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Remarks on the Rayleigh-Bénard Convection on Spherical Shells

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Abstract

The main objective of this article is to study the effect of spherical geometry on dynamic transitions and pattern formation for the Rayleigh-Bénard 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 \({S^{2l_c}}\) , where l c is the critical wave number corresponding to the critical Rayleigh number. Furthermore, it has shown in Ma and Wang (Physica D 239:3–4, 167–189, 2010) 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.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Shouhong Wang & Ping Yang

Authors
  1. Shouhong Wang
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  2. Ping Yang
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Correspondence to Shouhong Wang.

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Communicated by Chae

The work was supported in part by the US Office of Naval Research and by the US National Science Foundation.

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Wang, S., Yang, P. Remarks on the Rayleigh-Bénard Convection on Spherical Shells. J. Math. Fluid Mech. 15, 537–552 (2013). https://doi.org/10.1007/s00021-012-0128-8

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  • DOI: https://doi.org/10.1007/s00021-012-0128-8

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