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

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6. Velocity fields with non-constant Jacobi energy

Let us consider once more the situation described in Sect. 5. By far the greatest part of the star is covered with closed streamlines. Hence there is nothing to prevent us, in an ideal experiment, from introducing vorticity over this whole area. If viscosity is required for carrying out this experiment, it is of course removed immediately afterwards.

It is not difficult to show that, under the assumption that Eq. (18) is correct, the perpendicular component of the vorticity will, following the above experiment, remain constant on every streamline. We can imagine the vortices as moving around in an enclosed area ("vortex patch") thereafter.

The expression in brackets on the R.H.S. of Eq. (12) and Eq. (13) i.e. the total perpendicular vorticity, including the rotational contribution, is the relevant quantity in the above considerations. We note that, if this is non-zero, as it is following the above experiment, then the Jacobi energy on the L.H.S. of the above equations can not be constant. Outside the vortex patch, where the (total) vorticity is still zero, the Jacobi energy will remain constant as before.

In order to simplify matters as far as possible, let us assume that, in the enclosed region, the (perpendicular) vorticity is uniform. There is a vorticity jump (but not a velocity jump) along the edge of the vortex patch, including the stretch along the equator where the patches from the two hemispheres touch. The usual symmetry conditions across the equator are not affected by this.

In the enclosed region, therefore:

[EQUATION]

where F is some constant. It follows that F is also constant on each streamline in this region. Now, before the extra vorticity was introduced we had:

[EQUATION]

from which it follows that:

[EQUATION]

Using u to denote the change in velocity we then have from Eq. (14):

[EQUATION]

We look for solutions of the form:

[EQUATION]

so that Eq. (18) then requires that:

[EQUATION]

where we have used the condition that [FORMULA] vanishes at the equator.

Substituting from Eqs. (41) and (42) into Eq. (40) we then find, after some reduction:

[EQUATION]

where [FORMULA] denotes differentiation with respect to the complete argument [FORMULA].

The particular integral of Eq. (43) of interest for us is:

[EQUATION]

At very low latitudes this equation can only be regarded as an approximation since the assumption is made that for all higher latitudes than the one under consideration each parallel of latitude is filled with vortex elements. However since we shall only be interested in the qualitative consequence of introducing these vortex elements the approximation represented by Eq. (44) should be adequate.

Using Eqs. (25), (26), (41), (42), and (44) we now obtain for the total velocity:

[EQUATION]

The streamlines corresponding to this velocity field can be calculated by the same method as was used for the simple velocity field in Sect. 5. Once more using the identity given in Eq. (31) we find after integration that the limiting streamline is given by:

[EQUATION]

provided that

[EQUATION]

The analysis can be simplified if we consider only those cases for which, on the limiting streamline, the product [FORMULA] is everywhere sufficiently small to permit the expansion of the logarithm in Eq. (47) to first order. In these cases:

[EQUATION]

so that we see that the above approximation can be used when:

[EQUATION]

We further confirm, by comparing Eqs. (47) and (49), that Eq. (49) leads to exact results in the case [FORMULA].

As an illustrative example let us take [FORMULA] and [FORMULA] (as before) and compare the results with those for [FORMULA] (see Sect. 5) which will be given in brackets below. Then we find for the locations of the stagnation points ([FORMULA]):

[EQUATION]

and for the extremity of the limiting streamline:

[EQUATION]

We see that, relative to the case considered in Sect. 5 (in brackets) there has been a general increase in the area covered by the closed streamlines. Conversely a contraction of the region occupied by the `open' streamlines has occurred. We shall consider conditions in this latter region in the following section.

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© European Southern Observatory (ESO) 1999

Online publication: December 4, 1998
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