Asymptotic expressions for the angular dependence of low-frequency pulsation modes in rotating stars

Through the solution of Laplace's tidal equations, approximated to describe equatorially trapped wave propagation, analytical expressions are obtained for the angular dependence of pulsation modes in uniformly rotating stars. As the ratio between rotation and pulsation frequencies becomes large, these expressions approach the exact solutions of the governing low-frequency pulsation equations. Four classes of asymptotic solution are found, corresponding to g (gravito-inertial), r (Rossby), Kelvin and Yanai modes. The Kelvin modes arise through the conservation of specific vorticity, much like the r modes, but propagate in the same sense as the rotation; they are found to be the equivalents of prograde sectoral modes. The prograde Yanai modes behave like g modes, as do the retrograde ones if the rotation is sufficiently rapid; otherwise, the latter exhibit the character of r modes. Comparison between asymptotic and numerical solutions to the tidal equations reveals that the former converge rapidly towards the latter, for g and Yanai modes. The convergence is slower for Kelvin and r modes, as these become equatorially trapped only when the rotation is very rapid. It is argued that the utility of the asymptotic solutions does not rest on their accuracy alone, but also on the valuable physical insights that they are capable of providing.