Invariant measures for passive scalars in the small noise inviscid limit

TL;DR

This paper investigates invariant measures for passive scalars in fluid flows, showing their support relates to eigenfunctions of the flow operator, and characterizes these measures explicitly for certain flow types.

Contribution

It introduces a framework for understanding invariant measures as limits of viscous perturbations and characterizes these measures for shear, relaxation enhancing, and cellular flows.

Findings

Support of measures contains span of $H^1$ eigenfunctions

Unique characterization of measures for shear and relaxation enhancing flows

Regularity properties of functions in the support for cellular flows

Abstract

We consider a class of invariant measures for a passive scalar $f$ driven by an incompressible velocity field $\boldsymbol{u}$, on a $d$-dimensional periodic domain, satisfying $$ \partial_t f + \boldsymbol{u} \cdot \nabla f = 0, \qquad f(0)=f_0. $$ The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the $H^1$ eigenfunctions of the operator $\boldsymbol{u}\cdot\nabla$ contains the support of these measures. We also analyze several explicit examples: when $\boldsymbol{u}$ is a shear flow or a relaxation enhancing flow (a generalization of weakly mixing), we can characterize the limiting measure uniquely and compute its covariance structure. We also consider the case of two-dimensional cellular flows, for which further regularity properties of the functions in the support of the measure can be deduced. The main results are proved with the use of spectral theory results, in particular the RAGE theorem, which are used to characterize large classes of orbits of the inviscid problem that are growing in $H^1$.