Convection in a compressible fluid with infinite Prandtl number

<jats:p>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,<jats:italic>d</jats:italic>, to the temperature scale height of the liquid,<jats:italic>H</jats:italic><jats:sub>T</jats:sub>, governs the importance of the non-Boussinesq effects of compressibility, viscous dissipation, variable adiabatic temperature gradients and non-hydrostatic pressure gradients. When<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>[Lt ] 1 the Boussinesq equations result, but when<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>is<jats:italic>O</jats:italic>(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<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>is varied from 0·1 to 1·5. Both marginal stability and finite-amplitude convection are studied. For<jats:italic>d/H</jats:italic><jats:sub>T</jats:sub>[les ] 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<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>> 1·0 the density stratification significantly stabilizes the lower regions of the marginal state solutions. At all values of<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>[ges ] 0·25, an adiabatic temperature gradient proportional to temperature has a noticeable stabilizing effect on the lower regions. For<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>[ges ] 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,<jats:italic>F</jats:italic><jats:sub><jats:italic>u</jats:italic></jats:sub>, has an upper limit equal to<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>. This limit is achieved at high Rayleigh numbers, when heating is entirely from below, and, for sufficiently large values of<jats:italic>d</jats:italic>/<jats:italic>H</jats:italic><jats:sub>T</jats:sub>, Φ/<jats:italic>F</jats:italic><jats:sub><jats:italic>u</jats:italic></jats:sub>is greater than 1·00.</jats:p>