Icarus 247 (2015) 248–259

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Icarus

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Iron snow dynamo models for Ganymede

http://dx.doi.org/10.1016/j.icarus.2014.10.0240019-1035/� 2014 Elsevier Inc. All rights reserved.

E-mail address: christensen@mps.mpg.de

Ulrich R. ChristensenMax Planck Institute for Solar System Research, Göttingen, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 August 2014Revised 18 September 2014Accepted 14 October 2014Available online 22 October 2014

Keywords:GanymedeMagnetic fieldsInteriors

Ganymede’s internal magnetic field is dominated by the axial dipole. The measurements by the Galileospacecraft only place an upper limit on the quadrupole moment. Ganymede’s magnetic field has the low-est ratio of quadrupole power to dipole power for all known planetary dynamos, not only at the planetarysurface but possibly also at the top of the dynamo region. The dynamo operates in a fluid iron core thatprobably contains a significant amount of sulfur. Crystallization of the core will then proceed from the topby formation of iron snow in a layer that develops a stable compositional gradient. Remelting of the snowat the bottom of this layer enriches the underlying fluid in iron and drives compositional convection. Herewe explore the consequences for the dynamo process of this scenario by numerical modeling. Convectionis driven by an imposed buoyancy flux at the top of a convecting core region that is surrounded by a con-ducting fluid shell with a strongly stabilizing density gradient. Only horizontal flow is allowed in theouter shell. It is shown that this is a valid approximation in the case where the stabilizing density contrastin the upper shell exceeds by far the unstable density contrast in the convecting region. We vary the basiccontrol parameters, concentrating on the regime where the magnetic field is dominantly dipolar. Com-pared to reference cases without an extra layer above the dynamo, we find that a stable fluid conductinglayer with a thickness of 100 km or larger reduces the ratio of quadrupole power R2 to dipole power R1 bya factor of at least four. With a stable outer layer R2=R1 is compatible with the Galileo observations for alltested dipolar models, whereas in the absence of such layer R2=R1 is too large or at best marginally com-patible. For plausible values of the buoyancy flux the models reproduce Ganymede’s observed dipolemoment. A stable layer that is comparable in thickness to the unstable region is found to promote a hemi-spherical type of dynamo whose field in incompatible with observations. This may indicate that the snowlayer in Ganymede’s core has a moderate depth extent.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Ganymede is the only satellite in the Solar System which pres-ently has a dynamo-generated internal magnetic field (Kivelsonet al., 1996). It is dominated by the dipole with a moment of order700 nT� r3

G (rG ¼ 2634 km is Ganymede’s radius) and a tilt of 4�relative to the rotation axis (Kivelson et al., 2002). In terms of addi-tional internal field components, the Galileo magnetometer dataare equally consistent with two different models (Kivelson et al.,2002). In one model the data are fitted by a combination of internaldipole and quadrupole components. The quadrupole contributionwas found to be small in comparison to the dipole and most of itis described by the Gauss coefficient g21. In the other model thereis no quadrupole, but a time-variable induced field is assumed inaddition to the dipolar dynamo field. The oscillation of Jupiter’sfield at Ganymede due to the tilt of Jupiter’s dipole axis and theplanet’s rapid rotation leads to a significant induced field, provided

the electrical conductivity inside Ganymede is high enough at afairly shallow depth, most plausibly because of the existence of asalty water ocean below an outer ice shell. Kivelson et al. (2002)found a complete tradeoff, in the fit to the limited flyby data,between an induction signal and the g21-term. Recently, indepen-dent evidence has been presented for the existence of a stronginduced field component from Hubble space telescope observa-tions of the time-variable location of auroral emissions onGanymede, which are believed to indicate the location of theboundary between field lines that close in Ganymede and thosethat connect Ganymede with Jupiter (Saur et al., 2012). Thereforeit is reasonable to consider the quadrupole moment obtained bythe inversion of the Galileo magnetometer data in the model with-out induced field as an upper bound for Ganymede’s actual quad-rupole moment.

The hypothesis that Ganymede’s dynamo might operate in asalty ocean has been rejected, because in order to reach a magneticReynolds number that would be sufficient for a dynamo, the flowvelocity in the ocean must be implausibly large on the order of

0 200 400 600 8001300

1350

1400

1450

1500

Tem

pera

ture

[K]

15.0 %

15.5 %

16.0 %

0 200 400 600 80015

15.5

16

Radius [km]

C [%

]

Fig. 1. Schematic illustration of the iron snow regime in Ganymede’s core. Toppanel: temperature vs. radius; thin full lines are adiabats, broken lines meltingtemperature for different sulfur concentrations, bold line is actual temperature.Bottom panel: sulfur concentration vs. radius. The iron snow region is shaded.

U.R. Christensen / Icarus 247 (2015) 248–259 249

1 m s�1 (Schubert et al., 1996). Therefore a metallic core is consid-ered as the only possible environment in which a dynamo couldwork in Ganymede. Three-layer models of Ganymede that satisfyits mean density and moment of inertia, with a metallic core, a sil-icate mantle and an ice layer, find core radii between virtually zeroand half of Ganymede’s radius if a wide range of densities isallowed for each layer (Sohl et al., 2002; Hauck et al., 2006). Fixingthe silicate and ice shell denities to plausible values and allowingfor core compositions ranging from pure iron to pure FeS, Sohlet al. (2002) find core radii between 1/4 and 1/3 of Ganymede’sradius. This may represent the most likely range, although othercore sizes are possible.

The relative contribution of quadrupole components to themagnetic field can be quantified by the Mauersberger–Lowespower of the quadrupole, R2, to that of the dipole (R1), where

Rn ¼ ðnþ 1ÞXn

m¼0

g2nm þ h2

nm

� �; ð1Þ

with n spherical harmonic degree, m spherical harmonic order, andg and h the Gauss coefficients. Using the quadrupole coefficients inthe model by Kivelson et al. (2002) without induced field compo-nent (their Table III) we find R2=R1 ¼ 0:0025 at the surface of Gan-ymede. For a potential field of internal origin, the field strength atdegree n varies with radius r proportional to r�ðnþ2Þ, hence the ratioof dipole power to quadrupole power changes as R2=R1 / r�2. Itsvalue at the surface of the core, denoted here by R02=R01, is moremeaningful to characterize the dynamo. Using rc ¼ 0:25 rG for theradius of Ganymede’s core, the quadrupole to dipole power R02=R01is only 0.04. To put this into perspective, the value at the top ofEarth’s core in 2010 was 0.14. Making reasonable assumptionsabout the ratio between the radius of the conducting core to theouter radius of a planet, rc=rp, we find similar or higher values formost other planets with an active dynamo: 0.33 for Mercury withrc=rp ¼ 0:82 (Anderson et al., 2011), 0.10 for Jupiter withrc=rp ¼ 0:85 and 1.6 and 2.7 for Uranus and Neptune, respectively,with rc=rp ¼ 0:75 (Russell and Dougherty, 2010). Only Saturn withrc=rp ¼ 0:6 may have a lower R02=R01 ¼ 0:023 (Cao et al., 2011). How-ever, the value for Ganymede is probably smaller than 0.04. For acore with rc ¼ 0:30 rG the ratio R02=R01 would drop to 0.028 and inview of the very likely presence of an induced magnetic field com-ponent all these numbers are upper bounds. We can conclude thatGanymede has an anomalously low quadrupole moment relative toits dipole moment. It may fortuitously hold only at the presentepoch. However, this would require that all five independent quad-rupole coefficients are simultaneously small by chance, whichseems improbable. The main purpose of this work is therefore tostudy if properties of Ganymede’s dynamo that may distinguish itfrom other planetary dynamos can be the cause for the weak quad-rupole moment.

