Mixing and transport in the presence of a source

A common topic in mixing is the initial value problem \begin{equation*} \partial_t \theta + \bm{u}(\bm{x},t) \cdot \nabla \theta = D \Delta \theta + s(\bm{x},t) \end{equation*} where $\bm{u}$ is an incompressible flow stirring the mean-zero passive scalar concentration $\theta$ and $s$ is a mean-zero internal source. The initial concentration is unmixed and, for $s = 0$, will relax to the zero steady state. This convergence is typically maximized by a flow that increases the gradients of the passive scalar, dissipating variance via molecular diffusion. For $s \neq 0$, the passive scalar still relaxes to a steady state, but variance can now also be reduced --- sometimes optimally [3] depending on the distribution of the source --- by transporting hot spots to cold spots. We compare the optimality of gradient production to transport for various source distributions by employing multiscale asymptotics [1,2] and observing that the two mechanisms decouple when the length scale of the source is strongly separated from the length scale of the gradient producing stirring. [1] P. R. Kramer and S. R. Keating. Chin. Ann. Math., 30B(5):631-644, 2009. [2] A. J. Majda and P. R. Kramer. Physics Reports, 314:237-574, 1999. [3] J.-L. Thiffeault. Nonlinearity, 25(2):R1-R44, 2012.