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2 Earth and Planetary Science Letters (2013) Contents lists available at SciVerse ScienceDirect Earth and Planetary Science Letters journal homepage: Flow speeds and length scales in geodynamo models: The role of viscosity Eric M. King a,b,n, Bruce A. Buffett a a Department of Earth and Planetary Science, University of California, Berkeley, United States b Department of Physics, University of California, Berkeley, United States article info Article history: Received 25 September 2012 Received in revised form 27 March 2013 Accepted 2 April 2013 Editor: Y. Ricard Available online 4 May 2013 Keywords: geodynamo planetary magnetism geophysical fluid dynamics abstract The geodynamo is the process by which turbulent flow of liquid metal within Earth's core generates our planet's magnetic field. Numerical simulations of the geodynamo are commonly used to elucidate the rich dynamics of this system. Since these simulations cannot attain dynamic similarity with the geodynamo, their results must be extrapolated across many orders of magnitude of unexplored parameter space. For this purpose, scaling analysis is essential. We investigate the scaling behavior of the typical length scales, l, and speeds, U, of convection within a broad suite of geodynamo models. The model outputs are well fit by the scalings l E 1=3 and U C 1=2 E 1=3, which are derived from a balance between the influences of rotation, viscosity, and buoyancy (E is the Ekman number and C the convective power). Direct comparison with two previously proposed theories finds that the viscous scalings most favorably describe model data. The prominent role of viscosity suggested by these scaling laws may call into question the direct application of such simulations to the geodynamo, for which it is typically assumed that viscous effects are negligible. & 2013 Elsevier B.V. All rights reserved. 1. Introduction The Earth's magnetic field is generated by flowing liquid metal in its core. It is generally thought that this flow is driven by fluid buoyancy, as the core slowly cools and differentiates. The resulting convection generates electrical currents that maintain the geodynamo, and is subject to two significant forces: the Coriolis force, which stems from the Earth's daily rotation; and the Lorentz force, which accounts for the back-reaction of magnetic field on the flow from which it is generated. These key ingredients field generation by rotationally constrained convection are captured by selfconsistent geodynamo models. Conditions in the core, however, are more extreme than any current simulation can possibly replicate (e.g., Wicht and Tilgner, 2010). In particular, core fluid dynamics suffer from an extreme range of forces: Lorentz Coriolis inertia viscosity. Estimates of the relative magnitudes of these forces can be quantified by nondimensional parameters. The Rossby number characterizes the ratio between inertia and Coriolis forces in the core as Ro The Ekman number quantifies the ratio between viscous and Coriolis forces in the core as E Another extreme core parameter is the magnetic Prandtl number, which is the ratio n Corresponding author at: Department of Earth and Planetary Science, University of California, Berkeley, United States. Tel.: address: eric.king@berkeley.edu (E.M. King). between viscous and magnetic diffusion, Pm These and other important dimensionless numbers are defined in Table 1. The smallness of these parameters leads to an extreme range of anticipated time and length scales important for core dynamics, which cannot be resolved in present-day simulations (Davies et al., 2011). Since simulations are incapable of reaching the parameters necessary for true dynamic similarity with the geodynamo, we turn to scaling laws. Scaling laws depict the general behavior of one parameter with respect to others within a particular dynamical regime. There are two general purposes for developing and testing scaling laws. First, comparing theoretically founded scaling laws with observations in nature, experiments, or simulations tests our understanding of the basic physical processes responsible for producing these observations. Second, extrapolation of welltested scaling laws, even without firm theoretical basis, permits predictions of phenomena we cannot directly observe. Beginning with Glatzmaier and Roberts (1995), simulations of the geodynamo have proliferated such that there now exists an extensive population of results that permits the systematic scaling of their behavior (e.g., Christensen and Aubert, 2006; Olson and Christensen, 2006; Christensen, 2010; King et al., 2010; Jones, 2011). Here, we apply to such dynamo models a theoretical scaling law for average convective flow speeds. The scaling law, recently proposed for simpler, non-magnetic convection simulations (King et al., 2013), is based on a steady state balance between buoyant X/$ - see front matter & 2013 Elsevier B.V. All rights reserved.

