atlantic between the complex flow system at the equator and the North Atlantic Current. All these flows can be seen along 30"W, a meridional section that is often used to quantify numerical model results. The equatorial band contains a non-symmetric sequence of currents north and south of the Equator, with the near-surface North Equatorial Counter Current (NECC), the South Equatorial Current (SEC), the subsurface eastward Equatorial Undercurrent (EUC) and other seasonally variable flows (see, e.g., Stramma and Schott, 1996). While the large seasonal variability is almost linearly dependent on the wind field, the retroflection region where the North Brazil Current (NBC) turns into the NECC is a place of intense eddy generation. In general, a realistic representation of the western equatorial circulation is important for the interhemispheric exchange of heat and fresh water. Farther north, the North Equatorial Current (NEC) flows westward as the rather steady southern limb of the subtropical gyre. Within the gyre, at around 35"N, the Azores Current flows eastward from its formation area south of the Grand Banks to the Gulf of Cadiz. The cause for this zonal flow is still not completely understood, but it is well established that the Azores Current is a meandering jet that provides a major part of the transport of water into the Eastern Recirculation region in the Canary Basin (Kase and Krauss, 1996). Numerical models are typically unable to simulate this flow band, unless they use very high horizontal resolution (Section 7.6). The northern boundary of the subtropical gyre is located at around 50"N, where the North Atlantic Current (NAC) crosses the basin. The western boundary current turns zonal in the so-called Northwest Corner, the confluence region of the Gulf Stream and the Labrador Current. The NAC carries the majority of the eastward transport across the North Atlantic. It is associated with strong variability that is easily detected in observations (e.g., Beckmann et al., 1994b).

7.2.3 Eastern recirculation and ventilation The relatively quiet Sverdrup regime in the eastern basin is the place where most of the ventilation of the subsurface layers of the subtropical gyre occurs. Consequently, subduction is the dominant process in this area,


253

and numerical models need to capture this process to produce a realistic large-scale water mass structure. Observations of the water masses, and mixing processes (see, e.g., Siedler and Onken, 1996) are available for model validation. 7.2.4

Surface mixed layer

The surface mixed layer is important for the ocean-atmosphere interaction and consequently the formation of mode and deep waters of the North Atlantic. It can be as deep as 1500 meters in winter in the Labrador Sea, and as shallow as a few tens of meters in summer in the subtropical gyre. The main aspect is a realistic representation of the seasonal cycle of the mixed layer depth and temperature. Observations of these quantities are included in climatologial atlases (e.g., Levitus, 1982; Levitus and Boyer, 1994; Boyer and Levitus, 1997); a direct comparison of model results with these data is possible. From a numerical point of view, convection parameterization is one of the important numerical issues here. 7.2.5

Outflows and Overflows

The mid-depth and deep circulation of the North Atlantic is mainly determined by outflow processes from marginal seas, i. e., the Greenland-IcelandNorwegian (GIN) Seas and the Mediterranean Sea. Of the former, the Denmark Strait overflow is probably the most important pathway for dense water from the Nordic Seas into the deep North Atlantic to form North Atlantic Deep Water (NADW). Another major gap in the Greenland-Iceland-Scotland Ridge is the Faeroe Bank Channel, where another strong overflow of deep water masses occurs. Both overflows are bottom-trapped current systems which flow cyclonically around the North Atlantic basin, unite in the Irminger Sea and finally form the DWBC. Their representation in numerical models depends on details of the bottom topography, the bottom boundary layer and the nearbottom mixing. The integral effects of these outflows can be seen in the meridional overturning (see below), and determine in part the northward heat transport in the North Atlantic. The Mediterranean Water (MW) contributes significantly to the middepth water masses of the Atlantic. It can be traced as far north as 60"N and zonally all across the Atlantic due to the translation of Meddies (iso-

