Remarks on the Rayleigh-Benard Convection on Spherical Shells

TL;DR

This paper investigates how spherical geometry influences the pattern formation and dynamic transitions in Rayleigh-Benard convection, with implications for geophysical flows and atmospheric circulation modeling.

Contribution

It demonstrates that spherical geometry causes a continuous transition to specific pattern structures and derives explicit relations involving turbulent friction effects.

Findings

System undergoes a continuous transition to a 2l_c-dimensional sphere.

Turbulent friction terms influence pattern selection and transition dynamics.

Explicit formulas link critical wave number, aspect ratio, and friction coefficients.

Abstract

The main objective of this article is to study the effect of spherical geometry on dynamic transitions and pattern formation for the Rayleigh-Benard convection. The study is mainly motivated by the importance of spherical geometry and convection in geophysical flows. It is shown in particular that the system always undergoes a continuous (Type-I) transition to a $2l_c$-dimensional sphere $S^{2lc}$, where lc is the critical wave length corresponding to the critical Rayleigh number. Furthermore, it has shown in [12] that it is critical to add nonisotropic turbulent friction terms in the momentum equation to capture the large-scale atmospheric and oceanic circulation patterns. We show in particular that the system with turbulent friction terms added undergoes the same type of dynamic transition, and obtain an explicit formula linking the critical wave number (pattern selection), the aspect ratio, and the ratio between the horizontal and vertical turbulent friction coefficients.