Physics of the Earth and Planetary Interiors 187 (2011) 92–108

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Physics of the Earth and Planetary Interiors

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Planetary magnetic fields: Observations and models ⇑ G. Schubert , K.M. Soderlund

Department of Earth and Space Sciences, University of California, Los Angeles, CA 90095, USA article info abstract

Article history: The state of knowledge and understanding of planetary magnetic fields is reviewed. All of the planets, Available online 12 June 2011 with the possible exception of Venus, have had active dynamos at some time in their evolution. The prop- Edited by Keke Zhang erties and characteristics of the dynamos are as diverse as the planets themselves. Even within the sub- classes of terrestrial and giant planets, the contrasting compositions, sizes, and internal pressures and Keywords: temperatures of the planets result in strikingly different dynamos. As an example, the dynamos in Mer- Planetary magnetic fields cury and Ganymede are likely driven by compositional buoyancy distributions different from that in the Planetary cores Earth’s core. Dynamo models operate far from the parameter regimes appropriate to the real planets, yet Spacecraft observations provide insight into the dynamics of their interiors. While Boussinesq models are generally adequate for Numerical dynamo models simulating terrestrial planet dynamos, anelastic models that also account for large density and electrical conductivity variations are needed to simulate the dynamos in giant planets. Future spacecraft missions to planets with active dynamos are needed to learn about the character and temporal variability of the planetary magnetic fields. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction lution. Together with the gravitational field, the magnetic field provides a window into the interiors of bodies that we can only With the possible exception of Venus, all planets in the solar probe from a distance. The magnetic fields recorded and preserved system have or once had internally generated magnetic fields. in the crusts of planets and satellites also provide a window into The Moon also had its own magnetic field in the past, and Jupi- the histories of the bodies. ter’s satellite Ganymede has a magnetic field at present. Undoubt- Planetary magnetism therefore merits study from many per- edly, other bodies in the solar system have generated magnetic spectives, including that of understanding the physics of how fields in the past, e.g., the angrite meteorite parent body (Weiss dynamos work and what they imply about the past and present et al., 2008) and possibly some outer solar system objects other properties of the body containing the dynamo. In this paper we re- than Ganymede have a magnetic field at present. It is expected view what is known about magnetic fields in the solar system, that many of the extrasolar planets now being detected have what attempts have been made to simulate the working dynamos, magnetic fields and some observations support this conjecture and what we learn about the host bodies from their magnetic (e.g., Sanchez-Lavega, 2004; Shkolnik et al., 2005, 2008). fields. While many planets, satellites, and small bodies generate mag- netic fields by dynamo action in their interiors, not all objects with 2. Survey of planetary magnetism the requisite electrically conducting fluid layers have a dynamo. Prominent among such bodies in our own solar system are Venus The planets in our solar system can be classified into three types and Io, which are likely to have completely or at least partially – terrestrial, gas giant, or ice giant – based on their physical proper- molten metallic cores (e.g., Konopliv and Yoder, 1996; McEwen ties, and the magnetic fields show a large diversity in structure and et al., 1989; Khurana et al., 2011). The cores in Mars and the Moon strength. Table 1 summarizes the physical properties and magnetic are also likely partially molten at present (e.g., Yoder et al., 2003; field characteristics of the planets and select satellites. The mean Khan et al., 2004), yet the dynamos that once existed in these density and moment of inertia (MOI) of a planet provide constraints bodies have ceased to function. on the internal composition and structure. All of the planets and A planetary magnetic field, or the absence of such a field, in- many of the satellites show evidence of differentiation into layers. forms us about the internal structure of a body and its thermal evo- The terrestrial planets have an outer rocky crust, an intermediate silicate mantle, and a central iron-alloy core (e.g., Riner et al.,

⇑ Corresponding author. Tel.: +1 310 825 4577; fax: +1 310 825 2779. 2008; Zharkov, 1983; Dziewonski and Anderson, 1981; Longhi E-mail addresses: [email protected] (G. Schubert), [email protected] (K.M. et al., 1992). The gas giants, Jupiter and Saturn, have outer molecu- Soderlund). lar envelopes of hydrogen and helium, fluid mantles that are largely

0031-9201/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2011.05.013 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 93

Table 1 Summary of physical properties and magnetic field characteristics. Overbars indicate mean values. Physical property data taken from 1Lodders and Fegley (1998), 2de Pater and Lissauer (2001), 3Schubert et al. (2007) for the MOI of the Galilean satellites, and 4Iess et al. (2010) for the MOI of Titan. Magnetic field characteristics are calculated using Eqs. (2)– (4) and data in Uno et al. (2009) for Mercury, IAGA Division V-MOD geomagnetic field modeling website for Earth, Yu et al. (2010) for Jupiter, Kivelson et al. (2002) for Ganymede, Burton et al. (2009) for Saturn, and Holme and Bloxham (1996) for the ice giants. MOI is normalized with respect to the product of the mass and the square of the radius of the body.

1 1 1 24 Planet Mass Radius Density MOI Surface jBr j Dipolarity Dipole (1024 kg) (km) (kg/m3)(lT) tilt (°) Mercury 0.33 2440 5.4 0.33 0.30 0.71 3 Venus 4.87 6052 5.2 0.33 – – – Earth 5.97 6371 5.5 0.33 38 0.61 10 Moon 0.07 1738 3.3 0.39 [100 – – Mars 0.64 3390 3.9 0.37 [0.1 – – Jupiter 1900 69,911 1.3 0.25 550 0.61 9 Io 0.09 1821 3.5 0.38 – – – Europa 0.05 1565 3.0 0.35 – – – Ganymede 0.15 2634 1.9 0.31 0.91 0.95 4 Callisto 0.11 2403 1.9 0.35 – – – Saturn 570 58,232 0.7 0.21 28 0.85 <0.5 Titan 0.13 2575 1.9 0.34 – – – Uranus 87 25,362 1.3 0.23 32 0.42 59 Neptune 100 24,624 1.6 0.23 27 0.31 45

( composed of metallic hydrogen and helium, and perhaps central Xlmax Xl R lþ2 B ðrÞ¼rU ^r ¼ ðl þ 1Þ P gm cosðm/Þ rocky cores (Guillot, 1999a). The ice giants, Uranus and Neptune, r r l have outer molecular envelopes of hydrogen, helium, and ices, l¼1 m¼0 m m intermediate ionic oceans of water, methane, and ammonia, and þhl sinðm/Þ Pl ðcoshÞ ð2Þ possibly central rocky cores (Guillot, 1999a). However, the layer where lmax is the maximum harmonic degree of the observations. thicknesses are not well-constrained for planets other than the The two radial levels most frequently considered are the planet sur- Earth. Outer solar system satellites exhibit a wide variety of sizes, face, r = RP, and the top of the dynamo-generating region, r = RD. The from tens of kilometers to larger than the planet Mercury, and have strength of the dipole relative to the total field strength is the mean densities that are icy to rocky. Ganymede is the largest satel- dipolarity, lite in the solar system and also has the lowest moment of inertia of P all known planetary bodies (Schubert et al., 2007). This indicates 1 jgmjþjhmj f ¼ P mP¼0 1 1 : ð3Þ that the satellite is fully differentiated, with a thick water-ice shell lmax l m m l¼1 m¼0 jgl jþjhl j overlying an intermediate rocky mantle and a central metallic core (Schubert et al., 2007). Thus, Ganymede has a terrestrial-style inter- Perfectly-dipolar magnetic fields have f = 1. The fields are consid- nal structure beneath the ice shell. ered to be dipole-dominated when f > 0.5. The tilt of the dipole rel- Magnetic fields originate in the electrically conducting fluid re- ative to the rotation axis, hd, is given by 01 gions of planets and satellites and are likely driven by dynamo ac- 1=2 2 2 tion (e.g., Elsasser, 1939; Bullard, 1949; Stevenson, 2003). These B g1 þ h1 C B 1 1 C regions are expected to be unstable to thermochemical convection h ¼ tan1 B C: ð4Þ d @ 0 A in order to remove the heat of formation and radioactive decay jg1j (Stevenson, 2010). Convection of an electrically conducting fluid drives electrical currents and generates a magnetic field; the pro- cess whereby kinetic energy is converted into magnetic energy is 2.1. Active dynamos known as dynamo action. While convection is the most probable dynamo driver, it has been argued that precession, libration, nuta- 2.1.1. Earth tion, and tidal interactions may also promote dynamo action Earth’s magnetic field is at least 3.5 billion years old as recorded (Malkus, 1994; Tilgner, 2007; Dwyer et al., 2010). by crustal rocks that were magnetized as they cooled (Hale and Intrinsic planetary magnetic fields diffuse as a potential field Dunlop, 1984). These paleomagnetic records indicate that the dy- through the overlying electrically insulating or less electrically namo has almost always had a strong axial dipole component with conducting regions. This field can be expressed as B = U, where r a field strength that has not significantly varied over time (Hulot the magnetic potential U can be decomposed into spherical et al., 2010). The present field has a mean surface strength of about harmonics: 40 lT and the dominant dipole component is offset 10° from the ( rotation axis. However, episodic reversals in field polarity have X1 Xl R lþ1 Uðr; h; /Þ¼R P gm cosðm/Þ been recorded as magnetic stripes in oceanic crust over the past P r l l¼1 m¼0 hundred million years (Johnson et al., 2003). During the 10,000 years over which reversals occur, the total field strength decreases hm sin m/ Pm cosh 1 þ l ð Þ l ð Þ ð Þ and the field becomes multipolar (Bogue and Merrill, 1992). Observatories and satellites have greatly improved our under- where r is radius, h is colatitude, / is longitude, RP is the planet ra- standing of the Earth’s magnetic field and its evolution over short dius, l is spherical harmonic degree, m is spherical harmonic order, timescales. The Magsat (1979–1980), Ørsted (1999–), CHAMP m Pl are associated Legendre polynomials with partial Schmidt nor- (2000–2010), and SAC-C (2000–2004) satellites have provided con- m malization (see Appendix B of Merrill et al. (1996)), and gl and tinuous measurements of the field over a range of altitudes for m hl are the gauss coefficients. The gauss coefficients are used to de- more than ten years. The SWARM satellite constellation will be scribe planetary magnetic fields. The radial magnetic field for a gi- launched in 2011 to continue these efforts. Fig. 1 shows the radial ven radius r is obtained through magnetic field at the surface and the core-mantle boundary plotted 94 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 up to spherical harmonic degree 13. Higher order harmonics of the As shown in Fig. 2b, these measurements suggest that the field is internal field are masked by crustal remnant magnetism. The dy- nearly axially-aligned and dipole-dominated with a mean surface namic nature of the geodynamo is revealed through secular varia- strength of about 1 lT. However, it might also be appropriate to tion, where changes in the strength and spatial structure of the consider the value of Ganymede’s magnetic field at the ice-rock field occur over timescales of months (Olsen and Mandea, 2008) boundary since the body has a terrestrial-style internal structure to centuries (Bloxham et al., 1989) to millenia (Cande and Kent, beneath the ice shell. Assuming an ice shell thickness of about 1995). The most notable aspects of secular variation are polarity 900 km (Schubert et al., 2007), the mean magnetic field is 3 lTat reversals, westward drift of the non-axisymmetric field, anticy- the ice-rock boundary. This is about an order of magnitude larger clonic field motions at high latitudes, growth of the South Atlantic (smaller) than Mercury’s (Earth’s) surface field. The field resolu- anomaly where the field is unusually weak, and geomagnetic jerks tion, however, is limited to the dipole and quadrupole components (Finlay et al., 2010). The physical mechanisms that lead to these and additional data are needed to further characterize the mag- behaviors, however, are not yet understood. netic field. Since Ganymede is thought to have a metallic core with a fluid component, the field is likely driven by dynamo action 2.1.2. Mercury (Schubert et al., 1996; Sarson et al., 1997; Kivelson et al., 2002). Mercury has the weakest intrinsic magnetic field in the solar The size of the core is poorly constrained, and the energy source system with a mean surface strength of only about 0.3 lT. This for maintaining a liquid or partially liquid core in such a small body field was first measured during the Mariner 10 flybys in 1974 is not fully understood. and 1975 (Ness et al., 1975, 1976) and again during the MESSEN- GER flybys in 2008 and 2009 (Anderson et al., 2008, 2010; Uno 2.1.4. Jupiter et al., 2009; McNutt et al., 2010; Solomon, 2011). These observa- The magnetic field of Jupiter was discovered when Burke and tions suggest that the field is dipole-dominated and nearly aligned Franklin (1955) showed that the planet was a strong radio source. with the rotation axis. The surface radial magnetic field plotted up This was the first detection of a planetary magnetic field besides to spherical harmonic degree three is shown in Fig. 2a. Compared that of the Earth. The existence of the field was confirmed when to the Earth, the field is about 100 times weaker at the surface the planet was visited by the Pioneer 10 flyby in 1973 (Smith and about 500 times weaker at the core-mantle boundary, assum- et al., 1974). Additional measurements were obtained by the Pio- ing Mercury’s core has a radius of 1900 km. It is not yet understood neer 11 flyby in 1974 (Smith et al., 1975), the Voyager 1 and 2 fly- why Mercury has such a weak magnetic field. bys in 1979 (Ness et al., 1979), the Ulysses flybys in 1992 and 2004 The source of the field is not agreed upon in the literature (e.g., (Balogh et al., 1992), the Galileo orbiter between 1995 and 2003 Schubert et al., 1988; Stevenson, 2010), although dynamo action is (e.g., Connerney, 1993; Yu et al., 2010), and the New Horizons flyby the general consensus because libration measurements by Margot in 2007 (e.g., McComas et al., 2007; McNutt et al., 2007). These et al. (2007), gravity measurements of MESSENGER (Smith et al., measurements show that the magnetic field is dominated by the 2010), and thermodynamic phase equilibrium simulations of axial dipole component and is the strongest field in the solar sys- Malavergne et al. (2010) support the presence of a liquid core. Other tem with a mean surface strength of 550 lT. The surface of gaseous possibilities include remnant magnetism (Aharonson et al., 2005)or planets is taken to be the 1 bar pressure level. The surface radial thermoelectric currents (Stevenson, 1987; Giampieri and Balogh, magnetic field including up to the octupole component is shown 2002). Further, MESSENGER found that the internal field interacts in Fig. 2c. The spatial and temporal resolution of the field is a sig- with currents in the planet’s magnetosphere (McNutt et al., 2010), nificant limitation. While Jupiter’s magnetic field has been but the dynamical implications of this coupling are not well con- measured by several missions, the orbits do not extend inward of strained (Glassmeier et al., 2007; Kabin et al., 2008; Gomez-Perez 1.6 Jupiter radii and have longitudinal data gaps (Russell and and Solomon, 2010; Gomez-Perez and Wicht, 2010). Dougherty, 2010). Consequently, the observations are only re- solved to spherical harmonic degree three. Jupiter has not exhib- 2.1.3. Ganymede ited unambiguous secular variations in field strength or dipole The Galileo spacecraft revealed that the Jovian satellite Gany- tilt in the 20 years between the Pioneer and Galileo missions (Yu mede also has an intrinsic magnetic field (Kivelson et al., 1996). et al., 2010).

