Abstract: What is the long-time optimal mixing rate achievable under enstrophy-constrained flows when both advection and diffusion are active? To address this question, we investigate the long-time behavior of optimal mixing in an advection--diffusion equation using a shell model framework, a reduced model that captures the essential kinematics of advection and diffusion. Our focus is on quantifying the decay of scalar variance, measured by the shell-model analogue of the $H^{-1}$ mix-norm, under enstrophy-constrained stirring. We perform long-time computations using both local-in-time (maximizing the instantaneous mixing rate) and global-in-time (maximizing mixedness at a prescribed final time) optimization strategies. For mixing with diffusion ($\kappa>0$), the numerical results for both strategies show that the scalar length scale is eventually limited by a generalized Batchelor scale, in close agreement with theoretical predictions. In this regime, the shell-model analogue of the $H^{-1}$ mix-norm decays exponentially in time at a rate independent of the diffusivity $\kappa$. Compared with the purely advective case ($\kappa=0$), diffusion significantly enhances the long-time mixing rate; moreover, increasing diffusivity further improves mixing efficiency by reducing the prefactor. Guided by these numerical observations, we derive new conditional lower bounds on the decay rate of the $H^{-1}$ norm that are strictly independent of the diffusivity parameter $\kappa$ when $\kappa>0$. The computed decay rates are consistent with our theoretical bounds.

This is joint work with Baole Wen, Christian Seis, and Charles R. Doering.

Contact: Silas Alben