Mixing at the Batchelor Scale for White-In-Time Flows

TL;DR

This paper investigates the mixing properties of solutions to the advection-diffusion equation with white-in-time velocity fields, confirming the Batchelor scale conjecture and characterizing exponential mixing rates in both two and three dimensions.

Contribution

It provides the first verification of the Batchelor scale conjecture for white-in-time flows and extends the analysis to three-dimensional velocity fields.

Findings

Bounded exponential dissipation rate as diffusivity goes to zero

Verification of the Batchelor scale conjecture in white-in-time flows

Construction of a 3D velocity field with similar properties

Abstract

We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from below. Together with the corresponding upper bound established by Gess and Yaroslavtsev, this constitutes an example of a velocity field for which the Batchelor scale conjecture can be verified. In addition, we characterize the exponential mixing rate without diffusion of this system. Our results are not restricted to two dimensions, and we construct a three-dimensional white-in-time velocity field with the same properties.