Effects of Horizontal Gradients on Thermohaline Instabilities in a Porous Medium.

Thermohaline instabilities produced by horizontal gradients of temperature and salinity in a saturated, homogeneous, isotropic porous medium are studied using linear stability analysis. In the basic state the horizontal gradients of temperature and salinity are taken to be mutually compensating, so that the basic state fluid density does not vary horizontally. The fluid is assumed to be incompressible. Firstly a porous medium of infinite extent is considered. In this case the fluid is found to be unstable to horizontal concentration gradients of any magnitude. This is true even when the fluid density decreases with height. In a porous medium the effective advection rates of heat and dissolved salts are different. Because of this difference any disturbance with a horizontal component of displacement creates perturbation horizontal density gradients, and thus destabilizes the hydrostatic force balance. When the vertical Rayleigh number is positive, the typical disturbance velocity field consists of almost vertical layers of fluid sliding past each other in opposite directions (salt fingers). When the vertical Rayleigh number is negative the fluid layers are almost horizontal, similar to the interleaving observed in Newtonian fluids. Resulting perturbation fluxes of heat and salt always tend to reduce the basic state concentration gradients and the gravitational potential energy of the fluid. At the second stage of analysis the porous medium is assumed to be a horizontal layer of finite thickness. It is found that the geometric constraints imposed by the boundaries tend to eliminate the most effective of unstable modes. Thus the boundaries have a stabilizing effect. It is found that supercritical disturbances are qualitatively similar to the disturbances in an infinite medium. For the critical mode (i.e. neutrally stable disturbances corresponding to the minimum vertical Rayleigh number) the convection cells are almost horizontal and are about as thick as the porous layer. The critical vertical Rayleigh number, the critical horizontal wave number and the slope (from horizontal) of convection cells decrease monotonically as the horizontal Rayleigh number increases.