Robust and efficient identification of quasi-optimal mixing perturbations using proxy multiscale measures*

Understanding and optimizing passive scalar mixing in a diffusive fluid flow at finite P\'eclet number $Pe=U L/\kappa$ (where $U$ and $L$ are characteristic velocity  and length scales, and $\kappa$ is the molecular diffusivisity of the scalar) is a fundamental problem of interest in many environmental and industrial flows. Particularly when $Pe \gg 1$,   identifying initial perturbations of given energy which optimally and thoroughly mix fluids of initally different properties can be computationally challenging. To address this challenge, we consider the identification  of initial perturbations in an idealized two-dimensional flow on a torus that extremize various measures over finite time horizons. We identify such `optimal' initial perturbations using the `direct-adjoint looping' (DAL) method, thus requiring the evolving flow to satisfy the governing equations and boundary conditions at all points in space and time. We  demonstrate that minimising  multiscale measures commonly known as `mix-norms'  (i.e. Sobolev norms of negative index)  over short time horizons is a computationally efficient and robust way to identify  initial perturbations that thoroughly mix layered scalar distributions over relatively long time horizons, provided the magnitude of the mix-norm's index is not too large. Minimisation of such (bounded index) mix-norms triggers the development of quasi-coherent vortical flow structures which effectively mix such diffusive passive scalar distributions, with the particular properties of these flow structures depending on $Pe$ and also the time horizon of interest.