Paper detail

Dispersion in the large-deviation regime. Part II: cellular flow at large Péclet number

A standard model for the study of scalar dispersion through advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar in this flow can be characterised by an effective diffusivity that is a factor $\mathrm{Pe}^{1/2}$ larger than molecular diffusivity when the Péclet number $\mathrm{Pe}$ is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part I of this paper. We derive approximations to the rate function governing the scalar concentration at large time $t$ by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances from the release point that satisfy $|\boldsymbol{x}| = O(\mathrm{Pe}^{1/4} t)$ . The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenisation theory are recovered from our analysis. Regime II applies when $|\boldsymbol{x}|=O(\mathrm{Pe} \, t/\log \mathrm{Pe})$ and is characterised by an anisotropic concentration distribution that is localised around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.