Internally heated convection with rotation: bounds on heat transport

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, $\overline{⟨T⟩}$, 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-Bénard 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 $\overline{⟨T⟩}$ and ${\mathcal{F}}_B$ valid for arbitrary Prandtl numbers.

arXiv comment35 pages, 9 figures

References in corpus (32)

• Large scale dynamics in turbulent Rayleigh-Benard convection
• Scaling in thermal convection: A unifying theory
• Inside the Supernova: A Powerful Convective Engine
• The unifying theory of scaling in thermal convection: The updated prefactors
• Approaching the Asymptotic Regime of Rapidly Rotating Convection: Boundary Layers vs Interior Dynamics
• Radiative heating achieves the ultimate regime of thermal convection
• "Ultimate state" of two-dimensional Rayleigh-Benard convection between free-slip fixed temperature boundaries
• Subcritical thermal convection of liquid metals in a rapidly rotating sphere
• Theory and simulations of rotating convection
• A nonlinear model for rotationally constrained convection with Ekman pumping
• Neutrino-Driven Convection in Core-Collapse Supernovae: High-Resolution Simulations
• Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model
• Penetrative internally heated convection in two and three dimensions
• Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems
• Convection driven by internal heating
• Scaling in internally heated convection: a unifying theory
• Convection with Misaligned Gravity and Rotation: Simulations and Rotating Mixing Length Theory
• Frictional boundary layer effect on vortex condensation in rotating turbulent convection
• Internal heating driven convection at infinite Prandtl number
• The background method: Theory and computations
• Relationship between the methods of bounding time averages
• Bounds on heat transport for convection driven by internal heating
• Bounds for Rayleigh-Bénard convection between free-slip boundaries with an imposed heat flux
• Optimal bounds with semidefinite programming: an application to stress driven shear flows
• Bounds on Heat Transport in Rapidly Rotating Rayleigh-Bénard Convection
• Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks
• Bounds for internally heated convection with fixed boundary heat flux
• Analytical bounds on the heat transport in internally heated convection
• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Internal heating profiles for which downward conduction is impossible
• Rigorous bounds on the heat transport of rotating convection with Ekman pumping
• Rigorous scaling laws for internally heated convection at infinite Prandtl number