Icarus 196 (2008) 16–34 www.elsevier.com/locate/icarus

Models of magnetic field generation in partly stable planetary cores: Applications to Mercury and Saturn

Ulrich R. Christensen ∗, Johannes Wicht

Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany Received 14 August 2007; revised 18 February 2008 Available online 15 March 2008

Abstract A substantial part of Mercury’s iron core may be stably stratified because the temperature gradient is subadiabatic. A dynamo would operate only in a deep sublayer. We show that such a situation arises for a wide range of values for the heat flow and the sulfur content in the core. In Saturn the upper part of the metallic hydrogen core could be stably stratified because of helium depletion. The magnetic field is unusually weak in the case of Mercury and unusually axisymmetric at Saturn. We study numerical dynamo models in rotating spherical shells with a stable outer region. The control parameters are chosen such that the magnetic Reynolds number is in the range of expected Mercury values. Because of its slow rotation, Mercury may be in a regime where the dipole contribution to the internal magnetic field is weak. Most of our models are in this regime, where the dynamo field consists mainly of rapidly varying higher multipole components. They can hardly pass the stable conducting layer because of the skin effect. The weak low-degree components vary more slowly and control the structure of the field outside the core, whose strength matches the observed field strength at Mercury. In some models the axial dipole dominates at the planet’s surface and in others the axial quadrupole is dominant. Differential rotation in the stable layer, representing a thermal wind, is important for attenuating non-axisymmetric components in the exterior field. In some models that we relate to Saturn the axial dipole is intrinsically strong inside the dynamo. The surface field strength is much larger than in the other cases, but the stable layer eliminates non-axisymmetric modes. The Messenger and Bepi Colombo space missions can test our predictions that Mercury’s field is large-scaled, fairly axisymmetric, and shows no secular variations on the decadal time scale. © 2008 Elsevier Inc. All rights reserved.

Keywords: Magnetic fields; Mercury; Saturn

1. Introduction chemical differentiation process that leads to an unstable strati- fication. Planetary magnetic fields are generated in a self-sustained In many planets the entire conducting fluid region may be dynamo process associated with the circulation of an electri- convecting, but this is not necessarily the case. The iron cores cally conducting fluid in the core of the planet (Stevenson, of Mars and Venus could be stably stratified and completely 2003). Different sources of the flow may be possible, but most stagnant because of a subadiabatic temperature gradient and be- commonly it is believed to be driven by convection. Aside cause compositional convection may be unavailable for lack of from thermal convection, compositionally driven flow may oc- an inner core. This would explain the absence of a global mag- cur, for example by the rejection of light alloying elements netic field at these planets. More interesting is the case when from a growing solid inner core in the Earth and perhaps in some parts of the core convect while others are stable. other terrestrial planets. A prerequiste for thermal convection In the case of the Earth a stably stratified region may exist is that the radial temperature gradient is steeper than the adia- at the top of its fluid core. Estimates for the heat flow at the batic gradient. Compositional convection requires an ongoing Earth’s core–mantle boundary usually exceed the flux that can be conducted along an adiabatic gradient in the core, however, a slightly subadiabatic heat-flow cannot be ruled out (Labrosse * Corresponding author. Fax: +49 5556 979 219. et al., 1997), so that a top layer may be thermally stable. Al- E-mail address: [email protected] (U.R. Christensen). ternatively, a layer more enriched in light elements compared

