The rmodes of rotating fluids
Abstract
An analysis of the toroidal modes of a rotating fluid, by means of the differential equations of motion, is not readily tractable. A matrix representation of the equations on a suitable basis, however, simplifies the problem considerably and reveals many of its intricacies. Let Omega be the angular velocity of the star and (l, m) be the two integers that specify a spherical harmonic function. One readily finds the followings: 1) Because of the axial symmetry of equations of motion, all modes, including the toroidal ones, are designated by a definite azimuthal number m. 2) The analysis of equations of motion in the lowest order of Omega shows that Coriolis forces turn the neutral toroidal motions of (l, m) designation of the nonrotating fluid into a sequence of oscillatory modes with frequencies 2mOmega /l(l +1). This much is common knowledge. One can say more, however. a) Under the Coriolis forces, the eigendisplacement vectors remain purely toroidal and carry the identification (l,m). They remain decoupled from other toroidal or poloidal motions belonging to different l's. b) The eigenfrequencies quoted above are still degenerate, as they carry no reference to a radial wave number. As a result the eigendisplacement vectors, as far as their radial dependencies go, remain indeterminate. 3) The analysis of the equation of motion in the next higher order of Omega reveals that the forces arising from asphericity of the fluid and the square of the Coriolis terms (in some sense) remove the radial degeneracy. The eigenfrequencies now carry three identifications (s,l,m), say, of which s is a radial eigennumber. The eigendisplacement vectors become well determined. They still remain zero order and purely toroidal motions with a single (l,m) designation. 4) Two toroidal modes belonging to l and lpm 2 get coupled only at the Omega ^{2} order. 5) A toroidal and a poloidal mode belonging to l and lpm 1, respectively, get coupled but again at the Omega ^{2} order. Mass and masscurrent multipole moments of the modes that are responsible for the gravitational radiation, and bulk and shear viscosities that tend to damp the modes, are worked out in much detail.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 August 2001
 DOI:
 10.1051/00046361:20010866
 arXiv:
 arXiv:astroph/0105066
 Bibcode:
 2001A&A...375..680S
 Keywords:

 STARS: NEUTRON;
 STARS: OSCILLATIONS;
 STARS: ROTATION;
 Astrophysics
 EPrint:
 12 pages, 4 fiures, revised version to appear in A&