254


lated submesoscale lenses that contain MW). MW is injected into the Atlantic at its eastern boundary at about 35N, forming a tongue of warm and salty anomaly on the 1000-1200 meter depth horizon. This anomaly spreads a few hundred kilometers westward and is advected northward with the poleward undercurrent along the European continental slopes. For a recent review on MW observations, see K b e and Zenk (1996). Present in the Levitus climatology, this tongue disappears within a decade in a numerical simulation, unless specific measures are taken to renew this water mass. A standard method is t o use restoring t o climatology (usually in the Gulf of Cadiz), but this has proven to be too inefficient in many models of the North Atlantic circulation, unless the area of restoring is enlarged (to 5 by 5 degrees) or the time scale is very short (a few weeks). Gerdes et al. (1998) have recently shown for coarse resolution models that an inflow condition with about 3 Sverdrups of highly anomalous water is probably the best way to parameterize the interaction of the Atlantic with the Mediterranean Sea across the Straits of Gibraltar. 7.2.6

Meridional overturning and heat tmnsport

The zonally averaged meridional overturning streamfunction, a function of latitude and depth, is an important diagnostic quantity for basin-scale numerical models in that it characterizes the thermohaline circulation (its strength and latitudes of down-welling). It is computed from the twodimensional elliptic equation

-1- +d2- a = 8 2 ) ( r ; a42 8.22

(","I'

r;aTT.)

or by integrating the meridional velocities zonally and vertically:

a(4.2)= [*

(i;

'u

d h ) d.2

*

(Note that velocities from other than geopotential coordinate models need to be interpolated to a geopotential grid before this quantity can be computed.) The representation of a in the meridional-potential density plane [@(4,p,,t)] is very helpful in interpretation of the water mass transformation at high latitudes and the entrainment along path. This quantity is crucial for climate studies since it represents the oceanic heat transport from low to high latitudes,. The large-scale overturning is


255

not easily observable, but an annual-mean maximum overturning of about 20 Sverdrups between 40"N and 50"N in the depth range of 1000 to 1500 meters seems consistent with estimates of the corresponding heat transport. Similar considerations, and the results of numerical simulations, suggest a largescale structure of which is dominated by the NADW cell crossing the equator. Below, the counterrotating AABW cell transports 2-4 Sverdrups. A validation of the model results is possible based on (e.g.) the observations by Roemmich and Wunsch (1985) at 24" and 36"N. The meridional transport of heat is another quantity of relevance for climate studies. Independent estimates exist from both observations and atmospheric circulation models. It is defined as

HT($) = pocp

1' (l*,"

vt? a cos$ dX

-H

where cp is the specific heat of sea water. This curve shows a northward heat transport throughout the North Atlantic, with a maximum of about 1 PW between 20" and 40"N, and 0.1 to 0.3 PW at the Equator. For analysis and interpretation, the total heat transport can be decomposed in several ways. If the variables are split into their zonal means and deviations, the contributions by overturning and gyre transport can be examined. If the variables are split into a depth-averaged part and deviations, the contributions by barotropic and baroclinic motion can be discriminated. In addition, the Ekman contribution can be separated from the baroclinic part. If the variables are split into their time averages and deviations, the time-mean heat transport can be separated from the eddy contribution. Further details can be found in Boning and Bryan (1996).

7.2.7

Water masses

A somewhat neglected area of validation concerns the water mass evolution in a basin-wide circulation model. It is true that the time scales for water mass changes are much longer than the typical integration periods, but Klinck (1995) has shown that there are substantial trends related to the artificial boundaries and possibly incorrect surface forcing. It would seem that a volumetric water mass census can be very helpful to determine the role of forcing and buffer zones in these models.

