Forward models of torsional waves: dispersion and geometric effects

Abstract

Alfvén waves are a set of transverse waves that propagate in an electrically conducting fluid in the presence of an ambient magnetic field. Studies of such waves in the Earth's interior are important because they can be used to make inferences about the structure and physical properties of the core that would otherwise remain inaccessible. We produce 1-D forward models of cylindrical torsional Alfvén waves in the Earth's core, also known as torsional oscillations, and study their evolution in a full sphere and an equatorially symmetric spherical shell. Here, we find that travelling torsional waves undergo significant geometric dispersion that increases with successive reflections from the boundaries such that an initial wave pulse becomes unidentifiable within three transits of the core. Low amplitude wakes trail behind sharply defined pulses during propagation, a phenomenon that we interpret using the failure of Huygens' principle in even dimensions. We investigate the relationship between geometric dispersion and wavelength, concluding that long-wavelength features are more dispersive than short-wavelength features. This result is particularly important because torsional waves that have been inferred in the Earth's core from secular variation are relatively long wavelength, and are therefore likely to undergo significant dispersion within the core. When stress-free boundary conditions on angular velocity are applied, waves are reflected at the equator of the core-mantle boundary with the same sign as the incident wave. Waves that pass through the rotation axis undergo a pseudo-reflection and display a more complicated behaviour due to a phase shift. In an equatorially symmetric shell, we identify a weak reflection at the tangent cylinder due to geometric effects.

Propagation and reflection of diffusionless torsional waves in a sphere

“…We consider two different background magnetic fields: the constant one and a field that goes to zero at the equatorial boundary. The constant field has the advantage of supporting known normal modes: see Roberts & Aurnou (2012) and Cox et al (2014). The spatially varying field that we introduce in this study has an interesting behaviour at the equator that helps us illustrate the meaning of the boundary conditions for the torsional waves.…”

“…The aim of this study is to better understand the propagation of torsional waves, with particular focus on the reflections, both at the equator and at the rotation axis. In what follows we build on the work of Cox et al (2014), in which initial pulses of geostrophic velocity are evolved in time according to the torsional wave evolution equation with prescribed background magnetic field; their propagation and reflection properties are then analysed. For a constant background field Cox and coauthors suggested that at any reflection from the rotation axis a change in phase is introduced, therefore altering the shape of the initial pulse.…”

“…The phase shift resembles the π/2 phase shift introduced in seismic waves at any passage through a caustic, a phenomenon that is well known in seismology (Chapman 2004, section 7.2). In Cox et al (2014) only the reflection at the rotation axis (that we address as a pseudo-reflection) in cylindrical domains is rigorously considered. We propose a methodology to derive the reflection coefficients for both boundaries in a spherical domain.…”

Anelastic torsional oscillations in Jupiter's metallic hydrogen region

“…1]. Forward simulations of linear, nondissipative TWs in spherical geometry reported internal reflections where the gradient of a background magnetic field was steep [11]. The role of the background velocity on the reflections in our simulations has not yet been elucidated.…”

“…This could be explained through preferred excitation [39,40,41] near the tangent cylinder (TC, the imaginary cylinder aligned with the rotation axis that circumscribes the inner core) and dissipation beneath and above the core-mantle boundary (CMB) [36,35]. Studies ignoring dissipation showed that the effect of spherical geometry and variable internal magnetic fields can give rise to asymmetric reflections and hence weaken reflected waves [11]. We shall demonstrate that TWs in the gas giant's metallic region can be reflected from a magnetic TC, which is formed due to the transition to molecular hydrogen.…”