0019-1035/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2008.02.013 Magnetic field generation in Mercury and Saturn 17 to the bulk of the fluid core might have accumulated at the top thought to be indicative for a “magnetostrophic” balance of of the core (Braginsky, 1984; Lister and Buffett, 1998). So far Coriolis force and Lorentz force that presumably holds in plan- there is no compelling evidence that such a stable layer exists. If etary dynamos (Stevenson, 2003) and determines the internal it exists, its thickness may be small in comparison to that of the field intensity. Christensen and Aubert (2006) and Olson and convecting layer, implying that it would have limited influence Christensen (2006) suggested an alternative scaling law for the on the geodynamo. magnetic field strength and the dipole moment of planetary dy- Christensen (2006) suggested that in Mercury’s fluid core namos, based on the power that is available to balance ohmic the dynamo region is lying below a thick stable layer. Ther- dissipation. It leads to a relation between the Lorentz number mal evolution models for Mercury predict that the heat flow at B Lo = √ , (2) the core–mantle boundary is only a moderate fraction of the ρμΩD adiabatic heat flow, thus rendering the upper part of the core thermally stable. The evolution models also suggest that Mer- where D is the thickness of the fluid shell and μ is magnetic cury has nucleated a solid inner core. If the core contains some permeability, and a modified Rayleigh number that measures light element, most likely sulfur, inner core growth would lead the buoyancy flux available to drive the dynamo. The field to compositional convection in the deep parts of the fluid core strength predicted by this scaling law agrees well with the ob- that is augmented by thermal convection driven by the latent served field strengths for Earth and Jupiter. This is also the case heat of inner core freezing. In Section 2 we study what controls when applying the Elsasser number rule. the thickness of the unstable layer. Mercury’s field geometry has been characterized to a lim- Saturn is another candidate for a planet with a stably strati- ited degree by the measurements taken during two flybys of Mariner 10 in 1974/1975. During the first flyby Mariner 10 fied region at the top of its electrically conducting core. Saturn’s passed at low latitudes through the magnetotail and most of the core is composed of a mixture of hydrogen in a metallic state measured field is believed to be due to magnetospheric currents and of helium. Its upper boundary is at approximately half the (Connerney and Ness, 1988). The third flyby, where Mariner planetary radius. Under Saturn conditions helium is expected to 10 passed Mercury at 300 km above the surface at high North- become partly immiscible with hydrogen in some pressure in- ern latitudes, provided the best data for revealing the internal terval above the metal transition (Stevenson and Salpeter, 1977; magnetic field. The field seems to be large scaled and is per- Fortney and Hubbard, 2003). As a consequence, helium should haps dominated by a dipole component that is tilted slightly separate and sink as rain drops, depleting the upper layer and (14◦ ± 5◦) relative to the rotation axis (Ness, 1979). However, enriching the lower one, with a gradient zone at the top of the the measurements cannot discriminate between a dipolar and metallic region. The thickness of the stably stratified region a quadrupolar magnetic field (Connerney and Ness, 1988). All is uncertain and in the extreme case helium may be lost alto- field models in which the Gauss coefficients for the axial dipole gether from the hydrogen region and settle onto the rocky inner g0 and the axial quadrupole g0 satisfy the relation core (Fortney and Hubbard, 2003). In Jupiter temperatures are 1 2 0 + 0 =− higher than they are in Saturn, with the consequence that helium g1 1.52g2 320 nT (3) may be fully miscible. fit the Mariner 10 observations equally well (the field models In Uranus and Neptune an electrically conducting ionic fluid also involve non-axial dipole components and coefficients de- is assumed to extend to about 3/4 of the planetary radius. In scribing magnetospheric currents that are co-varied when the order to explain the low luminosity of these planets, Hubbard et 0 0 g2/g1 ratio is changed). al. (1995) proposed that only an outer shell of the ionic fluid is The enigmatic point about Mercury’s field is its weakness. convecting, whereas the deeper fluid layers are compositionally The mean strength at the surface is approximately 450 nT when stratified. Hence Uranus and Neptune may represent one class we take the field to be purely dipolar, i.e., it is hundred times of planets, where an unstable conducting fluid region lies above weaker than Earth’s surface field. Assuming the same value of a stable layer, and Mercury and Saturn may represent another the Elsasser number inside the dynamo regions of Earth and class, where the unstable region lies below a stable layer. Mercury we would expect a somewhat weaker field in the latter The magnetic fields of planets with active dynamos differ case because Mercury rotates ≈60 times slower than Earth. On substantially in terms of strength and geometry. The magnetic the other hand, Mercury’s core is larger relative to the size of fields of Earth and Jupiter might be considered as the “rule”: the planet, meaning that the geometric decrease in field strength they are dominated by the axial dipole, but the equatorial dipole from the core–mantle boundary to the surface is less than in and higher multipoles including non-axisymmetric parts make a case of the Earth. Taking both effects into account, the field significant contribution to the total field at the top of the dynamo intensity at Mercury’s surface should be ≈14,000 nT for an region. The magnetic field strength B in the dynamo regions of Earth-like dynamo. The observed intensity corresponds to an Earth and Jupiter, as far as it can be estimated from the observed Elsasser number of only 10−4 at the top of the core. field, is such that the Elsasser number The rate of rotation not only affects the field strength, but σB2 also the field geometry. Analyzing numerical dynamo simu- Λ = (1) lations, Christensen and Aubert (2006) and Olson and Chris- ρΩ tensen (2006) found that a local Rossby number is of order one, where σ is electrical conductivity, ρ den- urms sity and Ω rotation rate. An Elsasser number of one is often Ro = , (4) Ω 18 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34 where  is the characteristic length scale of the flow and urms Christensen (2006) presented two simulations in which the the characteristic velocity, controls into what class the dynamo dynamo operates beneath a stably stratified fluid layer. The sur- falls. This Rossby number measures the role of inertial forces face field strengths of the models bracket the observational relative to the Coriolis force (and other dominant forces) at the value. The magnetic field in the dynamo region is strong characteristic flow scale. At low values of Ro (fast rotation) (Λ ≈ 1), but is dominated by higher multipole components. the magnetic field is dominated by the axial dipole, whereas at They vary rapidly in time and are strongly damped by the skin high Ro (slow rotation) the dipole contribution is weak and effect in the stable layer. The axial dipole and quadrupole com- the field is dominated by higher multipole components with ponents are weak in the dynamo region, but vary on longer time a fairly white spectrum. According to Olson and Christensen scales. Hence they are less attenuated and dominate the field (2006) Mercury should clearly fall into the multipolar regime, geometry outside the core. in contrast to the dynamos of all other planets. While this may Saturn’s magnetic field strength is slightly on the weak side, explain the weakness of Mercury’s dipole moment, it is more both in terms of the Elsasser number at the top of the elec- difficult to reconcile with the seemingly large-scale structure of trically conducting layer or when the Lorentz number scaling the magnetic field observed in the Mariner 10 flybys. (Olson and Christensen, 2006) is applied, although the discrep- The problems that Mercury poses for dynamo theory have ancy is much less than in the case of Mercury. The particular led some authors to propose a different origin of the field. point about Saturn’s field is its extreme degree of axisymme- Stevenson (1987) suggested a thermoelectric dynamo, in which try (by axisymmetry we implicitly always mean symmetry with temperature differences that are associated with topographic respect to the rotation axis). The dipole tilt is indistinguish- variations of the core–mantle boundary lead to thermoelectric able from zero, and the observations by Pioneer 11 and the currents. Their (invisible) toroidal magnetic field is converted Voyagers are fitted very well by an expansion in purely zonal into an observable poloidal field by the α-effect of a flow in the harmonics up to the octupole term (Connerney et al., 1982; outer core that is too sluggish to drive a self-sustained dynamo. Connerney, 1993). Periodic variations of Saturn’s kilometric In order to explain the observed field, the electrical conductiv- radio emission, which were thought to indicate the presence ity of the lower mantle needs to be large and the topography of non-axisymmetric components of the internal magnetic field must have a significant long-wavelength component. The ob- and to constrain the rotation rate of the core (Giampieri et al., served strength of Mercury’s magnetic field is compatible with 2006), may be externally controlled (Gurnett et al., 2007). crustal remanent magnetization as the source, acquired from the Stevenson (1980, 1982) suggested that strong differential ro- field of a now extinct dynamo (Stephenson, 1976). In order to tation in the stably stratified helium-depleted layer overlying explain the long-wavelength structure of field, Aharonson et al. Saturn’s dynamo region attenuates the non-axisymmetric mag- (2004) invoke the large variability of Mercury’s surface tem- netic field components. Making the conservative assumption perature with longitude and latitude, which leads to a global that the field generated in the dynamo region is stationary, the variation of the thickness of the magnetized crust bounded by non-axisymmetric part of the field varies periodically with time the Curie temperature surface. To obtain a large-scale field from when seen in a reference frame moving with the overlying ro- remanent magnetization, this model also seems to require that tating shell. Hence it is attenuated by the skin effect when the the magnetizing field was stable (non-reversing) during the time magnetic Reynolds number characterizing the shell motion is period in which Mercury’s crust formed. sufficiently high. The axisymmetric part of the field remains Until recently there was no definitive evidence excluding stationary even in a differentially rotating reference frame and that Mercury’s core is entirely solid. In this case any dy- is not affected. namo model would be obsolete. The observed amplitude of The magnetic fields of Uranus and Neptune are charac- forced librations, obtained by ground-based radar interferom- terized by strong dipole tilts. At the planetary surface the etry, strongly suggests that Mercury’s core does not partic- dipole and multipole components contribute to a similar de- ipate in the librational motion, hence must be at least par- gree (Holme and Bloxham, 1996). Following the rule of the tially liquid (Margot et al., 2007). The final confirmation of local Rossby number Uranus and Neptune should have dipole- this conclusion has to await a more precise determination of dominated magnetic fields (Olson and Christensen, 2006). Mercury’s quadrupolar gravity moment, which enters into the However, this rule is based on models of a dynamo operating in analysis. a deep convecting shell. Stanley and Bloxham (2004, 2006) cal- The size of Mercury’s solid inner core, which is poorly con- culated dynamo models with a thin convecting shell overlying strained, may have an important effect on the dynamo. Stanley a stably stratified fluid region. In some of their models the mag- et al. (2005) and Takahashi and Matsushima (2006) present nu- netic power spectra as function of harmonic degree compare merical dynamo models for Mercury with a large solid inner favorably with those of Uranus and Neptune. core and Heimpel et al. (2005) calculated models with a very When the magnetic fields of Earth and Jupiter, characterized small inner core. All these models show a relatively low mag- by a dominant axial dipole, secondary non-axisymmetric con- netic field strength outside the core because most of the dynamo tributions and a core Elsasser number of one, are considered field is toroidal, or because it is small-scaled implying a rapid as the rule, those of Mercury, Saturn, Uranus and Neptune are decrease with radius. However, compared to observation the “exceptional” for various reasons of strength or geometry. Even field in these models is still too strong by a factor of ten or though it is awkward to have more exceptions than planets that more. follow the rule, the cause for the deviation may in each case re- Magnetic field generation in Mercury and Saturn 19 sult from the existence of a stable conducting layer above or controls the rate of inner core growth, mainly by the latent heat below the dynamo region. term which is given as The purpose of this paper is to further investigate dynamo = ˙ = 2 ˙ = ˙ models in which the conducting fluid is stably stratified near HL Lmi 4πLρri ri FLri, (5) the outer boundary and is unstable at depth. We do not con- where mi is the inner core mass, L is the latent heat per unit sider the opposite case. The effects of the presence of the stable mass, and ρ is a reference core density. The contribution from layer per se and of the toroidal motion in this layer on the ax- secular cooling to the CMB heat flow is given by isymmetric and non-axisymmetric magnetic field components ˙ generated in the dynamo are studied. The models are mainly HS = VcρcT, (6) applied to Mercury, to which we tune some of the model para- meters. In a more generic sense, some of the model results are where T is mean core temperature, Vc the core volume and c also applicable to the magnetic field of Saturn. the heat capacity. In order to relate the inner core growth rate and the com- 2. Buoyancy flux in Mercury’s core positional and thermal fluxes at different radii in the outer core, we average over the short-term convective fluctuations and con- The two sources of buoyancy in Mercurys core, thermal and sider the evolution of a mean reference state, which is a function compositional, are closely tied by the thermal power budget of of radius. For a fully convective core an adiabatic thermal ref- the core, which controls the rate of inner core growth (Lister erence state is usually chosen. For the case of Mercury we take and Buffett, 1995). In this section we derive the relative contri- a reference geotherm that is adiabatic in a deep convective sub- butions to the buoyancy flux and estimate the degree to which layer of the fluid core and is subadiabatic in its stable upper the core could be stably stratified for different values of the heat part. Although the CMB heat flow Ho and the neutral radius rn, flow at the core–mantle boundary, different inner core sizes, and which divides the core into a stable and a convecting part, will different sulfur concentrations. change over the age of the planet, we assume that at least on some intermediate time scale they remain nearly constant and Thermal evolution simulations suggest that the heat flux at ˙ Mercurys core–mantle boundary (CMB) may have been supera- that the core cools everywhere at the same rate T . diabatic only for a relatively brief period in the planet’s history The inner core radius is obtained from the crossing point of (Schubert et al., 1988; Hauck et al., 2004; Breuer et al., 2007). the reference geotherm (taken to be adiabatic close to the inner core boundary) and the melting curve. The inner core radius The present CMB heat flow ho is predicted to be of the order of afewmW/m2. This is significantly lower than the heat flow that changes due to the drop of temperature and because of the re- can be transported by conduction along an adiabatic gradient in duction of the melting point by the increasing enrichment of the 2 outer core in sulfur. As long as the initial sulfur concentration is the core, which is estimated to be about had,o = 12 mW/m at the core–mantle boundary (Stevenson et al., 1983; Hauck et modest and the inner core has not grown to very large size, the al., 2004). A subadiabatic CMB heat flux means that the ther- latter effect is much smaller and is ignored here. The relation ˙ ˙ mal gradient establishes a stable density stratification in the between T and the inner core growth rate ri depends on the upper part of the liquid core. The evolution models predict that variation of the potential melting temperature ΘM (i.e., melt- Mercury has a solid inner core growing with time. The associ- ing temperature minus adiabatic temperature) with pressure P ated compositional gradient has a destabilizing effect and the (Lister and Buffett, 1995). Using Eq. (6) the relation can be relative size of both contributions determines how deep the sta- written in terms of HS   ble layer may reach into the core. Note that the actual thermal r ∂Θ H = F r˙ = ρ2V g i c M r˙ , (7) gradient may exceed the adiabatic gradient deeper in the core S S i c o r ∂P i because the former steepens with depth when the latent heat of o ri inner core solidification is a major heat source, whereas the lat- where go is gravity at the CMB. From Eqs. (5) and (7) the inner ter flattens because of the decrease in gravity. core growth rate is obtained as The heat H = h S flowing from the core to the mantle o o o S h is supplied by different sources (S is the core surface area; r˙ = o o , o i + (8) in general the index o indicates a property on the outer core FL FS boundary and the index i at the inner core boundary). The main where ho = Ho/So is the CMB heat flux per unit area. contributors are secular cooling HS and the release of latent Having derived r˙i enables us to determine the absolute con- heat HL from a growing inner core. An additional source is the tributions of the different heat sources and the thermal and conversion of the compositional convective power into heat, but compositional flux on both boundaries (the compositional flux this is typically an order of magnitude smaller than the two main through the CMB is assumed to be zero). We refer to Lister and constituents (Wicht et al., 2007) and is neglected here as well as Buffett (1995) and Wicht et al. (2007) for details, and concen- a possible contribution from the decay of radioactive elements trate on an issue that is of particular interest for our Mercury in the core. The heat flow at the core–mantle boundary (CMB) dynamo models, i.e. the total buoyancy flux at the inner core is mainly controlled by the capability of the overlying mantle boundary and the thickness of the convecting layer. Before do- to transport heat by conduction and convection and its value is ing so, we introduce the concept of co-density, which we em- treated here as being set externally. The value of Ho = HL +HS ploy in our numerical models. 20 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34

Table 1 Model parameters and results Property Symbol Mercury value Saturn value Unit

Planetary radius rp 2440 58,200 km Core radius ro 1850 32,100 km Inner core radius ri 648–1110 km −2 Gravity at ro go 4.2 m s − Core density ρ 8200 1000 kg m 3 − − − Rotation rate Ω 1.24 × 10 6 1.60 × 10 4 s 1 − Magnetic diffusivity λ 1.0 4.0 m2 s 1 − Heat capacity c 675 J (kg K) 1 − − Thermal expansivity α 3 × 10 5 K 1 Compositional density coeff. β 0.38 – − Latent heat L 2.5 × 105 Jkg 1 −2 Adiabatic CMB flux had,o 12 mW m −9 a −1 Potential melting temp. gradient ∂ΘM /∂P (1.1–3.4) × 10 KPa a For ri /ro between 0.35 and 0.75 using the iron melting parameterization of Stevenson et al. (1983).