(a) Earth surface (b) Earth CMB

-80 µT0 80 µT -800 µT0 800 µT

Fig. 1. Radial magnetic field of Earth at (a) the surface and (b) the core-mantle boundary (CMB). The colors represent field intensity where purple (green) indicates outward (inward) directed field. A mollweide projection is used in which the horizontal lines indicate constant latitude. IGRF-11 coefficients are used and are available on the IAGA Division V-MOD geomagnetic field modeling website. G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 95

The magnetic field is produced by convectively-driven dynamo detecting short timescale secular variation, and constraining the action in the highly electrically conducting metallic hydrogen planet’s internal structure (Connerney, private communication). mantle and in the less electrically conducting region near the base of the molecular envelope (Stanley and Glatzmaier, 2010). The ex- 2.1.5. Saturn act location and extent of the dynamo region is poorly constrained. Saturn was revealed to have a magnetic field during the Pioneer Nellis (2000) argues that the dynamo can extend up to 95% of the 11 flyby in 1979 (Smith et al., 1980), and subsequent observations planet radius. Consequently, the dynamo region is expected to ex- were made by Voyager 1 in 1980 (Ness et al., 1981) and Voyager 2 hibit significant changes in density and electrical conductivity, the in 1981 (Ness et al., 1982). The Cassini spacecraft arrived at Saturn effects of which are relatively unknown. The upcoming Juno mis- in 2004 and is still collecting data (Dougherty et al., 2005; Burton sion will critically improve our understanding of Jupiter by resolv- et al., 2009). These observations have mapped the large-scale radial ing the magnetic field up to spherical harmonic degree 14, magnetic field, shown in Fig. 2d, and suggest that no resolved

(a) Mercury (b) Ganymede

-0.6 µT0 0.6 µT -1.5 µT0 1.5 µT

(c) Jupiter (d) Saturn

-1200 µT0 1200 µT -60 µT 0 60 µT (e) Uranus (f) Neptune

-100 µT0 100 µT -100 µT0 100 µT

Fig. 2. Radial magnetic field at the surfaces of (a) Mercury, (b) Ganymede, (c) Jupiter, (d) Saturn, (e) Uranus, and (f) Neptune. Data taken from Uno et al. (2009) for Mercury (with spectral resolution l, m 6 3), Kivelson et al. (2002) for Ganymede (l, m 6 2), Yu et al. (2010) for Jupiter (l, m 6 3), Burton et al. (2009) for Saturn (l, m 6 3), and Holme and Bloxham (1996) for the ice giants (l, m 6 3). 96 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108

(a) Mars surface field strengths near 30 lT, comparable to that of the Earth and Saturn. These multipolar magnetic fields were a surprise to 60 many scientists, and it is still not understood why these bodies produce remarkably different fields from the other planets. The fields are thought to be generated in the planets’ ionic oceans 30 (Cavazzoni et al., 1999; Lee et al., 2006). However, the internal 0 structures and dynamics of the ice giants are poorly constrained. Latitude -30 2.2. Extinct dynamos: Mars and the Moon

-60 Mars Global Surveyor (MGS) showed that Mars does not pres- 0 90180 270 0 ently have an active magnetic field, yet discovered magnetic anom- Longitude alies associated with the magnetization of crustal rocks due to an ancient dynamo (e.g., Ness et al., 1999; Acuña et al., 1998, 1999; -200 nT0 200 nT Connerney et al., 1999, 2001; Purucker et al., 2000; Langlais et al., 2004; Mitchell et al., 2007; Lillis et al., 2004, 2008a). Simi- (b) Moon larly, the Moon does not presently have an active magnetic field, but likely had one in the past. Crustal magnetization of lunar rocks 60 was first detected during the era of Apollo exploration (Coleman et al., 1972), and has since been mapped in detail by Lunar Pros- 30 pector in 1998 and 1999 (Binder, 1998; Purucker, 2008). The inter- nal structures of these bodies are consistent with the inferred 0 dynamo mechanism since there is strong evidence that both ob-

Latitude jects have metallic cores. The exact radii and densities of the cores -30 are not well-known, but are constrained by the measured values of the moment of inertia of both objects (e.g., Folkner et al., 1997; -60 Konopliv et al., 1998). The physical states of the cores are also lar- 0 90180 270 0 gely unknown, although there is evidence that both cores are at Longitude least partially molten (Yoder et al., 2003; Khan et al., 2004). The crustal magnetic fields of Mars and the Moon are mapped in -10 nT0 10 nT Fig. 3. The Martian crustal field is essentially confined to the south- ern hemisphere of the planet and is further concentrated into a rel- Fig. 3. Radial magnetic fields of (a) Mars and (b) the Moon. The colors represent atively small region of the southern highlands between about 120° field intensity where red (blue) indicates outward (inward) directed field. A and 240 East longitude (Terra Cimmeria and Terra Sirenum). Mercator projection is used, and circles indicate impact basins. Adapted from ° Langlais et al. (2010). Notably, many major impact basins lack magnetic signatures. Intensities of local flux patches at spacecraft altitudes tend to be one to two orders of magnitude larger than the Earth’s crustal mag- netization of about 20 nT (Langlais et al., 2010). The lunar crustal secular variation has occurred (Burton et al., 2009). The spatial res- field is distributed unevenly across the surface with flux patches olution is limited to spherical harmonic degree three. However, that tend to be at least an order of magnitude weaker than those this is due to uncertainty in the planet’s rotation rate (see Gurnett of Mars (Langlais et al., 2010). et al. (2007) and Anderson and Schubert (2007)) rather than insuf- The timing of the onset and cessation of dynamo activity in ficient measurements (Burton et al., 2009). The field is strongly Mars and the Moon is debated. For Mars, it has been argued that dipole-dominated with a mean surface (1 bar) strength of about the dynamo was active in the early Noachian (first Martian geo- 30 lT and has a remarkably small offset of less than 1°. No other logic period) and ceased to operate prior to the formation of the observed magnetic field has such a small dipole tilt. According to Hellas impact basin about four billion years ago (Acuña et al., Cowling’s theorem, no axisymmetric magnetic field can be gener- 1999; Acuña et al., 2001). However, Schubert et al. (2000) have ated by dynamo action, so the field must have unresolved non- proposed that the onset of dynamo activity could have postdated axisymmetric components. It has also been proposed that a the formation of Hellas with continued operation into the Hespe- secondary mechanism operating in the overlying regions may be rian or Amazonian periods (see also Milbury and Schubert responsible for axisymmetrizing the field (e.g., Stevenson, 1980, (2010)). Still another possibility is that the Martian dynamo was 1982, 1983). The mechanisms that lead to a strongly axisymmetric active for part of the Noachian prior to the Hellas impact and magnetic field, however, are not yet well-understood. was re-activated after the Hellas event (Lillis et al., 2005). There are similar uncertainties for the lunar dynamo. Paleoin- 2.1.6. Uranus and Neptune tensity data from Apollo samples imply that the dynamo was The magnetic fields of Uranus and Neptune were discovered by long-lived and turned off about four Gyr ago (Garrick-Bethell the Voyager 2 flybys in 1986 and 1989, respectively (Ness et al., et al., 2009). This early dynamo scenario is supported by modeling 1986, 1989; Holme and Bloxham, 1996). The spacecraft passed efforts. Stegman et al. (2003) propose that changes in mantle con- within 4.2 Uranian radii and 1.2 Neptunian radii, allowing their di- vection early in the Moon’s history enhanced core heat flux enough pole, quadrupole, and octupole components to be mapped as to power a dynamo, although short-lived. In addition, Takahashi shown in Fig. 2e and f. The ice giants’ magnetic fields are funda- and Tsunakawa (2009) argue that a heterogenous core-mantle mentally different from other known planetary fields since they boundary heat flux due to hemispherically-asymmetric mantle are multipolar, with f = 0.42 for Uranus and f = 0.31 for Neptune. convection can also power an early lunar dynamo. Alternatively, The dipole component is tilted about 59° and 47° from the rotation Dwyer et al. (2010) have suggested an early nutation-driven dyna- axis for Uranus and Neptune, respectively. Both planets have mean mo, while Russell et al. (1977) have proposed that the Moon’s G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 97 dynamo turned on only when the body cooled sufficiently for a so- among the fields. Given the diverse spectrum, the fundamental lid inner core to freeze out. goal is to determine what controls the strength, morphology, and Why then did the dynamos in Mars and the Moon turn off? For evolution of the magnetic fields. Since dynamos are intimately both bodies, one of two scenarios seems most likely. Either the linked to convective fluid motions driven by thermal and composi- cores of these bodies cooled to the point where thermal convection tional gradients, the coupling between magnetic fields, fluid flow, and dynamo action were no longer sustainable or they cooled to and heat/mass transfer must also be understood. the point of growing such large solid inner cores that dynamo ac- Further, it is important to understand why some large terrestrial tion could no longer take place in the surviving liquid outer core. bodies do not have, and may have not had, intrinsic magnetic fields. This raises the question of whether a purely thermally-driven dyna- 2.3. No dynamos mo can exist or if compositional buoyancy accompanying inner core freeze-out is needed. For example, Earth’s dynamo is at least 2.3.1. Venus partially driven by compositional buoyancy associated with growth Measurements by the Mariner spacecraft in the 1960s and Pio- of the solid inner core. Are the dynamos of Mercury or Ganymede neer Venus Orbiter in 1978 showed that Venus does not have an driven only by thermal gradients? Though this seems unlikely, we intrinsic magnetic field (Bridge et al., 1969; Russell et al., 1980; will probably not be able to observationally confirm or deny the Phillips and Russell, 1987). Why not? The planet likely has a mol- existence of solid inner cores in these bodies for the foreseeable ten metallic core similar in size to Earth’s core (Zhang and Zhang, future. 1995). Two plausible explanations have been discussed. One possi- bility is that Venus has not cooled sufficiently for an inner core to 3. Fundamentals of dynamo theory have yet formed but enough that a purely thermally-driven dyna- mo cannot operate. This is consistent with Venus’ slightly smaller 3.1. Basic equations size compared with Earth (Stevenson et al., 1983). It also attributes a major role to compositional buoyancy release upon inner core The governing equations of dynamo theory are the induction solidification in driving dynamo action. A second explanation is equation, conservation of mass, conservation of momentum, con- that Venus has no plate tectonics and, therefore, is inefficient in servation of energy, and the equation of state. The unknowns are cooling its core (Nimmo and Stevenson, 2000). A small heat flow the magnetic field B, the velocity field u, and three thermodynamic out of Venus’ core could mean that the core is not convective, a variables, e.g., pressure p, temperature T, and density q. There are state without the possibility of dynamo action. Venus may have nine equations for these nine variables. The induction equation is had a dynamo in the past, perhaps if a sudden mantle overturn resurfaced the planet and rapidly cooled the core as has been pro- @B ¼ r ðu BÞr ðgr BÞð5Þ posed for the Moon (Stegman et al., 2003). Venus’ high surface @t temperature makes it unlikely (though not impossible) for crustal 7 magnetization to have preserved the signature of a past dynamo. where g =1/lor is the magnetic diffusivity, lo =4p 10 H/m is the permeability of free space, and is the electrical conductivity. Absence of plate tectonics on Mercury and Ganymede, however, r has not inhibited these bodies from generating magnetic fields in The conservation of mass is their cores. @q þ r ðquÞ¼0; ð6Þ @t 2.3.2. Io Io, a moon of Jupiter about the size of Earth’s Moon, does not and the conservation of momentum is have a magnetic field (Jia et al., 2010). Yet it has a large metallic Du 1 core (Schubert et al., 2007). This is perhaps surprising in that its q þ 2qX u ¼rp qX ðX rÞþ ðr BÞB Dt l close neighbor, Ganymede, has a magnetic field and a virtually o 1 identical internal structure below its thick ice shell. What accounts þ qg þ qmrðr uÞþqmr2u; ð7Þ for the lack of dynamo activity in Io’s core? The answer appears to 3 be that the satellite is too hot. There is so much heat produced in where X is the rotation rate of the reference frame, r is the position the mantle by tidal dissipation that it cannot cool its core. With vector, g is the acceleration of gravity, and m is the kinematic viscos- no core cooling, there is no core thermal convection and no dyna- ity. In (7), terms from left to right are inertial acceleration, Coriolis mo activity (Wienbruch and Spohn, 1995; Khurana et al., 2011). acceleration, pressure gradient, centrifugal force, Lorentz force, This is basically the same explanation as offered above for why Ve- buoyancy, and two viscous diffusion terms where the Stokes nus has no magnetic field, though the reason for inefficient cooling assumption to neglect the bulk viscosity has been made. The con- of the core is different in the two cases. servation of energy equation is

2.3.3. Callisto and Titan DT aT Dp U þ r ðkrTÞ ¼ ; ð8Þ Callisto and Titan, satellites of Jupiter and Saturn, respectively, Dt qCp Dt qCp might be expected to have magnetic fields based on their compa- rable sizes to Ganymede. However, they do not (Jia et al., 2010). where a is the thermal expansivity, Cp is the specific heat at con- The lack of a magnetic field is easy to understand in the case of stant pressure, k is the thermal conductivity, and U is the dissipa- these bodies because they are only partially differentiated, rock tion function including both viscous and ohmic heating. The from ice, and do not have metallic cores (Schubert et al., 2004; Iess second term on the left of (8) represents heating (cooling) due to et al., 2010). adiabatic compression (expansion). The equation of state relates density to the temperature and/or pressure. 2.4. Comparative planetology 3.1.1. Boussinesq approximation While there are outstanding questions about each planetary In most geodynamo studies, Eqs. (5)–(8) are simplified using the magnetic field, it is also valuable to consider the comparative ap- Boussinesq approximation. Here, the material properties are as- proach which attempts to explain the similarities and differences sumed to be constant and the density is treated as constant except 98 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 in the buoyancy force of the momentum equation. The induction been studied using the Boussinesq approximation with some suc- equation simplifies to cess (e.g., Heimpel et al., 2005a; Stanley and Bloxham, 2006).

