Soderlund et al. Progress in Earth and Planetary Science (2015) 2:24 DOI 10.1186/s40645-015-0054-5 RESEARCH ARTICLE Open Access The competition between Lorentz and Coriolis forces in planetary dynamos Krista M. Soderlund1*, Andrey Sheyko2, Eric M. King3 and Jonathan M. Aurnou4 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. ∗ −1/2 Our results show that this ratio can be estimated by d iRm ( 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: Planetary dynamos; Core convection; Magnetic fields; Rotation; Magnetostrophic force balance Background fluid motions, and the Lorentz force which back-reacts on Magnetic fields are common throughout the solar sys- the fluid motions to equilibrate magnetic field growth. tem and provide a unique perspective on the internal In order to better understand the manifestation of dynamics of planetary interiors. Planetary magnetic fields Coriolis and Lorentz forces on core flow, let us consider are driven by the conversion of kinetic energy into mag- the Boussinesq momentum equation. In this commonly netic energy; this process is called dynamo action. Kinetic used approximation, the material properties are assumed energy is derived from thermo-compositional convection to be constant and the density is treated as constant except of an electrically conducting fluid although mechanical in the buoyancy force of the momentum equation (e.g., mechanisms, such as libration and precession, may also Tritton 1998): drive core flow in smaller bodies (e.g., Le Bars et al. 2015). Du 1 The geodynamo is the most studied planetary mag- + × =−∇ − α + ν∇2 + × 2 u T g u ρ J B.(1) netic field, yet the mechanisms that control its strength, Dt o morphology, and secular variation are still not well under- Here, u is the velocity vector, is the rotation rate of the stood. Towards explaining these observations, dimension- reference frame, is the non-hydrostatic pressure nor- less parameters are often used to characterize the force malized by the background density ρo, J is the electric balances present in the core and their link to processes current density, B is the magnetic induction, and g is the that govern convective dynamics and dynamo action. Two acceleration of gravity. Terms from left to right are iner- forces that are particularly influential on core processes tial acceleration, Coriolis acceleration, pressure gradient, are the Coriolis force, which tends to organize the core buoyancy, viscous diffusion, and Lorentz force. The iner- tial term consists of the temporal evolution of the velocity *Correspondence: [email protected] 1Institute for Geophysics, John A. & Katherine G. Jackson School of field as well as non-linear advection. The pressure gradi- Geosciences, The University of Texas at Austin, J. J. Pickle Research Campus, ent term absorbs the hydrostatic mean gravitational com- Building 196 (ROC), 10100 Burnet Road (R2200), Austin, Texas, 78758-4445, USA ponent as well as the centrifugal force. The buoyancy force Full list of author information is available at the end of the article arises from density differences with respect to the density © 2015 Soderlund et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Soderlund et al. Progress in Earth and Planetary Science (2015) 2:24 Page 2 of 10 of the motionless background state (assumed to be con- In contrast to Eq. 3, the curl of the magnetostrophic stant density ρo). These perturbations, ρ /ρo =−αT ,are balance Eq. 4 yields produced by temperature variations, T ,fromtheback- ∂u 1 ground state temperature profile; α is thermal expansivity. =− ∇×(J × B).(5) ∂z 2 ρ Viscous diffusion depends on the kinematic viscosity of o the fluid, ν. The Lorentz force is due to interactions This equation shows that magnetic fields can relax between the magnetic field and current density. the Taylor-Proudman constraint, allowing global-scale Planetary cores are rapidly rotating and the Coriolis motions that differ fundamentally from the small-scale force is expected to be large compared to the viscous, iner- axial columns typical of non-magnetic, rapidly rotat- tial, and buoyancy forces at global scales (the former two ing convection (e.g., Cardin and Olson 1995; Olson and forces are scale dependent quantities). If we neglect these Glatzmaier 1996; Zhang and Schubert 2000; Roberts and forces and magnetic fields in the Boussinesq momentum King 2013). equation (Eq. 1), the system is in geostrophic balance: Magnetic fields are typically considered to be important dynamically when the ratio of Lorentz to Coriolis forces 2 × u =−∇.