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# Electromagnetically driven westward drift and inner-core superrotation in Earth’s core

Edited by Gerald Schubert, Institute of Geophysics and Planetary Physics, Los Angeles, CA, and approved August 26, 2013 (received for review April 29, 2013)

## Significance

Seismic probing of the earth’s deep interior has shown that the inner core, the solid core of our planet, rotates slightly faster (i.e., eastward) than the rest of the earth. Quite independently, observations of the geomagnetic field provide evidence of westward-drifting features at the edge of the liquid outer core. This paper describes a computer model that suggests that the geomagnetic field itself may provide a link between them: The associated electromagnetic torque currently is westward in the outermost outer core, whereas an equal and opposite torque is applied to the inner core. Decadal changes in the geomagnetic field may cause fluctuations in both these effects, consistent with recent observations of a quasi-oscillatory inner-core rotation rate.

## Abstract

A 3D numerical model of the earth’s core with a viscosity two orders of magnitude lower than the state of the art suggests a link between the observed westward drift of the magnetic field and superrotation of the inner core. In our model, the axial electromagnetic torque has a dominant influence only at the surface and in the deepest reaches of the core, where it respectively drives a broad westward flow rising to an axisymmetric equatorial jet and imparts an eastward-directed torque on the solid inner core. Subtle changes in the structure of the internal magnetic field may alter not just the magnitude but the direction of these torques. This not only suggests that the quasi-oscillatory nature of inner-core superrotation [Tkalčić H, Young M, Bodin T, Ngo S, Sambridge M (2013) The shuffling rotation of the earth’s inner core revealed by earthquake doublets. *Nat Geosci* 6:497–502.] may be driven by decadal changes in the magnetic field, but further that historical periods in which the field exhibited eastward drift were contemporaneous with a westward inner-core rotation. The model further indicates a strong internal shear layer on the tangent cylinder that may be a source of torsional waves inside the core.

The slow westward drift of the geomagnetic field is one of the best- and longest-known features of the historical field, for which the most likely explanation is a latitudinally dependent westward flow in the outermost outer core (1, 2). Another westward-propagating and possibly related feature is equatorial waves (3), perhaps caused either by advection or instabilities on an equatorial westward jet. However, longer time series provide evidence of periods of eastward drift of the field over the past 3,000 y (4, 5). Although some geodynamo models have reproduced westward drift (6, 7), many rely on thermal winds whose structure is tied to the boundary conditions imposed by the lowermost mantle, which changes only on a timescale of 10–100 million years. A seemingly unrelated phenomenon is the superrotation (relative to the mantle) of the inner core, whose estimates vary from zero to several degrees per year (8). Links between core–surface flows and inner-core rotation are difficult to quantify because a coupling mechanism extending across the entire outer core has remained elusive. There is some evidence of such a coupling through thermal winds in polar regions (9); however, because their amplitude is linked to the mass and thermal flux at the inner-core boundary, unless the inner core is very young, the effect likely is small (10).

The fluid outer core, bounded by the solid inner core and overlying mantle, likely is in a quasi-magnetostrophic balance, in which the forces of pressure, buoyancy, Coriolis, and Lorentz are almost in equilibrium (6). Two key nondimensional parameters in any model are the Ekman and Rossby numbers, measures of viscosity and inertia, respectively, believed to be ∼10^{−15} and ∼10^{−6} in the core. However, because of the vast range of temporal and spatial scales in the rapidly rotating system, current models of the geodynamo struggle to reach values of these parameters lower than and (6, 7, 11⇓–13). Although scaling laws may attempt an extrapolation to the extreme Earth-like parameters (14), this is difficult over so many orders of magnitude. In the idealized regime of zero inertia and viscosity, the internal magnetic field **B** must satisfy Taylor’s constraint (15), in which the axial electromagnetic (EM) torque, ,*must integrate to zero over any concentric axially aligned cylinder , where are cylindrical polar coordinates and denotes the azimuthal component. A full understanding of the earth’s core likely is possible only by investigating the dynamics in the vicinity of this limit.