@B 2 ¼ r ðu BÞþg r B; ð9Þ 3.1.2. Anelastic approximation @t o The ratio of dynamo region thickness to density scale height is the conservation of mass simplifies to not small in the outer planets; consequently, the Boussinesq r u ¼ 0; ð10Þ approximation is not valid. Instead, the anelastic approximation should be adopted. This simplification involves replacing the con- and the conservation of momentum simplifies to tinuity equation with

Du 1 0 2 r ðquÞ¼0; ð13Þ þ 2X u ¼rP þ ðr BÞB aoT g þ mor u: ð11Þ Dt loqo where q is the basic density state. Here and below, the overbar re- The buoyancy force on the right side of (11) arises from density dif- fers to the basic state which can vary with radius. Approximating ferences with respect to the density of the motionless background (6) by (13) essentially eliminates sound waves from the problem. state (usually assumed to be constant density qo). These perturba- In this approximation, pressure variations from the basic state also tions, q0, are produced by temperature variations, T0, from the back- contribute to buoyancy in the momentum equation and adiabatic, ground state temperature profile. The equation of state is then given viscous, and ohmic heating must be accounted for in the energy 0 0 by q = qo + q = qo(1 aT ), and the gravity term becomes equation: qg = q g q aT0g. The mean gravitational component is hydrostatic o o Du 1 and can be absorbed into the pressure gradient term. Since the cen- þ 2X u ¼rP þ ðr BÞB þ vp0 aT0 g þ mr2u; ð14Þ trifugal force can be written as the gradient of a potential, this term Dt loq is also incorporated into the pressure gradient. As a result, P repre- DT a 0 2 U ðu gÞT ¼ jr T þ : ð15Þ sents the effective pressure relative to the hydrostatic pressure of a Dt Cp qCp motionless background state, i.e., P is associated with the velocity 0 field. The energy equation also simplifies in the Boussinesq approx- Here v is the isothermal compressibility and p is the pressure var- imation. Heating associated with adiabatic compression and due to iation from the basic state. The induction equation is given by (5) viscous and ohmic dissipation are assumed negligible (Hewitt et al., with g. For simplicity, a polytropic equation of state is often 1975), and (8) becomes adopted (e.g., Glatzmaier and Gilman, 1981; Clune et al., 1999; Jones and Kuzanyan, 2009). DT ¼ jr2T ð12Þ Dt 3.1.3. Boundary conditions where j is the thermal diffusivity. The dynamo equations are usually solved in the spherical shell The validity of the Boussinesq approximation requires that the geometry appropriate to planets. Velocity boundary conditions are relative density perturbations produced by temperature and pres- impenetrable and either stress-free or no-slip surfaces. No-slip sure variations be small compared with unity. The thermally in- boundaries are relevant to the liquid cores in terrestrial planets, duced relative density perturbations are proportional to aT0, while stress free boundaries may be more relevant to dynamos while the smallness of the pressure induced relative density per- in the outer planets. Temperature boundary conditions are either turbations requires that flow speeds be much less than the speed isothermal or insulating surfaces. The magnetic field must satisfy of sound. These requirements are likely satisfied in planetary cores, the continuity of the normal component of B and the continuity which are expected to be nearly-adiabatic with subsonic flows of the tangential component of B on the spherical shell boundaries. (e.g., Smylie and Rochester, 1980; Guillot, 2005). In addition, this The latter condition assumes no surface currents and no change in approximation further requires that the thickness of the convect- permeability across the boundaries. The magnetic field outside the ing region be small compared with the density scale height. dynamo region must therefore be determined or specified consis- Schubert et al. (2001) give a more detailed analysis of the Bous- tently with the field in the dynamo region. sinesq approximation and its applicability. The Boussinesq approximation is likely reasonable for the cores 3.2. Non-dimensional parameters of the terrestrial planets and moons of the outer solar system, but it is probably not adequate for the giant planets. In the Earth’s core, Non-dimensional parameters are meaningful ways to simply the density increases by 23% across the fluid layer (Boehler, 1996), represent the dynamics and geometry of a system. Table 2 summa- and the layer thickness corresponds to 0.2 density scale heights. rizes the parameters often used in the planetary dynamo commu- The number of density scale heights in a layer is given by nity. The relative influences of the terms in the governing

Nq = ln(qbot/qtop), where qbot (qtop) is the density at the bottom equations can be evaluated by taking their ratios and assuming (top) of the region (e.g., Evonuk, 2008). Since the Earth is likely typical scales for length D, velocity U, and magnetic field strength more strongly stratified than the other terrestrial bodies, the Bous- B. For example, the Ekman number, E, estimates the ratio of the vis- sinesq approximation is also practical for the cores of terrestrial cous to the Coriolis forces: E =(mr2u)/(2X u) (mU/D2)/ planets and moons of the outer solar system. However, density in- (XU)=m/XD2. The Rayleigh number, Ra = agDTD3/mj, character- creases rapidly with depth in the outer solar system planets. The izes the ratio of buoyancy to diffusion. The Prandtl numbers, gas giant internal structure model of Guillot (1999b) predicts the Pr = m/ j and Pm = m/g, characterize the ratio of viscous to thermal density to increase by a factor of about 100 between the 1 kbar and magnetic diffusivities, respectively. The Rossby number, and 10 Mbar levels; this region then extends about 5 density scale Ro = U/XD, characterizes the ratio of inertial to Coriolis forces. heights. The ice giant internal structure models of Hubbard et al. The Reynolds number, Re = UD/m = Ro/E, characterizes the ratio of (1995) predict their convecting regions to extend at least one den- inertial to viscous forces, and the magnetic Reynolds number, sity scale height. Because of the significant density stratification in Rm = UD/g = RePm, characterizes the ratio of magnetic induction 2 the outer solar system planets, it is likely necessary to adopt the to magnetic diffusion. The Elsasser number, K = B /qlog X, esti- anelastic approximation for these bodies. However, we note that mates the ratio of Lorentz to Coriolis forces. The depth of the dyna- deep convection and dynamo models of the giant planets have mo region relative to the planet radius is given by RD/RP, and the G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 99

Table 2 Lee et al., 2006); these values are used for the ice giants. The char- Non-dimensional parameter definitions and physical interpretations. Symbols are acteristic rms magnetic field strengths at the tops of the dynamo defined in the text. regions are calculated by downward continuing the measured ra- Parameter Definition Physical dial magnetic fields using (2). This provides a lower bound for interpretation the Elsasser number since the actual magnetic field strength will Ekman Number E = m/XD2 Viscous/Coriolis forces be stronger than the estimated value due to the unresolved field Rayleigh Number Ra = agoDT Buoyancy/Diffusion components. In addition, the Elsasser number will be larger still D3/mj within the dynamo region since toroidal field components are also Prandtl Number Pr = m/j Viscous/Thermal diffusivities Magnetic Prandtl Pm = m/g Viscous/Magnetic diffusivities present. Dynamo action occurs only when the fluid velocities are Number fast enough that magnetic induction overcomes magnetic diffu- Rossby Number Ro = U/XD Inertial/Coriolis forces sion; thus, the magnetic Reynolds number must exceed its critical Reynolds Number Re = UD/m Inertial/Viscous forces value: Rm J 102 (Christensen and Aubert, 2006). Using this crite- Magnetic Reynolds Rm = UD/g Magnetic induction/Magnetic Number diffusion rion, we infer the minimum flow speeds in terms of the Reynolds 2 Elsasser Number K = B /qlogX Lorentz/Coriolis forces and Rossby numbers. In the Earth’s core, secular variations suggest RD/RP Dynamo region depth core speeds of U 0.5 mm/s (Bloxham and Jackson, 1991), yielding 3 RDI/RD Dynamo region geometry Rm 10 . Assuming this value is valid for all of the planets, the flow speeds are likely at least an order of magnitude larger than the lower bound estimate. It is difficult to determine the superad- geometry of the dynamo region is given by RDI/RD. In these defini- iabatic heat flux and, thus, the Rayleigh number. For the Earth’s 22 30 tions, X is rotation rate, D = RD RDI is the thickness of the dynamo core, estimates range between 10 < Ra <10 (Jones, 2000; region, RD is the outer radius of the dynamo region, RDI is the inner Gubbins, 2001). Additional complications arise since compositional radius of the dynamo region, m is kinematic viscosity, j is thermal buoyancy should also be considered. Given this uncertainty, we do diffusivity, g is magnetic diffusivity, a is thermal expansion coeffi- not make Ra estimates for the other planets. cient, g is gravitational acceleration, DT is superadiabatic tempera- These parameters, given in Table 4, indicate that a range of dy- ture contrast, q is density, lo is magnetic permeability of free namo region geometries and convective dynamics are possible. The space, U is characteristic rms velocity, and B is characteristic rms geometries can range from a sphere for Ganymede to a thick spher- magnetic field strength. ical shell for Earth to possibly a thin spherical shell for Mercury. In order to estimate the non-dimensional parameters, we need Ganymede also has the most deep-seated dynamo. This may ex- to make assumptions about the dynamo regions, the fluid proper- plain the satellite’s strongly-dipolar surface magnetic field. How- ties, the flow speeds, and the magnetic field strengths. This is a dif- ever, recall that Ganymede consists of a water-ice shell, rocky ficult task since many of these values are poorly constrained; the mantle, and metallic core. In order to compare against other terres- best estimates are given in Table 3. Dynamo region rotation fre- trial bodies, we also consider the radius ratio of the top of the core quencies are taken from Lodders and Fegley (1998). Note that to the top of the mantle. This ratio is about 0.5 assuming a mantle the giant planet rotation rates are assumed to coincide with the thickness of 900 km (Schubert et al., 2007), similar to the RD/ rotation rates of their magnetic fields. The magnetic rotation rate RP = 0.55 ratio of the Earth. Interestingly, Uranus and Neptune ap- is inferred from the periodicity of radio emissions that result from pear to have different dynamo region geometries. Recent thermal charged magnetospheric particles (e.g., Carr et al., 1981, 1983). Ra- evolution models suggest that this difference occurs because Nep- dii of the dynamo regions are taken from Williams et al. (2007) for tune is fully convecting, while Uranus may only be convecting in a Mercury, from Dziewonski and Anderson (1981) for Earth, from thin layer (Fortney et al., 2011). Both of the gas giants have thick Nellis (2000) and Stevenson (1982) for the gas giants, from Schu- spherical shell geometries, assuming a central core is present. Fur- bert et al. (2007) for Ganymede, from Lee et al. (2006) and Podolak ther, the dynamo region of Jupiter extends up to a relatively shal- et al. (1991) for Uranus, and from Lee et al. (2006) and Stevenson low depth. This occurs because the electrical conductivity varies (1982) for Neptune. The inner dynamo region radii especially have strongly with radius and dynamo action can also occur in the a large degree of uncertainty. Based on the estimated physical semi-conducting region overlying the metallic hydrogen layer properties of the Earth’s core given in Stacey (2007), we assume (Liu et al., 2008). m 106 m2/s, g 1m2/s, and j 105 m2/s for all bodies with li- The low Ekman and Rossby numbers indicate that the Coriolis quid iron cores. Transport properties of metallic hydrogen are esti- force plays a strong role in the interiors of the planets, while vis- mated to be m 107 m2/s and j 106 m2/s (Stevenson et al., cous and inertial forces are substantially weaker. The inertial force 1977) and g 1m2/s (Nellis, 2000); these values are used for the may not be negligible, however, since all of the planets are strongly gas giants. Transport properties of high-pressure water are esti- turbulent. This is especially evident for the ice giants since they mated to be m 106 m2/s (Abramson, 2005), j 107 m2/s have the largest Reynolds numbers in the solar system. Inertial ef- (Abramson et al., 2001), and g 102 m2/s (Cavazzoni et al., 1999; fects have also been linked to multipolar magnetic field generation

Table 3

Summary of mean properties of the planets’ dynamo regions used to calculate the non-dimensional parameters given in Table 4. RP is planet radius given in Table 1. Molecular transport properties are assumed. References are given in the text. RDI is particularly uncertain for all bodies other than Earth.

Dynamo X RD RDI q mjgjBr(r = RD)j 1 3 2 2 2 (s )(RP)(RP) (kg/m )(m/s) (m /s) (m /s) (lT) Mercury 1.2 106 0.75 0.5 ? 8000 106 105 1 0.6 Earth 7.3 105 0.55 0.19 11,000 106 105 1 270 Jupiter 1.8 104 0.95 0.2 ? 2000 107 106 1 625 Ganymede 1.0 105 0.3 0 ? 6000 106 105 132 Saturn 1.6 104 0.5 0.25 ? 2000 107 106 145 Uranus 1.0 104 0.8 0.6 ? 2000 106 107 102 75 Neptune 1.1 104 0.8 0.3 ? 2000 106 107 102 73 100 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108

Table 4 tempt to review the geodynamo literature here. Instead we focus Order of magnitude estimates of key dimensionless parameters for the planets’ on recent modeling attempts to explain the magnetic fields of Mer- dynamo regions. Rm =102 is assumed to estimate lower bounds for Re = Rm/Pm and cury and Ganymede. Models of the ancient Martian dynamo will Ro = ReE. The K values neglect contributions from the toroidal field and the unresolved poloidal components, which likely increase the estimate by an order of also be discussed. See Kono and Roberts (2002), Glatzmaier magnitude. (2002), and Wicht et al. (2009) for reviews of geodynamo models.

