Singular limits for randomly forced Boussinesq system

Due to sensitivity with respect to initial data and parameters, individual solutions of the basic equations of fluid mechanics are unpredictable and seemingly chaotic. However, some of their statistical properties are robust. As early as the 19th century it was conjectured that turbulent flow cannot be solely described by deterministic methods, and indicated that a stochastic framework should be used. In this framework, invariant measures of the stochastic equations of fluid dynamics presumably contain the statistics posited by the basic theories of turbulence.

In this talk we investigate properties of invariant measures for the Boussinesq equations in the presence of a degenerate random forcing acting only in the temperature component. The main goal is to prove convergence of invariant measures in singular limits when Prandtl numbers approach infinity. As an application we recover estimates for the Nusselt number in stochastic setting.

More precisely, we show a general framework for converting the problem of convergence of measures to the question of finite time convergence of solutions. Then we analyze singular limit problems in a stochastic setting.

This is a joint work with S. Friedlander (U. of Southern California), N. Glatt-Holtz (Virginia Tech), G. Richards (U. of Rochester), and E. Thomann (Oregon State).