Mercury’s magnetic field in the MESSENGER era

J. Wicht1 and D. Heyner2

1Max-Planck Institut fur Sonnensystemforschung, Gottingen, Germany,wicht@mps.mpg.de

2Institut fur Geophysik und extraterrestrische Physik, TU Braunschweig, Braunschweig,Germany

January 19, 2017

Abstract

MESSENGER magnetometer data show that Mercury’s magnetic field is not only excep-tionally weak but also has a unique geometry. The internal field resembles an axial dipole thatis offset to the North by 20% of the planetary radius. This implies that the axial quadrupol isparticularly strong while the dipole tilt is likely below 0.8. The close proximity to the sun incombination with the weak internal field results in a very small and highly dynamic Hermeanmagnetosphere. We review the current understanding of Mercury’s internal and external mag-netic field and discuss possible explanations. Classical convection driven core dynamos havea hard time to reproduce the observations. Strong quadrupol contributions can be promotedby different measures, but they always go along with a large dipole tilt and generally rathersmall scale fields. A stably stratified outer core region seems required to explain not onlythe particular geometry but also the weakness of the Hermean magnetic field. New interiormodels suggest that Mercury’s core likely hosts an iron snow zone underneath the core-mantleboundary. The positive radial sulfur gradient likely to develop in such a zone would indeedpromote stable stratification. However, even dynamo models that include the stable layershow Mercury-like magnetic fields only for a fraction of the total simulation time. Large scalevariations in the core-mantle boundary heat flux promise to yield more persistent results butare not compatible with the current understanding of Mercury’s lower mantle.

1 Introduction

In 1974 the three flybys of the Mariner 10spacecraft revealed that Mercury has a globalmagnetic field. This was a surprise for manyscientists since an internal dynamo process wasdeemed unlikely because of the planet’s rel-ative small size and its old inactive surface[Solomon, 1976]. Either the iron core wouldhave already solidified completely or the heatflux through the core-mantle boundary (CMB)

would be too small to support dynamo action.The Mariner 10 measurements also indicatedthat Mercury’s magnetic field is special [Nesset al., 1974]. Being 100 times smaller thanthe geomagnetic field, it seems too weak tobe supported by an Earth-like core dynamo.And though the data were scarce, they never-theless allowed to constrain that the internalfield is generally large scale and dominated bya dipole but possibly also a sizable quadrupolecontribution. Both the Hermean field ampli-

1

arX

iv:1

701.

0506

0v1

[as

tro-

ph.E

P] 1

8 Ja

n 20

17

tude and its geometry are unique in our solarsystem.

Mercury is the closest planet to the Sun andtherefore subject to a particular strong anddynamic solar wind. Since Mercury’s mag-netic field is so weak, the solar wind plasmacan come extremely close to the planet andmay even reach the surface. Mariner 10 datashowed that Mercury’s magnetosphere is notonly much smaller than its terrestrial counter-part but also much more dynamic. Adaptedmodels originally developed for Earth failedto adequately describe the Hermean magneto-sphere which therefore remained little under-stood in the Mariner 10 era [Slavin et al., 2007].

Knowing a planet’s internal structure is cru-cial for understanding the core dynamo pro-cess. Mercury’s large mean density pointedtowards an extraordinary huge iron core anda relatively thin silicate mantle covering onlyabout the outer 25% in radius. Since littlemore data were available in the Mariner era,the planet’s interior properties and dynamicsremained poorly constrained.

Solving the enigmas about Mercury’s mag-netic field and interior where major incentivesfor NASA’s MESSENGER mission [Solomonet al., 2007]. After launch in August 2004and a first Mercury flyby in January 2008, thespacecraft went into orbit around the planet inMarch 2011. At the date of writing, more than2800 orbits have been completed. MESSEN-GER’s orbit is highly eccentric with a periap-sis between 200 to 600 km at 60 to 70northernlatitude and an apoapsis of about 15, 000 kmaltitude. This has the advantage that thespacecraft passes through the magnetosphereon each orbit but complicates the extractionof the internal field component because of astrong covariance of equatorially symmetricand anti-symmetric contributions [Andersonet al., 2012, Johnson et al., 2012]. The trade-off between the dipole and quadrupole fieldharmonics, that was already a problem with

Mariner 10 data, therefore remains an issue inthe MESSENGER era. The situation is fur-ther complicated by the fact that the classicalseparation of external and internal field contri-butions developed by Gauss [Olsen et al., 2010]does not directly apply at Mercury. It assumesthat the measurements are taken in a sourcefree region with negligible electric currents, anassumption not necessarily fulfilled in such asmall and dynamic magnetosphere.

In order to nevertheless extract informationon the internal magnetic field, the MESSEN-GER team analysed the location of the mag-netic equator where Bρ, the magnetic fieldcomponent perpendicular to the planet’s rota-tion axis, passes through zero [Anderson et al.,2011, Anderson et al., 2012]. Since the internalfield changes on a much slower time scale thanthe magnetosphere, the time-averaged locationshould basically not be affected by the magne-tospheric dynamics. The analysis not only con-firmed that the Hermean field is exceptionallyweak with an axial dipole of only 190 nT butalso suggested that the internal field is bestdescribed by an axial dipole that is offset by480 km to the north of the planet’s equator[Anderson et al., 2012]. This configuration,that we will refer to as the MESSENGER off-set dipole model (MODM) in the following, re-quires a strong axial quadrupole and a very lowdipole tilt, a combination that is unique in oursolar system.

This article tries to summarize the newunderstanding of Mercury’s magnetic field inthe MESSENGER era at the date of writing.MESSENGER is still orbiting it’s target andcontinues to deliver outstanding data that willfurther improve our knowledge of this uniqueplanet. Section 3 briefly reviews the currentknowledge of Mercury’s magnetosphere. Sec-tion 4 describes recent models for the planet’sinterior, focussing in particular on the possiblecore dynamics. The magnetic equator analysisand the offset dipole model MODM are then

2

discussed in section 4. Explaining the weak-ness of Mercury’s magnetic field already chal-lenged classical dynamo theory and the pecu-liar field geometry further raises the bar. Sec-tion 5 reanalysis several dynamo model can-didates in the light of the new MESSENGERdata. Some concluding remarks in section 6close the paper.

2 Mercury’s internalstructure

MESSENGER observations of Mercury’s grav-ity field [Smith et al., 2012] and Earth-basedobservations of the planet’s spin state [Margotet al., 2012] provide valuable information onthe interior structure. That fact that Mercuryis in a special rotational state (Cassini state 1)allows to deduce the polar moment of inertiaC from the degree two gravity moments andthe planet’s obliquity, the tilt of the spin axisto the orbital normal [Peale, 1969]. The mo-ment of inertia factor C/(MR2

M ), where M isthe planet’s total mass and RM its mean ra-dius, constrains the interior mass distribution.The factor is 0.4 for uniform density and de-creases when the mass is increasingly concen-trated towards the center. The Hermean valueof C/(MR2

M ) = 0.346 ± 0.014 [Margot et al.,2012] indicates a significant degree of differen-tiation.

The observation of the planet’s 88 day libra-tion amplitude g88, a periodic spin variationin response to the solar gravitational torqueson the asymmetrically shaped planet, allows toalso deduce the moment of inertia of the rigidouter part Cm. If the iron core is at least par-tially liquid, Cm is the moment of the silicateshell and thus smaller than C. The Hermanvalue of Cm/C = 0.431± 0.025 [Margot et al.,2012] confirms that the core remains at leastpartially liquid.

In addition to M and RM the ratios

C/(MR2M ) and Cm/C provide the main con-

straints for models of Mercury’s interior [Smithet al., 2012, Hauck et al., 2013]. Note thatRivoldini and Van Hoolst [2013] follow atsomewhat different approach, taking into ac-count the possible coupling between the coreand the silicate shell. The coupling has theeffect that Cm cannot be determined indepen-dently of the interior model and Rivoldini andVan Hoolst [2013] therefore directly use g88

rather than Cm as a constraint. The updatedinterior modelling indicates that the core ra-dius is relatively well constrained at 2020 ±30 km [Hauck et al., 2013] or 2004 ± 39km[Rivoldini and Van Hoolst, 2013]. This leavesonly the outer 16 to 19% of the mean planetaryradius RM = 2440 km to the mantle.

Hauck et al. [2013] find a mean mantle den-sity (including the crust) of 3380± 200 kg/m3.Measurements of MESSENGER’s X-Ray Spec-trometer (XRS) show that the volcanic surfacerocks have a low content of iron and other heav-ier elements [Nittler et al., 2011]. Smith et al.[2012] and Hauck et al. [2013] therefore spec-ulate that a solid FeS outer core layer maybe required to explain the mean mantle den-sity. Rivoldini and Van Hoolst [2013], however,argue that the mantle density is not particu-larly well contrained. Compositions compat-ible with XRS measurements are well withinthe allowed solutions and a denser lower man-tle layer is not required by the data.

Naturally, information about the core isof particular interest for the planetary dy-namo. There is a rough consensus on themean core density with Hauck et al. [2013]and Rivoldini and Van Hoolst [2013] suggesting6980± 280 km/m3 and 7233± 267 km/m3, re-spectively. However, the core composition andthe radius of a potential inner core are not wellconstrained. Admissable interior models coverall inner core radii from zero to very large val-ues with an aspect ratio of about a = ri/ro =0.9 [Rivoldini and Van Hoolst, 2013] where ri

3

and ro are the inner and outer core radii, re-spectively.

An additional constraint on the inner coresize relies on the observations of so-called lo-bate scarps on the planet’s surface which arelikely caused by global contraction. MESSEN-GER data based on 21% of the surface sug-gested a contraction between 1 and 3 km [DiAchille et al., 2012]. This sets severe boundson the amount of solid iron in Mercury’s corebecause of the density decrease associated withthe phase transition of the liquide core al-loy. Several thermal evolution models thereforefavour a completely liquide core or only a verysmall inner core [Grott et al., 2011, Tosi et al.,2014]. Recent more comprehensive MESSEN-GER observations, however, allow for a con-traction of up to 7 km. This somewhat releasesthe contraints [Solomon et al., 2014] thoughvery large inner cores may still be unlikely.

Sulfur has been found in many iron-nickelmeteorites and is therefore a prime candidatefor the light constituent in Mercury’s core.Rivoldini and Van Hoolst [2013] consider iron-sulfur core alloys and find a likely bulk sulfurconcentration of 4.5±1.8 wt%. Since this com-position lies on the iron rich side of the eutec-tic, iron crystalizes out of the liquid when thetemperature drops below the melting point.Where this happens first depends on the formof the melting curve and the adiabat describingcore conditions.

Since Mercury’s mantle is so thin it has likelycooled to a point where mantle convection isvery sluggish or may have stopped altogether[Grott et al., 2011, Michel et al., 2013, Tosiet al., 2014]. The heat flux through the core-mantle boundary is thus likely subadiabaticand therefore too low to support a core dy-namo driven by thermal convection alone. Therequired additional driving power may then ei-ther be provided by a growing inner core or byan iron snow zone. The solid inner core startsto grow as soon as the adiabat crosses the melt-

ing curve in the planetary center. Since thesolid iron phase can incorporate only a rela-tively small sulfur fraction, most of the sulfuris expelled at the inner core front and drivescompositional convection. The latent heat re-leased upon iron solidification provides addi-tional thermal driving power. Contrary to thesituation for Earth, freezing could also startat the core-mantle boundary (CMB) becauseof the lower pressures in Mercury’s core. Theiron crystals would then precipitate or snowinto the center and remelt when encounter-ing temperatures above the melting point ata depth rm. This process leaves a sulfur en-riched lighter residuum in the layer r > rm. Asthe planet cools, rm decreases and a stabilizingsulfur gradient is established that follows theliquidus curve and covers the whole snow zoner > rm [Hauck et al., 2006]. Since the heatflux through the CMB is likely subadiabatictoday, thermal effects will also suppress ratherthan promote convection in the outer part ofMercury’s core. A stably stratified layer under-neath the planet’s core mantle boundary andprobably extending over the whole iron snowregion therefore seems likely. The liquid ironentering the layer below rm serves as a com-positional buoyancy source. The latent heatbeing released in the iron snow zone diffuses tothe core mantle boundary. Today’s low CMBheat flux implies that this can be acchieved bya relatively mild temperature gradient.

