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

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.