We consider the nonlinear optimisation of irreversible mixing induced
by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity ν and density diffusivity κ. The initial diffusive error function density distribution varies continuously so that ρ∈[ρˉ−(1/2)ρ0,ρˉ+(1/2)ρ0]. A constant pressure gradient is imposed in a plane two-dimensional channel of depth 2h. We consider flows with a finite P\'eclet number Pe=Umh/κ=500 and Prandtl number Pr=ν/κ=1, and a range of bulk Richardson numbers Rib=gρ0h/(ρˉU2)∈[0,1] where Um is the maximum flow speed of the laminar parallel flow, and g is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimization problems, extending the optimal mixing
results of Foures, Caulfield \& Schmid (2014) to stratified flows, where
the irreversible mixing of the active scalar density leads to a conversion of
kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximize the time-averaged perturbation kinetic energy developed by the perturbations over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be T=10)
of a `mix-norm' as first introduced by Mathew, Mezic \& Petzold (2005), further discussed by Thi eault (2012) and shown by Foures et al. (2014) to be
a computationally efficient and robust proxy for identifying perturbations
that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged-kinetic-energy-maximising perturbations are significantly suboptimal at mixing compared to the mix-norm-minimising perturbations,
and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance
at long times. Although increasing stratification reduces the
mixing in general, mix-norm-minimising optimal perturbations can still
trigger substantial mixing for Rib≲0.3. By considering the
time evolution of the kinetic energy and potential energy reservoirs,
we find that such perturbations lead to a flow which, through Taylor
dispersion, very effectively converts perturbation kinetic energy into `available potential energy', which in turn leads rapidly and irreversibly to
thorough and efficient mixing, with little energy returned to the kinetic energy reservoirs