The Effects of Robin Boundary Condition on Thermal Convection in a Rotating Spherical Shell

Abstract:

Rayleigh-Bénard convection has been studied with many distinct setups along the last decades. In this article we focus on a rotating spherical shell filled with a constant viscosity newtonian fluid. The inner boundary of the shell is isothermal while the upper one is submitted to a Robin boundary condition, which linearly couples temperature and its radial derivative through the Biot number (Bi). This kind of boundary condition has several applications like modeling conducto-convective exchanges (Newton's law) or linearized radiative equilibria (Stefan-Boltzmann law). Tuning the Biot number allows us to switch from fixed temperature to fixed thermal flux boundary condition.

We present results of nonlinear simulations in a rotating spherical shell for which time averaged Nusselt (Nu) and Péclet (Pe) numbers are computed as functions of the control parameters like the Rayleigh number (Ra).
We show that increasing Bi from 0 (fixed heat flux) to infinite values (fixed boundary temperature) produces a monotonous and continuous transition from fixed flux to fixed temperature boundary condition behaviour in terms of time averaged Nu and Pe numbers.
The transition occurs for 0.1 < Bi < 100 independently from the Prandtl number (Pr) in our range of study. This implies that the Robin boundary condition can be safely replaced by the fixed flux boundary condition for Bi < 0.1 and by fixed temperature for Bi > 100.
We also show that an effective Rayleigh number -- built using the effective temperature difference across the shell -- is a fair tool to compare arbitrary Bi configurations in terms of global as well as local diagnostics.