Torsional Alfvén modes in a non-axisymmetric domain 

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Length-of-day variations and core-mantle inter-actions

Observed length-of-day variations

The length-of-day (LOD) on Earth is approximately 86400 s long and variations to it are on the order of a few milliseconds (Stephenson et al., 1995). Earth’s rotation measurements in the past mostly consisted of lunar occultations (Stephenson et al., 2016). Today it is measured by continuous observations of the night sky through Very-Long-Baseline-Interferometry and other advanced methods (see Gross, 2015, for a review on measurements). The observed variations show millennial trends at the longest periods, but also decadal, annual and even sub-diurnal periods (Stephen-son et al., 2016). The millennial linear trend corresponds to tidal friction (or tidal dissipation) of the Earth-moon system, gradually slowing Earth’s rotation, and a relatively small contribution is thought to be accounted to the post-glacial rebound of Earth, loosing its oblateness and thus accelerating its rotation (Stephenson et al., 2016). Other variations of periods around 1500 yr are accounted to coupling between the core and the mantle, but the relevant core dynamics are speculative (Stephenson et al., 1995; Dumberry and Bloxham, 2006).
At shorter time scales annual and seasonal variations are superimposed with inter-annual trends, as shown as the black line in Figure 1.2a. In this Figure the inter-annual trend is obtained by subtracting the modeled LOD variations of an oceanic and atmospheric loading model from the observed LOD variations (e.g. Gross et al., 2003; Holme and de Viron, 2005). Furthermore, an oscillation with a period around 6 yr and amplitude of 0.1–0.2 ms is found when analyzing the corrected signal (Von-drak, 1977; Abarca del Rio et al., 2000; Holme and de Viron, 2013; Gillet et al., 2015). An example of the 6 yr oscillation in the LOD variation time series is shown in Figure 1.2b. Such a change in the rotation rate corresponds to a change in an-gular momentum O p1016q Nm. Although atmospheric contributions to inter-annual frequencies are possible and not fully understood (e.g. Yu et al., 2020), it is generally assumed that the 6 yr oscillation originates from dynamics in the core that are cou-pled to the mantle. Some studies have proposed that these changes in the LOD are correlated with geomagnetic jerks, fast changes in the variations of the magnetic field (Holme and de Viron, 2005, 2013; Duan and Huang, 2020a). This suggests that both the geomagnetic field observations and the changes in LOD are caused by dynamics in the fluid outer core. Gillet et al. (2010) showed that torsional Alfvén modes in the outer core could be responsible for the angular momentum transport. How these dynamics may couple to the solid mantle is reviewed in the following subsections.

Torque balance

with L and Lm the angular momentum of the core and the rigid mantle, respectively. The core angular momentum may be further divided into that of the inner core and the outer core, but since the moment of inertia of the inner core is less than 1% that of the outer core, we neglect this separation. Assuming a rigid mantle and no variation in the orientation of the rotation axis, changes in Lm are tied directly to the net torque balance in the outer coreb torque , the hydrodynamic pressure torque p, the viscous torque , the Lorentzb torque L and the gravitational torque g. The latter arises from a possible buoy-ancy force FB. In (1.25) we have neglected the contribution of the non-linear term in the momentum equation, as it vanishes exactly for the non-penetrating boundary. The Lorentz torque can be split into the magnetic pressure torque pm and a magnetic tension torque b, so that Lpm (1.27)
It is seen that, similar to the non-linear term, for the conducting boundary condition (1.15) the magnetic tension torque b vanishes. For an insulating mantle, the Lorentz torque L vanishes entirely.
The estimated strength of these torques in Earth’s core is discussed in the following section. In Chapter 4 we investigate the torque balance of torsional Alfvén modes in a rotating ellipsoid assuming that viscosity is negligible, the mantle is perfectly conducting and no buoyancy is present. Then, the axial angular momentum must be balanced by the hydrodynamic and magnetic pressure torque.

