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Excerpts from the proposal for a 5 year Action de recherche concertées programmedevoted to the development of a

Second generation model of the ocean system

(research due to start in September 2004)

Eric Deleersnijder1,5, Thierry Fichefet2, Vincent Legat3 and Jean-François Remacle4

(1): SC/PHYS/ASTR, Institut d'astronomie et de Géophysique G. Lemaître, Universitécatholique de Louvain (UCL), 2 Chemin du Cyclotron, Louvain-la-Neuve, Belgium(2): SC/PHYS/ASTR, Institut d'astronomie et de Géophysique G. Lemaître, Universitécatholique de Louvain (UCL), 4 Avenue G. Lemaître, Louvain-la-Neuve, Belgium(3): FSA/MECA/MEMA, Unité de mécanique appliquée, Université catholique de Louvain(UCL), 4 Avenue G. Lemaître, Louvain-la-Neuve, Belgium(4): FSA/AUCE/GCE, Génie civil et environnemental, Université catholique de Louvain(UCL), 1 Place du Levant, Louvain-la-Neuve, Belgium(5): Spokesperson, Tel: +32-(0)10.47.23.63, E-mail: ericd@astr.ucl.ac.be

Contents:1. Motivation and objectives ...................................................................................22. The need for unstructured grid ocean models.....................................................43. A brief account of the state of the art..................................................................74. The development of the Louvain-la-Neuve ocean model ..................................85. The development of the Louvain-la-Neuve sea-ice model ..............................136. Europa: an extra-terrestrial oceanography test case .........................................167. Parallel computing .............................................................................................188. Synergy between the promoters ........................................................................199. References..........................................................................................................21

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1. Motivation and objectives

In 1969, the Journal of Computational Physics published a seminal article by K. Bryanpresenting the first ocean general circulation model (OGCM). Since then, many numericalstudies of the World Ocean, including those aiming at predicting climate change, usedOGCMs directly inspired by Bryan's work. A few of these models have emerged as highlymodular, well documented and widely used tools. These models are chiefly (in alphabeticalorder):

1. MICOM1 (and HYCOM): the Miami isopycnic coordinate ocean model;2 . MOM2: the modular ocean model of NOAA's Geophysical Fluid DynamicsLaboratory (Princeton);3. OPA3: the “océan parallélisé” model of the Laboratoire d'Océanographie Dynamiqueet de Climatologie (Paris).

Other OGCMs have been designed and used for specialised purposes, such as CLIO4, theCoupled Large-scale Ice-Ocean model (Deleersnijder and Campin 1995, Goosse and Fichefet1999), which was developed at UCL's Institut d'astronomie et de géophysique GeorgesLemaître (SC/PHYS/ASTR), and is now part of ECBILT-CLIO5, an intermediate complexitymodel6 of the Earth's climate system.

Over the last 3 decades, significant progress has been made in the parameterisation ofsubgrid-scale processes, data assimilation, as well as implementation on fast vector andsubsequently parallel computers, allowing high space resolution. However, today's OGCMs,such as those mentioned above, may still be regarded as members of the first generation ofocean models: rather similar geophysical fluid mechanics equations are solved numericallyusing a conservative finite-difference method on a structured grid.

Some aspects of OGCMs are now deemed to be out of date. In particular, the use of astructured grid (Figure 1a) leads to a marked lack of flexibility in the space resolution anddoes not allow taking advantage of the potential of modern numerical methods, such as finiteelements. The latter should be implemented on unstructured grids (Figure 1b), as wasrecommended by the participants of recent workshops7. Though OGCMs have evolvedsignificantly since Bryan's prototype, it is impossible to modify them step by step from astructured-grid approach to an unstructured-grid one. Therefore, a revolution in OGCMdesign is needed, paving the way for the second generation of ocean models. Thedevelopment of such an OGCM has been undertaken by only a few groups worldwide,including one at UCL8. 1 See http://oceanmodeling.rsmas.miami.edu/micom/micom.html2 See http://www.gfdl.gov/~smg/MOM/MOM.html3 See http://www.lodyc.jussieu.fr/opa/4 See http://www.astr.ucl.ac.be/tools/clio.html5 See http://www.knmi.nl/onderzk/CKO/ecbilt.html6 See http://www.pik-potsdam.de/data/emic/table_of_emics.pdf7 The 1st and 2nd International Workshops on Unstructured Grid Numerical Modelling of Coastal, Shelf andOcean Flows were co-organised by Eric Deleersnijder, and were held in Louvain-la-Neuve and Delft, TheNetherlands, on 4-5 November 2002 and 23-25 September 2003, respectively.8 At Louvain-la-Neuve, the Institut d'astronomie et de géophysique G. Lemaître (SC/PHYS/ASTR) and the Unitéde mécanique appliquée (FSA/MECA/MEMA) have joined forces to do the first steps towards a second-generation OGCM. This initiative was supported by UCL's FSR (Fonds spéciaux de recherche), under projects“Développement d'un modèle de circulation générale océanique de seconde génération pour l'étude du climat

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(a) (b)

Figure 1. Structured (a) (Deleersnijder et al. 1993, 1997) and unstructured (b) (Legrand et al.2000) grid discretisations of the Gulf of Mexico and a region of the North Atlantic9.

So far, the UCL group has dealt with some of the elements of an OGCM using the finite-element method on an unstructured grid, i.e. grid generation for the World Ocean andnumerical methods for treating the Poincaré wave equations and the advection terms. Manymore building blocks are needed, which pertain to sea-ice, parameterisation of subgrid-scaleprocesses, modern advection schemes, data assimilation10, adaptive grid, parallel computing,etc.

