The competition between Lorentz and Coriolis forces in planetary dynamos
- Krista M. Soderlund^{1}Email author,
- Andrey Sheyko^{2},
- Eric M. King^{3} and
- Jonathan M. Aurnou^{4}
https://doi.org/10.1186/s40645-015-0054-5
© Soderlund et al. 2015
Received: 1 April 2015
Accepted: 10 August 2015
Published: 2 September 2015
Abstract
Fluid motions within planetary cores generate magnetic fields through dynamo action. These core processes are driven by thermo-compositional convection subject to the competing influences of rotation, which tends to organize the flow into axial columns, and the Lorentz force, which tends to inhibit the relative movement of the magnetic field and the fluid. It is often argued that these forces are predominant and approximately equal in planetary cores; we test this hypothesis using a suite of numerical geodynamo models to calculate the Lorentz to Coriolis force ratio directly. Our results show that this ratio can be estimated by \(\Lambda _{d}^{*} \simeq \Lambda _{i} Rm^{-1/2}\) (Λ _{ i } is the traditionally defined Elsasser number for imposed magnetic fields and Rm is the system-scale ratio of magnetic induction to magnetic diffusion). Best estimates of core flow speeds and magnetic field strengths predict the geodynamo to be in magnetostrophic balance where the Lorentz and Coriolis forces are comparable. The Lorentz force may also be significant, i.e., within an order of magnitude of the Coriolis force, in the Jovian interior. In contrast, the Lorentz force is likely to be relatively weak in the cores of Saturn, Uranus, Neptune, Ganymede, and Mercury.
Keywords
Background
Magnetic fields are common throughout the solar system and provide a unique perspective on the internal dynamics of planetary interiors. Planetary magnetic fields are driven by the conversion of kinetic energy into magnetic energy; this process is called dynamo action. Kinetic energy is derived from thermo-compositional convection of an electrically conducting fluid although mechanical mechanisms, such as libration and precession, may also drive core flow in smaller bodies (e.g., Le Bars et al. 2015). The geodynamo is the most studied planetary magnetic field, yet the mechanisms that control its strength, morphology, and secular variation are still not well understood. Towards explaining these observations, dimensionless parameters are often used to characterize the force balances present in the core and their link to processes that govern convective dynamics and dynamo action. Two forces that are particularly influential on core processes are the Coriolis force, which tends to organize the core fluid motions, and the Lorentz force which back-reacts on the fluid motions to equilibrate magnetic field growth.
Here, u is the velocity vector, Ω is the rotation rate of the reference frame, Π is the non-hydrostatic pressure normalized by the background density ρ _{ o }, J is the electric current density, B is the magnetic induction, and g is the acceleration of gravity. Terms from left to right are inertial acceleration, Coriolis acceleration, pressure gradient, buoyancy, viscous diffusion, and Lorentz force. The inertial term consists of the temporal evolution of the velocity field as well as non-linear advection. The pressure gradient term absorbs the hydrostatic mean gravitational component as well as the centrifugal force. The buoyancy force arises from density differences with respect to the density of the motionless background state (assumed to be constant density ρ _{ o }). These perturbations, ρ ^{′}/ρ _{ o }=−α T ^{′}, are produced by temperature variations, T ^{′}, from the background state temperature profile; α is thermal expansivity. Viscous diffusion depends on the kinematic viscosity of the fluid, ν. The Lorentz force is due to interactions between the magnetic field and current density.
This is the Taylor-Proudman theorem and states that the fluid motion will be invariant along the direction of the rotation axis. However, in order for convection to occur, the Taylor-Proudman constraint cannot strictly hold as there must be slight deviations from two-dimensionality, at least in the boundary layers, to allow overturning motions in the fluid layer (e.g., Zhang 1992; Olson et al. 1999; Grooms et al. 2010). Further beyond the onset of convection, convective flows break down into an anisotropic rotating turbulence (e.g., Sprague et al. 2006; Julien et al. 2012; Stellmach et al. 2014; Cheng et al. 2015; Ribeiro et al. 2015).
