Entrainment and mixing dynamics of surface-stress-driven linearly stratified flow in a cylinder

We consider experimentally a linearly stratified fluid (with constant buoyancy frequency $N$) in a cylinder of depth $H$ subject to surface stress forcing from a disk spinning at constant angular velocity $\Omega$. A turbulent mixed layer develops bounded by a sharp interface of constant thickness. Its depth $h/H \sim (N/\Omega)^{-2/3} (\Omega t)^{2/9}$. We argue this is a consequence of: the kinetic energy of the mixed layer staying constant with time (as previously observed in a two layer flow by Shravat et al. 2012) the entrainment at the interface being governed entirely by local processes; and the rate of increase of the total potential energy of the fluid being dependent only on the global dissipation rate and the ratio $N^2/\Omega^2$. Below the moving primary interface, we also observe in some circumstances the formation of another partially mixed layer, separated by a secondary interface from the linearly stratified fluid below. Depending on the local flow properties, the secondary interfaces can exhibit rich time-dependent dynamics including drift towards or away from the primary interface, merger and/or decay. The secondary interfaces appear to develop due to the non-monotonic dependence of buoyancy flux on stratification as originally argued by Phillips (1972).