Putting mechanical forcing, e.g. by tidal or librational effects(Wicht and Tilgner, 2010), aside, flow in a metallic planetary corecan be driven by thermal convection or by compositional convec-tion. A condition for thermal convection to occur is that the heatflow out of the core is larger than the heat transported by conduc-tion along an adiabatic temperature gradient. Taking as lowerbound for the adiabatic gradient ð@T=@PÞS P 1:6� 10�8 K=Pa foran iron–sulfur alloy at a pressure of 6 GPa at the top of Ganymede’score (Williams, 2009) together with a thermal conductivityk P 30 W=ðm2 K) that is calculated from the electrical conductivity>106 S=m (Deng et al., 2013) using the Wiedemann–Franz law, adensity of q = 6500 kg/m3 and gravity g = 1.3 m/s2, the adiabaticconductive flux is >4 mW/m2. Thermal evolution models for Gany-mede predict a present core heat flow in the range 2–4 mW/m2

(Hauck et al., 2006), hence pure thermal convection is not verylikely to drive a dynamo in Ganymede. Kimura et al. (2009)

concluded that thermal convection may be possible if Ganymede’score is rather big and sulfur-rich, however, their assumed value forthe adiabatic conductive heat flux is very low. Compositional con-vection occurs when solidification is progressing in a core that con-tains alloying elements aside from iron, with sulfur being thefavorite candidate in small planetary bodies. In the Earth’s coresolidification starts from the center because the melting point gra-dient is steeper than the adiabatic temperature gradient which isestablished in a convecting fluid. However, this cannot necessarilybe considered as the prototype situation for smaller planets. Rathera variety of other scenarios are conceivable, e.g. crystallisation ofiron at the top of the core or at some intermediate depth (Haucket al., 2006). Williams (2009) pointed out that at the low pressuresin the cores of small bodies the adiabatic temperature gradientcould be steeper than the melting point gradient for a nearly pureiron composition. The addition of sulfur not only reduces the melt-ing temperature TM , but also leads to shallower pressure gradientsdTM=dP, particularly in the 6–10 GPa pressure range that is rele-vant for Ganymede’s core (Chen et al., 2008; Buono and Walker,2011). For compositions more sulfur-rich than �15 wt% sulfur,dTM=dP even becomes negative. Sulfur is a volatile element andis probably more abundant in the outer Solar System where lowercondensation temperatures have been reached in the protosolarnebula than in the region of the terrestrial planets. Hence Gany-mede’s core plausibly contains a significant amount of sulfur. Thesmall or negative dTM=dP implies that crystallization will start atthe core–mantle boundary and proceed downward as the corecools.

Fig. 1 illustrates the top-down iron snow scenario for Ganymede,assuming arbitrarily an initial sulfur concentration of�15.5% in thecore. In the top layer iron snow is formed and sinks gravitationally.The temperature gradient in this layer is set by the heat flux at thecore–mantle boundary, which is controlled by the heat transportcapability of the overlying shells and the thermal history. The tem-perature gradient at the top of the core is expected to be lower thanthe adiabatic gradient. As the core cools, the top layer becomesgradually more enriched in sulfur and the local sulfur concentrationis regulated in such a way that the temperature is everywhere at themelting point. A stable gradient in concentration is set up in the

250 U.R. Christensen / Icarus 247 (2015) 248–259

snow layer. At the bottom of the snow layer, or slightly below it, thesnow will melt, since here the temperature is above the local melt-ing temperature. This enriches the deep fluid in iron and increasesits density. Thus in the central region convection is driven by a com-positional buoyancy flux from above. If there is convective overturnin this region the temperature must follow an adiabat. If a substan-tially subadiabatic conductive thermal gradient existed in the corebefore iron precipitation started, it is not entirely clear how theregion below the snow layer evolves towards an adiabatic state,but for the purpose of the modeling studies we will assume thatis has been established. For a subadiabatic total heat flux convec-tion must then advect heat downward to compensate for theenhanced outward conductive heat transport. This implies a nega-tive thermal buoyancy flux. Rückriemen et al. (2014) present adetailed study of this scenario in evolutionary models. They findthat for a range of plausible assumptions the net buoyancy flux(mass anomaly flux) is positive, i.e. compositional buoyancy is lar-ger than the negative thermal buoyancy, and is of the order of104 kg=s. The snow layer grows with time and will eventuallyextend all the way to the center, convection will cease and the ironsnow starts to accumulate in a solid inner core. The time windowfor convection below a snow layer (without an inner core) is rela-tively short, from a hundred million years (Hauck et al., 2006) toat most a billion years (Rückriemen et al., 2014). Therefore Gany-mede’s dynamo should be a relative recent phenomenon. Theseauthors also used scaling laws by Christensen and Aubert (2006)and Aubert et al. (2009) to determine that the magnetic Reynoldsnumber in the convecting region would typically be in the rangeof several hundreds (i.e. sufficient for a dynamo) and that for plau-sible values of the buoyancy flux the dipole moment is on the orderof the observed value.

Our ability to explore planetary dynamos with convection-dri-ven magnetohydrodynamic models and the understanding of thelimitations of this approach has reached some degree of maturity(e.g. Jones, 2011). Several numerical dynamo models that involvestably stratified conducting layers have been published. They donot usually distinguish between thermal and compositional buoy-ancy and rather describe their combined buoyancy effects by a sin-gle variable called codensity, which is valid if the ‘turbulent’diffusivities for heat and composition can be assumed to be thesame (Braginsky and Roberts, 1995; Kutzner and Christensen,2004). In models for Mercury’s dynamo by Christensen (2006)and Christensen and Wicht (2008) convection in the deep partsof the fluid core was driven by latent heat release and a light ele-ment flux from a growing inner core. The upper part was stablystratified due to a subadiabatic temperature gradient. In agreementwith Mercury’s slow rotation, parameters were chosen such thatthe Coriolis force was not large compared to inertial forces. In thiscase the magnetic field generated inside the dynamo is not domi-nated by the axial dipole, but is rather multipolar. The modelsshowed, however, that the rapidly fluctuating multipole compo-nents were strongly attenuated in the stably stratified conductinglayer by a skin effect, whereas the weak but more slowly varyingaxial dipole and quadrupole components are much less attenuatedand dominate the magnetic field outside the core.

In this scenario for Mercury the upper part of the core would bethermally stable, but if there is a non-negligible light element fluxfrom the core, compositionally unstable. Manglik et al. (2010) stud-ied this in dynamo models that allow for double-diffusive convec-tion. Even though the net stratification in the top layer wasstabilizing, double-diffusive convection was associated with signif-icant radial overturning in this layer that reduced the filtering bythe skin effect described above. In the scenario considered herefor Ganymede double-diffusive convection in the snow layer isnot an issue, because both the compositional and the thermalstratification are stable.

The only work that explicitly attempted to model iron snowlayers, in the context of Mercury’s dynamo, is by Vilim et al.(2010). They accounted for two snow layers, one below thecore–mantle boundary and a second one at mid-depth by impos-ing inverse codensity gradients in the respective depth ranges.The double snow layer scenario is possible because of kinks inthe melting curve vs. pressure above 10 GPa for relativelysulfur-rich alloys (Chen et al., 2008). This is not relevant forGanymede, where the central pressure is less than 11 GPa (Sohlet al., 2002).

