  We consider the nonlinear optimisation of irreversible mixing induced by an
initial finite amplitude perturbation of a statically stable density-stratified
fluid. A constant pressure gradient is imposed in a plane two-dimensional
channel. We consider flows with a finite P\'eclet number $Pe=500$ and Prandtl
number $Pr=1$, and a range of bulk Richardson numbers $Ri_b \in [0,1]$. We use
the constrained variational direct-adjoint-looping (DAL) method to solve two
optimization problems, extending the optimal mixing results of Foures et al.
(2014) to stratified flows, where the mixing of the scalar density has an
energetic cost, and thus has an observable dynamic effect. We identify initial
perturbations of fixed finite kinetic energy which maximise the time-averaged
kinetic energy developed by the perturbations over a finite time interval, and
initial perturbations that minimise the value of a `mix-norm', as defined by
Thiffeault (2012) and shown by Foures et al. (2014) to be a computationally
efficient and robust proxy for identifying perturbations that minimise the
variance of a scalar distribution at a target time. 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. By
considering the time evolution of the kinetic energy and potential energy
reservoirs, we find that mix-norm-minimising optimal 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.
