On the analogy between gravity modes and inertial modes in spherical geometry

Analysing similarities and differences between gravity modes and inertial modes, we derive the analytic expression of gravity modes in an axisymmetric ellipsoid for a diffusionless fluid. We also give the expression of the dispersion relation for these modes in such a geometry. The case of a spherical shell is then considered. After deriving a class of analytical solutions, we exhibit the analogous structure of singular gravity modes and singular inertial modes. Finally, we show why singular modes should play an important part in the transport properties of a stratified or rotating fluid.