3 E.M. King, B.A. Buffett / Earth and Planetary Science Letters (2013) Table 1 Relevant dimensionless numbers, estimates of values in Earth's core, and the range of values reached in numerical simulations. Dimensional quantities are ν, viscosity; Ω, angular rotation rate; D, shell thickness; ρ =ρ 0, fractional density anomaly; g, gravity; κ, thermal diffusivity; η, magnetic diffusivity; P, convective power; A, outer boundary surface area; U, typical flow speed; B, magnetic field strength; μ 0, permittivity of free space. An estimate of Pr for the core depends on whether the buoyancy source considered is thermal (Pr o1) or compositional (Pr 41). An estimate of Ra in the core depends on the poorly constrained superadiabatic density contrast between inner core and mantle, and so a value is not specified. Symbol Name Definition Core Simulations E Ekman ν=ωd E 10 3 Ra Rayleigh ρ gd 3 =ρ 0 νκ Ra 2: Pr Prandtl ν=κ 0:1 Pr 30 Pm Magnetic Prandtl ν=η :06 Pm 20 C Convective power PD 3 =Aρ 0 ν C Re Reynolds UD=ν Re 2000 Ro Rossby U=2ΩD Ro 0:8 Λ Elsasser B 2 =ρ 0 μ 0 ηω 1 0:03 Λ 300 energy production and viscous dissipation. The important influence of rotation is revealed through its selection of the typical length scales of convective cells, which is a critical factor in setting the dissipation rate. Following King et al. (2013), we present hydrodynamic scaling laws for typical speeds and length scales of convection, U and l, in Sections 3 and 4, and test these predictions against a suite of geodynamo simulations, which are introduced in Section 2. In Section 5, we compare these results against other proposed scaling laws. Finally, in Section 6, we discuss the implications of the preceding results. 2. Numerical dynamo models The numerical dynamo model outputs analyzed here come from a suite of simulations carried out by Christensen using the MagIC numerical model (Christensen et al., 1999; Christensen and Aubert, 2006; King et al., 2010). In the model, the governing equations (momentum conservation equation, magnetic induction equation, heat advection diffusion equation, and mass conservation) are evolved using the spectral transform method of Glatzmaier (1984) in a spherical shell with Earth-core-like geometry. Conditions enforced on inner and outer boundaries are constant temperature and zero flow. The outer boundary is electrically insulating, and, for almost all of the models used here, the inner core is also taken to be an insulator. The data set used here is identical to the dynamo factory models of King et al. (2010), consisting of 159 individual models, many of which were also used for scaling analysis by Christensen and Aubert (2006), Olson and Christensen (2006), and Christensen (2010). The parameter ranges accessed by the dynamo suite are given in Table 1. Typical flow speeds are calculated as U ¼ u 2 1=2 ; ð1aþ where u is the fluid velocity, angled brackets represent averages over the entire spatial domain, and overlines represent averages in time. Characteristic length scales of flow are calculated as the mean scale for kinetic energy (Christensen and Aubert, 2006) l ¼ π u2 l u 2 l D; ð1bþ where u l is the velocity at harmonic degree l, and D is the shell thickness (see Table 1). 3. Flow speeds: the mean kinetic energy equation The equation governing fluid momentum in a rotating reference frame for Boussinesq magnetoconvection is ρ 0 ð t u þ u uþ þ 2ρ fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 0 Ω u fflfflfflfflfflffl{zfflfflfflfflfflffl} inertia Coriolis ¼ P þ ffl{zffl} pressure ρ g {z} buoyancy þ J B þ ρ ffl{zffl} 0 ν 2 u; fflfflfflffl{zfflfflfflffl} Lorentz viscosity where ρ 0 is the mean fluid density, Ω is the rotation vector, P is the pressure (which includes hydrostatic gravitation and the centrifugal force), ρ is the density anomaly, g ¼ g ^r is the gravitational acceleration, J is the electrical current density, B is the magnetic field, and ν is the viscous diffusivity. The labels under each term in (2) will be used to identify them below. The system is considered Boussinesq in that density, ρðr; tþ¼ρ 0 þ ρ ðr; tþ, is treated as variable only in its contribution to the buoyancy force. A particularly simple but exact relation for Boussinesq convection is the mean kinetic energy equation. Produced by the scalar product of velocity u with the momentum equation (2), averaging in time and integrating over the fluid volume V, the mean kinetic energy equation is Z u r ρ g þ 1 B ðb Þu ρ μ 0 νω 2 dv ¼ 0; ð3aþ 0 V where u r is the radial component of velocity and ω ¼ u is the vorticity. This particular form of the mean kinetic energy equation assumes no-slip boundaries. According to (3a), kinetic energy is produced by buoyant power Z P ¼ u r ρ g dv; ð3bþ V by a generic source(s) of buoyancy, which is expended by the Lorentz work Z 1 Q L ¼ B ðb Þu