Dispersion of passive tracers in velocity field with non delta-correlated noise

TL;DR

This paper investigates how non delta-correlated noise affects passive tracer dispersion in velocity fields, deriving analytical expressions for eddy diffusivity and showing noise decorrelation time enhances diffusion.

Contribution

It provides the first analytical expression for eddy diffusivity in 2D steady shear flows with non delta-correlated noise and introduces a renormalization approach for small noise decorrelation times.

Findings

Eddy diffusivity increases with noise decorrelation time τ.

Colored noise can be approximated by white noise with a renormalized velocity field.

Diffusion enhancement observed in 2D flow examples.

Abstract

The diffusive properties in velocity fields whose small scales are parameterized by non $\delta$-correlated noise is investigated using multiscale technique. The analytical expression of the eddy diffusivity tensor is found for a 2D steady shear flow and it is an increasing function of the characteristic noise decorrelation time $\tau$. In order to study a generic flow ${\bm v}$, a small-$\tau$ expansion is performed and the first correction ${\cal O}(\tau)$ to the effective diffusion coefficients is evaluated. This is done using two different approaches and it results that at the order $\tau$ the problem with a colored noise is equivalent to the white-in-time case provided by a renormalization of the velocity field ${\bm v} \mapsto \tilde{\bm v}$ depending on $\tau$. Two examples of 2D closed-streamlines velocity field are considered and in both the cases an enhancement of the diffusion is found.