Boundary-driven mixing

There are two separate mechanisms which can generate a boundary flow in a non-rotating, stratified fluid. The Phillips–Wunsch boundary flow arises in a stratified, quiescent fluid along a sloping boundary. Isopycnals are deflected from the horizontal in order to satisfy the zero normal mass flux condition at the boundary; this produces a horizontal density gradient which drives a boundary flow. The second mechanism arises when there is an independently generated turbulent boundary layer at the wall such that the eddy diffusion coefficients decay away from the wall; if the vertical density gradient is non-uniform the greater eddy diffusion coefficients near the wall result in a greater accumulation or diminution of density near the wall. This produces a horizontal density gradient which drives a boundary flow, even at a vertical wall. The turbulent Phillips Wunsch flow, in which there is a vigorous recirculation in the boundary layer, develops if the wall is sloping. This recirculation produces an additional dispersive mass flux along the wall, which also generates a net volume flux along the wall if the density gradient is non-uniform.We investigate the effect of these boundary flows upon the mixing of the fluid in the interior of a closed vessel. The mixing in the interior fluid resulting from the laminar Phillips–Wunsch-driven boundary flow is governed by \[ \rho_t = \frac{\kappa_{\rm m}}{A}(\rho z A)_z. \] The turbulence-driven boundary flow mixes the interior fluid according to \[ \rho_\frac{1}{A}\left(\kappa_{\rm e}\rho z\int\delta\,{\rm d}s\right)z. \] Here ρ is the density, κ_m and κ_e are the far-field (molecular) and effective boundary (eddy) diffusivities, including the dispersion, A is the cross-sectional area of the basin and ∫ δ ds is the cross-sectional area of the boundary layer. The interior fluid is only mixed significantly faster than the rate of molecular diffusion if there is a turbulent boundary layer at the sidewalls of the containing vessel which either (i) varies in intensity with depth in the vessel or (ii) is mixing a non-uniform density gradient. These mixing phenomena are consistent with published experimental data and we consider the effect of such mixing in the ocean.