The new model will be much more flexible than present day's OGCMs, so that it will beable to serve a wide range of purposes. For instance, adaptive grid resolution will allow theresolution of an extended range of time and space scales of motion: it is expected that thesame model configuration will be able to resolve the global ocean circulation and some of themeso-scale eddies11 — which are the most energetic processes in the World Ocean. Thoughshelf seas are believed to play, through their high biological productivity, an important role inthe carbon cycle, they are generally ignored in OGCMs; it will be possible to include themwith a relevant resolution in the domain of interest of our second-generation OGCM. Extra-terrestrial applications, chiefly the ice-covered global ocean of Europa12, will also beconsidered, offering a unique opportunity to test the model with parameters very differentfrom terrestrial ones. No single ocean, marine or environmental flow model available at the

terrestre I and II” (from 1 October 2000 until 30 September 2004). The promoters of these projects are E.Deleersnijder and V. Legat.9 Unstructured grids are not generated from a coordinate system, implying thate they are unlikely to exhibit asingularity at the Poles or anywhere else, while structured grid discretisations are generally plagued by such aproblem (Williamson 1979, Murray 1996).10 We are conducting research of an exploratory nature in this domain, which is funded by UCL's FSR (Fondsspéciaux de recherche), under project “Contribution of the assimilation of satellite data to sea-ice modelling”(from 1 October 2002 until 30 September 2004). The promoters of this research programme are T. Fichefet andE. Deleersnijder.11 Nowadays, these eddies are resolved in OGCMs running over periods of time much too short to addressclimate change problems.12 Europa is a moon of Jupiter.

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present time is capable of addressing all of the problems mentioned above with the accuracyand computer efficiency that our second-generation model will offer.

National and international collaboration, in an open-source mode, is crucial for our modelto thrive. A number a colleagues have already indicated that they are interested in using ourmodel or contributing to its development.

To summarise, the main objective of the present project is to build on the exploratory workthat has been done under the auspices of UCL's FSR to

develop an ocean model that will be based on an adaptive, unstructured grid,will rely on the finite element method and will resort to parallel computingalgorithms to solve the governing equations.

The model will include:• an embryonic unstructured-grid sea-ice component,• state-of-the-art parameterisations of subgrid-scale processes,• a flexible vertical discretisation system,• various advection schemes,• an elementary sea-ice data assimilation system.

The model will be ableto simulate the circulation in the World Ocean for studying the Earth's climate system,

and will also be tested for a range of geophysical flow problems, including the• assimilation of sea-ice data,• circulation in the shelf break region,• representation of an extended range of space and time scales,• exploration of an extra-terrestrial oceanography problem.

All aspects of the research work will be carried out in anopen-source mode, relying on the close collaboration with colleagues outside UCL.

2. The need for unstructured grid ocean models

In structured grids, vertices can be identified by incrementing or decremeting integer indices,which leads to computer programmes which are easy to develop, understand and use.Unstructured grids offer no such advantages. As a consequence, developing an unstructuredgrid model demands much more efforts. It was deemed worth doing it in many domains ofcomputational fluid dynamics, and ocean modellers are progressively realising that theadvantages of unstructured grid models by far offset the extra costs of development and use,as is explained below.

Global structured grids are usually based on geographical coordinates, implying thepresence of singular points at the geographical poles, or somewhere else. In the vicinity ofthese points, numerical schemes are unlikely to remain stable (Williamson 1979). Manytechniques were suggested to deal with this difficulty within the framework of structuredgrids. So far, essentially three types of solutions have been implemented in OGCMs:

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combined grids as equatorial transforms (Deleersnijder et al. 1993, Eby and Holloway 1994,Coward et al. 1994), grids generated semi-analytically (Madec and Imbard 1996) and gridsgenerated analytically (Murray 1996, Bentsen et al. 1999).

Figure 2. The space-time windows in which the most important marine and ocean processesoccur13. Figure courtesy Prof. Hans von Storch.

Some of the drawbacks due to the rigidness of all these structured grids cannot becircumvented. First, the staircase representation of coastlines causes some spurious formstress on model boundary currents (Adcroft and Marshall 1998) and grids based on boundary-fitted coordinate systems work only for regional simulations — while our main application isthe modelling of the global ocean for studying Earth's climate evolution. Second, the rigidnessof structured grid makes it difficult to achieve strong grid refinement in specific regions,which is necessary to deal properly with equatorial dynamics, western boundary currents,meso-scale eddies, continental slopes and shelves, etc. For instance, meso-scale eddies arepresent in a relatively small fraction of the World Ocean, but contain a very significant part ofthe kinetic energy of the World Ocean (Figure 2). Therefore it is desirable that they be wellresolved, which would demand a horizontal grid size of the order of 10 km. Nowadays, no 13 “Meso-scale eddies” and “geostrophic eddies” are expressions that can be considered as equivalent. The so-called “ocean general circulation” consists essentially of “circulation cells” and “thermohaline circulation”.

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computer — except, perhaps, Japan's Earth Simulator14 —, is able to run a model with such aresolution for a few hundreds to a few thousands of years, as is necessary to study the Earthclimate. The issue of resolving meso-scale eddies is only one aspect of a more generalproblem: the OGCMs, as well as other components of models of the Earth's climate system,should be able to represent a much larger range of space and time scales of motions (Figure2), which would make them less dependent on parameterisation of subgridscale processes, thereliability of which is more often than not quite questionable.

One might argue that relying on nested, structured grids would provide a flexible, variableresolution grid system while retaining the advantages inherent to structured grids (e.g. Spalland Holland 1991, Fox and Maskell 1995). This is not entirely so. Nested grids have beenused to represent localized phenomena, such as western boundary currents, or to findphysically realistic boundary conditions for a high resolution model. In the latter case, theonly purpose of the low resolution model is to give open sea boundary conditions to the highresolution model.

The main drawback of nesting techniques is that some interpolation procedure is needed totransfer information from one grid to the other. The transfer can be one- or two-way,depending on the model application. Interpolation between grids is not a straightforwardprocedure as it has to be conservative and should not introduce numerical errors that couldspoil the high resolution model accuracy. In particular, avoiding reflection of waves on theboundary between grids is far from easy. Moreover, since finite difference models are oftenexplicit, the stability constraint on the time step may be constraining as it is determined by thehigh resolution grid.

Unstructured grid models do not have those problems. No interpolation procedure isrequired and conservation issues do not arise. The stability constraint on the time step isusually less annoying as finite element models allow the use of implicit time integrationschemes at no additional cost. Moreover, unstructured grids remain much more flexible thatnested structured grid. Once a finite element code is implemented, the grid can be changed,refined and plugged back in the model very easily.

