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# Planetary Magnetic Fields and Dynamos

## Summary and Keywords

Since 1973 space missions carrying vector magnetometers have shown that most, but not all, solar system planets have a global magnetic field of internal origin. They have also revealed a surprising diversity in terms of field strength and morphology. While Jupiter’s field, like that of Earth, is dominated by a dipole moderately tilted relative to the planet’s spin axis, the fields of Uranus and Neptune are multipole-dominated, whereas those of Saturn and Mercury are highly symmetric relative to the rotation axis. Planetary magnetism originates from a dynamo process, which requires a fluid and electrically conducting region in the interior with sufficiently rapid and complex flow. The magnetic fields are of interest for three reasons: (i) they provide ground truth for dynamo theory, (ii) the magnetic field controls how the planet interacts with its space environment, for example, the solar wind, and (iii) the existence or nonexistence and the properties of the field enable us to draw inferences on the constitution, dynamics, and thermal evolution of the planet’s interior. Numerical simulations of the geodynamo, in which convective flow in a rapidly rotating spherical shell representing the outer liquid iron core of the Earth leads to induction of electric currents, have successfully reproduced many observed properties of the geomagnetic field. They have also provided guidelines on the factors controlling magnetic field strength and morphology. For numerical reasons the simulations must employ viscosities far greater than those inside planets and it is debatable whether they capture the correct physics of planetary dynamo processes. Nonetheless, such models have been adapted to test concepts for explaining magnetic field properties of other planets. For example, they show that a stable stratified conducting layer above the dynamo region is a plausible cause for the strongly axisymmetric magnetic fields of Mercury or Saturn.

# Introduction

The magnetic field of the Earth has been known for centuries. Its spatial structure is dominated by the dipole component aligned approximately with the rotation axis, similar to the magnetic field of a fictional bar magnet inside the Earth. The slow variation of the Earth’s field on a centennial timescale and, by indirect evidence, the occasional reversals of the dipole direction on geological timescales were known before magnetic fields were detected on other planets. Because of our detailed knowledge on the Earth’s magnetism, the geomagnetic field is often taken as a prototype for planetary magnetic fields in general.

With the advent of the space age, it turned out that magnetic fields are not uncommon on other planets in the solar system. However, there is substantial diversity. Planetary flyby and orbiting missions that carried magnetometers revealed that most planets, but not all of them (Venus, Mars), have global magnetic fields of internal origin. Some had a field that became extinct billions of years ago (e.g., Mars). In most cases the axial dipole dominates the field at the planetary surface, but Uranus and Neptune are exceptions. Saturn’s field is extremely symmetric with respect to the planet’s rotation axis. The field strengths at the planetary surfaces vary widely: Jupiter’s field is more than 1,000 times stronger than that of Mercury. The presence or absence of a magnetic field, and its properties where a field exists, provide important information on the structure and dynamics of the deep interior of planets. In order to make use of this information, a thorough understanding of the origin of magnetic fields is needed.

Global-scale planetary magnetic fields are generated by a dynamo process. Dynamos are based on the electromagnetic induction of currents due to the motion of an electrical conductor in an existing magnetic field. Geophysical observations have shown that one fundamental requirement for it, the existence of an electrically conducting fluid region, is met inside the Earth, whose outer core consists of a liquid iron alloy. It is very likely that all other planets also have fluid conducting cores. However, some planets may not conform with another basic condition for a dynamo, that is, sufficiently fast motion in the fluid layer. Convection is envisaged as the most likely source of a flow that can sustain a dynamo, but in some planets the fluid core may be essentially stagnant. Planetary dynamos are self-sustained, which means that the magnetic field necessary for induction is set up by the induced current itself. Historically, it required substantial theoretical efforts to clarify how a homogeneous dynamo can work, that is, a dynamo that operates in an unstructured volume of conducting fluid.

Since 1995, numerical dynamo models have been successful in reproducing many observed properties of the geomagnetic field. Looking beyond Earth, a clear understanding of the diversity in the morphology and amplitude of planetary magnetic fields is still lacking. Nonetheless, several promising ideas have been suggested and backed up by dynamo simulations. Some of the differences between planets can possibly be explained by a systematic dependence of dynamo properties on parameters such as rotation rate or energy flux. Others seem to require qualitative differences in the structure and dynamics of the planetary interior.

Knowledge on the structure and time dependence of the geomagnetic field is summarized, and the more limited knowledge on the fields of other solar system planets is presented. The internal structure and thermal budget of the planets will be discussed as far as it is essential for the understanding of planetary dynamos. A general introduction to the theory of cosmic dynamos is beyond the scope of this article, but conditions for magnetic field generation that are particular to planetary cores are discussed. Hypotheses, theories, and numerical models that attempt to explain the specific magnetic properties of the various planets are presented. Not covered here is the influence of the magnetic field on the way a planet interacts with charged particles in its space environment.

# Geomagnetic Field

In the year 1600 William Gilbert (Gilbert, 1958) showed convincingly that the force acting on a compass needle is intrinsic to the body of the Earth. Since his time the Earth’s magnetic field has been mapped in sufficient detail to determine its global structure. Most of the early measurements were purely directional and have been taken by mariners (Jackson, Jonkers, & Walker, 2000). In 1832 Carl Friedrich Gauss developed a method for measuring the field intensity in absolute terms. In his “General Theory of Terrestrial Magnetism” (Glassmeier & Tsurutani, 2014) Gauss introduced a mathematical description of the geomagnetic field in terms of spherical harmonic functions, which is also used for other planets. Different contributions to the field are described by the Gauss coefficients (as they are now called) $gnm$, $hnm$. Their physical unit, Tesla (T), is the same as that of magnetic field B. Increasing values of the harmonic degree n are associated with an increasing degree of spatial complexity; n = 1 stands for the dipole, n = 2 for the quadrupole, n = 3 for the octupole, and so forth. Larger values of n relate to smaller-scale structures in the field. The dominant contribution, that is, the dipole component aligned with the rotation axis, is given by $g10$. Gauss’s description of the field enables us to distinguish unambiguously between contributions from internal and external sources. By far the largest part of the magnetic field is of internal origin, but a small external contribution that varies on timescales ranging from minutes to days is caused by electrical currents in the Earth’s ionosphere and magnetosphere. The magnetosphere is the region extending roughly ten Earth radii into space where the Earth’s magnetic field has a dominant influence. Outside the so-called magnetopause the solar wind and the associated magnetic field prevail.

A new era started in 1980, when dedicated satellite missions carrying magnetometers in low Earth orbits were employed to map the geomagnetic field with unprecedented spatial resolution. The SWARM mission has operated since 2013 and consists of three satellites (Friis- Christensen, Lühr, & Hulot, 2006). As a result, representations of the Earth’s internal magnetic field are available for harmonic degrees n > 100.