To fully model the evolution of temperature T and sulfur flux emanating from the inner core gets mixed into the vol- concentration χ in the core, we have to solve two separate trans- ume of the outer core where it enhances the mean concentra- port equations that are characterized by diffusion coefficients tion. Flux boundary conditions for T and χ can also be com- which differ by a factor of order 1000 (Braginsky and Roberts, bined into a unified flux condition for C. In equilibrium the 1995). Unfortunately, simulations of planetary core dynamics co-density fluxes on the inner and outer boundaries, Qi and Qo, at realistic diffusivities are numerically impractical, since com- respectively, and the volume integral over the sink term  must putational resources do not allow to resolve the expected small add up to zero. Similar concepts of co-density have been em- scale structure. A common approach is to assume significantly ployed before in geodynamo models by Sarson et al. (1997) larger diffusivities that reflect the mixing due to small scale and Kutzner and Christensen (2002, 2004). While the assump- turbulence, which is a much more efficient transport mecha- tion of turbulent (equal) diffusivities may be justified in a fully nism than diffusion on the molecular level. Since the turbulent convective dynamo, its use in the context of a partially stable mixing acts in a similar way on all quantities, the effective dif- system is more questionable. Nonetheless, we retain this as- fusivities should be similar. Density differences of thermal and sumption for the sake of a simple and numerically tractable compositional origin can be lumped into a single variable C, model. named co-density (Braginsky and Roberts, 1995) Assuming an inner core of pure iron, the co-density flux on   the inner core boundary is obtained as C = ρ α[T − Tad]+β[χ − χ0] . (9) − Here Tad(r) is an adiabatic reference temperature chosen such ρFe ρ α Qi = m˙ i + (Lm˙ i + HS,i − had,iSi), (11) that it represents the average temperature in the convecting ρ c region, χ is its mean sulfur concentration, α is the thermal 0 with H = H (r /r )3 the inner core contribution to secular expansion coefficient and β its compositional counterpart β = S,i S i o − − ρ(ρ 1 − ρ 1) with the densities ρ and ρ of the core con- cooling, Si the inner core surface area and had,i the adiabatic S Fe Fe S heat flow at the inner core boundary. The first term in Eq. (11) stituents at core conditions. Note that Tad and χ0 change with time due to secular cooling and sulfur enrichment in the outer represents the compositional contribution and the second term core. the superadiabatic thermal contribution to the co-density flux When the thermal and compositional diffusivities are as- (or buoyancy flux). The co-density flux on the outer boundary sumed to have the same effective value κ, the two different is transport equations for T and χ can be combined into a sin- α Q = (h − h )S . (12) gle equation for C o c o ad,o o ∂C Note that for an adiabatic heat flow exceeding the actual heat + u ·∇C = κ∇2C − , (10) ∂t flow, Qo is negative, i.e. radially inward. The sink term  can where u is the velocity and  is a sink term. While in the ab- be obtained from the condition Qi − Qo + Voc = 0, where Voc sence of radiogenic heating the original equations for T and is the outer core volume. χ do not contain a volumetric source or sink term, it arises We have calculated the various co-density source and sink when transforming them into an equation for C because of terms for values of the inner core radius ranging from 0.35 the time dependence of Tad and χ0 and from a non-zero con- to 0.75 times the core radius and values of the core–mantle 2 tribution of the Laplacian acting on Tad(r).  will generally boundary heat flow between 1 and 10 mW/m . The values be positive, i.e., represent a sink, because the adiabat steep- of other relevant parameters are listed in Table 1. Because ens towards the core–mantle boundary and because the sulfur Mercury formed in a hot part of the protoplanetary nebula Magnetic field generation in Mercury and Saturn 21

3. Dynamo model setup

3.1. Basic equations and boundary conditions

We study convection and magnetic field generation in a ro- tating spherical shell with inner radius ri and outer radius ro. We vary the ratio η = ri/ro between 0.35 and 0.60. In the rest of the paper we use scaled variables. Length is scaled by the 2 −1 shell thickness D = ro −ri , time t by D /λ, where λ = (μσ ) is the magnetic diffusivity, and co-density C by qiD/κ, where qi is the co-density flux per unit area imposed at ri and κ is its diffusivity. Magnetic induction B is scaled by (ρμλΩ)1/2.We solve the following set of equations:   E ∂u + u ·∇u + 2zˆ × u +∇Π Pm ∂t RaEPm r = E∇2u + C + (∇×B) × B, (13) Pr ro ∂B −∇×(u × B) =∇2B, (14) ∂t ∂C Pm + u ·∇C = ∇2C − , (15) ∂t Pr ∇·u = 0, ∇·B = 0. (16) Here the unit vector zˆ indicates the direction of the rotation axis, Fig. 1. (a) Inner boundary co-density flux for sulfur contents of 0.3% (gray) Π is dynamic pressure, and gravity varies linearly with radius r. and 3% (black). ri /ro is 0.35 (solid lines), 0.50 (dotted), and 0.75 (dashed). (b) Relative thickness of unstable region. The non-dimensional control parameters are the Ekman number ν E = , (17) ΩD2 close to the Sun and sulfur is a volatile element, its abun- where ν is the kinematic viscosity, the Prandtl number dance is usually assumed to be less than in other terrestrial ν Pr = , (18) planets (Lewis, 1988). However, Mercury may have been ac- κ creted partly from planetesimals that formed at larger distance the magnetic Prandtl number from the Sun (Wetherill, 1988), so that the sulfur content of its ν core is difficult to estimate. Here we consider sulfur concen- Pm = , (19) trations in the outer core of 0.3 and 3% by mass. To obtain a λ rough estimate of the thickness of the unstable layer, we solve and the Rayleigh number a simple stationary radial diffusion equation for co-density at 4 goqiD given values of Q , Q and . When Q < 0 the solution Ra = , (20) i o o κ2νρ has a minimum at radius rn (ri

We point out that with our formulation of the problem we do Christensen and Aubert (2006) and Olson and Christensen not strictly impose a fixed boundary for the stable layer. In the (2006) derived from systematic dynamo model studies scaling convective state the actual thickness of the unstable layer, in laws that relate the characteristic flow velocity to a modified the sense of distance of the minimum of the horizontal average Rayleigh number. Using their results, we obtain the dependence [C](r) above the inner core boundary, may differ from dn. of the magnetic Reynolds number on the control parameters We assume impenetrable no-slip boundaries in most cases, employed here as i.e., u = 0atr and r . In a few cases we use free slip on the i o = 2/5 1/5 −4/5 outer boundary, for reasons explained below. The magnetic field Rm aPm(ηRa) E Pr , (23) at ro is matched continuously with a potential field lacking ex- where a is a numerical constant. Cast into physical properties ternal sources. The inner core is assumed to co-rotate with the this gives outer boundary (the mantle) when we use no-slip conditions. In   2 2 2 6 1/5 the case of free slip at r the inner core rotation rate relative η goq D o Rm = a i . (24) to the reference frame is controlled by viscous and magnetic λ5ρ2Ω torques. The inner core is assumed to have the same electri- Christensen and Aubert (2006) found a = 0.85 for dynamos cal conductivity as the fluid shell in all cases except one, and with η = 0.35 in which the whole spherical shell is convecting. at rSaturns dynamo should fall into the dipolar regime, values of Rm in the models. whereas Mercury’s dynamo is expected to be multipolar. Magnetic field generation in Mercury and Saturn 23

A caveat against the criterion of the local Rossby number is 99% that the flow length scale  (Eq. (4)), which is of the order 100 m > at the quoted value of Ro, is too small to affect in a direct way the magnetic field. At this scale the field is diffusion-dominated. However, in rotational flow an inverse cascade transports ki- netic energy from small scales to scales that are large enough to play a role magnetic field generation. Hence the small flow scales may affect the magnetic field in an indirect way. While 99% 77% > the merits of the local Rossby number rule still need further testing, its relevance at planetary conditions is supported by the finding that it seems to apply for dynamos at values of the mag- netic Prandtl number both larger than one and smaller than one, i.e., in situations where the magnetic length scale is smaller or where it is larger then the flow length scale (Christensen and Aubert, 2006). Equation (25) indicates that in numerical dynamo models for Mercury the Ekman number should not be too low, whereas the Rayleigh number must be large in order to reach a value of Ro that puts the dynamo into the multipolar regime. This is in con- flict with the desire for a low model value of E (whose planetary value is as low as 10−12 even in slowly rotating Mercury). The expected Mercury value of Ro cannot be reached without com- promising severely on the value of the Ekman number, with the possible consequence that viscous effects, which are thought to be unimportant in planetary dynamos, dominate the dynamics in the model. Christensen and Aubert (2006) found that viscos- 1.02 1.10 1.02 0.53 1.34 1.23 1.00 1.00 0.93 0.64 ity does not seem to play a big role in dynamo models when 2.60 2.59 3.40 6.04 5.30 3.47 5.72 2.19 1.95 2.08 E  3 × 10−4. Therefore, we took in most of our models the Ekman number to be either 10−4 or 3 × 10−5 and varied the Rayleigh number inbetween 2 × 108 and 8 × 108. Most of the dynamos fall into the multipolar regime. A value of one is taken for the Prandtl number. A magnetic Prandtl number of three to five puts the magnetic Reynolds number into the range that we estimated for Mercury. The magnetic Reynolds number in Saturn, which is of the order 104 (Stevenson, 2003), cannot be reached in numerical dynamo simulations. , 2—stable layer immobilized, 3—insulating inner core. The surface magnetic field strength has been scaled using Mercury parameters, except

4. Results o r = r We have calculated 20 model simulations, whose parameters and results are summarized in Table 2. We monitored several model properties and list their time-average values. The Nus- selt number, as a measure of the relative efficiency of advective transport of co-density in the convecting region, is defined as