Furthermore, several authors (Sreenivasan and Gubbins, 2008;Stanley and Mohammadi, 2008; Stanley, 2010; Nakagawa, 2011)studied models with thin stable layers above the dynamo region,applied to the Earth’s core or to Saturn. They often found limitedeffects on the magnetic field structure. A problem with the previ-ous implementations of stably stratified layers is that the stabiliz-ing density gradients were only moderately larger than theunstable gradients in the convecting region, typically by a factorof a few. This allows for a significant overshoot of convection intothe stable region. In the snow layer scenario the stabilizing gradi-ent is expected to be many orders of magnitude larger than theunstable density gradient. For example, for the conditions illus-trated in Fig. 1 the relative density contrast Dq across the stablelayer is approximately 10 kg/m3, whereas the density contraststhat drive the flow in planetary cores are expected to be of order10�4 kg/m3 (Aurnou et al., 2003; Christensen and Aubert, 2006).Hence it is expected that radial motion (aside from high-frequencygravity waves) is strongly suppressed in the snow layer and verylittle convective overshoot occurs. Obviously the sinking iron snowand the compensating upward flow (by mass conservation) repre-sent radial motion. However, even for a radial mass flux as high as106 kg=s the mean radial flow velocity associated with it (not thatof an individual snow flake) is less than 10�10 m=s, which is negli-gible compared to the expected typical flow velocity inside thedynamo region of >10�4 m=s (Hauck et al., 2006).

The present models treat the snow region as a separate layer inwhich only horizontal (toroidal) flow is possible. It is coupled tothe convection in the dynamo region beneath it by viscous andmagnetic forces. The technical implementation is described in Sec-tion 2. Model results obtained for a range of basic control parame-ters and different thicknesses of the snow layer are presented inSection 3. In Section 4 the implications of the model results forGanymede’s dynamo are discussed.

2. Model description

2.1. Basic setup

Our model consists of two concentric rotating and electricallyconducting spherical shells. The dynamo operates in the inner con-vecting shell with outer radius ro and inner radius ri. The aspectratio ri=ro is set to 0.2. A small inner core is kept only for technicalreasons as the code does not allow full-sphere simulations. To keepits influence small, it is electrically insulating with a stress-freeboundary and zero buoyancy flux at its surface.

Convection in the main shell is driven by a uniform codensityflux qC imposed at ro. It represents the mass flux of melting ironsnow at the bottom of the snow-forming outer layer that enrichesthe deep fluid in iron. For simplicity we ignore the negative ther-mal buoyancy flux, which must be subtracted. The flux is bal-anced by a volumetric sink � that represents remixing of theiron-enriched fluid and rehomogenization. In an overlying con-ducting fluid shell between the radii ro and ro þ do only horizontalflow is possible. It is coupled viscously and magnetically to theconvection below.

U.R. Christensen / Icarus 247 (2015) 248–259 251

2.2. Equations and parameters

We scale the Boussinesq equations of fluid flow and magneticfield generation by using the thickness D ¼ ro � ri of the main shellas the length scale, and D2=m with m the viscosity as the time scale.The scale for the magnetic field is

B� ¼ ðqlokXÞ1=2 ð2Þ

with q the density, lo the magnetic permeability, k the magneticdiffusivity and X the rotation rate. The sulfur concentration C rela-tive to its mean in the main shell is scaled by qCmD=ðqj2Þ, where j isthe compositional diffusivity.

In the main shell the flow velocity u, magnetic field B andconcentration C obey the following equations in terms ofnon-dimensional variables:

E@u@tþ u � ru

� �þ 2z� uþrP

¼ Er2uþ Rarro

C þ 1Pmðr � BÞ � B; ð3Þ

@B@t�r� ðu� BÞ ¼ 1

Pmr2B; ð4Þ

@C@tþ u � rC ¼ 1

Prr2C þ �; ð5Þ

r � u ¼ 0; r � B ¼ 0: ð6Þ

The unit vector z indicates the direction of the rotation axis. Weassume that gravity is a linear function of radius, gðrÞ ¼ gor=ro,where go is the value on the outer boundary of the convectingregion. P is the non-hydrostatic pressure. The four non-dimensionalcontrol parameters are the Ekman number

E ¼ mXD2 ; ð7Þ

the Rayleigh number

Ra ¼ ngoqCD4

qj2m; ð8Þ

the Prandtl number

Pr ¼ mj; ð9Þ

and the magnetic Prandtl number

Pm ¼ mk: ð10Þ

Here n relates the compositional flux to the buoyancy flux qB (con-vected mass anomaly flux per unit area on the outer boundary)

qB ¼Nu� 1

NunqC ; ð11Þ

where the Nusselt number Nu is the ratio of total compositional fluxto the diffusive flux. In a planet Nu� 1, but the moderate values ofNu in dynamo models warrant the correction factor depending onNu in Eq. (11).

Because of the small compositional diffusivity, the Prandtl num-ber based on molecular values is much larger than one, but turbu-lent mixing processes will shift the effective value towards unity(Kays and Crawford, 1993). Here we keep the Prandtl number fixedat one. The Ekman number is varied between 3� 10�4 and3� 10�5, the magnetic Prandtl number between one and five andthe Rayleigh number between 2.6 and 100 times the critical valuefor the onset of convection.

The fluid in the outer shell has (in the framework of the Bous-sinesq approximation) the same density and magnetic diffusivityand usually the same viscosity as in the main shell. For test pur-poses (see Section 3.1) the viscosity can also be fixed to differentvalues in the two regions. The fluid motion is governed by theNavier–Stokes Eq. (3) with the restriction that ur ¼ 0. On the hori-zontal flow inertial, Coriolis, viscous, pressure gradient and Lorentzforces act, whereas buoyancy does not play a role (other than sup-pressing ur). Eq. (5) does not need to be solved in the outer shell.The magnetic field here is governed by Eq. (4) for the special caseur ¼ 0.

At the interface r ¼ ro the horizontal velocity and the shearstress are continuous. The same holds for the magnetic field andthe horizontal electrical field. A homogeneous compositional gra-dient is imposed at ro. On the external boundary at rc ¼ ro þ do ano-slip condition is imposed on the velocity (u ¼ 0) and the mag-netic field is matched with a source-free potential field at r > rc .

2.3. Numerical technique and modeling strategy

Technically, the equations are solved by the spectral transformmethod described in Christensen and Wicht (2007). The velocityand magnetic field are expressed by poloidal and toroidal scalarpotentials, which are expanded in spherical harmonic functionsin the angular directions and in Chebychev polynomials in theradial direction. Separate sets of Chebychev polynomials are usedfor the main shell and the outer shell. The associated radial gridsare refined towards the edges of each layer. In the outer shell theflow is purely toroidal and only a toroidal flow equation must besolved. Depending on parameter values, a maximum spherical har-monic degree between 53 and 106 and a Chebychev expansionbetween 41 and 73 in the main shell and between 17 and 37 inthe outer shell has been used.

The non-dimensional outer shell thickness do has been variedbetween 0.1 and 0.8. This corresponds to 7.4–39% of the coreradius, or, for a nominal core radius of 770 km, to absolute valuesbetween 57 km and 300 km. For each combination of controlparameters the case without a stable layer (do ¼ 0) has also beenrun. This serves as reference for identifying the effect of the snowlayer and would represent a thermally driven dynamo with a sup-eradiabatic heat flow in Ganymede’s core. As a rule simulationshave been started with a strong and dipole-dominated magneticfield, usually from a simulation at different parameter values. Theyare run for at least 330 advection times (one advection time isD=urms, with urms the mean velocity in the main shell). Several prop-erties of interest are averaged over the simulation time, excludingat least the 30 first advection times or one magnetic diffusion timePmd2

o through the outer shell, whichever is longer, to avoid possi-ble transients. In a few cases, when a transition between distinctdynamo states occurred at a later time, the averaging was startedafter the transition was completed.

2.4. Validity of approach

Suppressing radial flow entirely in the stable outer layer andimposing a sharp boundary between the convecting and stableregions represents an end-member model. The transition betweenconvecting and stable layers has been studied in the context of solarconvection. Here convection penetrates to a significant degree fromthe outer convection zone into the deeper and stably stratified radi-ative zone. Based on two-dimensional convection simulationsRogers and Glatzmaier (2005) suggested that the penetration dis-tance depends only weakly on the degree of stability and that animpenetrable boundary is dynamically different from the transitioninto a stratified region irrespective of the degree of stability. Thesesimulations differed in some respects (compressible convection, no

101 102 10310−2

10−1

[dC/dr]stab

d pen

Fig. 3. Penetration depth of overshooting convection into stable layer with finitestabilizing codensity gradient. The straight line has a slope of �0.5.