When aiming at an understanding of the geodynamo, it is more meaningful to consider the magnetic field structure at the surface of the liquid iron core rather than at the Earth’s surface. To the extent that there are no significant sources of the magnetic field in the Earth’s crust and mantle, the mathematical formalism of Gauss enables the magnetic field to be “downward continued” from the surface to the top of Earth’s core. Basically, a component of degree n varies with radius as $r−(n+2)$, that is, smaller-scale components drop off more rapidly with r. In Figure 1 spatial power spectra of the magnetic field are compared for the Earth’s surface and the core-mantle boundary. At the surface the spectral power drops sharply up to harmonic degree 13, and is white beyond that. The spectrum of the field projected onto the core-mantle boundary is almost white up to n = 13, except for the dipole term, which stands out by a factor of five. The empirical finding of an approximately white power spectrum at the top of Earth’s core is tentatively used to infer the depth of the dynamo below the surface of other planets, by downward continuing the observed magnetic field until a white spectrum results. For n > 13 the spectrum at Earth’s core-mantle boundary rises steeply, which is a very unlikely property of the core field. The interpretation of these spectra is that the field at the Earth’s surface is dominated by the core field at large scales up to n ≈ 13. Shorter scales certainly exist in the core field, but their imprint on the observable field at the Earth’s surface is masked by contributions from the inhomogeneous remanent and induced magnetization of small amounts of ferromagnetic minerals in the Earth’s crust. Projecting this small-scale field onto the core-mantle boundary is unphysical and leads to the blue spectrum for n >13. Hence the magnetic field at the surface of Earth’s core is known only at large wavelengths.

Figure 1. Spatial power spectra of the geomagnetic field in 2016 according to the SIFmplus model (Olsen, Finlay, Kotsiaros, & Tøffner-Clausen, 2016) as a function of spherical harmonic degree n at Earth’s surface (diamonds) and at the core-mantle boundary (triangles; offset in amplitude). Units are µT2 for the surface field and mT2 for the core field.

Figure 2. Radial component of the geomagnetic field in 2016 at the Earth’s surface (a) and at the core-mantle boundary (b). Red for outward magnetic flux and blue for inward flux. Contour interval is 9 µT in (a) and 120 µT in (b).

Figure 2 shows the radial component of the geomagnetic field at the Earth’s surface and, truncated at n = 13, at the core-mantle boundary. At the surface the dipole part stands out. At the core-mantle boundary the dipole dominance is still visible, but there is significant structure at smaller scales. Most of the dipole field is formed by strong concentrations of magnetic flux into four lobes, two in each hemisphere, centered at ±65∘ latitude. The flux lobes in the Northern Hemisphere, under North America and Siberia, have counterparts in the Southern Hemisphere that lie at approximately the same longitudes. Close to the rotation poles, the flux is weak or even inverse with respect to the dominant polarity in the respective hemisphere. At low and midlatitudes patches of magnetic flux of both polarities are found.

The rms magnetic field strength at the core-mantle boundary is 0.39 mT for n=1-13. The field strength inside the core is difficult to estimate. Only the poloidal part of the magnetic field can be observed outside of the core, whereas the field lines of the toroidal part close inside the core and are unobservable. Speculations that the toroidal field in the Earth’s core would be much stronger than the poloidal part, as it is in the Sun, are not supported by geodynamo simulations. A range of 1–4 mT seems plausible for the field strength inside the core and is supported by indirect lines of observational evidence (Buffett, 2010; Gillet, Jault, Canet, & Fournier, 2010).

Figure 2 represents a snapshot in time. The internal magnetic field changes on a timescale of centuries, which is termed secular variation. The intensity of the dipole field has dropped by about 9% since 1840, when it was first measured in absolute terms. Maps of the core field based on the historical record of observations have been constructed back until the year 1590 (Jackson et al., 2000). Although the details of the field structure changed, some general traits remained the same. The Northern Hemisphere flux lobes are persistent and stay approximately in place, whereas some magnetic field structures at low latitudes are drifting westward. The field changes at the core-mantle boundary are used to infer the flow pattern and velocity of the liquid iron at the top of Earth’s core. This is possible because for a highly conducting medium the magnetic field lines behave approximately as if they are “frozen” into the moving fluid and are advected along with it (Holme, 2015). A typical velocity of the core fluid is 0.5 mm/s.

The magnetic field changes of the past 400 years are documented by direct measurements. Going further back in time is possible by accessing the archive of magnetized rocks, which recorded the magnetic field at the time of their formation. Most rocks contain small amounts of ferromagnetic minerals. A remanent magnetization is acquired, for example, when a magmatic rock cools in an ambient magnetic field below the Curie temperature (where a mineral becomes magnetic). From oriented and dated rock samples the field direction and the magnetic field strength at the time of their formation can be determined, although this is not straightforward but quite an art. Paleointensity measurements date back until 3.45 billion years and possibly even until 4.2 billion years ago (Tarduno, Cottrell, Davis, Nimmo, & Bono, 2015) and show that the Earth had a magnetic field very early in its history. Although the intensity of the geomagnetic field fluctuated, most of the time it was within a factor of two or three of the present field strength. The geometry of the field is more difficult to determine from paleomagnetic data. For rocks older than 5 million–10 million years continent drift becomes important, that is, the location of the rock at the time when it was formed is not the same as it is today. In fact, the movements of the continents are calculated from paleomagnetic data under the assumption that the geomagnetic field is that of a geocentric axial dipole when averaged over several ten thousands of years. Data from the past five million years, for which the effects of continental drift are small, strongly support this hypothesis. The dipole dominance is more difficult to prove for earlier times, but the available evidence is in support of it. In summary, the Earth’s magnetic field has not changed dramatically over the past four billion years in geometry or strength. A detailed account of the paleo-field can be found in Merrill, McElhinny, and McFadden (1996).

One of the earliest findings by paleomagnetism is the occurrence of reversals of the dipole field. During these events the magnetic north and south poles switch places. A detailed chronology of the geomagnetic polarity during the past couple of hundred million years has been established. Compared to the typical length of periods with stable dipole polarity of some hundred thousand years, reversals are fairly rapid. During reversals the dipole does not simply tip over, but it also weakens significantly, whereas the strength of higher multipole components does not seem to change much. On average, the geomagnetic field has reversed a few times in a million years during the recent geological past. In contrast to the quasi-cyclic behavior of the solar magnetic field, the timing of geomagnetic reversals is random. However, the reversal frequency itself changed drastically on timescales of 100 million years, which is comparable to the overturn time of the sluggish convection in the Earth’s mantle. For this reason it is assumed that the likelihood of reversals is controlled by the slowly changing conditions in the lowermost mantle, such as its thermal structure, which would affect convection in the liquid core. Details on reversals can be found in Glatzmaier and Coe (2015).

# Magnetic Fields of Other Planets

The magnetic fields of all major planets in the solar system have been characterized by space missions during flybys or from orbiting spacecraft, although in general rather crudely in comparison to the geomagnetic fields. For example, hardly anything is known about secular variation on other planets. Many planetary missions carried vector magnetometers that measured three components of the magnetic field. Balogh (2010) gives a detailed account of the instrumentation and an overview on the relevant space missions. Table 1 summarizes our knowledge of the magnetic field properties of the solar system planets (as of mid-2018). Figure 3 compares the magnetic field structures of the various planets. The most important observational findings on planetary magnetic fields are summarized object-by-object. A detailed account is given in Connerney (2015).

Table 1. Planetary Magnetic Fields

Planet

Active dynamo

Rc/Rp

Bs[nT]

Tilt

P2/P1

P3/P1

Ref.

Mercury

Yes

0.83

300

$<≈$

0.38

0.06

A12

Venus

No

0.55

PR87

Earth

Yes

0.55

43,700

9.6°

0.15

0.24

O16

Moon

No, yes in past

0.2

WT14

Mars

No, yes in past

0.5

A99

Jupiter

Yes

0.84

671,000

10.3°

0.09

0.11

C18

Ganymede

Yes

0.3

1,000

< 0.04

K02

Saturn

Yes

0.6

30,500

0.02

0.19

C11

Uranus

Yes

0.75

48,000

59°

1.3

1.5 ?

HB96

Neptune

Yes

0.75

47,000

45°

2.7

6 ?

