Buoyancy flux bounds for surface-driven flow

Abstract

I calculate the optimal upper bound, subject to the assumption of streamwise invariance, on the long-time-averaged buoyancy flux ${\cal B}^*$ within the flow of an incompressible stratified viscous fluid of constant kinematic viscosity $\nu$ and depth $h$ driven by a constant surface stress $\tau\,{=}\,\rho u^2_\star$, where $u_\star$ is the friction velocity with a constant statically stable density difference $\Delta \rho$ maintained across the layer. By using the variational ‘background method’ (due to Constantin, Doering and Hopf) and numerical continuation, I generate a rigorous upper bound on the buoyancy flux for arbitrary Grashof numbers $G$, where $G\,{=}\,\tau h^2/(\rho \nu^2)$. As $G \,{\rightarrow}\, \infty$, for flows where horizontal mean momentum balance, horizontally averaged heat balance, total power balance and total entropy flux balance are imposed as constraints, I show numerically that the best possible upper bound for the buoyancy flux is given by ${\cal B}^* \,{\leq}\, {\cal B}^*_{\hbox{\scriptsize max}}\,{=}\,u_{\star}^4/(4\nu)+ O(u_{\star}^3/h)$. This bound is independent of both the overall strength of the stratification and the layer depth to leading order. This bound is associated with a velocity profile that has the scaling characteristics of a somewhat decelerated laminar, linear velocity profile.