Effective diffusion of scalar fields in a chaotic flow

Abstract

The advection of a tracer field in a fluid flow can create complex scalar structures and increase the effect of weak diffusion by orders of magnitude. One tool to quantify this is to measure the flux of scalar across contour lines of constant scalar. This gives a diffusion equation in area coordinates with an effective diffusion that depends on the structure of the scalar field and, in particular, takes large values when scalar contours become very extended. The present paper studies the properties of this effective diffusion using a mixture of analytical and numerical tools. First the presence of hyperbolic stationary points, that is, saddles, in the scalar concentration field is investigated analytically, and it is shown that these give rise to singular spikes in the effective diffusion. This is confirmed in numerical simulations in which complex scalar fields are generated using a time-periodic flow. Issues of numerical resolution are discussed and results are given on the dependence of the effective diffusion on grid resolution and discretization in area or scalar values. These simulations show complex dependence of the effective diffusion on time as saddle points appear and disappear in the scalar field. It is found that time averaging (in the presence of an additional scalar source term) removes this dependence to leave robust results for the effective diffusion.