252 U.R. Christensen / Icarus 247 (2015) 248–259

rotation) from conditions relevant for Ganymede. Here I study for anon-magnetic case in a rotating sphere the penetration of convec-tion into an overlying region with different degrees of (finite) stabil-ity and compare it to the model without poloidal flow in the stableregion.

The control parameters, based on thickness D of the unstablelayer, are Ra ¼ 1:5� 106 and E ¼ 3� 10�4. The stable outer layerhas a non-dimensional thickness of do ¼ 0:4 and the codensity gra-dient here is varied from ½dC=dr�stab ¼ 10 to 640 (at the top of theconvecting region the nondimensional unstable gradient isdC=dr ¼ �1). In Fig. 2 the poloidal and toroidal velocities, averagedhorizontally and in time, are plotted as a function of radius for thecase with the highest tested stabilizing codensity gradient of 640and compared with the simulation for a separate outer layer with-out radial flow. The pattern are similar, but remaining differencesare on the order of 10%.

For technical reasons the change from unstable to stable gradi-ent had to be smoothed in the simulation (using a tanh-function tochange over from one gradient to the other) and occurs over adepth range of order 0.016. To facilitate a rather narrow transitioninterval, the location of the radial grid points has been condensedin the transition region by using a nonlinear mapping functionfrom the Chebychev definition space to the radius as described inTilgner (1999). Nonetheless, because of the finite width of the tran-sition region it is not straightforward to determine the scaling forthe penetration distance with the magnitude of the stabilizing den-sity gradient. The following procedure has been adopted, based onthe averaged velocities and codensities as functions of radius. Thepenetration radius of convection is defined as the point where thepoloidal velocity has dropped to one tenth of its mean value in theunstable region. We then take the difference between the codensi-ty at this radius and the codensity minimum at the boundarybetween the two layers, DC, and define as the effective penetrationdistance dpen ¼ DC=½dC=dr�stab. The actual penetration distance inthe simulations is slightly larger because of the finite transitioninterval, however, the effective penetration distance is meaningfulfor an assumed discontinuous change in the codensity gradient.The results (Fig. 3) show that for sufficiently large stabilizing gra-dient the penetration distance varies as

dpen ¼ 0:32ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½dC=dr�stab

q�: ð12Þ

This form of scaling is also suggested by a simple model where par-cels of rising fluid enter into the stable region with a fixed kineticenergy and come to rest when their gain in potential gravitationalenergy in the low-density environment is equal to the original

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

(r − ri)/(D+do)

Urm

s

Fig. 2. Mean velocity as function of radius, comparing non-magnetic convectionwith finite stability in the top layer (thin lines) with the case of zero poloidal flow inthe outer layer (thick lines). Full lines are for the poloidal flow component andbroken lines for the toroidal flow component.

kinetic energy. Applied to Ganymede, where the stabilizing densitygradient relative to the unstable gradient is expected to be of order105, a convective overshoot distance of order 3� 10�3 or �2.5 km ispredicted. This is not entirely negligible compared to possible thick-nesses of the snow layer, but ignoring it has presumably a smalleffect.

3. Model results

In Table 1 we list the dynamo model parameters and some char-acteristic model results based on time-average and, where applica-ble, spatially averaged values. These are in particular the magneticReynolds number as a measure of the mean flow velocity,Rm ¼ urmsD=k, and the mean magnetic field strength Brms, whereaverages are taken only in the convecting shell in both cases. TheNusselt number is calculated as the conductive difference ofcodensity between ri and ro, divided by the actual difference inthe mean values on the two boundaries. On the outer boundaryof the model at rc ¼ ro þ do the mean dipole field strength Bdip,the fraction of power in the zonal field components to the totalpower in the field Rz=R, the dipole tilt, and the ratio of quadrupolepower to dipole power R2=R1, are recorded (in this section R with-out the prime refers to the power on the outer boundary of themodel).

In the majority of the model runs the dipolar solution (imposedby the initial condition) turned out to be stable, i.e., persisted dur-ing the entire simulation time. The dipole is tilted on average onlyby a few degrees and does not reverse during the model run. Thesecases are marked as type ’’D’’ in Table 1. However, in several caseswith strong convective driving or a thick outer shell the dipolarmagnetic field broke down and was replaced by either a multipolarfield (type M) or a hemispherical field (type H). In some casesmixed types that vacillate between different states have beenfound. Because of their relevance for Ganymede we concentratehere on the dipolar solutions. Their breakdown and the propertiesof the non-dipolar cases are described in Section 3.3.

3.1. Dipolar solutions: flow structure and internal magnetic field

It is well established that in rapidly rotating dynamo simula-tions the flow is organized in helical convection columns alignedwith the rotation axis. This is explained by an approximate validityof the Proudman–Taylor theorem which excludes velocity changesin the z-direction when the force balance is between Coriolis andpressure gradient forces. Fig. 4 shows for a rather weakly drivencase the z-vorticity on a cut along the columns. At higher Rayleighnumber the structures become less organized and discontinuous,but maintain a strong anisotropy with elongation in the z-direc-tion. The columnar flow penetrates into the stable layer only to a

Table 1Model parameters and results.

