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 that varies on arbitrary timescales. 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. Within the model framework, the tracer gradient aligns with the compressive direction when the mean effective rotation in the flow is zero. At finite values of rotation, the tracer gradient aligns with a different direction, but the mean growth rate of the gradient is positive in all cases. In a certain limiting case, namely temporally decorrelated (rapidly varying) flows, exact, analytical expressions exist for the mean growth rate. 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.