dv; ð3cþ V μ 0 and viscous dissipation Z ϵ ν ¼ ρ 0 νω 2 dv: ð3dþ V Soderlund et al. (2012) argue that magnetic fields have a secondary influence on convective flow speeds in geodynamo models, citing a relatively weak Lorentz force in several simulations (which are similar to the simulations considered here). We follow from this suggestion and assume that Lorentz forces are unimportant for the typical magnitude of convective flow speeds in the models. The validity and geophysical relevance of this assumption are discussed in Section 6. Thus, in order to scale the flow speed, kinetic energy production is balanced with viscous dissipation P ϵ ν ð4þ Vorticity is taken to scale as U=l, such that ϵ ν can be scaled as ρ 0 νu 2 V=l 2. The viscous balance (4) then suggests flow speeds scale as! Pd 2 1=2 U : ð5þ νρ 0 D 3 In order to test this scaling against model outputs, we revert to dimensionless quantities for purely thermal convection. Total convective power P is related to the convective heat flow P T by P T ¼ ρ 0 c P Au r T c P αgd P; ð6aþ where c P is the specific heat, A is the surface area through which the heat power is fluxed, T is the anomalous temperature, and α is the thermal expansivity. We make P non-dimensional by ð2þ

4 158 E.M. King, B.A. Buffett / Earth and Planetary Science Letters (2013) the pressure gradient is eliminated from (2) by taking its curl, producing the vorticity equation ω t þ ðu uþþ2ω u¼ 1 ðρ gþ ρ 0 þ 1 ðj BÞþν 2 ω: ð8aþ ρ 0 For flows in which inertial effects and Lorentz forces are negligible, the radial component of the vorticity equation requires that 2Ω u r ν z ¼ ^r 2 ω: ð8bþ In the limit of vanishing viscosity, (8b) defines the Taylor Proudman theorem. Assuming that viscous effects are not negligible, we scale this balance in the following way. Vorticity is again taken to scale as ω U=l, and spatial derivatives as 1=l, except for the axial gradient, where the prevailing geostrophic balance suggests that = z 1=D. A viscously broken TP constraint (8b) therefore Fig. 1. Calculations of Re from the dynamo simulations are plotted versus C 1=2 l=d to test (7). Symbol shapes depict Ekman numbers, and symbol sizes are proportional to non-dimensional magnetic field strength, Λ 1=2, with minimum and maximum at Λ 1=2 ¼ 0:17 and Λ 1=2 ¼ 17. The solid line shows Re ¼ 0:1R 1=2 l=d. f introducing a convective power term for thermal convection C ¼ PD3 Aρ 0 ν 3 ¼ RaðNu 1Þ=Pr2 ; ð6bþ where Nu ¼ P T D=ðρ 0 c P κaþ is the Nusselt number. The dissipation based flow speed scaling (5) can now be written in a dimensionless form as Re C 1=2 l=d; where Re ¼ UD=ν is the Reynolds number. Fig. 1 shows calculations of Re from the simulations plotted against C 1=2 l=d. A best fit power law regression yields Re ¼ 0:21ð70:04Þ½C 1=2 l=dš 0:91ð 7 0:05Þ, in relative agreement with (7). Assuming proportionality, the data are the best fit byre ¼ 0:1C 1=2 l=d, shown as the solid line in Fig. 1,andwhichfits the dynamo data with a relative standard error of 23%. The data suggest a secondary dependence on the magnetic field strength, which is represented by the symbol sizes (sizes being proportional to Λ 1=2 ). Smaller symbols tend to lie above the plotted scaling law, and larger symbols below, which indicates that dynamos with stronger fields tend to convect slightly less vigorously. This weak Lorentz braking of convection has been observed in other studies (e.g., Christensen and Aubert, 2006; Soderlund et al., 2012). ð7þ Fig. 2. Calculations of l from the dynamo simulations are plotted versus E to test (9). Symbol shapes depict E as in Fig. 1. Symbol sizes are proportional to non-dimensional magnetic field strength, Λ 1=2. The solid line shows l ¼ 4:8E 1=3 D. 4. Length scales: the vorticity equation Interestingly, the flow speed scaling (7) shows no explicit dependence on Ω, despite our interest in systems with Ro 1. Hide (1974) noted that rotation affects U largely through its influence on l regarding flow within Jupiter's deep atmosphere. Below, we consider the selection of l in rapidly rotating convection. It is generally thought that the leading order force balance in the Earth's core is geostrophic. Geostrophic flows are constrained to two dimensions by the Taylor Proudman theorem, which dictates that flows be invariant in the axial ( ^z) direction. However, neither mean heat transport nor dynamo action can be accomplished by geostrophic flow in a closed system. The Taylor Proudman (TP) constraint must be broken, and the ensuing flow must be ageostrophic at second order. In order to isolate ageostrophic flow, Fig. 3. Calculations of Re from the dynamo simulations are plotted versus C 1=2 E 1=3 to test (10). Symbol shapes depict E as in Fig. 1. Symbol sizes are proportional to nondimensional magnetic field strength, Λ 1=2. The solid line shows Re ¼ 0:5C 1=2 E 1=3.