The possible core scenarios are illustratedin fig. (1) with melting curves for differentsulfur concentrations and core adiabats withCMB temperatures in the range between 1600and 2000 K suggested by interior [Rivoldini andVan Hoolst, 2013] and thermal evolution mod-els [Grott et al., 2011, Michel et al., 2013, Tosiet al., 2014]. Data on the melting behaviourof iron-sulfur alloys are few and the meltingcurves shown in fig. (1) therefore rely on simpleparametrizations [Rivoldini et al., 2011]. Theadiabats have been calculated by Rivoldini and

4

Van Hoolst [2013]. Mercury’s core pressure isonly grossly constrained, with CMB pressuresin the range 4 − 7 GPa and central pressuresin the range 30 − 45 GPa [Hauck et al., 2013].We adopt a central pressure of 40 GPa here.Fig. (1) suggests that iron starts to solidify inthe center for an inital sulfur concentrationsbelow about 4 wt%. Sulfur released from theinner core boundary increases the concentra-tion in the liquid core over time and therebyslows down the inner core growth and delaysthe onset of iron snow. For an initial sulfurconcentration beyond 4 wt% iron solidificationstarts with the CMB snow regime. A convec-tive layer that is enclosed by a solid inner coreand a stably stratified outer iron snow layerseems possible for sulfur concentrations be-tween about 2.5 and 7 wt%. For sulfur concen-tration beyond 7 wt% an inner core would onlygrow when the snow zones extends through thewhole core and the snow starts to accumulatein the center.

The adiabats and thin red lines in fig. (1) il-lustrate the evolution for an initial sulfur con-centration of 3 wt%. For the hot (red) adiabatwith Tcmb = 2000 K neither inner core growthnot iron snow would have started and therewould be no dynamo. When the temperaturedops, iron starts to solidify first at the cen-ter. For a CMB temperature of Tcmb = 1910 K(solid green adiabat), the inner core has al-ready grown to a radius of about 600 km whilethe outer snow layer is only about 160 kmthick. The sulfur released upon inner coregrowth has increased the bulk concentrationin the liquide part of the core to 3.4 wt% (firstthin red line from the top). The decrease inthe sulfur abundance due to the remelting ofiron snow has not been taken into account inthis model. When the CMB temperature hasdropped to Tcmb = 1890 K (dashed green adia-bat) the inner core and snow layer have grownby a comparable amount while the sulfur con-centration has increased to 4.4 wt% (second

thin red line from the top). At Tcmb = 1750 K(grey) there remains only a relatively thin con-vective layer between the inner core bound-ary at ri = 1440 km and the lower bound-ary of the outer snow layer at rm = 1650 km.For the coldest adiabat shown in fig. (1) withTcmb = 1890 K (blue) only the outer 300 km ofthe core remain liquid but belong to the ironsnow zone so that no dynamo seems possible.

Additional sometimes complex scenarioshave been discussed in the context ofGanymede by Hauck et al. [2006] and may alsoapply at Mercury since the iron cores of bothbodies cover similar pressure ranges. For ex-ample, fig. (1) illustrates a kink in the meltingcurve for pressures around 21 GPa and com-positions larger than 5 wt% sulfur. This couldlead to a double snow regime where not onlythe very outer part of the core precipitates ironbut also an intermediate layer around 21 GPa.This possibility has been explored in a dynamomodel by Vilim et al. [2010] that we will dis-cuss in section 2. Since the kink is not verypronounced, however, such a double snow dy-namo would not be very long lived.

Another interesting scenario unfolds whenthe light element concentration lies on the S-rich side of the eutectic. Under these condi-tions, FeS rather than Fe would crystalize outwhen the temperature drops below the FeSmelting curve. Since FeS is lighter then theresiduum fluid, the crystals would rise towardsthe core-mantle boundary. However, eutec-tic or even higher sulfur concentrations can-not represent bulk conditions since it would bedifficult to match Mercury’s total mass [Rivol-dini et al., 2011]. Inner core growth would in-crease the sulfur concentration in the remain-ing fluid over time but never beyond the eutec-tic point. This has likely not been reached inMercury because the eutectic temperature of1200− 1300 K [Rivoldini et al., 2011] is signif-icantly lower than today’s CMB temperaturesuggested by thermal evolution [Grott et al.,

5

2011, Tosi et al., 2014] and interior models[Rivoldini and Van Hoolst, 2013].

An alternative explanation for a locally highsulfur concentration was suggested by the XRSobservations. The low Fe but large S abun-dance in surface rocks indicates that Mercury’score could have formed at strongly reducingconditions. This promotes a stronger parti-tioning of Si into the liquid iron phase lead-ing to a ternary Fe-Si-S core alloy [Malavergneet al., 2010]. Experiments have shown thatSi and S are immiscible for pressures below15 GPa [Morard and Katsura, 2010] which isthe pressure range in the outer part of Mer-cury’s core. However, the immiscibility onlyhappens for sizable Si and S concentrations.Experiments by Morard and Katsura [2010],for example, demonstrate that at 4 GPa and1900 K abundances of 6 wt% S and 6 wt% Si arerequired to trigger the immiscibility and leadto the formation of a sulfur rich phase with acomposition of about 25 wt% S. For FeS crys-tallization to play a role at today’s CMB tem-peratures, the sulfur rich phase should lie sig-nificantly to the right of the eutectic where theFeS melting temperature increases with lightelement abundance. Thus even higher S andSi contributions are required but seem oncemore difficult to reconcile with the planet’stotal mass [Rivoldini and Van Hoolst, 2013].Since Si partitions much more easily into thesolid iron phase than sulfur, it’s contribution tocompositional convection and the stabilizationof the snow zone is significantly weaker.

Several numerical studies in the context ofEarth and Mars have shown that the CMBheat flux pattern can have a strong effecton the dynamo mechanism (see e.g. Wichtet al. [2011a] and Dietrich and Wicht [2013]for overviews). Like the mean heat flux out ofthe core, this pattern is controlled by the lowermantle structure. The Martian dynamo ceasedabout 4 Gyr ago but has left its trace in form ofa strongly magnetized crust. The fact that the

magnetization is much stronger in the southernthan in the northern hemisphere could reflect aspecial configuration of the planet’s ancient dy-namo. Impacts or large degree mantle convec-tion may have significantly decreased the heatflux through the northern CMB and thereforeweakened dynamo action in this hemisphere[Stanley et al., 2008, Amit et al., 2011, Dietrichand Wicht, 2013]. Mercury’s magnetic field isdistinctively stronger in the northern than inthe southern hemisphere and it seems attrac-tive to invoke an increased northern CMB heatflux as a possible explanation.

Clues about the possible pattern may oncemore come from MESSENGER observations.A combination of gravity and altimeter dataallowed to estimate the crustal thickness inthe northern hemisphere. On average, thecrust is about 50 km thicker around the equa-tor than around the pole [Smith et al., 2012]which points towards more lava productionand thus a hotter mantle at lower latitudes.This is consistent with the fact that the north-ern lowlands are filled by younger flood basaltssince melts more easily penetrate a thinnercrust [Denevi et al., 2013]. Missing altime-ter data and the degraded precision of gravitymeasurements does not allow to constrain thecrustal thickness in the southern hemisphere.The lack of younger flood basalts, however,could indicate a thicker crust and hotter man-tle. Since a hotter mantle would reduce theCMB heat flux, these ideas indeed translateinto a pattern with increased flux at highernorthern latitudes. However, Mercury’s vol-canism ceased more than 3.5 Gyr ago and to-day’s thermal mantle structure may look com-pletely different. Even simple thermal diffusionshould have eroded any asymmetry over such along time span. Thermal evolution simulationsshow that at least the lower part of the mantlemay still convect today [Smith et al., 2012, Tosiet al., 2014] which would change the structureon much shorter time scales. Since the active

6

10 20 30 40P [GPa]

2500

2000

1500

T [

K]

γ FeeutecticTcmb=2000 KTcmb=1910 KTcmb=1890 KTcmb=1750 KTcmb=1600 K

1

2

3

4

5

6

7

8

9

10

11

sulf

ur

wt

%

Figure 1: Melting curves for different initial sulfur concentrations and possible Mercury adiabatsfor different temperatures shown as thick red, green, turqoise, and blue lines. Thin red lines fromtop to bottom show the melting curves for the convecting part of the core for an initial sulfurconcentration of 3 wt% and a core state described by the solid green, dashed green, gray, andblue adiabats. The thick solid black line shows the melting curve for pure iron while the thickdashed black line shows the eutectic temperature. The figure, provided by Attilio Rivoldini, andhas been adapted from Rivoldini et al. [2011] to include the Mercury core adiabats calculatedin Rivoldini and Van Hoolst [2013]. A central pressure of 40 GPa is assumed for Mercury butthe adiabats are only drawn in the liquid part of the core.

7

shell is so thin, the pattern would be rathersmall scale without any distinct north/southasymmetry.

Because of Mercury’s 3:2 spin-orbit reso-nance, the high eccentricity of the orbit, andthe very small obliquity the time averaged inso-lation pattern shows strong latitudinal and lon-gitudinal variations. Williams et al. [2011] cal-culates that the mean polar temperature canbe 200 K lower than the equatorial. Longitudi-nal variations show two maxima that are about100 K hotter than the minima at the equator.If Mercury’s mantle convection has ceased longago, the respective pattern may have diffusedinto the mantle and could determine the CMBheat flux variation. Higher than average flux atthe poles and a somewhat weaker longitudinalvariation would be the consequence. We dis-cuss the impact of the CMB heat flux patternon the dynamo process in section 2.

3 Mercury’s externalmagnetic field

Planetary magnetospheres are the result of theinteraction between the planetary magneticfield and the impinging solar wind plasma. Be-cause of Mercury’s weak and asymmetric mag-netic field and the position close to the Sun,the Hermean and terrestrial magnetospheresdifferer fundamentally. Mercury experiencesthe most intense solar wind of all solar-systemplanets. Under average conditions, the ratio ofthe solar wind speed and the Alfven velocity,called the Alfvenic Mach-number, is compara-ble to the terrestrial one. With values of 6.6 forMercury [Winslow et al., 2013] and 8 for Earth,the solar wind plasma is super-magnetosonic atboth planets, i.e. the medium propagates fasterthan magnetic disturbances and a bow shocktherefore forms in front of the magnetosphere.Because of the weak Hermean magnetic field,the sub-solar point of the bow shock is located

rather close to the planet at an average po-sition of only 1.96 planetary radii [Winslowet al., 2013] compared to 14 planetary radii forEarth.

Behind the bow shock, the cold solar windplasma is heated up and interacts with theplanetary magnetic field, thereby creating themagnetosphere. To first order, the planetaryfieldlines form closed loops within the daysidemagnetosphere and a long tail on the nightside.The outer boundary of the magnetosphere, themagnetopause, is located where the pressureof the shocked solar wind and the pressureof the planetary magnetic field balance. Thesolar wind ram pressure, on average 14.3 nPaat Mercury [Winslow et al., 2013], is an or-der of magnitude higher than at Earth whilethe magnetic field is two orders of magnitudeweaker. Like the bow shock, the magnetopauseis therefore located much closer to the planetat Mercury than at Earth with mean standoffdistances of about 1.45 [Winslow et al., 2013]and 10 planetary radii, respectively. Both theHermean magnetosphere and magnetosheath,the region between bow shock and magne-topause, are thus much smaller than the terres-trial equivalents in relative and absolute terms.

Fig. (2) shows the current density in a nu-merical hybrid simulation that models the so-lar wind interaction with the planet [Mulleret al., 2012]. The location of the bow shockand the magnetosphere can be identified viathe related current systems. Along a spacecrafttrajectory these boundaries can be identifiedby the related magnetic field changes. Fig. (3)shows MESSENGER magnetic field measure-ments for a relatively quiet orbit (orbit number14) where both the bow shock and the magne-topause can be clearly classified on both sidesof the planet.

Another important element of the magne-tosphere is the neutral current sheet which isresponsible for the elongated nightside magne-totail and separates the northern and south-

8

Figure 2: Electrical currents in a numerical simulation of the Hermean magnetosphere. Theamplitude of the current density j is color-coded. An equatorial cross section is shown in ancoordinate system where X points towards the Sun (negative solar wind direction) and theY -axis lies in the Hermean ecliptic. The bow shock standing in front of the planet slows downthe solar wind. The magnetopause is the outer boundary of the magnetosphere. The neutralcurrent sheet is located in the nightside of the planet. An arc of electrical current visible closeto the flyby trajectory (January 14, 2008) could be interpreted as a partial ring current. Thisfigure is a snapshot from a solar wind hybrid simulation and is adapted from Muller et al.[2012].

