Comparison of Numerical Methods

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Comparison of numerical methods for modelling ocean

circulation in basins with irregular coasts

by

Frdric DUPONT

Department of Atmospheric and Oceanic SciencesMcGill University

Montreal

A thesis submitted to theFaculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree ofDoctorate in Philosophy

Frdric DUPONT, Juin 2001

'

ABSTRACT


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Comprehension of global oceanic currents and, ultimately, of climate variability requiresthe use of computer modelling. Although much effort has been spent on the accuracy oftraditional finite difference (FD) models used in ocean modelling, there are still concerns,especially since these models have a crude representation of the geometry of oceanicbasins. Such a crude representation may influence the accuracy of modelling boundary

currents, or unrealisticly represent the impinging of eddies or the propagation of Kelvinwaves along the coastline. This motivated the use of alternative modelling techniquesapplied on completely irregular geometries such as finite element (FE) and spectral element(SE) methods. In this thesis, we want to investigate the accuracy and cost-effectiveness ofthese three numerical methods in irregular domains and to understand to which extent theunstructured grid FE and SE methods constitute an improvement over the more traditionalFD methods. To accomplish this, we limit ourselves to modelling the shallow waterequations in presence of irregular coastlines with no bottom topography.

In the first part of the thesis, we compare the performances of FD methods on Cartesiangrids with FE and SE methods in various geometries for linear and nonlinear applications.

We argue that the SE method is to a certain extent superior to FD methods. In a secondpart, we study the influence of step-like walls on vorticity budgets for wind-driven shallowwater FD models. We show that vorticity budgets can be very sensitive to the FDformulation. This has certain implications for using vorticity budgets as a diagnostic tool inFD models. In the final part, we use a SE shallow water model for investigating the``inertial runaway problem'' in irregular domains for the single-gyre Munk problem.Ideally, one would like the statistical equilibrium observed at large Reynolds number to beinsensitive to model choices that are not well founded, e.g., the precise value of the viscouscoefficient, and choice of dynamic boundary condition. Simple models of geophysicalflows are indeed very sensitive to these choices. For example, flows typically converge tounrealisticly strong circulations, particularly under free-slip boundary conditions, even at

rather modest Reynolds numbers. This is referred to as the ``inertial runaway problem''. Weshow that the addition of irregular coastlines to the canonical problem helps to slowconsiderably the circulation, but does not prevent runway.

RSUM

La comprhension des courants occaniques globaux et, ultimement, de la varibilitclimatique requiert l'usage de la modlisation numrique. Bien que beaucoup d'effort ait tdpens dans l'amlioration des modles traditionnels aux differences finies (DF) utilissdans la modlisation ocanique, il reste des interrogations concernant la prcision de cesmodles, et ce d'autant plus que ces modles ont une reprsentation trs grossire de lagomtrie des bassins ocaniques. Une telle grossire reprsentation peut modifier laprcision des courants le long des frontires, ou mal reprsenter le choc des tourbillons sur,ou la propagation des ondes de Kelvin le long de la frontire. Ceci a motiv l'utilisation desmthodes numriques alternatives comme les lments finis (EF) ou les lments spectraux(ES) qui s'appliquent des gomtries compltement irrgulires. Dans cette thse, nousvoulons tudier la prcision et le cot de ces trois types de mthodes numriques dans desdomaines irrguliers et comprendre jusqu' quel point les mthodes EF et ES fonctionnantsur des grilles irrgulires constituent un progrs compar aux mthodes DF traditionnelles.


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Dans ce but, nous nous limitons modliser les quations en eaux peu profondes enprsence des ctes irrgulires sans topographie.