256

7.2.8

SIMULATION OF T H E NORTH A T L A N T I C

Mesoscale eddy variability

The role of mesoscale eddies deserves special attention. It has been reported from several numerical models (Cox, 1985; Boning and Budich, 1992; Beckmann et al., 1994a; Drijfhout, 1994) that there is no additional net heat transport in the presence of eddies because the meridional eddy flux of heat is almost completely compensated by an opposite heat flux associated with the eddy-induced mean flow (Bryan, 1991). It appears that the eddy effects are of minor importance for the heat transport, which is a welcome result for non-eddy-resolving climate studies. Nonetheless, the distribution of mean and eddy kinetic energy (EKE) are an important measure of model dynamics. Fields of time-mean EKE are known to be related in intimate ways to several model-specific issues such as horizontal resolution, subgridscale dissipation, formulation of advection operators, and others. The resulting fields can be compared to climatological maps of potential and kinetic eddy energy from in situ observations (e.g., Dantzler, 1977; Emery, 1983), and more recently from satellite observations (Stammer and Boning, 1992) and surface drifters (e.g., Briigge, 1995). To quantify the large-scale energy transformation mechanisms in a numerical simulation, the basin-averaged energy cycle is often diagnosed. Defining the time-mean and eddy components of kinetic and potential energy as

MKE

=

EKE

=

1 -(E2+G2) 2 1 - ( d 2+ d 2 ) 2

where fi is the horizontally averaged density, the energy transfer terms, per unit mass, in a closed volume are:

257


'

E K E + E P E : T3 = - / / / w d V EKE+MKE:

T4 =

711

+

u'(v' VTi) v ' ( 3 00)dV

.

The overbar represents a long-term time-average; the prime denotes the deviations from that mean. Note that the contributions from vertical velocity are usually neglected in T2 and T4, because the vertical eddy advection is several orders of magnitude smaller than the horizontal terms; hence:

Physically, positive transfer value of TZ are an indication of baroclinic instability; positive T4 suggests the occurrence of barotropic instability. (See Pedlosky, 1987, for more details on the theory of barotropic and baroclinic instability processes in geophysical fluids.) To obtain these quantities, three-dimensional fields of the time-mean of the prognostic variables (Ti,V, p ) , and time-mean correlations ('ltu, 55, TE, ;iiii, and u)p) are required. The time-mean vertical velocity TP can be inferred diagnostically. The eddy terms are then calculated according to

v,

- - - -

In an isopycnic model, the quantities h, hu, hv, hp replace some of the above. A description of energetics in isopycnic coordinates systems is given by Bleck (1978). 7.2.9

Sea surface height from a rigid lid model

Satellite altimeter measurements provide a large-scale quasi-synoptic data set that can be used for validation of near-surface properties of the ocean. An important example is the sea surface height (SSH) variability which can be inferred from altimeter measurements. While numerical models with a

258

SIMULATION OF THE NORTH A T L A N T l C

free sea surface can directly give an estimate of this quantity, rigid lid models need to compute these fields diagnostically. The surface pressure gradients can be reconstructed, if additional terms axe stored during the integration of the model. The vertically integrated momentum equations can be written as

where fi = (V,V ) are the depth-averaged velocities, ps is the surface pressure and 2 represents the vertical integral of all other RHS-terms. Since the surface pressure in a rigid-lid model is unknown, the solution procedure uses the gradients of the vertically integrated vorticity tendency (see Section 3.3)

a

-V2$ at

=

vxz

to determine the barotropic velocity tendencies:

z

If has been stored, the surface pressure gradients can then be inferred from equations 7.3 and 7.3. In principle, this leads to an elliptic problem:

subject to no-gradient boundary conditions. For diagnostic purposes, however, it is often sufficient to integrate p s in alternating directions from

Not only do satellite altimeter measurements provide the opportunity to estimate the regional distribution of near-surface eddy variability, they also enable inference of the typical scales of eddies. Based on one such

North Atlantic Modeling Projects

259

analysis, Stammer and Boning (1992) found a linear relation between the zonal wavelength L and RD. The representation of these scales in ocean models is an important aspect of model evaluation, especially through its close connection to horizontal resolution.

