Eigenoscillations of the differentially rotating Sun - II. Generalization of the Laplace tidal equation

The general partial differential equation governing linear adiabatic nonradial oscillations in a spherical, differentially and slowly rotating non-magnetic star is derived. This equation describes mainly low-frequency and high-degree g-modes, con- vective g-modes, rotational Rossby-like vorticity modes, and their mutual interaction for arbitrarily given radial and latitudinal gradients of the rotation rate. Applying to this equation the "traditional approximation" of geophysics results in a separation into radial- and angular-dependent parts of the physical variables, each of which is described by an ordinary differential equation. The angular parts of the eigenfunctions are described by the Laplace tidal equation generalized here to take into account differ- ential rotation. It is shown that there appears a critical latitude in the sphere where the frequencies of eigenmodes coincide with the frequencies of inertial modes. The resonant transformation of the modes into the inertial waves acts as a resonant damping mechanism of the modes. Physically this mechanism is akin to the Alfven resonance damping mechanism for MHD waves. It applies even if the rotation is rigid. The exact solutions of the Laplace equation for low frequencies and rigid rotation are obtained. The eigenfunctions are expressed by Jacobi polynomials which are polynomials of higher order than the Legendre polynomials for spherical harmonics. In this ideal case there exists only a retrograde wave spectrum. The modes are subdivided into two branches: fast and slow modes. The long fast waves carry energy opposite to the rotation direction, while the shorter slow-mode group velocity is in the azimuthal plane along the direction of rotation. It is shown that the slow modes are concen- trated around the equator, while the fast modes are concentrated around the poles. The band of latitude where the mode energy is concentrated is narrow, and the spatial location of these band depends on the wave numbers (l, m).