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Astron. Astrophys. 341, 567-573 (1999)

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8. General discussion

In K 97, Sect. 2.6 various examples (Nariai (1976), Webbink (1977), Zhou & Leung (1990)) are quoted as fitting in well with the author's proposed scheme of constant Jacobi energy plus a complete system of `open' surface streamlines to represent a thermally-driven circulation. Since the above series of references follows immediately after a critical comment directed at H 97 it is clear that we are obliged to take a position on these specific examples.

The situation regarding the first two examples listed above has already been discussed in H 97 - as long as Coriolis forces are neglected the condition of constant Jacobi energy is automatically satisfied; insofar the situation is not controversial. It remains to consider the third paper.

In the paper of Zhou & Leung Fig. 4 illustrates the equatorial flow field with the effect of the Coriolis forces clearly shown in the diagram. Now it is shown in H 97 and accepted in K 97 that constant Jacobi energy implies a very strong net retrograde motion in the surface layers. There is however no indication of such a net (i.e. phase-averaged) retrograde effect in the outer parts of the diagram, so that we must conclude that the Jacobi energy can not be a surface invariant. Furthermore the references to cyclones and anticyclones on opposite sides of the system would point to the presence of closed streamlines in the surface layers. Hence we can not argree that this model fits in well with the picture proposed in K 97.

We next consider the general argument brought in K 97 to support the view that, in a situation corresponding to thermally-driven circulation, a complete system of `open' surface streamlines is to be expected. Since this same topology also characterizes the case of zero Coriolis forces, this is at the same time an argument against these forces causing any topological changes.

The basic consideration is that if we multiply the Coriolis forces everywhere by the factor [FORMULA] and allow [FORMULA] to increase gradually from zero then there is really no reason why an abrupt change in the `solutions' should arise from this procedure. In principle, [FORMULA] = 1 could be (and, for the purposes of Sect. 2.6, can be) reached in this way.

Let us consider this argument in relation to the simple model of the present paper. Then we find that, as the parameter [FORMULA] is gradually increased, the flow velocity does indeed change continuously all the way to [FORMULA] = 1. However it is only necessary to consider the formation of the first closed streamline to see that topological changes can occur even when the velocity is changing continuously.

We therefore see that it is not permissible to assume that continuity of the velocity also implies preservation of the topology. This is the essential weakness in the `[FORMULA]-argument' described above.

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998