Spheroidal and Toroidal Normal Modes

Abstract

On the ground of the radial component of the vorticity equation, the eigenvalue problem of the linear, isentropic normal modes of a quasi-static star, which is of the third degree in the eigenvalue parameter, is split up into two partial eigenvalue problems: the eigenvalue problem of the normal modes for which the radial component of the curl of the Lagrangian displacement is equal to zero at all points in the equilibrium star, and that of the normal modes for which this condition is not satisfied. Moreover, the Lagrangian displacement field is resolved into a longitudinal and a transverse field, which itself consists of a toroidal and a poloidal field. The scalar functions involved in these three fields are expanded in terms of spherical harmonics of the colatitude and the azimuthal angle. It results that two types of uncoupled modes associated with a single spherical harmonic can be distinguished. The modes of the first type are solutions of the first eigenvalue problem, which is quadratic in the eigenvalue parameter. They are called spheroidal modes and their displacement field consist of a longitudinal and a poloidal field. The modes of the second type are solutions of the second eigenvalue problem, which is of the first degree in the eigenvalue parameter. They are time-independent toroidal modes with a purely horizontal displacement field.