Freddy Bouchet – Stochastic averaging and large deviations for stochastic partial differential equations describing atmosphere dynamics

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Date(s) - 21 mai 2013

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We consider the formation of large scale structures (zonal jets and vortices), in geostrophic turbulence forced by random forces, within the barotropic quasi-geostrophic model. This model includes as a special case the 2D stochastic Navier-Stokes equations. We study the limit of a time scale separation between inertial dynamics on one hand, and the effect of forces and dissipation on the other hand. Using a kinetic theory approach, we prove that stochastic averaging can be performed explicitly in this problem, which is unusual in turbulent systems. It is then possible to integrate out all fast turbulent degrees of freedom, and to get explicitly an equation that describes the slow evolution of zonal jets. The equation for this slow evolution, is a one dimensional stochastic differential equation with multiplicative noise. The average is described by a non-linear Fokker-Planck equation. It describes the attractors for the dynamics (alternating zonal jets, whose number depend on the force spectrum), and the relaxation towards those attractors. We describe regimes where the system has multiple attractors for the same physical parameters. We discuss possible transitions between attractors with either, three, four or more pairs of zonal jets in models of turbulent atmosphere dynamics. Those transitions are extremely rare, and occur over time scales of centuries or millennia. They are extremely hard to observe in direct numerical simulations, because this would require on one hand an extremely good resolution in order to simulate accurately the turbulence and on the other hand simulations performed over an extremely long time. Their study through numerical computations is inaccessible using conventional means. We present an alternative approach, based on instanton theory and large deviations. We discuss preliminary results on the computation of such instantons in the framework of the 2D Navier-Stokes equations and discuss instantons for atmosphere jet dynamics. J. Laurie, C. Nardini, T. Tangarife, O. Zaboronski have given contributions to one or several of the results discussed during this talk.