Core-mantle coupling mechanisms

The mechanical coupling, i.e. the mechanism to exchange angular momentum, could also be referred to as torquing (Hide, 1989). This paraphrasing emphasizes the direct correspondence of these coupling mechanisms to a torque exerted on the boundary. A short summary of the proposed mechanisms and their estimates in the Earth’s core as proposed by the literature is given. For a more thorough discussion of the topic the reader is referred to the review by Roberts and Aurnou (2012). As the axial torque balance (1.30) suggests, the change in angular momentum in the core may be caused by a viscous, gravitational, electromagnetic or topographic torque. The torque estimates are compared to the 1016 Nm change in angular momentum associated with the variations in the LOD at a 6 yr period.

Viscous

Viscous stresses between the viscous fluid outer core and a rigid mantle contribute to the torque balance. However, the viscous torque is generally assumed to be negligible as a coupling mechanism (Bullard et al., 1950; Rochester et al., 1984). It can be estimated to be 8 4 ? 14 Nm (by considering eq. (5) in ;z Rc Ek U .
Jault, 1995, without magnetic part) for a velocity U 5 10 6 m/s (Gillet et al., 2015). The amplitude of , unsurprisingly, depends strongly on the hard to estimate viscosity in the core (Wijs et al., 1998), but even for the highest plausible values the torque is still too small. Even in the case that turbulent viscosity is the relevant value to be taken into account, it is still an order of magnitude below the torque associated with changes of the LOD at inter-annual periods (Roberts and Aurnou, 2012).

Gravitational

To couple the inner core and the mantle gravitationally the so-called Mantle-Inner-Core-Gravitational (MICG) coupling has been proposed (Buffett, 1996a,b). In this scenario a deformed inner core exerts a torque on a deformed CMB. The mechanism is usually illustrated by considering an equatorial ellipticity of the inner core and the CMB with the semi-major axes out of phase. The phase lag between the two bulges can then lead to a gravitational force restoring to an equilibrium state. It is under debate if the deformations of the inner core are sustained long enough to make such a scenario effective, as it strongly depends on the viscosity of the inner core (Orman, 2004; Mound and Buffett, 2006; Deguen, 2012). The coupling of the flow in the liquid core to the conducting inner core is likely strong and the correct treatment of it in the proposed MICG models is important. Mound and Buffett (2003) consider the tangent cylinder, that is the cylinder drawn by the radius of the inner core and the height of the outer core, rigidly coupled to the inner core rotation, whereas Duan and Huang (2020b) propose that the inner and outer core are decoupled through a thin diffusive layer. Recent studies claim to have found a corresponding signal at a 5.8 yr period in GPS gravity signals (Ding and Chao, 2018). Other studies at a similar time have suggested that measurement errors are still too large to infer relevant gravitational signals from the core (Watkins et al., 2018).

Electro magnetic

If the lower mantle is not perfectly insulating it is possible that it is coupled electro-magnetically to flows in the outer core. The conductivity of the lowermost mantle is not well constrained today and depends strongly on the mineral physics at CMB conditions. There is some evidence for thin iron enriched, and thus conducting, layers near the CMB from seismological observations and experimental chemical partition-ing studies (Wicks et al., 2010; Otsuka and Karato, 2012), while other experiments suggest the opposite (Ozawa et al., 2009).
Initially introduced by Bullard et al. (1950) and put forward by Rochester and Bullard (1960), electromagnetic (EM) coupling of core flows with such a potential layer of high conductance has been discussed widely in the literature (e.g. Stix and Roberts, 1984; Holme, 1998; Jault, 2003; Dumberry and Mound, 2008). A conduc-tance of the lowermost mantle O p108q S was found to suffice to account for the LOD changes for optimized flows at the core surface (Holme, 1998). Similarly, Gillet et al. (2010) used a conductance on the same order of magnitude to explain the changes in the LOD by torsional Alfvén modes in the core. However, for example Ohta et al. (2010) give an upper limit O p107 q S, an order of magnitude below what is needed to have a sufficient EM coupling. Jault (2015) linked the damping influence of a conducting mantle to the spectrum geomagnetic field changes, showing that the ratio between conductances of the upper and lower mantle is the key factor. The complicated relationship of Earth’s nutation and the FCN might also be strongly in-fluenced by a conducting lowermost mantle (Buffett, 1992; Buffett and Christensen, 2007; Kuang et al., 2019). Dumberry and More (2020) have recently suggested that weak energy in the secular variations in the south pacific region point towards a strongly conducting area in the lowermost mantle. A lot of uncertainties remain, but if other coupling mechanisms can be ruled out a layer with enhanced conductivity at the bottom of the mantle is needed to transfer angular momentum to the mantle.