7.3

North Atlantic Modeling Projects

Three systematic comparisons among numerical models of the North Atlantic Ocean have been conducted over the past decade. These are the Community Modeling Effort (CME), the Dynamics of North Atlantic Models (DYNAMO), and the Data Analysis and Model Evaluation Experiment (DAMEE). As a result of these projects, we are beginning to understand the strengths and weaknesses of specific model classes, subgridscale parameterizations and numerical algorithms.

7.3.1

CME

By the mid-l980s, it had become clear that many details of model implementation and set-up have considerable effect on the representation of observed phenomena and that systematic sensitivity studies were necessary to understand the isolated results of prior simulations. As a consequence, the Community Modeling Effort (CME) was initiated at the National Center for Atmospheric Research (NCAR) to begin a systematic study of highresolution North Atlantic models. Shortly thereafter, the modeling group at the Institut fur Meereskunde (IfM) Kiel, Germany, joined the effort and added many experiments to the study. Starting from a central experiment (Bryan and Holland, 1989), the influence of forcing functions, resolution, and subgridscale parameterizations was evaluated. For an overview of model experiments see Boning and Bryan (1996). The CME reference experiment (Bryan and Holland, 1989) was carried out with the GFDL model and covers the region between 15"s and 65"N, with 1/3 x 2/5 degree resolution in the horizontal and 30 vertical levels (from 35 meters at the surface to 250 below 1000 m). It uses buffer zones at the northern and southern boundaries and in the Gulf of Cadiz. The model is forced by monthly mean values from Hellerman and Rosenstein (1983) and with thermal forcing according to Han (1984). The surface salinity is restored to the Levitus (1982) climatology. This simulation was able to

260


reproduce many of the observed features of the North Atlantic circulation, and to capture the physical processes responsible for both water mass formation and eddy formation (Bryan and Holland, 1989). First estimates of the contribution of mesoscale eddies to oceanic heat transport were also provided. Sensitivity studies on the forcing of the CME models have been performed in the areas of basin-wide wind climatologies, local thermohaline anomalies and buffer zone hydrographic fields. Within the CME framework, sensitivities were found to be wide-spread and to exist in often unexpected areas. One such example is the topography in the Caribbean (which has a large influence on the Western Boundary Current transport).

7.3.2

DYNAMO

The CME and similar simulations and sensitivity studies were performed with implementations of geopotential coordinate models (the GFDL model). In the early 199Os, models with different vertical coordinates had been developed and were ready to address large-scale circulation questions. Thus, the influence of model-specific design issues such as the vertical coordinate could be investigated. A number of two-way intercomparisons were carried out (Chassignet et al., 1996; Roberts et al., 1996; Marsh et al., 1996), but the first systematic attempts t o evaluate model performance across the full range of vertical coordinate treatments was the DYNAMO (Dynamics of North Atlantic Models) project jointly carried out by researchers from three European institutions using the MOM, MICOM and SPEM models, respectively. Details can be found in DYNAMO group (1997). The specific objectives of the DYNAMO project were the implementation of high-resolution models based on alternative numerical formulations of the physical system (the HPE), and an assessment of the models’ ability to reproduce the essential features of the hydrographic structure and velocity field in the North Atlantic. The latter was evaluated by comparison against synoptic-scale data. The DYNAMO strategy attempted to minimize overall model differences in order to isolate issues related to the vertical coordinate. Consequently, all three models used an identical horizontal grid, and identical parameterizations for vertical diffusion and bottom friction. Differences, however, exist in the areas of mixed layer model (Kraus-Thrner in MOM and MICOM, none in SPEM), lateral diffusion and

North Atlantic Modeling Pmjects

261

viscosity (harmonic in MICOM, biharmonic in SPEM and MOM), and the treatment of the southern boundary (closed with relaxation in SPEM and MICOM, open in MOM). For further details on the model configurations, the reader is referred to the DYNAMO group (1997). The DYNAMO configuration differs from the CME in three aspects: the domain extends to 70"N (including the Greenland-Iceland-ScotlandRidge and the important overflow regions), it uses an isotropic grid (thus adding resolution at higher latitudes) and it uses self-consistent forcing fields from the ECMWF atmospheric model. Further details can be obtained from the DYNAMO web site at http://www.ifm.uni-kiel.de/general/ocean.html

7.3.3

.

