On the Finite amplitude thermal instability of a rapidly rotating fluid sphere

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Abstract
The onset of instability for a self-gravitating, internally heated, fluid sphere of radius, r∗0, at large Taylor number, [Ttilde] (a measure of the rotation rate), is investigated. The pioneering work of Roberts (1968) and Busse (1970) has shown that at the onset of instability convection is concentrated in a layer of thickness, r∗0[Ttilde]−1/9, which forms a cylindrical surface, coaxial with the diameter parallel to the angular velocity. Inside this layer, Rossby waves of short length scale, r∗0 [Ttilde]− 1/6, propagate eastwards. For Rayleigh numbers, [Rtilde]∗, close to the critical value, [Rtilde]∗ c , determined by Busse (1970), a linear equation governing the evolution of the waves on the long length scale, r∗0[Ttilde]−1/9, and a long lime scale is derived., When [Rtilde]∗ is greater than [Rtilde]∗c some disturbances applied at time t∗ = 0 initially grow exponentially. Nevertheless the equation has the remarkable property that, as t∗ → ∞, all solutions decay exponentially irrespective of the value of [Rtilde]∗. Therefore, since no steady or periodic solution is possible, the critical Rayleigh number in the accepted sense is not close to [Rtilde]∗c. When ([Rtilde]∗—[Rtilde]∗c )/[Rtilde]∗ c is of order [Ttilde]−1/9, conditions are favourable for convection in a layer of thickness, r∗0[Ttilde]− 1/18. Here the paradoxical behaviour of the linear solutions is resolved by finite amplitude effects, which take account of modifications to the mean state caused by heat and angular momentum transport. Within the [Ttilde]−1/18-convection layer, the non-linear equations admit solutions in which convection is concentrated on the smaller r∗0[Ttilde]− 1/9 -length scale. These localised disturbances, in general drift with speed, c∗s , which varies slowly as the disturbance (a thermally driven solitary wave) moves across the convection layer. When [Rtilde]∗ is less than a certain value, [Rtilde]∗stat (say), all solitary waves move inwards towards the rotation axis. On the other hand, when [Rtilde]∗ exceeds [Rtilde]∗stat, some solitary waves move outwards away from the rotation axis in certain regions of the convection layer. In practice [Ttilde]− 1/18 is not particularly small so that the fine details of the solitary wave evolution are unimportant. Nevertheless, the general features of solitary wave structures is significant.

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