9

Figure 3: Magnetic field data recorded by the MESSENGER magnetometer (10s average)during orbit 14 on the DOY 84 in 2011. The upper panel, shows the time series of the absolutemagnetic field |B| (black), the negative absolute field (grey), the radial component Br (green),and the component Bρ perpendicular to the rotation axis. Time is measured in hours sincethe last apocenter passage. The plasma boundaries are marked with vertical dashed lines (BS:bow shock, MP: magnetopause). The location where Bρ vanishes defines the magnetic equator(MEQ). The lower panel shows the planetocentric distance r and the co-latitude θ. The dataare taken from the Planetary Data System / Planetary Plasma Interactions Node.

10

ern magnetotail lobes. Johnson et al. [2012]report, that the sheet starts at 1.41RM , whereRM is the mean Hermean radius, which is ap-proximately the standoff distance of the day-side magnetopause. Roughly the same propor-tion is also found at Earth.

The locations of bow shock, magnetopauseand neutral current sheet is not stationary butvary in time. The density of the average so-lar wind decreases with distance rS to the Sunlike 1/r2

S . Since Mercury orbits the Sun ona highly elliptical orbit (ellipticity: 0.21) thelocal solar wind pressure varies significantlyon the orbital time-scale of 88 days. The so-lar wind characteristics also changes constantlyon much shorter time scales because of spatialinhomogeneities due to, for example, coronal-mass ejections. As a result, the Hermean mag-netosphere is very dynamic. And since themagnetosphere is so small, the magnetic dis-turbance also propagate deep into the magne-tosphere and impede the separation of the fieldinto internal and external contributions [Glass-meier et al., 2010]. Reconnection processes inthe magnetotail are another source for varia-tions in the Hermean magnetosphere [Slavinet al., 2012].

The Hermean and terrestrial magneto-spheres differ in several additional aspects.Mercury’s surface temperature can reach sev-eral hundred Kelvin which means that theplanet’s gravitational escape velocity of 4.3km/s can easily be reached thermally. Thethermal escape rate is therefore significant andthe remaining atmosphere too thin to form anionosphere. At Earth, the ionosphere hostssubstantial current systems that significantlyaffect the magnetospheric dynamics, for exam-ple magnetic sub-storms. Field-aligned cur-rents that close via the ionosphere at Earthmust close within the magnetospheric plasmaor the planetary body at Mercury [Janhunenand Kallio, 2004].

When the planetary magnetic field on the in-

side of the magnetopause is nearly antiparallelto the magnetosheath field on the outside, therespective fieldlines can reconnect. This typ-ically happens when the interplanetary mag-netic field has a component parallel to theplanetary field. The reconnected fieldlines areadvected tail-wards by the solar wind, whichdrives a global scale magnetospheric convec-tion loop that ultimately replenishes the day-side field (Dungey-Cycle). Due to the smallsize of the Hermean magnetosphere, the typi-cal timescale of this plasma circulation is onlyabout 1− 2 minutes compared to 1 h at Earth[Slavin et al., 2012] which demonstrates thatthe Hermean magnetosphere can adapt muchfaster to changing solar wind conditions. Therate of reconnection, measured by the rela-tive amplitude of the magnetic field componentperpendicular to the magnetopause, is about0.15 at Mercury and thus 3 times higher thanat Earth [Dibraccio et al., 2013].

Charged particles that are trapped insidethe magnetosphere and drift around the planetin azimuthal direction form a major magneto-spheric current system at Earth, the so-calledring current. The drift is directed along iso-contours of the magnetic field strength. How-ever, since internal and magnetospheric fieldcan reach comparable values these contoursclose via the magnetopause at Mercury, as is il-lustrated in fig. (4). At Earth, the planetocen-tric distance Rrc,E of the ring current is aboutfour times the terrestrial radius. When assum-ing that the position scales linearly with theplanetary dipole moment, the distance can berescaled to the Hermean situation by

Rrc,M = Rrc,EmM

mE≈ 820km (1)

where mM and mE are the dipole moments ofMercury and Earth, respectively. The ring cur-rent would thus clearly lie below Mercury’s sur-face. Hybrid simulations by Muller et al. [2012]indicate that the solar wind protons entering

11

Figure 4: Equatorial isocontours of the total magnetic field in a Hermean model magnetosphere.The magnetopause is shown as a red line and the planet as a sphere. Figure from Baumjohannet al. [2010].

the magnetosphere can drift roughly half-wayaround the planet before being lost to the mag-netopause, as is illustrated in fig. (2). Thiscould be interpreted as a partial ring current.The protons create a diamagnetic current thatlocally decreases the magnetic field.

Because MESSENGER delivers only datafrom one location at a time inside a very dy-namic magnetosphere, it is not only challeng-ing to separate internal from external field con-tributions but also temporal from spatial varia-tions. Numerical simulations for the solar windinteraction with the planetary magnetic field,like the hybrid simulation used to investigatethe partial ring current (see fig. (2)), can im-prove the situation by constraining the pos-sible spatial structure for a given solar windcondition. However, as these codes are nu-merically very demanding, it becomes imprac-tical to perform simulation for all the differ-ent conditions possibly encountered by MES-SENGER. A more practical approach is to usesimplified models where a few critical proper-

ties like the shape of the magnetopause andthe strength and shape of the neutral currentsheet are described with a few free parameters.Johnson et al. [2012] demonstrate how the pa-rameters can be fitted to MESSENGER’s accu-mulated magnetic field data to derive a modelfor the time averaged magnetosphere.

The offset of Mercury’s magnetic field by20% of the planetary radius to the north cancause an equatorial asymmetry of the planet’sexosphere. Ground-based observations ofsodium emission lines suggest that there ismore sodium released from the southern thanthe northern planetary surface. Mangano et al.[2013] argue that precipitating solar wind pro-tons are the main player in the sodium re-lease and more likely reach the southern sur-face where the magnetic field is weaker.

The Hermean magnetosphere resembles itsterrestrial counterpart in several aspects butthere are also huge differences. Mercury’smagnetosphere is much smaller and signifi-cantly more dynamic, responding much faster

12

Figure 5: External fields from the paraboloid model based on measurements of the MESSEN-GER mission at the planetary surface. Top panel: amplitude of the magnetopause field. Bottompanel: amplitude of the neutral sheet magnetic field. Figure from Johnson et al. [2012].

13

to changing solar wind conditions. While theexternal field contributions are orders of mag-nitude smaller than internal contributions atEarth, they can become comparable at Mer-cury (see fig. (5)). This lead Glassmeier et al.[2007] to investigate the long-term effect of theexternal field on the internal dynamo process,as we will further discuss in section 5.

4 Mercury’s internalmagnetic field

The difficulties in separating internal and ex-ternal field and the strong covariance of dif-ferent spherical harmonic contributions causedby the highly elliptical orbit complicate a clas-sical field modelling with Gaussian coefficientsfor Mercury [Anderson et al., 2011]. Instead,the MESSENGER magnetometer team anal-ysed the location of the magnetic equator toindirectly deduce the internal field. The mag-netic equator is the point where the magneticfield component Bρ perpendicular to the plan-etary rotation axis vanishes. Changing solarwind conditions lead to variations in the equa-tor location on different time scales from sec-onds to months but should average out overtime, at least as long as the planetary bodyitself has no first order impact on the magne-tospheric current system. The mean locationof the magnetic equator is then primarily de-termined by the internal field.

Anderson et al. [2012] analysed the magneticequator for 531 descending orbits with alti-tudes between 1000 and 1500 km and 120 as-cending orbits with altitudes between 3500 and5000 km. They find that the equator crossingsare confined to a relatively thin band offset byabout Z = 480 km to the north of the planet’sequator. We adopt a planet-centered cylin-drical coordinate system here where ρ and zare the coordinates perpendicular to and alongthe rotation axis, respectively, and Φ is the

longitude. Anderson et al. [2012] minimizedthe effects of solar wind related magnetic fieldvariations by considering a mean where eachequator location is weighted with the inverseof the individual standard error σ. This pro-cedure yields a mean offset of Zd = 479 kmwith a standard deviation of ∆Zd = 46 km forthe descending orbits. The mean three stan-dard error in determining the individual equa-tor crossings is 3σd = 24 km. Because of theincreased solar wind influence and the closerproximity to the magnetosphere, the magneticequator is less well defined for the ascendingorbits with Za = 486 km, ∆Za = 270 km, and3σa = 86 km (see table 1 in Anderson et al.[2012]).

These observations suggest that the offsetof the magnetic equator has a constant valueof 480 km independent of the distance to theplanet. Such a configuration can readily beexplained by an internal axial dipole that isoffset by 480 km to the North of the equato-rial plane. This translates into an infinite sumof axisymmetric Gaussian field coefficients g`in the classical planet-centered representationwith

g`0 = ` g10Z`−1 , (2)

where Z = Z/RM is the normalized offset and` the spherical harmonic degree [Bartels, 1936,Alexeev et al., 2010]. Note that all contribu-tions have the same sign. In the Gaussian rep-resentation the planetary surface field is ex-panded into spherical surface harmonics Y`m ofdegree ` and order m [Olsen et al., 2010]. Thecoefficients g`m and h`m express the cos(mφ)and sin(mφ) dependence for a given degree `.Only coefficients g`0 contribute to an axisym-metric field.

Anderson et al. [2012] report that coeffi-cients up to ` = 4 suffice to explain the meanmagnetic equator locations in the MESSEN-GER offset dipole model (MODM). To illus-trate the characteristics of MODM, we experi-

14

c)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

s/RM

0.05

0.10

0.15

0.20

0.25

0.30 tilt= 0.8tilt= 2.0tilt= 5.0

Z

a)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.05

0.10

0.15

0.20

0.25

0.30

s/RM

Z

l=1,2l=1,2,3l=1,2,3,4

Z

b)

d)

0 45 90 135 180 225 270 315 360

longitude (arbitrary)

0.05

0.10

0.15

0.20

0.25

0.30

Z

-0.2 -0.1 0.0 0.1 0.2 0.3

g30/g10

0.15

0.18

0.20

0.21

0.24

0.25

0.16

0.17

0.22

0.23

0.19

Figure 6: Illustration of the offset dipole model by Anderson et al. [2012]. Panel a) demonstrateshow the location of the magnetic equator for the descending (left box) and ascending (rightbox) orbits is explained by combining axial Gauss coefficients up to degree ` = 4. Light greyboxes illustrate the standard deviation, middle grey boxes the mean three sigma error (see text),and the horizontal black line corresponds to the mean offset. Panel b) illustrates the impactof different relative octupole amplitudes g30/g10. Coloured dots in panels c) and d) show theequator locations found on a dense spherical longitude/latitude grid when an equatorial dipolecomponent g11 has been added that corresponds to a dipole tilt of 0.8. In panel c) the dashed,dotted, and dash-dotted lines show the mean equator offset for each spherical surface of radiuss/RM plus and minus the standard deviation.

15

ment with different combinations of the spher-ical harmonic contributions and perform a nu-merical search for the magnetic equator ona dense longitude/latitude grid for sphericalsurfaces with radii up to 4RM . Panel a) infig. (6) illustrates how the different axisym-metric contributions in the MODM team upto yield an offset that is nearly independentof the distance to the planet. A large ax-ial quadrupole contribution which amounts tonearly 40% of the axial dipole guarantees a re-alistic offset for ρ > 2RM . Additional higherharmonic contributions are required to achievea consistent offset at closer distances. Panel b)in fig. (6) demonstrates that already the rela-tive axial octupole g30/g10 is not particularlywell constrained and values between 0.05 and0.12 seem acceptable. Anderson et al. [2012],however, suggest a surprisingly tight range of0.116± 0.009. Contributions beyond ` = 3 cannot be particularly large to retain a nearly con-stant offset value in the observed range. Con-straining them further, however, would requiredata closer to the planet than presently avail-able. The analysis shows that the mean offsetZ further away from the planet can serve asa proxy for the ratio of the axial quadrupoleto axial dipole contribution while the depen-dence of Z on the distance closer to the planetprovides information on higher order axial con-tributions.

Anderson et al. [2012] estimate an upperlimit for the dipole tilt of Θ = 0.8. A tiltof the planetary centered dipole causes a lon-gitudinal variation of the magnetic equatorlocation that increases with distance to theplanet, as is demonstrated in panels b) and c)of fig. (6). A tilt as large as 2 seems still com-patible with the data but the more complexlongitudinal dependence of the offset [Ander-son et al., 2012] indicates that either higherorder harmonics or more likely the solar windinteraction contributes to the variation aroundthe mean offset. A tilt below < 0.8 is also

consistent with a more complete field analysisby Johnson et al. [2012] that includes a param-eterized magnetospheric model.

