Effects of vertical boundaries on infinite Prandtl number thermal convection

The physics of thermomechanical coupling of the continental lithosphere and its surrounding mantle is studied using a simple hydrodynamic model in which a fluid is heated from below and is bounded by rigid side walls. Rayleigh numbers up to 10 are considered. A series of finite-element stability analyses is employed to characterize systematically the nature of convective instability in such a system. The marginal stability of a bounded fluid is increased not only by the rigid boundary condition on side walls, but also by heat conduction across side walls. The planform of convection at the marginal state is 3-D, having convective rolls aligned perpendicular to side walls. The characteristics of non-linear convection at supercritical Rayleigh numbers are investigated by first calculating 2-D steady-state solutions and then applying 2-D and 3-D stability analyses. The 2-D stability analysis shows that the presence of side walls significantly reduces the transition Rayleigh number for time-dependent convection, and the 3-D stability analysis shows that heat conduction through side walls strongly prefers 3-D convective motion. Finally, the characteristics of 3-D convection are analysed through the 3-D single-mode approximation. Both 2-D and 3-D single-mode solutions demonstrate that conducting side walls reduce the strength of convection at low Rayleigh numbers but have a negligible influence at higher Rayleigh numbers. We therefore propose that the basal topography of the continental lithosphere can modulate the convective planform efficiently in the upper mantle.