Internally heated convection with rotation: bounds on heat transport

TL;DR

This paper establishes bounds on heat transport in rotating internally heated convection, revealing how rotation and internal heating influence temperature and flux, and deriving scaling laws for different regimes.

Contribution

It provides the first rigorous bounds on heat transport in rotating internally heated convection at infinite Prandtl number, incorporating effects of rotation and internal heating.

Findings

Bounds on mean temperature and heat flux derived

Critical Rayleigh number scales as E^{-4/3} for small E

Scaling laws for temperature and flux established

Abstract

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, $\overline{\langle T \rangle }$, and the heat flux out of the bottom boundary, $\mathcal{F}_B$ 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 $\overline{\langle T \rangle }$ and $\mathcal{F}_B$ valid for arbitrary Prandtl numbers.