4. The g and p components of the displacement field
Using a gauged version of Helmholtz theorem (Sobouti 1981), one may decompose a general displacement vector into an irrotational and a "weighted" solenoidal component. Thus
Here is the unit vector in direction, and and are two scalars. Evidently both components are poloidal and mutually orthogonal, =0. Next, we define the dimensionless parameter Schwarzschild's criterion for convective neutrality is . In such a fluid one readily sees that and . That is, the oscillatory motions of the fluid are of purely p -type and are driven mainly by the compression forces . The g -type motions are neutral, It can also be shown (Sobouti & Silverman 1978) that to the first order of smallness in , the p oscillations retain their pure p nature of Eq. (8a), while the g motions develop into a sequence of new and long period oscillations of the type of Eq. (8b). The latter are driven by the buoyancy forces, For larger values of the two types get more and more coupled. One last remark: If , then ; the fluid is stable to convective motions and the oscillatory g modes develop. If then and convective motions are set up. The imaginary frequency, , then indicates the rate of the exponential growth of the convective motions.
Note added in revision: Toroidal modes of the fluid in the present scheme have the form , scalar. In the Newtonian models considered in this paper they are neutral and remain neutral upon exposure to gravitational waves. The referee, however, has informed us of a recent work of Kokkotas (in press) where he shows that in relativistic stars they also are excited by gravitational waves.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998