  Understanding and optimizing passive scalar mixing in a diffusive fluid flow
at finite P\'eclet number $Pe=U h/\kappa$ (where $U$ and $h$ 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' 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 mix-norms triggers the
development of coherent vortical flow structures which effectively mix, with
the particular properties of these flow structures depending on $Pe$ and also
the time horizon of interest.

Fluid Dynamics;Optimization and Control