Linear asymptotic analyses of quasigeostrophic convection predict that the azimuthal wavenumber of these columns relative to the thickness of the fluid shell varies as ℓ _{ U }/D∝E ^{1/3}, where the Ekman number, E=ν/(2Ω D ^{2}), represents the ratio of viscous to Coriolis forces (e.g., Roberts 1968; Julien et al. 1998; Jones et al. 2000; Dormy et al. 2004). Thus, in rapidly rotating systems such as planetary cores (where E≲10^{−12}), it is often predicted that quasigeostrophic convection occurs as tall, thin columns (e.g., Kageyama et al. 2008; cf. Cheng et al. 2015).
This equation shows that magnetic fields can relax the Taylor-Proudman constraint, allowing global-scale motions that differ fundamentally from the small-scale axial columns typical of non-magnetic, rapidly rotating convection (e.g., Cardin and Olson 1995; Olson and Glatzmaier 1996; Zhang and Schubert 2000; Roberts and King 2013).
indicating that the Lorentz to Coriolis force ratio is length scale dependent. In the latter equality, R m=U D/η is the global ratio of magnetic induction to magnetic diffusion. This parameter must exceed approximately 10 in order for dynamo action to occur (e.g., Roberts 2007). Thus, on global scales, where ℓ _{ B }≃D, the dynamic Elsasser number will always be less than Λ _{ i } by at least an order of magnitude for active dynamos. However, the influence of magnetic fields will become increasingly important at smaller scales since \(\Lambda _{d} \propto \ell _{B}^{-1}\) in (Eq. 8).
This scaling estimate will be referred to as the modified dynamic Elsasser number.
In this paper, we will investigate the relative magnitudes of Lorentz and Coriolis forces to better understand the dynamics of planetary cores. Towards this end, we calculate directly the Lorentz to Coriolis force ratio in a suite of geodynamo models described in the “Methods” section, test Eq. 11 to predict this ratio in the “Results” section, and extrapolate the results to Earth and other planetary cores in the “Discussion” section.
Methods
We analyze 34 dynamo models presented in Soderlund et al. (2012, 2014) and five dynamo models presented in Sheyko (2014); these datasets will be referred to as SKA and S14, respectively. These simulations of Boussinesq convection and dynamo action are carried out in thick, rotating spherical shells with thickness D, rotation rate Ω, and no-slip boundaries. Gravity is assumed to vary linearly with radius; g _{ o } is acceleration at the top of the fluid shell. The outer boundary at radius r _{ o } is assumed electrically insulating and the solid inner core at radius r _{ i } has the same electrical conductivity as the convecting fluid region.