HB96

Rp—radius of planetary surface; Rc—radius of core or assumed dynamo surface; Bs—rms field strength at planetary surface, Tilt—of dipole axis with respect to the rotation axis, P1, P2, P3—power in dipole, quadrupole, octupole field, respectively, at radius Rc, References: A12—Anderson et al. (2012), PR87—Phillips and Russell (1987), O16—Olsen et al. (2016), WT14—Weiss and Tikoo (2014), A99—Acuña et al. (1999), C18—Connerney et al. (2018), K02—Kivelson, Khurana, and Volwerk (2002), C11—Cao, Russell, Christensen, Dougherty, and Burton (2011), HB96—Holme and Bloxham (1996).

Figure 3. Radial component of the magnetic fields at the planetary surfaces (blue = inward magnetic flux). Color scales differ. References for field models are given in Table 1.

## Mercury

The discovery of Mercury’s internal magnetic field during a flyby of Mariner 10 in 1975 came as a surprise. Before, it was believed that internal activity, including the working of a dynamo, had ceased in the small planet. Mariner 10 found that the field is global and probably dipolar but very weak in comparison to Earth’s field. The MESSENGER spacecraft orbited Mercury from 2011 to 2015, which allowed a better characterization. However, the orbit was highly elliptical and at southern latitudes outside the magnetosphere (because of the weakness of the internal field, the magnetopause is close to Mercury’s surface). Also, magnetospheric currents contribute rather strongly to the observed field inside the magnetosphere, and the separation of internal and external field contributions poses a problem. Nonetheless, besides confirming the dipole dominance, the data revealed some intriguing properties of the internal magnetic field (Anderson et al., 2012). The dipole tilt with respect to the spin axis is less than a degree. Furthermore, the dipole seems to be offset toward the north from Mercury’s center by 20% of the planet’s radius. This is equivalent to a strong contribution of the axial quadrupole in addition to the centered dipole, with a ratio of quadrupole power to dipole power much larger than in case of the Earth. During the final mission stage MESSENGER approached Mercury’s surface to within a few tens of kilometers, which allowed the identification of a weak contribution from crustal magnetization (Johnson et al., 2015). Because Mercury’s crust is old, this testifies to the existence of a dynamo field some 3.8 billion years ago.

## Venus

No intrinsic magnetic field has been observed on Venus. The upper limit for the dipole moment is $10−5$ of Earth’s value (Phillips & Russell, 1987). No small-scale magnetic field due to remanent magnetization of crustal rocks, which would be indicative of an ancient dynamo field, has been detected.

This is not unexpected given that the surface temperature of Venus is close to, or above, the Curie temperature of ferromagnetic minerals. The answer to the question of whether Venus once had an operating dynamo remains elusive.

## Moon

The Moon has no global field at present. The Apollo missions found small-scale magnetic fields of crustal origin that were mapped out in more detail by the Lunar Prospector mission. Locally the surface field strength reaches values around 100 nT (Mitchell et al., 2008). Lunar rock samples brought to Earth show remanent magnetization. Originally their analysis did not result in a coherent picture of the strength of the magnetizing field. A more recent analysis of Apollo samples with modern paleomagnetic methods has improved this situation and suggests that the Moon had a global magnetic field between 4.25 and 3.55 billion years ago, with a surface strength close to the present strength of the Earth’s field (Weiss and Tikoo, 2014).

## Mars

The first orbiter missions to Mars that carried magnetometers did not find conclusive evidence for an internal magnetic field. This changed in 1997 when the Mars Global Surveyor approached the planet’s surface to within 100 km during aerobraking maneuvers and found locally strong fields of crustal origin. Their amplitude is in excess of 1000 nT at satellite altitude in some regions (Acuña et al., 1999), implying local surface field strengths of several thousands of nT. This is significantly more than typical crustal field contributions on Earth, the Moon, or Mercury. The distribution is uneven, with strong magnetization in parts of the southern highlands of Mars and weak or absent magnetization in the northern lowlands. The only plausible cause for its acquisition is the existence of a strong global field generated by an early dynamo that is now extinct. From the magnetization associated with large dated impact basins (or its absence) it has been estimated that the dynamo ceased to operate 4.1 billion years ago (Lillis, Frey, & Manga, 2008).

## Jupiter

The detection of Jupiter’s global magnetic field predated the planet’s exploration by spacecraft. It has been inferred from the observation of emissions of decameter radio waves. These are generated by energetic electrons that gyrate around the field lines of Jupiter’s magnetic field and emit synchrotron radiation (Barrow & Carr, 1992). Later, observations from the flybys of the Pioneer and Voyager spacecrafts allowed the derivation of the magnetic field structure up to spherical harmonic degree three or four. Jupiter’s field is about ten times stronger at the surface than Earth’s field, but the structure at the largest scales is similar. The dipole tilt is around ten degrees, and the ratio of the quadrupole and octupole components to the dominant dipole is not dissimilar for both planets. Since 2016 the Juno mission is orbiting Jupiter on an elliptical polar orbit that brings the spacecraft to 0.06 Jupiter radii above the surface during closest approaches. A magnetic field model that is considered to be reliable up to harmonic degree 10 has been derived from magnetometer measurements of the first nine orbits (Connerney et al., 2018). It shows strong magnetic flux concentrations superimposed on the global dipole field; however, in contrast to the geomagnetic field at the core-mantle boundary, they are few and restricted to northern and midlatitudes.

## Saturn

Saturn’s magnetic field has been probed during flybys of Pioneer 11 and the two Voyager spacecraft, and between 2005 and 2017 by the Cassini orbiter. It is slightly weaker at the surface than Earth’s field. The dipole tilt is indistinguishable from zero, with an upper limit of 0.06° (Cao et al., 2011). Furthermore, only zonal (spin-axisymmetric) quadrupole and octupole components are needed in addition to the axial dipole to fit the measurements. The lack of non-axisymmetric magnetic field components defies Cowling’s theorem, which states that a dynamo cannot generate a purely axisymmetric field. Furthermore, it prevents us from determining the precise rotation rate of the planet. Tracking of clouds does not provide a unique rotation rate because of strong latitude-dependent jet streams. In the case of Jupiter, the rotation of the tilted magnetic dipole represents the bulk rotation rate of Jupiter’s interior, because in electrical conducting regions magnetic forces prevent strong differential motions. The lack of detectable non-axisymmetric field components at Saturn means that the rotation rate remains uncertain within 1–2%, which has implications for models of the internal structure.

## Uranus and Neptune

These two planets can be dealt with jointly, because their magnetic fields are similar, but distinct from those of other planets. Uranus’s and Neptune’s fields have been characterized during a single flyby by Voyager 2 at each of these planets, and uncertainties remain concerning structural details. However, while the surface field strength is comparable to that of Earth, the geometry is clearly different. The dipole axis is strongly inclined with respect to the rotation axis and quadrupole and probably octupole contributions are comparable in magnitude to the dipole at the surface (Holme & Bloxham, 1996). At the probable radius of the top of the dynamo region, the quadrupole and octupole fields are stronger than the dipole field (Table 1). While all other dynamo-generated planetary magnetic fields in the solar system are dipole-dominated, those of Uranus and Neptune are multipolar.

## Ganymede

Jupiter’s largest moon, Ganymede, is the only satellite in the solar system for which a global field with a probable dynamo origin has been found. Flybys of the Galileo spacecraft showed that the field is dipolar with a tilt of the dipole axis of approximately 4° and a surface strength of 1,000 nT (Kivelson et al., 1996, Kivelson, Khurana, & Volwerk, 2002). The quadrupole contribution could not be constrained unambiguously because of the limited amount of data and uncertainties about an induced magnetic field component, but it is in any case weak compared to the dipole.

