Hydromagnetic quasi-geostrophic modes in rapidly rotating planetary cores

Abstract

The core of a terrestrial-type planet consists of a spherical shell of rapidly rotating, electrically conducting, fluid. Such a body supports two distinct classes of quasi-geostrophic (QG) eigenmodes: fast, primarily hydrodynamic, inertial modes with period related to the rotation time scale and slow, primarily magnetic, magnetostrophic modes with much longer periods. Here, we investigate the properties of these hydromagnetic quasi-geostrophic modes as a function of non-dimensional parameters controlling the strength of the background magnetic field, the planetary rotation rate, and the amount of magnetic dissipation.We first present analytic solutions that illustrate the essential parameter dependences of the modes and provide a convenient benchmark for our numerical scheme. A comparison between known three-dimensional inertial modes in a sphere and our axially invariant QG modes shows encouraging agreement at low azimuthal wavenumbers, particularly for the slowest modes. The container geometry and background magnetic field structure are found to influence the radial structure of the modes, but not the scaling of their frequency with the control parameters. When the background magnetic field decreases toward the outer boundary in a spherical shell, QG modes tend to be compressed towards the outer boundary. Including magnetic dissipation, we find a continuous transition from diffusionless slow magnetic modes into quasi-free decay magnetic modes. During that transition (which is controlled by the magnitude of the Elsasser number), we find that slow magnetic modes weakly modified by diffusion exhibit a distinctive spiralling planform. When magnetic diffusion is significant (Elsasser number much smaller than unity), we find quasi-free decay slow magnetic modes whose decay time scale is comparable to, or shorter than, their oscillation time scale.Based on our analysis, we expect Mercury to be in a regime where the slow magnetic modes are of quasi-free decay type. Earth and possibly Ganymede, with their larger Elsasser numbers, may possess slow modes that are in the transition regime of weak diffusion, depending on the details of their poorly known internal magnetic fields. Fast QG modes, that are almost unaffected by the background magnetic field, are expected in the cores of all three bodies.

Slow magnetic Rossby waves in the Earth's core

“…Indeed the geostrophic modes for the Malkus fieldB = B 0 sê [Malkus, 1967] generally imply faster propagation. Recently, Canet et al [2014] examined the modes in the spherical geometry, assuming several morphologies of (11) is calculated at s = 0.77r°for wave numbers m = 5 and 8, given that = 1.13 × 10 4 kg m −3 , r°= 3.485 × 10 6 m, and Ω = 7.29 × 10 −5 s −1 . Black dotted and dashed-dotted lines indicate the observed drift speed in the geomagnetic model gufm1 [Finlay and Jackson, 2003] and the drift speed where the zonal flow speed obtained in the mean QG flow model inverted from gufm1 over the period 1840-1990[Pais et al, 2015 is extracted, respectively.…”

“…Nonaxisymmetric oscillations of the core on secular variation timescales are crucially affected by both rotation and magnetic field, and travel in the azimuthal direction [e.g., Jones, 2007;Finlay et al, 2010;Canet et al, 2014;Hori et al, 2014]. Rotating MHD waves split into two classes, fast waves where the primary balance is between 10.1002/2015GL064733 inertia and a combination of Coriolis and Lorentz force, and slow waves where inertia is negligible and the core evolves slowly through a sequence of magnetostrophic states.…”

“…This class of waves, derived by Hide [1966] in a local plane model, have the property that the flow is quasi-geostrophic; that is, there is very little variation in the z direction. They propagate westward, and they may be derived as a limit of the full spectrum of global modes for the special case of the Malkus field B = B 0 sê [Malkus, 1967;Canet et al, 2014].…”

Recent geomagnetic secular variation from Swarm and ground observatories as estimated in the CHAOS-6 geomagnetic field model

“…Quasi-geostrophic models in a thin-shell (β -plane) geometry, as is relevant for the atmosphere and oceans, are known to break down at the equator. However, the outer core is a thick shell and recent tests of the quasi-geostrophic approximation in this geometry (comparing inertial modes in quasi-geostrophic models against full 3D solutions) show encouraging agreement, even for equatorially confined modes (Canet et al 2014;Labbé et al 2015). Further work is needed to better understand the dynamics of the low-latitude non-axisymmetric jets.…”

Planetary gyre, time‐dependent eddies, torsional waves, and equatorial jets at the Earth's core surface

“…Under the incompressible QG hypothesis (see section 2.2), the flow in the whole volume can be represented through a stream function ψ ( s , φ ) [ Jault and Finlay , ; Canet et al , , equations (14) and (15)]: …”