DAMEE

The DAMEE (Data Assimilation and Model Evaluation Experiment) project is a collaborative effort among seven U.S. modeling groups whose goal is to contribute to the development of a global ocean nowcasting capability with basin-wide forecasting skill (three-dimensional ocean structure, the locations of mesoscale eddies and fronts, etc.) superior to climatology and persistence. The DAMEE project encompasses several primary technical efforts including the acquisition, quality control, and distribution of a basinscale database for model initialization, forcing and validation; the conduct of basin-scale prognostic simulations using multiple ocean circulation models; the characterization and evaluation of the results via model-model and model-data comparisons; and finally the implementation and exploration of advanced methodologies for model/data blending (data assimilation). While DYNAMO tried to minimize model differences, the DAMEE strategy, though controlled, has allowed a wider range of inter-model differences. For example, though average horizontal resolution, surface forcing, and open boundary condition treatments were fixed across model classes, the details of horizontal gridding and subgridscale mixing were left to the discretion of the individual DAMEE groups. Additional differences from CME and DYNAMO included domain size (6"N to 50"N for the primary comparative simulation) and surface forcing datasets (COADS; Da Silva et al., 1994). The DAMEE project included multiple representatives of all four model classes described in Chapter 4 - geopotential (two versions of MOM and

262


DieCAST), a-coordinate (SCRUM and POM), isopycnal (MICOM and NLOM), and finite element (SEOM). Further background on the DAMEE program can be obtained from the DAMEE web site : http://www.coam.usm.edu/damee

.

Since the results obtained in the DAMEE project are only now being prepared for publication, the following summary of North Atlantic modeling emphasizes the CME and DYNAMO programs.

7.4

Sensitivity to Surface Forcing

As the driving by the atmosphere is a major component in the dynamics of the ocean, a strong sensitivity to the forcing data can be expected. It is common practice to drive large-scale ocean circulation models with climatological (monthly mean) forcing data sets, which are based on observations over the last few decades. Measurement errors, large differences in data coverage for different regions of the ocean, and uncertainties in the parameterization of surface fluxes result in substantial error margins even in these long-term climatological fields. For example, different wind climatologies (Hellerman and Rosenstein, 1983 and Isemer and Hasse, 1987) differ by almost by a factor of two over the subtropical North Atlantic, and the corresponding transport differences can reach 10 Sverdrups near the western boundary (Boning et al., 1991b). Heat and fresh water flux data are similarly uncertain and will influence the climatologically relevant net meridional heat and fresh water fluxes in the ocean. An alternative, which has become available recently, is to use the products of numerical weather prediction centers (ECMWF, NCEP), which have the advantage of global coverage, consistency between different atmospheric variables and high temporal resolution. They also allows studies of interannual variability. Systematic comparisons between various forcing data sets are currently under way. First results indicate that the main effect of daily winds is an increase in near surface variability (with consequences for the mixed layer depth and related quantities), especially in high latitudes.

