Bounds on stratified mixing with a mixing coefficient constraint

We derive non-trivial upper bounds for the long-time averaged vertical buoyancy flux ${\cal B}^* \!:=\! \langle \rho u_3 \rangle g/\rho_{0} $ for stably stratified Couette flow, with reference density $\rho_0$, kinematic viscosity $\nu$, thermal diffusivity $\kappa$, plate separation $d$, driven by constant relative velocity $\Delta U$, maintained at a statically stable density difference $\Delta \rho$. We numerically solve the variational problem using the ``background method'', and require that the mean flow is streamwise independent and statistically steady. We impose a coupling constraint such that a fixed fraction $\Gamma_c$ of the energy input into the system leads to enhanced irreversible mixing. We calculate the bound up to asymptotically large Reynolds numbers for a range of choices of $\Gamma_c$ and bulk Richardson numbers $J$. For any $Re$, the calculated upper bound increases with $J$, until a maximum possible value $J_{\max}(Re,\Gamma_c)$ at which the new constraint cannot be imposed, and the density field and velocity field become decoupled. The value of the bound at $J_ {\max}$ is a non-monotonic function of $\Gamma_c$, with $\Gamma_c=1/2$ leading to the largest possible values as $Re \! \to\! \infty$, consistently with the findings in Caulfield, Tang \& Plasting (2004) where this coupling constraint was not imposed. In fact, at any particular $Re$, the previous solution may be associated with a specific value of $\Gamma_c$. Imposing the coupling constraint with that $\Gamma_c$, as $J \! \to\! J_{\max}$, the new bound approaches from below the previous bound exactly.