Case E Pm Ra do Type Rm Nu Brms Bdip Rz=R Tilt (�) R2=R1

1.0 3� 10�4 5 1:5� 106 0.0 D 131 1.86 2.13 0.461 0.46 2.5 0.077

1.2 3� 10�4 5 1:5� 106 0.2 D 130 1.87 2.95 0.271 0.93 2.6 0.0144

1.4 3� 10�4 5 1:5� 106 0.4 D 133 1.91 3.03 0.192 .993 1.7 0.0047

2.0 3� 10�4 5 3:0� 106 0.0 D 206 2.65 2.93 0.448 0.26 7.1 0.186

2.2 3� 10�4 5 3:0� 106 0.2 D 200 2.71 3.87 0.369 0.91 3.5 0.0181

3.0 3� 10�4 4 6:0� 106 0.0 DMH 279 3.79 1.87 0.169 0.28 19 0.276

3.2 3� 10�4 4 6:0� 106 0.2 D 232 3.90 4.10 0.407 0.93 3.2 0.0141

3.4 3� 10�4 4 6:0� 106 0.4 H 277 3.36 2.81 0.071 0.94 5.6 1.165

4.0 3� 10�4 4 9:0� 106 0.0 DMH 316 5.22 3.36 0.416 0.30 11 0.299

4.2 3� 10�4 4 9:0� 106 0.2 H 350 4.19 2.96 0.127 0.85 8.7 0.814

5.0 10�4 3 5:0� 106 0.0 D 93 1.61 1.43 0.461 0.76 0.1 0.029

5.8 10�4 3 5:0� 106 0.8 D 99 1.95 2.20 0.098 .999 0.5 0.0021

6.0 10�4 3 6:0� 106 0.0 D 108 1.83 1.77 0.523 0.69 1.0 0.030

6.8 10�4 3 6:0� 106 0.8 H 109 1.69 1.55 0.054 .996 1.6 0.298

7.0 10�4 3 1:0� 107 0.0 D 144 2.15 2.57 0.639 0.42 4.1 0.070

7.2 10�4 3 1:0� 107 0.2 D 139 2.78 3.20 0.379 0.93 3.1 0.0119

7.4 10�4 3 1:0� 107 0.4 D 141 2.87 3.25 0.263 .987 1.8 0.0031

7.8 10�4 3 1:0� 107 0.8 H 150 1.92 2.73 0.050 .998 1.2 1.028

8.0 10�4 1 2:5� 107 0.0 D 91 3.97 1.54 0.423 0.26 6.8 0.038

8.2 10�4 1 2:5� 107 0.2 D 83 4.31 3.20 0.379 0.84 2.4 0.0082

9.0 10�4 3 2:5� 107 0.0 D 248 4.31 3.30 0.612 0.26 6.8 0.106

9.1 10�4 3 2:5� 107 0.1 D 232 4.47 3.84 0.435 0.80 3.9 0.0233

9.2 10�4 3 2:5� 107 0.2 D 231 4.60 3.95 0.435 0.93 3.2 0.0117

9.2a 10�4 3 2:5� 107 0.2 H 334 3.20 1.34 0.031 0.74 19 1.611

9.4 10�4 3 2:5� 107 0.4 D 232 4.64 3.96 0.295 .985 2.1 0.0033

9.8 10�4 3 2:5� 107 0.8 H 279 3.77 4.06 0.993 1.42 1.9 1.415

10.0 10�4 3 5:0� 107 0.0 D 372 6.36 3.39 0.545 0.19 6.3 0.109

10.2 10�4 3 5:0� 107 0.2 D 340 6.35 4.35 0.439 0.93 3.9 0.0143

10.4 10�4 3 5:0� 107 0.4 H 402 6.15 4.60 0.074 0.967 7.2 2.402

11.0 10�4 3 9:0� 107 0.0 MH 601 9.27 2.39 0.114 0.08 21 1.01

11.2 10�4 3 9:0� 107 0.2 H 598 7.44 2.85 0.090 0.90 11 2.15

12.0 3� 10�5 3 1:0� 108 0.0 D 296 4.34 3.98 0.783 0.28 6.2 0.074

12.2 3� 10�5 3 1:0� 108 0.2 D 283 5.00 4.11 0.452 0.95 2.7 0.0093

13.0 3� 10�5 1 2:5� 108 0.0 D 177 7.03 2.29 0.578 0.43 4.4 0.039

13.2 3� 10�5 1 2:5� 108 0.2 D 161 7.70 2.59 0.379 0.95 2.9 0.0089

14.0 3� 10�5 3 2:5� 108 0.0 D 500 7.22 4.33 0.788 0.20 6.1 0.053

14.2 3� 10�5 3 2:5� 108 0.2 D 474 7.84 4.86 0.443 0.94 4.5 0.0125

14.4 3� 10�5 3 2:5� 108 0.4 D 476 7.85 4.90 0.294 .983 2.7 0.0036

15.0 3� 10�5 1 5:0� 108 0.0 D 262 10.3 2.34 0.730 0.33 4.5 0.055

15.2 3� 10�5 1 5:0� 108 0.2 D 232 10.7 2.96 0.405 0.96 2.8 0.0066

16.0 3� 10�5 1 9:0� 108 0.0 MH 518 13.2 0.75 0.021 0.09 20 2.088

16.2 3� 10�5 1 9:0� 108 0.2 D 316 13.9 3.30 0.430 .963 3.0 0.0064

Solution types: D = dipolar, M = multipolar, H = hemispherical; combinations indicate vacillation between types. Critical Rayleigh numbers for convection onset without astable layer are 4:74� 105, 1:94� 106 and 9:22� 106 for E ¼ 3� 10�4; 10�4 and 3� 10�5, respectively. The mean dipole tilt has been determined via time-averaging the sineof the instantaneous tilt angle (using the sine characterizes the mean tilt relative to either rotation pole independent of possible polarity reversals).

U.R. Christensen / Icarus 247 (2015) 248–259 253

limited degree. It should be noted that some weak flow in z-direc-tion is required for the Proudman–Taylor theorem to apply approx-imately under incompatible boundary conditions, such as slopingboundaries at the ends of the columns. While columnar flow canextend through regions with moderately stable density stratifica-tion, Zhang and Schubert (2000) estimated that it is preventedfrom doing so when the ratio S between stable and unstable den-sity gradients becomes larger than OðE�1=3Þ. The Ekman numberin Ganymede’s core is of order 10�12 and the stability parameterin Ganymede’s snow layer, S � 105, exceeds the critical value of104. Hence the assumption of an essentially infinite S in the simu-lations is compatible with the flow dynamics in Ganymede’s snowlayer.

In Fig. 5 the time-average axisymmetric flow and magnetic fieldare compared for a model with an outer layer and the correspond-ing case without additional layer (cases 14.4 and 14.0, highlightedin bold print in Table 1). The zonal flow in the convecting shell,

which contributes 6% to the total kinetic energy here, is not muchaffected by the presence of the outer layer. In the stable shell, thezonal flow is westward at low latitudes and eastward at high lati-tudes. It has a similar amplitude as in the convection zone and con-tributes 34% to the kinetic energy in the outer shell. Here, the zonalflow is governed by the balance between Lorentz forces and vis-cous forces, as shown in Fig. 6. Given this balance one may ques-tion if the model provides an appropriate description of the flowin a stratified part of a planetary core, were viscous forces arethought to be many orders of magnitude smaller than they are indynamo models. However, it is obvious from Fig. 6 that both forcesbecome strong only in boundary layers. In the bulk of the stablelayer they are tiny, at least on time-average. This contrasts withthe situation in the main shell, where the time-average zonal Lor-entz force is found to be of order one, more than an order of mag-nitude larger than in the interior of the stable layer. In the stablelayer the magnetic field and the flow seem to configure themselves

Fig. 4. Snapshot of z-component of vorticity on a cut along the rotation axis for case1.4. Contour step is 75.

a b

dc

Fig. 5. Time-average zonal velocity (a, c) and axisymmetric magnetic field (b, d).Color countour step for velocity is 5 and for toroidal field is 0.5. Case 14.0 in (a, b)and case 14.4 in (c, d). (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

0 0.1 0.2 0.3 0.4−1

0

1

2

3

4

5

r−ro

U φ/1

0 φF

Fig. 6. Time-average zonal velocity (full line, scaled down by a factor 10), zonalLorentz force (dots) and viscous force (broken line) vs. radius at 30� colatitude inthe outer layer of case 14.4.

254 U.R. Christensen / Icarus 247 (2015) 248–259

in such a way that the time-average Lorentz force almost vanishes.This could work in the same way in the inviscid limit. In order totest this assumption, the viscosity in the stable layer of case 14.4has been reduced by a factor of 1/2 and the model has been runfor an additional 50 advection times. The mean velocity of the flowin the outer shell increased by 9% and the zonal part of the flow by6%. These modest changes suggest that the viscosity in the stablelayer has only a weak influence on the flow in the models.

The axisymmetric toroidal magnetic field near the outer bound-ary at low latitudes is intensified and diffuses across the boundaryinto the stable layer (Fig. 5b,d). This field is generated through ana-effect (cf. Olson et al., 1999). An analysis of the case in Fig. 5c,d shows that the X-effect has a destructive (but small) effect onthese field bundles. The rather sharp bending of poloidal field linesat the interface between the two layer (Fig. 5d) hints at the pres-ence of strong sheet currents, which result in the strong localizedLorentz forces in the boundary layers seen in Fig. 6.

The rms field strength in the dynamo region is moderately lar-ger for the model with a stable shell, i.e., the dynamo seems towork more efficiently. This is generally the case for all models witha dipolar field geometry (see Table 1). The mean flow velocity (ormagnetic Reynolds number) is slightly reduced in most modelswith an outer shell, which may be a consequence of larger Lorentzforces associated with the stronger field that quench the flow morestrongly.