## Induced Fields at the Galilean Satellites

The Galilean satellites Europa, Ganymede, and Callisto consist of water (ice) in their outer layers. They orbit inside Jupiter’s magnetosphere and are permeated by Jupiter’s magnetic field, which, because of the tilt of Jupiter’s dipole relative to the spin axis, varies at the place of the satellite periodically with Jupiter’s synodic rotation rate. In a conducting interior of the satellite the time-variable part of Jupiter’s field induces electrical currents. According to Lenz’s rule the magnetic field of these currents opposes the change of the external field. During flybys at Europa and Callisto, the Galileo orbiter found clear evidence for induced magnetic fields (Saur, Neubauer, & Glassmeier, 2010). Because (salty) water is a much better electrical conductor than solid ice, the presence of liquid water oceans beneath the outer ice shell has been inferred. At Ganymede an induced field component could not be identified unambiguously because of the superposition with the intrinsic magnetic field, but it appears very likely based on other observations (Saur et al., 2015).

## Small Solar System Bodies

Meteorites, which are mostly fragments of larger objects in the asteroid belt, have been analyzed with paleomagnetic methods. In many cases the existence of a primary remanent magnetization, that is, one acquired on the parent body of the meteorite and indicative of magnetic fields in the early solar system, is doubtful. However, a primary magnetization is well established for a few meteorites (Weiss, Gattacceca, Stanley, Rochette, & Christensen, 2010). A likely interpretation is that they represent pieces of the crust of protoplanets of sufficient size (> 100 km) that formed a metallic core. A dynamo could have operated in the core until it solidified, probably within some tens of million years after formation.

## Synopsis

Space missions have characterized the magnetic properties of all large solar system bodies. Many have a global magnetic field of internal origin and for others there are strong indications that a global field existed in the past. Existing fields differ vastly in strength. Concerning their morphology (Figure 3) they can be grouped into three classes: (1) dipole-dominated fields, with moderate tilt of the dipole axis relative to the planet’s spin axis and moderate contributions of non-axisymmetric multipole components (Earth and Jupiter fall into this class); (2) dipole-dominated magnetic fields with small dipole tilt and very weak non-axisymmetric field contributions in general (Saturn, Mercury, and possibly Ganymede are members of this group); and (3) multipole-dominated dynamos with large dipole tilts (Uranus and Neptune).

# Structure and Energy Budget of Planets

In this section the internal structure and the energy budget of planetary interiors are discussed as far as they are relevant for the operation of a dynamo. We must distinguish between the rocky (terrestrial) planets of the inner solar system and the outer planets. Both types of planets can host dynamos, although their structures and their energetics are different. A schematic overview of planetary internal structures is given in Figure 4.

## Earth

The Earth’s interior structure is known from seismology and serves as the prototype for the terrestrial planets. Below the solid silicate mantle there is a mostly liquid core. A small innermost part of the core is solid. From its seismic properties and from cosmo-chemical element abundances the core must consist mainly of iron plus some nickel. The density in the outer core is slightly less than that of pure iron-nickel and must contain ≈ 10% of light chemical elements. Silicon, oxygen, and sulfur are the top candidates and probably all contribute to some degree. In the solid inner core the light elements are more depleted than in the liquid outer core. It is believed that the inner core has grown in the cooling Earth, probably over the past 0.5 billion–2 billion years, by freezing of iron onto its outer surface. Because the melting temperature in the Earth’s core increases more steeply with pressure (depth) than the adiabatic temperature profile established in a convecting fluid, the core freezes from the bottom upward rather than from the top down.

The best understood mechanism for powering a dynamo in planets is convection, which is directly or indirectly driven by the internal heat. The total internal heat flow at the Earth’s surface is 46 TW. Roughly half of it is balanced by the heat generated from the decay of radioisotopes in the silicate part of the Earth, the remainder resulting from whole Earth cooling. How much of the Earth’s heat flow comes from the core is rather uncertain. Recent estimates that are based on different lines of evidence mostly fall into the range of 7–15 TW (Lay, Hernlund, & Buffett, 2008, Olson, 2016). A small amount of radioactive potassium may be present in the core, but the majority of the core heat must be due to cooling, which includes the latent heat of freezing of the inner core. The heat loss from the core is regulated by the slow solid-state convection in the mantle. The core, which convects vigorously in comparison to the mantle and which is thermally well-mixed, delivers as much heat as the mantle is able to carry away.

Figure 4. Interior structure of planets with active or extinct dynamos. Relative sizes are not drawn to scale. Gradients in shading indicate a hypothetical gradient in composition.

Liquid metal is a good thermal conductor. Based on quantum-mechanical calculations, estimates for the thermal conductivity of iron at core pressures and temperatures have recently been revised upward by a factor of two or three (Pozzo, Davies, Gubbins, & Alfè, 2012). The heat flux that can be transported by conduction along an adiabatic temperature gradient is called the “adiabatic heat flow.” For high values of the thermal conductivity, the adiabatic heat flow at the top of Earth’s core can be a large fraction of the actual heat flow or may even exceed it. For thermal convection to occur, the actual heat flow must be larger than the adiabatic heat flow. If the heat flow at the core-mantle boundary is less, at least the top layers of the core would be thermally stable, whereas the deeper parts would be convecting. This is because the latent heat from inner-core solidification is an effective heat source, which contributes strongly to the heat budget of the core. For geometrical reasons the heat flux per unit area that originates at the inner-core boundary decreases with radius as $r−2$ in the fluid core. The adiabatic gradient depends on gravity g, which drops to zero at the center. Because the actual heat flux decreases with $r−2$ and the adiabatic heat flux increases approximately proportional to r, the top layer of the core may be thermally stable whereas the deeper part of the fluid core is convectively unstable. Such a scenario may be more important in Mercury’s core than in the Earth. A second effect of inner-core growth is that the light elements in the outer core are preferentially rejected when iron freezes onto the inner core. Hence they become concentrated in the residual fluid near the inner core boundary and reduce its density. This layering is gravitationally unstable and leads to compositional convection, which homogenizes the light elements in the bulk of the fluid core. Compositional convection is estimated to contribute even more than thermal convection to the driving of the present geodynamo.

A serious problem associated with high thermal conductivity in the core, called the “new core paradox” by Olson (2013), may exist for the time before an inner core nucleated, that is, the first 2.5 billion–4 billion years of Earth’s history. Neither the latent heat of inner-core freezing nor the associated compositional convection would be available at that time and unless the cooling rate of the core had been larger than estimated for today, it would not convect. However, the geomagnetic field existed during most of Earth’s history. As a possible way out it has been suggested that mantle components (magnesium silicates) dissolved in appreciable amounts in liquid iron at the high temperatures during formation of the Earth. Since then, as the Earth cools, they exsolve gradually near the top of the core (O’Rourke & Stevenson, 2016; Badro, Siebert, & Nimmo, 2016). The loss of this component enhances the density of the residual fluid and causes a different form of compositional convection, driven from above.

## Other Terrestrial Planets and Ganymede

For terrestrial planets other than Earth the internal structure and thermal budget are difficult to constrain in a detailed and unambiguous way. Inferences on their interior can be drawn from geodetic data (shape, gravity field, rotational state, reaction to tides), the composition of surface rocks, and to some extent magnetic properties (Sohl & Schubert, 2015). These data strongly suggests that all terrestrial planets are differentiated into crust, mantle, and core. The reaction of Mars to solar tides indicates that the core must be at least partially liquid. The observation of forced librations, that is, the slightly uneven rotation under the influence of a solar torque, shows the same for Mercury. Because of Mercury’s high mean density its core must be very large in relation to the size of the planet, about 80% in terms of radius. The Earth’s moon is at the opposite end with a core radius of about 20%. A poorly constrained parameter in structural models is the amount of light elements in the cores of the planets.