The governing equations15 of an ocean model can be solved on an unstructured grid byvarious techniques, which belong to three classes, i.e. the finite volume, finite element andspectral methods. In our opinion, the finite volume method may be regarded as a special typeof finite elements, so that there is no real need for drawing a boundary between these twotechniques. For instance, this is readily seen when a discontinuous Galerkin technique isbeing used (e.g. Hanert et al. 2004). We opted for the finite element method for two distinctreasons. First, the Louvain-la-Neuve unstructured grid OGCM is being developed byscientists from essentially 2 research groups of UCL; in one of them (SC/PHYS/ASTR) therewas no prior expertise in unstructured grid methods, whereas the other group(FSA/MECA/MEMA) has been using finite elements for decades. Therefore, the finiteelement method was the obvious choice. On the other hand, finite elements enjoy a rigorous

14 See http://www.es.jamstec.go.jp/esc/eng/15 Boussinesq approximation equations for the overall budget of mass, momentum budget, heat budget, andequations for the concentrations of all the relevant constituents of seawater which can include tracers / scalarvariables, such as CFCs, pollutants, nutrients, plankton, etc.

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mathematical formulation based on a weighted residual formulation, rendering it possible tocontrol the accuracy, paving the way for adaptive grid techniques.

3. A brief account of the state of the art

Besides the Louvain-la-Neuve group, there are several research teams developingunstructured grid models for marine and oceanic applications. Only a few of these modelsfocus specifically on large-scale, deep-ocean problems, i.e. FENA and SEOM. All of them arestill in their infancy, with the noticeable exception of QUODDY and, to a certain extent,FENA and SEOM. These models may be classified according to the type of numericalmethod resorted to:

Finite Element models:1. FENA: The FENA model is developed at the Bremerhaven site of the Alfred Wegener

Institute for Polar and Marine Research. It is a three-dimensional finite element model thatsolves the primitive equations and advection-diffusion equations for scalar quantities. P1

linear shape functions are used for horizontal velocity, elevation and tracers, and thevertical velocity is interpolated with constant shape functions. Such a finite elementscheme requires a stabilization procedure to avoid numerical oscillations. The model hasbeen assessed in simulations of the North Atlantic circulation (Danilov et al. 2004).

2. QUODDY: The QUODDY model is developed at Dartmouth College. It is a 3D, fullynon-linear finite element model with advanced turbulence closure, which has been usedfor relatively small-scale problems only. Horizontal velocity, elevation and tracers areinterpolated with P1 linear shape functions. Spurious oscillations problem is avoided bysolving a wave equation for the elevation. However, by treating a wave equation for theelevation, overall mass conservation is lost. This might not represent a major problem forcoastal applications, but it is a very serious issue for large-scale and long-termsimulations. The QUODDY model does not require the use of a linear solver as masslumping is performed (Lynch et al. 1996), an aspect of the QUODDY which may be seenas questionable.

3. Imperial College model: The research team of Dr Chris Pain at Imperial College iscurrently developing a three-dimensional, non-hydrostatic, finite element model. Themain feature of this model is that it will offer sophisticated adaptive mesh refinement16.

4. Walters and Casulli (1998) model: They developed a 2D finite element shallow watermodel. The model is based on the RT0 finite element pair. A semi-Lagrangian method isused to represent advection. A wave equation for the elevation is dealt with at the discretelevel to decouple the problem. This model is expected to exhibit drawbacks somewhatsimilar to those of QUODDY.

5. Miglio and Quarteroni (1999) model: Like Walters and Casulli, they also developed a 2Dfinite element shallow water model based on the RT0 finite element pair. A semi-Lagrangian method is also used to represent advection effects. They solve the discrete

16 See http://amcg.th.ic.ac.uk/research/

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system of equations without building a wave equation for the elevation. Lumping ishowever performed to avoid having to use a linear solver.

Finite Volume models:1. FVCOM: This model is developed at the University of Massachusetts-Dartmouth. It is a

finite volume, 3D, primitive equation ocean model with a Mellor-Yamada turbulenceclosure. It has been used for coastal and estuarine applications (Chen et al. 2003).

2 . TU-Delft model: The research team of Prof. Julie Pietrzak at Delft University ofTechnology17 is currently developing a three-dimensional finite volume model to dealwith coastal and shelf problems. A semi-Lagrangian scheme is used for transport terms.

3. Casulli and Zanolli (2002) model: They developed a three-dimensional, non-hydrostaticNavier-Stokes model able to use unstructured grids. The model is based on a finitedifference-finite volume algorithm.

Spectral Element models:1. SEOM: The SEOM model is developed by the research team of Prof. Dale Haidvogel at

Rutgers University and Prof. Mohamed Iskandarani at the University of Miami. SEOM isa high order spectral element model. It solves the 3D primitive equations and advection-diffusion equations for scalar quantities on an unstructured grid made of quadrilaterals(Iskandarani et al. 2003). Using higher order interpolations provides very good accuracy ifthe grid is regular, and the domain boundaries and forcing are smooth. In an irregulardomain with realistic forcings, a spectral model requires an important amount of diffusionto prevent the occurrence of spurious oscillations.

4. The development of the Louvain-la-Neuve ocean model

What has been done

The development of the Louvain-la-Neuve unstructured grid OGCM is by no means behindothers. A step-by-step strategy has been adopted, in which every building block of the modelis studied in detail. A first version of the complete OGCM does not exist yet, but should beassembled 2 to 3 years after the start of the present research programme — if funded.

Unstructured grid generation on the sphere was the first element of the Louvain-la-NeuveOGCM that was dealt with (Legrand et al. 2000). A Watson incremental method wasdeveloped to generate automatically “2D horizontal” boundary-fitted Delaunay triangulationsof the World Ocean. Earth's curvature was taken into account, and local mesh refinement wasmade available so as to resolve topological or dynamic features such as mid-ocean ridges,equatorial dynamics, western boundary currents, etc. Special attention was paid to the qualityfactor of the triangulation.

The discretisation of the linearised shallow water equations was then considered, as thelatter are a simplified version of the external mode of a free-surface, three-dimensional —hydrostatic — OGCM (Hanert et al. 2002). Thanks to their inherent simplicity, those

17 See http://fluidmechanics.tudelft.nl/

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equations are the traditional test case problem to select spatial discretisation of the externalmode: they have been widely used for assessing finite difference or finite element schemes.