5 E.M. King, B.A. Buffett / Earth and Planetary Science Letters (2013) prescribes a characteristic length scale l νd Ω! 1=3 ¼ E 1=3 D: ð9þ This scaling is identical to the predicted scaling for non-axial wavenumbers of critical modes for the onset of rapidly rotating, non-magnetic convection (Roberts, 1968; Jones et al., 2000). Fig. 2 shows calculations of l from the dynamo models plotted versus the Ekman number. A best fit power law regression yields l=d ¼ 2:8ð70:5ÞE 0:29ð 7 :02Þ. If we restrict our data to simulations with the lowest Ekman numbers, Eo10 4, this fit becomes l=d ¼ 4ð72ÞE 0:32ð 7 :05Þ. Fixing the exponent at 1/3, misfit is minimized by l=d ¼ 4:8E 1=3, which fits all the dynamo models with a relative standard error of 17% and is shown as a solid line in Fig. 2. We observe that neither strong fields nor highly supercritical convection produce fundamental changes in the viscously set length scales. Substituting for l in the flow speed scaling (7) using (9) gives (e.g., Aubert et al., 2001; Gillet and Jones, 2006; King et al., 2013) Re C 1=2 E 1=3 : ð10þ Fig. 3 shows calculations of Re from the dynamo models plotted against C 1=2 E 1=3. The solid line shows Re ¼ 0:5C 1=2 E 1=3,whichfits the dynamo data with a relative standard error of 14%. 5. Comparison with the previous work 5.1. Non-magnetic convection The scaling laws presented here have also been applied to simulations of non-magnetic rotating convection (King et al., 2013), for which the steady state balance between buoyant power and viscous dissipation (4) holds exactly. Fig. 4 shows comparisons between the non-magnetic simulations and the scaling laws (9) and (10), as in Figs. 2 and 3, respectively. The dynamo simulations and non-magnetic simulations produce flow speeds and length scales that behave similarly. This observation suggests that our assumption that magnetic fields are not critically important for flow speeds and length scales within the dynamo models is reasonable Inertial theory: MAC balance Two other theoretical scalings for flow speeds in convective dynamos have been put forward and should be compared with the results shown here. The first is based on the so-called inertial theory beginning with Hide (1974) and Ingersoll and Pollard (1982), and is developed further by Cardin and Olson (1994), Aubert et al. (2001), Gillet and Jones (2006) and Christensen and Aubert (2006). The second is proposed by Starchenko and Jones (2002) under the assumption that the rapidly rotating convective dynamo naturally settles into a triple balance between Coriolis, buoyancy, and Lorentz forces, the so-called MAC balance (for magneto-archimedean-coriolis). Both arguments begin with a balance between Coriolis and buoyancy forces in the vorticity equation, and differ in their treatment of the characteristic length scales of convection. For a more complete review, see Christensen (2010) and Jones (2011). A balance between Coriolis and buoyancy terms in (8) is called the thermal wind balance, and can be scaled to give U ρ gd 2ρ 0 Ωl : ð11aþ In dimensionless parameters, this is Re Ra E Pr 1 D=l: ð11bþ Next, ρ is eliminated by multiplying (11a) by U and assuming that Uρ g ¼ u r ρ g, such that! PD 4 1=2 U or Re ðced=lþ 1=2 : ð11cþ 2ρ 0 Ωl This additional step is implemented primarily because estimates for the magnitude of the convective power within natural dynamos are generally better constrained than those for the typical density anomaly. The difference between the inertial scaling (sometimes called the CIA scaling), the MAC scaling, and the viscous scaling presented in the previous sections (which may be called a VAC scaling) is entirely determined by their treatment of l in (11c). The CIA scaling assumes that the TP constraint is broken by inertia, resulting in a Rhines scale for convection (Cardin and Olson, 1994) l Ro 1=2 D: ð12þ Fig. 4. Calculations of l and Re from the non-magnetic, plane layer rotating convection simulations of King et al. (2013). Compare panel (a) with Fig. 2 and panel (b) with Fig. 3. The solid lines in each comparison show the scaling relation from (9) and (10).