Dynamo EPrPmRmRoReK RDI/ RD/

RD RP 4.1.1. Mercury Dynamo modeling of Mercury’s magnetic field has focused on Mercury 1012 0.1 106 102 104 108 105 0.6 0.75 Earth 1015 0.1 106 102 107 108 0.1 0.35 0.55 explaining why it is so weak. Mercury’s magnetic dipole moment 19 2 Jupiter 1019 0.1 107 102 1010 109 1 0.2 0.95 is about 3.3 4.2 10 Am (Anderson et al., 2008). Compared Ganymede 1013 0.1 106 102 105 108 103 0 0.2 with Earth’s magnetic dipole moment of 7.8 1022 Am2, Mercury’s 18 7 2 9 9 Saturn 10 0.1 10 10 10 10 0.01 0.5 0.5 moment is only about 5 104 that of Earth. Further, Mercury’s Uranus 1016 10 108 102 106 1010 104 0.6 0.8 Neptune 1016 10 108 102 106 1010 104 0.4 0.8 magnetic dipole moment is about half that of Ganymede, which is 1.32 1020 Am2, even though Ganymede, stripped of its ice shell, has a radius only about 75% of Mercury’s radius. It has been proposed that a stably-stratified layer at the top of the liquid part in dynamo models (e.g., Christensen and Aubert, 2006; Olson and of Mercury’s core might be responsible for weakening the external Christensen, 2006). As indicated by the Elsasser number, the Lor- dipole field (Christensen, 2006; Stanley and Mohammadi, 2008). In entz force is significant for the Earth and the gas giants, but may the Christensen (2006) model, this weakening is due to attenuation be less so for the other planets. It is often assumed that the planets of the internally generated field by the electrically conducting sta- are in magnetostrophic balance where the Lorentz and Coriolis ble layer. The Stanley and Mohammadi (2008) model emphasizes forces are comparable (e.g., Stevenson, 2003). The low K estimates the importance of thermal winds in the stable layer that induce for Mercury, Ganymede, and the ice giants, however, do not sup- unfavorable zonal flows throughout the liquid part of the core. In port this hypothesis. Recent dynamo modeling results further dis- their models, the magnetic field is strongly time variable and the pute the prevalence of magnetostrophic balance (e.g., Christensen external dipole can be either weak or strong. Another possible and Aubert, 2006; Soderlund et al., in review). The magnetic Pra- explanation is the thickness of the liquid part of Mercury’s core ndtl numbers much less than unity suggest a separation of scales within which the dynamo operates. Dynamo action in a thin shell between the magnetic and velocity fields where the magnetic flux can produce weak external dipole fields (Stanley et al., 2005; patches are large compared to the flow eddies. However, the Takahashi and Matsushima, 2006), while Heimpel et al. (2005b) dynamical implications of low viscosity and rapid rotation on con- have produced single plume dynamos in a thick shell that are also vection and dynamo action are not well understood since it is dif- consistent with Mercury’s weak magnetic field. ficult to study low Pm, low E systems. Yet another explanation for Mercury’s weak magnetic field in- volves the self-interaction of its internal dynamo with the magne- 3.3. Methods tosphere created around the planet by the dynamo itself, a type of feedback effect (Glassmeier et al., 2007; Gomez-Perez and Numerical models and laboratory experiments are valuable Solomon, 2010; Gomez-Perez and Wicht, 2010). Mercury is effec- tools to better understand the dynamics of planetary interiors. tively embedded in an external magnetic field generated by Chap- Both approaches, however, are unable to reach the extreme plane- man-Ferraro currents in its magnetopause. This self-generated tary parameter values due to computational and technological lim- ambient magnetic field influences the dynamics of the internal itations. Fortunately, many systems exhibit asymptotic behavior at dynamo, similar to the way the Jovian magnetic field influences large or small values of the control parameters (e.g., Christensen dynamo action in Jupiter’s moon Ganymede (Sarson et al., 1997). and Aubert, 2006; Aurnou, 2007; King et al., 2010). Large surveys Chapman-Ferraro currents generate a magnetic field that enhances are carried out to map the behavioral regimes and to determine the magnetospheric field and tends to cancel the field outside the how the control parameters influence the dynamics. With these magnetosphere. In the dynamo region, the Chapman-Ferraro field surveys, it is possible to develop scaling laws to quantify how opposes the dynamo-generated field so that the dynamo is embed- the dynamics (e.g., magnetic field strength, flow speeds, heat trans- ded in an ambient field of opposite polarity. Glassmeier et al. fer efficiency) depend on the control parameters. Researchers then (2007) use a kinematic a–X-dynamo model to show that the feed- attempt to develop asymptotic scaling laws which can be extrapo- back dynamo indeed has a Mercury-type solution with a weak lated to planetary conditions. For a recent review of scaling laws, magnetic field. Recent self-consistent dynamo models with im- see Christensen (2010). Moreover, numerical models have shown posed external magnetic fields also support strong magnetospheric that convectively-driven dynamos in rotating spherical shells are feedback on weak internal fields (Gomez-Perez and Solomon, able to generate some of the key features of the planetary magnetic 2010; Gomez-Perez and Wicht, 2010). fields (e.g., Stanley and Bloxham, 2004; Sakuraba and Roberts, The possibility that solid iron precipitation in Mercury’s core oc- 2009; Stanley and Glatzmaier, 2010). Recent planetary dynamo curs differently than it does in Earth’s core provides another feasi- models are summarized in the next section. ble explanation for Mercury’s weak magnetic field (Vilim et al., 2010). Laboratory experiments have shown that the Fe-S system 4. Planetary dynamo models behaves in a non-ideal way at temperatures and pressures likely encountered in Mercury’s core for sulfur concentrations between 4.1. Terrestrial planets and Ganymede about 7 and 12 wt% (Chen et al., 2008). As a consequence, iron could precipitate at different places in the core depending on the In this section we discuss the modeling that has been carried sulfur concentration; an Fe snowfall could originate from the top out to understand the dynamos in planets with molten iron-alloy or the mid-point of the core or from both locations. When Fe cores. The objects include the terrestrial planets, except for Venus, freezes out at both the top and midway through the core, there and the outer planet satellite Ganymede. There is an enormous lit- are two sources of compositional buoyancy, the heavy iron sinking erature on the modeling of Earth’s dynamo and with our primary from the top of the core and from below mid-depth and the light S- focus being the magnetic fields of the other planets, we do not at- rich fluid rising above mid-depth. Vilim et al. (2010) modeled G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 101 dynamo action driven by this unusual source of compositional deep core. These alternative scenarios for the thermochemical buoyancy, a double snow layer, and found separate dynamo re- state of the core arise because the eutectic melting temperatures gions above and below the mid-depth freezing layer that produce and eutectic sulfur contents in the binary Fe–FeS system decrease magnetic fields of opposite sign with a consequent weakened sur- with increasing pressure in the range of core pressures in Gany- face field. Conditions in Ganymede’s core, as discussed below, mede (14 GPa). Iron snow or flotation of solid FeS occur upon might also lead to a distinctly non-earthlike form of Fe precipita- cooling of Ganymede’s core depending on whether the core com- tion (Hauck et al., 2006). position is more or less iron-rich than eutectic. Hauck et al. The generation of Mercury’s weak magnetic field is not yet (2006) did scaling calculations and concluded that for iron snow understood, although there is no lack of ideas to explain it. The cooling scenarios, typical Rossby and magnetic Reynolds numbers MESSENGER spacecraft is now in orbit around Mercury and data are consistent with dynamo action occurring in the core and esti- from three flybys are published to date (Uno et al., 2009). With mates of the generated magnetic field agree with the strength of careful analysis of these observations, particularly with respect to Ganymede’s field. separating the externally driven effects from the internally pro- Though Ganymede is larger than Mercury, the silicate-metal duced magnetic field, we may be able to distinguish among the part of the satellite is only about the size of Io and the Moon, above explanations. For example, Fig. 4 compares the observed and Ganymede’s core is at most about half that large (Schubert surface magnetic spectra against the spectra predicted by dynamo et al., 2004). It is perhaps surprising that such a small core could models. This comparison shows that the model of Christensen support dynamo action over a geologically long time. What sup- (2006) is in best agreement, supporting his conclusion that an elec- plies the energy to drive the dynamo? Why is the core not almost trically conducting stable layer at the top of the dynamo region completely solidified? Bland et al. (2008) and Kimura et al. (2009) may be responsible for the weak, dipole-dominated field at the pla- have investigated thermal history models for Ganymede with the net surface. aim of identifying scenarios that allow for present day dynamo activity. Bland et al. (2008) conclude that a present day dynamo could exist only if the sulfur mass fraction in Ganymede’s core is 4.1.2. Ganymede very low (<3%) or very high (>21%), and only if the silicate mantle Dynamo action in Ganymede’s core was first studied by Sarson could cool rapidly (e.g., a mantle with a wet olivine rheology). Kim- et al. (1997) who investigated how Jupiter’s magnetic field might ura et al. (2009) conclude that a Ganymede dynamo could cur- influence magnetic field generation in the satellite. The Jovian rently be operative for a sufficiently large sulfur-rich metallic magnetic field at Ganymede’s orbit is much weaker than the satel- core in which dynamo activity started late in the satellite’s evolu- lite’s intrinsic magnetic field and, therefore, is not expected to play tion. In contrast, Hauck et al. (2006) inferred that a dynamo could a significant role in the operation of Ganymede’s dynamo. The dy- exist at present for a broad range of core sulfur concentrations. namo model of Sarson et al. (1997) confirmed this, but left open Bland et al. (2008) also studied whether tidal heating during pas- the possibility that in the initial stages of dynamo activity, the Jo- sage through an eccentricity pumping Laplace-like resonance in vian magnetic field could provide a seed field that would explain Ganymede’s past could enable present day dynamo action in the the anti-alignment of Jupiter’s and Ganymede’s dipole moments. metallic core. They found that such an event would not facilitate Similar to Mercury, Ganymede’s dynamo acts in a relatively low the operation of a present day dynamo. Of course, Ganymede does pressure, low temperature environment compared with the Earth. have a dynamo at present which may indicate that its core formed Solidification of Ganymede’s core and the associated compositional relatively late in the satellite’s thermal evolution. driving of its dynamo might then be completely different from the To date, there are no published quantitative models of Gany- freezing out of Earth’s core and the compositional buoyancy driv- mede’s dynamo. The studies of Hauck et al. (2006), Bland et al. ing the geodynamo. According to Hauck et al. (2006), Ganymede’s (2008), and Kimura et al. (2009) use scaling arguments and quali- dynamo can be driven in part by compositional buoyancy released tative criteria to determine whether dynamo action could occur in through the sinking of Fe snow formed below the core-mantle their Ganymede models. Progress in understanding how Gany- boundary or by the upward flotation of solid FeS formed in the mede generates its magnetic field requires at least the numerical modeling of dynamo activity in Ganymede models derived consis- tently from thermal-orbital evolution models of the satellite.

100 4.1.3. Mars Mars crustal magnetization preserves a record of its ancient dy- namo. The record has been exploited to infer the nature of the Mar- tian dynamo, but only a few attempts have been made to model 10-1 dynamo action in the special circumstances of ancient Mars. As discussed in Section 2.2, one of the main questions about the Mar- tian dynamo is the period in which it was active. When did dyna- -2 10 mo action begin and when did it end? Why did the Martian dynamo stop working? There is an observed absence of magnetic Uno et al., 2009 anomalies over the major impact basins, perhaps due to impact Power / Dipole power Power 10-3 Heimpel et al., 2005a demagnetization (Hood et al., 2003; Mohit and Arkani-Hamed, Takahashi and Matsushima, 2006 2004). This has led many investigators to the conclusion that the Christensen, 2006 Vilim et al., 2010 Martian dynamo had an early onset and ceased before the forma- 10-4 tion of the major impact basins (Acuña et al., 1999; Frey, 2003; 1 2 3 4 5 6 Arkani-Hamed, 2004). The only constraint on the timing of the dy- Spherical Harmonic Degree, l namo comes from the magnetized Allan Hills meteorite ALH84001. This meteorite is the oldest known Martian rock, dated between Fig. 4. Normalized surface magnetic power spectra for Mercury from Uno et al. 3.9 and 4.1 Ga (Weiss et al., 2002). The onset of dynamo activity (2009) and select dynamo models: the v = 0.15 case of Heimpel et al. (2005b), the Ra =2 105 case of Takahashi and Matsushima (2006), case II of Christensen in Mars, therefore, occurred around 4 Ga or earlier. Bibring et al. (2006), and model 1 of Vilim et al. (2010). (2006) presented a global mineralogical map of Mars and 102 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 concluded that a rapid drop in atmospheric pressure near the end 2006; Ke and Solomatov, 2006). The endogenic and exogenic of the Noachian (first Martian geologic period) could have been mechanisms can both not only remove magnetized crust from triggered by a rapid decrease in dynamo field strength. It has also the northern hemisphere, but can also create conditions deep in been suggested that the major impacts could have altered the the Martian mantle that influence convection and dynamo genera- internal temperature and convective character of Mars and may tion in the Martian core (Watters et al., 2009). Both mechanisms, have led to the early demise of the dynamo (Frey et al., 2007; Lillis upwelling in the northern hemisphere and energy deposition by et al., 2008b). impact, imply a hotter deep mantle in the northern hemisphere Early dynamo cessation has been challenged by Schubert et al. of Mars and, therefore, a reduced heat flux from the core into the (2000) (see Section 2.2). More recently it has been recognized that northern hemisphere mantle. Stanley et al. (2008) investigate the some volcanic edifices which postdate the formation of the major effect of this core-mantle boundary heat flux dichotomy on mag- impact basins on Mars are magnetized. Hood et al. (2010) demon- netic field generation in the core by imposing a large outward strated that the Apollinaris Mons construct, dated at 3.7 Ga core-mantle boundary heat flux in the southern hemisphere of a (Werner, 2009), is the most likely source of the magnetic field dynamo model and an inward core-mantle boundary heat flux in anomaly centered on the volcano. Milbury and Schubert (2010) the northern hemisphere. The heat flux pattern imposed on the did a statistical analysis of the global magnetic field and found very outer shell boundary of the model and the resulting magnetic field small differences between Noachian and Hesperian geologic units, upward continued to the planet surface are shown in Fig. 5. Their suggesting that the dynamo persisted into the Hesperian. Further, dynamo model generates a magnetic field concentrated in and volcanic edifices at both Hadriaca Patera and Tyrrhenus Mons were strongest in the southern hemisphere. This largely occurs because built in the period 3.7–4.0 Ga and show signs of magnetization the core-mantle heat flux boundary condition made the northern (Greeley and Crown, 1990; Williams et al., 2008; Werner, 2009; hemisphere of the core subadiabatic and focused core convection Robbins et al., 2011). Lillis et al. (2006) used the electron reflectom- in the southern hemisphere. Thus, it is possible that the concentra- etry data from Mars Global Surveyor and found a magnetic signa- tion of Martian magnetization to its southern hemisphere could ture associated with Hadriaca Patera, an edifice located on the rim have been the result of a peculiar southern hemisphere Martian of the Hellas impact basin that postdates basin formation. Milbury core dynamo or the destruction of magnetized northern hemi- et al. (2011) modeled a large number of magnetic anomalies collo- sphere crust by post-magnetization processes. cated with gravity anomalies within the region surrounding Tyrrh- While the hemispheric dynamo of Stanley et al. (2008) shows enus Mons. They found a correlation of paleopole locations with that it is possible to confine the Martian magnetic field to the units of different geologic age showing a progression of pole posi- southern hemisphere of the planet, it does not explain the further tions from initial magnetization to pole reversal through polar concentration of the observed magnetization to the Terra Cimme- wander. ria-Terra Sirenum region of the southern highlands. It is not appar- The numerical dynamo simulations of Kuang et al. (2008) sug- ent that impact demagnetization of the southern hemisphere crust gest possible behaviors of the Martian dynamo toward the end of can account for this localization of magnetization either. Perhaps its activity. Near its demise, the Martian dynamo could have re- an even more restrictive condition on heat flow at the core-mantle versed frequently, and it could have operated as a subcritical dyna- boundary or some azimuthal dependence of the condition could mo since the energy to sustain a dynamo is significantly less than explain the observation. what is required to excite it. Subcritical dynamo activity would lengthen the life of the magnetic field. In addition, it would enable 4.2. Giant planets an abrupt termination of dynamo activity. Once turned off, it would be difficult to reactivate the dynamo without a substantial The magnetic fields of the giant planets are likely coupled to increase of the buoyancy force in the Martian core. Glatzmaier their surface zonal flows and convective heat transfer patterns. et al. (1999) and Stanley and Glatzmaier (2010) suggest another These planets have strong winds that are organized into zonal jets. possible termination scenario for the Martian dynamo involving The wind velocities are measured with respect to the planet’s rota- a critical role of reversals. Mars’ dynamo could have ended by tion rate, assumed to correspond with the rotation rate of the deep going through a series of dipole reversals and not completely interior and magnetic field (e.g., Russell et al., 2001; Giampieri and recovering after each one. Evidence for dipole reversal in the Tyr- Dougherty, 2004). Jupiter and Saturn have strong prograde (east- rhenus Mons region of Mars (Milbury et al., 2011) might indicate ward) equatorial jets and multiple smaller-scale jets at higher lat- the termination of the Martian dynamo in the Hesperian. itudes (e.g., Porco et al., 2003; Sanchez-Lavega et al., 2000). The The spatial distribution of Mars crustal magnetic field confronts high latitude jets of Jupiter alternate in direction, while those of us with another major puzzle (Fig. 3). Is the southern hemisphere Saturn are exclusively prograde assuming the Voyager-derived confinement of the magnetic anomalies a consequence of post dy- rotation rate (Desch and Kaiser, 1981) and alternate in direction namo modification of the magnetized crust by endogenic processes assuming the planetary rotation rate estimates of Anderson and or impacts, or did the Martian dynamo produce a field that reflects Schubert (2007) and Read et al. (2009). In contrast, the zonal winds the current magnetization distribution? Since the asymmetry of of Uranus and Neptune are retrograde (westward) at low latitudes the crustal magnetic field coincides with the crustal dichotomy, and prograde at high latitudes (e.g., Sromovsky et al., 2001; Ham- it is possible that whatever process created the dichotomy also de- mel et al., 2005). The wind speeds, however, can vary up to 50% due stroyed the crustal magnetization in the northern hemisphere. to uncertainties in the rotation rates (Helled et al., 2010). It is not Plausible causes of the dichotomy include endogenic removal of presently known to what depths the winds of the giant planets ex- crust in the northern hemisphere or exogenic disruption of the tend, although the Galileo probe suggested that the Jovian winds northern hemisphere crust by a giant impact (Wilhelms and Squy- penetrate to at least the 20 bar pressure level (Atkinson et al., res, 1984; Frey and Schultz, 1988; Andrews-Hanna et al., 2008; 1998). The latitudinal internal heat flux profiles peak in the polar Marinova et al., 2008; Nimmo et al., 2008). An endogenic process regions with a minimum near the equator on the gas giants (Pirra- that could have removed crust from the northern hemisphere is glia, 1984) and peak in the equatorial and polar regions with min- degree-1 mantle convection or overturn (Schubert and Lingenfel- ima at mid-latitudes on Neptune (Pearl and Conrath, 1991). The ter, 1973; Lingenfelter and Schubert, 1973; Weinstein, 1995; heat flux pattern of Uranus is not well constrained (Pearl et al., Harder, 1998; Breuer et al., 1998; Zhong and Zuber, 2001; Elkins- 1990). Modelers attempt to self-consistently generate these obser- Tanton et al., 2003; Elkins-Tanton et al., 2005; Roberts and Zhong, vations, to understand the underlying mechanisms, and to explain G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 103

