Homogenisation of a Passive Scalar Transported by Locally Supported White Noise

Abstract

Stochastic perturbations of transport type are a common and widely accepted way of representing turbulent effects in fluid dynamics models. In many known examples, it even leads to improved solution theory, a phenomenon known as \emph{regularization by noise}. A common thread in the recent literature on the topic is the so-called \emph{It\^o-Stratonovich diffusion limit}. By selecting Stratonovich transport noise with carefully arranged vector fields, one can show that the solution of certain SPDEs are close, in an appropriate topology, to an effective, deterministic, equation with a new effective second order elliptic operator, linked to the Ito-Stratonovich corrector. In this work, we deal with a passive scalar model with molecular diffusivity $\kappa$. Starting from the results in [Flandoli \emph{et al.}, 2022, \emph{Philos. Trans. Roy. Soc. A}, 380(2219)], we consider a transport noise made by a sum of independent and compactly supported vector fields. This setting is relevant for models of stratified turbulence which naturally occur in boundary layers and Boussinesq models. Due to the anisotropic nature of the noise, the identification of the limit equation is not straightforward as in all other examples known in literature, as the Ito-Stratonovich corrector is a generic second order elliptic operator with non-constant coefficients. Using tools from Homogenisation theory, we obtain a representation for the limiting effective diffusivity matrix. Exploiting this representation, we study asymptotics, in the $\kappa \rightarrow 0$ regime, of the effective diffusivity across a number of vector field regimes parametrised by the radius of their support. Finally, we provide a careful numerical analysis of the effective diffusivity, discovering a nonlinear behavior for $\kappa \rightarrow 0$, in some regimes.