Dans la premire partie de cette thse, nous comparons les performances des mthodes DFsur des grilles cartsiennes avec les mthodes EF et ES dans des gomtries diffrentes

pour des problmes linaires et non-linaires. Nous argumentons que la mthode ES est,dans une certaine mesure, suprieure aux mthodes DF. Dans la seconde partie, noustudions l'influence des marches d'escalier prsentes le long des murs sur les budgets devorticit pour des modles DF en eaux peu profondes forcs par le vent. Nous montronsque les budgets de vorticit peuvent tre trs sensibles la formulation DF utilise. Ceci acertaines implications concernant l'utilisation des budgets de vorticit comme outil dediagnostique dans les modles DF. Dans la dernire partie, nous utilisons un modle ES eneaux peu profondes pour tudier le problme de ``fuite inertielle'' dans des domainesirrguliers pour le problme de Munk non-linaire. Idalement, on voudrait que l'quilibrestatistique observ grand nombre de Reynolds soit insensible au choix fait concernant desapproximations mal--ou peu--fondes du modle, comme la valeur du coefficient de

viscosit turbulente ou le type de condition frontire. Les modles simples de fluidesgophysiques ont en effet tendance tre trs sensibles. Par exemple, l'coulementconverge vers des circulations totalement irralistes, particulirement pour une conditionfrontire de glissement libre, et ce mme pour des nombres raisonnables de Reynolds. C'estce que l'on nomme ``fuite inertielle''. Nous montrons que l'inclusion de ctes irrguliresdans ce problme canonique permet de ralentir considrablement la circulation, maisn'limine pas pour autant le problme de la fuite inertielle.

CONTRIBUTIONS

In Chapter 2, we develop our own adaptive spectral element method which is an automatic

procedure for assessing local errors and increasing, accordingly, the resolution of the mesh.Test and use of this procedure are made in Chapters 3 and 5. We also develop our owncurved spectral element method for better representing smoothly varying coastlines.

In Chapter 3, we develop a series of test cases in order to study the convergence withresolution of the accuracy and cost-effectiveness of each scheme in regular and irregulardomains. The originality of this approach stems from the variety of numerical methods wetest and compare, and the thorough study of the influence of the resolution on them. Weexplore the limitations of each numerical scheme.

In Chapter 4, we use vorticity budgets as a way to assess the accuracy of differentnumerical formulations for modelling wind driven ocean gyres in a rectangular basin. Inparticular, we demonstrate that, for finite difference formulations, the advective terms inthe vorticity budget do not integrate to zero. This error can be exacerbated by the presenceof near step-like structures along the boundary, such as those that occur when a straightcoastline lies at an angle to the coordinate axes used for discretization. It is further foundthat this problem is minimized for certain numerical choices relating to the treatment of theadvective and viscous terms.


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In Chapter 5, we use a spectral element model to investigate the inertial runaway problem(i.e., that models produce unrealistically strong flows as dissipative parameters are reducedtowards what are considered realistic values) in irregular domains. In particular, we showthat small scale (but resolved) features in the coastlines lead to the generation of fine scalestructure in the vorticity field, where the Rossby number can become of order unity, and

the quasi-geostrophic approximation becomes suspect. That these occur under free-slipboundary condition contrasts the classic, rectangular basin case. We find that small scalestructures in the coastline act to slow, but not to stop inertial runaway.

Contents List of Figures List of Tables

Introduction Presentation of the Numerical Methods

o The Time Discretizationo Finite Difference Models

Introduction The Three Staggerings Used

The C-grid Formulation The B-grid The A-grid

o Finite Element Models Introduction

The Galerkin Formulation The Different Finite Element Models Tested

The Quoddy Model The Peraire et al. (1986) Model The Hua and Thomasset (1984) Model The Le Roux et al. (2000) Model

o The Discontinuous Spectral Element Method Introduction The Model Formulation Adaptive Mesh Refinement Curved Spectral Element Method

o Summary Testing the Different Numerical Methods

o Gravity Waves in a Square Domaino The Wind-driven Circulation in a Circular Domaino Conservative Properties of the Different Numerical Formulations for a

Nonlinear Problemo The Munk Problem in a Square Domaino Conclusions
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Finite Difference Methods in Rotated Basinso Introductiono Vorticity Budgets in a C-grid SW Model

The General Form of the Discretized Vorticity Budget Numerical Formulations

Resultso Vorticity Budgets in SW B-grid Modelso The Quasi-Geostrophic Model

Discretization Results

o Discussion and Conclusion Single Gyre Circulation in Irregular Domains

o Review of the Single Gyre Problem with Free-Slip Boundary Conditionso Model Selection and Experimental Designo Results

General Results for all Geometries

Role of the Transients for Geometry V at High Reynolds Numbero Scale Analysis and Discussiono Adaptivityo Conclusions

Conclusions an A-grid energy conserving formulation Model vorticity budget on a B-grid Bibliography About this document ...