6 160 E.M. King, B.A. Buffett / Earth and Planetary Science Letters (2013) The MAC balance assumes that the TP constraint is broken by Lorentz forces, resulting in large scale flow (Starchenko and Jones, 2002) l D: ð13þ The VAC scaling again assumes that the TP constraint is broken by viscous forces, resulting length scales given by (9). Substituting these scalings for l into (11c) gives a CIA scaling (Aubert et al., 2001) Re C 2=5 E 1=5 ; a MAC scaling (Starchenko and Jones, 2002) Re C 1=2 E 1=2 ; and the VAC scaling ð14þ ð15þ Re C 1=2 E 1=3 ; ð16þ as in (10). In Fig. 5a c, we compare each of these scaling laws against calculations from the dynamo models. To promote fair comparison, in each panel is plotted as Re versus CE γ, where only γ differs corresponding to differences between the scalings (14) (16).Best-fit power-law regressions yield: (a) Re ¼ 0:36ð70:06ÞðCE 1=2 Þ 0:46ð 7 0:01Þ ; (b) Re ¼ 2:6ð70:5ÞðCEÞ 0:48ð 7 0:02Þ ; and (c) Re ¼ 0:60ð70:07Þ ðce 1=3 Þ 0:48ð 7 0:01Þ. Assuming scaling laws of forms (14) (16), thedata are fit byre ¼ 0:8C 2=5 E 1=5 to within an average standard error of 22%, by Re ¼ 2:4C 1=2 E 1=2 to within 31%, and by Re ¼ 0:5C 1=2 E 1=3 to within 14%, and these scalings are shown in Fig. 5a c, respectively. For quantitative comparison of quality of fit between pairs of scaling laws, we turn to statistical f-tests. An f-test assesses whether two scalings are statistically distinguishable by testing their misfits against the null hypothesis that they have equal variance to within 5% significance. That is, the ratio of the residual variances from the two scalings is compared with 95% confidence bounds from an f-distribution with the same degrees of freedom as these residual populations (Snedecor and Cochran, 1980). Applying f-tests to (14) (16) using the simulation data, we find that the quality of fit of the VAC scaling (16) is significantly better than that of the CIA scaling (14), which is significantly better than the MAC scaling (15). It can be observed in Fig. 5b that the MAC balance incorrectly scales the E-dependence of Re, as small-e dynamos tend to lie above the best-fit scaling, and higher E below. Fig. 5a reveals that the CIA scaling may not correctly capture the C dependence of Re, the data appearing to trend slightly steeper. To isolate the analysis Fig. 5. Comparisons of different theoretical flow speed scaling laws with dynamo simulation data. Symbol shapes depict E as in Fig. 1. Symbol sizes are proportional to Pr, with minimum and maximum of 0.1 and 30. Panel (a) shows Re plotted versus R f E 1=2 to test (14). The solid line shows Re ¼ 0:8C 2=5 E 1=5. Panel (b) shows Re plotted versus R fe to test (15). The solid line shows Re ¼ 2:4C 1=2 E 1=2. Panel (c) shows Re plotted versus R f E 2=3 to test (16). The solid line shows Re ¼ 0:5C 1=2 E 1=3. Panel (d) shows Re normalized by C 1=2 for Pr¼1 and three different Ekman numbers. The solid and dashed lines illustrate Re C 1=2 and Re C 2=5 scalings, respectively.