Input (E, Pm, Ra, \({Ro}_{c}=\sqrt {Ra E^{2} / Pr}\)) and output (Λ _{ i }, Λ _{ d }, \(\Lambda _{d}^{*}\), Λ ^{ I }, Λ ^{ P }) parameters for our dataset. P r=1 is fixed for all simulations. Additional simulation information is given in Soderlund et al. (2012, 2014) for cases with E≥10^{−5} and Sheyko (2014) for cases with E≲10^{−6}
E | Pm | Ra | R o _{ c } | Λ _{ i } | Λ _{ d } | \(\Lambda _{d}^{*}\) | Λ ^{ I } | Λ ^{ P } |
---|---|---|---|---|---|---|---|---|
10^{−3} | 5 | 6.50×10^{4} | 0.26 | 4.34 | 0.44 | 0.47 | 0.39 | 0.26 |
10^{−3} | 5 | 9.70×10^{4} | 0.31 | 3.91 | 0.30 | 0.35 | 0.43 | 0.32 |
10^{−3} | 5 | 1.12×10^{5} | 0.33 | 3.25 | 0.20 | 0.25 | 0.40 | 0.29 |
10^{−3} | 5 | 1.32×10^{5} | 0.36 | 0.61 | 0.03 | 0.04 | 0.05 | 0.01 |
10^{−3} | 5 | 5.00×10^{5} | 0.71 | 4.14 | 0.15 | 0.17 | 0.29 | 0.16 |
10^{−3} | 5 | 1.96×10^{6} | 1.40 | 23.0 | 0.51 | 0.64 | 0.79 | 0.46 |
10^{−3} | 5 | 1.96×10^{7} | 4.43 | 248 | 2.62 | 4.00 | 4.20 | 3.12 |
10^{−4} | 2 | 1.42×10^{6} | 0.12 | 1.30 | 0.14 | 0.12 | 0.18 | 0.12 |
10^{−4} | 2 | 2.12×10^{6} | 0.15 | 2.10 | 0.17 | 0.16 | 0.21 | 0.14 |
10^{−4} | 2 | 2.83×10^{6} | 0.17 | 2.42 | 0.17 | 0.16 | 0.24 | 0.16 |
10^{−4} | 2 | 3.54×10^{6} | 0.19 | 2.37 | 0.15 | 0.14 | 0.27 | 0.19 |
10^{−4} | 2 | 3.68×10^{6} | 0.19 | 2.36 | 0.15 | 0.13 | 0.21 | 0.17 |
10^{−4} | 2 | 3.75×10^{6} | 0.19 | 2.27 | 0.14 | 0.12 | 0.21 | 0.15 |
10^{−4} | 2 | 3.82×10^{6} | 0.20 | 0.14 | 0.01 | 0.007 | 0.01 | 0.006 |
10^{−4} | 2 | 3.96×10^{6} | 0.20 | 0.18 | 0.01 | 0.009 | 0.02 | 0.009 |
10^{−4} | 2 | 4.11×10^{6} | 0.20 | 0.22 | 0.01 | 0.01 | 0.02 | 0.009 |
10^{−4} | 2 | 4.24×10^{6} | 0.21 | 0.24 | 0.02 | 0.01 | 0.02 | 0.006 |
10^{−4} | 2 | 6.00×10^{6} | 0.24 | 0.68 | 0.04 | 0.03 | 0.06 | 0.03 |
10^{−4} | 2 | 8.50×10^{6} | 0.29 | 1.53 | 0.07 | 0.06 | 0.11 | 0.06 |
10^{−4} | 2 | 1.42×10^{7} | 0.38 | 3.44 | 0.13 | 0.11 | 0.21 | 0.11 |
10^{−4} | 2 | 2.10×10^{7} | 0.46 | 5.48 | 0.18 | 0.16 | 0.28 | 0.17 |
10^{−4} | 2 | 4.24×10^{7} | 0.65 | 11.4 | 0.29 | 0.26 | 0.44 | 0.27 |
10^{−4} | 2 | 8.00×10^{7} | 0.89 | 20.7 | 0.40 | 0.39 | 0.59 | 0.37 |
10^{−4} | 2 | 1.42×10^{8} | 1.19 | 36.3 | 0.58 | 0.58 | 0.81 | 0.53 |
10^{−4} | 2 | 4.24×10^{8} | 2.06 | 106 | 0.84 | 1.28 | 1.18 | 0.77 |
10^{−4} | 2 | 8.48×10^{8} | 2.91 | 197 | 1.11 | 2.10 | 1.63 | 1.07 |
10^{−5} | 2 | 3.10×10^{7} | 0.06 | 4.84 | 0.30 | 0.30 | 0.39 | 0.33 |
10^{−5} | 2 | 5.89×10^{7} | 0.08 | 7.07 | 0.31 | 0.31 | 0.45 | 0.37 |