Ganymede can be considered as a terrestrial planet with an additional thick ice shell above the deeper layers of rock and metal. Geodetic data do not allow an unambiguous determination of whether rock and metal have separated, but the existence of an internal magnetic field is taken as proof that a liquid iron core must exist.

No direct evidence on the existence or nonexistence of a solid inner core is available for any planet other than Earth. The possible absence of an inner core could explain why Venus and Mars do not have an active dynamo. On Earth mantle convection reaches the surface in the form of plate tectonics, which is an efficient mode of removing heat from the interior. None of the other rocky planets have plate tectonics. Without plate tectonics the heat flow is expected to be significantly lower, not only at the surface, but also at the top of the core, where it is probably sub-adiabatic. If no inner core exists to provide latent heat, the entire core will be sub-adiabatic and compositional convection is unavailable to drive a dynamo. The slower cooling of the planetary interior in the absence of plate tectonics fits with the idea that an inner core has not (yet) nucleated in Mars or Venus. Early in the Martian history the cooling rate was probably much higher and the associated core heat flow large enough for thermal convection to drive a dynamo. The demise of the dynamo must have occurred when the declining heat flow dropped below the conductive threshold.

The cores of some planets could be in a regime different from that of the Earth’s core, where crystallization of iron starts from the center. The controlling factor is the steepness of the melting point gradient relative to the adiabatic temperature gradient, which depends on pressure and on the nature and concentration of the light alloying elements. In particular sulfur strongly reduces the melting temperature gradient at the moderate pressures that are relevant for the cores of smaller terrestrial planets. As a consequence, crystallization of iron starts at the core-mantle boundary. Different scenarios are conceivable once core crystallization has started in such a planet (Hauck, Aurnou, & Dombard, 2006). A particular interesting one that could be relevant for Ganymede and perhaps for Mercury is an “iron snow” regime.

Aside from convection, flows driven inside planetary cores by tides, precession of the spin axis or libration, that is, the slightly uneven rotation caused by external torques, have also been considered for driving a dynamo. Such mechanisms draw on rotational energy, which is in principle available in sufficient amounts to sustain a dynamo over geological time. The laminar flow response to the periodic forcing is too weak for driving a dynamo; however, instability of the primary flow can possibly lead to strong turbulent motions (Le Bars, 2016). While the conditions for instability are probably not satisfied for present-day Earth, they may well be in other celestial objects and they may have more generally been met in the early history of the solar system. Open questions are whether the saturation of the turbulent flow occurs at a sufficiently high level for driving a dynamo and whether the strength and structure of the generated magnetic fields agree with the observed fields.

# Planetary Dynamos: Some Basics

The basic physical concept for describing planetary dynamos is that of magnetohydrodynamic flow in a rotating spherical shell combined with the associated magnetic induction effects. Here mainly convection-driven flow is considered. The general principles of magnetic field generation by a dynamo in the context of the flow dynamics in planetary interiors are described in Jones (2011). Here a brief overview is given on the necessary requirements for planetary dynamos and specific conditions for the flow in planetary cores.

When a metallic fluid moves in a preexisting magnetic field, electrical currents are induced. In a self-sustained dynamo the magnetic field associated with the induced currents has a sufficient strength and a suitable geometry so that it can replace the field that is necessary for the induction process in the first place, making an external field superfluous. Under which conditions can this work? Consider as a model for a planetary dynamo a fluid-filled spherical shell of thickness D with electrical conductivity σ‎. In order to have a self-sustained dynamo, the fluid must move with a sufficiently large velocity U, so that the magnetic Reynolds number Rm = UD/λ‎ exceeds a critical value Rmcrit. Here λ‎ = 1/(µoσ‎) is the magnetic diffusivity, with µo the magnetic permeability. The condition of suitable geometry of the magnetic field is achieved in a technical dynamo by the complex arrangement of wires that guide the electrical currents. In contrast, a planetary core represents an unstructured volume of conducting fluid. For such a homogeneous dynamo to work, the flow pattern must have a certain complexity. For example, simple differential rotation alone cannot support a dynamo. In particular helical (corkscrew-type) motion with a large-scale order in the distribution of right-handed and left-handed helices is very suitable.

Figure 5. Columnar convection in a rotating spherical shell near onset. The inner core tangent cylinder is shown by broken lines; Ω‎ indicates the direction of rotation, g that of gravity, grey arrows indicate the sense of fluid motion (Christensen, 2011; reproduced with permission of Cambridge University Press).

The Coriolis force plays a significant role for the fluid motion inside planets and facilitates helical motion. For this kind of flow, self-sustained dynamo action is possible above ≈40−50 (Christensen & Aubert, 2006). For the geodynamo, Rm is estimated to be of the order 1,000, well above the critical limit.

The thinking on planetary dynamos has been shaped by the theory for the onset of convection and by theoretical arguments on the dominant force balance for the flow in rotating planets (Jones, 2015). In a convecting planetary core, the Coriolis force, the electromagnetic Lorentz force, and the buoyancy force due to thermal or compositional differences are believed to be dominant (in addition to pressure forces) and to balance to first order. This has been termed a MAC balance (from Magnetic, Archimedean, Coriolis), in which inertial forces and viscous forces are insignificant. Ignoring also magnetic forces, the primary balance is between the pressure gradient force and the Coriolis force (geostrophic balance), as in the case of large- scale weather systems in the Earth’s atmosphere. In a deep fluid layer then the Proudman-Taylor theorem, which states that the flow velocity is invariable in the direction of the rotation axis, holds approximately. As a consequence, convection in a rotating spherical shell such as the Earth’s outer core starts in the form of columns aligned with the rotation axis (Figure 5). The columns surround the inner-core tangent cylinder like pins in a roller bearing (the tangent cylinder is coaxial with the rotation axis and touches the inner core at the equator). The primary circulation is around the axes of these columns. In addition there is a net flow along the column axes, which diverges from the equatorial plane in anticyclonic vortices and converges toward the equatorial plane in columns with a cyclonic sense of rotation, where “cyclonic” means anticlockwise when viewed from the north. The combination implies a coherently negative (left-handed) flow helicity in the Northern Hemisphere and positive helicity in the Southern Hemisphere. When the motion becomes more vigorous for highly supercritical convection and when a strong magnetic field is generated, the flow becomes less well structured and the Lorentz force modifies the flow. However, as long as the Coriolis force plays an essential role, the flow pattern remains anisotropic, that is, elongated along the direction of the rotation axis, and shows a statistical segregation of helicity between the two hemispheres. These properties of the flow are the basis for the preferred alignment of the magnetic dipole axis with the rotation axis at many planets.

# Numerical Models of Planetary Dynamos

Numerical modeling of planetary dynamos came of age around 1995 with several milestone papers (Glatzmaier & Roberts, 1995; Kageyama & Sato, 1995; Kuang & Bloxham, 1997). Since then direct numerical simulations have become a standard tool for exploring the principles of dynamos, for explaining properties of the geomagnetic field, and for testing hypotheses on the origin of specific magnetic field structures of other planets. Numerical modeling is complemented by laboratory experiments on rotating convection or on magnetohydrodynamic flow, using for example liquid sodium as working fluid (Olson, 2011). In several experiments self-sustained dynamos have been realized. On the one hand, so far dynamo experiments have been more remote from conditions in planetary cores than numerical simulations in several respects, such as geometry and the absence of bulk rotation. On the other hand, in experiments a more turbulent regime can be reached than in simulations, which is more realistic for flow in planetary cores. Laboratory experiments can elucidate fundamental physical processes operating in natural dynamos and help to clarify to what extent results from simulations can be extrapolated to the relevant parameter regime in planets.

