On the development of geometric integrators for geophysical fluid dynamics.

 F. Gay-Balmaz (ENS Paris)

We present structure-preserving numercical schemes for several fluid models used in oceanic and atmospheric circulations, such as the Boussinesq and anelastic equations. The numerical schemes are based on a finite dimensional approximation of the group of volume preserving diffeomorphisms and are derived via a discrete version of the Euler-Poincare variational formulation of rotating stratified fluids. The resulting variational integrators allow for a discrete version of Kelvin circulation theorem, are applicable to irregular meshes and, being symplectic, exhibit excellent long term energy behavior. We then present a series of preliminary tests for rotating stratified flows such as hydrostatic and geostrophic adjustments, and inertial instability. Recent extensions of this geometric approach to compressible fluids willbe also presented.