A bound on scalar variance for the advection–diffusion equation

Abstract

We study the statistics of a passive scalar $T({\bm{x},t)$ governed by the advection–diffusion equation with variations in the scalar produced by a steady source. Two important statistical properties of the scalar are the variance, $\sigma^2\equiv \langle T^2 \rangle$, and the entropy production, $\chi \equiv \kappa \langle|\bm{\nabla} T|^2\rangle$. Here $\langle\rangle$ denotes a space–time average and $\kappa$ is the molecular diffusivity of $T$. Using variational methods we show that the system must lie above a parabola in the $(\chi, \sigma^2)$-plane. The location of the bounding parabola depends on the structure of the velocity and the source. To test the bound, we consider a large-scale source and three two-dimensional model velocities: a uniform steady flow; a statistically homogeneous and isotropic flow characterized by an effective diffusivity; a time-periodic model of oscillating convection cells with chaotic Lagrangian trajectories. Analytic solution of the first example shows that the bound is sharp and realizable. Numerical simulation of the other examples shows that the statistics of $T({\bm{x},t)$ lie close to the parabolic frontier in the $(\chi, \sigma^2)$-plane. Moreover in the homogenization limit, in which the largest scale in the velocity field is much less than the scale of the source, the results of the simulation limit to the bounding parabola.