Eigenoscillations of the differentially rotating Sun: II. Generalization   of Laplace's tidal equation
  The general PDE governing linear, adiabatic, nonraradial oscillations in a spherical, differentially and slowly rotating non-magnetic star is derived. This equation describes mainly low-frequency and high-degree g-modes, convective g-modes, and rotational Rossby-like vorticity modes and their mutual interaction for arbitrarily given radial and latitudinal gradients of the rotation rate. In "traditional approximation" the angular parts of the eigenfunctions are described by Laplace's tidal equation generalized here to take into account differential rotation. From a qualitative analysis of Laplace's tidal equation the sufficient condition for the formation of the dynamic shear latitudinal Kelvin-Helmholtz instability (LKHI) is obtained. The exact solutions of Laplace's equation for low frequencies and rigid rotation are obtained. There exists only a retrograde wave spectrum in this ideal case. 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. The eigenfuncions are expressed by Jacobi's polynomials which are polynomials of higher order than the Legendre's for spherical harmonics. The solar 22-year mode spectrum is calculated. It is shown that the slow 22-year modes are concentrated around the equator, while the fast modes are around the poles. The band of latitude where the mode energy is concentrated is narrow, and the spatial place of these band depends on the wave numbers (l, m). 
N. S. Dzhalilov (1), J. Staude (2) ((1)Institute of Terrestrial   Magnetism, Ionosphere and Radio Wave Propagation of the Ru...