In recent years, the problem of the asteroseismology of magnetic stars has been thoroughly studied from the point of view of acoustic oscillations pertubated by a permanent magnetic field. Motivated by the conclusion that regular perturbation methods are not valuable in stellar atmospheres where low values of the parameter of the plasma are found, we consider a simpler model involving a spherical layer of incompressible resistive plasma and apply a strong dipolar magnetic field on it.
We carry out the investigation of axisymmetric shear Alfvén waves in such a geometry. A radial decomposition on the Gauss-Lobato grid and an angular decomposition on the spherical harmonics basis are used to compute the eigenmodes of the system. Numerical results involving up to L=1000 spherical harmonics and Nr=300 Chebyschev polynomials show that the least-damped Alfvénic modes concentrate near the magnetic polar axis where they are trapped owing to the dipolar structure of the field. This feature remains present in the adiabatic limit.
We demonstrate that the selection rule on the frequencies of the oscillations is closely related to the thickness of the layer, and argue that the eigenfrequencies become of the same order as those observed on the polar caps of roAp stars in the thin-layer limit. The oscillation patterns also exhibit internal shear/magnetic layers associated with resonant field lines which appear to be the main numerical limitation.
The model may be applied to planetary cores sustaining a dynamo, or to the magnetic layer of roAp stars. In this latter case, it shows that Alfvén waves are an unavoidable feature of the oscillations of roAp stars and more generally of magnetic stars.

 
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