Remarks on the Rayleigh-Bénard convection on spherical shells. (English) Zbl 1273.35216
Summary: 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 T. Ma and Sh. Wang [Physica D 239, No. 3–4, 167–189 (2010; Zbl 1190.37084)] 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.
MSC:
| 35Q30 | Navier-Stokes equations |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 86A10 | Meteorology and atmospheric physics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
Rayleigh-Bénard convection on spherical shells; dynamic transition; center manifold reduction; spherical harmonics; pattern formationCitations:
Zbl 1190.37084References:
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