The r-modes of rotating fluids

An analysis of the toroidal modes of a rotating fluid, by means of the diff erential equations of motion, is not readily tractable. A matrix representa tion 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 spherica l harmonic function. One readily finds the followings: 1) Because of the ax ial symmetry of equations of motion, all modes, including the toroidal ones , are designated by a definite azimuthal number m. 2) The analysis of equat ions of motion in the lowest order of Omega shows that Coriolis forces turn the neutral toroidal motions of (l, m) designation of the non-rotating flu id into a sequence of oscillatory modes with frequencies 2m Omega /l(l + 1) . This much is common knowledge. One can say more, however. a) Under the Co riolis forces, the eigendisplacement vectors remain purely toroidal and car ry 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 nu mber. As a result the eigendisplacement vectors, as far as their radial dep endencies go, remain indeterminate. 3) The analysis of the equation of moti on in the next higher order of Omega reveals that the forces arising from a sphericity of the fluid and the square of the Coriolis terms (in some sense ) remove the radial degeneracy. The eigenfrequencies now carry three identi fications (s, l, m), say, of which s is a radial eigennumber. The eigendisp lacement vectors become well determined. They still remain zero order and p urely toroidal motions with a single (l, m) designation. 4) Two toroidal mo des belonging to l and l +/-2 get coupled only at the Omega (2) order. 5) A toroidal and a poloidal mode belonging to l and l +/-1, respectively, get coupled but again at the Omega (2) order. Mass and mass-current multipole m oments of the modes that are responsible for the gravitational radiation, a nd bulk and shear viscosities that tend to damp the modes, are worked out i n much detail.