Global Magnetohydrodynamic Waves in Stably Stratified Rotating Layers

The 2D shallow water approximation in magnetohydrodynamics is solved, numerically
and analytically, for a perfectly conducting fluid on a rotating sphere with a basic state
for the toroidal magnetic field: Bφ = B0 sin θ. The results are given in terms of the
parameters ε=(2Ω_0 R_0)^2/gH_0 and α^2=v_A^2/(2Ω_0 R_0)^2,
where Ω_0 is the rotation rate, R0 is
the radius, g is the gravity, H0 is the height of the layer and vA is the Alfv´en speed. Five
types of solution have been found: Magneto-inertial gravity waves (MIG), Kelvin waves,
fast and slow magnetic Rossby waves and a slow anomalous mode travelling westward.
A comprehensive numerical study describes the modes in a full range of parameters.
As α → 0, the eigenfunctions are the Associated Legendre polynomials, if ε → 0. When ε → ∞ the eigenfunctions describing MIG and Fast magnetic Rossby waves are defined
by the parabolic cylinder functions for waves confined to the equator. The slow magnetic
Rossby waves are not equatorially trapped.
When α ≥ 0.5 there is a transition for magnetic Rossby waves. The slow and fast
modes coalesce and an unstablemode emerges, but only when the azimuthal wavenumber
m = 1. After this transition point (α = 0.5) the fast magnetic Rossby waves turn into
subalfv´enic waves and tend to be trapped at the poles.
As α → ∞, theMIG waves become equatorially trapped Alfv´en waves. These modes are
always stable. The slow and fast magnetic Rossby waves (real and complex) are polar
trapped with eigenfunctions described by Laguerre polynomials multiplied by a factor that gives the confinement.
The antisymmetric configuration for the field Bφ = B0 sin θ cos θ, produces similar
results to the previous case but the main difference is that the slow magnetic Rossby
waves are absent. Also, magnetic Rossby waves become unstable for certain values of
α and ε, then become real again by interacting with another mode and so on, weaving a
net. On the other hand, when α is large, there is a critical layer which absorbs the MIG
waves.