10^{−5} | 2 | 8.20×10^{7} | 0.09 | 8.28 | 0.32 | 0.30 | 0.45 | 0.34 |
10^{−5} | 2 | 8.50×10^{7} | 0.09 | 8.54 | 0.32 | 0.31 | 0.47 | 0.39 |
10^{−5} | 2 | 9.50×10^{7} | 0.10 | 9.02 | 0.32 | 0.30 | 0.45 | 0.35 |
10^{−5} | 2 | 1.05×10^{8} | 0.10 | 9.63 | 0.33 | 0.31 | 0.47 | 0.34 |
10^{−5} | 2 | 1.50×10^{8} | 0.12 | 12.4 | 0.36 | 0.35 | 0.52 | 0.37 |
10^{−5} | 2 | 2.00×10^{8} | 0.14 | 14.8 | 0.38 | 0.37 | 0.56 | 0.36 |
1.2×10^{−6} | 1 | 1.86×10^{8} | 0.02 | 5.73 | 0.21 | 0.28 | 0.39 | 0.30 |
1.2×10^{−6} | 0.2 | 1.86×10^{8} | 0.04 | 0.07 | 0.005 | 0.007 | 0.01 | 0.009 |
1.2×10^{−6} | 0.2 | 9.28×10^{8} | 0.09 | 0.61 | 0.03 | 0.03 | 0.08 | 0.04 |
1.2×10^{−6} | 0.2 | 5.57×10^{9} | 0.02 | 9.92 | 0.24 | 0.25 | 0.47 | 0.30 |
3.0×10^{−7} | 0.05 | 2.23×10^{10} | 0.04 | 3.61 | 0.13 | 0.17 | 0.32 | 0.24 |
The SKA simulations are carried out on numerical grids that range from 37 fluid shell radial levels and 42 spherical harmonic degrees (near onset at E=10^{−3}) to 73 radial levels and 213 spherical harmonic degrees (highest Ra for E=10^{−5}). Hyperdiffusion is used for the three most extreme cases (R a≥2×10^{8} for E≤10^{−4}); additional details are given in Soderlund et al. (2012). The S14 simulations are carried out on numerical grids with 512–528 radial levels and 256 spherical harmonic degrees. All simulations are carried out using pseudo-spectral methods with no azimuthal symmetries assumed (Wicht 2002; Christensen and Wicht, 2007; Sheyko 2014). Once the initial transient has subsided, global diagnostic quantities (Λ _{ i }, Rm, ℓ _{ B }, Λ _{ d }, and \(\Lambda _{d}^{*}\)) are time-averaged to minimize statistical errors.
Here, B _{ l } and B _{ m } are magnetic induction at spherical harmonic degree l and order m.
Results
A Lorentz to Coriolis force ratio histogram is evaluated for a representative dipole-dominated dynamo case at a snapshot in time in Fig. 1. The histogram indicates that the Coriolis force is dominant for the majority of grid points, most frequently by an order of magnitude. Moreover, the points with Lorentz to Coriolis force ratios greater than unity tend to have relatively low kinetic energies, denoted by coloration of the bins. Thus, the Lorentz force is expected to have a secondary influence on the convective (non-zonal) dynamics (Soderlund et al. 2012). This does not mean, however, that the magnetic field cannot have an important, local-scale dynamical impact (cf. Sreenivasan and Jones 2011).