3.2. Dipolar solutions: external magnetic field

In Fig. 7 snapshots of the radial magnetic fields above thedynamo region are compared for the two models in Fig. 5. Forthe model with a stable layer (Fig. 7b) Br is plotted at the outerboundary of that layer (radius rc). For the reference model withoutadditional layer (Fig. 7a) the field has been upward continued as apotential field to the equivalent level (ro þ 0:4) for better compar-ison. At the outer boundary of this model, r ¼ ro, the field showsmuch more small-scale structure. Without a stable layer the basi-cally dipolar field is modulated at rather large scale and has at this

instant in time a significant quadrupolar contribution of sphericalharmonic order m ¼ 1, which results in weaker flux at high lati-tudes in the western hemisphere than in the eastern hemisphere.In contrast, with the stable layer the magnetic field is much morezonal with little large-scale modulation. The much higher degree ofzonality on the outer boundary of the model in the presence of astable layer is a general trait, even when the dynamo field becomes

a

b

Fig. 7. Snapshot of the radial magnetic field. (a) Case 14.0, do ¼ 0, field upwardcontinued to r ¼ ro þ 0:4, (b) Case 14.4, do ¼ 0:4, field shown at rc ¼ ro þ do. Contourstep is 0.1.

U.R. Christensen / Icarus 247 (2015) 248–259 255

non-dipolar. The fraction of power in the zonal part of the field, Rz,of the total power in the field on the outer boundary, R ¼

PRn, is

larger than 90% for all dipolar dynamos with a stable shell thick-ness do P 0:2 and larger than 98% for do P 0:4. In comparison, inthe dipolar cases without a stable shell the fraction of zonal poweris typically in the range 20–50% (cf. Table 1). The time-average tiltof the dipole axis with respect to the rotation axis is also reducedby the presence of the stable shell, however, the reduction in tiltangle is moderate, typically by a factor of 1.4–2.0.

Fig. 8 compares the time-average spatial power spectra at thesurfaces of the two models of Fig. 7, normalized by the respectivepower in the dipole. Without stable layer the spectrum is similar tothat of the geomagnetic field at the core–mantle boundary. Thepresence of the stable layer reduces the power of all higher multi-poles relative to the dipole by more than an order of magnitude.Some part of the attenuation can simply be associated with the

1 2 3 4 5 6 7 8

10−3

10−2

10−1

100

Harmonic degree n

Rn

/ R1

Fig. 8. Time-average magnetic power spectra normalized by the dipole power.Circles are for case 14.0 without a stable layer and triangles for case 14.4 with astable layer thickness do ¼ 0:4. Filled symbols are at the top of the conductingregion. Open circles show the spectrum of the model without a stable layer, upwardcontinued over a non-dimensional distance of 0.4. For comparison the geomagneticspectrum at the core mantle boundary averaged from 1840 to 1990 (Jackson et al.,2000) is also shown by small crosses.

geometric decrease of the field components from the top of theactive dynamo region to the top of the stable layer. For a potentialfield the attenuation factor relative to the dipole would be½ro=ðro þ doÞ�ð2n�2Þ. For a comparison, the spectrum of the modelwithout a stable layer, upward continued to a radius of ro þ 0:4,is also shown. The geometric effect explains the reduction of therelative quadrupole power in the model with a stable layer onlyto a small degree. In addition there is a further reduction by anorder of magnitude. For multipoles with n P 5 the geometric effectis the dominant one with an additional reduction by a factor in therange 3–4.

As discussed in Christensen (2006) and Christensen and Wicht(2008), the drop in power of the multipoles relative to that in thedipole is caused by the skin effect. All multipole components fluc-tuate in time with a zero mean value, hence they are damped bya skin effect as they penetrate through the stable layer. In con-trast, the axial dipole coefficient g10ðtÞ has a strong DC compo-nent (for dipole-dominated dynamos) that is unaffected by theskin effect. In Fig. 9 time series of the poloidal field coefficientg21 taken at ro and at rc are shown for the model with a stableshell. On top of the stable layer the rapid fluctuations are almostentirely filtered out and the slower fluctuations with time scales100–200 yr are severely attenuated. Also, there is a phase lag as isexpected for the skin effect. In addition to the skin effect,which would also act in a completely stagnant layer, the differen-tial rotation in the stable layer (Fig. 5c) shears the nonzonalquadrupole field apart and transfers its energy to higher multi-poles. The rms-value of g21 at rc is nearly a factor of ten smallerthan it is at ro (the simple geometric decrease would be by afactor of three).

A substantial reduction of the ratio of quadrupole power todipole power by the presence of a stable layer is found in all caseswhere the dynamo is dipole-dominated. In Fig. 10 the ratio R2=R1 isplotted for all these models. For an outer layer thickness of 0.2 thereduction relative to the reference case without a stable shell is inthe range four to ten and for a thickness of 0.4 it is by a factor ofabout 15. Because the time scale of secular variation tends to beinversely related to the magnetic Reynolds number (Christensenand Tilgner, 2004) a stronger skin effect and more attenuationwould be expected at a higher value of Rm. However, a systematicdependence of the reduction factor on Rm is not visible in Fig. 10,nor is there an obvious dependence on E or Pm.

0 2000 4000

−8000

−6000

−4000

−2000

0

2000

4000

6000

8000

Time [yr]

g 21 [nT]

Fig. 9. Time series of the magnetic field coefficient g21 at the top of the dynamoregion (thin line) and at the top of the stable layer (bold line) for case 14.4. Time andmagnetic field have been scaled to dimensional values as described in Section 3.4.

50 100 200 500

10−2

10−1

Rm

R2 /

R1

E=3x10−4

E=1x10−4

E=3x10−5

Fig. 10. Ratio of quadrupole power to dipole power at the outer boundary of themodel plotted vs. magnetic Reynolds number for the dipolar dynamo models.Models with the same control parameters and different stable layer thicknesses areconnected by lines; open symbols for do ¼ 0, light gray fill for do ¼ 0:1, mid gray fordo ¼ 0:2, dark gray do ¼ 0:4 and black for do ¼ 0:8. The Ekman number is keyed bythe symbol shape. The horizontal broken lines indicates the probable upper limit ofR2=R1 at the top of Ganymede’s iron core.

a

b

Fig. 11. Snapshot of the radial magnetic field. (a) Case 11.0, do ¼ 0, field upwardcontinued to r ¼ ro þ 0:2, (b) Case 11.2, do ¼ 0:2, field shown at rc ¼ ro þ do . Contourstep is 0.06.

10−2 10−1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ra E9/4

d o

dipolar

hemispherical

Fig. 12. Regime diagram for type of dynamo solution. Open symbols dipolar, filledsymbols hemispherical or mixed-mode nondipolar. Symbol shape keyed to Ekmannumber as in Fig. 10.

256 U.R. Christensen / Icarus 247 (2015) 248–259

3.3. Non-dipolar solutions

With increased driving of convection, typically a breakdown ofthe dipolar dynamo mode in favor of a multipolar one is found inmodels with fixed temperature boundary conditions (Kutznerand Christensen, 2002). Multipolar dynamos show a flat spatialpower spectrum of the magnetic field, sometimes with lowerpower in the dipole compared to higher multipoles. Flux patchesof both polarities are scattered over the outer boundary. Whenconvection is driven by a buoyancy flux from above rather thanfrom an inner core and the more realistic fixed flux rather thanfixed codensity condition is used, a hemispherical dynamo can befound as a third mode (Grote and Busse, 2000; Landeau andAubert, 2011). It is characterized by strong differences in codensitybetween the northern and southern hemisphere, strong differentialrotation between the hemispheres and concentration of dynamoaction and strong magnetic surface flux into one of them. Thismode also requires enhanced driving of convection; at moderatelysupercritical Rayleigh number the magnetic field is dipolar(Landeau and Aubert, 2011).

In the simulations without a stable outer layer the breakdownof the dipolar mode results in dynamos that have characteristicsof both the multipolar and the hemispherical mode, changing backand forth in time between these two modes or even all threemodes. A snapshot of the magnetic field for such a case is shownin Fig. 11a at a time when it is multipolar. In contrast, when a sta-ble layer exists, the preference is always for a purely hemisphericalmode in cases where the dipolar solution breaks down. Further-more, these hemispherical magnetic fields are more strongly axi-symmetric than the multipolar or mixed modes (cf. Fig. 11b)found without a stable layer. The ratio R2=R1 is of order one inthe hemispherical cases, with most of the power being in themodes of order m ¼ 0.