We have compared the triangular finite element counterparts of the finite difference A-, B-,C- and CD-grids. The corresponding finite element pairs are respectively denoted P1P1, P1P0,RT0 and P1

NCP1. Numerical experiments have first been performed to simulate the propagationof inertia-gravity and Rossby waves. Mass and energy conservation, as well as stabilisationissues has been considered. By performing the test case introduced by Batteen and Han(1981), it was seen that the P1P1 and P1P0 finite element schemes allow the existence ofspurious pressure modes. As for the finite difference B-grid spurious modes (e.g. Batteen andHan 1981, Killworth et al. 1991, Deleersnijder and Campin 1995), the finite element pressuremodes are high frequency oscillations in the elevation field that are not “seen” by the scheme.If they are not filtered out, they may significantly spoil the solution. Typical stabilizationprocedures consist in adding a diffusion term to the mass equation. Those numerical artifactsfilter oscillations but are physically questionable. The RT0 finite element pair allows theexistence of spurious velocity modes. They may be filtered out by adding diffusion to themomentum equations. This makes more sense as, in any case, there must be momentumdiffusion in the flows we are interested in. However, the amount of diffusion needed toprevent the occurrence of oscillations is generally bigger than the one dictated by the physicsof the flow. Finally, the P1

NCP1 finite element pair is free of spurious pressure and velocitymodes. As a result, it does not require any stabilization procedures. This discretisation hasthus been selected for our model.

Some aspects of unstructured grid advection schemes ocean processes have also beenstudied (Hanert et al. 2004). The circulation strongly depends on the density contrasts, whichin turn are due to gradients of temperature and salinity. The latter variables are advected anddiffused. Therefore, the discretisation of the advection-diffusion diffusion is a key issue, inwhich the advective part is by far the most difficult to deal with. Four advection schemes havebeen compared and assessed in the context of ocean modelling by solving a scalar transportequation. Schemes under consideration included continuous, non-conforming anddiscontinuous finite elements and finite volumes. From the test case suggested by Hecht et al.(1995), it appeared that discontinuous and non-conforming finite elements were the mostpromising approaches.

Numerical scheme developments

Three-dimensional grids will be generated. Since the ocean aspect ratio is very small andsince the gravitational acceleration plays a prominent role in oceanography, it is believed thatthe vertical direction should be preserved. This is why we will generate 3D meshes of theglobal ocean by extrusion of our surface triangulations, leading to prismatic elements whoselateral walls belong to vertical planes. In each prism, the lower and upper interfaces will bedefined in a flexible manner, that will allow a smooth representation of the ocean bottom andsurface and, if needed, will be fitted as much as possible to isopycnal surfaces. Such a systemwill be the unstructured grid counterpart of the generalised vertical coordinate which tends tobe used nowadays in finite difference models (e.g. Griffies et al. 2001 and the refs. therein).

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Considerable efforts will be devoted to advection schemes, according to a discontinuousGarlerkin approach, in order to obtain schemes that are monotonic whatever the spacevariations of the variable to be advected. A set of advection modules will be made available,allowing the users to select the one that is the most appropriate for the flow to be simulated.Clearly, designing a satisfactory advection scheme is as difficult for an unstructured grid as itis for a structured one. Transport processes, as simulated over a wide range of time and spacescales by the model, will be diagnosed using numerical tools such as the Constituent-orientedAge Theory (CAT18), which was developed by Delhez et al. (1999) and Deleersnijder et al.(2001).

A range of sophisticated 3D turbulence closures now exist for geophysical fluid flowproblems (Burchard 2002a), which include the k-ε and Mellor-Yamada models. Significantefforts were devoted to the finite difference discretisations of the related equations, so as toobtain turbulence variables that are positive and free of spurious oscillations (Deleersnijderand Luyten 1994, Burchard and Deleersnijder 2001, Burchard 2002b). Similar issues will beaddressed in finite element modelling. However, it is not yet clear whether the solutionsretained in finite difference schemes can be adapted in a straightforward manner to finiteelement discretisations. Every finite element discretisation will be assessed by means of someof the test cases suggested by the GOTM19 team.

In ocean models, small-scale, three-dimensional turbulence leads essentially to verticaldiffusion. There are however motions with much larger horizontal scales and small aspectratio which are not taken into account in the turbulence models mentioned above. Thesesubgrid-scale processes are usually parameterised as horizontal or quasi-horizontal diffusion,with eddy coefficient much larger than those prevailing for three-dimensional turbulence. It isnow clear that horizontal diffusion with constant eddy coefficients is not appropriate, even inmodels resolving meso-scale eddies (Roberts and Marshall 1998). Isopycnal diffusion andbolus velocity terms are needed (Redi 1982, Gent et al. 1995). The finite differencediscretisation of the related differential operators poses monotonicity problems, for which noentirely satisfactory solution exist to date (Mathieu and Deleersnijder 1997, Beckers et al.1998, Beckers et al. 2000). One may expect that designing an accurate, monotonic andconservative finite element discretisation of isopycnal diffusion and bolus velocity terms willdemand considerable efforts.

A common problem in numerical modelling is the interpolation of data from one mesh toanother. This issue arises for instance when initialising a model with reinterpretedclimatological data. Seeking inspiration in inverse modelling and data assimilation (Rixen etal. 2000), we are developing a mesh-to-mesh interpolation procedure based on a variationaltechnique. The cost function to minimise includes an elliptic regularisation operator and adata proximity constraint. Other constraints can be added to the cost function, for instance, inorder to remap an incompressible velocity field. This is being tested in line with the studies ofDeleersnijder (2001) and Legrand et al. (2003).

18 Unfortunatley, the expression “Constituent-oriented Age Theory” and the related acronym ”CAT” was notused since the inception of this theory; in fact, CAT was introduced in articles which are yet to be published.19 GOTM = General Ocean Turbulence Model; see http://www.gotm.net

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Interpolation or field remapping is an essential ingredient in adaptive grid techniques. Theefficient use of parallel computers is unlikely to be sufficient to capture all the scales wewould like to resolve, such as meso-scale eddies and seasonal or inter-annual displacementsof polar fronts and western boundaries currents. Numerical schemes that produce variableresolution in both space and/or time are called adaptive (e.g. Remacle et al. 2002a, Remacle etal. 2003). In various domains of computational physics, adaptivity has been dealt with since along time. Automatic Mesh Refinement (AMR) schemes have proved quite successful inastrophysics (Calder et al. 2000), fluid dynamics (Remacle et al. 2003), reactive flows (Mirinet al 1999), etc. The AMR techniques are based on non-conforming refinement of uniformstructured grids. Though efficient, these methods cannot be extended to unstructured grids.The use of more complicated unstructured mesh adaptation technologies will enable us toproduce adaptive, spatially varying discretisations. In the time domain, techniques of localtime stepping will be investigated in order to resolve the multiple time scales adaptively.

