A&A 375, 680-690 (2001)

DOI: 10.1051/0004-6361:20010866

## The *r*-modes of rotating fluids

**Y. Sobouti**

^{1, 2}and V. Rezania^{1, 3}^{1}Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195, Iran

^{2}Center for Theoretical Physics and Mathematics, AEOI, PO Box 11345-8486, Tehran, Iran

^{3}Department of Physics, University of Alberta, Edmonton AB, T6G 2J1 Canada (

*Present address*)

e-mail: sobouti@iasbs.ac.ir

(Received 1 February 2001 / Accepted 19 June 2001)

** 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 be the
angular velocity of the star and () 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
shows that Coriolis
forces turn the neutral toroidal
motions of () designation of the non-rotating fluid into
a sequence of oscillatory modes with
frequencies
.
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 (). They remain decoupled from other toroidal
or poloidal motions belonging to different '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 identifications (),
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 () designation.
4) Two toroidal modes belonging to and get coupled only at the
order.
5) A toroidal and a poloidal mode belonging to and , respectively, get coupled but again at the 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.

**Key words:**stars: neutron

**--**stars: oscillations

**--**stars: rotation

Offprint request: V. Rezania, vrezania@phys.ualberta.ca

**©**

*ESO 2001*