Convection in a compressible fluid with infinite Prandtl number
An approximate set of equations is derived for a compressible liquid of infinite Prandtl number. These are referred to as the anelastic-liquid equations. The approximation requires the product of absolute temperature and volume coefficient of thermal expansion to be small compared to one. A single parameter defined as the ratio of the depth of the convecting layer, d, to the temperature scale height of the liquid, HT, governs the importance of the non-Boussinesq effects of compressibility, viscous dissipation, variable adiabatic temperature gradients and non-hydrostatic pressure gradients. When d/HT [double less-than sign] 1 the Boussinesq equations result, but when d/HT is O(1) the non-Boussinesq terms become important. Using a time-dependent numerical model, the anelastic-liquid equations are solved in two dimensions and a systematic investigation of compressible convection is presented in which d/HT is varied from 0·1 to 1·5. Both marginal stability and finite-amplitude convection are studied. For d/HT [less-than-or-eq, slant] 1·0 the effect of density variations is primarily geometric; descending parcels of liquid contract and ascending parcels expand, resulting in an increase in vorticity with depth. When d/HT > 1·0 the density stratification significantly stabilizes the lower regions of the marginal state solutions. At all values of d/HT [gt-or-equal, slanted] 0·25, an adiabatic temperature gradient proportional to temperature has a noticeable stabilizing effect on the lower regions. For d/HT [gt-or-equal, slanted] 0·5, marginal solutions are completely stabilized at the bottom of the layer and penetrative convection occurs for a finite range of supercritical Rayleigh numbers. In the finite-amplitude solutions adiabatic heating and cooling produces an isentropic central region. Viscous dissipation acts to redistribute buoyancy sources and intense frictional heating influences flow solutions locally in a time-dependent manner. The ratio of the total viscous heating in the convecting system, φ, to the heat flux across the upper surface, Fu, has an upper limit equal to d/HT. This limit is achieved at high Rayleigh numbers, when heating is entirely from below, and, for sufficiently large values of d/HT, Φ/Fu is greater than 1·00.(Published Online April 19 2006)
(Received August 22 1978)
(Revised April 6 1979)
p1 Present address: Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7.