Contents

Contents List of Figures List of Tables Introduction Presentation of the Numerical Methods

o The Time Discretizationo Finite Difference Models

Introduction The Three Staggerings Used

o Finite Element Models Introduction The Galerkin Formulation The Different Finite Element Models Tested

o The Discontinuous Spectral Element Method Introduction The Model Formulation Adaptive Mesh Refinement
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Curved Spectral Element Methodo Summary

Testing the Different Numerical Methodso Gravity Waves in a Square Domaino The Wind-driven Circulation in a Circular Domain

o Conservative Properties of the Different Numerical Formulations for aNonlinear Problemo The Munk Problem in a Square Domaino Conclusions

Finite Difference Methods in Rotated Basinso Introductiono Vorticity Budgets in a C-grid SW Model

The General Form of the Discretized Vorticity Budget Numerical Formulations Results

o Vorticity Budgets in SW B-grid Models

o The Quasi-Geostrophic Model Discretization Results

o Discussion and Conclusion Single Gyre Circulation in Irregular Domains

o Review of the Single Gyre Problem with Free-Slip Boundary Conditionso Model Selection and Experimental Designo Results

General Results for all Geometries Role of the Transients for Geometry V at High Reynolds Number

o Scale Analysis and Discussiono Adaptivityo Conclusions

Conclusions an A-grid energy conserving formulation Model vorticity budget on a B-grid Bibliography

List of Figures

o Effect of the rotation on the discretization of a square domain.o Effect of a poor resolution on the geometry of a strait.o Elevation field for the Kelvin retardation problem in presence of steps along

the walls at two different resolutions.o The three major horizontal staggerings for the primitive equations.o Triangulation of the domain.

o , the basis function related to the nodeMi.

o is the basis function related to the nodex2=0.
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o The discontinuous linear non conforming basis function for theP1NC -P1element of Hua and Thomasset hua84.

o The discontinuous constant basis function for pressure over the macro-element of LLS.

o Local non-orthogonal coordinates in a given triangle

o Example of Legendre polynomialso Remeshing strategies. The triangle to be refined is in grey.o Transformation of one triangle intro a curved triangleo Transformation of one triangle intro a curved triangle with the coordinate

system used in the computation of the integralso The wave test experimento Convergence with resolution of the normalized error in the u-component for

second order C-grid formulation (FDM), O-FDM4 and R-FDM4 models.o CPU cost with the normalized error in u-component for the second order C-

grid (FDM) and R-FDM4 models.

o The four FE models (LW, HT, PZM, LLS) are tested against the analyticalsolution with increasing resolution.o Convergence of the normalized error in v with respect to the resolution for

the LW-FE and SE modelso Convergence of the normalized error in u with respect to the resolution for

the C-grid FD, LW-FE and SE models.o Variation in the u-component normalized error as a function of CPU cost

for five models.o Grids for the circular domain for the FD models.o Convergence with resolution of the normalized elevation error for the

second order C-grid FD, O-FDM4 and R-FDM4 models in a circular

domain.o Elevation error in a circular domain for four FE models.o Normalized elevation error for the C-grid FD, LW-FE and SE models for a

circular domain.o Total energy after 18 days of simulation for the C-grid FD and the lumped

LW, delumped LW, PZM and LLS FE models and the SE model for thegeostrophically balanced eddy.

o Elevation field after a six year simulation in a non-rotated basin using O-FDM4 and R-FDM4.

o As for Figure 3.13but for rotated basin.o Kinetic energy during a 6 year spin-up for the C-grid FD, the lumped LW,

HT, PZM and LLS FE models.o Elevation field after a 6 year spin-up for the C-grid FD, the lumped LW,

delumped LW, HT, PZM and LLS FE models for the single gyre windforcing problem.

o Single gyre wind forcing experiments for the delumped LW FE modelcompared to the C-grid FD model.

o Kinetic energy during spin-up for the single gyre Munk problem withm2s-1 for the C-grid FD, the delumped LW FE and SE models.
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o Elevation field for the SE model after 6 years from spin-up for the singlegyre Munk problem with m2s-1, nc=5.