In this study we use a method (*Methods*) that accesses a regime very close to a magnetostrophic balance by seeking steady, inertia-free, ultra–low-viscosity solutions of the geodynamo equations with a prescribed poloidal magnetic field (16) that matches the xCHAOS model (17) to degree 4 at the core–mantle boundary. This technique suppresses the short timescales that prevent current modeling strategies from reaching low viscosities, and we can compute models in the range , two orders of magnitude lower than any previous calculation. By slowly decreasing *E* in small discrete steps, we could identify well-defined limiting behavior at small *E*. In numerical models at larger Ekman numbers (18), it is acknowledged that inertia, particularly in a turbulent regime, may play an important role in balancing EM torques. Our model excludes this possibility on the grounds that the Rossby number for the earth is small,^{†} and we achieve an “Ekman-state” balance in which viscosity must absorb any unbalanced torque and, indeed, may itself play an important role in the determination of the geostrophic flow (19, 20).

The total axial EM torque integrated over the core may be subdivided into four parts: the three contributions from each of the fluid regions, labeled I, II, and III in Fig. 1, which partition the fluid outer core (FOC), and the torque over the solid inner core (SIC). Surrounded by an electrically insulating mantle, in a steady state this net torque must sum to zero (15); therefore, the axial EM torque on the inner core may be expressed asFig. 1 shows how the magnetic field approaches the Taylor limit of in terms of the regionally integrated EM torques. The torques do not scale uniformly as a function of *E*: that in the outermost outer core (region I) is consistent with a scaling of , dominating the contributions from regions II and III within the tangent cylinder that are bounded by . These different scalings may arise as a result of the increased influence of the boundary conditions on cylinders of largest cylindrical radii, which impart constraints on the magnetic field, not just at either end of the cylinder but everywhere on its surface. Because the westward-directed torque in region I dominates the contributions to the FOC, the SIC experiences an eastward-directed EM torque that scales also as . In our model, the inner core is assumed to be gravitationally locked (21) to the mantle and so cannot move. However, if this constraint is relaxed (22), the dynamical response of the inner core would be a tendency for an (eastward) superrotation: In this sense, our model is simply an end-member case of a spectrum of models in which the locking gets progressively stronger.

An important quantity in the weak-viscosity limit is the geostrophic flow, , the azimuthal component of flow averaged on a cylinder . In the absence of inertia but in the presence of weak viscosity, is linked to how quickly decreases with *E*, in region I by the relation (6)Fig. 2*A* shows the magnitude of the flow decomposed into its geostrophic and ageostrophic components as a function of *E*. The toroidal-dominated ageostrophic part is almost independent of *E*, whereas the geostrophic flow (both rms and maximum value) scales as , as anticipated from Eq. **1** and the scaling in region I. For values of *E* greater than 10^{−7}, the geostrophic flow is a subdominant feature of the model. However, as *E* is decreased below the threshold at approximately , the geostrophic flow becomes dominant (in rms). Using the scaling of , extrapolating to the earth’s core gives dimensional flow velocities of m/s, consistent with values derived from studies of core flow (3, 23). Following the derived scalings, the geostrophic flow would dominate the ageostrophic component by a factor of about 100 at . However, thermal wind effects that are not included in our model may drive a much stronger ageostrophic flow with an amplitude consistent with typical estimates coming from large-scale core flow inversions (24); such flows may have a significant zonal component.

Although the westward flow is of broad structure (Figs. 3 and 4), it rises to a peak close to , creating an equatorial jet (25). Fig. 2*B* shows the location of the maximum geostrophic flow in colatitude (derived from the fit to the data). This scaling of is derived using empirical means but suggests a connection to the lateral extent of the equatorial viscous boundary layer, a distance from the equator that scales identically as . The scaling of , although occurring in other quantities in the model, does not fit the data well. Extrapolation to shows that in the Earth-like regime, the maximum geostrophic flow would lie less than 5° from the equator. Analysis of secular variation (3) shows there is rapid change within a narrow equatorial belt (latitude −5° to +10°), suggesting that these changes may be caused by instabilities on an equatorial jet. Furthermore, because the geostrophic flow is axisymmetric, this model predicts that the same westward jet, responsible for the westward drift in Atlantic regions, must be present in Pacific regions where secular variation is low (26). Inverse studies that determine the magnetic field strength inside the core from the motion of torsional waves show there is a local minimum in its cylindrical radial component in the outermost FOC (27). Because magnetic fields in general quench shearing motions, such a minimum is consistent with the existence of a strong equatorial jet.

We perturb the poloidal magnetic field structure to reverse its (dominant) contribution to the EM torque in region I by changing the sign of its degree 3 and 4 components (28). Our model shows that an eastward flow (of a magnitude comparable to that of the westward flow of Fig. 4*A*) was then driven in region I with an associated westward-directed torque on the inner core.

