Large Prandtl Number Asymptotics in Randomly Forced Turbulent Convection

TL;DR

This paper proves the convergence of invariant states and heat transport measures in stochastic Boussinesq equations as the Prandtl number becomes large, with implications for mantle convection and high-pressure gases.

Contribution

It establishes the convergence of invariant measures and solutions for stochastic Boussinesq equations in the infinite Prandtl number limit, including phase space extension methods.

Findings

Invariant measure is unique for infinite Prandtl number equations.

Solutions converge on finite time intervals as Prandtl number increases.

Derived well-posed equations approximating dynamics for large Prandtl numbers.

Abstract

We establish the convergence of statistically invariant states for the stochastic Boussinesq Equations in the infinite Prandtl number limit and in particular demonstrate the convergence of the Nusselt number (a measure of heat transport in the fluid). This is a singular parameter limit significant in mantle convection and for gasses under high pressure. The equations are subject to a both temperature gradient on the boundary and internal heating in the bulk driven by a stochastic, white in time, gaussian forcing. Here, the stochastic source terms have a strong physical motivation for example as a model of radiogenic heating. Our approach uses mixing properties of the formal limit system to reduce the convergence of invariant states to an analysis of the finite time asymptotics of solutions and parameter-uniform moment bounds. Here, it is notable that there is a phase space mismatch between the finite Prandtl system and the limit equation, and we implement methods to lift both finite and infinite time convergence results to an extended phase space which includes velocity fields. For the infinite Prandtl stochastic Boussinesq equations, we show that the associated invariant measure is unique and that the dual Markovian dynamics are contractive in an appropriate Kantorovich-Wasserstein metric. We then address the convergence of solutions on finite time intervals, which is still a singular perturbation. In the process we derive well-posed equations which accurately approximate the dynamics up to the initial time when the Prandtl number is large.