Concluding remarks and discussions

We have applied the linear stability theory to a two-dimensional convective motion in a rotating cylindrical annulus in order to investigate the effect of a fixed heat flux condition on the characteristics of convection, especially on the horizontal scale of convection cells. The results show that a convection cell with a large horizontal size always emerges at a critical state regardless of the rotation rate. The neutral curves at large rotation rates have another local minimum at k¹0 in addition to the critical point at k = 0. The characteristics of the mode corresponding to this minimum coincide with those obtained under the fixed temperature boundary condition in the limit of large rotation rates. A finite amplitude convection solution is obtained by the numerical time integration, where the large-scale convection cells are superimposed on the small-scale convection cells. This result is consistent with that expected from the profiles of the neutral curve.

Our results suggest that the Taylor-column type convection structure appearing in rapidly rotating spherical shells can be greatly altered by the difference in thermal boundary conditions. A convection cell with a large horizontal scale is expected to appear as a critical mode under the fixed heat flux condition. The small-scale convection cells which will appear during the linear developing stage at super-critical Rayleigh numbers must have similar characteristics as those with fixed temperature boundaries.

We have discussed fixed heat flux boundary cases as the extreme situation opposite to the fixed temperature cases. In order to consider convective motion in actual astronomical bodies, for example, in the earth's central core or atmospheres of Jovian planets and the Sun, we must examine the effect of boundaries with finite thermal conductivity. Such studies were performed for the convection in non-rotating systems (Sparrow et al.,1964; Hurle et al.,1967; Busse and Riahi, 1980). The boundary condition can be now described by the additional non-dimensional number, the Biot number, which is a function of the ratio of the conductivities and the ratio of the thicknesses of the convection layer and the boundary material layer. However, in the cases of rotating systems, the convection modes generally become oscillatory, and hence the phase speed of the mode appears in the expression of the thermal boundary condition. Qualitatively, the thermal boundary condition for a rotating system is considered to be governed by the time scales of the propagation of a convection pattern and the penetration of the temperature disturbance into the boundary material. When the phase speed is sufficiently small and a convection pattern can be regarded to be spatially fixed, the boundary condition can be approximated by that of the non-rotating system. When the phase speed is large enough, the convection pattern moves before the temperature disturbance penetrates into the boundary material, and hence, the boundary condition will approach the fixed temperature condition.

We have utilized the truncated system up to the vertical wave number 1 to investigate the system analytically. The truncated system holds the qualitative features of the original full non-linear system. By applying a weak non-linear method to the truncated system, we will also be able to investigate the behavior of finite amplitude convection in rotating systems with fixed heat flux boundaries.