Sensitivity to Resolution

7.5

263


Due to the existence of a highly energetic eddy field in the ocean, sensitivity to numerical resolution is perhaps not unexpected. A systematic study of this dependency was attempted within the CME framework, wherein a sequence of non-eddy-resolving, coarse-resolution experiments were performed in parallel. Although high-resolution realizations are now becoming more routine, non-eddy-resolving studies are still important for climate research because coupled atmosphere-ocean models will have to rely on coarse resolution for the foreseeable future. A clean study of the dependency on horizontal resolution would keep all other model parameters unchanged. This, however, is inconvenient for a realistic basin. For example, higher resolution changes the coastline and topography in a possibly significant way. In addition, holding the subgridscale parameterization (e.g., values of viscous/diffusive coefficients) fixed is in conflict with the usual practice of reducing values of subgridscale mixing parameters as resolution is improved (thereby reducing lateral viscous effects to the minimum necessary at a given horizontal resolution). Also, it has been traditional for coarse-resolution models to use harmonic mixing operators, while high-resolution models have preferred the use of biharmonic terms because of their enhanced scale selectivity. For these reasons, no clean resolution sensitivity study exists with identical mixing assumptions. Examples drawn from CME that illustrate the dependence on resolution are shown in Figs. 7.2,7.3, and 7.4 for the time-mean barotropic Aow. Aside from the crude representation of the coastlines in the one degree (approximately 100 km) resolution case, the major current systems, the cyclonic subpolar gyre, the anticyclonic subtropical gyre and a weak equatorial circulation can be identified. The Gulf Stream appears as a broad band of flow along the North American coast, with 35 Sverdrups in the Florida Straits and about 50 Sverdrups off Cape Hatteras. Clear deficiencies are the width and location of the North Atlantic Current (5-10 degrees too far south), and the lack of separation of the Gulf Stream from the coast. In contrast, the next higher resolution (1/3 degree, 30 km, Figure 7.3), has greatly improved flow fields. Both the subpolar and the subtropical gyres are more energetic, the North Atlantic Current is located at the right latitude (- 50"N), and the equatorial circulation is much more realistic. Both the more accurate coastline/bottom topography and the existence N

264

SIMULATION OF T H E N O R T H A T L A N T I C

Fig. 7.2 Time-mean vertically integrated transport streamfunction from a 6/5 x 1 degree resolution CME run (Doscher, 1994). The contour lines range from -15 to 50 Sverdrups, with an interval of 5 Sverdrups.

of an eddy field contribute to this improvement. Still, some deficiencies remain: the Gulf Stream separation is unsatisfactory, as is the non-existence of an Azores Current in the Eastern North Atlantic. The eddy variability (defined as the squared deviation from the long-term time-mean velocity fields) is too weak (Treguier, 1992). The analysis, however, does show a close correspondence with the major mean flow bands. This suggests that a major source of variability in the numerical ocean is baroclinic instability. Lastly, the highest horizontal resolution in this series (1/6 degree, -15 km; Figure 7.4), is surprisingly similar to the 1/3 degree results. The horizontal scales of the multi-year average are not significantly reduced, probably because the effects of higher resolution are compensated by more eddy activity that smears out fronts in the time-mean fields. On the contrary, some of the unrealistic features of the 1/3 degree simulation persist and are even worse (the anticyclonic cells in the Loop Current of the Gulf of Mexico and north of Cape Hatteras). Further similarities and differences of the two high-resolution experiments are discussed in Beckmann et al. (1994a, 1994b). Two possible explanations exist for this unexpected result: first, the (fixed) vertical resolution might inhibit further improvements (especially for a geopotential coordinate model with its step-like representation of to-


265

Fig. 7.3 Time-mean vertically integrated transport streamfunction from a 2/5 x 1/3 degree resolution CME run (Baning and Bryan, 1996.) The contour lines range from -25 to 80 Sverdrups, with an interval of 5 Sverdrups.

pography). The other is that significantly higher resolution is needed for truly eddy-resolving models. There are recent indications that both contribute: even higher resolution simulations with both geopotential (Chao et al., 1996) and isopycnic (Chassignet, pers. comm.) models show improved EKE fields and Gulf Stream separation patterns and seem to indicate that the answer to the separation problem lies in the need for strong nonlinearities of the near surface Gulf Stream at the separation point. It is interesting to note that coarse-resolution simulations with terrainfollowing models show a different representation of the Gulf Stream separation (see Dengg et al., 1996). Due to a different interaction with topography, terrain-following models tend to show a stronger Northern Recirculation cell, which leads to a more southernly path of the Gulf Stream. Numerical models seem to be able to successfully generate eddy variability in areas where the ratio of grid spacing to Rossby radius is smaller than about one. Nonetheless, even “high-resolution” simulations with 1/3 degree zonal and meridional resolution are only marginally eddy resolving. This is confirmed by two experiments with the POP model (Dukowicz and Smith, 1994) at 1/5 and 1/10 degree resolution. (See also Smith et al. (1999).) The sea surface height variability field is used as a measure of model fluctuations, which can be validated against satellite observations.

