Section: New Results

Mathematical Modelling of the Ocean Dynamics

Coupling Methods for Oceanic and Atmospheric Models

Participants : Eric Blayo, Mehdi-Pierre Daou, Laurent Debreu, Florian Lemarié, Antoine Rousseau, Manel Tayachi.

Interface conditions for coupling ocean models

Many physical situations require coupling two models with not only different resolutions, but also different physics. Such a coupling can be studied within the framework of global-in-time Schwarz methods. However, the efficiency of these iterative algorithms is strongly dependent on interface conditions. As a first step towards coupling a regional scale primitive equations ocean model with a local Navier-Stokes model, a study on the derivation of interface conditions for 2-D $x-z$ Navier-Stokes equations has been performed in D. Cherel PhD thesis. It has been shown theoretically that several usual conditions lead to divergent algorithms, and that a convergent algorithm is obtained when using transmission conditions given by a variational calculation.

D. Cherel has implemented a Schwarz-based domain decomposition method, for which he developed optimized absorbing boundary conditions that mix the velocity and pressure variables on an Arakawa-C grid. The numerical results confirm the rate of convergence that has been obtained theoretically, thanks to a Fourier analysis of the semi-discretized problem. New optimized conditions offer much better convergence than classical Dirichlet-Dirichlet conditions, even when domains overlap. A paper is now ready for submission.

Coupling dimensionally heterogeneous models

The coupling of different types of models is gaining more and more attention recently. This is due, in particular, to the needs of more global models encompassing different disciplines (e.g. multi-physics) and different approaches (e.g. multi-scale, nesting). Also, the possibility to assemble different modeling units inside a friendly modelling software platform is an attractive solution compared to developing more and more global complex models. More specifically one may want to couple 1D to 2D or 3D models, such as Shallow Water and Navier Stokes models: this was the framework of our partnership with EDF in the project MECSICO, now extended with ARTELIA Group. In her PhD (defended Oct. 28th, 2013) [4] , M. Tayachi built a theoretical and numerical framework to couple 1D, 2D and 3D models for river flows.

In [103] (now accepted for publication), we propose and analyze an efficient iterative coupling method for a dimensionally heterogeneous problem. We consider the case of a 2-D Laplace equation with non symmetric boundary conditions with a corresponding 1-D Laplace equation. We first show how to obtain the 1-D model from the 2-D one by integration along one direction, by analogy with the link between shallow water equations and the Navier-Stokes system. Then we focus on the design of a Schwarz-like iterative coupling method. We discuss the choice of boundary conditions at coupling interfaces. We prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and to the control of the difference between a global 2-D reference solution and the 2-D coupled one. These theoretical results are illustrated numerically. The extension of this work to shallow water equations has been started in 2013 with the PhD thesis of Medhi Pierre Daou (funded by ARTELIA). An extension to primitive equations is envisaged: a post-doc position has been proposed in 2013 (not funded) and will renewed in 2014.

Ocean-Atmosphere coupling

Coupling methods routinely used in regional and global climate models do not provide the exact solution to the ocean-atmosphere problem, but an approached one [75] . This finding has motivated a deep numerical analysis of multi-physics coupling problems, first on simplified academic cases based on diffusion equations. In this context, Schwarz-like iterative domain decomposition methods have been analyzed and efficient interface conditions have been determined to optimize the convergence rate of the method [19] , [20] , [53] , [79] . This method has then been applied to the coupling of realistic oceanic and atmospheric models to simulate the propagation of a tropical cyclone (cyclone erica, Fig. 1 ). Sensitivity tests to the coupling method have been been carried out in an ensemblist approach. We showed that with a mathematically consistent coupling, compared to coupling methods en vogue in existing coupled models, the spread of the ensemble is reduced, thus indicating a much reduced uncertainty in the physical solution [24] , [75] .

The next step is now to complete the theoretical work done on a diffusion problem by including the formulation of physical parameterizations to tackle a problem more representative of the realistic models: a PhD thesis should start on this subject in fall 2014. In parallel, an important perspective is to assess the impact of our work on IPCC-like climate models, this task will be initiated in 2014 through a collaboration between the MOISE project-team and the LSCE (Laboratoire des Sciences du Climat et de l’Environnement).

In collaboration with geophysicists, number of studies to investigate important small-scales air-sea feedbacks are in progress [45] . Through those studies, the aim will be to mathematically derive a metamodel able to represent important processes in the marine atmospheric boundary layers. The medium term objective will be to use this metamodel to force high-resolution oceanic operational models for which the use of a full atmospheric model is not possible due to a prohibitive computational cost.

Numerical schemes for ocean modelling

Participants : Laurent Debreu, Jérémie Demange, Florian Lemarié.

Reducing the traditional errors in terrain-following vertical coordinate ocean models (or sigma models) has been a focus of interest for the last two decades. The objective is to use this class of model in regional domains which include not only the continental shelf, but the slope and deep ocean as well. Two general types of error have been identified: 1) the pressure-gradient error and 2) spurious diapycnal diffusion associated with steepness of the vertical coordinate. In ( [92] ,[46] ) we have studied the problem of diapycnal mixing. The solution to this problem requires a specifically designed advection scheme. We propose and validate a new scheme, where diffusion is split from advection and is represented by a rotated biharmonic diffusion scheme with flow-dependent hyperdiffusivity. The main numerical development was to render the biharmonic diffusion operator scheme unconditionally stable. This is particularly needed when the slopes between coordinates lines and isopycnal surfaces are important so that the rotation of the biharmonic leads to strong stability condition along the vertical coordinate where the grid size is relatively small. This work also extends more classical results on the stability of Laplacian diffusion with mixed derivatives.

In his PhD, Jérémie Demange works on advection-diffusion schemes for ocean models (Supervisors : L. Debreu, P. Marchesiello (IRD)). His work focuses on the link between tracers (temperature and salinity) and momentum advection and diffusion in the non hyperbolic system of equations typically used in ocean models (the so called primitive equations with hydrostatic and Boussinesq assumptions). We also investigated the use of a depth dependent barotropic mode in free surface ocean models. When most ocean models assume that this mode is vertically constant, we have shown that the use of the true barotropic mode, derived from a normal mode decomposition, allows more stability and accuracy in the representation of external gravity waves.

Salinity at 1000 m in the Southwest Pacific ocean is shown in figure 2 . The use of traditional upwind biased schemes (middle) exhibits a strong drift in the salinity field in comparison with climatology (left). The introduction of high order diffusion rotated along geopotential surfaces prevents this drift while maintaining high resolution features (right).