Gravito-inertial waves in a rotating stratified sphere or spherical shell

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq a pproximation. In the perfect fluid limit, these modes obey a second-order p artial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-peri odic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by on e of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give s upport to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenva lues, probably dense in some subintervals of [0,N], N being the Brunt-Vaisa la frequency. Second, with periodic orbits are associated singular modes wh ich feature a shear layer following the periodic orbit; as the zero-diffusi on limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the po int spectrum is empty in some subintervals of [0,N]. It is also shown that these internal shear layers contain the two scales E-1/3 and E-1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with char acteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectr um is also empty.