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Boundary-driven mixing

Published online by Cambridge University Press:  26 April 2006

Andrew W. Woods
Andrew W. Woods
Affiliation:
Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, La Jolla, CA 92093, USA Present address: The Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK.
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Abstract

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, κm and κe 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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 226 , May 1991 , pp. 625 - 654
Copyright
© 1991 Cambridge University Press

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