Nondipolar dynamos are found at high Rayleigh number, butlowering the Ekman number stabilizes the dipolar solution. A suit-able ordering parameter, which clearly separates the two regimesfor all three values of the Ekman numbers, is the combinationRaE9=4. A regime diagram as function of this parameter and of the

outer shell thickness do is shown in Fig. 12. For do ¼ 0:2 dipolarsolutions persist to a value of RaE9=4 that is about 40% larger thanthe maximum at do ¼ 0. Interestingly, while a stable layer of mod-erate thickness stabilizes the dipolar solution, at larger values of do

this trend is reversed and for do ¼ 0:8 dipolar solutions are onlyfound at a value of RaE9=4 that is an order of magnitude smallerthan the maximum value at do ¼ 0:2. The exponent for E is empir-ical and any value between 2.1 and 2.4 separates dipolar and non-dipolar cases in different domains without overlap.

As a rule simulations have been started with a strong dipolarfield. For two models it has been tested if the same dipolar solutionwould stabilize when the run is initialized from a weak field with awhite power spectrum. This was the case for model 7.2, however,

U.R. Christensen / Icarus 247 (2015) 248–259 257

in model 9.2 with a higher Rayleigh number it resulted in a hemi-spherical solution (case 9.2a in Table 1). Bistability, i.e. the coexis-tence of different dynamo states that are selected by the initialcondition, has been found to be common in dynamo simulationswith free-slip mechanical boundaries (Simitev and Busse, 2009;Gastine et al., 2012; Yadav et al., 2013). In the present simulationsthe existence of a fluid shell above the dynamo region may be moresimilar to a free-slip than to a no-slip condition for the flow in theconvecting region.

Ganymede’s dynamo operates in Jupiter’s magnetic field. At thesurface of Ganymede’s core the intrinsic dipole field is more than afactor hundred stronger than Jupiter’s field and Sarson et al. (1997)concluded from mean-field simulations that Ganymede’s dynamoworks in its own right rather than to depend critically on the sup-plied external field. Nonetheless, Gómez-Pérez and Wicht (2010)found in MHD dynamo simulations that even an external field with2% of the intrinsic field strength at the dynamo’s surface has a non-negligible influence. Jupiter’s field served most likely as a seed fieldat the start of Ganymede’s dynamo. The uniform (at the scale ofGanymede) imposed field might have been instrumental for select-ing the dipolar solution branch. In additional experiments, case 9.2was therefore run with an imposed uniform axial field and startedwithout any additional perturbation. With an external Bz ¼ 0:05,which is approximately 10% of the intrinsic rms dipole fieldstrength at rc for the case on the dipolar branch, the dynamo indeedsettled in the dipolar state. However, when Bz was set to 0.005 or 1%of the saturated dipole field, the dynamo went into the hemispher-ical state, which persisted for the simulation duration of 950 advec-tion times or 13 dipole free decay times of the full system.

3.4. Scaling to Ganymede

Since dynamo simulations cannot be performed at the true val-ues of the control parameters, scaling of the results to physicalunits is ambiguous. There are several inherent time scales in thedynamo problem that could be used to rescale model time to phys-ical units: the rotation time, the magnetic diffusion time and theadvection time. Comparing the spectrum of dipole fluctuations ingeodynamo models with the observed changes of the geomagneticfield, Olson et al. (2012) identified the advection time scale as themost appropriate one. Assuming that the magnetic Reynolds num-ber in the present models are in the correct range for Ganymede,which has been estimated to be between several hundreds and1000 by Rückriemen et al. (2014), re-scaling with the magnetic dif-fusion time gives an identical result. The applicable scaling factor is

t� ¼ 1Pm

D2

k: ð13Þ

The electrical conductivity for pure liquid iron at the temperaturesand pressures in Ganymede’s core is in the range ð1—2Þ � 106 S=m,corresponding to k = 0.4–0.8 m2/s (Secco and Schloessin, 1989;Deng et al., 2013). The addition of light elements is expected toreduce the conductivity somewhat, hence we pick a value tok = 0.7 m2/s near the upper end of the range.

In view of the expected magnetometer data from the plannedESA JUICE mission it is of interest to consider how much secularchange may have occurred since the visit by Galileo. A change inthe tilt or location of the dipole axis probably offers the largestchance for detection. For cases 14.0, 14.2 and 14.4 with a magneticReynolds number of order 500 the secular variation time scale forthe equatorial dipole components has been calculated. It is definedas

seqdipSV ¼ hg2

11 þ h211i

h _g211 þ _h2

11i

!1=2

; ð14Þ

where the dot over the Gauss coefficients indicates the time deriv-ative and h. . .i the temporal average. seqdip

SV is the time after whichthe geographic location of the dipole axis is decorrelated with itsprevious location. Using Eq. (13) to express the result in physicaltime, a time scale of 60 yr is obtained for the model without a stableouter layer. This goes up to �200 yr for the models with a stablelayer. Note that even though the time scale is roughly the samefor do ¼ 0:2 and do ¼ 0:4, the secular change in absolute terms isless in the case of a thicker stable layer because here the meandipole tilt is substantially lower.

The basis for re-scaling magnetic field strength is the findingthat for dipolar dynamos the field strength is nearly independentof rotation rate and viscosity and varies with the cubic root ofthe power generated by buoyancy forces (Christensen andAubert, 2006; Aubert et al., 2009; Christensen, 2010). In numericaldynamo models this holds aside from a correction factor thataccounts for the fraction of energy flux that is not dissipated byJoule heating (which in planets is assumed to be small). In all mod-els with a stable shell and a dipolar field the fraction of non-ohmicdissipation is less than 40%, hence the correction has no strongeffect on the result. Also, a weak dependence of the field strengthon Pm has been found in numerical models (Christensen andAubert, 2006; Stelzer and Jackson, 2013), which is ignored here.The strategy for rescaling magnetic field is to select reasonable val-ues for the buoyancy flux, core radius, density and gravity and tocompromise on the viscosity and rotation rate, because of theirsupposed small influence on the field strength. The physical valuesof m and X are calculated, for each model, such that the model val-ues of Ra and E are matched, given the choice of the other physicalparameters.

Using the integrated buoyancy flux Q B ¼ 4p~r2oqB, replacing go by

the gravity at the top of Ganymede’s core gc ¼ goðro þ doÞ=ro andusing Eqs. (9) and (11), the Rayleigh number (Eq. (8)) can beexpressed as

Ra ¼ cNuPr2

Nu� 1gcQ B~r2

c

qm3 : ð15Þ

Here the tilde denotes radii in physical units and

c ¼ D4

4p~ro~r3c

ð16Þ

is a geometry factor. Solving for the viscosity results in

m ¼ cNuPr2

Nu� 1gcQB~r2

c

qRa

!1=3

: ð17Þ

Subsequently, the values for rotation rate and magnetic diffusivityare obtained from

X ¼ mED2 ; k ¼ m

Pm: ð18Þ

Using the values of X and k derived in this way in Eq. (2) finallyleads to the prescription for relating the non-dimensional field unitsto physical units:

Bscale ¼ q1=6 lo

EPm

� �1=2 D4p~ro

NuPr2

Nu� 1gcQ B

~rc Ra

!1=3

: ð19Þ

A characteristic value for the buoyancy flux at the interfacebetween Ganymede’s snow layer and the interior region is 104 kg/s(Rückriemen et al., 2014). Taking ~rc ¼ 770 km, gc ¼ 1:3 m=s2 andq = 6500 kg/m3 for a sulfur concentration of about 16 wt% in thecore, with the control parameters of case 14.4 a nominal viscosityof 4.3 m2/s, a magnetic diffusivity of 1.4 m2/s and a nominal rota-tion period of 112 d are calculated. We note that the magnetic dif-fusivity is slightly higher but of the same order as the

258 U.R. Christensen / Icarus 247 (2015) 248–259

experimentally determined values. The dipole moment calculatedfrom the model result is 460 nT� r3

G, 65% of Ganymede’s observeddipole moment of approximately 700 nT� r3

G. For the various mod-els with a stable shell thickness of 0.4 (187 km for a core radius of770 km) a range of 460—710 nT� r3

G is found and for do ¼ 0:2(106 km) the range is 625—1100 nT� r3

G. These values are for thenominal buoyancy flux of QB ¼ 104 kg=s. Since the magnetic fieldstrength scales with Q 1=3

B , the precise value of QB is not very critical,e.g., a change in Q B by a factor of three changes the dipole momentby a factor of 1.4.

