Icarus 196 (2008) 16–34 www.elsevier.com/locate/icarus Models of magnetic field generation in partly stable planetary cores: Applications to Mercury and Saturn Ulrich R. Christensen ∗, Johannes Wicht Max-Planck-Institut für Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany Received 14 August 2007; revised 18 February 2008 Available online 15 March 2008 Abstract A substantial part of Mercury’s iron core may be stably stratified because the temperature gradient is subadiabatic. A dynamo would operate only in a deep sublayer. We show that such a situation arises for a wide range of values for the heat flow and the sulfur content in the core. In Saturn the upper part of the metallic hydrogen core could be stably stratified because of helium depletion. The magnetic field is unusually weak in the case of Mercury and unusually axisymmetric at Saturn. We study numerical dynamo models in rotating spherical shells with a stable outer region. The control parameters are chosen such that the magnetic Reynolds number is in the range of expected Mercury values. Because of its slow rotation, Mercury may be in a regime where the dipole contribution to the internal magnetic field is weak. Most of our models are in this regime, where the dynamo field consists mainly of rapidly varying higher multipole components. They can hardly pass the stable conducting layer because of the skin effect. The weak low-degree components vary more slowly and control the structure of the field outside the core, whose strength matches the observed field strength at Mercury. In some models the axial dipole dominates at the planet’s surface and in others the axial quadrupole is dominant. Differential rotation in the stable layer, representing a thermal wind, is important for attenuating non-axisymmetric components in the exterior field. In some models that we relate to Saturn the axial dipole is intrinsically strong inside the dynamo. The surface field strength is much larger than in the other cases, but the stable layer eliminates non-axisymmetric modes. The Messenger and Bepi Colombo space missions can test our predictions that Mercury’s field is large-scaled, fairly axisymmetric, and shows no secular variations on the decadal time scale. © 2008 Elsevier Inc. All rights reserved. Keywords: Magnetic fields; Mercury; Saturn 1. Introduction chemical differentiation process that leads to an unstable strati- fication. Planetary magnetic fields are generated in a self-sustained In many planets the entire conducting fluid region may be dynamo process associated with the circulation of an electri- convecting, but this is not necessarily the case. The iron cores cally conducting fluid in the core of the planet (Stevenson, of Mars and Venus could be stably stratified and completely 2003). Different sources of the flow may be possible, but most stagnant because of a subadiabatic temperature gradient and be- commonly it is believed to be driven by convection. Aside cause compositional convection may be unavailable for lack of from thermal convection, compositionally driven flow may oc- an inner core. This would explain the absence of a global mag- cur, for example by the rejection of light alloying elements netic field at these planets. More interesting is the case when from a growing solid inner core in the Earth and perhaps in some parts of the core convect while others are stable. other terrestrial planets. A prerequiste for thermal convection In the case of the Earth a stably stratified region may exist is that the radial temperature gradient is steeper than the adia- at the top of its fluid core. Estimates for the heat flow at the batic gradient. Compositional convection requires an ongoing Earth’s core–mantle boundary usually exceed the flux that can be conducted along an adiabatic gradient in the core, however, a slightly subadiabatic heat-flow cannot be ruled out (Labrosse * Corresponding author. Fax: +49 5556 979 219. et al., 1997), so that a top layer may be thermally stable. Al- E-mail address: [email protected] (U.R. Christensen). ternatively, a layer more enriched in light elements compared 0019-1035/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2008.02.013 Magnetic field generation in Mercury and Saturn 17 to the bulk of the fluid core might have accumulated at the top thought to be indicative for a “magnetostrophic” balance of of the core (Braginsky, 1984; Lister and Buffett, 1998). So far Coriolis force and Lorentz force that presumably holds in plan- there is no compelling evidence that such a stable layer exists. If etary dynamos (Stevenson, 2003) and determines the internal it exists, its thickness may be small in comparison to that of the field intensity. Christensen and Aubert (2006) and Olson and convecting layer, implying that it would have limited influence Christensen (2006) suggested an alternative scaling law for the on the geodynamo. magnetic