C (r ) − C (r ) Nu = d i d n , (27) [C](ri) −[C]min where Cd refers to the diffusive solution of Eq. (15) and [C] to the horizontal and time average in the convective state. [C]min is the minimum of [C](r). The Nusselt number is inbetween two and seven in our models, indicating that convection in the unstable layer is well above the critical point of onset. The El- 1.300001203311110 17.63 1010101003 40317.631010101051.2103333 33 3 1.16 33333 333330.35 3333 5 501 40 2 03001 20000 0.57 5 0.41 0000 0 185 20 0.35 1 2 811.38 85 0.5 20 41 0.41 1.01 0.5 85 40 0.44 1.00 81 0.5 0.44 10637% 1.00 60 6535% 0.5 0.44 19% 1.10 8520% 2.29 33%39% 49 0.44 93% 1.06 0.69 25%24 106 60 3% 87% 1 97% 1.09 0.5 0% 65 106 330 1% 1% 80 37% 1.00 0.35 1% 0.44 65 85 0 88% 1% 22% 0.87 80 0.35 0.41 25% 86% 41 5% 106 1.02 41% 0.35 0.30 0 40 5% 50% 85 3% 6% 65 0.35 0.50 6% 53% 40 16 3% 6% 85 0.60 61 0.6 4% 31% 4 96% 20% 60 85 0.35 0.5 14% 81 48% 78% 51% 0.36 13% 25% 50 9 85 0.35 49 4% 24% 24% 0.41 97% 0.5 25 39% 460 49 5% 23% 0.44 134 92% 2% 0.35 20 17 1% 28% 0.41 85 85% 65 2% 0.57 1% 25 45 10% 1 64 6% 61 2% 6% 0.35 17 5% 9% 106 0.725 61 0.41 86% 3% 2% 33 64 25 92% 65 5% 0% 0 0% 85 61 1% 0% 21% 0 0% 64 41 1% 0% 6 24% 41 0% 8 0% 0 0 5 0 2912.25 211 2.47 131 2.48 348 4.33 342 4.33 5.70 480 6.53 591 6.81 423 6.47 429 3.38 324 273 315 261 525 462 357 466 205 211 145 sasser number Λ represents the strength of the magnetic field in 0.74 1.3 5.9 2.3 1.4 4.8 8.3 13 9.4 2.1 1.0 0.98 6.8 7.0 4.8 1.4 32 9.4 5.3 12

the dynamo region and has been calculated with its rms-value 5 7 − 0 10

r s n max run E E/ Pm η BC d  N t E E N oe1ab234457 91011121313a13b Table 2 Model parameters and results Model11a1b22a344a4b5678 Ra Code for boundary condition (BC): 0—standard, 1—free slip at in cases 8, 13, 13a and 13b, where Saturn parameters have been used. E Rm Nu in most models, i.e., the internal dynamo field is strong. Λ B 24 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34

Other values in Table 2 refer to the magnetic field upward continued from ro to the planetary surface and scaled to plane- tary values. In most cases this has been done using the Mercury properties of Table 1, except for a few cases in the dipolar dy- namo regime, where Saturn parameters have been taken. Bs is the mean surface field strength in harmonic modes n = 1to n = 8. Edip, Equad and Eoct are the contributions of magnetic energy at harmonic degree n equal to one, two, three, respec- tively, relative to the energy of the surface field in degrees one to eight. Em>0 is the relative energy in non-axisymmetric modes. Nrev is the number of reversals during one magnetic diffusion time τD of the full sphere, which is 108,500 yr for our adopted Mercury parameters. Each change of sign of the Gauss coeffi- 0 cient g1 has been counted as a reversal. However, in those cases 0 where Equad >Edip the coefficient g2 has been taken instead. Model cases 1 and 2, with a ratio η = ri/ro of 0.35 and 0.5, respectively, have been discussed in Christensen (2006).The surface field strength is one fifth of the observed strength at Mercury in case 1 and twice the observed strength in case 2. In case 1, the dipole and the quadrupole contribute on time- average equally to the surface field and reverse on a time scale of 5000 yr, whereas in case 2 the surface field is strongly dom- Fig. 2. Snapshot of radial magnetic field in case 3, (a) at 0.43D ≈ dn above inated by the axial dipole, which did not reverse during the the inner core boundary and (b) at the planetary surface. The contour step is 60,000 nT for the interior field and 100 nT for the surface field. model run time.

4.1. Models fitting Mercury’s field

Here we begin by discussing cases 3 and 4 in some detail. Most parameters are the same as in case 2, in particular the ratio of inner-core radius to core radius is η = 0.5, but the Rayleigh number is larger by a factor of two and three, respectively. The field properties of case 3 are in excellent agreement with what is presently known of Mercury’s field. Fig. 2 compares in a typ- ical snapshot the radial magnetic field at the planetary surface with that inside the dynamo region. At the surface, the field is dominated by the axial dipole. The axial quadrupole makes a secondary but significant contribution, as seen by the difference in field strength at the two poles. The field in the dynamo region is small-scaled, and the dipole or quadrupole contributions are Fig. 3. Time averaged power spectra of the magnetic field at the planetary sur- not obviously discernible. This contrast between the dynamo face (circles) and of the volume-averaged field inside the fluid shell (crosses) field and the outside field can also be seen by comparing the as function of harmonic degree n for case 3. The interior field power has been 4 magnetic power spectrum at the surface with that averaged over scaled down by a factor of 10 . the fluid shell (Fig. 3). In the interior the poloidal magnetic field has a broad maximum in power around harmonic degree 15, at in Fig. 4 because (i) a dynamo generating the field B may a level three times higher than the dipole energy. At the planet’s equally well produce −B and (ii) as long as there is no external surface, the dipole stands out and the quadrupole and the octu- influence breaking the symmetry of the dynamo (which could 0 pole components contribute significantly on time average. The arise from mantle heterogeneity), the sign of g2 may also be 0 spectral power drops rapidly beyond n = 3. flipped relative to that of g1 . The coefficients of the model field The quantitative agreement with what is known about Mer- have the correct magnitude. Moreover, at several instances dur- cury’s field is illustrated in Fig. 4, where the Gauss coefficients ing the run the model field would actually fit the observations 0 0 g1 for the axial dipole and g2 for the axial quadrupole are made at the third Mariner 10 encounter. tracked over 100,000 yr of the model run. The shaded regions In addition to being large-scaled, the surface magnetic field indicate the range of possible solutions for Mercury’s magnetic is also very axisymmetric. On time-average only 3% of the field in terms of these two coefficients that satisfy the field ob- magnetic energy resides in non-axisymmetric modes. This is servations of the Mariner 10 flybys within 5 nT according to in strong contrast to the magnetic energy inside the fluid Eq. (3). While the solution for the actual Mercury field is re- shell, where harmonic modes with m>0 contribute more than stricted to one of the branches, mirror images have been added 97%. The strong damping of higher harmonics and of non- Magnetic field generation in Mercury and Saturn 25 axisymmetric modes in the field outside the core has been ex- layer (which we will treat for the moment as if it were com- plained in Christensen (2006) by the different time-dependent pletely stagnant). However, the low-order axisymmetric modes, behavior of the various modes generated in the dynamo region. such as the axial dipole, have a component that varies with long All field components show fast fluctuations on time-scales of periods up to 100,000 yr. These low-frequency components can 200 yr (Fig. 5). These are almost completely filtered out by the escape through the stable layer with little or no attenuation, skin effect when the field diffuses through the stably stratified aside from the geometric decrease of the field strength accord- ing to r−(n+2). The non-axisymmetric modes have very little energy at low frequencies and are largely lost in the outside field (Fig. 5b). The relative power at low frequencies tends to become less with increasing harmonic degree also for the axisymmetric modes (Fig. 5c). This agrees with the finding from observations of the geomagnetic field (Olsen et al., 2006) and geodynamo modeling (Christensen and Tilgner, 2004) that the characteristic time scale of secular variation depends on harmonic degree as n−1 or n−1.35. Higher harmonics are sup- pressed compared to low-order components in the outside field not only by their faster geometric decay, but also by a stronger skin effect. Christensen and Tilgner (2004) found that, aside from the dependence on n, the secular variation time scale is controlled by the magnetic Reynolds number. Since the model values of Rm are arguably in the correct range for Mercury, the filtering by the skin effect should be correctly represented in the models. For fluctuations of intermediate periods, T ≈ 5000 yr, a phase lag of the outside field relative to the interior field of about 1500 yr is obvious in Fig. 5d. This compares well with the predicted phase lag for the diffusion√ through a plane layer of thickness D/2, which is TD/(2 4πλT)≈ 1600 yr. The Rayleigh number is increased stepwise going from 0 0 case 2 to case 4 for otherwise identical parameters. The mag- Fig. 4. Plot of Gauss coefficient g1 versus g2 during the model run of case 3. Shaded regions indicate acceptable models for Mercury’s actual field as de- netic field strength in the dynamo region increases with the scribed in the text. Rayleigh number; the Elsasser number Λ rises from 2.3 to

Fig. 5. Time series of Gauss coefficients (full line) for case 3. The gray line shows an equivalent coefficient describing the poloidal magnetic field at r = ri + D/2 inside the fluid shell, scaled down by 0.1. (a) Axial dipole, (b) equatorial dipole component, (c) axial quadrupole, (d) closeup for the axial dipole. 26 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34

