4. Dynamical description of the flow
The motion of an inviscid fluid under the action of the total potential is given by:
where denotes the velocity in the rotating frame and stationarity is assumed.
We now confine our considerations to motions in the surface layer. As in Sect. 3 we shall treat these as if they occurred over a spherical surface (see, however, below). We shall use spherical polars (with on the surface) and the coordinate axis aligned in the source-sink direction i.e. perpendicular to the rotation axis. Taking the vector product of Eq. (11) with leads to a further vector equation the components of which yield, after some reduction:
where we have again used Eq. (1). The quantity
represents the component of vorticity perpendicular to the surface.
As in Sect. 3 and as assumed earlier (H97 and K97) the pressure has been taken to be constant on the boundary. If this condition is not fulfilled, Eqs. (12) and (13) must be modified accordingly (for details see Appendix A).
It follows from Eqs. (12) and (13) that:
So far we have treated all motions as if they occurred over a spherical surface. The surface will however in practice adjust its form in order that the potential there can satisfy the above equations. As we have noted in H 97 this does not represent an inconsistency since the spherical approximation can still be used to tell us in which way the adjustment must occur. As in Tassoul 1992, the gravitational component of the potential can be taken to be spherically symmetrical in order to support the pseudospherical approach.
We see from the above equations that a necessary condition for constant Jacobi energy everywhere is:
for non-zero velocities. This will also represent a sufficient condition, assuming that the continuity equation (see Sect. 3) can also be satisified.
We shall first consider the case where Eq. (16) is satisfied. This will serve as a useful basis for the discussion later of the more general case.
In order to see what Eq. (16) implies we substitute for from Eq. (14) into Eq. (16) to give:
This must now be combined with the continuity equation. Following the arguments of Sect. 3 we adopt this in the simplified form (k 0):
We look for solutions of the form:
where is restricted to be a non-singular function . Hence the solution includes, but does not entirely consist of, a source-sink pair. The additional function is included in order to allow us to satisfy the full set of equations. A solution of the form given in Eq. (19) should therefore permit us to study mathematically the simple model proposed in Sect. 2. Substituting from Eq. (19) into Eq. (18) we find:
We next attempt a separation of variables:
where the integration constant vanishes since must be zero at the equator .
Using Eqs. (17), (19), (21) and (22) we obtain:
for which the only non-singular solution is:
Hence Eqs. (19), (21), (22) and (24) give:
We must next consider the source strength A. This should not be taken to be so large that the pseudospherical approximation breaks down. Let us suppose, somewhat arbitrarily, that the object introduced in Sect. 2 can be considered roughly spherical if for 99% of the surface the r.m.s. radius variation does not exceed 10% of the average radius. We find this criterion leads to:
where we have taken to be equal to one third of the quadratic angular velocity needed for rotational break-up (Whelan 1972). In order to lie reasonably safely within the limit given by Eq. (27) we shall assume that
in what follows.
© European Southern Observatory (ESO) 1999
Online publication: December 4, 1998