In this talk, I will present two types of results related to equilibrium statistical behavior of fluid systems.
The first type is the emergence and persistence of large scale coherent structures in a two dimensional fluid system under small scale random bombardments with relatively small forcing and appropriate scaling assumptions. The analysis shows that the large scale structure emerging out of the small scale random forcing at statistical equilibrium is not the one predicted by equilibrium statistical mechanics. But the error is very small which explains earlier successful prediction of the large scale structure based on equilibrium statistical mechanics.This is a joint work with Andrew Majda of New York University.
Back to Small Scales and Extreme Events: The Hurricane
The second is related to convection at large Prandtl number within the Boussinesq model. We show that the invariant measures of the Boussinesq system for Rayleigh-B\'enard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection.