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# Approaching a realistic force balance in geodynamo simulations

Edited by Peter L. Olson, Johns Hopkins University, Baltimore, MD, and approved September 13, 2016 (received for review June 6, 2016)

## Significance

Flow in Earth’s liquid core is expected to be primarily governed by the Coriolis force due to Earth’s rotation, the buoyancy force driving the convection, and the Lorentz force due to the geomagnetic field. A relevant geodynamo model has to be in such a force balance. Contemporaneous simulations invoke high viscosities to suppress flow turbulence to keep the computational costs manageable. The unrealistically large viscosity in these simulations is a major concern. Here we show that the state-of-the-art simulations with a viscosity that is lower than in most simulations, but still much larger than in Earth’s core, can approach a realistic force balance. Our simulations produce many properties that have been theoretically predicted in the past.

## Abstract

Earth sustains its magnetic field by a dynamo process driven by convection in the liquid outer core. Geodynamo simulations have been successful in reproducing many observed properties of the geomagnetic field. However, although theoretical considerations suggest that flow in the core is governed by a balance between Lorentz force, rotational force, and buoyancy (called MAC balance for Magnetic, Archimedean, Coriolis) with only minute roles for viscous and inertial forces, dynamo simulations must use viscosity values that are many orders of magnitude larger than in the core, due to computational constraints. In typical geodynamo models, viscous and inertial forces are not much smaller than the Coriolis force, and the Lorentz force plays a subdominant role; this has led to conclusions that these simulations are viscously controlled and do not represent the physics of the geodynamo. Here we show, by a direct analysis of the relevant forces, that a MAC balance can be achieved when the viscosity is reduced to values close to the current practical limit. Lorentz force, buoyancy, and the uncompensated (by pressure) part of the Coriolis force are of very similar strength, whereas viscous and inertial forces are smaller by a factor of at least 20 in the bulk of the fluid volume. Compared with nonmagnetic convection at otherwise identical parameters, the dynamo flow is of larger scale and is less invariant parallel to the rotation axis (less geostrophic), and convection transports twice as much heat, all of which is expected when the Lorentz force strongly influences the convection properties.

Sustained magnetism in astrophysical objects is due to the dynamo mechanism, which relies on the generation of electrical currents by fluid motion (1). The secular cooling of Earth’s interior and the release of light elements at the boundary of the solid inner core provide buoyancy sources that drive convection, leading to the generation of electrical currents (2). It has been more than two decades since the idea of modeling the geomagnetic field using computer simulations was successfully demonstrated (3, 4). These pioneering simulations were able to reproduce the dipole-dominant nature of the geomagnetic field and showed reversals of the geomagnetic dipole. Since then, computer simulations have become a primary tool for studying the properties of the geomagnetic field (5⇓⇓⇓–9).

The range of flow length scales present in the liquid outer core is enormous due to the very small viscosity of the fluid. To model this aspect in geodynamo simulations would require tremendous computing power that is not available even in the foreseeable future. Therefore, all geodynamo simulations must use unrealistically large viscosity to reduce the level of turbulence. One quantity that epitomizes this discrepancy is the Ekman number *ν* is the viscosity, *D* is the thickness of the liquid outer core), which roughly quantifies the ratio of the viscous force

The Coriolis force tends to suppress changes of the flow in the direction of the rotation axis, i.e., makes the flow nearly geostrophic (10, 11). This is known as the “Proudman−Taylor constraint” (PTC). Because the boundary of the fluid core is inclined relative to the direction of rotation (except at the poles), convective motions cannot be purely geostrophic, and therefore the PTC impedes convection (12). In the absence of a magnetic field, viscous force or inertial force

In Earth’s core, the buoyancy force and the Lorentz force *B* is mean magnetic field, *ρ* is density, *μ* is magnetic permeability, and *λ* is magnetic diffusivity) (14, 15). Note that here we use the term MAC balance in the sense that

Although a MAC state has long been expected from theoretical considerations, its existence in geodynamo simulations has not been demonstrated so far. A recent study of geodynamo models at an Ekman number of

## Method

We carry out a detailed study of geodynamo models where we analyze data from our recent study (18) and carry out new simulations at more extreme values of the control parameters. The basic setup is geodynamo-like, and we consider a spherical shell where the ratio of the inner (*D* of the shell is given by *D* as standard length scale,

We use the Boussinesq approximation, and the equations governing the velocity *T* are

where *P* is the pressure. The control parameters that govern the system are

where *α* is the thermal expansivity, *κ* is the thermal diffusivity.

Both boundaries have fixed temperature, are no-slip, and are electrically insulating. The open-source code MagIC (available at https://www.github.com/magic-sph/magic) is used to simulate the models (19). The code uses spherical harmonic decomposition in latitude and longitude and Chebyshev polynomials in the radial direction. MagIC uses the SHTns library (20) to efficiently calculate the spherical harmonic transforms. Because we use nondimensional equations, the relative influence of viscosity is mainly expressed by the value of the Ekman number. To explore the effect of the magnetic field, we perform hydrodynamic (HD) simulations, i.e., without a magnetic field, in parallel to the dynamo models.

The results of simulations with

## Results

We begin our analysis by explicitly calculating the various forces involved in the system, namely, Coriolis force *m* = 0 component from the force values. The PTC implies that the Coriolis force is largely compensated by the pressure gradient. For our purpose, we only concentrate on that part of

Because we use no-slip boundary conditions, Ekman layers are formed at the boundaries (12). Within these layers, the viscous force is dominant. Due to the larger viscosity, contemporary geodynamo simulations have much thicker Ekman boundary layers than those present in Earth’s core. This leads to a rather substantial contribution of the boundary layer viscous force to the total viscous force (e.g., see refs. 22 and 23). Therefore, we choose to exclude thin boundary layers, one below the outer boundary and one above the inner boundary, from the force calculation. The thickness of the excluded layers is 1%, 2%, and 3% of the shell thickness for

The various forces calculated from the simulations are portrayed in Fig. 1 *A*−*C* as a function of the convective supercriticality *E*. The Lorentz force

We plot the ratio of *D*. In simulations with *E* to *E*, the ratio *F* shows, such a comparison provides a succinct way of highlighting the overall dominance of the Lorentz force. In this context, it is worth pointing out that assuming a higher magnetic Prandtl number may help to increase the strength of the magnetic field and, in turn, its influence on the flow (21, 25, 29). However, whether such an approach is justified or not remains to be tested.

The trends in the forces highlighted above have important consequences for the properties of convection. When a VAC balance holds in rapidly rotating convection, the characteristic flow length scale *E* (2, 13, 30). As shown in Fig. 2 *A*−*D*, the convective structures in our HD simulations do follow this trend qualitatively as *E* decreases. On the other hand, in the MAC regime, *E* (2, 16, 31). For simulation with *E* and *F*). At *G*) compared with the HD case (Fig. 2*C*). At *H*) than the corresponding HD setup (Fig. 2*D*). This increased influence of the magnetic field is also reflected in the total magnetic energy, which exceeds the total kinetic energy more and more as *E* is decreased (Fig. S1). Another interesting feature in the

In Fig. 3, we present the three-dimensional morphology of the convection in the HD and in the dynamo case for the lowest-viscosity simulation with the largest ratio of Lorentz force to viscous and inertial forces. The HD setup has small axially aligned tube-like convection columns. In the dynamo case, however, the convection occurs in the form of thin sheets stretched in the cylindrically radial direction. It is also clear that, compared with the HD case, the convective structures vary more along the rotation axis. Both features demonstrate the influence of the Lorentz forces on the convention morphology.

Another way to quantify the relaxed Proudman−Taylor constraint in the dynamo cases is to analyze the total heat transferred from the bottom boundary to the top; this stems from the notion that rotation quenches the efficiency of convection by suppressing motions along the rotation axis (12). Any relaxation of this constraint will lead to a gain in heat transfer efficiency. We use the ratio of the Nusselt number *E* to *D*−*F* highlights that the gain in the heat transfer efficiency in the dynamo cases is largest when the Lorentz force is maximally dominant over viscous and inertial forces.

## Discussion

To summarize, we used a systematic parameter study to test the existence of a dynamical state in dynamo simulations where magnetic forces play a crucial role together with Coriolis and buoyancy forces (MAC state), as is expected to be present in Earth’s core. We lowered the viscosity to a small value, close to the limit allowed by today’s computational resources, and found that Lorentz forces become equal in strength to (uncompensated) Coriolis and buoyancy forces and, for a limited range of Rayleigh numbers, far exceed viscous and inertial forces. The increased influence of the Lorentz force leads to large-scale convection, substantial axial variation in the convection structures, and a 100% increase in the heat transfer efficiency compared with the corresponding HD setup. All of these features are expected theoretically (2). For higher viscosity values, the convection is much less affected by the magnetic field (17).

We note that, in our simulations the Lorentz force is substantially smaller than the Coriolis force or the pressure force (taken individually). Hence, the state can be called quasi-geostrophic (26). Nonetheless, a completely geostrophic state is impossible, and the essential question is what balances the residual Coriolis force. Because these are the Lorentz and Archimedean forces, with an insignificant role for viscosity and inertia, it is also justified to speak of a MAC balance. We also note that, although a MAC balance is satisfied globally, this does not imply that the residual Coriolis force, Lorentz force, and buoyancy force are pointwise of the same magnitude. For example, strong Lorentz forces seem to be rather localized (Fig. S2), as found in previous studies (e.g., ref. 34). In regions where the Lorentz force is weak, the balance could be almost perfectly geostrophic, or buoyancy alone could balance the residual Coriolis force.

Our results show some similarities with earlier studies done in a similar context. Larger-scale convection in dynamo simulations compared with their HD counterparts has been reported in rotating convection in Cartesian geometry (35); there, the dynamo simulation with

In the context of geodynamo simulations, studies at Ekman numbers comparable to the lowest value used in our study have been reported before. A substantial change in the convection length scale due to the dynamo-generated magnetic field was found, but it only occurred in cases with constant heat flux boundary conditions (37). In contrast, we find the same enlargement of flow length scales also for fixed temperature conditions. Differences in the model setup and parameter values prevent us from elucidating the exact cause for these differences. Miyagoshi et al. (38, 39) also performed geodynamo simulations with

Our parameter study has shown that, at an Ekman number of

## Acknowledgments

We thank the two anonymous referees for very constructive comments. Funding from NASA (through the Chandra Grant GO4-15011X) and Deutsche Forschungsgemeinschaft (through SFB 963/A17) is acknowledged. S.J.W. was supported by NASA Contract NAS8-03060. Simulations were performed at Gesellschaft für wissenschaftliche Datenverarbeitung mbH Göttingen (GWDG) and Rechenzentrum Garching der Max-Planck-Gesellschaft (RZG).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: rakesh.yadav{at}cfa.harvard.edu.

Author contributions: R.K.Y., T.G., and U.R.C. designed research; R.K.Y. performed research; R.K.Y. analyzed data; and R.K.Y., T.G., U.R.C., S.J.W., and K.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1608998113/-/DCSupplemental.

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