(a) Imposed Heat Flux (b) Radial Magnetic Field

-15 µT0 µT 15 µT

Fig. 5. (a) Imposed outer boundary heat flux pattern in the Stanley et al. (2008) model. Red (blue) indicates inward (outward) directed heat flux. (b) Simulated radial magnetic field upward continued to the surface of Mars. Red (blue) indicates outward (inward) directed field. the fundamental differences in behavior between the gas and ice giants. Deep convection modeling efforts are typically channelled into non-magnetic rotating convection models and into planetary dy- namo models. Non-magnetic models are also important to con- sider because the electrically insulating regions are likely coupled to the dynamo region and the mechanisms responsible for driving zonal flows and controlling the heat transfer are more easily understood without the added complexity of the Lorentz force. The results of recent gas giant-like and ice giant-like models are summarized below.

4.2.1. Gas giants Busse (1976) first proposed that deep convection in the molec- ular envelopes may be responsible for the zonal winds of Jupiter. He envisioned that the alternating jets are the surface expressions of differentially-rotating cylinders aligned with the rotation axis. In this hypothesis, the fluid motions are quasigeostrophic and do not strongly vary along the direction of the rotation axis, in agreement Fig. 6. Instantaneous azimuthal winds on the inner and outer boundaries of the Heimpel et al. (2005a) model. Red (blue) indicates prograde (retrograde) flow. with the Taylor–Proudman theorem (Taylor, 1917; Proudman, 1916). Gas giant-like zonal flows with multiple jets were first ob- tained in rapidly-rotating Boussinesq convection models by Christensen (2001, 2002). More recently, Heimpel et al. (2005a), heat transfer (Aurnou et al., 2008). In the equatorial plane, the con- Heimpel and Aurnou (2007), and Aurnou et al. (2008) have shown vective flows are less efficient at transferring heat outward since that turbulent, rapidly-rotating convection in thin spherical shells they must compete with the strong azimuthal forcing. The convec- produces zonal flow and heat flux patterns that are consistent with tive flows in the polar regions are most efficient because the zonal those observed on Jupiter. The simulated zonal winds of the flows do not inhibit radial motions. Heimpel et al. (2005a) model are shown in Fig. 6. The equatorial Density stratification must also be considered to accurately jets in these models result from Reynolds stresses due to correla- simulate the dynamics of the gas giants. Flow speeds will tend to tions between the fluctuating cylindrical and azimuthal velocity decrease with increasing density and the convection will not tend components (e.g., Zhang, 1992). Outside the tangent cylinder, the to manifest as axially-invariant columns (Glatzmaier et al., 2009). imaginary right cylinder tangent to the inner shell at the equator, Thus, the vortex-stretching mechanism of driving the zonal flow the height of the convection columns decreases with increasing is no longer appropriate. Rather, compressional torques produce cylindrical radius. As a result, fluid moving away from the rotation negative vorticity in the expanding, uprising plumes and positive axis must speed up and fluid moving toward the rotation axis must vorticity in the contracting, sinking plumes. Since fluid moving slow down in order to conserve potential vorticity. This leads to radially outward will tend to become faster and fluid moving radi- spiraling convection cells in which cylindrically outward flows cor- ally inward will tend to become slower due to density effects, the relate with prograde azimuthal flows, and vice versa. These Rey- plumes tilt and correlations again develop between the fluctuating nolds stresses transport positive angular momentum outward cylindrical and azimuthal motions (Glatzmaier and Gilman, 1981; and negative angular momentum inward, driving zonal flows that Evonuk, 2008; Glatzmaier et al., 2009). Reynolds stresses drive are prograde near the outer boundary and retrograde near the in- the zonal flow by transporting positive angular momentum out- ner boundary. Inside the tangent cylinder, quasigeostrophic turbu- ward and negative angular momentum inward, resulting in equa- lence leads to multiple jets whose widths are determined by the torial zonal flows that have the same organization as models Rhines scale (Rhines, 1975; Heimpel and Aurnou, 2007). This style without density stratification. Jones and Kuzanyan (2009) contrast of zonal flow inhibits equatorial heat transfer and promotes polar anelastic and Boussinesq convection in rotating spherical shell 104 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 models and find that the zonal flow pattern is not strongly affected this flow can suppress rapidly-varying, higher-order magnetic field by the density stratification, but the convective flow speeds are components through a skin effect mechanism. suppressed at depth in the anelastic models. Electrical conductivity stratification makes the models even 4.2.2. Ice giants more realistic since it allows coupling between the lower conduc- Convection in the molecular envelopes of Uranus and Neptune tivity molecular envelope and the metallic mantle. Stanley and has also been proposed to drive their observed zonal flows and Glatzmaier (2010) have carried out dynamo simulations with both to be dynamically coupled to their dynamo regions (Suomi et al., density stratification and radially-varying electrical conductivity. 1991; Aurnou et al., 2007). Further, since the ice giants are esti- Rapidly-rotating, turbulent convection in a thin shell geometry mated to have the most turbulent fluid flows in the solar system produces zonal flows and a radial magnetic field that agree to first (compare Re in Table 4), inertial effects may be important in these order with the observations of the gas giants (Fig. 7). The zonal planets. The non-magnetic Boussinesq models of Aurnou et al. flows are maintained by the compressional torque mechanism. (2007), non-magnetic anelastic models of Brun and Palacios The magnetic field is dipole-dominated, but has significant power (2009), and Boussinesq dynamo models of Soderlund and Aurnou in the higher order modes since the axisymmetric radial field (2010) show that strongly-forced convection in rotating spherical exhibits a banded structure with latitude. Additional measure- shells, regardless of shell thickness, can drive zonal flows that qual- ments are needed to improve the spatial resolution of the planets’ itatively agree with the observations of Uranus and Neptune. The magnetic fields in order to compare against the model predictions. simulated zonal winds and radial magnetic field of the Soderlund For Saturn, modelers must also attempt to explain the strong and Aurnou (2010) model are shown in Fig. 8. In these models, axisymmetry of the magnetic field. Stevenson (1980, 1982) pro- the Taylor–Proudman constraint is not valid due to strong inertial posed that differential rotation in a stably-stratified layer created effects, leading to three-dimensional convection. This convective by helium rain-out near the molecular-metallic hydrogen transi- turbulence strongly mixes the fluid and homogenizes the temper- tion could diminish the non-axisymmetric components of the ature and absolute angular momentum (Scorer, 1966; Gough and magnetic field. Recent dynamo models support this hypothesis. Lynden-Bell, 1968; Bretherton and Turner, 1968). Aurnou et al. The kinematic dynamo models of Schubert et al. (2004) show that (2007) find that retrograde equatorial jets and strong prograde jets the morphology of the dynamo is determined by the characteris- at high latitudes develop in order to conserve angular momentum. tics of the zonal thermal winds in the overlying stable layer. Soderlund and Aurnou (2010) show that this convective regime Christensen and Wicht (2008) and Stanley (2010), respectively, also generates heat flux patterns and magnetic field morphologies show that thick and thin stably-stratified layers overlying the dy- that are consistent with observations of the ice giants. The impor- namo region develop differential rotation through thermal winds; tance of inertia for the transition from dipolar to multipolar

(a) Zonal Flows (b) Radial Magnetic Field 90 90

60 60

30 30

0 0 Latitude Latitude -30 -30

-60 -60

-90 -90 0 200 400 -3000 0 3000 Velocity, m/s Intensity, µT

Fig. 7. (a) Zonal flow and (b) radial magnetic field snapshots on the outer boundary of the Stanley and Glatzmaier (2010) model. In (a), red (blue) indicates prograde (retrograde) flow. In (b), red (blue) indicates outward (inward) directed fields. The latitudinal profiles are averaged over all longitudes.

(a) Zonal Flows (b) Radial Magnetic Field 90 90

60 60

30 30

0 0 Latitude Latitude -30 -30

-60 -60

-90 -90 -500 0 500 -103 0103 Velocity, m/s Intensity, µT

Fig. 8. Snapshots of (a) zonal flow and (b) radial magnetic field up to harmonic three on the outer boundary of the Soderlund and Aurnou (2010) model. In (a), red (blue) indicates prograde (retrograde) flow. In (b), purple (green) indicates outward (inward) directed fields. The latitudinal profiles are averaged over all longitudes. G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 105 magnetic field generation has also been shown in the geodynamo mo models that can account for these extreme property variations models of Grote et al. (1999, 2000), Kutzner and Christensen need to be developed. To understand dynamos that operated in the (2000), Kutzner and Christensen (2002), Simitev and Busse past, and even some presently active dynamos, connections with (2005), Aubert (2005), Sreenivasan and Jones (2006); and thermal evolution models of the planets and satellites are required. Soderlund et al. (in review). The internal structures of the outer planets are especially uncertain Other groups have also produced ice giant-like magnetic fields. and need to be better known if the dynamos in these planets are to Gomez-Perez and Heimpel (2007) find that dynamo models tend to be modeled and understood. Even the rotation rates of the giant become multipolar as the electrical conductivity is decreased and planet interiors need to be determined with more confidence be- as strong zonal flows develop. In addition, Stanley and Bloxham fore we can really know the significance of the observed zonal cir- (2004), Stanley and Bloxham (2006) find multipolar dynamos culations. This is especially true of Saturn and possibly the ice when moderately strong convection occurs in a thin spherical shell giants. Gaps in our knowledge of planetary properties place serious overlying a stably-stratified fluid. They suggest that solid, electri- limitations on our ability to understand the variety of planetary cally conducting inner cores act to stabilize the magnetic field dynamos found in our solar system. Despite these obstacles, we and promote dipolar dynamos, while stably-stratified fluid cores are optimistic about future discoveries and modeling efforts that do not have this anchoring effect. While this model may be appro- will progress the study of planetary magnetic fields. priate for Uranus, it is inconsistent with recent internal structure models of Neptune (Fortney et al., 2011). Further, the zonal flows Acknowledgments in each group’s models have strong prograde equatorial jets, oppo- site to those observed. G.S. was supported by NASA grant NNX09AB57G from the Plan- etary Atmospheres Program and by NSF Grant AST-0909206 from the Planetary Astronomy Program. K.M.S. was supported by NASA 5. Conclusions and outlook Grant NNX09AB61G from the Planetary Atmospheres program.