8.3. However, the time-average surface magnetic field becomes weaker, from an rms-value of 890 nT in case 2, to 420 nT in case 3, and to 285 nT in case 4. Compared to Mercury field models based on Mariner 10 data, the last value is slightly on the low side, but at some instances in time the combination of axial dipole and axial quadrupole moments falls into the range given by Eq. (3) also in model 4. The reason for the opposite trends of the internal and the surface field strength is that with increasing Rayleigh number relatively less magnetic energy is carried by the low-degree modes, and they tend to fluctuate at shorter periods. For example, the axial dipole shows no re- versals in case 2, four reversals per magnetic diffusion time in case 3, and nine reversals in case 4. An interesting question is why only the axisymmetric modes in the dynamo field have a low-frequency component that can pass the stable layer, whereas non-axisymmetric modes at the same harmonic degree do not. A fundamental reason for the preference for axisymmetric modes is the structuring of the flow by the Coriolis force, which tends to create a statistical align- ment with the rotation axis. There is no preferred longitude (barring coupling of the dynamo to a heterogeneous exterior structure), and non-axisymmetric modes should be able to drift freely in longitude. For the axisymmetric modes both polari- ties are possible in principle, and in some dynamo models with a weak relative dipole contribution the dipole reverses errati- cally on short time scales (Kutzner and Christensen, 2002). In the present models the dipole polarity is stabilized by the stag- nant conducting regions of the solid inner core. For η = 0.5the 2 2 dipole decay time of the inner core ri /(π λ) is 2700 yr. It is Fig. 6. Variation of components of the velocity (a) and of the magnetic field expected that statistical polarity fluctuations in the fluid core (b) with radius in case 4. rms-values averaged in the angular directions and only lead to a complete reversal if they persist for a longer time in time are shown. Full line—poloidal component, broken line—toroidal, dot- (Hollerbach and Jones, 1993), thus stabilizing a given polar- ted—axisymmetric toroidal, dash-dotted—axisymmetric poloidal. The axisym- metric poloidal velocity is tiny and not shown. The vertical line indicates the ity. We tested this hypothesis by branching off from the non- nominal radius of neutral stability. reversingcase2asimulationwheretheinner core conductivity is set to zero (case 2a). After a transitional period it settled into components have similar amplitude, as expected for columnar a state with frequent dipole reversals. As a consequence, the convection. In the stable layer the flow is mainly toroidal and surface magnetic field is significantly weaker than in case 2 and weaker than in the convecting region, but not much weaker. In the axial dipole is no longer as dominant as before (Table 2). a transitional region that extends roughly to 0.65D above the In similar tests for geodynamo-like models Wicht (2002) found inner core boundary, overshooting of convection is the main little evidence for a stabilizing effect of inner core conductivity. contributor to the poloidal and the non-axisymmetric toroidal The relatively larger size of the inner core and the additional velocity. As shown in Christensen (2006), convection columns stabilizing influence of the stably stratified conducting layer in that align with the rotation axis penetrate into the stable layer, the fluid shell may provide sufficient magnetic inertia to make the effect significant in the present models. although the velocity decreases in magnitude and becomes in- While inner core conductivity can explain the relative stabil- creasingly toroidal with distance from the equatorial plane. This ity of the axial dipole polarity even in a dynamo whose dipole is akin to “teleconvection” described by Zhang and Schubert field is weak compared to higher multipole components, it can (2000). In the convective state the thickness of the unstable be argued that the same effect could likewise stabilize the equa- layer, in the sense of the distance of the minimum in the radi- torial dipole against rapid variations. This does not happen in ally averaged co-density profile from the inner core boundary our models. In order to understand the reasons, we must con- (Fig. 7a) is less than the nominal thickness dn based on the sider the flow in the stably stratified layer and its effect on the minimum in the diffusive profile (0.32 versus 0.441, respec- magnetic field. tively). However, the radius at which the advective co-density flux (buoyancy flux) reaches zero is almost identical to the nom- 4.2. Flow in the stable layer inal value of the neutral radius (Fig. 7b). In the overshoot region some downward advection of co-density occurs in addition to Fig. 6 shows the variation of velocity and magnetic field with the downward diffusive flux. This enables mixing in this region radius. In the dynamo region, poloidal and toroidal velocity and implies that work is done against buoyancy forces. The en- Magnetic field generation in Mercury and Saturn 27

Fig. 7. Stability and buoyancy flux in case 4. (a) Radial variation of co-density. Full line—co-density averaged horizontally and in time, broken line—diffusive co-density profile. (b) Time-average radial co-density flux. Full line—negative diffusive flux, broken line—negative total flux, dash-dotted line—positive ad- vective flux scaled up by a factor 3, obtained as difference between total and conductive flux. ergy for this must come from the power generated in deeper layers. Fig. 8. Various properties averaged in azimuth and in time for case 4. In the upper part of the stable layer the mean poloidal veloc- (a) uφ , contour step 75, (b) C, contour step 0.012, (c) ∂uφ/∂z,(d)RaEPm/ ity is of order 20, which is small but not negligible (Fig. 6a). (2Pr)∂C/∂θ. Broken contour lines for negative values, shading indicates ab- solute amplitude. Monitoring ur at shallow depth shows that it fluctuates rapidly and changes sign at any given location on a time scale of a few years. The fluctuations are too rapid for convective flow; rather ing this average by square brackets, we obtain they represent wave motion, most probably gravity waves. The ∂[u ] RaEPm ∂[C] φ = . (28) Brunt-Väisälä frequency in the upper part of the stable layer ∂z 2Pr ∂θ is approximately four years for the given co-density profile, In Figs. 8c and 8d we have calculated the left-hand side and in good agreement with the periods found in the model. The right-hand side of Eq. (28), respectively, for the time-average gravity waves are excited by the high-frequency part of the con- zonal velocity and co-density field. The agreement in pattern vective motion in the dynamo region and propagate upward into and amplitude is good. Our finding that the zonal flow is mainly the stable layer. driven by a thermal wind agrees with that of Aubert (2005) for The toroidal flow in the upper part of the stable layer is fully convective dynamo models. almost entirely axisymmetric, i.e., represents zonal wind or dif- We have tested the influence of the stable conducting layer ferential rotation (Fig. 6a). Its magnitude decreases towards the and of the flow in this layer on the magnetic field by calculat- surface but is quite large overall, with an average value of 250 ing model variants of case 4. In case 4a we removed this layer in the stable layer. One effect of the zonal flow is the generation entirely and simulate a “naked” dynamo layer in isolation. The of toroidal magnetic field by the shearing of poloidal field lines upper boundary in this case coincides with the neutral radius (ω-effect). In the upper part of the stable region the strength of in the original model. Because the thickness of the fluid shell is the toroidal field exceeds that of the poloidal field by far, and the our basic length scale, the non-dimensional parameters Ra, E,  field is predominantly axisymmetric (Fig. 6b). Fig. 8 shows the and η that involve the shell thickness must be recalculated with pattern of the zonal flow and demonstrates that it mainly rep- respect to their values in case 4. To have mechanical conditions resents a thermal wind, i.e., a circulation driven by latitudinal close to those of the deep dynamo layer, which is bounded from temperature variations. The thermal wind balance is obtained above by a mobile fluid, we have set the boundary condition at by neglecting viscous, Lorentz and inertia forces in Eq. (13), the outer radius to free slip. The same kind of comparison is taking the curl and average its φ-component in azimuth. Denot- done with case 1a which corresponds to model 1. The magnetic 28 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34

strong field inside the dynamo. The simple fact that the active dynamo is at greater depth, hence its field strength lower at the planetary surface than that of a dynamo which reaches up to the core–mantle boundary, is insufficient to explain the weak surface field. Next we addressed the question of the role of motion inside the stable layer for the magnetic field by calculating models where we artificially suppress the flow at radii r>ri + D/2 (cases 1b and 4b). The outer shell is a passive conductor in these models, much like the solid inner core. If the properties of the surface field could be explained mostly by a “passive” skin effect, which results from diffusion through a stagnant con- ductor, the results should be similar to those of cases 1 and 4, respectively. However, we find strong differences. Compared to case 1, the dynamo properties change fundamentally in case 1b, which falls into the dipolar class. Inside the fluid shell the dipole component is now stronger than higher harmonic com- ponents of the poloidal field and it does not reverse. In case 1 the dipole is comparatively weak and reverses on time scales of a few thousand years. This shows that flow in the stagnant layer can act back on the dynamo. However, case 1 seems to lie close to the transition point between the dipolar and multi- dipolar regimes and hence the solution may be very sensitive Fig. 9. Snapshots of magnetic field at the planetary surface. (a) Case 4 with contour step 50 nT, (b) case 4a with 2000 nT contour step. to changes in any condition. Because of the fundamental differ- ence in the dynamos, a direct comparison of the surface fields of cases 1 and 1b is of little value. field strength inside the “naked” dynamos 1a and 4a is compa- In contrast to the pair 1/1b, the spectra of the internal mag- rable to that of their counterparts, being slightly larger, and the netic field are very similar for cases 4 and 4b. However, the spectra of the interior fields are roughly similar, suggesting that surface field is stronger in case 4b by a factor of three. This is the internal properties of the dynamo are not strongly affected mostly due to the addition of non-axisymmetric modes, which by the presence or absence of the stable layer. now carry 78% of the power compared to 3% in case 4. For ex- For calculating the magnetic field at the planetary surface in ample, the axial dipole has similar strength in both cases, but cases 1a and 4a we took the outer radius of the model as the while the equatorial dipole is small in case 4, it exceeds on time radius of Mercury’s core (Table 1). With this assumption these average the axial dipole in case 4b. The equatorial dipole and cases represent Mercury models with a fully convective fluid other non-axisymmetric field components show a strong low- core and a solid inner core that is larger than in the correspond- frequency component at the top of the dynamo region in case ing models 1 and 4. Fig. 9 compares snapshots of the surface 4b and can therefore penetrate the stable conducting layer. magnetic fields of cases 4 and 4a. As expected, in the absence Stevenson (1982) has presented a simplified analytical the- of a stable layer the surface magnetic field is much stronger and ory of how differential rotation in the stable layer affects the is characterized by non-axisymmetric modes, which dominate non-axisymmetric field. According to his result, a mode of har- inside the active dynamo region anyway. On time average the monic order m = 1 will be attenuated by a factor of more then surface field is 25 times stronger in case 4a compared to case 4. 2 ten when the parameter αRm = UφL /(Rλ) exceeds the range Non-axisymmetric modes carry 97% of the energy in 4a (a sig- 20–30, where Uφ is the characteristic velocity of differential ro- nificant part of it is in the equatorial dipole) compared to 3% in tation, L the thickness of the stable layer, and R the radius. The case 4. Similar results hold for the comparison between cases 1 dynamo field itself is assumed to be stationary. For our case 4 2 and 1a (see Table 2). we find αRm = uφ(1 − dn) (1 − η) ≈ 80, which confirms that An alternative way of scaling the outside magnetic field of non-axisymmetric modes should be strongly damped. Case 4b cases 1a and 4a is to assume a hypothetical planet, which has the with a stagnant outer layer shows that also non-axisymmetric same size as Mercury but a smaller and fully convective core. components of the dynamo-generated field can vary on in- Here the outer core radius coincides with the neutral stability trinsically long time-scales. However, when horizontal flow is radius rn of case 1 or case 4, respectively. Compared to the first allowed in the stable layer, the differential rotation that we typ- method of scaling, the geometric field decay is stronger and ically find in our models will strongly suppress these modes in leads to a weaker surface field with a redder spectrum. Still, the the outside field. surface field is far less axisymmetric and is seven times stronger The fraction of non-axisymmetric magnetic energy in the in case 1a compared to case 1 and is 18 times stronger in case surface spectrum is 5% in our Mercury-like models, with the 4a compared to 4. This highlights the importance of the stable exception of case 1 where it is 39%. In that case we find that the conducting layer for obtaining a weak surface field in spite of a toroidal flow near the core–mantle boundary, which is predom- Magnetic field generation in Mercury and Saturn 29

Fig. 10. Snapshot of the radial magnetic field at the planetary surface of case 5. Contour interval is 100 nT. inantly zonal in other models, has a large non-zonal component representing the penetration of columnar flow far into the sta- ble region (Christensen, 2006). Advection of the dipole field in latitudinal direction near the core surface modulates the field structure such that it deviates from axisymmetry.

