## Section: New Results

### Mathematical Modelling of the Ocean Dynamics

#### Beyond the traditional approximation on the Coriolis force

Participant : Antoine Rousseau.

Recently, A. Rousseau and C. Lucas have performed some theoretical and numerical studies around the derivation of viscous Shallow Water equations. They proved that it is sometimes necessary to take into account the cosine part of the Coriolis force (which is usually neglected, leading to the so-called Traditional Approximation).

After a first paper published in 2008 [47] , they presented their new results in an international conference on mathematical oceanography, [27] . They now pursue the work with M. Petcu (Poitiers University) in order to investigate the influence of the traditional approximation in more complicated models such as the viscous Primitive Equations of the ocean.

#### Coupling Methods for Oceanic and Atmospheric Models

Participants : Éric Blayo, David Cherel, Laurent Debreu, Antoine Rousseau.

##### Open boundary conditions

The implementation of high-resolution local models can be performed in several ways. An usual way consists in designing a local model, and in using some external data to force it at its open boundaries. These data can be either climatological or issued from previous simulations of a large-scale coarser resolution model. The main difficulty in that case is to specify relevant open boundary conditions (OBCs).

In collaboration with V. Martin (LAMFA Amiens), we have started a work on the analysis of the impact of the imperfection of the external data on the design of efficient OBCs.

In order to avoid a modal decomposition in the vertical direction for the inviscid primitive equations, a complete x-z finite volume discretization has been proposed in collaboration with R. Temam, with a special treatment of the boundaries in order to match the required boundary conditions underlined in [48] . This work is under progress.

##### Interface conditions for coupling ocean models

Other 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, we started a study on the derivation of interface conditions for 2-D x-z Navier-Stokes (D. Cherel's PhD thesis). It has been shown that several usual conditions lead to divergent algorithms, and that a convergent algorithm is obtained when using transmission conditions obtained by a variational calculation.

##### Ocean-atmosphere coupling

Many applications in regional oceanography and meteorology require high resolution regional models with accurate air-sea fluxes. Separate integrations of oceanic and atmospheric model components in forced mode (i.e. without any feedback from one component to the other) may be satisfactory for numerous applications. However, two-way coupling is required for analyzing energetic and complex phenomena (e.g. tropical cyclones, climate studies, ... ). In this case, connecting the two model solutions at the air-sea interface is a difficult task, which is often addressed in a simplified way from a mathematical point of view. In this context, domain decomposition methods provide flexible and efficient tools for coupling models with non-conforming time and space discretizations.

F. Lemarié, in his PhD thesis (2008), addressed the application of such methods to this ocean-atmosphere coupling problem. In the continuity of this work, we have improved some results on the convergence of the Schwarz coupling method for two non-stationary 1-D diffusion equations with different coefficients.

These works are partially supported by the ANR (COMMA project).

#### Numerical schemes for ocean modelling

Participant : Laurent Debreu.

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 a recent paper [15] , 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 satisfying the Peclet constraint.

Work in this area will continue and is also supported by a contract with IFREMER.