Physical interpretation

According to Dowling (1988), when the rotation axis is vertical (parallel to the gravity axis), a convection cell with a large horizontal scale does not emerge if the rotation of the system is sufficiently large. However, our results show that when the rotation axis is horizontal (perpendicular to the gravity axis), the horizontal scale of convection remains large even when the rotation rate increases. The difference of the convection features between these two rotating systems is qualitatively understood by the difference of the characteristics of waves existing in each rotating system. The waves existing in the system with the vertical rotation axis are inertia waves, whose dispersion relation is

$\begin{displaymath} \omega = \frac{2\Omega m}{\sqrt{k^2+m^2}}. \end{displaymath}$ (14)

where k and m are the wave numbers perpendicular to and parallel to the axis of rotation, respectively. Since the frequency indicates the magnitude of the restoring force which is responsible for the waves, waves with large frequencies prevent the occurrence of convection. Since the frequencies of inertia waves approaches $\omega \rightarrow 2\Omega$ as $k \rightarrow 0$ , that is, the frequency does not vanish as $k \rightarrow 0$ , inertia waves prevent the occurrence of convection at k~ 0. Therefore, the critical wave number departs from k=0 as the rotation rate increases.

On the other hand, waves existing in the system with a horizontal rotation axis are Rossby waves, whose dispersion relation is

$\begin{displaymath} \omega = \frac{\eta k}{\sqrt{k^2+m^2}}, \end{displaymath}$ (15)

Since the frequency of Rossby waves approaches w=0 as $k \rightarrow 0$ , they do not prevent convection with horizontal wave number 0. Therefore, the critical wave number is always zero regardless of the rotation rate.