4.3. Variation of inner core radius and stable layer thickness

We have varied the fractional radius η of the inner core be- Fig. 11. Time series of Gauss coefficients for case 5. See Fig. 5 for explanations. tween 0.35 and 0.6 and the relative thickness of the unstable layer between 0.3 and 0.6. The surface magnetic field in the dynamos with η  0.5 is dominated by the axial dipole (ex- with parameters similar to those of case 3 led to a dynamo in cluding cases without a stable layer). With a small inner core, the strongly dipolar regime. We found a Mercury-like dynamo for a higher Rayleigh number and smaller value of d (case 9). η = 0.35, we find a tendency towards a strong contribution, or n The dynamo layer is only 260 km thick in this model. The even dominance, of the axial quadrupole. In case 1, the quadru- dipole-dominated surface magnetic field has a mean strength pole and dipole contribute equally on time average, with one of 240 nT. component dominating at some times and the other compo- The main effect of varying the thickness of the unstable layer nent at other times. In case 5 we doubled the Rayleigh num- with all other parameters held constant is an increase of the ber compared to case 1. This makes the surface magnetic field surface field strength with d (see Table 2). This is quite pre- strongly quadrupolar (Fig. 10). The time average mean surface n dictable because both the geometric field decay with r and the field strength is 380 nT, in agreement with the Mariner 10 ob- filtering of higher frequencies by the skin effect become weaker servations. In addition to the axial quadrupole, the harmonic for a thinner stable layer. Furthermore, in most cases there is component (n, m) = (4, 0) is rather large, corresponding to a a trend for the dipole to contribute more strongly to the sur- concentration of magnetic flux into the two polar caps. The face field in a relative sense when dn is smaller. As long as the quadrupolar field reverses nearly periodically on a 5000 yr time dynamo operates in the multipolar regime, the quadrupole con- 0 g0 scale (Fig. 11). The g4 coefficient changes in phase with 2 . tribution to the surface field (which is more strongly affected When we reduce the relative thickness of the unstable layer by geometric attenuation then the dipole) goes up for a thinner from 0.41 in case 5 to 0.30 in case 6, the dominance of the axial stable layer. dipole is restored, but the quadrupole still makes a significant contribution. In this case the surface field strength is almost a 4.4. Quadrupole or dipole dominance? factor 10 below the Mercury value. Case 7 has the same para- meters as case 1, except that the thickness of the unstable layer We already reported that the field at the planetary surface is is increased from 0.41 to 0.50. At times still the dipole domi- dominated by the axial quadrupole contribution rather than by nates, but the axial quadrupole is stronger on average. A further the dipole in some of our models that have a small inner core. increase of dn to 0.6 (case 8) causes a change from the multi- Here we further investigate the conditions for the dominance of polar to the dipolar dynamo regime. Case 8 is discussed further one of the two harmonic contributions. in Section 4.5. Most of the dynamos with η = 0.35 have been calculated Obviously the inner core radius cannot be too large for the for an Ekman number of 3 × 10−5, whereas we use E = 10−4 filtering effect of the stable layer to be effective, because a very in cases with η  0.5. To clarify if the cause for the quadrupole large inner core leaves insufficient space for a stable region. The preference is the smaller size of the inner core or the lower value maximum value of η = 0.6 for which we performed model sim- of the Ekman number, we calculated case 11 with η = 0.35, −4 ulations corresponds to an inner core of 1110 km radius (using E = 10 and dn = 0.41. It can be compared to the similar our nominal Mercury properties). At this value of η a model run cases 1 and 5, which have a lower Ekman number. The Rayleigh 30 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34 number in case 11 is less in absolute terms, but in terms of its ratio to the critical Rayleigh number it is slightly larger than in case 5. The axial quadrupole dominates the surface field even more clearly than in case 5, demonstrating that the lower value of E in the models with small inner core is not the cause for the quadrupole preference. The local Rossby number in the Mercury-like models dis- cussed so far is in the range of 0.09–0.44 when calculating Ro from Eq. (25) and ignoring the pre-factor, i.e., setting a = 1. This is not far above the threshold value for the transition between dynamos with dipolar and multipolar internal fields: when we reduce the Rayleigh number in case 1 somewhat, we obtain at Ro = 0.06 a dynamo with a strongly dipolar internal field (case 13, discussed in Section 4.5). We note that the sur- face magnetic field generally tends to become more quadrupo- lar with increasing local Rossby number. This becomes evident when comparing cases 1, 5 and 11 with Ro = 0.085, 0.12, 0.27 and quadrupole contributions of 35, 51 and 86%, respec- tively. The Mercury value of Ro is of order 25 when we take −1 −6 2 −1 Qi = 6000 kg s and ν = 4 × 10 m s in Eq. (26).This is hard to reach in dynamos models with a decently low value of the Ekman number. We reach the largest value of 3.6 for the local Rossby number in case 12, which basically uses the Fig. 12. Radial magnetic field at the surface (scaled with Saturn parameters same parameters as model 2 except for a larger value of the Ek- and ignoring surface flattening). (a) Case 13 with contour interval 5000 nT, (b) case 13a with contour interval 15,000 nT. man number E = 10−3. While in all other cases with η  0.50 the dipole dominates the surface magnetic field, in case 12 it is characterized by a non-reversing axial quadrupole and has on when scaled with Mercury parameters the difference would be time-average only a very small dipole contribution (Table 2). at least by a factor 20. The components (n, m) = (2, 0) and (4, 0) together contribute These two cases represent Saturn’s dynamo in a less direct 98% to the power in the surface field. In case 12 the Rossby sense than the other models may represent Mercury’s dynamo, number is an order of magnitude larger than in the other mod- because the magnetic Reynolds number in the gas giants is ex- els but still falls short of the estimated Mercury value. pected to be much larger than it is in the models. Also, the We conclude that in Mercury-like dynamos both the geom- relative thickness of the stable layer may be too large in the etry and the local Rossby number control whether the axial models compared to what it is in Saturn, at least in case 13. dipole or the axial quadrupole dominates the surface field. The Furthermore, we do not take into account that electrical con- dipole is more preferred for a larger inner core and a thin- ductivity increases gradually in a transition region between the ner unstable layer. Larger values of the local Rossby number, molecular and the metallic form of hydrogen (Nellis, 2000), and i.e. stronger driving of convection or slower rotation, favor a that Saturn’s dynamo is probably not driven by a co-density quadrupolar field at the planet’s surface. flux from below. Nonetheless, a direct comparison is tempting because the scaled magnetic dipole moments agree with that 4.5. Strongly dipolar dynamos of Saturn within a factor of 1.5. The time average Gauss co- 0 efficient g1 is 14,400 nT in case 8 and 15,600 nT in case 13, As discussed in Section 3.2, Saturn’s dynamo is likely to op- compared to 21,000–22,000 nT from the Voyager observations erate in the regime where the dipole dominates the spectrum of Saturn (Connerney et al., 1982). In case 13, the surface mag- inside the dynamo region. Hence we have used Saturn’s physi- netic field is extremely dipolar, with an average dipole tilt of ◦ cal parameters (Table 1) to scale those two models with a stable 1.5 . Non-axisymmetric modes carry only 0.2% of the surface ◦ upper layer that generate a dipole-dominated interior field. Both magnetic energy (Fig. 12a). In case 8 the mean dipole tilt is 3.6 cases are derivates of case 1. In case 13 the Rayleigh number and 2% of the energy is in non-axisymmetric modes. is reduced to 5/8 of its value in case 1. However, it turned out For comparison with case 13 we have run a “naked” dy- that the magnetic Prandtl number had to be increased from 3 to namo model (case 13a), in which the stable layer is removed 5 to avoid the decay of the magnetic field. In case 8 the relative (see the previous description of case 4a for more details). When thickness of the unstable layer dn is increased to 0.6 from 0.41 the surface field is scaled by assuming a fully convective core in case 1. Because the relative dipole contribution to the total extending outward to the nominal core radius of Saturn, it is field in the dynamo region is much larger than it is in the case still dominated by the dipole in case 13a, but non-axisymmetric of multipolar dynamos, and because the dipole does not reverse components now contribute significantly (Fig. 12b). On time- on a magnetic diffusion time scale, the surface magnetic field is average, they contain 24% of the surface magnetic energy. This considerably stronger than in the cases discussed before. Even demonstrates the importance of the stable layer for suppress- Magnetic field generation in Mercury and Saturn 31 ing non-axisymmetric modes also in the case when the internal only non-axisymmetric modes m>0inEq.(29) reduces the dynamo field is dipole-dominated. A test for the role of differ- time scale to 250 yr in case 4. However, these all represent mi- ential rotation in the stable layer, by immobilizing the region nor contributions to the surface field. Because it is unlikely that r>ri + D/2 (case 13b), did not show significant differences the upcoming space mission can resolve the spatio-temporal in the geometry of the surface magnetic field compared to case character of planetary magnetic fields in detail, it seems suffi- 13. This is in contrast to the result of the same test for model cient to consider the global time constant for secular variation. 