4. Discussion and conclusions

At present the higher multipoles of Ganymede’s internal mag-netic field are poorly constrained. A reliable determination of thequadrupole moment must probably await the arrival of the JUICEmission at Ganymede. However, a reanalysis of the Galileo magne-tometer data with improved models of the plasma environment ofGanymede (Jia et al., 2009; Duling et al., 2014) that also take a fieldcomponent induced in a salty internal ocean into account mighthelp to put more robust limits on the quadrupole. For the timebeing the assumption that the quadrupole-to-dipole ratio of themodel by Kivelson et al. (2002) without an induced field representsan upper limit to the actual ratio appears reasonable.

The simulations show that the iron snow scenario provides aviable explanation for the known and inferred properties of Gany-mede’s magnetic field. In particular, a stably stratified iron snowlayer above the dynamo region attenuates the quadrupole compo-nents of the magnetic field sufficiently in relation to the dipole tomake it compatible with the likely upper limit for Ganymede’squadrupole moment (Fig. 10). In models without such layer, thequadrupole-to-dipole ratio is above this limit or at best marginallycompatible with the limit. While the presence of a snow layer axi-symmetrizes the field at the surface of the core, this is mainly byattenuating higher multipole components that supply most ofthe non-axisymmetric field components in the absence of suchlayer. The equatorial dipole is only moderately affected and the tiltof 4� observed by Galileo is compatible with the snow layer model.

Changes of the tilt and the location of the dipole axis betweenthe Galileo result and the measurements to be taken by the JUICEmission nearly 35 years later may serve as a criterion to evaluatedynamo models. Without a stable layer the characteristic time scaleof change of the equatorial dipole is around 60 yr for a plausiblevalue of 500 for the magnetic Reynolds number, i.e. a significantchange in the location of the dipole axis could be expected. Witha stable layer the time scale is 200 yr and the differences shouldbe considerably less. However, because secular variation is intrinsi-cally a stochastic process which proceeds at a different pace at dif-ferent times, and because there is a trade-off with the inversedependence of the secular variation time scales on the magneticReynolds number (Christensen and Tilgner, 2004; Christensenet al., 2012), it is not clear if the difference in the predicted timescales is large enough so that the amount of movement of the dipoleaxis could serve as a useful discriminator between possible models.

The dipole moment of the models, rescaled to physical unitsunder the assumption that magnetic field strength does not dependon rotation rate and viscosity, is of the same order of magnitude asGanymede’s observed dipole moment for plausible values of thebuoyancy flux, in agreement with earlier estimates based only onscaling laws (Hauck et al., 2006). In the models with a thicker snowlayer of order 200 km the dipole moment is slightly lower than theobserved one, whereas a 100 km layer gives a good match. How-ever, uncertainties on the buoyancy flux and on possible deviationsfrom the scaling assumptions as well as the temporal fluctuations ofthe dipole moment in planetary dynamos prevent us from usingthis as a constraint on the snow layer thickness.

Ganymede’s core size is not tightly constrained by the observedmean density and moment of inertia. For a core radius rc ¼ 385 km,half of the nominal size assumed in Section 3.4, the ratio of quad-rupole power to dipole power at the top of the core that is compat-ible with the Galileo observations would go up to 0.12. This is inthe range of values found at other planetary dynamos and wouldremove the need for a special mechanism of damping the quadru-pole. However, while for the nominal core size the observed dipolemoment is obtained with a plausible value of the buoyancy flux,this is hardly possible with a much smaller core. The field strengthB inside the dynamo scales with the buoyancy flux per unit areaaccording to B / ðFqBÞ

1=3 (Christensen, 2010), where F is an ‘effi-ciency factor’, that is proportional to both g and D, hence to r2

c .For a given value of the dipole moment the dipole field strengthat the surface of the core (and presumably also the mean fieldstrength inside the core) varies as B / r�3

c . Taken together thebuoyancy flux required to produce a given value of the dipolemoment depends on core size as qB / r�11

c . Even if we assume thatqB in Ganymede’s core could be up to an order of magnitude largerthan the value used in Section 3.4 this would only allow for a mod-erate reduction of core radius from 770 km to 625 km in order tokeep the same dipole moment.

Do the model results support that Ganymede’s dynamo oper-ates in the dipolar mode? The parameter study suggests that therelevant condition is RaE9=4 < 0:1 for a snow layer thickness ofthe order of 100 km (Fig. 12). This is tentative because the influ-ence of the two Prandtl numbers has not been studied and becausethe precise exponent for the Ekman number is uncertain. If we usethe values for k; gc; rc; QB; q and X from Section 3.4 and molec-ular values for the viscosity m � 10�6 m2=s; RaE9=4 is of order oneor larger in Ganymede’s core, in particular for a compositional dif-fusivity j that is smaller than the viscosity. One may argue, though,that under the conditions of planetary cores ‘turbulent’, values form and j apply that are at least two or three orders of magnitude lar-ger than molecular values. This would bring RaE9=4 to values lessthan 0.1, consistent with a dipole-dominated magnetic field.Whether or not such reasoning makes sense could only be testedby simulations at much lower Ekman number, which are currentlyout of reach. The observed magnetic field of Ganymede clearlyexcludes a hemispherical dynamo, but the question whether thiswould be more compatible with a different scenario than thesnow-layer dynamo, e.g. one with driving convection from a grow-ing inner core, remains open. The preference for hemisphericaldynamos in the case of a very thick stable layer (Fig. 12) may indi-cate that the snow layer in Ganymede’s core is not significantlythicker than 200 km at present.

If the bistable regime found for at least one of the present sim-ulations applies under planetary conditions it becomes relevanthow Ganymede’s dynamo was started. According to the evolutionmodels (Rückriemen et al., 2014) the formation of a snow layer inGanymede’s core must have been relatively recent and the associ-ated net buoyancy flux increased approximately linearly with time,probably starting from negative values if the heat flux out of thecore is below the adiabatic conductive heat flux. Therefore thedynamo would have started with relative weak convective driving(low Rayleigh number), which is favorable for establishing thedipolar mode. In case of bistability, the dynamo would then haveremained on the dipolar solution branch as the Rayleigh numberincreased. Also, the ability of a moderately thick snow layer toextend the stability of the dipolar mode to higher Rayleigh number(Fig. 12) could contribute to explaining the dipolarity of Gany-mede’s magnetic field. On the other hand, tests with an imposedfield suggest that the influence of Jupiter’s field on Ganymede’sdynamo is too weak to foster the selection of the dipolar mode.

Even though our present understanding of the criteria forselecting the dynamo mode under conditions relevant in planets

U.R. Christensen / Icarus 247 (2015) 248–259 259

is rather incomplete, there are no compelling arguments preclud-ing that a snow-layer dynamo in Ganymede generates a dipole-dominated magnetic field. As shown, such a dynamo can explainnot only the observed dipole moment but also the relative weak-ness of the quadrupole.

Acknowledgments

I thank T. Rückriemen for sharing her results prior to publica-tion. Comments by S. Hauck and an anonymous reviewer helpedto improve the paper.

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