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Heating a salinity gradient from a vertical sidewall: linear theory

Published online by Cambridge University Press:  26 April 2006

Oliver S. Kerr
Oliver S. Kerr
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
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Abstract

When a body of fluid with a vertical salinity gradient is heated from a single vertical wall, instabilities have sometimes been observed experimentally (Thorpe, Hutt & Soulsby 1969; Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986). We present a linear stability analysis for this configuration and show that for strong salinity gradients the stability of the fluid to infinitesimal disturbances is governed by a single non-dimensional parameter, \[ Q = \frac{(1-\tau)^6g(\alpha\Delta T)^6}{\nu\kappa_Sl^2(-\beta\overline{S}_z)^5} \] where g is the acceleration due to gravity, α the coefficient of thermal expansion, β the density change due to a unit change in the salinity, ΔT the change of temperature at the wall, $\overline{S}_z$ the vertical salinity gradient, l the horizontal lengthscale (kTt)½, v the kinematic viscosity, (κTt)½ the diffusivity of heat, kS the diffusivity of salt and τ the salt/heat diffusivity ratio. This non-dimensional parameter is related to the Rayleigh number, however, it involves two different lengthscales; the penetration depth of the thermal front, l, and the height by which a heated element of fluid would rise in the salinity gradient, $g\alpha \Delta T/(- \beta \overline{S}_z)$. This analysis is valid when the ratio of the vertical lengthscale to the horizontal lengthscale is small. This analysis gives good agreement with the published experimental results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 207 , October 1989 , pp. 323 - 352
Copyright
© 1989 Cambridge University Press

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