4, where non-axisymmetric components become much more prominent when flow in the stable layer is suppressed. We note 5. Discussion and conclusions that in case 13 the toroidal flow in the stable layer is weaker and less dominated by axisymmetric modes than it is in case 4. Our numerical simulations consolidate the findings of Chris- Perhaps non-axisymmetric modes show an intrinsically higher tensen (2006) on the effect of a stable conducting layer in plan- degree of time dependence in dynamos with a strong internal di- etary dynamos and elucidate the mechanisms of how it controls pole field. In this case the passive skin effect would be sufficient the strength and structure of the observable field outside the to suppress them outside the electrically conducting region. core. The main effect of the stable shell is to act as a frequency 4.6. Secular variation filter, suppressing rapidly fluctuating field components. There is a relation between the spatial scale of the field and its time It is obvious that the filtering effect of the stable layer results variability. Higher multipoles tend to vary more rapidly than in much slower secular variation of the surface magnetic field low-degree components, hence the former are more strongly than what is characteristic for the rate of field change inside the suppressed by the skin effect. dynamo region. With a view on future observations of planetary While a “passive” skin effect, that arises also in a completely magnetic fields it is useful to quantify the rate of secular change stagnant shell, basically explains the strong preference for large in our models. We have calculated the characteristic time scale scales in the field above the core surface, it may be insuffi- of change of the surface magnetic field, defined as cient to explain the preferred selection of zonal modes. Our   simulations confirm the important role of differential rotation [B2] 1/2 τ = , (29) in the stable layer for suppressing non-axisymmetric field com- [ 2] (∂B/∂t) ponents. This was first envisioned by Stevenson (1980, 1982) where []stands for the average taken over the planetary surface in the context of Saturn’s magnetic field, but it may also play an and over time. This follows the convention used by Hulot and important role for Mercury. We find that the differential rota- Le Mouël (1994) and Olsen et al. (2006). Here, we do not dis- tion is maintained by a thermal wind effect. Stevenson assumed tinguish between the variability at different harmonic degrees. externally imposed temperature differences from the latitudi- The time scale τn, which is specific for harmonic degree n,is nal differences in insolation at Saturn’s surface. In our models defined in an analogous way, using only the contribution at de- the temperature (or co-density) differences arise from the dy- gree n to B and ∂B/∂t in Eq. (29). namics of the convective layer. However, in some dynamos the In all models with a stably stratified layer we find that τ non-axisymmetric modes may have a higher intrinsic variabil- exceeds 1000 yr, in some cases by far. The only exception ity than zonal field components, in which case a passive skin is the quadrupolar case 5 with quasi-periodic reversals, where effect would be sufficient to preferentially select the zonal com- τ = 800 yr. It is instructive to compare case 4, with a weak ponents. dipole-dominated surface field, where τ = 2000 yr, and its vari- Saturn and Mercury are the two planets in the Solar System ants. In case 4b we artificially suppressed the flow in the stable where we have reasons to assume that a stable conducting layer layer, which led to much larger non-axisymmetric contribu- of sufficient thickness shields a dynamo operating below this tions in the surface field. Here the secular variation time is only layer. The uncertainties on the thickness of the stable shell are slightly shorter (1850 yr). In the case of a fully convective core large in both cases. For Mercury we have shown that the subdi- without a stable layer (case 4a), we find τ = 114 yr. Comparing vision into an active dynamo region and a thick stably stratified cases 1 and 1a gives a similar result, τ = 1300 yr and 180 yr, region would occur for a plausible range of core heat fluxes and respectively. The small values of τ for the dynamos in a fully core compositions. convective core are comparable to the secular variation time The consequences for the observable field differ somewhat, scale of harmonics with n>1 in the geomagnetic field (Olsen depending on whether the dynamo is intrinsically in a regime et al., 2006). We do not find a systematic difference in τ be- where a strongly dipolar field is generated or if it is in the multi- tween cases with dipolar or quadrupolar surface field. Cases 8 polar regime. According to the study by Olson and Christensen and 13 with a strong dipole field inside the dynamo region have (2006) this is to a large degree controlled by the planetary rota- exceptionally long time scales of 6000 yr and 30,000 yr, respec- tion rate, which puts Saturn into the dipolar regime and Mercury tively. into the non-dipolar regime. The variation time scales of harmonic modes other than the In the case of Saturn, the strong axial dipole generated in the dominant (dipole or quadrupole) mode in the surface field are dynamo region is hardly attenuated by the stable layer, except often significantly shorter than the global time scale. For exam- for the geometric decrease of field strength proportional to r−3, ple in case 4, τn ≈ 500 yr for n = 2 and 3. Also, considering which is larger when the outer edge of the active dynamo re- 32 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34 gion is at greater depth. Therefore, the magnetic field at Saturn’s of magnitude and a more complete understanding of how the surface is fairly strong, but not as strong as it would be if the dy- various parameters control the pattern selection is needed. namo region extended up to the top of the conducting core. The An interesting question concerns the maximum radius of the main effect of the stable layer is in this case to suppress non- inner core that would still be compatible with our explanation axisymmetric modes, either by a “passive” or by an “active” of Mercury’s weak field. We found a dynamo with a fractional skin effect due to differential rotation in the stable layer, as sug- inner core radius of 0.6 that matches Mercury’s field strength. gested by Stevenson (1980). Although our models in the dipolar This may not be the upper limit, provided the active dynamo dynamo regime are remote from Saturn conditions in several re- is restricted to a thin but vigorously convecting sublayer of the spects, they nonetheless show highly axisymmetric fields with fluid core, so that the time scales of variation of the dominant a strength similar to Saturn’s field. However, the model field is multipole modes are short enough to be efficiently damped even also extremely dipolar, whereas the axial quadrupole and octu- by a moderately thin stable layer. An upper limit for the inner pole contribute significantly to Saturn’s field (Connerney et al., core radius is hard to estimate, but might lie around 75% of the 1982). core radius. Our models are better adapted to Mercury, capturing prob- Our predictions for Mercury can be tested with the new mag- ably the correct values of the magnetic Reynolds number, the netic field data obtained by the Messenger and Bepi Colombo geometry of the fluid core, and the driving of the dynamo by a missions. According to our models Mercury’s field should be buoyancy flux emanating from the inner core boundary. How- large-scaled and fairly axisymmetric. An superimposed field ever, our description of the combination of compositional and contribution from remanent crustal magnetizations could blur thermal buoyancy by the co-density formulation is simplified. the picture. However, if the spatial character of the crustal field While the co-density approach may correctly describe the large- is similar to what we know for Earth and Mars, it may be dis- scale dynamics in the region of turbulent convection, it is more tinguishable by its fairly white spectrum. problematic in the stably stratified region. The dynamics of the In January 2008 Messenger passed 200 km above the sur- stable layer could be more complex than in our models. For ex- face of Mercury at low latitudes in a first flyby. In combination ample, it is conceivable that episodic overturns may occur when with the magnetometer data from the third Mariner 10 flyby it the light element in the convective layer has become enriched should be possible to resolve the ambiguity expressed in Eq. (3) enough to overcome the stabilizing thermal buoyancy in the up- and to distinguish a dipole-dominated field from a dominantly per layer. However, the density stratification in the upper layer quadrupolar field. The new data seem to support a dipolar solu- should on time-average still be stabilizing and permit little ra- tion (http://messenger.jhuapl.edu). dial motion. A more rigorous treatment of the problem in terms Messenger and Bepi Colombo in orbit will be able to charac- of double diffusive convection (e.g., Turner, 1973) is desirable terize the magnetic field structure unambiguously. Combining and remains a task for future work. results from the different missions, it may also be possible to The cause for the weakness of Mercury’s surface field is detect a secular variation of the internal field, or more likely, to that the dipole and quadrupole components are only minor con- put an upper limit on its rate. A challenge is to distinguish be- tributors to the total field in the dynamo region, but they are tween changes of the internal field and variations in the field the only components that show variations at long enough time contribution from magnetospheric currents. Our models with a scales to allow passage of the stable layer filter. The sluggish stably stratified layer in the outer core typically have secular magnetic reaction of the conducting solid inner core seems to variation time scales in excess of 1000 yr, which means that the play a significant role for stabilizing these components against field changes by less than 1% in the ten years time interval be- rapid variations. Non-axisymmetric field components are sup- tween the observations to be made by Messenger and those to pressed for the same reasons as discussed for Saturn; in particu- be taken by Bepi Colombo. It is very unlikely that such a small lar the “active” skin effect caused by differential rotation in the change in the internal field can be observed or discriminated stable layer may be essential. However, the non-axisymmetric