## General Model Setup and Parameters

Most modern planetary dynamo models are direct numerical simulations of the equations for convection-driven magnetohydrodynamic flow in a rotating spherical shell and of the magnetic induction equation (Christensen, 2015). In the iron cores of terrestrial planets the variability of density due to pressure effects is small, and the models assume incompressible flow. In the dynamo regions of gas planets, the density can vary by a factor of a few (much more when the outer poorly conducting layers are also included in the model), and some form of compressible flow equation is appropriate. It is not possible to run the simulations at the actual planetary values for some of the dimensionless control parameters, most notably the Ekman number $Ek=ν/(ΩD2)$, which is a measure of the ratio of viscous forces to Coriolis forces (Ω‎ is the rotation frequency), and the magnetic Prandtl number P m = ν‎/λ‎, the ratio between the kinematic viscosity ν‎ and the magnetic diffusivity. Planetary values are approximately $Ek=10−15$ and $Pm=10−6$, which contrasts with model values of $Ek=10−4−10−6$ and $Pm≈1$. Physically, viscous forces play a too large role in the models.

Model results, such as the characteristic flow velocity and characteristic field strength B, can be expressed in terms of (diagnostic) non-dimensional parameters. The magnetic Reynolds number is such a parameter. Geodynamo models can match the value of Rm ≈ 1,000 inferred for the Earth’s core. A non-dimensional measure for the magnetic field strength (or for the ratio of Lorentz forces to the Coriolis force) is the Elsasser number Λ‎, which is estimated to be of order one in the Earth’s core and also assumes such values in the numerical models. The agreement in Rm and Λ‎ gives some credibility to the relevance of the dynamo models despite the strong mismatch in some of the control parameters.

## Gas Planets

Jupiter and Saturn are similar in composition to the Sun (Guillot & Gautier, 2015). Shells of a hydrogen-helium mixture surround a small rocky core. In the outer envelope, where hydrogen forms H2 molecules, the electrical conductivity is poor. At high pressure, hydrogen becomes a metallic liquid with free electrons. Shock-wave experiments and quantum-mechanical calculations show that there is no first-order phase transition, but the electrical conductivity rises gradually and assumes metallic values at around 1.3 Mbar pressure (Nellis, Weir, & Mitchell, 1999; Lorenzen, Holst, & Redmer, 2011). This is reached at a depth corresponding to 84% of Jupiter’s radius and 62% of Saturn’s.

Uranus and Neptune also have an envelope rich in hydrogen and helium, but the bulk of their mass consists of a water-rich mixture of H2O, NH3 and CH4, termed “ices” in planetology, even when in a fluid state (Guillot & Gautier, 2015). The ice layer extends to approximately 75% of the radius. It has ionic electrical conductivity, which is two orders of magnitude lower than metallic conductivity, but seems sufficient for sustaining a dynamo.

The internal heat flow of the gas planets has been determined by monitoring their infrared luminosity in excess of the re-emission of absorbed sunlight. On Jupiter and Saturn the internal heat flux is comparable to the solar flux hitting the atmosphere. The results of simple evolution models agree with the observed luminosity of Jupiter, but underpredict it in the case of Saturn and overpredict it for Neptune and in particular for Uranus. The He/H-ratio in Saturn’s atmosphere is less than the solar ratio. Stevenson (1980) proposed that helium becomes immiscible with hydrogen in the upper part of the metallic layer in Saturn, resulting in a downward segregation in the form of a “helium rain.” The gravitational energy released by the ongoing internal differentiation increases the luminosity. The radially varying helium depletion in the metallic shell results in a stable compositional stratification that suppresses convection. It is unclear if helium rains down all the way to the rocky core, but possibly at higher pressure and temperature He becomes miscible again and the droplets dissolve, resulting in a deep homogeneous metallic H-He layer. For Uranus and Neptune it has been suggested that stratification in deeper parts of the ice layers inhibits convection and explains the reduced ability of these planets to loose internal heat.

## Generic and Earth Dynamo Models

Many published dynamo models show a magnetic field that is strongly dominated by the axial dipole on the outer boundary. Other numerical dynamos have a spatially complex multipolar field in which the dipole contribution is small and frequently changes polarity. Figure 6 shows the radial magnetic field at the dynamo surface for examples of the two types. The analysis of a large number of numerical models with different values of the control parameters suggested that when the ratio of inertial forces acting on the flow to the Coriolis forces exceeds a certain threshold, the magnetic field geometry changes from dipolar to multipolar (Christensen, 2010). Basically, in a geodynamo-like setting, rapid rotation favors a dipolar magnetic field and slow rotation favors a multipolar one. Dynamo models with a dipolar field geometry that show occasional dipole reversals are often found close to the transition region in parameter space between dipolar and multipolar dynamos. However, it is uncertain if inertial forces, which are usually assumed to be insignificant in planetary cores, can play a role for dipole reversals.

The magnetic field geometry in Figure 6a shows, aside from the dipole dominance, some further traits that are similar to the field structure at the Earth’s core-mantle boundary (compare with Figure 2b). Among these are magnetic flux concentrations at high northern and southern latitudes, but offset from the poles, at nearly matching longitudes in the two hemispheres.

Figure 6. Snapshots of the radial field component on the outer boundary of numerical dynamo models, filtered to spherical harmonic degree n ≤ 13. Model with a dipolar field in (a), model with multipolar field in (b). The magnetic Reynolds number is 900 in both cases, but inertial forces have a stronger influence on the flow relative to the Coriolis force in (b).

These flux patches are located close to the boundary of the inner core tangent cylinder. Comparison with the flow pattern in the models shows that they are associated with downwelling flow in convection columns with a cyclonic sense of rotation that stretches through the core parallel to the rotation axis (compare with Figure 5). Another similarity between Figure 6a and Figure 2b is the low magnetic flux in the vicinity of the rotation poles, where a pure axial dipole field would be strongest. The models show that this is associated with rising plumes close to the rotation axis that disperse magnetic field lines at the outer boundary.

While the agreement of the model magnetic field with structures of the Earth’s field is encouraging, it has been disputed whether the models capture the correct dynamical regime of the Earth’s core. In particular, it has been argued that viscous forces play a dominant role in the models in contrast to their negligible influence in the Earth (e.g., King & Buffett, 2013). Recent work that explicitly monitored the magnitude of the various forces suggested that a balance between magnetic, Lorentz, and buoyancy forces, which is expected for planetary cores, is approached in the more advanced contemporary models (Yadav, Gastine, Christensen, Wolk, & Poppenhaeger, 2016; Aubert, Gastine, & Fournier, 2017; Schaeffer, Jault, Nataf, & Fournier, 2017).