Table 1 compares primary magnetic fieldcharacteristics of the MODM with models forother planets and fig. (7) shows the respectiveradial magnetic surface fields. MODM’s largequadrupole contribution is comparable to thatinferred for Uranus or Neptune. Unlike thefields of the ice giants, however, Mercury’s fieldis also very axisymmetric, a property it shareswith Saturn. The seemingly perfectly axisym-metry of Saturn’s field is also the reason for thesmall spread ∆Z of magnetic equator locationsfor this planet. Saturn’s relative quadrupolecontribution, however, and thus the relativeoffset is much smaller than at Mercury.

Magnetic harmonics where the sum of degree` and order m is odd (even) represent equatori-ally anti-symmetric (symmetric) field contribu-tions. The axial dipole field is thus equatoriallyanti-symmetric while the axial quadrupole fieldis symmetric. Mercury’s field has a significantequatorially symmetric contribution because ofthe strong axial quadrupole. Another measurerelated to the equatorial symmetry breaking isthe hemisphericity

H =BN −BSBN +BS

(3)

where BN and BS are the rms surface fieldamplitudes in the northern and southern hemi-spheres, respectively. Due to the offset dipolegeometry, the Hermean magnetic field is sig-nificantly stronger in the northern than in thesouthern hemisphere so that the hemispheric-ity reaches a relatively large value of 0.2. Inconclusion, Mercury’s magnetic field is notonly very weak but also has a peculiar ge-ometry unlike any other planet in our solarsystem that combines a relatively large ax-ial quadrupole contribution with a very smalldipole tilt.

16

Uranus

Mercury

Earth

Jupiter

Saturn

Figure 7: Comparison of different radial magnetic fields at planetary surface. Blue (red andyellow) indicates radially inward (outward) field. See table 1 for information on the differentfield models.

17

Quantity MODM Earth Jupiter Saturn Uranus

g10 [nT] −190± 10 −29 560 420 500 21 191 11 855tilt [] < 0.8 10.2 9.5 < 0.06 58.8g20/g10 0.392± 0.010 0.079 −0.012 0.075 −0.496g30/g10 0.116± 0.009 −0.045 −0.004 0.112 0.353g40/g10 0.030± 0.005 −0.031 −0.040 0.003 0.034

H 0.20 0.017 0.045 0.050 0.251

Z 2.0×10−1 2.6×10−2 3.5×10−3 3.8×10−2 5.3×10−2

Zd 2.0×10−1 3.6×10−3 2.8×10−2 4.0×10−2 1.6×10−1

∆Z 1.7×10−2(1.1×10−1) 2.2×10−1 1.6×10−1 2.9×10−3 1.0

∆Zd 7.5×10−3(1.9×10−2) 1.3×10−1 9.3×10−2 8.5×10−4 6.6×10−1

Table 1: Comparison of Mercury’s offset dipole model MODM [Anderson et al., 2012] withmagnetic field models for other planets: the Grimm model [Lesur et al., 2012] for Earth, theVIP4 model [Connerney et al., 1998] for Jupiter, the model by Cao et al. [2012] for Saturn,and the model by Holme and Bloxham [1996] up to degree ` = 4 for Uranus. The neptunianmagnetic field is similar to the field of Uranus and has therefore not been included. The lastfour lines list mean offset values Z for all spherical surfaces up to 4R and the mean offset Zd forthe distances between 1.3R and 1.5R covered by MESSENGER’s descending orbits. R refersto the planetary radius (1 bar level for gas planets). ∆Z and ∆Zd are the related standarddeviations. For Mercury, we list the deviation caused by an 0.8 tilt and also the observedstandard deviations in brackets.

The time averaged residual field after sub-tracting the internal and external field modelsby Johnson et al. [2012] from the observationaldata is surprisingly strong with amplitudes ofup to 45 nT at 300 km altitude above Mercury’ssurface [Purucker et al., 2012]. The fact thatthe residual field is concentrated at high north-ern latitudes, is relatively small scale, and cor-relates with the boundary of the northern vol-canic plains to a fair degree points towardscrustal remanent magnetization, though an in-ternal field contribution can also not be ex-cluded. A crustal origin would suggest thatMercury’s dynamo is long lived and probablyolder than 3.5 Gyr. Since the residual field op-poses the current dipole direction, the dynamomust have reversed its polarity at least once.This would put valuable constraints on ther-mal evolution models and dynamo simulationsfor Mercury.

5 Modelling Mercury’sInternal Dynamo

5.1 Dynamo Theory

Numerical dynamo simulations solve for con-vection and magnetic field generation in aviscous, electrically conducting, and rotatingfluid. Since the solutions are very small distur-bances around an adiabatic, well mixed, non-magnetic, and hydrostatic background state,only first order terms are taken into account.For terrestrial planets, the mild density andtemperature variations of the background stateare typically neglected in the so called Boussi-nesq approximation [Braginsky and Roberts,1995]. The mathematical formulation of thedynamo problem is then given by the Navier-

18

Stokes equation

EdU

dt= −∇P − 2z×U + Ra

r

roC r (4)

+1

Pm(∇×B)×B + E∇2U ,

the induction or dynamo equation

∂B

∂t= (B · ∇) U +

1

Pm∇2B , (5)

the codensity evolution equation

dC

dt=

1

Pr∇2C + q , (6)

the flow continuity equation

∇ ·U = 0 , (7)

and the magnetic continuity equation

∇ ·B = 0 . (8)

Here, d/dt stands for the substantial timederivative ∂/∂t+U·∇, U is the convective flow,B the magnetic field, P is a modified pressurethat also contains centrifugal effects, and C isthe codensity.

The equations are given in a non-dimensional form that uses the thicknessof the fluid shell d = ro − ri as a length scale,the viscous diffusion time d2/ν as a time scale,the codensity difference ∆C across the shellas the codensity scale, and (ρµλΩ)1/2 as themagnetic scale. Here, ri and ro are the radii ofthe inner and outer boundary, respectively, νis the kinematic viscosity, ρ the reference statecore density, µ the magnetic permeability, λthe magnetic diffusivity, and Ω the rotationrate.

The problem is controlled by five dimension-less parameters: the Ekman number

E =ν

Ωd2, (9)

the Rayleigh number

Ra =goα∆c d3

κν(10)

the Prandtl number

Pr =ν

κ, (11)

the magnetic Prandtl number

Pm =ν

λ, (12)

and the aspect ratio

a = ri/ro . (13)

These five dimensionless parameters replacethe much larger number of physical propertiesof which the thermal and/or compositional dif-fusivity κ, the thermal and/or compositionalexpansivity α, and the outer boundary refer-ence gravity g0 have not been defined so far.

Convection is driven by density variationsdue to super-adiabatic temperature gradients— only this component contributes to con-vection — or due to deviations from a ho-mogeneous background composition. Possiblesources for thermal convection are secular cool-ing, latent heat, and radiogenic heating. Pos-sible sources for compositional convection arethe light elements released from a growing in-ner core and iron from an iron snow zone. Tosimplify computations, both types of densityvariation are often combined into one variablecalled codensity C despite the fact that themolecular diffusivities of heat and chemical el-ements differ by orders of magnitude. Theapproach is often justified with the argumentthat the small scale turbulent mixing, whichcan not be resolved in the numerical simula-tion, should result in larger effective turbu-lent diffusivities that are of comparable magni-tude [Braginsky and Roberts, 1995]. This hasthe additional consequence that the ‘turbulent’

19

Prandtl number and magnetic Prandtl num-ber would become of order one [Braginsky andRoberts, 1995]. The codensity evolution equa-tion (6) contains a volumetric source/sink termq that can serve different purposes dependingon the assumed buoyancy sources. For con-vection driven by light elements released fromthe inner core, q acts as a sink that compen-sates the respective source. When modellingsecular cooling, the outer boundary is the sinkand q the balancing volumetric source [Kutznerand Christensen, 2000]. For iron snow thatremelts at depth q should be positive in thesnow zone but negative in the convective zoneunderneath.

Typically, no-slip boundary conditions areassumed for the flow. For the condensity, ei-ther fixed codensity or fixed flux boundary con-ditions are used. The latter translates to afixed radial gradient and requires a modifica-tion of the Rayleigh number (10) where ∆Cthen stands for the imposed gradient timesthe length scale d. For terrestrial planets,the much slower evolving mantle controls howmuch heat is allowed to leave the core, sothat a heat flux condition is more appropri-ate. Lateral variations on the thermal lowermantle structure translate into an inhomoge-neous core-mantle boundary heat flux [Aubertet al., 2008]. Since the electrical conductiv-ity of the rocky mantle in terrestrial planets isorders of magnitudes lower than that of thecore, the magnetic field can be assumed tomatch a potential field at the interface r = ro.This matching condition can be formulated asa magnetic boundary condition for the indi-vidual spherical harmonic field contributions[Christensen and Wicht, 2007]. A simplifiedinduction equation (5) must be solved for themagnetic field in a conducting inner core whichhas to match the outer core field at ri. We re-fer to Christensen and Wicht [2007] for a moredetailed discussion of dynamo theory and thenumerical methods employed to solve the sys-

tem of equations.Explaining the weakness of Mercury’s mag-

netic field proved a challenge for classical dy-namo theory. In convectively driven core dy-namos, the Lorentz force and thus the mag-netic field needs to be sufficiently strong to in-fluence the flow and thereby saturate magneticfield growth. The impact of the Lorentz forceis often expressed via the Elsasser number

Λ = B2/ρµλΩ (14)

where B is the typical magnetic field strength.The Elsasser number estimates the ratio of theLorentz to the Coriolis force which is knownto enter the leading order convective force bal-ance. For Earth, Λ is of order one which sug-gests that the Lorentz force is indeed signifi-cant. For Mercury, however, extrapolating themeasured surface field strength to the planet’score mantle boundary yields Λcmb ≈ 10−5,a value much too low to be compatible withan Earth-like convectively driven core dynamo[Wicht et al., 2007]. Several authors thereforepursued alternative theories like crustal mag-netization or a thermo-electric dynamo (seeWicht et al. [2007] for an overview).

However, convectively driven core dynamosremain the preferred explanation since differ-ent modifications of the numerical models orig-inally developed to explain the geodynamo suc-cessfully reduced the surface field strength to-wards more Mercury like values (for recentoverviews see Wicht et al. [2007], Stanley andGlatzmaier [2010], Schubert and Soderlund[2011]). We revisit several of these models inthe following and test whether they are consis-tent with MESSENGER magnetic field data.

20

Mod

elM

OD

ME

5R

6E

5R

36

E5R

45

CW

2C

W3

CW

4Y10

BD

Y10

IDY20

Ra

—2×

107

1.2×

108

1.5×

108

2×

108

4×

108

6×

108

4×

107

4×

107

4×

107

E10−13

3×

10−5

3×

10−5

3×

10−5

10−4

10−4

10−4

10−4

10−4

10−4

Pm

10−6

11

13

33

22

2P

r0.1

11

11

11

11

1a

—0.3

50.3

50.3

50.5

00.5

00.5

00.3

50.3

50.2

0R

o`

80.0

20.1

00.1

80.4

22.7

3.6

0.1

10.0

60.0

5Λcm

b10−5

2.2×

10−2

1.8×

10−1

1.6×

10−2

4.9×

10−4

1.9×

10−4

4.7×

10−5

1.8×

10−1

1.9×

10−1

4.0×

10−2

|g10|[n

T]

190

8.8×

103

2.6×

104

1.4×

103

1.4×

103

924

432

2.1×

104

1.7×

104

8.3×

103

tilt

[]

<0.8

02.5

38.1

3.5

4.2

8.6

3.4

10.7

8.2

|g20/g10|

0.3

90

0.3

43.8

0.0

80.1

60.3

10.0

60.2

50.5

2H

0.2

00

0.0

20.1

10.0

50.0

90.0

90.0

40.2

00.2

3

|Z|

2.0×

10−1

02.2×

10−2

2.1×

10−1

4.0×

10−2

8.1×

10−2

8.4×

10−2

3.8×

10−2

1.5×

10−1

2.0×

10−1

∆Z

1.7×

10−2

(1.1×

10−1)

06.9×

10−2

8.6×

10−1

8.3×

10−2

2.4×

10−1

1.1×

10−1

2.2×

10−1

2.3×

10−1

Tab

le2:

Com

par

ison

ofM

ercu

ry’s

dyn

amo

par

amet

eran

dp

rop

erti

esw

ith

nin

ed

yn

amo

mod

els

that

hav

eb

een

(re)

an

aly

sed

her

e.F

orM

ercu

ryw

eli

stth

eM

OD

Mw

hil

eti

me

aver

aged

loca

lR

ossb

ynu

mb

ers

Ro`

and

mag

net

icfi

eld

pro

per

ties

are

list

edfo

rth

enu

mer

ical

sim

ula

tion

s.M

ercu

ry’s

core

Ray

leig

hnu

mb

eran

das

pec

tra

tio

are

bas

ical

lyu

nco

nst

rain

edb

ut

an

dw

eh

ave

assu

med

anE

arth

-lik

eva

lue

ofa

=0.3

5to

calc

ula

teth

eE

km

annu

mb

er.

