Kinematic validation of a quasi-geostrophic model for the fast dynamics in the Earth’s outer core

We derive a quasi-geostrophic (QG) system of equations suitable for the description of the Earth’s core dynamics on interannual to decadal timescales. Over these timescales, rotation is assumed to be the dominant force and fluid motions are strongly invariant along the direction parallel to the rotation axis. The diffusion-free, QG system derived here is similar to the one derived in Canet et al. but the projection of the governing equations on the equatorial disc is handled via vertical integration and mass conservation is applied to the velocity field. Here we carefully analyse the properties of the resulting equations and we validate them neglecting the action of the Lorentz force in the momentum equation. We derive a novel analytical solution describing the evolution of the magnetic field under these assumptions in the presence of a purely azimuthal flow and an alternative formulation that allows us to numerically solve the evolution equations with a finite element method. The excellent agreement we found with the analytical solution proves that numerical integration of the QG system is possible and that it preserves important physical properties of the magnetic field. Implementation of magnetic diffusion is also briefly considered

Plesio-geostrophy for Earth’s core: I. Basic equations, inertial modes and induction

An approximation is developed that lends itself to accurate description of the physics of fluid motions and motional induction on short time scales (e.g. decades), appropriate for planetary cores and in the geophysically relevant limit of very rapid rotation. Adopting a representation of the flow to be columnar (horizontal motions are invariant along the rotation axis), our characterization of the equations leads to the approximation we call plesio-geostrophy, which arises from dedicated forms of integration along the rotation axis of the equations of motion and of motional induction. Neglecting magnetic diffusion, our self-consistent equations collapse all three-dimensional quantities into two-dimensional scalars in an exact manner. For the isothermal magnetic case, a series of fifteen partial differential equations is developed that fully characterizes the evolution of the system. In the case of no forcing and absent viscous damping, we solve for the normal modes of the system, called inertial modes. A comparison with a subset of the known three-dimensional modes that are of the least complexity along the rotation axis shows that the approximation accurately captures the eigenfunctions and associated eigenfrequencies

Propagation and reflection of diffusionless torsional waves in a sphere

We consider an inviscid and perfectly conducting fluid sphere in rapid rotation and permeated by a background magnetic field. Such a system admits normal modes in the form of torsional oscillations, namely azimuthal motions of cylinders coaxial with the rotation axis. We analyse this system for a particular background magnetic field that provides a new closed form normal mode solution. We derive Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximations to the normal modes, and focus particularly on the reflections that take place on the rotation axis and at the equator. We propose a procedure to calculate the reflection coefficients and we discuss the analogy of our findings with well-known seismological results. Our analytical results are tested against numerical calculations and show good agreement