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

TL;DR

This study investigates how Robin boundary conditions influence thermal convection in rotating spherical shells, revealing effects on flow structure, onset, and heat transfer, with implications for planetary and stellar modeling.

Contribution

It provides a detailed analysis of Robin boundary conditions in spherical shell convection, showing their impact on flow dynamics and establishing when they can be approximated by fixed flux or fixed temperature conditions.

Findings

Onset of convection is unaffected by Robin conditions in non-rotating cases.

Universal scaling laws for heat transfer and flow are independent of boundary conditions.

Flow structure varies with Robin parameter, showing more vigorous large scales for smaller Bi*.

Abstract

Convection in a spherical shell is widely used to model fluid layers of planets and stars. The choice of thermal boundary conditions in such models is not always straightforward. To understand the implications of this choice, we report on the effects of the thermal boundary condition on thermal convection, in terms of instability onset, fully developed transport properties and flow structure. We use the Boussinesq approximation, and impose a Robin boundary condition at the top. This enforces the temperature anomaly and its radial derivative to be linearly coupled with a proportionality factor $\beta$. Using the height H of the fluid layer, we introduce the non-dimensional Biot number Bi* = $\beta$H. Varying Bi* allows us to transition from fixed temperature for Bi* = +$\infty$, to fixed thermal flux for Bi* = 0. The bottom boundary of the shell is kept isothermal. We find that the onset of convection is only affected by Bi* in the non-rotating case. Far from onset, considering an effective Rayleigh number and a generalized Nusselt number, we show that the Nusselt and P{\'e}clet numbers follow standard universal scaling laws, independent of Bi* in all cases considered. However, the large-scale flow structure keeps the signature of the boundary condition with more vigorous large scales for smaller Bi* , even though the global heat transfer and kinetic energy are the same. Finally, for all practical purposes, the Robin condition can be safely replaced by a fixed flux when Bi* < 0.03 and by a fixed temperature for Bi* > 30.