o As for Fig. 3.18but with m2s-1.o As for Fig. 3.19but with m2s-1.o Convergence with resolution for the nonlinear Munk problem of the

normalized kinetic energy error for the solution from the C-grid FD and theSE models.o As for Fig. 3.22but for the convergence of the normalized error with CPU

cost.o Solutions after a 6 year spin-up for the Munk problem using the adaptive SE

model with nc=5.o Locations of variables near a step for the SW C-grid model and the QG

model.o Local advective flux along the boundary at 20 km resolution in a square

basin for the enstrophy conserving formulation.o Northward flow past a forward step. The shaded area is the model domain.

o Elevation fields in meters after a 6 year spin-up for 20 km and 10 kmresolution. Shown are results from the A and B combination with or withouta 3.44o rotation angle of the basin.

o (a) Kinetic energy after spin-up and (b) ratio of to for the fourcombinations combinations.

o Kinetic energy after spin-up for the B combination in 1010 m5/s2.

o Ratio of to for the B combination.

o Convergence of with resolution for 0o, 20o, 45o rotation angle for the Bcombination.

o Kinetic energy during spin-up for six runs using theJ1 Jacobian at 6different rotation angles.

o Kinetic energy after spin-up for (a)J3 at 0o rotation, (b)J7 at 0o, (c)J3 at30o, (d)J7 at 30o, (e)J3 at -30o, (f)J7 at -30o.

o Ratio of to the wind input for (a)J3 at 0o rotation, (b)J7 at 0o, (c)J3 at30o, (d)J7 at 30o.

o Ratio of to the wind input. (a-d) as described in Fig. 4.10.

o Ratio of to the wind input. (a-d) as described in Fig. 4.10.

o Semi-advective flux, < , and beta contribution, , to the vorticitybudget for theJ3 Jacobian at -30o rotation angle.

o Notation corresponding to the curvilinear coordinates.o The five geometries used for our application of the SE method.o Elevation fields in the Geometry V for the C-grid model after 3 years of

spin-up.
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9/50

o Total energy during spin-up for the A and B combinations of the FD C-gridmodel and for the SE model at nc=5 (SPOC 5) in Geometry V.m2s-1.

o Mean elevation fields for the five geometries using the SE model.o Mean vorticity field for the Geometries IV and V using the SE model.

o Power input by the wind using the mean fields with respect to the boundaryReynolds number.o Mean standard deviation of the elevation.o Mean standard deviation of the elevation for frequencies with period above

200 days.o Mean standard deviation of the elevation for frequencies with period

between 17 and 200 days.o Mean standard deviation of the elevation for frequencies with period

between .6 and 17 days.o Hovmller diagram of the filtered elevation with respect to time and

location along the boundary.o Amplitude of the Kelvin wave in meters along the boundary averaged over6 years.o Time series of the amplitude of the oscillations at (x=500 km, y=0 km).o Total energy for the last 6 years of simulation.o Instantaneous vorticity field in the vicinity of the recirculation.o (a) Local wind input to the vorticity in Geometry V. (b) Local divergence of

the eddy transport of vorticity in Geometry V. (c) Vector plot of the eddytransport of vorticity normal to the streamlines in Geometry V.

o The region in grey represents where the absolute vorticity is approximatelyconserved.

o Maximum of the mean elevation for the three geometries. The maximum

elevation is a good proxy for the strength of the recirculation.o Kinetic energy of the mean fields with respect to the boundary Reynolds

number.o Mesh for the original and the refined runs.o Total energy for the last 6 years of simulation for the original and refined

meshes.o Amplitude of the fast oscillations at (x=500 km, y=0 km) along the

boundary for the original and refined meshes.o Mean elevation fields for the original mesh and the refined mesh.o Mean vorticity fields for the original mesh and the refined mesh.

List of Tables

o List of variables in (2.1-2.2)o Refinements parameters used in the simulations unless otherwise specified.
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o Convergence order for the different models for the linear wave experimentin a square domain. For all models, the order is fairly close to theirtheoretical value. Models using unstructured grids lost almost an order forthe error in v compared to the error in u.

o Convergence order in elevation, for the different models for the linear wind-

driven experiment in a circular domain without Coriolis terms.o Convergence order for the different models for the nonlinear Munk problem

in a square domain.o The four combinations of advection formulations and stress tensor

formulations.o Summary of the vorticity budget and kinetic energy diagnostics for the B-

grid after a spin-up of 6 years.o Notations for the finite volume method

ACKNOWLEDGEMENTS

Many thanks go to my supervisors, Charles Lin and David Straub, for their support and theproof-reading of the thesis. David Straub is in particular acknowledged for his importantcontribution to Chapters 4 and 5.