Oth

erM

ercu

ryp

aram

eter

sfo

llow

Sch

ub

ert

an

dS

od

erlu

nd

[201

1]an

dO

lson

and

Ch

rist

ense

n[2

006]

.T

wo

mod

els

wit

haY

10

CM

Bh

eat

flu

xp

atte

rar

eli

sted

,on

eis

bott

om

dri

ven

(BD

)w

hil

eth

eot

her

isin

tern

ally

dri

ven

(ID

).T

he

mod

elw

ith

aY

20

pat

tern

isin

tern

ally

dri

ven

.

21

All dynamo simulations have the problemthat numerical limitations prevent the use real-istic diffusivities. For example, the viscous dif-fusivity is many orders of magnitude too largeto damp the very small scale convection mo-tions that cannot be resolved with the availablecomputer power. Dynamo modelers typicallyfix the Ekman number E, the ratio of viscousto Coriolis forces, to the smallest value acces-sible with the numerical resources. The mostadvanced computer simulations reach down toE = 10−7 which is still many orders of mag-nitude larger than the planetary value of E ≈10−12 (see table 2). The Prandtl number Prcan assume realistic values but the magneticPrandtl number Pm has to be set to a valuethat guarantees dynamo action. Because ofthe increased viscous diffusivity, Pm is also or-ders of magnitudes too large. The Rayleighnumber is then adjusted to a value that yieldsthe desired dynamics. The fact that numericalDynamo simulations are very successful in re-producing many aspects of planetary dynamossuggest that at least the large scale dynamicsresponsible for producing the observable mag-netic field is captured correctly.

The simulation results must be rescaled tothe planetary situation. For simplicity, wewill rescale the magnetic field strength by as-suming that the Elsasser number would notchange when pushing the parameter towardsrealistic values. Assuming Mercury’s rotationrate, mean core density, magnetic permeabilityand magnetic diffusivity then allows the de-duce the dimensional magnetic field strengthvia eqn. (14). Note, however, that other scal-ings have been proposed [Christensen, 2010]and may lead to somewhat different answers.

5.2 Standard Earth-like DynamoModels

To highlight the difficulties of classical dynamosimulations to reproduce the Hermean mag-

netic field, we start with analysing three mod-els that have been explored in the geomagneticcontext by Wicht et al. [2011b]. All have thesame Ekman (E = 3×10−5), Prandtl number(Pr = 1), magnetic Prandtl number (Pm = 1),aspect ratio (a = 0.35), use rigid and fixed co-density boundary conditions and are driven bya growing inner core. They differ only in theRayleigh number: Model E5R6 has the lowestRayleigh number of Ra = 2×107, six times thecritical value for onset of convection. ModelE5R36 has an intermediate Rayleigh numberof Ra = 1.2×108 while model E5R45 has thelargest Rayleigh number at Ra = 1.5×108. Allmodel parameters are listed in table 2.

Fig. (8) shows the time evolution of the axialdipole strength, the dipole tilt, the mean mag-netic equator offset Z for up to four planetaryradii (assuming Mercury’s thin crust), and therelated standard deviation ∆Z. At the low-est Rayleigh number, convective driving is toosmall to break the equatorial symmetry. Themagnetic field is therefore perfectly equatori-ally anti-symmetric and very much dominatedby the axial dipole contribution. Dipole tilt,offset and standard deviation therefore vanish.At the intermediate Rayleigh number the so-lution is sufficiently dynamic and asymmetricto be considered very Earth-like [Wicht et al.,2011b, Christensen et al., 2010]. While the ax-ial quadrupole and other equatorially symmet-ric field contributions have grown, the strongaxial dipole still clearly dominates. The meanoffset Z therefore remains small but oscillatesaround zero since neither the northern nor thesouthern hemisphere are preferred in the dy-namo setup. The spread ∆Z is of the sameorder as the offset itself mainly because of theEarth-like dipole tilt. The inertial contribu-tions in the flow force balance have increasedto a point where magnetic field reversals canbe expected [Christensen and Aubert, 2006,Wicht et al., 2011b].

Christensen and Aubert [2006] introduced

22

the local Rossby number

Ro` =U

LΩ(15)

to quantify the ratio of inertial to Coriolisforces. Here, U is the rms flow amplitude andL is a typical flow length scale defined by

L = d π

∑U`∑`U`

. (16)

U` is the rms flow amplitude of spherical har-monic contributions with degree `. The Cori-olis force is responsible for organizing the flowinto quasi two-dimesional convective columnswhich tend to produce the larger scale dipoledominated magnetic field. Inertia and in par-ticular the non-linear advective term, on theother hand, is responsible for the mixing of dif-ferent scales and therefore the braking of flowsymmetries. At Ro` = 0.10 inertia is likelylarge enough in model E5R36 to trigger rever-sals though no such event has been observed inthe relatively short period we could afford tosimulate.

Christensen and Aubert [2006] report thatthis typically happens for Ro` ≈ 0.1, a limitclearly exceeded at Ro` = 0.18 in modelE5R45. Smaller scale contributions dominatethe now multipolar magnetic field which alsobecomes very variable in time and constantlychanges its polarity [Wicht et al., 2011b]. Con-sequently, the offset also varies rapidly andmay even exceed Mercury’s offset value attimes where the axial dipole is particularly low(see fig. (8)b). While axial dipole and offsetcan assume Mercury-like values during briefperiods in time, this is not true for the dipoletilt and ∆Z. The larger Rayleigh number pro-motes not only the axial quadrupole but higherharmonics and non-axial field contribution ingeneral. The tilt is therefore typically ratherlarge and the magnetic equator covers a widelatitude range. Closer to the planet, even two

or more closed lines with Bρ = 0 can be foundat a given radius.

Olson and Christensen [2006] estimate alarge local Rossby number of Ro` ≈ 8 for Mer-cury, mainly because of the planet’s slow ro-tation rate. This suggests that the dynamoproduces a multipolar field at least as com-plex as in the large Rayleigh number modelE5R45. This is at odds with the observationsunless we could add a physical mechanism tothe model that would filter out smaller scalefield contributions while retaining the strongaxial quadrupole. As we discuss in the fol-lowing, the stably stratified layer underneathMercury’s core-mantle boundary (See section2) may meet these requirements.

5.3 Dynamos with a stably stratifiedouter layer

The idea of a stably stratified layer in the outerpart of a dynamo region was first proposed byStevenson [1980] to explain Saturn’s very ax-isymmetric magnetic field. The immiscibilityof Helium and Hydrogen in Saturn’s metal-lic envelope [Lorenzen et al., 2009] may causeHelium to precipitate into the deeper interior.Similar to the iron snow scenario discussed insection 2 this process may establishes a stabi-lizing Helium gradient in the rain zone.

Christensen [2006] and Christensen andWicht [2008] adopt this idea for Mercury.They propose that the subadiabatic heat fluxthrough the CMB leads to the stable stratifica-tion but since they use a condensity approachthe model is not able to distinguish betweenthermal and compositional effects. The mag-netic field that is produced in the convectingdeeper core region has to diffuse through thelargely stable outer layer so that the magneticskin effect applies here. The time variabil-ity of the magnetic field increases with spa-tial complexity [Christensen and Tilgner, 2004,Lhuillier et al., 2011]. The higher harmonic

23

0.00 0.04 0.08 0.12 0.16 0.20

time [magn. diff. time]

0.4

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

Z

0.00 0.04 0.08 0.12 0.16 0.20

time [magn. diff. time]

10-2

10-1

100

∆Z

d

100

101

102

103

105

104

g10 [nT

]

0.00 0.04 0.08 0.12 0.16 0.20

time [magn. diff. time]

000 00 0044 00 0088 00 12

MercuryE5R6, Ra=2x107

E5R36, Ra=1.2x108

E5R45, Ra=1.5x108

a) b)

c)

0

10

20

30

40

50

60

dip

ole

til

t [d

egre

e]

0.00 0.04 0.08 0.12 0.16 0.20

time [magn. diff. time]

d)

Figure 8: Time evolution of three standard dynamo models with different Rayleigh numbers.The thick black horizontal lines indicate the MESSENGER offset dipole model. Panel a) showsthe axial dipole coefficient, panel b) the dipole tilt, panel c) the mean offset Z averaged overall radii up to 4RM , and panel d) shows the standard deviation for the offset in the distancerange of the descending orbits ∆Zd. For the numerical simulations time is given in units of themagnetic diffusion time τλ = d2/λ. When assuming an Earth-like aspect ratio of 0.35 and amagnetic diffusivity of λ = 1 the Hermean magnetic diffusion time amounts to τλ ≈ 54 kyr.

24

field contributions are therefore more signifi-cantly damped by the skin effect than for ex-ample dipole or quadrupole. Zonal motionsthat may still penetrate the stable layer cannotlead to significant dynamo action but furtherincrease the skin effect for non-axisymmetricfield contributions. Thanks to this filtering ef-fect, the multipolar field of a high Ro` dynamoshould look more Mercury-like when reachingthe planetary surface.

Testing different dynamo setups, Chris-tensen [2006] and Christensen and Wicht[2008] demonstrate that the surface field is in-deed weaker and less complex when a sizablestable layer is included. We reanalyse the mod-els 2,3, and 4 published in Christensen andWicht [2008] to test whether they are consis-tent with the new MESSENGER data. Allthree models, that we will refer to as CW2,CW3, and CW4 in the following, have a solidinner core that occupies the inner 50% in ra-dius and a stable region that occupies the outer28%. Thus only a relatively thin region is leftto host the active dynamo. Like for the stan-dard models explore above, all three cases havethe same Ekman number (E = 10−4), Prandtlnumber (Pr = 1), and magnetic Prandtl num-ber (Pm = 3) but differ in Rayleigh number.Once more, they use rigid flow boundary con-ditions and are driven by a growing inner core.The model parameters are listed in table 2.

Fig. (9) shows the time evolution of theaxial dipole contribution, the dipole tilt, themean offset Z, and of its standard deviation∆Z. At the lowest Rayleigh number of Ra =2×108 in CW2, the magnetic field strengthis already significantly weaker and the dipoletilt and offset standard deviation can actu-ally reach Mercury-like small values. The ax-ial dipole component, however, is still some-what strong and dominant and the offset valuetherefore too small. Increasing the Rayleighnumber to Ra = 4×108 in model CW3 de-creases the axial dipole in absolute and rela-

tive terms. The mean tilt, offset, and spreadincrease, but there are times when Mercury-like field geometries are approached. The ax-ial dipole is still by a factor four too strong.At Ra = 4×108 in model CW4, however, theaxial dipole can even become smaller than atMercury. The field is very time dependentduring these episodes and is characterized bylarge dipole tilts and ∆Z values since higherharmonic and non-axisymmetric field contribu-tions dominate. Very Mercury-like fields, thatcombine small dipole tilts with larger offset val-ues but small offset standard deviations, canbe found during brief periods when the axialdipole is somewhat stronger than the Mercuryvalue.

Fig. (10) illustrates the location of the mag-netic equator for two particularly Mercury-likesnapshots in the two larger Rayleigh numbermodels CW3 and CW4. Fig. (11) directly com-pares the respective radial magnetic fields withthe MESSENGER model. Both figures demon-strate that solutions very similar to the offsetdipole field proposed for Mercury can be foundwith a stably stratified outer core layer and asufficiently high Rayleigh number. However,the magnetic field varies considerably in timeand since neither hemisphere is preferred theoffset can switch from north to south and back.The particular offset dipole configuration en-countered by MESSENGER would thus onlybe transient and representative for only a fewpercent of the time at best.

Fig. (12) compares the time averaged spheri-cal harmonics surface spectrum of models CW3and CW4 with MODM, confirming that therelative quadrupole contribution and thus theequator offset is typically too low. The relativeenergy in spherical harmonic degrees ` = 3 and4, however, agrees quite well with MESSEN-GER observations.