Elsasser number definitions: (1) Imposed Λ _{ i }, (2) Dynamic Λ _{ d }, (3) Modified dynamic \(\Lambda _{d}^{*}\), (4) Integrated Λ ^{ I }, and (5) Probability Λ ^{ P }
Λ _{ i } | Λ _{ d } | \(\Lambda _{d}^{*}\) | Λ ^{ I } | Λ ^{ P } |
---|---|---|---|---|
\(\frac {B^{2}}{2 \rho _{o} \mu _{o} \eta \Omega }\) | \(\frac {\Lambda _{i}}{Rm} \frac {D}{\ell _{B}}\) | \( 0.77 \frac {\Lambda _{i}}{Rm^{1/2}}\) | \(\frac {\int _{V}{|\mathbf {F_{L}}| dV}}{\int _{V}{|\mathbf {F_{C}}| dV}}\) | \(\frac {|\mathbf {F_{L}}|}{|\mathbf {F_{C}}|}\), max probability |
This pre-factor is determined by minimizing the least squares residual between the calculated and predicted values across our dataset. Roberts and King (2013) also investigated the length scale of magnetic field variations in geodynamo models and found a similar dependence on R m ^{−1/2} with a pre-factor of 3 due to their slightly modified definition of ℓ _{ B }/D. More importantly, they also show that the magnetic length scale is weakly dependent on the magnetic field strength Λ _{ i }, which may explain some of the spread in our model-prediction deviations.
In order to apply the modified dynamic Elsasser number to planetary cores, its applicability to core conditions must first be evaluated. Our datasets cover a broad range of diagnostic physical parameters: magnetic field strengths of 10^{−1}≲Λ _{ i }≲10^{2}, magnetic induction to diffusion ratios of 10^{2}≲R m≲10^{4}, and Lorentz to Coriolis force ratios of 10^{−2}≲Λ _{ d }≲1. However, these ranges span only a fraction of the planetary parameter estimates, which vary over 10^{−5}≲Λ _{ i }≲10^{10}, 10^{2}≲R m≲10^{5}, and \(10^{-6} \lesssim \Lambda _{d}^{*} \lesssim 10^{7}\) as detailed in the “Discussion” section. The most fundamental deviation is for Λ _{ d }>1, which implies a different dominant force balance. While our dataset includes cases with Λ _{ d }>1, these simulations are strongly driven with inertia exceeding both Lorentz and Coriolis forces (see Fig. 4 of Soderlund et al. 2012), which is not expected for most planets (cf. Soderlund et al. 2013).
Figure 5 shows the ratio of the dynamic to modified dynamic Elsasser numbers as a function of the convective Rossby number, R o _{ c }=(R a E ^{2}/P r)^{1/2}, which characterizes the ratio of buoyancy-driven inertial forces to Coriolis forces (e.g., Gilman 1977, Aurnou et al. 2007). The \(\Lambda _{d}^{*}\) approximation works well when the Coriolis force dominates; in contrast, the largest discrepancy occurs when R o _{ c }>1. Geodynamo simulations from Dormy (2014) provide a \(\Lambda _{d}^{*}\) test for cases with Λ _{ d }>1 and R o _{ c }<1 at an Ekman number of E=1.5×10^{−4} and a Prandtl number of unity. Here, large magnetic Prandtl numbers (P m≥12) are required to obtain dynamo action since the inertial forces are relatively weak (i.e., quasi-laminar flows). Application of Eq. 18 to his dataset yields \(0.85 \leq \Lambda _{d}/\Lambda _{d}^{*} \leq 1.3\), implying that our results are applicable to strong field systems with dominant Lorentz forces.
A discrepancy between input parameters also exists between the simulations and planetary cores. In particular, all numerical dynamo models are necessarily limited to massively overestimated kinematic viscosities, which prohibits simulations with realistic Ekman and magnetic Prandtl numbers: 10^{−19}≲E≲10^{−12} and 10^{−8}≲P m≲10^{−6} are estimated for planets with active dynamos (e.g., Schubert and Soderlund 2011). In contrast, our dataset considers 10^{−7}≲E≲10^{−3} and 10^{−1}≲P m≲10^{1}. Importantly, no clear Ekman or magnetic Prandtl number dependence is identified in Fig. 5. We, therefore, hypothesize that the modified dynamic Elsasser number (Eq. 18) can be extrapolated to extreme planetary parameters.
