Propagation and Reflection of Diffusionless Torsional Waves in a Sphere

Abstract:

The magnetohydrodynamics of stars and planetary cores is usually dominated by the overwhelming importance of rotation compared to other forces. Under these conditions the fluid motions are characterized by a strong invariance along the rotation axis. In the presence of a background magnetic field, magnetohydrodynamic oscillations can be triggered. Among these, of particular interest are the torsional waves, azimuthal perturbations of the fluid that are axisymmetric and invariant along the vertical direction. Their periods depend solely on the intensity of the magnetic field component aligned with the radial direction of propagation.

As the detection of the fundamental period could constrain the magnetic field intensity in the Earth’s outer core there is a long history of attempted detection of torsional waves from geomagnetic data. There is however a fundamental lack of knowledge concerning the propagation and reflection properties of these waves, as observational studies suggests behaviors that are different from theoretical expectations. In particular, recent findings (Gillet et al., 2011) suggest the lack of reflection at the equator and at the rotation axis.

Through numerical simulation and analytical techniques we analyze the temporal evolution of diffusionless torsional waves in spherical geometry, with particular attention on the reflection at the equator and the pseudo-reflection at the rotation axis. We develop a novel analytical solution to the torsional wave eigenvalue problem whose behavior at the boundaries helps us to illustrate the meaning of the boundary conditions. Furthermore we find that for any acceptable magnetic background field, reflections at both boundaries are allowed and we illustrate how the WKBJ approximation is an efficient tool for investigating them.