PROBABILISTIC DYNAMICS OF TWO-LAYER GEOPHYSICAL FLOWS

Abstract

The two-layer quasigeostrophic flow model is an intermediate system between the single-layer two-dimensional barotropic flow model and the continuously stratified three-dimensional baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf stream and subtropical gyres. We consider the two-layer quasigeostrophic flow model under stochastic wind forcing on the top layer. The fluctuating part of the wind forcing is modeled as the generalized time derivative of a Wiener process. We first transform this stochastic two-layer fluid system into a coupled system of random partial differential equations. Then we prove that the stochastic two-layer fluid system has finite sets of asymptotically determining functionals (such as determining modes and determining nodes) in probability. Furthermore, we show that the asymptotic probabilistic dynamics of this system depends only on the top fluid layer. Namely, in the probability sense and asymptotically, the dynamics of the two-layer quasigeostrophic fluid system is determined by the top layer. In other words, the bottom layer is slaved by the top layer. This conclusion is true provided that the Wiener process and the fluid parameters satisfy a certain condition. In particular, this latter condition is satisfied when the trace of the covariance operator of the Wiener process is small enough and the Ekman constant r is sufficiently large.