(2)is of order unity. This ratio is represented by the Elsasser number, , which can be approximated by assuming a This balance requires horizontal fluid motions to fol- characteristic current density, magnetic induction, den- low lines of constant pressure. Further insight can be sity, rotation rate, and velocity scale: obtained by taking the curl of Eq. 2 and using the continu- ity equation (∇·u = 0) to obtain ·∇u = 0. If we assume 1 J × B Lorentz ρo JB = zˆ where is constant, this equation simplifies to = = ≈ .(6) Coriolis |2 × u| 2ρoU ∂u = 0. (3) In systems where the magnetic field is not strongly time ∂z variant (i.e., ∂B/∂t =−∇×E ∼ 0), the current den- ∼ σ This is the Taylor-Proudman theorem and states that sity can be approximated via Ohm’s Law, J UB where σ = (μ η)−1 η the fluid motion will be invariant along the direction o is electrical conductivity and is magnetic of the rotation axis. However, in order for convection diffusivity, such that to occur, the Taylor-Proudman constraint cannot strictly 2 = B hold as there must be slight deviations from two- i .(7) 2ρoμoη dimensionality, at least in the boundary layers, to allow overturning motions in the fluid layer (e.g., Zhang 1992; This formulation is thus appropriate for steady imposed Olson et al. 1999; Grooms et al. 2010). Further beyond the magnetic fields (e.g., Olson and Glatzmaier 1996; King and onset of convection, convective flows break down into an Aurnou 2015). In dynamos, which tend to exhibit signifi- anisotropic rotating turbulence (e.g., Sprague et al. 2006; cant temporal variability, it is more appropriate to approx- Julien et al. 2012; Stellmach et al. 2014; Cheng et al. 2015; imate the current density using Ampere’s law under the ∼ /μ Ribeiro et al. 2015). MHD approximation, J B o B where B is the char- Linear asymptotic analyses of quasigeostrophic con- acteristic length scale of magnetic field variations (e.g., vection predict that the azimuthal wavenumber of these Cardin et al. 2002; Soderlund et al. 2012). In this dynamic case, Eq. 6 becomes columns relative to the thickness of the fluid shell varies as / /D ∝ E1 3, where the Ekman number, E = ν/ 2D2 , 2 U = B = i D represents the ratio of viscous to Coriolis forces (e.g., d ,(8) 2ρoμoU B Rm B Roberts 1968; Julien et al. 1998; Jones et al. 2000; Dormy et al. 2004). Thus, in rapidly rotating systems such as plan- indicating that the Lorentz to Coriolis force ratio is length = /η etary cores (where E 10−12), it is often predicted that scale dependent. In the latter equality, Rm UD is the quasigeostrophic convection occurs as tall, thin columns global ratio of magnetic induction to magnetic diffusion. (e.g., Kageyama et al. 2008; cf. Cheng et al. 2015). This parameter must exceed approximately 10 in order In planetary dynamos, however, magnetic fields are also for dynamo action to occur (e.g., Roberts 2007). Thus, thought to play an important dynamical role on the con- on global scales, where B D, the dynamic Elsasser vection and zonal flows. When strong imposed magnetic number will always be less than i byatleastanorder fields and rapid rotation are both present, the dominant of magnitude for active dynamos. However, the influence force balance is magnetostrophic—a balance between the of magnetic fields will become increasingly important at ∝ −1 Coriolis, pressure gradient, and Lorentz terms: smaller scales since d B in (Eq. 8). The quantity B/D cannot be measured for planetary × =−∇ + 1 × cores, however. It is, therefore, important to provide a 2 u J B.(4) ρo scaling estimate for d in terms of quantities that can Soderlund et al. Progress in Earth and Planetary Science (2015) 2:24 Page 3 of 10 2 be estimated from observations, such as i and Rm.The Pr = ν/κ, is unity. Here, E = ν/ 2D is the ratio of vis- length scale B can be approximated using the magnetic cous to Coriolis forces; Pm = ν/η is the ratio of viscous 3 induction equation: to magnetic diffusivities, and Ra = αgoTD /(νκ) is the ∂B ratio of buoyancy due to superadiabatic temperature con- =∇×(u × B) + η∇2B (9) trast T to diffusion.