Optimized Schwarz Methods in the Context of Ocean-Atmosphere Coupling

International audienceMany applications in coastal and operational oceanography require high resolution local models, for which ocean-atmosphere interactions must be properly taken into account. In order to address the ocean-atmosphere coupling problem in a mathematically consistent way we intend to design an adequate Global-in-Time Schwarz-like domain decomposition method. We rst show how the usual ad hoc ocean-atmosphere coupling methods can be described in the formalism of the Schwarz methods. Then we propose a rst real-case study (genesis and propagation of a tropical cyclone) with the WRF (Weather Research and Forecasting) and the ROMS (Regional Ocean Modeling System) numerical models. We introduce some numerical results obtained with a non-optimized method. However when one wants to optimize the convergence speed of the corresponding algorithm by looking for a good approximation of the absorbing boundary conditions we face some problem that have been relatively few studied so far : the in uence of the presence of turbulent boundary layers on both sides of the interface on the convergence speed. In order to illustrate this point we propose to study an idealized form of the problem: a coupling between two di usion equations modeling the turbulent mixing in the boundary layers. We consider rst the case of coe cients which are discontinuous at the air-sea interface but constant on each subdomain. In this case the optimized transmission conditions are determined analytically. In the more general case with discontinuous and spatially variable coe cients the convergence of the Schwarz algorithm is studied using an ad hoc expansion into eigenfunctions of an associated Sturm-Liouville problem. We nally discuss how to extend those results found on a simpli ed problem to the full ocean-atmosphere coupling problem