Geodynamo-like models covering a wide parameter range have been employed to test scaling laws that relate the magnetic field strength to fundamental parameters such as electrical conductivity, size of the dynamo, rotation rate, and driving energy flux (Christensen, 2010). Since no dynamo simulation can be performed at actual planetary parameters, scaling laws are important to relate model results quantitatively to planetary properties. For some time the “Elsasser number rule,” which holds that in a rapidly rotating planet the magnetic field strength settles at such a value that Lorentz forces and Coriolis forces balance, was popular (Stevenson, 1979). It predicts for the characteristic magnetic field strength inside the dynamo $B∝σΩ$, with Ω‎ the rotation rate and σ‎ electrical conductivity. While this rule gives plausible values for the magnetic field inside the Earth or Jupiter, it is not supported by the results of dynamo simulations. They rather suggest that B depends on the available energy flux per unit area q, according to $B∝ρ1/6(Fq)1/3$, with little influence of rotation rate or conductivity (Christensen, 2010). Here ρ‎ is density and F a thermodynamic efficiency factor for conversion of heat into other forms of energy. This rule likewise gives reasonable values for the magnetic fields of Earth and Jupiter. Furthermore, its predictions agree with observed field strengths of rapidly rotating low-mass stars, which also host convection-driven dynamos (Christensen, Holzwarth, & Reiners, 2009). This finding strengthens the case for a general applicability of the rule.

Figure 7. Comparison of field strength predicted by a power-based scaling law (solid line) with observationally inferred magnetic energy inside the dynamo region (scale on left y-axis) and associated mean field strength Bsd at the surface of the dynamo (right y-axis). E = Earth, J = Jupiter, S = Saturn, S* = Saturn assuming a deep-seated dynamo, U = Uranus, N = Neptune.

The scaling laws predict magnetic field strength inside the dynamo. A comparison with the field observed at the surface of the planet is not straightforward. Since the dipole field decreases with $r−3$, a good estimate for the depth of the dynamo inside the planet is needed. In addition, the field strength on the top of the dynamo is probably weaker than the strength in its interior. Geodynamo-like models suggest a factor around 3–4. Furthermore, for comparing with the power-based scaling law, an estimate on the energy flux and of the efficiency factor F is necessary. Estimates for the heat flow from the Earth’s core are uncertain within 50%. The internal heat flux of the gas planets is well known (except for Uranus). For Mercury and Ganymede we do not have observational constraints on the heat flux in their cores. Therefore these planets are omitted from Figure 7, which shows a comparison of the field strength inferred from observation with the prediction of the power-based scaling law. The predictions agree well for Earth and Jupiter, moderately well for Neptune, and within large uncertainties for Uranus. Saturn’s magnetic field is substantially overpredicted when we assume that the dynamo surface coincides with the radius where hydrogen becomes a metallic conductor (S in Figure 7). S* shows the result for an alternative Saturn model in which the dynamo operates at greater depth.

## Mercury’s Dynamo

Challenges for a Mercury dynamo model are that it must explain the relative weakness of the magnetic field, the high degree of axisymmetry including the small dipole tilt, and the large quadrupole contribution. The energy flux in Mercury’s core is very uncertain. However, trying to explain the low field amplitude simply by very weak driving of convection is not viable. It would imply such sluggish flow that the magnetic Reynolds number remained below critical for dynamo action. Various non-dynamo or “exotic dynamo” models have been proposed as explanation for the weakness of the field: a large-scale coherent magnetization of Mercury’s crust (Aharonson, Zuber, & Solomon, 2004), a “thermoelectric dynamo” (Stevenson, 1987), and a dynamo that is strongly damped by a negative feedback from the induced magnetospheric field (Heyner et al., 2011). These models either require special assumptions to hold, or have difficulties reproducing the geometry of the field. In other models it has been proposed that Mercury’s dynamo operates in a thin liquid shell above a large solid inner core (Stanley, Bloxham, & Hutchison, 2005; Takahashi & Matsushima, 2006). The magnetic fields in these models are weaker than in Earth-like models, but are still too strong or disagree with other observations, for example, the dipole dominance. Of all models, that by Christensen (2006) is arguably closest in agreement with our current knowledge of Mercury’s field. Based on thermal evolution models it assumes that the heat flow at Mercury’s core-mantle boundary is distinctly subadiabatic and the top of Mercury’s liquid core is thermally stable to great depth (see discussion in the section Structure and Energy Budget of Planets). The dynamo only operates in a deep shell surrounding a solid inner core, driven by the latent heat of inner core crystallization. Inside the convective shell the magnetic field is strong, but the externally observable dipole field is weak. Because of Mercury’s long rotation period of 56 days the dynamo generates a dominantly multipolar magnetic field, whose dipole component is weak in the first place. However, the multipole components fluctuate rapidly in time. As a consequence, they are attenuated by a skin effect in the stagnant but electrically conducting outer layer of the core. The weak axial dipole component varies much more slowly, therefore it can pass the stable layer and dominates the observable field above the core. As discussed for Saturn, non-axisymmetric field components are preferentially filtered out by the stable layer, which would explain the observed high degree of axisymmetry. A strong axial quadrupole (or dipole offset) is not a persistent property of this model, but some of the simulations in Christensen and Wicht (2008) show at least episodically a large quadrupole contribution. Models by Cao et al. (2014) suggest that higher heat flow in the equatorial regions of Mercury’s core mantle boundary could foster a persistent dipole offset.

## Models for the Ancient Dynamo of Mars

Models for the early dynamo of Mars have focused on explaining the dichotomy in the crustal magnetization, which is strongly concentrated in the southern hemisphere. A prime candidate is a hemispherical dynamo, where significant magnetic field is generated only in a single hemisphere. Such dynamos can be found even for homogeneous conditions, such as uniform heat flow at the core-mantle boundary (Landeau & Aubert, 2011), but it is unclear if this still holds for actual planetary parameter values. The crustal dichotomy (northern lowlands vs. southern highlands) makes it plausible that a hemispherical difference in mantle convection pattern and core-mantle boundary heat flow existed in the early Mars. Such a heat flow pattern favors a hemispherical dynamo (Stanley, Elkins-Tanton, Zuber, & Parmentier, 2008). However, the hemispherical dynamo shows a tendency for frequent polarity reversals on a 10,000 year timescale (Dietrich & Wicht, 2013). It is unclear if this is compatible with a strong and coherent (in direction) magnetization of significant parts of the Martian crust, which is required to produce the observed high local field intensities. An alternative explanation for the dichotomy is that processes like large impacts or thermal alteration after the cessation of the dynamo removed the magnetization from parts of the Martian crust (e.g., Milbury & Schubert, 2010).

## Jupiter Dynamo Models

At first glance, Jupiter’s dynamo seems to be similar to the geodynamo from the likeness of the large-scale magnetic field geometries. The higher field strength on Jupiter is readily explained by the higher internal energy flux. However, on closer inspection important differences exist, both in the magnetic field structure at smaller scale and concerning the internal structure of the dynamo region. In the Earth there is a distinct boundary between the highly conducting fluid core with approximately constant density and the poorly conducting solid mantle. In Jupiter the electrical conductivity rises gradually until it reaches metallic values, the density varies strongly with depth, and it is unclear if barriers exist for radial flow between different layers in the interior. At Jupiter’s surface strong jet streams are observed, alternating in east-west direction, with velocities up to 140 m/s. It has been debated whether these jets are confined to a shallow weather layer; however, recent gravity data from the Juno mission show that they must extend to substantial depth (Kaspi et al., 2018). The interaction of deep-reaching zonal winds with the dynamo is a matter of discussion. The surface winds cannot persist unabated down to the depth where conductivity becomes significant because the induced currents would imply excessive ohmic dissipation (Liu, Goldreich, & Stevenson, 2008). Numerical dynamo models for Jupiter that include compressible flow and radially variable conductivity including a poorly conducting outer layer (e.g., Jones, 2014) can produce a dipolar magnetic field and at the same time a strong eastward jet in equatorial regions. This jet is geostrophic, that is, the velocity is constant along a chord that stretches parallel to the rotation axis from the northern to the southern hemisphere. At low latitudes this is possible without reaching the depth where the conductivity becomes large. The jets observed at higher latitudes of Jupiter are not reproduced in these models. Cao and Stevenson (2017) employed a simplified model to estimate that magnetic field perturbations in the range of 0.1–1% are caused by the interaction of the zonal winds with the main dipole field. In the dynamo models of Gastine, Wicht, Duarte, Heimpel, and Becker (2014) and Duarte, Wicht, and Gastine (2018) the interaction of the equatorial jet with the dynamo field in the deeper, moderately conducting regions leads to the generation of isolated strong magnetic flux concentrations, somewhat similar to the “great blue spot” in Jupiter’s observed field at low latitude (Figure 3).