Manglik et al. [2010] explore what happensto the stable layer when giving up the coden-sity formulation. They use a so-called double-

25

0.4

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

Z

10-2

10-1

100

∆Z

d

100

101

102

103

105

104

g10 [n

T]

0.0 0.1 0.2 0.3 0.5 0.60.4

time [magn. diff. time]

0 00 11 00 22 00 33

MercuryCW2, Ra=2x108

CW3, Ra=4x108

CW4, Ra=6x108

a) b)

c)

0

10

20

30

40

50

60

dip

ole

til

t [d

egre

e]

d)

0.0 0.1 0.2 0.3 0.5 0.60.4

time [magn. diff. time]

0.0 0.1 0.2 0.3 0.5 0.60.4

time [magn. diff. time]

0.0 0.1 0.2 0.3 0.5 0.60.4

time [magn. diff. time]

Figure 9: Time evolution of three dynamo models with a stably stratified layer. See fig. (8) formore explanation. The dashed vertical red and green lines marke the times for the snapshotsillustrated in fig. (10) and fig. (11).

26

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

s/RM

Z

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

s/RM

Z

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Z

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 45 90 135 180 225 270 315 360

longitude (arbitrary)

Z

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 45 90 135 180 225 270 315 360

longitude (arbitrary)

a) b)

c) d)

Figure 10: Magnetic equator location for two snapshots in dynamo models CW3 (top panels)and CW4 (bottom panels). The respective snapshot times have been marked by the verticaldashed lines in fig. (9). Coloured dots show the equator locations found on a dense sphericallongitude/latitude grid. The curved solid lines in panels a) and b) show the mean equator offsetfor each spherical surface with radius s/RM , the dashed lines show the mean offset plus andminus the standard deviation. Thick horizontal lines illustrate the mean offset measured bythe MESSENGER magnetometer while mid gray and light grey boxed show mean three sigmaerror and standard deviation for descending (left) and ascending orbits (right), respectively.

27

Mercury

CW3

CW4

Figure 11: Comparison of the MODM radial magnetic field for Mercury with the two particu-larly Mercury-like snapshots in models CW3 and CW4 already depicted in fig. (10). Blue (redand yellow) indicates radially inward (outward) field.

28

1 2 3 4 5 6

100

10-1

10-2

10-3

nom

aliz

ed m

agne

tic e

nerg

y

spherical harmonic degree

Anderson et al. 2012CW3CW4Y10, internally drivenY20, internally driven

Figure 12: Comparison of the normalized MODM surface spectrum by Anderson et al. [2012]with time averaged spectra for four different dynamo models: the dynamo models CW3 andCW4 that incorporate a stably stratified outer layer [Christensen and Wicht, 2008] and modelswith an inhomogeneous core-mantle boundary heat flux following a spherical harmonic Y10 orY20 pattern, respectively.

diffusive approach where two equations of theform of eqn. (6) separately describe the evolu-tion of temperature and composition. Whenassuming a compositional diffusivity that isone order of magnitude lower than the ther-mal diffusivity, the compositional plumes thatrise from the inner core boundary already staysignificantly narrower than their thermal coun-terparts. This allows them to more easilypenetrate and destroy the stable outer layer.The desirable filtering effect is greatly lost un-less the sulphur concentration is below 1 wt%where compositional convection starts to playan inferior role. For such a low light elementconcentration, however, Mercury’s core wouldlikely be completely solid today.

The iron snow mechanism discussed in sec-tion 2 offers an alternative scenario where thestable stratified layer is likely to persist even ina double-diffusive approach. The sulfur gradi-ent that develops in the iron snow zone is po-

tentially much more stabilizing than the sub-adiabatic thermal gradient assumed by Chris-tensen and Wicht [2008] and Manglik et al.[2010]. Furthermore, the additional convec-tive driving source represented by the remelt-ing snow would counteract the effects of themore sulfur rich plumes rising from a growinginner core.

5.4 Inhomogeneous boundary condi-tions

As already discussed in section 1, an inhomoge-neous heat flux through the CMB is an obviousway to break the north/south symmetry andinforce a more permanent offset of the mag-netic equator. To explain the stronger magne-tization of the southern crust on Mars severalauthors explored a variation following a spher-ical harmonic function Y10 of degree ` = 1 andorder m = 0 [Stanley et al., 2008, Amit et al.,

29

2011, Dietrich and Wicht, 2013]. The totalCMB heat flux is then given by

q = q0 (1− q?10 cos θ) (17)

where q0 is the mean heat flux, q?10 the relativeamplitude of the lateral variation, and θ thecolatitude. Positive values of q?10 are requiredat Mars and negative values should enforce thestronger northern magnetic field observed onMercury.

To explore the impact of the CMB heat fluxpattern we use dynamo simulations in the pa-rameter range discussed by Dietrich and Wicht[2013] and Cao et al. [2014]. The parametersare E = 10−4, Ra = 4×107, Pr = 1, Pm = 2,and a = 0.35. Once more, rigid boundary con-ditions are used and we impose the heat fluxat the outer boundary. The Rayleigh num-ber is then defined based on the mean CMBheat flux [Dietrich and Wicht, 2013]. Fig. (13)demonstrates that a relative variation ampli-tude of q10 = −0.10 is nearly sufficient toenforce the observed offset when the dynamois driven by homogeneously distributed inter-nal sources. These may either model secularcooling, radioactive heating, or the remeltingof iron snow. We have used a codensity for-mulation here and set the codensity flux fromthe inner core boundary to zero. Significantlylarger heat flux variations are required for theother end member when the dynamo is drivenby bottom sources that mimic a growing innercore. This is consistent with the findings by[Hori et al., 2012] who report that the impactof thermal CMB boundary conditions is gen-erally larger for internally driven than bottomdriven simulations.

Another not so obvious method to promote anorth/south asymmetry is to increase the heatflux through the equatorial region. Cao et al.[2014] explore a Y20 pattern, which means thatthe total CMB flux is given by

q = q0

(1− q?20

1

2(3 cos θ − 1)

). (18)

The green line in fig. (13) illustrates that a vari-ation amplitude of q?20 = 1/3 causes a more orless persistent Mercury-like offset value. Thistranslates into an increase of the equatorialflux by 17% and a decreases to polar flux by33%. Cases with an increased heat flux atthe poles, i.e. negative values of q?20, did notyield the desired result. Except for the CMBheat flux pattern and a smaller inner core thatonly occupies 20% of the radius, the modelsare identical to the Y10 cases explored above.Though the Y20 pattern is equatorially sym-metric, it promotes an equatorially asymmet-ric flow and therefore an asymmetric magneticfield production. A preliminary analysis ofthe system suggest that the Y20 pattern sig-nificantly decreases the critical Rayleigh num-ber for the onset of equatorially anti-symmetricconvection modes, which is very large when theCMB heat flux is homogenous [Landeau andAubert, 2011].

The inhomogenous CMB heat flux mainlyhelps to promote a Mercury-like mean offsetof the magnetic equator while other importantfield characteristics seem not consistent withthe observations. The field is generally muchtoo strong and the often large dipole tilt andoffset spread ∆Z testify that higher harmonicand non-axisymmetric field contributions re-main too significant. This is confirmed bythe time averaged spectra shown in fig. (12).Adding a stably stratified outer layer, proba-bly in combination with a larger Rayleigh num-ber to bring down the too stong axial dipolecontribution, seems like an obvious solution tothis problem. This was confirmed by the firstresults presented by Tian et al. [2013] who ex-plore the combiation of the stable layer withthe Y10 heat flux pattern.

5.5 Alternatives

Several authors varied the inner core size to ex-plore its impact on the dynamo process. Heim-

30

0.4

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

Z

10-2

10-1

100

∆Z

d

100

101

102

103

105

104

g1

0 [nT

]

0.00 0.05 0.10 0.15 0.250.20

time [magn. diff. time]

MercuryY10, bottom drivenY10, internally drivenY20, internally driven

a) b)

c)

0

5

10

15

20

25

30

dip

ole

til

t [d

egre

e]

d)

0.00 0.05 0.10 0.15 0.250.20

time [magn. diff. time]

0.00 0.05 0.10 0.15 0.250.20

time [magn. diff. time]

0.00 0.05 0.10 0.15 0.250.20

time [magn. diff. time]

Figure 13: Time evolution of three dynamo models with inhomogeneous core-mantle boundaryheat flux. A Y10 pattern with increased heat flux through the northern hemisphere but also aY20 pattern with a larger heat flux in the equatorial region promotes a Mercury-like offset. Seetext and fig. (8) for more explanation.

31

1 2 3 4 5 6

101

100

10-1

10-2

10-3

no

mal

ized

mag

net

ic e

ner

gy

spherical harmonic degree

Anderson et al. 2012Heimpel et al. 2005Takahashi et al. 2006Vilim et al. 2010Heyner et al. 2012

Figure 14: Comparison of the normalized MODM surface spectrum by Anderson et al. [2012]with spectra for different dynamo models. A time averaged spectrum is shown for the modelsby Vilim et al. [2010] and Heyner et al. [2011a] while the spectra for Heimpel et al. [2005],Takahashi and Matsushima [2006] represent snapshots.

pel et al. [2005] analyse models with aspect ra-tios between a = 0.65 and a = 0.15 that are alldriven by a growing inner core. They reportthat the smallest inner core yields a particu-larly weak magnetic field with a CMB Elsassernumber of Λcmb = 10−2 when the Rayleighnumber is close to onset for dynamo action.This is still more than two orders of magni-tude too large for Mercury. Convection anddynamo action are mainly concentrated at onlyone convective column attached to the innercore. Such localized magnetic field produc-tion is not very conducive to maintaining alarge scale magnetic field which is confirmedby the magnetic surface spectrum of a modelsnap shot shown in fig. (14). The relativequadrupole contribution nearly matches theMODM value but the higher harmonic contri-butions can reach a similar level and are thustoo strong. This is also true for the dipole tiltwhich has a mean value of 8 for this model.

Takahashi and Matsushima [2006] find thatthe magnetic field strength is also reducedwhen using a large inner core with a = 0.7in combination with a large Rayleigh num-ber. Once more, the field is still too strongfor Mercury with an Elsasser number aroundΛcmb = 10−2 and is also much too small inscale with ` = 3 and 5 contributions dominat-ing the spectrum (see fig. (14)). Stanley et al.[2005] explore even larger inner cores with as-pect ratios up to a = 0.9 and report particu-larly weak fields at rather low Rayleigh num-bers. The use of stress-free flow boundary con-ditions set this dynamo model apart from allthe other cases discussed here. Field strength,dipole tilt, and offset are highly variable butseem to assume Mercury-like values at times.Little more is published about the field geome-try and it seems worth to explore these modelsfurther.

Vilim et al. [2010] explore the double snowzone regime that may develop when the sulfur

32

content in Mercury’s core exceeds 10 wt%, asbriefly discussed in section 4. They considera thin outer snow zone and a thicker zone inthe middle of the liquid core in addition to agrowing inner core. Since both snow zones arestably stratified the dynamo action is concen-trated in the two remaining shells. The mag-netic fields that are produced in these two dy-namo regions tend to oppose each other whichleads to a reduced overall field strength thatmatches the MESSENGER observation. How-ever, the octupole component is generally toostrong while the quadrupole contribution is tooweak, as is demonstrated in fig. (14).

Since internal and external magnetic fieldcan reach similar magnitudes at Mercury thelatter may actually play a role in the core dy-namo. The idea of a feedback between inter-nal and external dynamo processes was firstproposed by Glassmeier et al. [2007] for Mer-cury and further developed in a series of pa-pers [Heyner et al., 2010, 2011a,b]. Becausethe internal dynamo process operates on timescales of decades to centuries, only the longtime-averaged magnetospheric field needs to beconsidered. This can be approximated by anexternal axial dipole that opposes the direc-tion of the inernal axial dipole within the core.The ratio of the external to internal dipolefield depends on the distance of the magne-topause to the planet and thus on the intensityof the internal field. Heyner et al. [2011a] findthat the feedback quenches the dynamo fieldto Mercury-like intensities when the simulationis started off with an already weak field andthe Rayleigh number is not too high. Theseconditions can, for example, be met when dy-namo action is initiated with the beginning ofiron snow or inner core growth at a period inthe panetary evolution where mantle convec-tion is already sluggish and the CMB heat fluxtherefore low. The feedback process modifiesthe dipole dominated field by concentrating theflux at higher latitudes. The result is a spec-

trum where the relative quadrupole (and otherequatorially anti-symmetric contributions) istoo weak while the octupole (and other equa-torially symmetric contributions) is too strong(see fig. (14)).

6 Conclusion

The MESSENGER data have shown that Mer-cury has an exceptional magnetic field [Ander-son et al., 2012, Johnson et al., 2012]. The in-ternal field is very weak and has a simple butsurprising geometry that is consistent with anaxial dipole offset by 20% of the planetary ra-dius to the North. This implies a very strongaxial quadrupole but at the same time alsosmall higher harmonic and non-axial contribu-tions, a unique combination in our solar sys-tem.