Adaptive mesh procedures can be divided into three stages (e.g. Remacle et al. 2002b).First, a relevant measure of the accuracy of the solution is required. This error estimate, whichwill be used to define a remeshing criterion, is usually based on the Hessian matrix of themodelled fields. However, when discontinuous discretisations are used, another promisingerror indicator can be derived from the solution jump between two adjacent elements. In thesecond stage, a new mesh is generated in order to obtain a uniform distribution of therefinement criterion for all elements. This is usually achieved by nodes displacement (r-adaptivity) or nodes insertion/destruction (h-adaptivity). Finally, essentially for the r-adaptivity, all the modelled fields have to be transferred from the old mesh to the new one,which should be achieved under appropriate constaints such as preserving the total mass of atracer or the fact that the velocity is divergence-free. Although we already have developedsome of the tools required for dealing with adaptive mesh methods, we still have to define anerror estimator and a refinement criterion relevant to oceanic flows.

Our model must be able to simulate a wide range of time and space scales in a singleimplementation. As a consequence, at least in some regions of the domain of interest, it is notunlikely that the model will have to deal with phenomena exhibiting an aspect ratio that is notsmall, by contrast to most large-scale ocean processes. For such processes, which may beconvection or flows over continental slopes, it is desirable that the hydrostatic assumption berelaxed. For simulating non-hydrostatic processes at an affordable computational cost,inspiration may be found in the studies of Marshall et al (1997) and Marshall et al. (1998).

Applications of the unstructured grid OGCM

Numerous tests will be made to assess every building block of the model, or the model itself— once it will be assembled. In addition to the test cases alluded to above, we will use thoseavailable on the web20, and, perhaps, develop a certain number of test cases by ourselves.

The main objective of our OGCM is to simulate the circulation in the World Ocean forstudying the Earth's climate system. At the end of the research programme our unstructuredgrid OGCM should be able to run for thousands of years over the World Ocean, which will

20 See e.g. http://www.ocean-modeling.org/frames.php

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demand an appropriate parallel computing implementation. The circulation and location of thewater masses should be comparable to those obtained in a finite-difference model with aresolution of about 1 1° × ° resolution in latitude and longitude, and 30 levels over the verticaldirection, i.e. the standard — non-eddy resolving — finite-difference model used nowadaysfor climate purposes. Other applications will be selected as a function of their potential fortesting the ability of the model to deal with motions ranging over a wide spectrum of time andspace scales.

Adaptive grid techniques will be used to simulate an extended range of space and timescales of motions. This might be applied to a variety of problems — including locally non-hydrostatic flows. One of them will be the representation of meso-scale eddies in a basin- orglobal-scale model. Most of the domain will be discretised by a relatively coarse grid, but thegrid generation system will have to refine the resolution when and where eddies are likely todevelop. This type of numerical experiment should demonstrate the flexibility of the modeland its advantages over nesting techniques.

In the long run, we believe that the domain of interest of unstructured grid OGCMs willinclude the continental shelves with an appropriate resolution. It is our intention to do steps inthis direction, mainly by attempting to simulate complex flows arising in the shelf breakregion, which are responsible for the exchanges of matter, momentum and energy betweenoceans and shelf seas. The detailed description and simulation of the hydrodynamics of oceanmargins is critical for the accurate assessment of nutrient fluxes between productive shelfareas and nutrient-rich oceanic waters — at depth. It is also vital for the downscaling of largescale climate change scenarios and the prediction of the local impacts of environmentalchanges. Finally, the shelf seas are likely to play a significant role in the carbon cycle (Walsh1988). These issues were addressed in depth in several inter-disciplinary studies (e.g. Walsh1988, Monaco et al. 1990, Biscaye et al. 1994, Brink and Cowles 1991, Souza et al. 2001,Wollast and Chou 2001).

The interaction of barotropic and baroclinic processes — tides, slope currents, filaments,non-hydrostatic processes and meso-scale instabilities of the shelf break fronts — with thesteep and intricate topography is responsible for the very complex hydrodynamics of the sloperegions (e.g. Huthnance et al., 2002). The shelf break region is, by definition, at the interfacebetween the deep ocean and the shelf seas. While specific models are available and appliedroutinely for these two different environments, their interaction and the quantitativeassessment of the associated ocean-margin exchanges cannot be taken into account by thesimple exchange of boundary conditions between these models. Thus, the challenge is todesign models that not only cover both type of environments separately, deep ocean and shelfsea, but also take into account the particular dynamics of the shelf break. In this context, wepropose to examine the benefits of an adaptive unstructured grid approach. The adaptivecharacter of the method, its ability to increase the resolution locally or implement specificparameterisations for local processes will be investigated.

The study will be carried out in a simplified framework or applied to specific locations inwhich members of MARE21 have been conducting research for a long time: the Northwest

21 Two of our external partners, Jean-Marie Beckers and Eric J.M. Delhez, are members of MARE, theinterfacultary center for marine research of the University of Liège. See http://www.ulg.ac.be/oceanbio/MARE/

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European Atlantic margin, or the Northwest Iberian seasonal upwelling region of the Gulf ofLions. MARE will support the project by providing related — public domain — data, as wellas the necessary scientific guidance and assistance.

5. The development of the Louvain-la-Neuve sea-ice model

The Earth's climate system is made up of several interacting constituents, the atmosphere, theWorld Ocean, the cryosphere, the lithosphere and the biosphere. To date, no climate modelcontains a detailed description of all of the aforementioned constituents. However, every“decent” climate models results from the coupling of a number of models of the constituentsof the climate system. Clearly, the coupling of these models is much easier if the models to beput together have been designed while bearing in mind that they will have to be coupled toeach other. At Louvain-la-Neuve, CLIO has demonstrated the usefulness of jointlydeveloping an OGCM and a sea-ice model. The new unstructured grid model developmentsshould remain true to this successful tradition.

