## Appendix A: some comments on the assumptionsWe first comment on the assumption of uniform surface pressure (see Sect. 3). This was made principally to preserve uniformity with previous papers (H97, K97 and references therein). Nevertheless it may reasonably be asked whether this simplified boundary condition is a) permissible and b) representative. Let us see how this simplified boundary condition would fit in
with, for example, the geostrophic approximation (see e.g. Tassoul
1992). Here the pressure gradient is regarded as being non-zero Nevertheless we must ask whether a departure from our simple boundary condition would be likely to change the flow pattern so much as to invalidate our general conclusions. To fix matters, let us assume that the specific entropy is uniform over the surface (barotropy); this will give us a situation which is thermally similar to that obtained by setting k = 0 in Sect. 3. If, as previously, the gas is assumed perfect, with constant molecular weight, then the thermal equation of state can be written: Rather than returning with this equation to Sect. 3, with P now variable over the surface, it is more useful to go directly to the equations of motion. We then find that there is an extra term inside the brackets on the L.H.S. of Eqs. (12) and (13): where the above triple sum is sometimes referred to as Bernoulli's integral. There is however no change on the R.H.S. of Eqs. (12) and (13). For small surface pressures and supersonic motions Somewhat more serious than the assumption of constant surface pressure is perhaps the neglect of the contribution of the radial velocity component to the velocity divergence; this is our 2-dimensional approximation of setting everywhere on the surface except of course at the singular points where the source and sink are located. We can improve on this highly idealized situation by allowing both
source and sink to have a finite lateral extent where is a small positive
constant; the previous idealized case corresponds to
, Keeping closely to the methods used in the text, but now assuming
It is worth noting that Eqs. (A5) and (A6) predict (for small
© European Southern Observatory (ESO) 1999 Online publication: December 4, 1998 |