Dynamo models yield a diverse spectrum of dynamical behav- iors and are able to reproduce aspects of the planets’ magnetic References fields, fluid flows, and thermal emissions. Thus, through these Abramson, E.H., 2005. Viscosity of water measured to pressures of 6 GPa and models, we learn about the dynamics and underlying mechanisms temperatures of 300 °C. Phys. Rev. E 76, 051203. that may be occurring in planetary interiors. Further modeling pro- Abramson, E.H., Brown, J.M., Slutsky, L.J., 2001. The thermal diffusivity of water at gress and additional constraints on the magnetic fields and the high pressures and temperatures. J. Chem. Phys. 115, 10461–10463. Acuña, M.H. et al., 1998. Magnetic field and plasma observations at Mars: initial internal structures and dynamics derived from upcoming space- results of the Mars Global Surveyor Mission. Science 279, 1676–1680. craft missions will likely allow us to develop sophisticated models Acuña, M.H. et al., 1999. Global distribution of crustal magnetism discovered by the that capture the relevant physics and to make predictions about Mars Global Surveyor MAG/ER experiment. Science 284, 790–793. Acuña, M.H. et al., 2001. The magnetic field of Mars: summary of results from the the evolution of planetary magnetic fields. For example, technolog- aerobraking and mapping orbits. J. Geophys. Res. 106, 23403–23417. ical advances will enable models to reach lower Ekman and mag- Aharonson, O., Auber, M., Solomon, S., 2005. Crustal remnance in an internally netic Prandlt numbers, to develop asymptotic scaling laws, and magnetized non-uniform shell: a possible source for Mercury’s magnetic field? Earth Planet. Sci. Lett. 218, 261–268. to incorporate significant stratification in density and electrical Anderson, B.J., Acuña, M.H., Korth, H., Purucker, M.E., Johnson, C.L., Slavin, J.A., conductivity. By understanding the dynamos of our solar system, Solomon, S.C., McNutt, R.L., 2008. The structure of Mercury’s magnetic field we may also use these models and associated scaling laws to pre- from MESSENGER’s first flyby. Science 321, 82–85. Anderson, B.J., Acuña, M.H., Korth, H., Slavin, J.A., Uno, H., Johnson, C.L., Purucker, dict the magnetic field strengths and morphologies of exoplanetary M.E., Solomon, S.C., Raines, J.M., Zurbuchen, T.H., Gloeckler, G., McNutt, R.L., dynamos. 2010. The magnetic field of Mercury. Space Sci. Rev. 152, 307–339. As for observations, the MESSENGER and Bepi-Columbo mis- Anderson, J.D., Schubert, G., 2007. Saturn’s gravitational field, internal rotation, and sions will help constrain the structure of Mercury’s dynamo region interior structure. Science 371, 1384–1387. Andrews-Hanna, J.C., Zuber, M.T., Banerdt, W.B., 2008. The Borealis basin and the and the behavior of the body’s magnetic field. In addition, a possi- origin of the martian crustal dichotomy. Nature 453, 1212–1215. ble future mission to the Jupiter system would help to characterize Arkani-Hamed, J., 2004. Timing of the Martian core dynamo. J. Geophys. Res. 109, Ganymede’s dynamo and internal structure. The upcoming Juno E03006. Atkinson, D.H., Pollack, J.B., Seiff, A., 1998. The Galileo probe doppler wind mission will critically improve our understanding of Jupiter by experiment: measurement of the deep zonal winds on Jupiter. J. Geophys. resolving the magnetic field up to spherical harmonic degree 14, Res. 103, 22911–22928. detecting short timescale secular variation, and constraining the Aubert, J., 2005. Steady zonal flows in spherical shell dynamos. J. Fluid Mech. 542, 53–67. depth of the zonal winds (Connerney, private communication). Aurnou, J.M., 2007. Planetary core dynamics and convective heat transfer scaling. The Cassini spacecraft may closely orbit, and eventually impact, Geophys. Astro. Fluid Dyn. 101, 327–345. Saturn in the possible extended-extended phase of the mission, Aurnou, J.M., Heimpel, M.H., Wicht, J., 2007. The effects of vigorous mixing in a convective model of zonal flow on the ice giants. Icarus 190, 110–126. allowing the magnetic and gravitational fields to be mapped in fur- Aurnou, J.M., Heimpel, M., Allen, L., King, E.M., Wicht, J., 2008. Convective heat ther detail (Cassini Mission Plan, Revision N). Missions to Uranus transfer and the pattern of thermal emission on the gas giants. Geophys. J. Int. and/or Neptune are necessary to better constrain the internal 173, 793–801. Balogh, A., Dougherty, M.K., Forsyth, R.J., Southwood, D.J., Tsurutani, B.T., Murphy, structures and dynamics of the ice giants and to map the magnetic N., Burton, M.E., 1992. Magnetic field observations in the vicinity of Jupiter fields globally and with higher resolution. during the Ulysses flyby. Science 257, 1515–1518. Improved understanding of planetary dynamos will come not Bibring, J.P. et al., 2006. Global mineralogical and aqueous Mars history derived only from additional observations of planetary magnetic fields by from OMEGA/Mars express data. Science 312, 400–404. Bland, M.T., Showman, A.P., Tobie, G., 2008. The production of Ganymede’s future spacecraft missions, but also from new modeling efforts to magnetic field. Icarus 198, 384–399. address the diversity of physical settings in which planetary dyna- Bloxham, J., Gubbins, D., Jackson, A., 1989. Geomagnetic secular variation. Philos. mos operate. The release of compositional buoyancy in the cores of Trans. R. Soc. Lond. Ser. A 329, 415–502. Bloxham, J., Jackson, A., 1991. Fluid flow near the surface of Earth’s outer core. Rev. Ganymede and Mercury can be fundamentally different from the Geophys. 29, 97–120. source of compositional buoyancy in the Earth’s core, and numer- Binder, A.B., 1998. Lunar prospector: overview. Science 281, 1475–1476. ical dynamo models that focus on these different buoyancy distri- Boehler, R., 1996. Melting temperature of the Earth’s mantle and core: Earth’s thermal structure. Annu. Rev. Earth Planet. Sci. 24, 15–40. butions are needed. Strong density and electrical conductivity Bogue, S.W., Merrill, R.T., 1992. The character of the field during geomagnetic stratifications occur in the interiors of the giant planets and dyna- reversals. Ann. Rev. Earth Planet. Sci. 20, 181–219. 106 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108

Bretherton, F.P., Turner, J.S., 1968. On the mixing of angular momentum in a stirred bombardment in the inner solar system. Seventh Intl. Conf. on Mars, Calif. Inst. rotating fluid. J. Fluid Mech. 32, 449–464. of Technol., Pasadena, CA, 9–13 July. Breuer, D., Yuen, D.A., Spohn, T., Zhang, S., 1998. Three dimensional models of Martian Garrick-Bethell, I., Weiss, B.P., Shuster, D.L., Buz, J., 2009. Early lunar magnetism. mantle convection with phase transitions. Geophys. Res. Lett. 25, 229–232. Science 323, 356–359. Bridge, H.S., Lazarus, A.J., Snyder, C.W., Smith, E.J., Davis Jr., L., Coleman Jr., P.J., Jones, Giampieri, G., Balogh, A., 2002. Mercury’s thermoelectric dynamo model revisited. D.E., 1969. The Venus Atmosphere, 533. Planet. Space Sci. 50, 757–762. Brun, A.S., Palacios, A., 2009. Numerical simulations of a rotating red giant star. I. Giampieri, G., Dougherty, M., 2004. Rotation rate of Saturn’s interior from magnetic Three-dimensional models of turbulent convection and associated mean flows. field observations. Geophys. Res. Lett. 31, L16701. Astrophys. J. 702, 1078–1097. Glassmeier, K.H., Auster, H.U., Motschmann, U., 2007. A feedback dynamo Bullard, E.C., 1949. The magnetic field within the Earth. Proc. R. Soc. Lond. A 197, generating Mercury’s magnetic field. Geophys. Res. Lett. 34, L22201. 433–453. Glatzmaier, G.A., 2002. Geodynamo simulations – How realistic are they? Annu. Burke, B.F., Franklin, K.L., 1955. Observations of a variable radio source associated Rev. Earth Planet. Sci. 30, 237–257. with the planet Jupiter. J. Geophys. Res. 60, 213–217. Glatzmaier, G.A., Gilman, P.A., 1981. Compressible convection in a rotating spherical Burton, M.E., Dougherty, M.K., Russell, C.T., 2009. Model of Saturn’s internal shell. III. Analytic model for compressible vorticity waves. Astrophys. J. Suppl. planetary magnetic field based on Cassini observations. Planet. Space Sci. 57, 45, 381–388. 1706–1713. Glatzmaier, G.A., Coe, R.S., Hongre, L., Roberts, P.H., 1999. The role of the Earth’s mantle Busse, F.H., 1976. A simple model of convection in the Jovian atmosphere. Icarus 20, in controlling the frequency of geomagnetic reversals. Nature 401, 885–890. 255–260. Glatzmaier, G.A., Evonuk, M., Rogers, T.M., 2009. Differential rotation in giant Cande, S.C., Kent, D.V., 1995. A new geomagnetic polarity time scale for the Late planets maintained by density-stratified turbulent convection. Geophys. Cretaceous and Cenozoic. J. Geophys. Res. 97, 13917–13951. Astrophys. Fluid Dyn. 103, 31–51. Carr, T.D., Schauble, J.J., Schauble, C.C., 1981. Pre-encounter distributions of Saturn’s Gomez-Perez, N., Heimpel, M.H., 2007. Numerical models of zonal flow dynamos: low frequency radio emission. Nature 292, 745–747. an application to the ice giants. Geophys. Astrophys. Fluid Dyn. 101, 371–388. Carr, T.D., Desch, M.D., Alexander, J.K., 1983. Phenomenology of magnetospheric Gomez-Perez, N., Solomon, S., 2010. Mercury’s weak magnetic field: a result of radio emissions. In: Dessler, A.J. (Ed.), Physics of the Jovian Magnetosphere. magnetospheric feedback? Geophys. Res. Lett. 37, L20204. Cambridge Univ. Press, Cambridge, UK. Gomez-Perez, N., Wicht, J., 2010. Behavior of planetary dynamos under the Cavazzoni, C., Chiarotti, G.L., Scandolo, S., Tosatti, E., Bernasconi, M., Parrinello, M., influence of external magnetic fields: application to Mercury and Ganymede. 1999. Superionic and metallic states of water and ammonia at giant planet Icarus 1, 53–62. conditions. Science 283, 44–46. Gough, D.O., Lynden-Bell, D., 1968. Vorticity expulsion by turbulence: astrophysical Chen, B., Li, J., Hauck II, S.A., 2008. Non-ideal liquidus curve in the Fe-S system and implications of an AlkaSeltzer experiment. J. Fluid Mech. 32, 437–447. Mercury’s snowing core. Geophys. Res. Lett. 35, L07201. Greeley, R., Crown, D.A., 1990. Volcanic geology of tyrrhena patera. Mars. J. Christensen, U.R., 2001. Zonal flow driven by deep convection on the major planets. Geophys. Res. 95 (B5), 7133–7149. Geophys. Res. Lett. 28, 2553–2556. Grote, E., Busse, F.H., Tilgner, A., 1999. Convection-driven quadrupolar dynamos in Christensen, U.R., 2002. Zonal flow driven by strongly supercritical convection in rotating spherical shells. Phys. Rev. E 60, 5025–5028. rotating spherical shells. J. Fluid Mech. 470, 115–133. Grote, E., Busse, F.H., Tilgner, A., 2000. Regular and chaotic spherical shell dynamos. Christensen, U.R., 2006. A deep dynamo generating Mercury’s magnetic field. Phys. Earth Planet. Int. 117, 259–272. Nature 444, 1056–1058. Guillot, T., 1999a. Interiors of giant planets inside and outside the solar system. Christensen, U.R., 2010. Dynamo scaling laws and applications to the planets. Space Science 286, 72–77. Sci. Rev. 152, 565–590. Guillot, T., 1999b. A comparison of the interiors of Jupiter and Saturn. Planet. Space Christensen, U.R., Aubert, J., 2006. Scaling properties of convection-driven dynamos Sci. 47, 1183–1200. in rotating spherical shells and application to planetary magnetic fields. Guillot, T., 2005. The interiors of giant planets: models and outstanding questions. Geophys. J. Int. 166, 97–114. Annu. Rev. Earth Planet. Sci. 33, 493–530. Christensen, U.R., Wicht, J., 2008. Models of magnetic field generation in partly Gubbins, D., 2001. The Rayleigh number for convection in the Earth’s core. Phys. stable planetary cores: applications to Mercury and Saturn. Icarus 196, 16–34. Earth Planet. Int. 128, 3–12. Clune, T.C., Elliot, J.R., Miesch, M., Toomre, J., Glatzmaier, G.A., 1999. Computational Gurnett, D.A. et al., 2007. The variable rotation period of the inner region of Saturn’s aspects of a code to study rotating turbulent convection in spherical shells. plasma disk. Science 316, 442–445. Parallel Computing 25, 361–380. Hale, C., Dunlop, D., 1984. Evidence for an early archean geomagnetic field: a Coleman Jr., P.J., Schubert, G., Russell, C.T., Sharp, L.R., 1972. Satellite measurements paleomagnetic study of the Komati Formation, Barberton Greenstone Belt. of the Moon’s magnetic field: a preliminary report. Moon 4, 419–429. South Africa Geophys. Res. Lett. 11, 97–100. Connerney, J.E.P., 1993. Magnetic fields of the outer planets. J. Geophys. Res. 98, Hammel, H.B., de Pater, I., Gibbard, S., Lockwood, G.W., Rages, K., 2005. Uranus in 18659–18679. 2003: zonal winds, banded structure, and discrete features. Icarus 175, 534– Connerney, J.E.P. et al., 1999. Magnetic lineations in the ancient crust of Mars. 545. Science 284, 794–798. Harder, H., 1998. Phase transitions and the three-dimensional planform of thermal Connerney, J.E.P. et al., 2001. The global magnetic field of Mars and implications for convection in the Martian mantle. J. Geophys. Res. 103, 16775–16797. crustal evolution. Geophys. Res. Lett. 28, 4015–4018. Hauck II, S.A., Aurnou, J.M., Dombard, A.J., 2006. Sulfur’s impact on core evolution Desch, M.D., Kaiser, M.L., 1981. Voyager measurement of the rotation period of and magnetic field generation on Ganymede. J. Geophys. Res. 111, E09008. Saturn’s magnetic field. Geophys. Res. Lett. 8, 253–256. Heimpel, M.H., Aurnou, J.M., Wicht, J., 2005a. Simulation of equatorial and high- de Pater, I., Lissauer, J.J., 2001. Planetary science. Cambridge Univ. Press, Cambridge, latitude jets on Jupiter in a deep convection model. Nature 438, 193–196. UK. Heimpel, M.H., Aurnou, J.M., Al-Shamali, F., Gomez-Perez, N., 2005b. A numerical Dougherty, M.K. et al., 2005. Cassini magnetometer observations during Saturn study of dynamo action as a function of spherical shell geometry. Earth Planet. orbit insertion. Science 307, 1266–1270. Sci. Lett. 236, 542–557. Dwyer, C.A., Stevenson, D.J., Nimmo, F., 2010. An early nutation-driven lunar Heimpel, M.H., Aurnou, J.M., 2007. Turbulent convection in rapidly rotating dynamo. American Geophysical Union, Fall Meeting, Abstract #GP31A-02. spherical shells: a model for equatorial and high-latitude jets on Jupiter and Dziewonski, A.M., Anderson, D.L., 1981. Preliminary reference earth model. Phys. Saturn. Icarus 187, 540–557. Earth Planet. Int. 25, 297–356. Helled, R., Anderson, J., Schubert, G., 2010. Uranus and Neptune: shape and rotation. Elkins-Tanton, L.T., Parmentier, E.M., Hess, P.C., 2003. A model for martian magma Icarus 210, 446–454. ocean crystallization and overturn. Meteoritics and Planetary 38, 1753–1772. Hewitt, J.M., McKenzie, D.P., Weiss, N.O., 1975. Dissipative heating in convective Elkins-Tanton, L.T., Hess, P.C., Parmentier, E.M., 2005. Possible formation of ancient flows. J. Fluid Mech. 68, 721–738. crust on Mars through magma ocean processes. J. Geophys. Res. 110, E12S01. Holme, R., Bloxham, J., 1996. The magnetic fields of Uranus and Neptune: methods Elsasser, W.M., 1939. On the origin of the Earth’s magnetic field. Phys. Rev. 55, 489– and models. J. Geophys. Res. 101, 2177–2200. 498. Hood, L.L., Richmond, N.C., Pierazzo, E., Rochette, P., 2003. Distribution of crustal Evonuk, M., 2008. The role of density stratification in generating zonal flow magnetic fields on Mars: shock effects of basin-forming impacts. Geophys. Res. structures in a rotating fluid. Astrophys. J. 673, 1154–1159. Lett. 30, 1281. Finlay, C.C., Dumberry, M., Chulliat, A., Pais, M.A., 2010. Short timescale core Hood, L.L., Harrison, K.P., Langlais, B., Lillis, R.J., Poulet, F., Williams, D.A., 2010. dynamics: theory and observations. Space Sci. Rev. 152, 177–218. Magnetic anomalies near Apollinaris Patera and the Medusae Fossae Formation Folkner, W.M., Yoder, C.F., Yuan, D.N., Standish, E.M., Preston, R.A., 1997. Interior in Lucus Planum, Mars. Icarus 208, 118–131. structure and seasonal mass redistribution of Mars from radio tracking of Mars Hubbard, W., Podolak, M., Stevenson, D., 1995. Interior of Neptune. In: Cruickshank, Pathfinder. Science 278, 1749–1752. D. (Ed.), Neptune and Triton. Univ. Arizona Press, Tucson. Fortney, J.J., Ikoma, M., Nettelmann, N., Guillot, T., Marley, M.S., 2011. Self- Hulot, G., Finlay, C.C., Constable, C.G., Olsen, N., Mandea, M., 2010. The magnetic consistent model atmospheres and the cooling of the solar system’s giant field of planet Earth. Space Sci. Rev. 152, 159–222. planets. Astrophys. J. 729, 32–46. Iess, L., Rappaport, N.J., Jacobson, R.A., Racioppa, P., Stevenson, D.J., Tortora, P., Frey, H.V., Schultz, R.A., 1988. Large impact basins and the mega-impact origin for Armstrong, J.W., Asmar, S.W., 2010. Gravity field, shape, and moment of inertia the crustal dichotomy on Mars. Geophys. Res. Lett. 15, 229–232. of Titan. Science 327, 1367–1369. Frey, H.V., 2003. Buried impact basins and the earliest history of Mars. Sixth Intl. Jia, X., Kivelson, M.G., Khurana, K.K., Walker, R.J., 2010. Magnetic fields of the Conf. on Mars, Calif. Inst. of Technol., Pasadena, CA, 20–25 July. satellites of Jupiter and Saturn. Space Sci. Rev. 152, 271–305. Frey, H.V., Edgar, L., Lillis, R., 2007. Very large visible and buried impact basins on Johnson, C., Constable, C., Tauxe, L., 2003. Mapping long-term changes in Earth’s Mars: implications for internal and crustal evolution and the late heavy magnetic field. Science 300, 2044–2045. G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108 107