The Louvain-la-Neuve Ice Model (LIM)

Sea-ice, which results from the freezing of seawater, is a key element of the cryosphere as itprofoundly modifies the surface heat, mass, and momentum exchanges at high latitudes. Dueto its high albedo and low thermal conductivity, sea-ice alters the radiative and turbulentcomponents of the surface heat balance, cutting the absorption of short-wave radiation by asmuch as 80% in the case of snow-covered ice and reducing the turbulent heat fluxes by one totwo orders of magnitude. In addition, sea-ice intercepts most of the snow falling duringwinter, thus preventing it from immediately contributing to the ocean freshwater balance.Finally, a thick and compact ice pack also hinders the free exchange of momentum betweenatmosphere and ocean. Another facet of the influence of sea-ice on climate is related to thefact that it alters the seasonal cycle of the atmospheric and oceanic fields in polar regions. Inthe first place, the release and absorption of latent heat that respectively accompany thegrowth and decay phases of the ice cover tend to delay the seasonal surface temperatureextremes. In the second place, the influx of salt into the ocean during ice formation and thefreshwater input during ice melting tend to alter the density structure of the upper ocean andhence the ocean thermohaline circulation. The coupling of the ice drift to the aforementionedprocesses introduces new important phenomena. On the one hand, ice motion is responsiblefor the formation within the ice cover of open water areas — called leads — through whichair-sea exchanges are much enhanced compared to ice-covered regions. On the other hand, icemotion combines with ice production and destruction to create a net annual polewardtransport of heat in the atmosphere and of salt in the ocean.

It is believed that a number of positive and negative feedbacks linking changes in sea-iceto variations in surface albedo, cloud cover, atmospheric water vapour, and oceanthermohaline circulation play an important part in determining the Earth’s climate and itssensitivity to external and internal perturbations. Indeed, climate change scenarios over the

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21st century carried out with atmosphere-ocean general circulation models suggest that thelargest surface warmings will occur over polar regions, especially in the NorthernHemisphere. However, these regions are also those where inter-model discrepancies aremaximum (IPCC, 2001), so that many scientists agree on the necessity of refining therepresentation of polar regions, and notably sea-ice, in climate models.The sea-ice processes that need to be taken into account in a comprehensive sea-ice model canbe divided into three categories: (1) thermodynamic processes, (2) dynamic processes, and (3)processes that couple ice thermodynamics and dynamics. The thermodynamic component ofthe model determines the freezing/melting rate of ice given the atmospheric and oceanic heatand mass fluxes. Ice thermodynamics can be subdivided into two components: one associatedwith vertical processes and one dealing with lateral processes. The former aims at calculatingthe accretion/ablation rates at the top and bottom boundaries of the ice, whereas the lattercomputes the changes in ice concentration that result from ice-ocean heat fluxes in leads. Thedynamic component of the model consists of a momentum balance equation, whichdetermines the ice drift as a function of the atmospheric and oceanic dynamical forcings, andan ice rheology, which prescribes the dependence of internal ice stresses on ice deformationand ice thickness characteristics. The thermodynamic-dynamic coupling is done via (1) anadvection scheme, which works out the transport of volumetric ice variables, such as icevolume and sensible and latent heat contents, and (2) the ice thickness distributionformulation. The ice thickness distribution describes the evolution of the different thicknesscategories into which ice breaks as a result of dynamical processes. Such as a model has beendeveloped in Louvain-la-Neuve during the nineties. A brief description of this model may befound in Fichefet and Morales Maqueda (1997, 1999).

Sea-ice model developments

It is necessary to first improve the finite-difference version of the Louvain-la-Neuve Ice Model(LIM), so that it remains at the forefront of large-scale sea-ice modelling22. Then, anunstructured version of LIM will be developed; in this respect, the expertise in unstructuredgrid rheology of one of the promoters of the present project, Vincent Legat, will be a majoradvantage. Finally, the unstructured ocean and sea-ice models will be coupled, leading to theunstructured version of CLIO.

The thermodynamic component of the model — which today has three layers only — willbe replaced by a multi-layer model that explicitly takes into account the effects of sea-icesalinity on the specific heat, thermal conductivity and latent heat of the ice. For this, we willbasically follow the approach of Bitz and Lipscomb (1999), who recently built such a model.In addition, we will incorporate in the model the formulation proposed by Vancoppenolle(2003) for the spatial and temporal evolution of the sea-ice salinity. In this scheme, theevolution of the sea-ice salinity is modelled following simple treatments of the physical

22 LIM is used in several institutes in Europe, as may be seen from the web sites http://www.lodyc.jussieu.fr/opa/and http://www.knmi.nl/onderzk/CKO/workshop.html. This is why it is deemed to be appropriate to continue thedevelopment of its finite-difference version until the finite-element one will be validated and demonstrablybetter. Then, it is conceivable that the finite-difference version will be progressively abandoned. But, this willtake about a decade, or perhaps even more.

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mechanisms of salt entrapment during ice formation and brine drainage within the ice. Giventhe strong dependence of the ice growth/melt rate on the ice thickness and since the physicalproperties of sea-ice vary widely from one ice type to the other, the inclusion of various icetypes and ice thickness categories in LIM is imperative. The relevant ice types that can bepresent in 100-km-wide oceanic areas — which is the typical size of grid cells of large-scalesea-ice models — are: open water, frazil ice, pancake ice, level and ridged first year ice, andlevel and ridged multiyear ice. We will modify LIM so that each grid cell can accommodatethese various ice types. Each ice type will be characterised by its own snow and ice thicknessdistributions and surface and bottom properties — e.g., surface albedo and drag coefficients.Evolution equations for each ice type and thickness category will have to be formulated, andredistribution functions between the different ice types and thickness categories will have tobe designed. This work will be done in collaboration with Dr. J. Haapala from the Universityof Helsinki, Finland, who has developed a model of this type for the Baltic Sea (Haapala,2000).

