Linearr‐Mode Oscillations in a Differentially Rotating Star

Abstract

Differential rotation of a stellar envelope lifts the degeneracy of r-mode rotation rates and can prohibit some modes. It also reduces the physical distinction between the two branches of r-modes: slow (retrograde) and fast (geostrophic). One mode in each branch is dominated by a given spherical harmonic (l, m), and, for increasing rotational shear in the star, their characteristics approach each other until both modes simultaneously cease to exist. Only about half of the 60 r-modes with l ≤ 5 and a given radial harmonic can survive as global, linear modes in the Sun. Each of these has at least 70%, and typically 90%, of its energy in a single toroidal component of motion. Among solar survivors are all modes where | m | = l > 1. All results apply to linear, adiabatic oscillations of a spherical fluid shell exhibiting steady, axisymmetric rotation, Ω(r, θ). Since terms ~Ω2 are small in the Sun, spheroidal motions are neglected. A generalization of classical perturbation theory was used to treat perturbations of large size and to avoid the need for special treatment of oscillation frequencies that are zero or degenerate.