Analysis of a stratified Kraichnan flow

Abstract

We consider the stochastic convection–diffusion equation \[ \partial _{t} \theta (t\,,\bm {x}) =\nu \Delta \theta (t\,,\bm {x}) + V(t\,,x_{1})\partial _{x_{2}} \theta (t\,,\bm {x}), \] for $t>0$ and $\bm {x}=(x_{1}\,,x_{2})\in \mathbb {R}^{2}$, subject to $\theta _{0}$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure \[ \Cov [V(t\,,a)\,,V(s\,,b)]= \delta _{0}(t-s)\rho (a-b)\qquad \text {for all $s,t\geqslant 0$ and $a,b\in \mathbb {R}$}, \] where $\rho $ is a continuous and bounded positive-definite function on $\mathbb {R}$. We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $\theta $ as an expectation functional of an exogenous infinite-dimensional Brownian motion. We use that probabilistic representation in order to study the Itô/Walsh solution, when it exists, and relate it to the Stratonovich solution which is shown to exist for all $\nu >0$. Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, \[\label {DR} \mathrm {P}\left \{ \sup _{|x_{1}|\leqslant m}\sup _{\mathstrut x_{2}\in \mathbb {R}} |\theta (t\,,\bm {x})| = O\left ( \frac {1}{\sqrt {t}}\right )\qquad \text {as $t\to \infty $} \right \} =1\qquad \text {for all $m>0$},\tag {0.1} \] and the $O(1/\sqrt {t})$ rate is shown to be unimproveable. Our probabilistic (Lagrangian) representation is malleable enough to allow us to analyze the Stratonovich solution in two physically-relevant regimes: As $t\to \infty $ and as $\nu \to 0$. Among other things, our analysis leads to a “macroscopic multifractal analysis” of the rate of decay in (0.1) in terms of the reciprocal of the Prandtl (or Schmidt) number, valid in a number of simple though still physically-relevant cases.