THERMOSOLUTAL CONVECTION BETWEEN POORLY CONDUCTING PLATES

Thermosolutal convection is considered in a fluid layer between poorly conducting horizontal boundaries. The horizontal scale of the motions is much greater than the depth of the fluid layer, provided the motion is not too vigorous, and this disparity between horizontal and vertical scales provides the basis for an asymptotic expansion of the solution. Under the assumption of near-constant solute-flux at the horizontal boundaries, a pair of evolution equations is derived for the depth-averaged temperature and solute concentration fields. These long-wave equations are investigated for two-dimensional convection by numerical integrations, and the results are compared with linear and weakly non-linear theory. The asymptotic expansion is shown to break down for large values of R, when the form assumed for the convection becomes inappropriate. The onset of three-dimensional convection is analysed. Steady square convection cells are stable. Oscillatory convection in the form of two-dimensional travelling-wave rolls is stable to three-dimensional disturbances near onset.