The C2GRC group at McGill was a great place for meeting and listening to world-renowned scientists. I must thank in particular Lawrence Mysak for inviting me along onnumerous occasions for after-seminar beers or suppers. Many McGill PhD students left thedepartment in the course of my studies and I am indebted to them for engrossing scientificconversations we had around a few beers. Thanks therefore go to Rob Scott, HalldorBjornsson, Martin Charon, Daniel Le Roux, Bruno Tremblay, Daniel Bourgault, StephenNewbigging and three cornerstones of the department, Rick Danielson, Werner Wintels

and Marco Carrera.

Deep thanks go to Catherine Mavriplis for discussion about error-estimators in adaptivespectral element methods and Julien Dompierre for the remeshing strategies. StephenNewbigging, Ted Tedford, Drew Peterson and Jason Chaffey corrected the spelling and theEnglish grammar. CERCA (Centre de recherche en calcul appliqu) provided me with anexcellent working environment. Many thanks go to the CERCA staff, scientists and otherPhD students in general for their support and discussions.

I must also thank my father for motivating me to do a PhD in North-America despite thegreat distance and stretched family bounds implied by doing so.

This research was supported by NSERC and CCGCR. VU (developed at CERCA) GMT,Gnuplot and Ferret are the graphical packages used in this thesis.

Introduction

Modelling the ocean has become an essential component of coastal and navigational hazardprevention (beach erosion, pollutant transport, tidal or storm surge, ice drift, wave height).
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Moreover, the ocean being a large component of the global climate, to model its circulationis essential to obtaining a better understanding of the dramatic climatic changes whicheither have occurred or might occur. Finally, some theoretical studies require the use ofmodels in order to understand fundamental physical processes which involve nonlineardynamics and/or complex geometries, and which are beyond analytical approaches.

The first attempts to model the ocean circulation were made in the middle of the 20thcentury, after the work of Ekman Ekman05 who recognized the importance of the wind asthe major source of mechanical forcing. Sverdrup sverdrup47 derived a simple law thatrelates the ocean currents to the curl of the wind friction. Stommel stom48 and Munkmunk50 derived analytical models of the wind induced ocean circulation in closedrectangular basins using simplified dissipative laws. Both the Stommel and Munk modelsafford simple explanation of the westward intensification of oceanic currents, such asobserved for the Gulf Stream or the Kuroshiwo. An important threshold in computerperformances was reached in the late sixties, and this allowed for the first full prognosticthree-dimensional studies of the ocean circulation [Bryan and Cox, 1967,Bryan, 1969].

These models were driven by mechanical forcing (the winds and a bottom drag) and byfluxes of salt and heat exchanged with the atmosphere. They could take into account thecomplexity of the geometry and the nonlinear nature of the oceanic currents. Ideally, thesemodels should be able to fill the gaps in the data and give reasonable estimates of the oceancirculation. However, because of their inherent complexity, the poor knowledge ofnumerous physical processes and problems with the definition of coastlines, straits and sea-mounts, they drift easily from any reasonable state if no restoring terms are added to theequations for temperature and sanility. Thus, prognostic three-dimensional models maysometimes look like expensive interpolators and yield no very different results than simplerinverse models do. Nonetheless, they have produced useful estimates of the role of theoceans in the thermal global budget and the importance of the so-called conveyor belt.

Our main concern is the representation of irregular domains in numerical ocean models,and their influence on the dynamics of the currents. Models, so far, have only crudelyrepresented these irregular boundaries, either in the vertical (the topography) or in thehorizontal (the coastlines). Our objective in this thesis is to evaluate the accuracy ofconventional numerical methods in the presence of irregular coastlines and to introducemore accurate alternatives.

