Internally heated convection with rotation: bounds and scaling laws for heat transport

This work investigates heat transport in rotating internally heated convection, for a horizontally periodic fluid between parallel plates under no-slip and isothermal boundary conditions. The main results are the proof of bounds on the mean temperature, T ̄, and the heat flux out of the bottom boundary, FB at infinite Prandtl numbers where the Prandtl number is the nondimensional ratio of viscous to thermal diffusion. The lower bounds are functions of a Rayleigh number quantifying the ratio of internal heating to diffusion and the Ekman number, E, which quantifies the ratio of viscous diffusion to rotation. We utilise two different estimates on the vertical velocity, w, one pointwise in the domain (Yan 2004, J. Math. Phys., vol. 45(7), pp. 2718-2743) and the other an integral estimate over the domain (Constantin et al . 1999, Phys. D: Non. Phen., vol. 125, pp. 275-284), resulting in bounds valid for different regions of buoyancy-to-rotation dominated convection. Furthermore, we demonstrate that similar to rotating Rayleigh-B\'enard convection, for small E, the critical Rayleigh number for the onset of convection asymptotically scales as E-4/3.This result is combined with heuristic arguments for internally heated and rotating convection to arrive at scaling laws for T ̄ and FB valid for arbitrary Prandtl numbers.