Two new sea-ice rheologies have been recently developed: the elastic-viscous-plastic(EVP) rheology of Hunke and Dukowicz (1997) and the granular material (GM) rheology ofTremblay and Mysak (1997). Compared to the popular viscous-plastic rheology (Hibler1979), the EVP one has the advantage that it allows the use of a fully explicit numericalscheme, which improves the model computational efficiency and makes it easier toimplement the model on parallel computers. Regarding the GM rheology, it has thepeculiarity of modelling explicitly the formation of leads by shearing deformation. Moreover,it has been coded on C-grid, which permits a better simulation of the ice drift through straitsand along coastlines. Within this project, we will implement both the EVP and GM rheologiesin LIM and test their performance at the global scale.

All of the above-mentioned modifications will be assessed one by one by conductingsimulations of the Arctic and Antarctic sea-ice evolution over the last 50 years by means ofLIM coupled to the French oceanic general circulation model ORCA — which is the globalimplementation of the model OPA mentioned above. In these experiments, the coupled modelwill be driven by the most accurate atmospheric reanalysis data available at the beginning ofthe project. Results from these runs will be thoroughly compared to in-situ and satelliteobservations. Sensitivity experiments will also be performed in order to find the optimalmodel configuration for climate studies.

Another way of improving the performance of LIM is to assimilate data into the model.This takes advantage of the great advances that have been made in polar observationalcapabilities during the last two decades. These advances have led to a rich collection of dataon sea-ice, including, among others, satellite passive microwave observations of ice motionand concentration. Since it is only very recently that the assimilation of data into large-scalesea-ice models has been initiated, we propose here to test some of simplest and cheapest dataassimilation techniques such as nudging, optimal interpolation and an elementary version ofKalman filtering. In order to reduce as much as possible the computational cost, we willevaluate these techniques in a simplified version of LIM using twin experiments, i.e.numerical experiments that assimilate model outputs instead of real observations. On the basisof these experiments, we will select the best method and will implement it into the full

16

version of LIM coupled to the French oceanic general circulation model ORCA. Previousstudies on the assimilation of data into sea-ice models showed that problems can arise whenassimilating ice motion data alone or ice concentration data alone (e.g. Thomas et al. 1996,Meier et al. 2000). Therefore, we will assimilate both types of data. Also, in order to have thebest possible space-time coverage over the recent past, we will choose the ScanningMultichannel Microwave Radiometer (SMMR) and Special Sensor Microwave/Imager(SSM/I) satellite data. Two experiments will then be conducted with the coupled model overthe last two decades23: one with data assimilation and the other without data assimilation.Comparison between results of the two experiments and observations will allow us toevaluate the performance of the data assimilation system. It is noteworthy that theseexperiments should also contribute to answering the important question: “Is the Arctic sea-icethinning?” (Rothrock et al. 1999). All this work will be performed in collaboration with Drs.M. Drinkwater from the European Space Agency, Noordwijk and R. Ezraty from IFREMER,Brest, who have a great deal of expertise in collecting and analysing satellite sea-ice data. It isalso worth stressing that this study will be of great use for the French project of operationaloceanography MERCATOR24, in which LIM in an essential component.

Finally, we will examine the feasibility of solving the ice momentum and transportequations on unstructured grids with finite element techniques. In particular, the finiteelement formulation of sea-ice dynamics in the Lagrangian description recently developed byWang and Ikeda (2003) will be assessed on various test cases and its adaptation to the physicsof LIM will be considered. This step constitutes a prerequisite for coupling LIM to the finiteelement oceanic general circulation model to be built within the project. The sea-ice dataassimilation system will also be adapted to the unstructured grid sea-ice code.

6. Europa: an extra-terrestrial oceanography test case

There is a growing demand for extra-terrestrial ocean/ice modelling, in particular forEuropa. The latter is the second Jovian moon — starting from Jupiter. Its size is close to thatof the Earth’s Moon, but it is lighter — about 0.65 of the Moon’s mass. Europa is at about671,000 km from Jupiter, with an orbital period of 85.2 hours. Its state is described as“Cassini” state, which implies that its rotation period is about the same period as its revolutionperiod; as for the Earth's Moon, only one side is visible from Jupiter. From 1973, it has beenobserved by 5 space missions studying Jupiter, leading to some surprising results (e.g.Erickson et al. 2000):• Europa probably has a H2O layer at its surface, about 300 km deep;• Europa’s surface is very young;• part of the water seems to be liquid, under a thick ice layer;• there is an atmosphere at Europa’s surface.

The probable existence of liquid water at the surface of Europa is very exciting for theplanetary science community. Indeed, the search for extra-terrestrial life is presently the

23 The period during which relevant sea-ice satellite data are available.24 See http://www.mercator-ocean.fr/

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hottest topic of planetary science, and the presence of liquid water is considered as one of thekey ingredients of life (Kargel et al. 2000). Feasibility analysis is now in progress for futuremissions entirely devoted to the study of the ocean of Europa (Gershman et al. 2003).

It could appear surprising to have liquid water on Europa, as its surface is at a temperatureof about -170 °C; the radiogenic decomposition and the tidal friction bring heat in thethermodynamic budget, and the ice layer at the top reduces the loss of energy at the surface(Hussmann et al. 2002, Spohn and Schubert 2003). Theoretical thermodynamic models haveshown that the ice layer can be expected to be 30 km thick, leaving a liquid ocean of about270 km in depth.

The rotation of Europa presents librations, which are oscillations of Europa around itsrotation axis, due to the gravitational interaction between its equatorial elliptical bulge andJupiter. It has been observed that the rotation of Europa is not perfectly regular. At the RoyalObservatory of Belgium, the rotation of the Earth and other celestial bodies is being studied,focusing on the effect of fluid layers on the rotation. For the Earth, the atmosphere, the oceanand the fluid core are the major sources of rotation fluctuations. Studying the dynamics of thefluid is essential in the understanding of the rotation of the Earth and of other bodies.Conversely, the study of the rotation has been used in order to provide constraints on the fluiddynamics. For instance, the value of the magnetic field at the boundary between the solidinner-core and the liquid outer-core has been evaluated from the nutation modelling.

Preliminary results — as yet unpublished — have been obtained by analysing the angularmomentum budget of Europa, with very simple hypotheses on its dynamics, assuming threehomogeneous layers, i.e. ice-ocean-solid, rotating rigidly. It has been shown that (1) thelibrations of Europa are very sensitive to the presence of an ocean, (2) the characteristics ofthe libration motion depend on the ocean depth, (3) it is also unfortunately very sensitive tothe method used to include the rigid rotation of the ocean in the angular momentum budget.The latter findings implies that we need to drastically improve the ocean modelling, in orderto add a dynamically consistent variation of the angular momentum of the ocean in order toget a relevant model of the rotation dynamics.

