Core Eigenmodes and their Impact on the Earth's Rotation

Abstract

Changes in the Earth's rotation are deeply connected to fluid dynamical processes in the outer core. This connection can be explored by studying the associated Earth eigenmodes with periods ranging from nearly diurnal to multi-decadal. It is essential to understand how the rotational and fluid core eigenmodes mutually interact, as well as their dependence on a host of diverse factors, such as magnetic effects, density stratification, fluid instabilities or turbulence. It is feasible to build detailed models including many of these features, and doing so will in turn allow us to extract more (indirect) information about the Earth's interior. In this article, we present a review of some of the current models, the numerical techniques, their advantages and limitations and the challenges on the road ahead.

Keywords: Core modes; Earth rotation; Rotational modes.

Fig. 1

(a) Geometry of the Earth’s core model. The CMB has been represented by a triaxial ellipsoid of semi-axes [a, b, c]. (b) Density in the Earth’s core as a function of the radius (normalized by equatorial radius $R_o=3480$ km of the CMB). Open circles: PREM values (Dziewonski and Anderson 1981). Red curve: isentropic model (Labrosse 2015). Gray area illustrates the inner core

Fig. 2

Illustration of viscously driven layers and flows in a spherical shell. Modified from figure 1 in Calkins et al. (2010). The dotted blue lines near the CMB and ICB represent the Ekman layer thickness. The two black dots on the inner and outer boundaries represent the critical colatitudes. Oblique red and blue beams represent oscillatory shear layers resulting from the eruption of the Ekman boundary layer at the critical colatitudes (Kerswell 1995). The scaling laws for the Ekman boundary layer at the ICB are identical to those at the CMB, except for the velocity amplitude in the shear layer $\propto \mathcal{O}(E^{\mathfrak{p}})$ where the exponent is still disputed (Kerswell ; Le Dizès and Le Bars 2017)

Fig. 3

Schematic diagram of the (dimensionless) angular frequency $\omega$ for MAC modes in the outer core, as a function of the Lehnert number Le. Adapted from Labbé et al. (2015), Vidal et al. (2019), Gerick et al. (2020), Gerick et al. (2021). GIM: gravito-inertial modes (red area). IGM: inertia-gravity modes (yellow area). Other colored regions illustrate the typical frequency range of the largest-scale magnetic modes, and their scaling law as a function of Le. TM: torsional modes (hatched area). MCM: magneto-Coriolis modes (blue area). Typical forcing frequencies ${\omega }_0$ for orbital forcings and core convection are also indicated (see Sect. 6)

Fig. 4

Domains of existence (colored areas) of the inertia-gravity modes in (a) and gravito-inertial modes in (b). Sketch in a meridional plane, where the solid arrow indicates the axis of rotation. Oblique dashed line shows the critical colatitude $\omega =2cos{\theta }_c$. Horizontal dashed line shows $z=\omega /N_{max}$. Top left panel: Modes ${\mathcal{H}}_1$ with $N_{max}\le {\omega }^2\le 4$. Top right panel: Modes ${\mathcal{H}}_2$ with $0\le {\omega }^2\le min(4,N_{max}^2)$. Bottom left panel: Modes ${\mathcal{E}}_1$ with $max(4,N_{max}^2)\le {\omega }^2\le 4+N_{max}^2$. Bottom right panel: Modes ${\mathcal{E}}_2$ with $4\le {\omega }^2\le N_{max}^2$

Fig. 5

Inertial modes at $E={10}^{-8}$ in a spherical shell with ratio $\eta =0.35$, computed with an open-source code (Vidal and Schaeffer 2015). Meridional slices for the local kinetic energy (one-sided, logarithmic scale) in panel (a), and the three cylindrical components of $\mathit{u}$ (double-sided, linear scale) in panel (b)

Fig. 6

Here the vertical axis represents the difference between the theoretical spin-over frequency ${\omega }_{\mathrm{so}}^{\mathrm{an}}$ and the frequencies $\omega$ of nearby eigenmodes (characterized by their angular wavenumber $\overline{\ell }$). Red dots correspond to numerical eigenfrequencies computed when the mantle is free to wobble. Open blue circles represent numerical eigenfrequencies when the mantle rotates uniformly. The FCN frequency (continuous red line) converges to the spin-over frequency in a planet with a spherical CMB. Other inertial mode frequencies remain unaltered when the mantle is free to wobble (i.e., open blue circles have a matching red dot)

Fig. 7

Frequency and damping of nearly diurnal eigenmodes as the parameter q is varied. The FCN’s frequency crosses over nearby inertial eigenmodes when $q\ll 1$ until its damping becomes too close to the damping of the modes it is crossing (around $q\sim 9$) and an avoided crossing takes place. The $q=0$ case reduces to the spectrum of inertial modes in uniformly rotating planets