Discussion
Order of magnitude estimates of the magnetic Reynolds number Rm, the imposed Elsasser number Λ _{ i }, and the Lorentz to Coriolis force ratio as estimated by \(\Lambda _{d}^{*}\) for planetary cores with active dynamos
Mercury | Earth | Ganymede | Jupiter | Saturn | Ice Giants | |
---|---|---|---|---|---|---|
Rm | 75 | 10^{3} | 500 | 10^{5} | 10^{4} | 10^{2} |
Λ _{ i } | 10^{−3} | 10^{2} | 10^{−1} | 10^{2} | 10^{0} | 10^{−2} |
\(\Lambda _{d}^{*}\) | 10^{−4} | 10^{0} | 10^{−3} | 10^{−1} | 10^{−2} | 10^{−3} |
Earth: Geodynamo core flow inversions based on secular variation of the magnetic field imply velocities of U∼5×10^{−4} m/s (e.g., Holme 2007; Aubert 2013), which corresponds to R m∼10^{3} assuming η∼1 m^{2}/s for liquid metals. Ensemble inversions further suggest RMS magnetic field strengths in the cylindrical direction of B _{ S }∼2 mT (Gillet et al. 2010), while toroidal magnetic field constraints of 1<B _{ T }/B _{ S }<100 have been inferred by Shimizu et al. (1998). Thus, B∼10B _{ S }∼20 mT appears to be a reasonable estimate of the core’s total field strength such that Λ _{ i }∼10^{2}. In contrast, direct magnetic field measurements suggest a lower bound of Λ _{ i }∼1 and application of the ω-effect suggests an upper bound of Λ _{ i }∼10^{6}. We, therefore, predict a Lorentz to Coriolis force ratio of \(\Lambda _{d}^{*} \sim 1\) for the geodynamo, with a possible range of \(10^{-2} \lesssim \Lambda _{d}^{*} \lesssim 10^{4}\). This result suggests that the Lorentz force cannot be neglected dynamically in the core and, more likely, that the core is in magnetostrophic balance.
Mercury and Ganymede: No observation-based velocity constraints are available currently for Mercury or Ganymede, so dynamo models that capture many of their magnetic field characteristics may serve as a basis for determining the magnetic Reynolds number. For example, Cao et al. (2014) obtain a Mercury-like dynamo when convection is driven by volumetrically distributed buoyancy sources and the core-mantle boundary heat flux peaks at low latitudes; their best fitting case has R m=75. Similarly, Christensen (2015) find Ganymede-like dynamos driven by iron snow with a stably stratified layer below the outer shell boundary; R m≈500 in his optimal simulation. Using the upper and lower bound constraints derived from the ω-effect and observations, respectively, the imposed Elsasser number is estimated to vary between Λ _{ i }∼10^{−5} and ∼10^{−1} for Mercury, which corresponds to \(\Lambda _{d}^{*} \lesssim 10^{-2}\) and implies that magnetostrophic balance is unlikely in the Hermian core. Conversely, the possible magnetic field strengths for Ganymede range from Λ _{ i }∼10^{−3} to ∼10^{2}, yielding \(10^{-5} \lesssim \Lambda _{d}^{*} \lesssim 1\). However, the best estimate for the imposed Elsasser number is Λ _{ i } ∼ 10^{−1} such that \(\Lambda _{d}^{*} \sim 10^{-3}\). As a result, magnetostrophic balance is possible, but unlikely, in Ganymede’s core.
Gas Giants: Jovian velocities can be estimated through scaling arguments and potentially secular variation. Jones (2014) suggests flow velocities of \(U \gtrsim 10^{-3}\) m/s, leading to values of \(Rm \gtrsim 10^{5}\) given the planet’s large size. In this case, magnetic field strengths range from Λ _{ i }∼1 via core field measurements to Λ _{ i }∼10^{10} when the ω-effect is included, with a best estimate of Λ _{ i }∼10^{2}. The corresponding Lorentz to Coriolis force ratio then spans the range \(10^{-3} \lesssim \Lambda _{d}^{*} \lesssim 10^{7}\), with \(\Lambda _{d}^{*} \sim 0.1\) being the best estimate. Thus, barring the large uncertainties, the Lorentz force is predicted to be sub-dominant, but not negligible, in Jupiter’s core.
