Large Prandtl number asymptotics in randomly forced turbulent convection

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 wellposed equations which accurately approximate the dynamics up to the initial time when the Prandtl number is large.

Asymptotic Analysis for Randomly Forced MHD

“…Since D is of order |k| 8 whereas the numerator of M u is of order |k| 6 , the two degrees of smoothing of M is evident from (1.9)-(1.10). It is worth emphasizing that the singular limit of (1.1)-(1.3) to (1.5)-(1.7) in this work bears some significant formal similarities to the infinite Prandtl limit for the stochastic Boussinesq equations which we have recently considered in [FGHR15]. Here however our problem involves multiple small parameters and both the linear and non-linear structure structure of the governing equations is quite different.…”

“…[Mof08]. Our analysis will focus on the convergence of such statistically stationary states in a certain singular parameter limit suggested by [ML94] and which bears some formal similarities to a large Prandtl limit considered in our recent work [FGHR15]. See also [Wan04,Wan05,Wan07,Wan08,Par10] for other formally analogous limits considered within a deterministic framework.…”

On the convergence of stationary solutions in the Smoluchowski-Kramers approximation of infinite dimensional systems

“…In the case the constant λ in Hypothesis 2 belongs to (1,3], the uniform condition (4.9) holds. By proceeding as in [15,Lemma 4.1] it is possible to show that (4.9) implies that for every ǫ > 0 there exists t ⋆ = t ⋆ (ǫ) > 0 such that inf x,y∈ H sup Γ∈ C(P ⋆ t δx,P ⋆ t δy)…”

“…Proof. The method used here is analogous to the one used for example in [15]. Due to the invariance of ν µ and ν, for every t ≥ 0 we have…”

“…By proceeding again as in[15, Lemma 4.1], it is possible to show that (4.8) implies that for every M, ǫ > 0 there exists t ⋆ = t ⋆ (M, ǫ) > 0 such that inf x,y∈ B H (M ) sup…”