Jones, C.A., 2000. Convection-driven geodynamo models. Philos. Trans. R. Soc. Lond. McNutt Jr., R.L. et al., 2010. The MESSENGER mission: results from the first two Ser. A 204, 227–238. Mercury flybys. Acta Astronautica 67, 681–687. Jones, C.A., Kuzanyan, K.M., 2009. Compressible convection in the deep atmospheres Merrill, R.T., McElhinny, M.W., McFadden, P.L., 1996. The magnetic field of the of giant planets. Icarus 358, 873–897. Earth: paleomagnetism, the core, and the deep mantle. Academic Press, London. Kabin, K., Heimpel, M.H., Rankin, R., Aurnou, J.M., Gomez-Perez, N., Paral, J., Milbury, C., Schubert, G., 2010. Search for the global signature of the Martian Gombosi, T.I., Zurbuchen, T.H., Koehn, P.L., DeZeeuw, D.L., 2008. Global MHD dynamo. J. Geophys. Res. 115, E10010. modeling of Mercury’s magnetosphere with applications to the MESSENGER Milbury, C., Schubert, G., Raymond, C., Smrekar, S., 2011. Modeling gravity and mission and dynamo theory. Icarus 195, 1–15. magnetic anomalies at Tyrrhena Patera, Mars: implications for early polar Ke, Y., Solomatov, V.S., 2006. Early transient superplumes and the origin of the wander and a late dynamo. In review. J. Geophys. Res.. Martian crustal dichotomy. J. Geophys. Res. 111, E10001. Mitchell, D.L., Lillis, R.J., Lin, R.P., Connerney, J.E.P., Acuña, M.H., 2007. A global map Khan, A., Mosegaard, K., Williams, J.G., Lognonné, P., 2004. Does the Moon possess a of Mars’ crustal magnetic field based on electron reflectometry. J. Geophys. Res. molten core? Probing the deep lunar interior using results from LLR and lunar 112, E01002. prospector. J. Geophys. Res. 109, E09007. Mohit, P.S., Arkani-Hamed, J., 2004. Impact demagnetization of the Martian crust. Khurana, K.K., Jia, X., Kivelson, M.G., Nimmo, F., Schubert, G., Russell, C.T., 2011. Icarus 168, 305–317. Evidence of a global magma ocean in Io’s interior. Science. doi:10.1126/ Nellis, W.J., 2000. Metallization of fluid hydrogen at 140 GPa (1.4 Mbar): science.1201425. implications for Jupiter. Planet. Space Sci. 48, 671–677. Kimura, J., Nakagawa, T., Kurita, K., 2009. Size and compositional constraints of Ness, N.F., Behannon, K., Lepping, R., Whang, Y., 1975. Magnetic field of Mercury Ganymede’s metallic core for driving an active dynamo. Icarus 202, 216–224. confirmed. Science 255, 204–206. King, E.M., Soderlund, K.M., Christensen, U.R., Wicht, J., Aurnou, J.M., 2010. Ness, N.F., Behannon, K., Lepping, R., Whang, Y., 1976. Observations of Mercury’s Convective heat transfer in planetary dynamo models. Geochem. Geophys. magnetic field. Icarus 28, 479–488. Geosyst. 11, Q06016. Ness, N.F., Acuña, M.H., Lepping, R.P., Burlaga, L.F., Behannon, K.W., Neubauer, F.M., Kivelson, M.G., Khurana, K.K., Russell, C.T., Walker, R.J., Warnecke, J., Coroniti, F.V., 1979. Magnetic field studies at Jupiter by Voyager 1: preliminary results. Polanskey, C., Southwood, D.J., Schubert, G., 1996. Discovery of Ganymede’s Science 204, 982–987. magnetic field by the Galileo spacecraft. Nature 384, 537–541. Ness, N.F. et al., 1981. Magnetic field studies by Voyager 1 – preliminary results at Kivelson, M.G., Khurana, K.K., Volwerk, M., 2002. The permanent and inductive Saturn. Science 212, 211–217. magnetic moments of Ganymede. Icarus 157, 507–522. Ness, N.F., Acuña, M.H., Behannon, K.W., Burlaga, L.F., Connerney, J.E.P., Lepping, Kono, M., Roberts, P.H., 2002. Recent geodynamo simulations and observations of R.P., Neubauer, F.M., 1982. Magnetic field studies by Voyager 2: preliminary the geomagnetic field. Rev. Geophys. 40, 1013. results at Saturn. Science 215, 558–563. Konopliv, A.S., Yoder, C.F., 1996. Venusian k(2) tidal Love number from Magellan Ness, N.F., Acuña, M.H., Behannon, K.W., Burlaga, L.F., Connerney, J.E.P., Lepping, and PVO tracking data. Geophys. Res. Lett. 23, 1857–1860. R.P., Neubauer, F.M., 1986. Magnetic fields at Uranus. Science 233, 85–89. Konopliv, A.S., Binder, A.B., Hood, L.L., Kucinskas, A.B., Sjogren, W.L., Williams, J.G., Ness, N.F., Acuña, M.H., Burlaga, L.F., Connerney, J.E.P., Lepping, R.P., Neubauer, F.M., 1998. Improved gravity field of the Moon from lunar prospector. Science 281, 1989. Magnetic fields at Neptune. Science 246, 1473–1478. 1476–1480. Ness, N.F. et al., 1999. MGS magnetic fields and electron reflectometer Kuang, W., Jiang, W., Wang, T., 2008. Sudden termination of Martian dynamo? investigation: discovery of paleomagnetic fields due to crustal remanence. Implications from subcritical dynamo simulations. Geophys. Res. Lett. 35, L14204. Adv. Space Res. 23, 1879–1886. Kutzner, C., Christensen, U.R., 2000. Effects of driving mechanisms in geodynamo Nimmo, F., Stevenson, D.J., 2000. Influence of early plate tectonics on the thermal models. Geophys. Res. Lett. 27, 29–32. evolution and magnetic field of Mars. J. Geophys. Res. 105, 11969–11979. Kutzner, C., Christensen, U.R., 2002. From stable dipolar to reversing numerical Nimmo, F., Hart, S.D., Korycansky, D., Agnor, C.B., 2008. Implications of an impact dynamos. Phys. Earth Planet. Int. 131, 29–45. origin for the martian hemispheric dichotomy. Nature 453, 1220–1223. Langlais, B., Purucker, M.E., Mandea, M., 2004. Crustal magnetic field of Mars. J. Olsen, N., Mandea, M., 2008. Rapidly changing flows in the Earth’s core. Nature 1, Geophys. Res. 109, E02008. 390–394. Langlais, B., Lesur, V., Purucker, M.E., Connerney, J.E.P., Mandea, M., 2010. Crustal Olson, P., Christensen, U.R., 2006. Dipole moment scaling for convection-driven magnetic fields of terrestrial planets. Space Sci. Rev. 152, 223–249. planetary dynamos. Earth Planet. Sci. Lett. 250, 561–571. Lee, K.K.M. et al., 2006. Laser-driven shock experiments on precompressed water: Pearl, J.C., Conrath, B.J., 1991. The albedo, effective temperature, and energy balance implications for ‘‘icy’’ giant planets. J. Chem. Phys. 125, 014701. of Neptune as determined from Voyager data. J. Geophys. Res. Suppl. 96, 18921– Lillis, R.J., Mitchell, D.L., Lin, R.P., Connerney, J.E.P., Acuña, M.H., 2004. Mapping 18930. crustal magnetic fields at Mars using electron reflectometry. Geophys. Res. Lett. Pearl, J.C., Conrath, B.J., Hanel, R.A., Pirraglia, J.A., 1990. The albedo, effective 31, L15702. temperature, and energy balance of Uranus as determined from Voyager IRIS Lillis, R.J., Manga, M., Mitchell, D.L., Lin, R.P., Acuña, M.H., 2005. Evidence for a data. Icarus 84, 12–28. second Martian dynamo from electron reflection magnetometry. Lunar Planet. Phillips, J.L., Russell, C.T., 1987. Upper limit on the intrinsic magnetic field of Venus. Sci. XXXVI 1578. J. Geophys. Res. 92, 2253–2263. Lillis, R.J., Manga, M., Mitchell, D.L., Lin, R.P., Acuña, M.H., 2006. Unusual magnetic Pirraglia, J.A., 1984. Meridional energy balance of Jupiter. Icarus 59, 169–176. signature of the Hadriaca Patera Volcano: implications for early Mars. Geophys. Podolak, M., Hubbard, W.B., Stevenson, D.J., 1991. Models of Uranus’ interior and Res. Lett. 33, L03202. magnetic field. In: Bergstralh, J.T., Miner, E.D. (Eds.), Uranus, Matthews, M.S., Lillis, R.J., Frey, H.V., Manga, M., 2008a. An improved crustal magnetic field map of Univ. of Ariz. Press, Tucson. Mars from electron reflectometry: highland volcano magmatic history and the Porco, C.C. et al., 2003. Cassini imaging of Jupiter’s atmosphere, satellites and rings. end of the martian dynamo. Icarus 194, 575–596. Science 299, 1541–1547. Lillis, R.J., Frey, H.V., Manga, M., 2008b. Rapid decrease in Martian crustal Proudman, J., 1916. On the motion of solids in a liquid possessing vorticity. Proc. R. magnetization in the Noachian era: implications for the dynamo and climate Soc. Lond. A 92, 408–424. of early Mars. Geophys. Res. Lett. 35, L14203. Purucker, M.E., Ravat, D., Frey, H., Voorhies, C., Sabaka, T., Acuña, M., 2000. An Lingenfelter, R.E., Schubert, G., 1973. Evidence for convection in planetary interiors altitude-normalized magnetic map of Mars and its interpretation. Geophys. Res. from first-order topography. Earth Moon Planets 7, 172–180. Lett. 27, 2449–2452. Liu, J., Goldreich, P.M., Stevenson, D.J., 2008. Constraints on deep-seated zonal Purucker, M.E., 2008. A global model of the internal magnetic field of the Moon winds inside Jupiter and Saturn. Icarus 196, 653–664. based on lunar prospector magnetometer observations. Icarus 197, 19–23. Lodders, K., Fegley Jr., B., 1998. The Planetary Scientist’s Companion. Oxford Univ. Read, P.L., Dowling, T.E., Schubert, G., 2009. Saturn’s rotation period from its Press, Oxford. atmospheric planetary-wave configuration. Nature 460, 608–610. Longhi, J., Knittle, E., Holloway, J.R., Waenke, H., 1992. The bulk composition, Rhines, P.B., 1975. Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417– mineralogy and internal structure of Mars. In: Kieffer, H.H. et al. (Eds.), Mars. 443. Univ. of Ariz. Press, Tucson. Riner, M.A., Bina, C.R., Robinson, M.S., Desch, S.J., 2008. Internal structure of Malavergne, V., Toplis, M.J., Berthet, S., Jones, J., 2010. Highly reducing conditions Mercury: implications of a molten core. J. Geophys. Res. 113, E08013. during core formation on Mercury: implications for internal structure and the Robbins, S.J., Achille, G.D., Hynek, B.M., 2011. The volcanic history of Mars: high- origin of a magnetic field. Icarus 206, 199–209. resolution crater-based studies of the calderas of 20 volcanoes. Icarus 211, Malkus, W.V.R., 1994. Energy sources for planetary dynamos. In: Proctor, M.R.E., 1179–1203. Gilbert, A.D. (Eds.), Lectures on Solar and Planetary Dynamos. Cambridge Univ. Roberts, J.H., Zhong, S., 2006. Degree-1 convection in the Martian mantle and the Press, Cambridge, UK. origin of the hemispheric dichotomy. J. Geophys. Res. 111, E06013. Margot, J., Peale, S., Jurgens, R., Slade, M., Holin, I., 2007. Large longitude libration of Russell, C.T., Weiss, H., Coleman, P.J., Jr., Soderblom, L.A., Stuart-Alexander, D.E., Mercury reveals a molten core. Science 316, 710–714. Wilhelms, D.E., 1977. In: Proc. 8th Lunar Sci. Conf., Pergamon, New York, pp. Marinova, M.M., Aharonson, O., Asphaug, E., 2008. Mega-impact formation of the 1171–1185. Mars hemispheric dichotomy. Nature 453, 1216–1219. Russell, C.T., Elphic, R.C., Slavin, J.A., 1980. Limits on the possible intrinsic magnetic McComas, D.J., Allegrini, F., Bagenal, F., Crary, F., Ebert, R.W., Elliott, H., Stern, A., field of Venus. J. Geophys. Res. 85 (A13), 8319–8332. Valek, P., 2007. Diverse plasma populations and structures in Jupiter’s Russell, C.T., Yu, Z.J., Kivelson, M.G., 2001. The rotation period of Jupiter. Geophys. magnetotail. Science 318, 217–220. Res. Lett. 28, 1911–1912. McEwen, A.S., Lunine, J.I., Carr, M.H., 1989. Dynamic Geophysics of Io. In Time Russell, C.T., Dougherty, M.K., 2010. Magnetic fields of the outer planets. Space Sci. Variable Phenomena in the Jovian System, NASA-SP-494, pp. 11–46. Rev. 152, 251–269. McNutt Jr., R.L. et al., 2007. Energetic particles in the Jovian magnetotail. Science Sakuraba, M., Roberts, P.H., 2009. Generation of a strong magnetic field using 318, 220–222. uniform heat flux at the surface of the core. Nature Geol. 2, 802–805. 108 G. Schubert, K.M. Soderlund / Physics of the Earth and Planetary Interiors 187 (2011) 92–108

