Modelling the growth rate of a tracer gradient using stochastic differential equations

TL;DR

This paper presents a stochastic differential equation model to analyze the growth rate of a tracer gradient in a rapidly varying, statistically homogeneous flow, providing insights into the alignment and mixing efficiency.

Contribution

The paper introduces a novel two-dimensional stochastic model linking flow properties to tracer gradient growth, with a rigorous proof of alignment in a specific flow limit.

Findings

Model accurately predicts tracer gradient alignment with flow compression direction.

Numerical simulations validate the model's applicability to real mixing scenarios.

Mapping of stochastic parameters to flow parameters enhances practical relevance.

Abstract

We develop a model in two dimensions to characterise the growth rate of a tracer gradient mixed by a statistically homogeneous flow with rapid temporal variations. % % The model is based on the orientation dynamics of the passive-tracer gradient with respect to the straining (compressive) direction of the flow, and involves reducing the dynamics to a set of stochastic differential equations. The statistical properties of the system emerge from solving the associated Fokker--Planck equation. In a certain limiting case, and within the model framework, there is a rigorous proof that the tracer gradient aligns with the compressive direction. This limit involves decorrelated flows whose mean vorticity is zero. % % % Using numerical simulations, we assess the extent to which our model applies to real mixing protocols, and map the stochastic parameters on to flow parameters.