Topographic

The idea of a pressure torque exerted from the flow onto a deformed mantle reaches back to Munk and MacDonald (1960), who extends the concept of inertial coupling developed in the framework of nutation and precession theories of Poincaré (1910) to changes in the rotation speed. In an axisymmetric domain, the axial pressure torque is exactly zero and cannot contribute to the torque balance, as seen in (1.26d). A departure from axisymmetry is referred to as topography and this departure can be on the very largest scales, e.g. ellipticity of the CMB. Then, a pressure torque, or topographic torque, can act on the deformed boundary. Hide (1969) and Hide and Weightman (1977) estimated that such a torque could be very effective in Earth’s core. By considering a typical height of the topography (named h in the article), it was proposed that the pressure acting on the boundary should scale as U , with U the characteristic velocity. Values of U have been inferred from tangential geostrophic (TG) flows, leading to a pressure as large as 103 Pa (Jault and Mouël, 1990). Tangential geostrophy relies on the assumption that the flow field close to the surface is in a geostrophic balance and the associated pressure is easily obtained (Le Mouël et al., 1985; Jault and Mouël, 1989). Most flows, including QG flows, are however not exactly represented by TG flows and the assumption breaks down at the equator. The estimate of a pressure of a generic flow from the TG pressure is not necessarily correct.

Table of contents :

1 Introduction 
1.1 Overview
1.2 Earth’s core
1.2.1 Earth’s interior structure
1.2.2 Properties of the liquid outer core
1.2.3 Geodynamo and convection in the outer core
1.2.4 Core-mantle boundary topography
1.3 The equations governing flows in planetary cores
1.3.1 Fluid description in the rotating frame
1.3.2 Magnetohydrodynamic equations
1.3.3 Dimensional analysis
1.3.4 Reduced models in planetary fluid dynamics
1.4 Length-of-day variations and core-mantle interactions
1.4.1 Observed length-of-day variations
1.4.2 Torque balance
1.4.3 Core-mantle coupling mechanisms
1.5 Geomagnetic field changes and core flow inversions
2 Quasi-geostrophic models 
2.1 Small slope approximation
2.2 Quasi-geostrophic approximation
2.3 Vorticity equation
2.4 Galerkin approach
2.5 Lagrangian formalism
2.6 Columnar magnetic field
2.7 Non-axisymmetric core volume
2.7.1 Cartesian basis in the ellipsoid
2.7.2 Non-orthogonal coordinate systems
3 Modes in a planetary core 
3.1 Inertial modes
3.2 Quasi-geostrophic inertial modes
3.3 Magneto-Coriolis modes
3.4 Taylor’s constraint and torsional Alfvén modes
3.5 Excitation and presence of hydromagnetic modes in Earth’s core
3.6 Numerical calculation of modes
3.6.1 Solving the generalized eigen problem
3.6.2 Code examples
4 Torsional Alfvén modes in a non-axisymmetric domain 
4.1 Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle
4.2 Pressure in a quasi-geostrophic model
4.3 Towards more complex geometries using non-orthogonal coordinates
5 On quasi-geostrophic Magneto-Coriolis modes 
5.1 Fast quasi-geostrophic Magneto-Coriolis modes in the Earth’s core
5.2 Diffusive Magneto-Coriolis modes
5.3 Towards an insulating magnetic field basis in the ellipsoid
6 Conclusions & Perspectives

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