Internally heated convection with rotation: bounds on 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 lower bounds on the mean temperature, $\overline {{\langle {T} \rangle }}$ , and the heat flux out of the bottom boundary, ${\mathcal {F}}_B$ , at infinite Prandtl number, where the Prandtl number is the non-dimensional ratio of viscous to thermal diffusion. The lower bounds are functions of the 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, J. Math. Phys. , vol. 45(7), 2004, pp. 2718–2743) and the other an integral estimate over the domain (Constantin et al. , Phys. D : Non. Phen. , vol. 125, 1999, pp. 275–284), resulting in bounds valid for different regions of buoyancy-to-rotation dominated convection. Furthermore, we demonstrate that similar to rotating Rayleigh–Bénard convection, for small $E$ , the critical Rayleigh number for the onset of convection asymptotically scales as $E^{-4/3}$ .