Furthermore, we suspect that irregular coastlines have important but sometimes under-estimated influences of the dynamics of the currents flowing along them. The energetics ofthese currents are controlled by the transfer of energy to smaller scales by nonlinearinteractions. These interactions are likely to take place along the western part of theoceanic basins where the currents are the strongest. In particular, as the geostrophic balance(the main assumption governing the dynamics of the currents) in these regions breaksdown, we hint at possible interactions between the geostrophic and ageostrophic modes.We therefore focus on models which are simple enough to represent geostrophic-ageostrophic interactions on a large range of scales. A shallow water model seemsappropriate for these goals. The dynamics are only two dimensional which allows for veryhigh resolutions in the horizontal directions, as opposed to more complex three-
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dimensional models where the same level of resolution would be too expensive. In oneparticular context, we introduce a quasi-geostrophic (QG) model for comparison. Thisparticular type of model represents only the geostrophic motions. Previous studies focusingon nonlinear interactions were usually done in very idealized and regular domains usingQG models. Hence, we hope to make an interesting contribution by conducting relatively

simple experiments involving idealized but still irregular coastlines and somewhat morecomplex dynamics compared to QG models.

The problem of accurately representing the geometry of the domain in ocean models isdivided in two sub-problems: representing the bottom boundary (the topography) andrepresenting the lateral boundaries (the coastline). Topographical features (sills or sea-mounts) are essential to the mixing of waters of different properties, origins and depths,and, therefore, their influence extends to the largest scales. Coastlines partially enclose theoceanic basins. Their presence is essential to the comprehension of the oceanic currents(such as the westward intensification of currents). It was early realized that a crude verticalrepresentation of the topography (z- or geopotential vertical coordinate) could be

detrimental to an accurate modelling of the ocean circulation. In particular, waters ofdifferent properties tend to mix over sills with dramatic consequences for the global oceancirculation if the vertical discretization is too crude. To remedy this problem, verticalterrain following coordinates were proposed, despite various known limitations. However,the horizontal discretization has not received the same level of scrutinity. Most of themodern oceanic models still crudely represent coastlines. A crude horizontal discretizationhas several consequences. Straits may be under-resolved and the associated exchange ofwater modified: The strait of Gilbratar controls the Mediterranean salt input into theAtlantic; the Bering strait controls the freshwater input between the Arctic and the Pacificand the Indonesian archipelago is notorious for being the location of the so-called returnflow of the vast thermo-haline conveyor belt which circles the globe. A crude horizontal

representation has also retardation effects for fast oceanic modes (Kelvin waves) whichpropagate along coastlines [Schwab and Beletsky, 1998]. For the wind-driven circulation,little is known about the influence of crude horizontal representations.

In order to study the influence of the choice of the numerical method, we propose to testseveral of them, and investigate which one best handles irregular coastlines. We thereforepropose to test different staggerings of the finite difference (FD) method and several finiteelement (FE) formulations against a spectral element method. The test-cases we choose arevery idealized in order to focus only on the dynamical aspects of two-dimensional flows(no physical parameterizations except for constant dissipative coefficients) and range fromsimple linear and non-linear test-cases in square domains, to linear and non-linear test-cases in smoothly irregular domains. The finite difference models range from theconventional Arakawa C-grid (preferentially used for regional studies -- e.g. Bleck andBoundra, 1981; Blumderg and Mellor, 1983), to the conventional Arakawa B-grid(preferentially used for global studies as in Bryan-Cox derived models -- e.g. Bryan, 1969;Cox, 1984), to the unconventional A-grid (Dietrich et al., 1993).

We choose to use the FE method because it has the decisive characteristic of representingboundaries more efficiently than the more conventional FD method and because it presents
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variable resolution capabilities. Oceanographers, especially the tidal community use FEmodels to represent tidal interactions and resonances which occur at different scales, fromocean basin to coastal inlets [Connor and Wang, 1974,Lynch and Gray, 1979,Walters andCheng, 1979]. Some of the modern FE models are derived from the earlier modelsformulated by Lynch and Werner lynch87 and Le Provost et al. lepro94. Others used the