Starchenko and Jones (2002) predict similar flow speeds for the Saturnian interior, which corresponds to a lower R m∼10^{4} value due to the smaller size of the dynamo region. Saturn’s magnetic field is also weaker than that of Jupiter; the best estimate imposed Elsasser number is Λ _{ i }∼1 with a feasible range of 10^{−2}≲Λ _{ i }≲10^{6}. In the most likely scenario, the Lorentz force is predicted to be two orders of magnitude smaller than the Coriolis force (\(\Lambda _{d}^{*} \sim 10^{-2}\)) with uncertainties extending the possible range to \(10^{-4} \lesssim \Lambda _{d}^{*} \lesssim 10^{4}\).
Ice Giants: Uranus and Neptune have relatively weak core magnetic fields with Λ _{ i }∼10^{−4} based on spacecraft observations, but there are few constraints on the flow speeds. We, therefore, assume U∼10^{−3} m/s based on the other planetary estimates such that R m∼10^{2} to give an upper bound of Λ _{ i }∼1 when the toroidal estimate is included. For these Λ _{ i } estimates, the predicted Lorentz to Coriolis force ratio is always less than unity: \(10^{-5} \lesssim \Lambda _{d}^{*} \lesssim 10^{-1}\). These results thus imply that the ice giants are not in magnetostrophic balance.
Conclusions
The Lorentz to Coriolis force ratio is an important parameter for the dynamics of planetary cores since dynamos with dominant Coriolis forces at global scales are expected to be driven by fundamentally different archetypes of fluid motions than those with dominant (or co-dominant) Lorentz forces (Roberts and King 2013, Calkins et al. 2015). We hypothesize that a representative global estimate of the Lorentz to Coriolis force ratio can be predicted by \(\Lambda _{d}^{*} = 0.77 \Lambda _{i} Rm^{-1/2}\). An advantage of this formulation is that it depends on quantities that can be estimated for planetary cores (Table 3). Our results suggest that the Earth’s core is likely to be in magnetostrophic balance where the Lorentz and Coriolis forces are comparable. The Lorentz force may also be substantial in Jupiter’s core, where it is predicted to be a factor of ten less than the Coriolis force. Magnetic fields become increasingly sub-dominant for the other planets: the Coriolis force is predicted to exceed the Lorentz force by at least two orders of magnitude within the cores of Saturn, Uranus/Neptune, Ganymede, and Mercury.
These conclusions are subject to large uncertainties, however. Core flow speeds are difficult to estimate, while total magnetic field strengths cannot be measured directly. The applicability of Eq. 18 for the Lorentz to Coriolis force ratio may also break down at extreme planetary core conditions that cannot be explored numerically or in the laboratory due to technological limitations. In order to mitigate these uncertainties, we have considered a range of possible core magnetic field strengths (Λ _{ i }) and included state-of-the-art simulations (Sheyko 2014; cf. Nataf and Schraeffer 2015).
Abbreviations
Declarations
Acknowledgements
The authors thank two anonymous referees for their thoughtful reviews, Hao Cao for helpful suggestions, and Wolfgang Bangerth for enlightening discussions at the 2014 Study of Earth’s Deep Interior (SEDI) meeting in Kanagawa, Japan. KMS gratefully acknowledges Japan Geoscience Union the National Science Foundation to attend this symposium as well as research support from the National Science Foundation (grant AST-0909206). JMA acknowledges the support of the National Science Foundation Geophysics Program (grant EAR-1246861). Computational resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center and by the Swiss National Supercomputing Centre (CSCS) under project ID s225. This is UTIG contribution 2858.
Authors’ Affiliations
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