Sanchez-Lavega, A., Rojas, J.F., Sada, P.V., 2000. Saturn’s zonal winds at cloud level. Stevenson, D.J., 1980. Saturn’s luminosity and magnetism. Science 208, 746–748. Icarus 147, 405–420. Stevenson, D.J., 1982. Reducing the non-axisymmetry of a planetary dynamo and an Sanchez-Lavega, A., 2004. The magnetic field in giant extrasolar planets. Astrophys. application to Saturn. Geophys. Astrophys. Fluid Dyn. 21, 113–127. J. 609, L87–L90. Stevenson, D.J., 1983. Planetary magnetic fields. Rep. Prog. Phys. 46, 555–620. Sarson, G.R., Jones, C.A., Zhang, K., Schubert, G., 1997. Magnetoconvection dynamos Stevenson, D.J., 1987. Mercury’s magnetic field: a thermoelectric dynamo? Earth and the magnetic fields of Io and Ganymede. Science 276, 1106–1108. Planet. Sci. Lett. 82, 114–120. Schubert, G., Lingenfelter, R.E., 1973. Martian centre of mass – centre of figure Stevenson, D.J., 2003. Planetary magnetic fields. Earth Planet. Sci. Lett. 208, 1–11. offset. Nature 242, 251–252. Stevenson, D.J., 2010. Planetary magnetic fields: achievements and prospects. Space Schubert, G., Ross, M.N., Stevenson, D.J., Spohn, T., 1988. Mercury’s thermal history Sci. Rev. 152, 651–664. and the generation of its magnetic field. In: Vilas, F., Chapman, C.R., Matthews, Stevenson, D.J., Salpeter, E.E., 1977. The phase diagram and transport properties for M.S. (Eds.), Mercury. Univ. of Ariz. Press, Tucson. hydrogen–helium fluid planets. Astrophys. J. Suppl. 35, 221–237. Schubert, G., Zhang, K., Kivelson, M.G., Anderson, J.D., 1996. The magnetic field and Stevenson, D.J., Spohn, T., Schubert, G., 1983. Magnetism and thermal evolution of internal structure of Ganymede. Nature 384, 544–545. the terrestrial planets. Icarus 54, 466–489. Schubert, G., Russell, C.T., Moore, W.B., 2000. Geophysics – timing of the Martian Stacey, F., 2007. Core properties, physical. In: Gubbins, D., Herrero-Bervera, E. (Eds.), dynamo. Nature 408, 666–667. Encyclopedia of Geomagnetism and Paleomagnetism. Springer-Verlag, Schubert, G., Turcotte, D.L., Olson, P., 2001. Mantle Convection in the Earth and Netherlands. Planets. Cambridge Univ. Press, Cambridge, UK. Suomi, V.E., Limaye, S.S., Johnson, D.R., 1991. High winds of Neptune: a possible Schubert, G., Chan, K., Liao, X., Zhang, K., 2004. Planetary dynamos: effects of mechanism. Science 251, 929–932. electrically conducting flows overlying turbulent regions of magnetic field Takahashi, F., Matsushima, M., 2006. Dipolar and non-dipolar dynamos in a thin generation. Icarus 172, 305–315. shell geometry with implications for the magnetic field of Mercury. Geophys. Schubert, G., Anderson, J.D., Spohn, T., McKinnon, W.B., 2007. Interior composition, Res. Lett. 33, L10202. structure and dynamics of the Galilean satellites. In: Bagenal, F., Dowling, T.E., Takahashi, F., Tsunakawa, H., 2009. Thermal core-mantle coupling in an early lunar McKinnon, W.B. (Eds.), Jupiter. Cambridge Univ. Press, Cambridge, UK. dynamo: implications for a global magnetic field and magnetosphere of the Scorer, R.S., 1966. Origin of cylcones. Sci. J. 2, 46–52. early Moon. Geophys. Res. Lett. 36, L24202. Shkolnik, E., Walker, G.A.H., Bohlender, D.A., Gu, P.-G., Kuerster, M., 2005. Hot Taylor, G., 1917. Motion of solids in fluids when the flow is not irrotational. Proc. R. Jupiters and hot spots: the short- and long-term chromospheric activity on stars Soc. Lond. A 93, 99–113. with giant planets. Astrophys. J. 622, 1075–1090. Tilgner, A., 2007. Kinematic dynamos with precession driven flow in a sphere. Shkolnik, E., Bohlender, D.A., Walker, G.A.H., Collier Cameron, A., 2008. The on/off Geophys. Astrophys. Fluid Dyn. 101, 1–9. nature of star–planet interactions. Astrophys. J. 676, 628–638. Uno, H., Johnson, C.L., Anderson, B.J., Korth, H., Solomon, S.C., 2009. Modeling Simitev, R., Busse, F.H., 2005. Prandtl-number dependence of convection-driven Mercury’s internal magnetic field with smooth inversions. Earth Planet. Sci. dynamos in rotating spherical fluid shells. J. Fluid Mech. 532, 365–388. Lett. 285, 328–339. Smith, E.J., Davis, L., Jones, D.E., Coleman, P.J., Colburn, D.S., Pyal, P., Sonett, C.P., Vilim, R., Stanley, S., Hauck II, S.A., 2010. Iron snow zones as a mechanism for Frandsen, A.M.A., 1974. The planetary magnetic field and magnetosphere of generating Mercury’s weak observed magnetic field. J. Geophys. Res. 115, Jupiter: Pioneer 10. J. Geophys. Res. 79, 3501–3513. E11003. Smith, E.J., Davis, L., Jones, D.E., Coleman, P.J., Colburn, D.S., Pyal, P., Sonett, C.P., Watters, W., Zuber, M., Hager, B., 2009. Thermal perturbations caused by large 1975. Jupiter’s magnetic field, magnetosphere, and interaction with the solar impacts and consequences for mantle convection. J. Geophys. Res. 114, E02001. wind: Pioneer II. Science 188, 451–454. Weinstein, S.A., 1995. The effects of a deep mantle endothermic phase change on Smith, E.J., Davis, L., Jones, D.E., Coleman, P.J., Colburn, D.S., Sonett, C.P., 1980. the structure of thermal convection in silicate planets. J. Geophys. Res. 100 (E6), Saturn’s magnetic field and magnetosphere. Science 207, 407–410. 11719–11728. Smith, E.J. et al., 2010. The equatorial shape and gravity field of Mercury from Weiss, B.P., Vali, H., Baudenbacher, F.J., Kirschvink, J.L., Stewart, S.T., Shuster, D.L., MESSENGER flybys 1 and 2. Science 209, 88–100. 2002. Records of an ancient Martian magnetic field in ALH84001. Earth Planet. Smylie, D.E., Rochester, M.G., 1980. Compressibility, core dynamics and the Sci. Lett. 201, 449–463. subseismic wave equation. Phys. Earth Planet. Int. 24, 308–319. Weiss, B.P., Berdahl, J.S., Elkins-Tanton, L., Stanley, S., Lima, E.A., Carporzen, L., 2008. Solomon, S., 2011. A new look at the planet Mercury. Phys. Today 64, 50–55. Magnetism on the angrite parent body and the early differentiation of Soderlund, K.M., Aurnou, J.M., 2010. Simulation of an ice giant-style dynamo. planetesimals. Science 322, 713–716. American Geophysical Union Fall Meeting 2010. San Francisco. Werner, S.C., 2009. The global martian volcanic evolutionary history. Icarus 201, Soderlund, K.M., King, E.M., Aurnou, J.M., in review. The weak influence of magnetic 44–68. fields in planetary dynamo models. Earth Planet. Sci. Lett. Wicht, J., Stellmach, S., Harder, H., 2009. Numerical models of the geodynamo: from Sreenivasan, B., Jones, C.A., 2006. The role of inertia in the evolution of spherical fundamental cartesian models to 3D simulations of field reversals. In: dynamos. Geophys. J. Int. 164, 467–476. Geomagnetic Field Variations, Springer, p. 107. Sromovsky, L.A., Fry, P.M., Dowling, T.E., Baines, K.H., Limaye, S.S., 2001. Neptune’s Wienbruch, U., Spohn, T., 1995. A self-sustained magnetic field on Io? Planet. Space atmospheric circulation and cloud morphology: Changes revealed by 1998 HST Sci. 43, 1045–1057. imaging. Icarus 150, 244–260. Wilhelms, D.E., Squyres, S.W., 1984. The martian hemispheric dichotomy may be Stanley, S., 2010. A dynamo model for axisymmetrizing Saturn’s magnetic field. due to a giant impact basin. Nature 309, 138–140. Geophys. Res. Lett. 37, L05201. Williams, J.-P., Aharonson, O., Nimmo, F., 2007. Powering Mercury’s dynamo. Stanley, S., Bloxham, J., 2004. Convective-region geometry as the cause of Uranus’ Geophys. Res. Lett. 34, L21201. and Neptune’s magnetic fields. Nature 428, 151–153. Williams, D.A., Greeley, R., Werner, S.C., Michael, G., Crown, D.A., Neukum, G., Stanley, S., Bloxham, J., 2006. Numerical dynamo models of Uranus’ and Neptune’s Raitala, J., 2008. Tyrrhena Patera: geologic history derived from Mars express unusual magnetic fields. Icarus 184, 556–572. high resolution stereo camera. J. Geophys. Res. 113, E11005. Stanley, S., Bloxham, J., Hutchison, W.E., Zuber, M.T., 2005. Thin shell dynamo Yoder, C.F., Konopliv, A.S., Yuan, D.N., Standish, E.M., Folkner, W.M., 2003. Fluid core models consistent with Mercury’s weak observed magnetic field. Earth Planet. size of Mars from detection of the solar tide. Science 300, 299–303. Sci. Lett. 234, 27–38. Yu, Z.J., Leinweber, H., Russell, C.T., 2010. Galileo constraints on the secular variation Stanley, S., Mohammadi, A., 2008. Effects of an outer thin stably stratified layer on of the Jovian magnetic field. J. Geophys. Res. 115, E03002. planetary dynamos. Phys. Earth Planet. Sci. 168, 179–190. Zhang, K., 1992. Spiraling columnar convection in rapidly rotating spherical fluid Stanley, S., Elkins-Tanton, L., Zuber, M., Parmentier, E.M., 2008. Mars’ paleomagnetic shells. J. Fluid Mech. 236, 535–554. field as the result of a single-hemisphere dynamo. Science 321, 1822–1825. Zhang, C.Z., Zhang, K., 1995. On the internal structure and magnetic fields of Venus. Stanley, S., Glatzmaier, G.A., 2010. Dynamo models for planets other than Earth. Earth Moon Planets 69, 237–247. Space Sci. Rev. 152, 617–649. Zharkov, V.N., 1983. Models of the internal structure of Venus. Earth Moon Planets Stegman, D.R., Jellinek, A.M., Zatman, S.A., Baumgardner, J.R., Richards, M.A., 2003. 29, 139–175. An early lunar core dynamo driven by thermochemical mantle convection. Zhong, S., Zuber, M.T., 2001. Degree-1 mantle convection and the crustal dichotomy Nature 421, 143–146. on Mars. Earth Planet. Sci. Lett. 189, 75–84.