field strength and the dipole moment of planetary dy- Christensen (2006) suggested that in Mercury’s fluid core namos, based on the power that is available to balance ohmic the dynamo region is lying below a thick stable layer. Ther- dissipation. It leads to a relation between the Lorentz number mal evolution models for Mercury predict that the heat flow at B Lo = √ , (2) the core–mantle boundary is only a moderate fraction of the ρμΩD adiabatic heat flow, thus rendering the upper part of the core thermally stable. The evolution models also suggest that Mer- where D is the thickness of the fluid shell and μ is magnetic cury has nucleated a solid inner core. If the core contains some permeability, and a modified Rayleigh number that measures light element, most likely sulfur, inner core growth would lead the buoyancy flux available to drive the dynamo. The field to compositional convection in the deep parts of the fluid core strength predicted by this scaling law agrees well with the ob- that is augmented by thermal convection driven by the latent served field strengths for Earth and Jupiter. This is also the case heat of inner core freezing. In Section 2 we study what controls when applying the Elsasser number rule. the thickness of the unstable layer. Mercury’s field geometry has been characterized to a lim- Saturn is another candidate for a planet with a stably strati- ited degree by the measurements taken during two flybys of Mariner 10 in 1974/1975. During the first flyby Mariner 10 fied region at the top of its electrically conducting core. Saturn’s passed at low latitudes through the magnetotail and most of the core is composed of a mixture of hydrogen in a metallic state measured field is believed to be due to magnetospheric currents and of helium. Its upper boundary is at approximately half the (Connerney and Ness, 1988). The third flyby, where Mariner planetary radius. Under Saturn conditions helium is expected to 10 passed Mercury at 300 km above the surface at high North- become partly immiscible with hydrogen in some pressure in- ern latitudes, provided the best data for revealing the internal terval above the metal transition (Stevenson and Salpeter, 1977; magnetic field. The field seems to be large scaled and is per- Fortney and Hubbard, 2003). As a consequence, helium should haps dominated by a dipole component that is tilted slightly separate and sink as rain drops, depleting the upper layer and (14◦ ± 5◦) relative to the rotation axis (Ness, 1979). However, enriching the lower one, with a gradient zone at the top of the the measurements cannot discriminate between a dipolar and metallic region. The thickness of the stably stratified region a quadrupolar magnetic field (Connerney and Ness, 1988). All is uncertain and in the extreme case helium may be lost alto- field models in which the Gauss coefficients for the axial dipole gether from the hydrogen region and settle onto the rocky inner g0 and the axial quadrupole g0 satisfy the relation core (Fortney and Hubbard, 2003). In Jupiter temperatures are 1 2 0 + 0 =− higher than they are in Saturn, with the consequence that helium g1 1.52g2 320 nT (3) may be fully miscible. fit the Mariner 10 observations equally well (the field models In Uranus and Neptune an electrically conducting ionic fluid also involve non-axial dipole components and coefficients de- is assumed to extend to about 3/4 of the planetary radius. In scribing magnetospheric currents that are co-varied when the order to explain the low luminosity of these planets, Hubbard et 0 0 g2/g1 ratio is changed). al. (1995) proposed that only an outer shell of the ionic fluid is The enigmatic point about Mercury’s field is its weakness. convecting, whereas the deeper fluid layers are compositionally The mean strength at the surface is approximately 450 nT when stratified. Hence Uranus and Neptune may represent one class we take the field to be purely dipolar, i.e., it is hundred times of planets, where an unstable conducting fluid region lies above weaker than Earth’s surface field. Assuming the same value of a stable layer, and Mercury and Saturn may represent another the Elsasser number inside the dynamo regions of Earth and class, where the unstable region lies below a stable layer. Mercury we would expect a somewhat weaker field in the latter The magnetic fields of planets with active dynamos differ case because Mercury rotates ≈60 times slower than Earth. On substantially in terms of strength and geometry. The magnetic the other hand, Mercury’s core is larger relative to the size of fields of Earth and Jupiter might be considered as the “rule”: the planet, meaning that the geometric decrease in field strength they are dominated by the axial dipole, but the equatorial dipole from the core–mantle boundary to the surface is less than in and higher multipoles including non-axisymmetric parts make a case of the Earth.