The r-modes of rotating fluids

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 in a suitable basis, however, simplifies the problem considerably and reveals many of its intricacies. Let Ω 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 Ω shows that Coriolis forces turn the neutral toroidal motions of (l, m) designation of the non rotating fluid into a sequence of oscillatory modes with frequencies 2mΩ/l(l + 1). This much is a 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. For 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 equation of motion in the next higher order of Ω 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 l ± 2 get coupled only at Ω order. Send offprint requests to: V. Rezania ∗ Present address 5) A toroidal and a poloidal mode belonging to l and l± 1, respectively, get coupled but again at Ω 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 details.