against external field changes. Our dynamo models with a fully contribution to the surface field is not necessarily as weak convective outer core show more rapid secular variations. These as in the case when the dynamo field is intrinsically dipole- models are not suitable for Mercury, because the surface field dominant. is far too strong. But let us assume that their secular variation In some of our Mercury-like models the surface field is time scales, on the order of 100–200 yr, are representative for dipolar and in others it is dominated by the axial quadrupole. a Mercury dynamo operating in the whole volume of the core. A larger size of the inner core or a thinner convecting layer The field change during a ten years time interval would then be seem to favor the dipolar solution, whereas stronger driving of 5–10%, which is probably resolvable with the magnetometer the dynamo, leading to a larger value of the local Rossby num- data expected from the two space missions. The unambiguous ber, favors the quadrupolar solution. It is often assumed that detection of such changes in the internal magnetic field would the Mercury’s inner core is relatively large, which would argue clearly falsify our model of a deep dynamo. for a dipolar surface field. Once Mercury’s field geometry has Evidence on the size of the outer core and perhaps the in- been clearly established, it may be used as indirect evidence for ner core will come from informations on Mercury’s rotation the inner core size and/or the thickness of the convecting layer. state, gravity field, and reaction to solar tides. These are ob- However, the value of the local Rossby number in our models tained by laser altimetry on board of Messenger and Bepi falls short of the estimated Mercury value by one or two orders Colombo, from Doppler tracking of the spacecrafts’ radio sig- Magnetic field generation in Mercury and Saturn 33 nal, and from radar echos received at Earth. The forced libration Holme, R., Bloxham, J., 1996. The magnetic fields of Uranus and Neptune: of Mercury together with its obliquity and quadrupole grav- Methods and models. J. Geophys. Res. 101, 2177–2200. ity moment, and independently the tidal Love numbers, con- Hubbard, W.B., Podolak, M., Stevenson, D.J., 1995. The interior of Neptune. strain the state of the core and its radius (Peale et al., 2002; In: Cruikshank, D.P. (Ed.), Neptune and Triton. Univ. of Arizona Press, Tucson, pp. 109–138. Van Hoolst and Jacobs, 2003). Free librations of Mercury, pro- Hulot, G., Le Mouël, J.-L., 1994. A statistical approach to the Earth’s main vided they have been recently excited sufficiently to make them magnetic field. Phys. Earth Planet. Inter. 82, 167–183. observable, carry information on the radius of the inner core Kutzner, C., Christensen, U.R., 2002. From stable dipolar to reversing numeri- (Peale et al., 2002). If both Love numbers can be determined cal dynamos. Phys. Earth Planet. Inter. 131, 29–45. precisely, h2 by laser altimetry and k2 from the observation Kutzner, C., Christensen, U.R., 2004. Simulated geomagnetic reversals and pre- of the spacecrafts’ motion in Mercury’s gravity field, this may ferred virtual geomagnetic pole paths. Geophys. J. Int. 157, 1105–1118. also allow to put constraints on the size of the inner core Labrosse, S., Poirier, J.-P., LeMouël, J.-L., 1997. On cooling of the Earth’s core. Phys. Earth Planet. Inter. 99, 1–17. (Van Hoolst, 2004, personal communication). Both methods are Lewis, J., 1988. Composition of Mercury. In: Vilas, F., Vilas, C.R., Matthews, likely to lead to a positive detection of an inner core only if its M.S. (Eds.), Mercury. Univ. of Arizona Press, Tucson, pp. 429–460. radius is very large, perhaps more than 80% of the core radius. Lister, J.R., Buffett, B.A., 1995. The strength and efficiency of thermal and Our model would be falsified if an inner core of such size can compositional convection in the geodynamo. Phys. Earth Planet. Inter. 91, be found, whereas a null result would leave it as a viable option, 17–30. provided it stands the other tests. Lister, J.R., Buffett, B.A., 1998. Stratification of the outer core at the core– mantle boundary. Phys. Earth Planet. Inter. 105, 5–19. Margot, J.L., Peale, S.J., Slade, M.A., Holin, I.V., 2007. Large longitude libra- References tion of Mercury reveals a molten core. Science 316, 710–714. Nellis, W.J., 2000. Metalization of fluid hydrogen at 140 MPa (1.4 Mbar): Im- Aharonson, O., Zuber, M.T., Solomon, S.C., 2004. Crustal remanence in an plications for Jupiter. Planet. Space Sci. 48, 671–677. internally magnetized non-uniform shell: A possible source for Mercury’s Ness, N.F., 1979. The magnetic field of Mercury. Phys. Earth Planet. Inter. 20, magnetic field? Earth Planet. Sci. Lett. 218, 261–268. 209–217. Aubert, J., 2005. Steady zonal flows in spherical shell dynamos. J. Fluid Olsen, N., Lühr, H., Sabaja, T.J., Mandea, M., Rother, M., Tøffner-Clausen, Mech. 542, 53–67. L., Choi, S., 2006. Chaos—A model of the geomagnetic field derived from Braginsky, S.I., 1984. Short-period geomagnetic secular variation. Geophys. CHAMP, Ørsted and SAC-C magnetic satellite data. Geophys. J. Int. 166, Astrophys. Fluid Dyn. 30, 1–78. 67–75. Braginsky, S.I., Roberts, P.H., 1995. Equations governing convection in Earth’s Olson, P., Christensen, U.R., 2006. Dipole moment scaling for convection- core and the geodynamo. Geophys. Astrophys. Fluid Dyn. 79, 1–97. driven planetary dynamos. Earth Planet. Sci. Lett. 250, 561–571. Breuer, D., Hauck II, S.A., Buske, M., Pauer, M., Spohn, T., 2007. Interior evolution of Mercury. Space Sci. Rev. 132, 229–260. Peale, S.J., Phillips, R.J., Solomon, S.C., Smith, D.E., Zuber, M.T., 2002. Christensen, U.R., 2006. A deep dynamo generating Mercury’s magnetic field. A procedure for determining the nature of Mercury’s core. Meteorit. Planet. Nature 444, 1056–1058. Sci. 37, 1269–1283. Christensen, U.R., Aubert, J., 2006. Scaling properties of convection-driven dy- Sarson, G.R., Jones, C.A., Longbottom, A.W., 1997. The influence of boundary namos in rotating spherical shells and applications to planetary magnetic layer heterogeneities on the geodynamo. Phys. Earth Planet. Inter. 101, 13– fields. Geophys. J. Int. 166, 97–114. 32. Christensen, U.R., Tilgner, A., 2004. Power requirement of the geodynamo Schubert, G., Ross, M.N., Stevenson, D.J., Spohn, T., 1988. Mercury’s thermal from ohmic losses in numerical and laboratory dynamos. Nature 429, 169– history and the generation of its magnetic field. In: Vilas, F., Chapman, C.R., 171. Matthews, M.S. (Eds.), Mercury. Univ. of Arizona Press, Tucson, pp. 651– Christensen, U.R., Wicht, J., 2007. Numerical dynamo simulations. In: Schu- 666. bert, G. (Ed.), Treatise of Geophysics, vol. 8. Elsevier, Amsterdam, pp. 245– Stanley, S., Bloxham, J., 2004. Convective-region geometry as the cause of 282. Uranus’ and Neptune’s unusual magnetic fields. Nature 428, 151–153. Connerney, J.E.P., 1993. Magnetic fields of the outer planets. J. Geophys. Stanley, S., Bloxham, J., 2006. Numerical dynamo models of Uranus’ and Nep- Res. 98, 18569–18679. tune’s magnetic fields. Icarus 184, 556–572. Connerney, J.E.P., Ness, N.F., 1988. Mercury’s magnetic field and interior. In: Stanley, S., Bloxham, J., Hutchison, W.E., 2005. Thin shell dynamo models Vilas, F., Chapman, C.R., Matthews, M.S. (Eds.), Mercury. Univ. of Ari- consistent with Mercury’s weak observed magnetic field. Earth Planet. Sci. zona Press, Tucson, pp. 494–513. Lett. 234, 341–353. Connerney, J.E.P., Ness, N.F., Acuna,˜ M.H., 1982. Zonal harmonic models of Stephenson, A., 1976. Crustal remanence and the magnetic moment of Mer- Saturn’s magnetic field from Voyager 1 and 2 observations. Nature 298, cury. Earth Planet. Sci. Lett. 28, 454–458. 44–46. Stevenson, D.J., 1980. Saturn’s luminosity and magnetism. Science 208, 746– Fortney, J.J., Hubbard, W.B., 2003. Phase separation in giant planets: Inhomo- 748. geneous evolution of Saturn. Icarus 164, 228–243. Stevenson, D.J., 1982. Reducing the non-axisymmetry of a planetary dynamo Giampieri, G., Dougherty, M.K., Smith, E.J., Russell, C.T., 2006. A regular and an application to Saturn. Geophys. Astrophys. Fluid Dyn. 21, 113–127. period for Saturn’s magnetic field that may track its internal rotation. Na- Stevenson, D.J., 1987. Mercury’s magnetic field: A thermoelectric dynamo? ture 441, 62–64. Earth Planet. Sci. Lett. 82, 114–120. Gurnett, D.A., Persoon, A.M., Kurth, W.S., Averkamp, T.F., Dougherty, M.K., Southwood, D.J., 2007. The variable rotation period of the inner region of Stevenson, D.J., 2003. Planetary magnetic fields. Earth Planet. Sci. Lett. 208, Saturn’s plasma disk. Science 316, 442–445. 1–11. Hauck, S.A., Dombard, A.J., Phillips, R.J., Solomon, S.C., 2004. Internal and Stevenson, D.J., Salpeter, E.E., 1977. The dynamics and helium distribution in tectonic evolution of Mercury. Earth Planet. Sci. Lett. 222, 713–728. helium–hydrogen fluid planets. Astrophys. J. Suppl. 35, 239–261. Heimpel, M., Aurnou, J.M., Al-Shamali, F.M., Gomez Perez, N., 2005. A nu- Stevenson, D.J., Spohn, T., Schubert, G., 1983. Magnetism and thermal evolu- merical study of dynamo action as a function of spherical shell geometry. tion of terrestrial planets. Icarus 54, 466–489. Earth Planet. Sci. Lett. 236, 542–557. Takahashi, F., Matsushima, M., 2006. Dipolar and non-dipolar dynamos in thin Hollerbach, R., Jones, C.A., 1993. Influence of the Earth’s inner core on the spherical shell geometry with implications for the magnetic field of Mer- geomagnetic fluctuations and reversals. Nature 365, 541–543. cury. Geophys. Res. Lett. 33, doi:10.1029/2006GL02579. L10202. 34 U.R. Christensen, J. Wicht / Icarus 196 (2008) 16–34

Turner, J.S., 1973. Buoyancy Effects in Fluids. Cambridge Univ. Press, Cam- Wicht, J., 2002. Inner-core conductivity in numerical dynamo simulations. bridge. Phys. Earth Planet. Inter. 132, 281–302. Van Hoolst, T., Jacobs, C., 2003. Mercury’s tides and interior structure. J. Geo- Wicht, J., Mandea, M., Takahashi, F., Christensen, U.R., Matsushima, M., phys. Res. 108, doi:10.1029/2003JE002126. 5121. Langlais, B., 2007. The origin of Mercury’s internal magnetic field. Space Wetherill, G.W., 1988. Accumulation of Mercury from planetesimals. In: Vi- Sci. Rev. 132, 261–290. las, F., Chapman, C.R., Matthews, M.S. (Eds.), Mercury. Univ. of Arizona Zhang, K., Schubert, G., 2000. Teleconvection: Remotely driven thermal con- Press, Tucson, pp. 670–691. vection in rotating stratified spherical layers. Science 290, 1944–1947.