266


Fig. 7.4 Time-mean vertically integrated transport streamfunction from a 1/5 X 1/6 degree resolution CME run (Beckmann et al., 1994). The contour lines range from -30 to 80 Sverdrups, with an interval of 5 Sverdrups.

The reference field is the TOPEX/POSElDON five-year mean (Fig. 7.5). Observed features include enhanced variability in the Gulf Stream System and in the North Atlantic Current, and a smaller maximum along the main axis of the Azores Current. The numerical simulations of the CME era (with resolutions from 1/3 to 1/6 degrees) were notoriously poor in their representations of eddy variability, especially in the eastern basin (Beckmann et al., 1994a). As an example for these experiments, the 1/5 degree results are shown in Fig. 7.6. The signatures of the Gulf Stream and North Atlantic Current are clearly seen, but the amplitudes are still too small by about 25%. The most striking deficiencies are the absence of an Azores Current maximum and the generally low variability in the Eastern basins. Several eddy-related features are improved in the 1/10" resolution case (Fig. 7.7). Not only is the maximum at the level of the observations, but there is also a n Azores Current. The Northern Recirculation of the NAC is now also characterized by increased levels of variability. The remaining differences may be due to the forcing of the ocean model, or the method of data processing of the satellite data. The rms sea surface height (SSH) shows that at least 0.1" resolution is necessary in this class of ocean models to obtain major features of the circulation.

Effects of Vertical Coordinates

--

-80

-0U

-70

-63

-50

-40

-30

-20

267

-10

0

10

Fig. 7.5 TOPEX-POSEIDONvariability figure (courtesy Y. Chao).

7.6


Even though the configuration of the three DYNAMO models is much more similar than for any other previous model comparison, some intrinsic differences in the model concepts remain. The different horizontal grids (“B” vs. “C”) have to be mentioned, as well as the different advection schemes (centered finite difference vs. MPDATA, see Chapters 2 and 4) and the treatment of the external mode (rigid lid vs. free surface). hrthermore, the three vertical coordinates are intrinsically different in their representation of bathymetry, as discussed above. Also, the smoothing and editing of the bathymetry was done in accordance with the specific needs of each model, with widening of straits and passages for MOM and smoothing the


268

-80

-80

0

-e4

-70

10

-50

40

20

-30

-20

30

-10

0

10

40

lcml

Fig. 7.6

1/5 degree variability figure (courtesy R. Smith).

slopes (and introducing a minimum depth of 200 m) for SPEM. Despite these and a few other dynamical and algorithmic differences (mixed layer, order of the lateral subgridscale operators and southern boundary condition), a fundamental result of this model comparison study was that all three models were successful in simulating the North Atlantic circulation with a considerable degree of realism. Differences among the three high-resolution models were found in many aspects to be smaller than those found in comparisons of coarse-resolution models. This reassuring result indicates some convergence between the models at this resolution. Many problems remain (DYNAMO group, 1997), however, and all three models will require continuing development. It is often difficult to attribute model differences unequivocally to the


269

Fig. 7.7 1/10 degree variability figure (courtesy R. Smith).

vertical coordinate system. The different numerical algorithms and parameterizations may preclude definite statements. Three areas, however, can be identified wherein we can safely assume that the coordinate is the main contributing factor. These are: the large-scale meridional overturning (and its dependence on the diapycnal mixing in critical small-scale regions), the near-bottom circulation (in particular the strength and path of the deep western boundary current), and the ventilation of the thermocline (subduction in the eastern basin). The maximum overturning in all three models is roughly in the range of accepted values; however, some notable differences do exist which are likely to be related to the model concept. MOM (Fig. 7.8a) has the typical structure, but with a smaller amplitude, probably due to insufficient