QG approximation and proposed a finite element formulation of the vorticity equation forthe general ocean circulation [Fix, 1975,Dumas et al. , 1982,Myers, 1995]; results werevery encouraging. Unfortunately, no general circulation model based on the primitiveequations (explained below) has been proposed based on the FE method and we try todetermine the reasons for this relative failure. On the other hand, the spectral elementmethod (an extension of the finite element method) has been used with a relative successby Iskandarani et al. iskan95. Their method is based on quadrangular elements; instead, wefavor the use of triangular elements which offer increased geometrical flexibility.Specifically, we propose to test a spectral element model based on this discretizationtechnique. The apparent advantage of the spectral element method lies in the acceptedadvantage of spectral methods (the accuracy and the fast convergence with increasing

resolution for regular problems) and the flexibility of an irregular grid. However, as withthe spectral method, there is always the possibility that Gibbs oscillations appear when thefields being approximated are too irregular or under-resolved (the classical example is thestep-function). We try to solve this problem by use of an adaptive method which increasesthe resolution (the number of triangles) in regions where the largest errors in the solutionare observed (to be defined later). Finally, since finite element methods are potentiallymore costly than conventional finite difference methods due to the need for more matrixinversions, the spectral element method may be a good alternative because its enhancedaccuracy (compared to finite elements) is not severely offset by an excessive cost. In orderto verify this statement, we give an accuracy-to-cost function for all models.

Modelling the ocean is very challenging due to the coexistence of many physical processesat various spatial and time scales, from the lowest scales (salt intrusion and viscousboundary layers of a few centimeters), to surface waves induced by wind or wavebreakings, tides, geostrophic eddies, to the general ocean circulation. Since all theseprocesses can interact with each other, it is virtually impossible to reproduce and isolatewith a high degree of realism any of these processes. Most often, approximations andparameterizations are used to represent the small scale (or sub-grid) phenomena and limitthe explicit motions of the model to the scales of interest. In this hierarchy ofapproximations, the barotropic QG model represents the leading largest scale and lowestfrequency approximation. There is no vertical structure and no horizontal divergentmotions such as gravity waves. Then, somewhat more complex is the shallow water model.

The variables are the horizontal velocity, (u,v) and the elevation of the free-surface, . Itallows for divergent motions but still does not permit vertical structure. Layered models areextended versions of shallow water models and allow for crude vertical (baroclinic)variations. For more realistic vertical structures, the so-called primitive equations are used.They are based in the incompressible Navier-Stokes equations and use the Boussinesq andhydrostatic assumptions. Further improvement can be gained by using an non-hydrostaticmodel which can represent the small scale convection. However, the limitation imposed bycomputer performance fixes the length scales and the physical processes which can be
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explicitly resolved. Some features, such as synoptic eddies, are very difficult to resolve inglobal circulation models. These eddies are of the order of 10 to 100 km and are relativelysmall compared to the basin scale (10,000 km). Nonetheless, some authors [Holland andLin, 1975,Treguier, 1992] stress the importance of representing explicitly the role of theeddies in the transfer of energy between the different scales and their positive influence on

the (more realistic) mean fields. These processes can not be perfectly mimicked by thealternative strategy of using eddy-parameterizations and, therefore, this strategy is arguedto be flawed [Lesieur, 1997]. Moreover, these parameterizations use coefficients difficult toadjust to real observations when these coefficients are not simply ``cosmetic''. Therefore, agood general ocean model should be eddy-resolving. However, since the requiredresolution for a general circulation model of the ocean is too high (10 km at mid-latitudes),models should have, at least, variable resolution capabilities, in the sense that they shouldhave capabilities to follow and resolve isolated eddies or westward boundary currents,while the rest of the domain is discretized at a coarser resolution.

It may be important to represent other physical processes. The Topex-Poseidon satellite

mission, for instance, renewed interest in the surface ocean large scale activity: tides,Kelvin and Rossby waves and synoptic eddies whose signature were measurable on thesurface elevation field of the ocean. Hence, a good general or regional ocean model shouldalso have a moving free-surface which allows for fast barotropic gravity waves. This wasalso pointed out by studies of the vertical eddy-viscosity over the rough topography of theAtlantic ridge [Polzin et al. , 1997]. The large values of the eddy-viscosity observed overthe ridge, probably induced by external tides, imply that the general circulation mustinteract with the tides, i.e. the QG physics interacts with the large scale gravity waves(ageostrophic horizontally divergent dynamics). From a theoretical point of view, it seemsalso more and more necessary to include ageostrophic motions in numerical models, evenin the extra-tropics. The difficulty comes from explaining the cascade of energy down to

