Dispersion in the large-deviation regime. Part II: cellular flow at large P\'eclet number

摘要

A standard model for the study of scalar dispersion through advection andmolecular diffusion is a two-dimensional periodic flow with closed streamlinesinside periodic cells. Over long time scales, the dispersion of a scalar inthis flow can be characterised by an effective diffusivity that is a factor$\mathrm{Pe}^{1/2}$ larger than molecular diffusivity when the P\'eclet number$\mathrm{Pe}$ is large. Here we provide a more complete description ofdispersion in this regime by applying the large-deviation theory developed inPart I of this paper. We derive approximations to the rate function governingthe scalar concentration at large time $t$ by carrying out an asymptoticanalysis of the relevant family of eigenvalue problems. We identify twoasymptotic regimes and make predictions for the rate function and spatialstructure of the scalar. Regime I applies to distances from the release pointthat satisfy $|\boldsymbol{x}| = O(\mathrm{Pe}^{1/4} t)$ . The concentration inthis regime is isotropic at large scales, is uniform along streamlines withineach cell, and varies rapidly in boundary layers surrounding the separatricesbetween adjacent cells. The results of homogenisation theory are recovered fromour analysis. Regime II applies when $|\boldsymbol{x}|=O(\mathrm{Pe} \, t/\log\mathrm{Pe})$ and is characterised by an anisotropic concentration distributionthat is localised around the separatrices. A novel feature of this regime isthe crucial role played by the dynamics near the hyperbolic stagnation points.A consequence is that in part of the regime the dispersion can be interpretedas resulting from a random walk on the lattice of stagnation points. The tworegimes overlap so that our asymptotic results describe the scalarconcentration over a large range of distances. They are verified againstnumerical solutions of the family of eigenvalue problems yielding the ratefunction.