7 E.M. King, B.A. Buffett / Earth and Planetary Science Letters (2013) Fig. 6. Calculations of l from the dynamo simulations are plotted versus Ro to test (12). Symbol shapes depict E as in Fig. 1. Symbol sizes are proportional to non-dimensional magnetic field strength, Λ 1=2. The solid line shows l ¼ 2Ro 1=2 D. to C dependence, we separate data into subsets, each with a given E and Pr. Choosing Pr¼1, we have 111 data points for six different Ekman numbers in the range E Collectively, these subsets give a best fit flow speed scaling of Re C 0:47ð 7 0:04Þ. This fitting favors flow speed scaling laws of the form Re C 1=2, rather than the weaker C 2=5 dependence predicted by the CIA balance scaling. Three of these subsets are shown in Fig. 5d, where the solid and dashed lines illustrate the difference between scaling exponents of 1/2 and 2/5, respectively. We can further test the veracity of each theory (as they apply to the simulations) by testing the assumptions on which they are based, namely the scaling behavior of l. Fig. 2 clearly illustrates that the presence of magnetic fields does not lead to predominantly large scale convection in the models (cf. (15)), as assumed by the MAC balance scaling. Fig. 6 shows l plotted versus the Rossby number. The inertial theory scaling assumes a Rhines length scale (12), but we observe in Fig. 6 that the simulation data do not scale well with Ro; the Rhines scale gives a minimum standard error of 78% for l=d ¼ 2:0Ro 1=2. Although there does appear to be a correlated trend between l and Ro in Fig. 6, we observe that much of the variation in l can be attributed to varying E (indicated by symbol shape; see symbol key in Fig. 1), as proposed by the viscous scaling. An f-test reveals that the viscous scaling shown in Fig. 2 provides a statistically superior fit to the Rhines scale. The assumption that convection occurs on Rhines scales (12) is central to the inertial theory, but the simulation data are not well described by this scaling, which calls into question the applicability of the inertial scaling laws to the simulations. 6. Discussion The persistent influence of viscosity in these models allows application of simple scaling relations such as (9) and (10), but may simultaneously raise doubts about the extrapolation of such results to the geodynamo (and the dynamos of other planets). The viscous scaling law for the characteristic size of convection cells (9) would predict that the typical scale of convection in Earth's core is 100 m. Such small scale core convection is typically disfavored, since Pm 1 and so Ohmic dissipation is thought to dominate viscous dissipation, effectively eliminating the dynamical role of viscosity. If the typical scale of convection is as small as 100 m, however, viscous dissipation would still likely be a small relative contribution to the core power budget. We can estimate viscous dissipation as ϵ ν ρ 0 νu 2 V=l 2, where V 1: m 3 is the volume of the fluid core, and we take ρ kg m 3, ν 10 6 m 2 s 1, U ms 1, and l 100 m, such that ϵ ν W in the core. The core power budget, in contrast, is estimated to be ϵ ¼ Oð10 12 Þ W (e.g., Jones, 2011), and so even such very small convection cells cannot be ruled out by energy constraints. Even if we accept that core flow can exist at such small scales, the magnetic field must be maintained against diffusion. The simulations typically dissipate between 20% and 80% of convective power viscously, largely because Pm ¼ Oð1Þ by computational necessity, whereas Pm 10 6 in the core. In contrast, most estimates for Ohmic dissipation within the core are ϵ η ¼ Oð10 11 ÞW (e.g., Roberts and Glatzmaier, 2000). Even with the generous estimate for viscous dissipation given above, we find ϵ ν ϵ η in the core, which renders the viscous scaling for flow speeds (10) no more than a generous upper bound. We are then confronted with a difficult question: is it reasonable to expect that the Lorentz force remains secondary as one approaches the extreme planetary values of E and Pm that are currently unattainable numerically? The l E 1=3 scaling observed in Fig. 2 suggests that either convection occurs at very small scales in the core, or the relevant dynamical balance has not yet been reached in the models. Some cartesian dynamo models have, however, shown the emergence of large scale convection when E o10 6 (Stellmach and Hansen, 2004), which suggests that a magnetostrophic balance may be reached at slightly lower E than most geodynamo simulations typically achieve. There is also some evidence that Lorentz forces permit large scale flows in low E dynamo models when fixed flux, rather than fixed temperature, thermal boundary conditions are imposed (Sakuraba and Roberts, 2009; Takahashi and Shimizu, 2012), an effect which is not yet well understood. In contrast with E, little has been done to understand how geodynamo simulations behave as Pm-0. It seems likely to us that as E and Pm are reduced beyond the bounds of the present work, viscous dissipation will succumb to Ohmic dissipation such that (4) no longer holds, and numerical geodynamo models will transition to a new dynamical regime in which the influence of Lorentz forces is paramount and the role of viscosity is no longer important. Acknowledgments We are grateful to reviewers Julien Aubert and Chris Jones for their helpful suggestions. We thank Uli Christensen for kindly providing us with the simulation data. E.M.K. was supported by the Miller Institute for Basic Science Research. 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