the molecular viscosity scale where the energy can be dissipated. Indeed, the twodimensional (and QG) dynamics tend to cascade the energy up to the Rhines arrest scale(50 to 200 km) in typical basins and not down. Therefore, there is no clear mechanism thatcascades down and dissipates the energy in QG dynamics. This mechanism may come fromthe non-linear interactions between the geostrophic and ageostrophic modes. This may bevisible from spectral analysis [Stammer, 1997] which show no particular cut-offfrequencies or wave-numbers separating geostrophic and ageostrophic modes 1.1.

The presence of irregular coastlines may also be important for the interactions of thegeostrophic and ageostrophic modes. First, because it provides a forcing source at variouswave numbers and, moreover, because the westward side of ocean basins is the locationwhere the geostrophic approximation is most likely to break down, i.e., where the transferof energy is most likely to occur. These preceding arguments imply that general circulationmodels should allow for the interactions between the geostrophic and ageostrophicmotions. The simplest system that allows for such interactions is the shallow waterequation system.

Also for physical reasons, even eddy-resolving models need an explicit parameterization ofdissipation. Although very crude, this is usually done through an explicit eddy-viscosity
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(Laplacian operator) parameterization. Such a parameterization requires an arbitrary choicefor the dynamical boundary condition at the walls (a problem which is exacerbated whenhigher order dissipations are employed). We consider herein only two boundary conditions.One is the free-slip boundary condition and corresponds to fluids being free to slip alonglateral boundaries. There is some ambiguity as to the precise definition of free-slip.

Pedlosky (1987, p. 183) takes the point of view that it corresponds to there being noviscous flux of tangential momentum across the boundary (i.e., atthe wall). We take a less stringent definition by simply forcing the normal derivative of the

tangential velocity to be zero (for instance, on a meridionally oriented wall).The latter choice is the one generally found in the literature. On straight walls, the twodefinitions are equivalent, and correspond to vertical vorticity vanishing at the wall. Oncurved boundaries, either definition mentioned above results in non-zero relative vorticity1.2. This bears some dynamical consequences that we discuss below. The second boundarycondition is the so-called no-slip boundary condition which corresponds to fluids that donot slip along walls (the tangential velocity is zero), and leads to strong shear along walls.

This boundary condition is considered to be the ``real'' one because it is the one observed inlaboratory experiments at microscales. However, at the resolution used in modern oceanmodels (1 to 100 km), it is not clear which, if either, boundary condition is appropriate. Thepresent trend in ocean modelling is to go towards higher Reynolds number (smaller lateralviscosity) and higher resolution along with no-slip boundary conditions. But the level ofresolution is still very far from sufficient to represent realistic viscous boundary layers,although inertial boundary layers are definitely becoming more realistic. On the other hand,authors experienced problems with the free-slip boundary condition. In an idealized squarebasin and barotropic ocean, it is observed that under single-gyre wind forcing anddecreasing eddy-viscosity, the oceanic currents tend to jump to unrealisticly high valueswith no signs of transient eddies, whatever the resolution of the model is. Hence, the free-

slip boundary condition prevents the transient activity which usually allows for reasonablemean currents by transporting the excess of negative vorticity from the interior, through theinertial layer to the walls, where it is dissipated [Pedlosky, 1996]. Using a barotropic QGapproximation, Dengg dengg92 showed that free-slip flows tend not to separate from acape compared to the clear separation observed with no-slip flows, whatever the value ofthe wind forcing. For all of these reasons, it seems safer to use the no-slip boundarycondition, even if it requires unrealisticly large viscosity values in order to resolve theboundary viscous sub-layer. Nonetheless, these studies fail to realize some importantissues. The absence of transients and separation, under the free-slip boundary condition, isconnected to the fact that most of those models fail to produce relative vorticity at thewalls. This is because the relative vorticity at the boundary is specified to be zero for

reasons of simplicity. Therefore, those studies fail to note that, even under free-slip, flowscan produce relative vorticity simply because of the coastline curvature. Hence, absence oftransients in a square basin is due to the idealized straig

Comparison of Numerical Methods

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