The r-modes of rotating fluids

Abstract

An analysis of the toroidal modes of a rotating fluid, by means of the dierential equations of motion, is not readily tractable. A matrix representation of the equations on a suitable basis, however, simplies the problem considerably and reveals many of its intricacies. Let be the angular velocity of the star and ('; m) be the two integers that specify a spherical harmonic function. One readily nds the followings: 1) Because of the axial symmetry of equations of motion, all modes, including the toroidal ones, are designated by a denite azimuthal number m. 2) The analysis of equations of motion in the lowest order of shows that Coriolis forces turn the neutral toroidal motions of ('; m) designation of the non-rotating fluid into a sequence of oscillatory modes with frequencies 2m='(' + 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 identication ('; m). They remain decoupled from other toroidal or poloidal motions belonging to dierent ''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 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 identications (s; '; 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 ('; m) designation. 4) Two toroidal modes belonging to ' and '2 get coupled only at the 2 order. 5) A toroidal and a poloidal mode belonging to ' and '1, respectively, get coupled but again at the 2 order. Mass and mass-current 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.