Numerical dynamo models have a hard timeto explain these observations. Strong axialquadrupole contributions and thus a significantmean offset of the magnetic equator can be pro-moted by different measures. Very small andvery large inner cores or strong inertial forcesare three possibilities that lead to a sizable butalso very time dependent axial quadrupole con-tribution.

A more persistent Mercury-like mean offsetcan be enforced by imposing lateral variationsin the core-mantle boundary heat flux. Pat-tern with either an increased heat flux in thenorthern hemisphere or in the equatorial regionyield the desired result. They are particularlyeffective when the dynamo is not driven by agrowing inner core but by homogeneously dis-tributed buoyancy sources [Cao et al., 2014].New models for Mercury’s interior, however,suggest that neither pattern is likely to persisttoday.

Unfortunately, the measures that promote astronger axial quadrupole also tend to promotenon-dipolar and non-axisymmetric field con-

33

tributions in general. The offset of the mag-netic equator therefore strongly depends onlongitude and distance to the planet, which isat odds with the MESSENGER observations.Dynamo simulations by Christensen [2006] andChristensen and Wicht [2008] have shown thata stably stratified outer core layer helps tosolve this problem. The magnetic field thatis produced in the deeper core regions has todiffusive through this largely passive layer toreach the planetary surface. And since themagnetic field varies in time it is damped bythe magnetic skin effect during this process.Higher harmonic and non-axisymmetric con-tributions are damped more effectively thanaxial dipole or quadrupole because the vari-ation time scale decreases with increasing spa-tial complexity. When reaching the surface,the field is therefore not only more Mercury-like in geometry but also similarly weak.

Recent interior models for Mercury suggestthat a stable outer core layer may indeed ex-ist. Because of the low pressures in Mercury’souter core, an outer iron snow zone should de-velop underneath the CMB for mean core sul-fur concentration beyond about 2 wt%. As theplanet cools, the snow zone extends deeper intothe core and a stably stratifying sulfur gradi-ent develops. Since the mean heat flux out ofthe Hermean core is likely subadiabatic today,thermal effects would further contribute to sta-bilizing the outer core region. Such a layer isalso likely to persist when double-diffusive ef-fects are taken into account [Manglik et al.,2010]. Additional work on the FeS melting be-haviour, on Mercury’s interior properties, andthe planet’s thermal evolution is required tobetter understand and establish this scenario.The possible presence of Si in the Hermean corecould further complicate matters [Malavergneet al., 2010].

Dynamo simulations that more realisticallymodel the iron snow stratification and the con-vective driving in the presence of an iron snow

zone and possibly also a growing inner coreseem a logical next step. Lateral variationsin the core-mantle boundary heat flux and afeedback with the magnetospheric field are twoother features that may play an important rolein Mercury’s dynamo process.

The Hermean magnetospheric field remainsa challenging puzzle despite the wealth of datadelivered by the MESSENGER magnetome-ter. It’s small size and high variability com-plicates the separation of internal and exter-nal field contributions, of temporal and spa-tial variations, and of solar wind dynamics andMercury’s genuine field dynamics. The Bepi-Colombo mission, scheduled for launch in 2016,will significantly improve the situation sincetwo spacecrafts will orbit the planet at thesame time, a planetary orbiter build by ESAand a magnetospheric orbiter build by JAXA.

Acknowledgement

Johannes Wicht was supported by theHelmholz Alliance ”Planetary Evolution andLive” and by the Special Priority Programm1488 ”Planetary Magnetism” of the GermanScience Foundation. D. Heyner was supportedby the German Ministerium fur Wirtschaftund Technologie and the German Zentrumfur Luft- und Raumfahrt under contract 50QW 1101. We thank Attilio Rivoldini, TinaRuckriehmen, Wieland Dietrich, Hao Cao,Brian Anderson, Karl-Heinz Glassmeier, andUlrich R. Christensen for helpful discussions.Attilio Rivoldini also kindly provided figure 1.

References

S. C. Solomon. Some aspects of core forma-tion in Mercury. Icarus, 28:509–521, August1976. doi: 10.1016/0019-1035(76)90124-X.

N. F. Ness, K. W. Behannon, R. P. Lepping,

34

Y. C. Whang, and K. H. Schatten. Mag-netic Field Observations near Mercury: Pre-liminary Results from Mariner 10. Science,185:151–160, July 1974. doi: 10.1126/sci-ence.185.4146.151.

J. A. Slavin, S. M. Krimigis, M. H. Acuna, B. J.Anderson, D. N. Baker, P. L. Koehn, H. Ko-rth, S. Livi, B. H. Mauk, S. C. Solomon,and T. H. Zurbuchen. MESSENGER: Ex-ploring Mercury’s Magnetosphere. SpaceSci. Rev., 131:133–160, August 2007. doi:10.1007/s11214-007-9154-x.

S. C. Solomon, R. L. McNutt, R. E. Gold,and D. L. Domingue. MESSENGER MissionOverview. Space Sci. Rev., 131:3–39, August2007. doi: 10.1007/s11214-007-9247-6.

Brian J Anderson, Catherine L Johnson,Haje Korth, Reka M Winslow, Joseph EBorovsky, Michael E Purucker, James ASlavin, Sean C Solomon, Maria T Zuber,and Ralph L McNutt. Low-degree structurein Mercury’s planetary magnetic field. Jour-nal of Geophysical Research: Planets (1991–2012), 117(E12), 2012.

C. L. Johnson, M. E. Purucker, H. Ko-rth, B. J. Anderson, R. M. Winslow,M. M. H. Al Asad, J. A. Slavin, I. I.Alexeev, R. J. Phillips, M. T. Zuber, andS. C. Solomon. MESSENGER observa-tions of Mercury’s magnetic field struc-ture. Journal of Geophysical Research (Plan-ets), 117:E00L14, December 2012. doi:10.1029/2012JE004217.

N. Olsen, K.-H. Glassmeier, and X. Jia. Sepa-ration of the Magnetic Field into Externaland Internal Parts. Space Sci. Rev., 152:135–157, May 2010. doi: 10.1007/s11214-009-9563-0.

B. J. Anderson, C. L. Johnson, H. Korth, M. E.Purucker, R. M. Winslow, J. A. Slavin, S. C.

Solomon, R. L. McNutt, J. M. Raines, andT. H. Zurbuchen. The Global Magnetic Fieldof Mercury from MESSENGER Orbital Ob-servations. Science, 333:1859–, September2011. doi: 10.1126/science.1211001.

D. E. Smith, M. T. Zuber, R. J. Phillips,S. C. Solomon, S. A. Hauck, F. G. Lemoine,E. Mazarico, G. A. Neumann, S. J. Peale, J.-L. Margot, C. L. Johnson, M. H. Torrence,M. E. Perry, D. D. Rowlands, S. Goossens,J. W. Head, and A. H. Taylor. Grav-ity Field and Internal Structure of Mer-cury from MESSENGER. Science, 336:214–,April 2012. doi: 10.1126/science.1218809.

J.-L. Margot, S. J. Peale, S. C. Solomon,S. A. Hauck, II, F. D. Ghigo, R. F.Jurgens, M. Yseboodt, J. D. Giorgini,S. Padovan, and D. B. Campbell. Mer-cury’s moment of inertia from spin and grav-ity data. Journal of Geophysical Research(Planets), 117:E00L09, October 2012. doi:10.1029/2012JE004161.

S. J. Peale. Generalized Cassini’s Laws.Astronom. J., 74:483, April 1969. doi:10.1086/110825.

S. A. Hauck, J.-L. Margot, S. C. Solomon,R. J. Phillips, C. L. Johnson, F. G. Lemoine,E. Mazarico, T. J. McCoy, S. Padovan, S. J.Peale, M. E. Perry, D. E. Smith, and M. T.Zuber. The curious case of Mercury’s inter-nal structure. Journal of Geophysical Re-search (Planets), 118:1204–1220, June 2013.doi: 10.1002/jgre.20091.

A. Rivoldini and T. Van Hoolst. The in-terior structure of Mercury constrained bythe low-degree gravity field and the ro-tation of Mercury. Earth Planet. Sci.Lett., 377:62–72, September 2013. doi:10.1016/j.epsl.2013.07.021.

35

L. R. Nittler, R. D. Starr, S. Z. Weider,T. J. McCoy, W. V. Boynton, D. S. Ebel,C. M. Ernst, L. G. Evans, J. O. Goldsten,D. K. Hamara, D. J. Lawrence, R. L. Mc-Nutt, C. E. Schlemm, S. C. Solomon, andA. L. Sprague. The Major-Element Com-position of Mercury’s Surface from MES-SENGER X-ray Spectrometry. Science, 333:1847–, September 2011. doi: 10.1126/sci-ence.1211567.

G. Di Achille, C. Popa, M. Massironi, E. Maz-zotta Epifani, M. Zusi, G. Cremonese, andP. Palumbo. Mercury’s radius change esti-mates revisited using MESSENGER data.Icarus, 221:456–460, September 2012. doi:10.1016/j.icarus.2012.07.005.

M. Grott, D. Breuer, and M. Laneuville.Thermo-chemical evolution and global con-traction of mercury. Earth Planet. Sci.Lett., 307:135–146, July 2011. doi:10.1016/j.epsl.2011.04.040.

N. Tosi, M. Grott, A.C. Plesa, and D. Breuer.Thermochemical evolution of Mercury’s in-terior. acc. for pub. at Geophys. Res. Lett.,2014.

S. C. Solomon, P. K. Byrne, C. Klimczak,A. M. Cegal Sengor, T. R. Watters, and S. A.Hauck. Geological evidence that Mercurycontracted by more than previously recog-nized. presented at AGU, Dec. 2013, 2014.

N. C. Michel, S. A. Hauck, S. C. Solomon,R. J. Phillips, J. H. Roberts, and M. T.Zuber. Thermal evolution of Mercury asconstrained by MESSENGER observations.J. Geophys. Res., 118:1033–1044, May 2013.doi: 10.1002/jgre.20049.

S. A. Hauck, J. M. Aurnou, and A. J. Dom-bard. Sulfur’s impact on core evolution andmagnetic field generation on Ganymede. J.

Geophys. Res., 111:E09008, September 2006.doi: 10.1029/2005JE002557.

A. Rivoldini, T. Van Hoolst, O. Verhoeven,A. Mocquet, and V. Dehant. Geodesy con-straints on the interior structure and com-position of Mars. Icarus, 213:451–472, June2011. doi: 10.1016/j.icarus.2011.03.024.

R. Vilim, S. Stanley, and S. A. Hauck. Ironsnow zones as a mechanism for generatingMercury’s weak observed magnetic field. J.Geophys. Res., 115(E14):E11003, November2010. doi: 10.1029/2009JE003528.

V. Malavergne, M. J. Toplis, S. Berthet, andJ. Jones. Highly reducing conditions duringcore formation on Mercury: Implications forinternal structure and the origin of a mag-netic field. Icarus, 206:199–209, March 2010.doi: 10.1016/j.icarus.2009.09.001.

G. Morard and T. Katsura. Pressure-temperature cartography of Fe-S-Siimmiscible system. Geoch. Cosmoch.Acta, 74:3659–3667, June 2010. doi:10.1016/j.gca.2010.03.025.

J. Wicht, S. Stellmach, and H. Harder. Numer-ical dynamo simulations: From basic con-cepts to realistic models. In W. Freeden,M.Z. Nashed, and T. Sonar, editors, Hand-book of Geomathematics, sec. ed., pages459–502. Springer, Berlin - Heidelberg -NewYork, 2011a.

W. Dietrich and J. Wicht. A hemisphericaldynamo model: Implications for the Mar-tian crustal magnetization. Phys. EarthPlanet. Inter., 217:10–21, April 2013. doi:10.1016/j.pepi.2013.01.001.

S. Stanley, L. Elkins-Tanton, M. T. Zuber, andE. M. Parmentier. Mars’s PaleomagneticField as the Result of a Single-HemisphereDynamo. Science, 321:1822–, September2008. doi: 10.1126/science.1161119.

36

H. Amit, U. R. Christensen, and B. Langlais.The influence of degree-1 mantle heterogene-ity on the past dynamo of Mars. Phys. EarthPlanet. Inter., 189:63–79, November 2011.doi: 10.1016/j.pepi.2011.07.008.

B. W. Denevi, C. M. Ernst, H. M. Meyer, M. S.Robinson, S. L. Murchie, J. L. Whitten,J. W. Head, T. R. Watters, S. C. Solomon,L. R. Ostrach, C. R. Chapman, P. K. Byrne,C. Klimczak, and P. N. Peplowski. Thedistribution and origin of smooth plains onMercury. J. Geophys. Res., 118:891–907,May 2013. doi: 10.1002/jgre.20075.