Studies of core–surface secular variation show that changes in the internal field occur on the decadal–centennial timescale of core convection (29). Assuming that the magnetic field deep inside the core changes on similar timescales, the model then predicts concomitant changes in drift direction and the rotation of the inner core relative to the mantle and, plausibly, even episodic reversals in direction. Gravitational coupling of the inner core to the mantle will dampen the rotational response of the EM torques and although poorly constrained in magnitude (30), likely has an associated inner-core deformation time of decades. We suggest that both the directions of drift and of the inner-core rotation then would be effectively enslaved to decadal–centennial secular changes in the magnetic field; in the absence of any obvious periodicity within the internal magnetic field, the inner-core rotation rate and drift velocities likely will have a nonconstant and probably complex time dependence. Of particular relevance here is a recent study (31) reporting inner core rotation rates from 1961 to 2007 that are quasi-oscillatory with an y period, superimposed on a small constant positive (eastward) trend. Our model suggests that decadal changes in the magnetic field itself may be responsible for the fluctuations in inner-core rotation.^{‡} With regard to longer timescales, our model has significant bearing on the seismically observed degree 1 structure of the inner core (8). Assuming the magnetic field had a comparable time-variation in the past as it does in the present day, the unavoidable time-dependent torques and ensuing quasi-oscillatory rotation rate likely would smear any mantle-induced texturing that otherwise would have taken place with a gravitationally locked inner core over a million-year timescale. This suggests an alternative mechanism, such as convection (33, 34), is required to explain the hemispheric structure.

It is of additional interest to identify the structure of the flow close to the tangent cylinder (the cylinder aligned with the rotation axis and tangent to the SIC), a likely location of shear layers (35). Fig. 3 shows such an asymmetric shear layer in , which requires high resolution to resolve. If such shear layers exist in Earth’s core, instabilities could trigger torsional oscillations, which have been observed to propagate away from the tangent cylinder (27).

Finally, we remark that the magnitude of the geostrophic flow depends on *E*, in contrast to scaling laws derived for other quantities based on much more viscous models that are independent of viscosity (14). Ultimately, the kinetic energy of the flow must be controlled by a global energy budget and therefore independent of *E* as , suggesting that some physics that we have neglected become important in the low-E regime. One prime candidate is inertia, whose estimated size (given by the Rossby number ) is comparable to in the core and thus likely is of comparable importance to viscosity. Our scalings therefore may be a valid description only for the dynamics of the modeled core in which viscosity dominates inertia, i.e., . We speculate that the macrodynamics of at core conditions then would be given by the Ekman-state description, and the dominance of inertia over viscosity at would remove the singularity in as .

## Methods

In a spherical shell using coordinates , we evolve the coupled dimensionless Navier–Stokes and the toroidal component of the induction equation

to a steady state. The SIC (radius of 0.35) is a nonslip conductor of identical electrical conductivity to the outer core; the mantle (radius 1) is modeled as a nonslip electrical insulator. In a steady state, the magnetic field satisfies in the SIC whose solutions supply a boundary condition at the inner core boundary (ICB) for the toroidal field: evolving the magnetic field within the SIC therefore is not required. The equations have been nondimensionalized based on the core–mantle boundary radius 3,485 km (*L*), the magnetic diffusion timescale (where η is the magnetic diffusivity), and velocity scale ms^{−1}; and then take the approximate values 10^{−9} and 10^{−15}, where is the earth’s rotation rate and ν the viscosity of the fluid core. We focus only on the axial torques, associated with the manner with which the magnetic field achieves the balance required of Taylor’s constraint; because a radial buoyancy force does not contribute to these torques, we ignore any thermal or compositional driving. Following Aurnou et al. (16), the system is driven instead by prescribing the poloidal field structure: only the toroidal magnetic field and the flow are evolved; this is equivalent to the artificial insertion of a time-dependent forcing into the induction equation that renders constant the poloidal field.

The poloidal field is chosen to match the xCHAOS model (17) at epoch 2004 to degree 4, extrapolated inside the core using a smooth minimum-Ohmic-dissipation profile (36):

higher degrees of the poloidal field are zero. This particular truncation is adopted to minimize the bandwidth of the poloidal-generated Lorentz force (degree and order 8) and therefore of the flow that must be resolved, while keeping the large-scale features of the earth’s field. Calculations using the poloidal field prescribed nonzero to degree 14 are more difficult but show the same scaling and drift direction and similar threshold Ekman number as when using degree 4.