An ocean model for Europa

At the present time, we have almost no information about the ocean on Europa, and none ofthem are direct. Consequently, it may seem to be difficult to build anything useful to study itsdynamics. Nevertheless, the observations we have and the thermodynamic models, togetherwith our knowledge of the Earth Ocean, provide us with a set of necessary constraints:• The ocean is probably about 270 km deep.• Pure water is not the only constituent of the ocean, as there is a variable magnetic field at

its surface; NH3 is one of the likely candidates from cosmogenic considerations.• The gravitational forcing, and consequently an important part of its short term dynamics

are well known from celestial mechanics consideration.• Unlike the Earth, the ocean is not much heated by the Sun, and receives internal heating.• The ocean is most likely global.

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Using those constraints, it is possible to study the ocean dynamics with a not-too-broad setof parameters. This would be enough to get useful information on the rotation dynamics ofEuropa, and thus on the perturbations of its librations. A flexible ocean model such as thatbeing built in Louvain-la-Neuve will be implemented to the restrained set of parameters,allowing a more precise libration modelling. Conversely, when the librations will be betterobserved, we will be able to test it with the simulated ocean dynamics, which will provide uswith restriction to the possible dynamics. So far, it is not clear how the ice layer should bedealt with and how it affects the ocean dynamics.

Modelling the ocean-ice layer of Europa entails a number of highly speculative aspects, butthis should be no reason for doing nothing25. In any case, Europa provides a uniqueopportunity for testing our model with a set of parameters that are completely different fromthose of the Earth. This would allow us to demonstrate the flexibility of our model, which isjust its main intended strength.

7. Parallel computing

Resolving a wide range of spatial and temporal scales in oceanic flows — while keeping thecomputational cost at an acceptable level — presents a formidable challenge for algorithmicdevelopment. Only parallel computing is able to take up such a challenge. The past few yearshave seen a revolution in high performance scientific computing26. Clock rates of processorshave increased from about 40 MHz (e.g. a MIPS R3000 in 1988) to over 3 GHz (e.g. aPentium 4 in 2003). Parallel computation is now a well-established tool for large scalescientific computations (Mirin et al. 1999, Calder et al. 2000, Shingu et al. 2002).

PC clusters have become a popular tool in parallel computing, essentially because of theirrelative cheapness. Such tools are available at an affordable price in many universities,including UCL27. While some problems may be parallelised using compiler technology(Lester 1993), the vast majority of them, including ours, must rely on explicit parallelisation.The related issues lead to an exciting and promising research subject per se.

Here are the key rules that have to be taken into account upon designing an algorithm forparallel computation:1. The most crucial rule is that the serial algorithm, i.e. the algorithm that runs on a single

processor, has to be optimal. It is actually useless to render parallel an inefficient serialalgorithm. The effective performance of a programme on a computer relies not just on thespeed of the processor but also on the ability of the memory system to feed data to the

25 In his address opening the international symposium organised to celebrate the 50st anniversary of UCL'sGeorges Lemaître big-bang cosmology (Louvain-la-Neuve, October 1983), André Deprit, from the Center forApplied Mathematics of the U.S. National Bureau of Standards, gave students the following piece of advice:“Jeunes filles, jeunes gens qui vous êtes arrêtés ici sur le chemin de vos classes et de vos laboratoires, quand onvous demande qui sont les grandes figures de cette maison, laissez entendre dans un sourire que de plus grandesencore sont à venir. Tout comme il est un cratère sur la face invisible de la Lune qui porte le nom de Lemaître, ily aura des montagnes qui porteront le nom de certains d'entre vous sur des planètes dans le système solaire deVéga”.26 See http://www.top500.org27 See http://www.mapr.ucl.ac.be/serveurs/serveur.html

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processor. Improving effective memory latency using caches is essential in order toachieve serial efficiency (Grama et al. 2003).

2. Computational load has to be equally distributed on processors. Things get tricky when thegrid evolves — e.g. with adaptivity, load changes in time and has to be balanced — and/orwhen time scales vary greatly in space; the load may depend on the physics that issimulated in a given region (Karpyris and Kumar 1999).

3. The ratio computation/communication should be optimised. Inter-processorcommunications are just an overhead and, therefore, should be minimised.

4. Computations and communications have to be alternated in order to use the network in anefficient manner.

Failing to take into account the first and second rule can have a huge negative impact. Rule 3is usually easier to deal with, while rule 4 is very difficult to handle in practice — as processsynchronisation, even if not optimum, leads usually to an easier algorithmic. An adequateorganisation of simulation data, an effective load balancing strategy as well as the choice of acost effective numerical scheme are some of the key ingredients that will lead to an effectiveimplementation.

Clearly, parallel implementation of algorithms is not a “technical task”. As may be seen inFigure 3, the need to obtain efficient computer codes will be kept in mind for most of theintended model developments and applications. In addition, it is worth underscoring that UCLhas excellent facilities and expertise in high performance computing (see, for instance,footnote # 27), which will be very useful for the present research programme.

8. Synergy between the promoters

Roughly speaking, the chief area of expertise of each promoter is as follows:• Eric Deleersnijder: ocean modelling• Thierry Fichefet: sea-ice modelling• Vincent Legat: finite element techniques• Jean-François Remacle: parallel computing

However, every promoter is competent in more than one field of research. In fact, the areas ofexpertise of the four promoters of the present project are complementary and, to a certainextent, present overlaps. In addition, the model developments and applications will be carriedout in an open-source mode with the close collaboration of external partners. All this isillustrated in Figure 3.

Four PhD students will need to be hired, whose main task will be as follows:• sea-ice model developments and applications: 1 PhD student;• ocean model developments and applications: 2 PhD students;• parallel computing algorithm design and implementation: 1 PhD student.

Neither the promoters nor the external partners are highly competent in data assimilation.This is the reason why a research scientist — at the post-doc level —, possessing theexpertise we are missing, will be hired for a period a three-years. Heshe will be in charge ofthe development of the sea-ice data assimilation system.

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