J.-P. Williams, J. Ruiz, M. A. Rosenburg,O. Aharonson, and R. J. Phillips. Insolationdriven variations of Mercury’s lithosphericstrength. J. Geophys. Res., 116:E01008, Jan-uary 2011. doi: 10.1029/2010JE003655.

R. M. Winslow, B. J. Anderson, C. L. John-son, J. A. Slavin, H. Korth, M. E. Purucker,D. N. Baker, and S. C. Solomon. Mer-cury’s magnetopause and bow shock fromMESSENGER Magnetometer observations.J. Geophys. Res., 118:2213–2227, May 2013.doi: 10.1002/jgra.50237.

J. Muller, S. Simon, Y.-C. Wang,U. Motschmann, D. Heyner, J. Schule,W.-H. Ip, G. Kleindienst, and G. J. Pringle.Origin of Mercury’s double magnetopause:3D hybrid simulation study with A.I.K.E.F.Icarus, 218:666–687, March 2012. doi:10.1016/j.icarus.2011.12.028.

K.-H. Glassmeier, H.-U. Auster, D. Heyner,K. Okrafka, C. Carr, G. Berghofer, B. J.Anderson, A. Balogh, W. Baumjohann,P. Cargill, U. Christensen, M. Delva,M. Dougherty, K.-H. Fornacon, T. S.Horbury, E. A. Lucek, W. Magnes,M. Mandea, A. Matsuoka, M. Matsushima,U. Motschmann, R. Nakamura, Y. Narita,H. O’Brien, I. Richter, K. Schwingenschuh,

H. Shibuya, J. A. Slavin, C. Sotin, B. Stoll,H. Tsunakawa, S. Vennerstrom, J. Vogt,and T. Zhang. The fluxgate magnetometerof the BepiColombo Mercury Planetary Or-biter. Planet. Space Sci., 58:287–299, Jan-uary 2010. doi: 10.1016/j.pss.2008.06.018.

J. A. Slavin, B. J. Anderson, D. N. Baker,M. Benna, S. A. Boardsen, R. E. Gold,G. C. Ho, S. M. Imber, H. Korth, S. M.Krimigis, R. L. McNutt, Jr., J. M. Raines,M. Sarantos, D. Schriver, S. C. Solomon,P. Travnıcek, and T. H. Zurbuchen. MES-SENGER and Mariner 10 flyby observationsof magnetotail structure and dynamics atMercury. J. Geophys. Res., 117:A01215,January 2012. doi: 10.1029/2011JA016900.

P. Janhunen and E. Kallio. Surface conductiv-ity of Mercury provides current closure andmay affect magnetospheric symmetry. An-nales Geophysicae, 22:1829–1837, May 2004.doi: 10.5194/angeo-22-1829-2004.

G. A. Dibraccio, J. A. Slavin, S. A. Board-sen, B. J. Anderson, H. Korth, T. H. Zur-buchen, J. M. Raines, D. N. Baker, R. L.McNutt, and S. C. Solomon. MESSENGERobservations of magnetopause structure anddynamics at Mercury. Journal of Geophysi-cal Research (Space Physics), 118:997–1008,March 2013. doi: 10.1002/jgra.50123.

W. Baumjohann, M. Blanc, A. Fedorov, andK.-H. Glassmeier. Current Systems in Plane-tary Magnetospheres and Ionospheres. SpaceSci. Rev., 152:99–134, May 2010. doi:10.1007/s11214-010-9629-z.

V. Mangano, S. Massetti, A. Milillo, A. Mura,S. Orsini, and F. Leblanc. Dynamical evolu-tion of sodium anisotropies in the exosphereof Mercury. Planet. Space Sci., 82:1–10, July2013. doi: 10.1016/j.pss.2013.03.002.

37

K.-H. Glassmeier, H.-U. Auster, andU. Motschmann. A feedback dynamogenerating Mercury’s magnetic field. Geo-phys. Res. Lett., 34:22201–+, November2007. doi: 10.1029/2007GL031662.

J. Bartels. The eccentric dipole approximatingthe Earth’s magnetic field. J. Geophys. Res.,41:225–250, 1936.

I. I. Alexeev, E. S. Belenkaya, J. A. Slavin,H. Korth, B. J. Anderson, D. N. Baker,S. A. Boardsen, C. L. Johnson, M. E. Pu-rucker, M. Sarantos, and S. C. Solomon.Mercury’s magnetospheric magnetic fieldafter the first two MESSENGER flybys.Icarus, 209:23–39, September 2010. doi:10.1016/j.icarus.2010.01.024.

V. Lesur, P. Kunagu, S. Asari, M. Hamoudi,M. Rother, and I. Wardinski. GRIMM–3,Third version of the GFZ Reference Internalmagnetic Model. In preparation, 2012.

J. E. P. Connerney, M. H. Acuna, N. F. Ness,and T. Satoh. New models of Jupiter’smagnetic field constrained by the Io fluxtube footprint. J. Geophys. Res., 103:11929–11940, June 1998. doi: 10.1029/97JA03726.

H. Cao, C. T. Russell, J. Wicht, U. R. Chris-tensen, and M. K. Dougherty. Saturn’s highdegree magnetic moments: evidence for aunique planetary dynamo. Icarus, 221:388–394, 2012.

R. Holme and J. Bloxham. The magnetic fieldsof Uranus and Neptune: Methods and mod-els. J. Geophys. Res., 101:2177–2200, 1996.doi: 10.1029/95JE03437.

M. E. Purucker, C. L. Johnson, R. M. Winslow,J. B. Nicholas, B. J. Anderson, H. Korth,J. W. Head, M. T. Zuber, S. C. Solomon,J. A. Slavin, I. I. Alexeev, R. J. Phillips, and

D. A. Paige. Evidence for a Crustal Mag-netic Signature on Mercury from MESSEN-GER Magnetometer Observations. In Lunarand Planetary Institute Science ConferenceAbstracts, volume 43 of Lunar and PlanetaryInstitute Science Conference Abstracts, page1297, March 2012.

S.I. Braginsky and P.H. Roberts. Equationsgoverning convection in Earth’s core andthe geodynamo. Geophys. Astrophys. FluidDyn., 79:1–97, 1995.

C. Kutzner and U.R. Christensen. Effects ofdriving mechanisms in geodynamo models.Geophys. Res. Lett., 27:29–32, 2000.

J. Aubert, H. Amit, G. Hulot, and P. Olson.Thermochemical flows couple the Earth’sinner core growth to mantle heterogeneity.Nature, 454:758–761, August 2008. doi:10.1038/nature07109.

U. Christensen and J. Wicht. Numerical dy-namo simulations. In Olson P., editor, CoreDynamics, volume 8 of Treatise on Geo-physics, pages 245–282. Elsevier, 2007.

J. Wicht, M. Mandea, F. Takahashi,U. R. Christensen, M. Matsushima,and B. Langlais. The Origin of Mer-cury’s Internal Magnetic Field. Space Sci.Rev., 132:261–290, October 2007. doi:10.1007/s11214-007-9280-5.

S. Stanley and G. A. Glatzmaier. DynamoModels for Planets Other Than Earth. SpaceSci. Rev., 152:617–649, May 2010. doi:10.1007/s11214-009-9573-y.

G. Schubert and K. M. Soderlund. Planetarymagnetic fields: Observations and models.Phys. Earth Planet. Inter., 187:92–108, Au-gust 2011. doi: 10.1016/j.pepi.2011.05.013.

38

P. Olson and U. Christensen. Dipole momentscaling for convection-driven planetary dy-namos. Earth Planet. Sci. Lett., 250:561–571, 2006.

U. R. Christensen. Dynamo scaling laws: ap-plications to the planets. Space Sci. Rev.,152:565–590, 2010.

J. Wicht, S. Stellmach, and H. Harder. Numer-ical dynamo simulations: From basic con-cepts to realistic models. In W. Freeden,M.Z. Nashed, and T. Sonar, editors, Hand-book of Geomathematics, pages 459–502.Springer, Berlin - Heidelberg - NewYork,2011b.

U. R. Christensen, J. Aubert, and G. Hu-lot. Conditions for Earth-like geo-dynamo models. Earth Planet. Sci.Lett., 296:487–496, August 2010. doi:10.1016/j.epsl.2010.06.009.

U. Christensen and J. Aubert. Scaling proper-ties of convection-driven dynamos in rotat-ing spherical shells and applications to plan-etary magnetic fields. Geophys. J. Int., 116:97–114, 2006.

D. J. Stevenson. Saturn’s luminosity and mag-netism. Science, 208:746–748, 1980.

W. Lorenzen, B. Holst, and R. Red-mer. Demixing of Hydrogen and He-lium at Megabar Pressures. Phys. Rev.Lett., 102(11):115701, March 2009. doi:10.1103/PhysRevLett.102.115701.

U. R. Christensen. A deep dynamo generatingMercury’s magnetic field. Nature, 444:1056–1058, 2006.

U. R. Christensen and J. Wicht. Models ofmagnetic field generation in partly stableplanetary cores: Applications to Mercuryand Saturn. Icarus, 196:16–34, July 2008.doi: 10.1016/j.icarus.2008.02.013.

U. Christensen and A. Tilgner. Power require-ment of the geodynamo from Ohmic losses innumerical and laboratory dynamos. Nature,429:169–171, 2004.

F. Lhuillier, A. Fournier, G. Hulot, andJ. Aubert. The geomagnetic secular varia-tion timescale in observations and numeri-cal dynamo models. Geophys. Res. Lett., 38:L09306, 2011.

A. Manglik, J. Wicht, and U. R. Christensen.A dynamo model with double diffusive con-vection for Mercury’s core. Earth and Plan-etary Science Letters, 289:619–628, January2010. doi: 10.1016/j.epsl.2009.12.007.

H. Cao, J. Aurnou, J. Wicht, W. Dietrich,K. M. Soderlund, and C. T. Russell. A dy-namo explanation for the large offset of Mer-cury’s magnetic equator. submitted to Geo-phys. Res. Lett., 2014.

K. Hori, J. Wicht, and U. R. Christensen. Theinfluence of thermo-compositional bound-ary conditions on convection and dynamosin a rotating spherical shell. Phys. EarthPlanet. Inter., 196:32–48, April 2012. doi:10.1016/j.pepi.2012.02.002.

M. Landeau and J. Aubert. Equatoriallyasymmetric convection inducing a hemi-spherical magnetic field in rotating spheresand implications for the past martian dy-namo. Physics of the Earth and Plane-tary Interiors, 185:61–73, April 2011. doi:10.1016/j.pepi.2011.01.004.

Z. Tian, M. T. Zuber, and S. Stanley. Explain-ing Mercurys magnetic field observables us-ing dynamo models with stable layers andlaterally variable heat flux. presented atAGU, Dec. 2013, 2013.

D. Heyner, J. Wicht, N. Gomez-Perez,D. Schmitt, H.-U. Auster, and K.-H. Glass-

39

meier. Evidence from Numerical Experi-ments for a Feedback Dynamo GeneratingMercury’s Magnetic Field. Science, 334:1690–, December 2011a. doi: 10.1126/sci-ence.1207290.

M. H. Heimpel, J. M. Aurnou, F. M. Al-Shamali, and N. Gomez Perez. A numer-ical study of dynamo action as a functionof spherical shell geometry. Earth Planet.Sci. Lett., 236:542–557, July 2005. doi:10.1016/j.epsl.2005.04.032.

F. Takahashi and M. Matsushima. Dipolar andnon-dipolar dynamos in a thin shell geome-try with implications for the magnetic fieldof Mercury. Geophys. Res. Lett., 33:L10202,2006. doi: 10.1029/2006GL025792.

S. Stanley, J. Bloxham, W.E. Hutchison, andM.T. Zuber. Thin shell dynamo models con-sistent with Mercury’s weak observed mag-netic field. Earth Planet. Sci. Lett., 234:341–353, 2005.

D. Heyner, D. Schmitt, J. Wicht, K.-H. Glass-meier, H. Korth, and U. Motschmann. Theinitial temporal evolution of a feedback dy-namo for Mercury. Geophys. Astrophys.Fluid Dyn., 104:419–429, August 2010. doi:10.1080/03091921003776839.

D. Heyner, D. Schmitt, K.-H. Glassmeier, andJ. Wicht. Dynamo action in an ambient field.Astronom. Nach., 332:36, January 2011b.doi: 10.1002/asna.201011466.

40