| This thesis derives upper bounds on the bulk heat transport due to convection in a rotating Rayleigh-Bénard system governed by the infinite Prandtl number equation for an incompressible fluid coupled with the advection-diffusion equation for temperature. Taking the limit as the Prandtl number goes to infinity simplifies the analysis of the equations considerably. Indeed, a global existence and uniqueness theorem is obtained. Applying a background method developed by Doering and Constantin in recent years to this system, a rotation dependent upper bound on the Nusselt number is obtained. This bound also depends upon the boundary conditions considered. For no-slip boundary conditions the bound applies and for stress-free boundary conditions the bound is , where E is the Ekman number, a parameter inversely proportional to the rotation rate. These bounds imply a nonlinear stability result. In addition, bounds that have been obtained for the nonrotating infinite Prandtl number system are shown to be valid for nonzero rotation as well. In particular, the best known bound in the case of no-slip boundary conditions, is shown to be valid for large enough values of E. |