Holding fixed the poloidal field not only allows us to impose an Earth-like structure on the system, but also suppresses the short-timescale dynamics (e.g., Alfvén waves) that prevent standard models from reaching very low viscosities. Empirical tests at show that allowing the poloidal magnetic field of degrees 5 and upward to evolve subject to the insulating boundary conditions (while keeping degrees 1–4 fixed) results in only very small differences in the eventual steady state but requires a significant drop in time step for stability. Because we seek only a steady state, stability may be improved further by taking the Rossby number to be 1 rather than ; in so doing, we could use a time step no smaller than 0.05 (and this at the smallest *E*). There is no a priori guarantee that steady states exist, but if they do, it is automatic that they are independent of and approach a steady Taylor state as .

The poloidal and toroidal components of velocity, along with the toroidal component of magnetic field, are expanded in solid angle using spherical harmonics and in radius using Chebyshev polynomials, and the equations for their coefficients are evolved to a steady state using the second-order exponential time-differencing evolution method (37), which preserves fixed points of the equations. The nonlinear terms were evaluated using standard pseudospectral transforms; to ensure accuracy, all time-evolution matrices and spherical harmonics were computed to quadruple precision. Considerable care was taken to ensure that the calculations were converged adequately, principally in terms of the decay of their spectra. The highest resolution used was spherical harmonic degree 700, order 26 and 300 in Chebyshev degree. The code, parallelized using the Message Passing Interface (MPI), was developed especially to handle these large resolutions.

Six models were produced with decreasing values of ; *k* = 4, 5, 6, 7, 8; and . The first run had zero initial conditions; each subsequent run took the final steady state of the previous as an initial condition. Because the system is nonlinear, multiple steady solutions may exist, although this process seeks only a single branch. The small generated toroidal field (taking rms values of only 2–3% of the poloidal field) is sufficient to perturb the system toward a Taylor state.

To extrapolate from our models to the earth, we note that our poloidal field is roughly three to five times weaker than expected inside the earth; this may be quantified in one of two ways: first, by the rms value of on the core–mantle boundary, which for our models is 0.26 mT, three times smaller than the anticipated value of 0.69 mT from nutation studies (38); second, by the volumetric rms value of the magnetic field being 0.5 mT, two to eight times smaller than expected values of 1–4 mT (27). Because depends quadratically on **B** (principally on its poloidal component, the toroidal component being much weaker), our prescribed poloidal magnetic field will drive a geostrophic flow at least 10 times too small. Scaling up the solution using the factor of 10 discussed above, along with the dependence from 10^{−7} to , produces a dimensional flow of m/s. Furthermore, our toroidal field is relatively small (2–3% of the poloidal field) compared with a toroidal field expected in the core of equal or larger magnitude than the poloidal field: a stronger toroidal field may drive even stronger geostrophic flows.

Further details of the model may be found in a manuscript planned for submission elsewhere.

## Acknowledgments

The models were run on Monte Rosa, a massively parallel Cray XE6 at the Swiss National Supercomputing Center (CSCS), Lugano, Switzerland, for which we acknowledge CSCS Grant s225. P.W.L. was supported by Natural Environment Research Council Grant NE/G014043/1. We acknowledge funding from European Research Council (ERC) Grant 247303 Magnetostrophic Flow in Experiments and the Core of the Earth (“MFECE”).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: p.w.livermore{at}leeds.ac.uk.

Author contributions: P.W.L., R.H., and A.J. designed research; P.W.L. and R.H. performed research; P.W.L. analyzed data; and P.W.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵*Note that the symbol is not to be confused with the Taylor integral, a closely related quantity that differs by a factor of

*s*from the axial EM torque we consider here.↵

^{†}Because viscosity enters into boundary-layer scalings as rather than*E*, and further that , arguably both viscosity and inertia may be equally important in the earth’s core. However, our goal was to explore the magnetostrophic limit, and as such, we chose to focus attention on the Ekman-state balance at very small viscosity.↵

^{‡}Although our model provides a consistent description of the dynamics of inner-core rotation over the 10–100-y timescale, different dynamics may be important on millennial timescales (32).

Freely available online through the PNAS open access option.

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