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

Abstract

The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one 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 support 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 eigenvalues, probably dense in some subintervals of [0, N], N being the Brunt–Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0, N]. It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not square-integrable and the corresponding point spectrum is also empty.

Low-frequency internal waves in magnetized rotating stellar radiation zones

“…In the general case, the Poincaré operator for the pressure fluctuation, which governs the spatial structure of the waves, is of mixed type (elliptic and hyperbolic) and not separable (for a detailed discussion we refer the reader to Friedlander & Siegman 1982;Dintrans et al 1999;Ballot et al 2010). In the hyperbolic regime, this may lead to the appearance of detached shear layers associated with the underlying singularities of the adiabatic problem, which could be crucial for transport and mixing processes in stellar radiation zones since they are the seat of strong dissipation (Stewartson & Richard 1969;Stewartson & Walton 1976;Dintrans & Rieutord 2000;Maas & Harlander 2007).…”

“…In the hyperbolic regime, this may lead to the appearance of detached shear layers associated with the underlying singularities of the adiabatic problem, which could be crucial for transport and mixing processes in stellar radiation zones since they are the seat of strong dissipation (Stewartson & Richard 1969;Stewartson & Walton 1976;Dintrans & Rieutord 2000;Maas & Harlander 2007). Let us first introduce the two dimensionless numbers, which compare the stratification restoring force with the Coriolis acceleration and the Lorentz force…”

“…This corresponds to the elliptic modes family (the E 1 modes in Dintrans et al 1999), which propagate in the whole sphere. In the other regime where |ν M;m | ≥ 1, which corresponds to equatorially trapped hyperbolic modes (the H 1 modes in Dintrans et al 1999), the MHD TA fails to reproduce the wave behaviour rigourously and the full dynamical equation has to be solved (a detailed discussion is given in Gerkema & Shrira 2005a;Gerkema et al 2007). For a strong stratification, which is considered here, Miles (1974) proposed an asymptotic solution to the problem, for which it is necessary to construct a boundary layer about the trapping latitude,…”

“…when σ 2 s − m 2 ω 2 A < 0). We also display the corresponding wave families (extending the classification by Dintrans et al 1999, to the hydromagnetic case): the elliptic modes (E family) for which the MHD TA applies, the equatorially trapped hyperbolic modes (H family), and the vertically trapped modes (T family), for which σ 2 s − m 2 ω 2 A < 0, which cannot propagate. Let us now examine the consequences on wave properties using their dispersion relation.…”

A Comparison of the Anelastic and Subseismic Approximations for Low-Frequency Stellar Oscillations: an Application to Rapidly Rotating Stars

“…The anelastic equations need to be solved in a co-rotating frame; they read (Dintrans & Rieutord, 2000, hereafter DR2000)…”