The importance of being fractional in mixing: optimal choice of the index $s$ in $H^{-s}$ norm

A natural measure of homogeneity of a mixture is the variance of the concentration field, which in the case of a zero-mean field is the $L^2$-norm. Mathew \emph{et.al.} (Physica D, 2005) introduced a new multi-scale measure to quantify mixing referred to as the mix-norm, which is equivalent to the $H^{-1/2}$ norm, the Sobolev norm of negative fractional index. Unlike the $L^2$-norm, the mix-norm is not conserved by the advection equation and thus captures mixing even in the non-diffusive systems. Furthermore, the mix-norm is consistent with the ergodic definition of mixing and Lin~\emph{et al.} (JFM, \ 2011) showed that this property extends to any norm from the class $H^{-s}, \ s>0$. We consider a zero-mean passive scalar field organised into two layers of different concentrations advected by a flow field in a torus. We solve two non-linear optimisation problems. We identify the optimal \emph{initial perturbation} of the velocity field with given initial energy as well as the \emph{ optimal forcing} with given total action (the time integral of the kinetic energy of the flow) which both yield maximal mixing by a target time horizon. We analyse sensitivity of the results with respect to $s$-variation and thus address the importance of the choice of the fractional index