|Thermal convection of Boussinesq fluids has been researched vigorously
for understanding the fundamental features of fluid motions of geophysical and
astronomical bodies such as the atmospheres and the interiors of planets and stars. Benard
convection is one of the oldest frameworks used for this purpose. It is a convection
driven by fixed temperatures given at the upper and lower boundaries of the fluid layer.
It is well known that convection cells
with the aspect ratio of 2 to 3 emerge when the Rayleigh number is not very large.
Thermal convection in a rotating system with a fixed temperature boundary condition has also been investigated in relation to global circulations of the interiors of astronomical bodies. Nakagawa and Frenzen (1955) and Chandrasekhar(1961) investigated thermal convections in rotating systems with vertical rotation axes (in the direction of gravity). Based on the linear theory and laboratory experiments, they showed that the convection cells which appear as critical modes have smaller horizontal scales as the rotation rate increases.
On the other hand, it has been recognized in the non-rotating cases that the horizontal scale of convection cells becomes larger when the flux-fixed thermal boundary condition is used instead of the fixed temperature condition. Jakeman (1968) shows that with the linear theory, a convection cell with a large horizontal scale appears as the critical state of the fixed heat flux boundary condition. In the case of finite amplitude convection, it is also shown by the weak nonlinear theory (Chapman and Proctor,1980) and by numerical calculations (Hewitt et al.,1980 ; Ishiwatari et al.,1994) that convection cells with small horizontal scales are unstable and convection cells with the largest horizontal scale emerge in the final stage. The fixed heat flux boundary condition and the rotation of the system have the opposite effects on the horizontal size of the convection cells.
Few studies have been conducted so far on thermal convection in a rotation system with a fixed heat flux boundary condition. Dowling (1988) is the only example of such a study. By using the variational method and the asymptotic expansion in respect to the small horizontal wave number k, Dowling showed that convection cells with the largest horizontal scale appear as the critical state when the Taylor number is small, while for large Taylor numbers, convection cells with large horizontal scales do not emerge.
(a) The case with fixed temperature boundary condition
(b) The case with fixed heat flux boundary condition
Fig.1 Convection patterns of non-rotating systems