## Saturn’s Dynamo

The challenge in the case of Saturn is to explain the extreme degree of axisymmetry of the magnetic field. Furthermore, the observed field strength falls short of the expectation (S in Figure 7). A model that can explain both has been proposed by Stevenson (1980) and has so far passed the test of time. It relies on the existence of a stably stratified region in the upper part of the metallic hydrogen region. In this electrically conducting stratified layer horizontal flow, such as differential rotation, is possible but radial flow is strongly inhibited; therefore, a dynamo cannot operate here. The magnetic field is generated in the deep homogeneous metallic hydrogen region (Figure 4). This is somewhat akin to the model for Mercury’s dynamo but with the difference that Saturn is rotating rapidly, which favors the generation of a strong dipolar field in the first place. The thickness of the stable layer in Saturn is highly uncertain, but placing the outer boundary of the deep homogeneous region to 0.4 Saturn radii would bring the magnetic field strength at the top of the dynamo region in line with the expectation of the scaling law (S* in Figure 7). To understand how the stable layer eliminates non-axisymmetric components of the dynamo field, let us assume for simplicity that the entire layer rotates as a whole with respect to the underlying dynamo region and that the dynamo field is stationary on the rotation timescale. As seen by an observer who rotates with the shell, the axial dipole part as well as other axial multipole components remain stationary. However, non-axisymmetric components appear to oscillate with the rotation frequency. They are therefore damped by the skin effect. For differential rotation with a velocity comparable to predicted velocities of the convective flow a relatively thin stable layer of 1,000 km is sufficient to practically eliminate non-axisymmetric components in the outside field. It is not entirely clear what causes a differential rotation. One suggestion has been that the equator-to-pole differences in insolation drive a “thermal wind” in the stable layer. Even without differential rotation a skin effect in the stable layer can damp non-axisymmetric field components, although less efficiently, provided they have a tendency to fluctuate rapidly in time with a zero mean, in contrast to axial components that vary more slowly (Christensen, Kao, Dougherty, and Khurana, 2018). Dedicated numerical models for Saturn that include a stably stratified layer (Christensen & Wicht, 2008; Stanley, 2010) show a significant reduction of the dipole tilt compared to the case without such a layer. Because of the inability to run models at appropriate values of the magnetic Reynolds number, which is in the range $104–105$ in Saturn, they cannot reach tilt angles as small as the upper limit derived from observation.

## Dynamo Models for Uranus and Neptune

In the case of Uranus and Neptune it must be explained why their dynamos generate a multipolar field in contrast to the dipolar fields observed at all other planets. In numerical dynamo models multipolar geometries are not uncommon, but usually they are found in cases with comparatively weak influence of rotational forces, which is not expected for the two rapidly rotating planets. Special conditions in their interior structure may be essential. Stanley and Bloxham (2004) present a dynamo model with a thin convecting shell that surrounds a conducting, but convectively stable, fluid core region. Some of their models generate magnetic fields that agree well with the spectral power distribution in the low-degree harmonic field components. Potentially also the lower electrical conductivity in the interior of Uranus and Neptune, compared to that of other planetary dynamos, could play a role (G´omez-P´erez & Heimpel, 2007). However, a widely accepted explanation for the magnetic field structure of Uranus and Neptune is still missing.

## Lunar Dynamo

Although it is plausible that a convection-driven dynamo has operated in the Moon’s metallic core early in the lunar history, the rather high inferred strength and the long persistence of the magnetic field that magnetized the Apollo samples is difficult to understand. The small size of the lunar core, whose radius is at most 25% of the lunar radius, implies a much stronger decrease of field strength (proportional to $r−3$) from the core to the surface than in case of the Earth. Non-core dynamo explanations have been put forward, such as magnetization by short-lived fields generated in the plasma clouds produced by basin-forming impacts, but have been largely discarded. As an alternative to convection, a dynamo powered by mechanical stirring, either by Earth-driven precession of the lunar spin axis (Dwyer, Stevenson, & Nimmo, 2011) or as consequence of a re-orientation of the axis after very large impacts (LeBars, Wieczorek, Karatekin, Cébron, & Laneuville, 2011), has been proposed.

## Ganymede Dynamo Models

Provided that Ganymede’s iron core contains a sufficient amount of sulfur, the “iron snow” regime is a plausible scenario for driving a dynamo. Once the temperature has dropped to the liquidus at the top of the core, a snow-forming layer will grow downward in time. Flakes of solid iron form and sink due to their higher density. The layer becomes enriched in sulfur and is stably stratified. The iron flakes dissolve again at the bottom of the layer because of the higher temperature. The enrichment in iron by the influx from above drives compositional convection in the central homogeneous region of the core. Eventually, when the snow layer has grown to the center of the core, convection ceases and the sinking iron crystals accumulate in a growing solid inner core. According to thermal evolution models, the stage during which a dynamo can be active would last only several hundred million years (Ru¨ckriemen, Breuer, & Spohn, 2015). Hence at present Ganymede would be in a special era of its history. The stably stratified snow-forming layer renders the external magnetic field more axisymmetric than it would be otherwise, although in Ganymede the effect would be milder than in Mercury or Saturn. Dedicated numerical dynamo models for the snow-layer scenario (Christensen, 2015) give results in agreement with the observed strength and the limited information on the structure of Ganymede’s magnetic field.

# Perspectives

Ongoing and upcoming space missions will refine the knowledge on the magnetic fields of the solar system planets. Additional data from Juno will reveal finer-scale structures of Jupiter’s magnetic field and may allow the detection of variations of the internal field. The Bepi Colombo mission of the European Space Agency is scheduled to go into orbit around Mercury in 2025 and will measure the internal magnetic field in the so far unmapped southern hemisphere of the planet. The Juice mission is planned to enter into orbit around Ganymede in 2033, which will allow determination of the magnetic field structure in much more detail than was possible during the previous flybys. Looking beyond the solar system, there are prospects that magnetic fields of exoplanets will be detected by the synchrotron radiation emitted at radiofrequencies (Zarka, 2007). This might expand the sample size for planetary magnetic fields tremendously.

The new data on magnetic field properties, together with improvements in the understanding of structural properties of planetary interiors, will certainly foster theoretical and numerical efforts to explain the differences between dynamos in different planets. Systematic numerical studies that extend the accessible parameter space toward more realistic values, together with laboratory experiments on conducting rotating fluids, will clarify to what extent dynamo models capture the essential physics that controls the flow and magnetic field generation inside planets. Finally, convection-driven dynamos exist in many stars, and, while there are certainly differences, investigating the common ground between dynamos in the two classes of objects can also contribute to our understanding of how planetary magnetic fields are generated.

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