SIMULATION OF TH E NORTH ATLANTIC

270 0 -250

-500 -750

- 1000 -2000 -3000 -4000 -5000

- 6000 0 --250

-500 -750

- 1000 -2000

- 3000 -4000

-5000

-6000~

20

-10

0

10

-- 6000 2 0

-10

0

10

20

30

40

50

60

70

20

30

40

50

60

70

0 -250

-500

- 750 .- i no0 -2000 -3000 -4000

-5000

L a t i t u d e I d e g N1

Fig. 7.8 Time-mean meridional overturning streamfunction ip for DYNAMO-MOM, DYNAMO-SPEM and DYNAMO-MICOM (DYNAMO Group, 1997).


271

overflow in the Denmark Strait and across the Iceland-Scotland Ridge. MICOM (Fig. 7.8b) lacks the deep counterrotating cell, due to the inability of a purely isopycnic model to discriminate AABW from lower NADW. (Both have the same potential density relative to the surface.) Consequently, the NADW cell extends all the way down to the bottom. But there is a 4 Sverdrup overflow in Denmark Strait. The SPEM results (Figure 7.8~)differ mostly by relatively large-amplitude, small-scale cells. Also, there is significant down-welling at 47"N, which has not been observed. Both these features are also found in another sigma-coordinate simulation (Ezer and Mellor, 1997). On the other hand, SPEM produces the strongest overflow (6 Sverdrups). These model results confirm that the large-scale meridional overturning is controlled by rather local processes in critical regions, namely by the amount and water mass properties of the overflow across the GreenlandScotland region, and by the details of mixing within a few hundred kilometers south of the sills. All models have difficulties here in representing the processes in the bottom boundary layer. The parameterization of diapycnal mixing in this regime should be of high priority in the future development of all models. The near-bottom flow in the three models differs dramatically. Some caution in comparing Figs. 7.9 is advised, because the topographies differ and the flow may not occur at the same depth in all locations. But it is quite clear that MOM has the weakest and most variable bottom flows (probably due to the lateral no slip boundary condition), SPEM produces an extremely strong deep western boundary current at relatively shallow depths, and MICOM is somewhere in between. The large-scale potential vorticity distribution (Fig. 7.10) is helpful in analyzing model ability to correctly represent subduction and ventilation processes. The meridional section at 30"W is chosen, as it crosses a subduction region and is a standard section. Here, the isopycnic model excels in the vertical water mass structure (as compared to the climatological fields), and it also is the only model to generate an Azores current. The general problems and sensitivities of each of the model concepts can be summarized as follows. For geopotential coordinate models, the pathways of the boundary currents (such as the Florida Current and the Antilles Current, but also the North Atlantic Current off the Flemish Cap) are strongly influenced by details of topography. The causes are not well understood, but most likely related to the staircase nature of the topogra-

272


[ -:,w

c I----T=

-71111

'I'.O

'71s

HI.,

LIYCI

28r11~ ~ i c i

Fig. 7.9 Time-mean bottom layer flow in the northern North Atlantic for DYNAMOMOM, DYNAMO-SPEM, DYNAMO-MICOM (DYNAMO Group, 1997).

273


0.0

- 100.0 -200.0

-300.0 -400.0

-500.0 -600.0

--700.0 -800.0 -!300.0

- 1000.0 - 1100.0 - 1200.0

- 1300.0 - 1.1-00.0 Latitude [ d e g N

-5

L

A K . - z m x z3

01

20

30

40

01

70

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30

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80

90

100

200

100

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-200 0 -300 0 -400 0

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Fig. 7.10 Time-mean potential vorticity (in units of lO-"rn-'s-') along 3OoW from (a) climatological data (Levitus, 1982) and (b) DYNAMO-MOM (DYNAMO Group, 1997).

274


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