Powering Ganymede s dynamo

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2012je004052, 2012 Powering Ganymede s dynamo X. Zhan 1 and G. Schubert 1 Received 12 January 2012; revised 3 July 2012; accepted 6 July 2012; published 22 August [1] It is generally believed that Ganymede s core is composed of an Fe-FeS alloy and that convective motions inside it are responsible for generating the satellite s magnetic field. Analysis of the melting behavior of Fe-FeS alloys at Ganymede s core pressures suggests that, besides the growth of a solid inner core, convection can be driven by two novel mechanisms: Fe snow and FeS flotation. To advance our understanding of magnetic field generation in Ganymede, we construct dynamo models in which deep inner core growth, Fe-snow and FeS flotation drive convection. Although a dynamo can be found in each of these cases, the dynamos have different characteristics. For example, some dynamos are dipole dominant and others are not. It is found that multipole-dominant magnetic fields are generated in all Fe-snow cases, while dipole dominant dynamos are found in FeS flotation cases and in inner core growth cases. Ganymede s present dipole-dominant magnetic field suggests that the Fe-snow process does not play a primary role in driving Ganymede s core convection. The reason that Fe-snow driven convection does not produce a dipole-dominant dynamo can be related to the buoyancy flux. In Fe-snow cases, the buoyancy source is located at the core-mantle boundary (CMB), and the buoyancy flux peaks there, while in the other two cases, the buoyancy source is located at the inner core boundary where the buoyancy flux peaks. Citation: Zhan, X., and G. Schubert (2012), Powering Ganymede s dynamo, J. Geophys. Res., 117,, doi: /2012je Introduction [2] Ganymede is the only known satellite in the solar system with an intrinsic magnetic field. The small value of the non-dimensional moment of inertia of Ganymede suggests a fully differentiated interior and the existence of a metallic core [Schubert et al., 2004]. Active dynamo action within the liquid part of Ganymede s core is believed to be the most plausible origin of its magnetic field [Schubert et al., 1996; Kivelson et al., 2002]. [3] Ganymede s core is likely to be composed of an Fe- FeS alloy. Other light elements could exist and alter the melting behavior of Ganymede s core alloy, but the Fe-FeS alloy is a reasonable first approximation, since sulfur, a highly siderophile element, is readily available in the protosolar system. Due to the unknown oxidation state of Ganymede s interior during differentiation, we are unable to constrain whether the core composition is on the Fe-rich or FeS-rich side of the eutectic composition. Experimental studies indicate that the eutectic melting temperature in the Fe-FeS system decreases with increasing pressure when pressures are less than 19 GPa [Fei et al., 1995, 1997, 2000]. 1 Department of Earth and Space Sciences University of California, Los Angeles, California, USA. Corresponding author: G. Schubert, Department of Earth and Space Sciences, University of California, 2810 Geology, 595 Charles E. Young Dr. E., Los Angeles, CA , USA. (schubert@ucla.edu) American Geophysical Union. All Rights Reserved /12/2012JE After thoroughly analyzing the complex melting behavior of Fe-FeS alloys at Ganymede s core pressures, Hauck et al. [2006] put forward two novel mechanisms to drive core convection: Fe snow and FeS flotation. Fe snow can occur for bulk core sulfur contents greater than 6%; in this case Fe first precipitates at Ganymede s core-mantle boundary, and then falls toward the center of the core, releasing gravitational energy to drive convection. FeS flotation occurs when the bulk core sulfur content is high (>22.5%); in this case solid FeS can form either at mid-core depths or deep near the center, and then buoyantly rise, generating fluid motions along the way. Of course, when the concentration of sulfur is rather small, Fe may first condense in the deep core, leading to solid inner core growth that can also drive convection. [4] Based on a thermal evolution model of Ganymede with high initial core and mantle temperatures, Bland et al. [2008] suggest that the dynamo process can only occur if the sulfur mass fraction in Ganymede s core is very low (<3%) or very high (>22.5%), because only the associated inner core growth (sulfur mass fraction <3%) or solidification of FeS deep within the core (sulfur mass fraction >22.5%) can significantly lower the power requirement for dynamo action so that the core power output exceeds the power required to drive the dynamo. For sulfur mass fractions between 3% and 22.5%, Fe snow can take place and drive convection, but the latent heat released by Fe precipitation at the top of the core is readily removed by the mantle above and is not available to power the dynamo; therefore the power output of the core falls short of driving the dynamo. The analysis of Bland et al. [2008] doesn t take 1of7

2 into account the satellite s internal structure, which still has a large uncertainty based on gravity data alone. By varying the size of the core and the volume of the mantle in a thermal evolution model, Kimura et al. [2009] demonstrate that a much wider range of sulfur contents can sustain a dynamo, which means that inner core growth, Fe snow and FeS flotation are all possible power sources of Ganymede s dynamo. [5] Many studies have been carried out to understand how the dynamics of a planet s interior affect its dynamo s dipole moment, the most important dynamo property that can be directly observed. Control parameters, which are derived from basic physical properties such as radius, rotation rate, viscosity, electrical conductivity and so on, play the most important role in determining whether a dynamo has a dominant dipole component. For dynamos heated from below, increasing the Rayleigh number could eventually lead to the transition from dipolar to multipolar states [Kutzner and Christensen, 2002]. The magnetic Prandtl number also can significantly affect the morphology of dynamo generated magnetic fields, i.e., when the magnetic Prandtl number is very large, dynamos tend to have a strong dipole moment. However, aside from control parameters, numerical simulations indicate that the convection driving mechanism is another important factor which can affect the relative strength of a dynamo s dipole with respect to its multipole components [Kutzner and Christensen, 2000, 2002; Olson and Christensen, 2006; Wicht and Tilgner, 2010]. Dipole dominance is usually observed in heated from below dynamos, whereas internally heated dynamos usually have systematically weaker dipole moments than their comparable counterparts and are mostly non-dipolar. There are also a few exceptions, such as the dipole dominant dynamos found by Aubert et al. [2009] in models with only internal heating. However, their models adopt a very small inner core size (close to zero) which might account for their dipolarity, and dynamo models with a very small inner core or without an inner core are more suitable for modeling planets in their early stage of evolution when the interior is fully convective. Olson and Christensen [2006] present several dipole dominant internally heated dynamos as well. The magnetic Prandtl number is in the range in their models. It is likely that the dipole dominant internally heated dynamos found by them are the result of relatively large magnetic Prandtl number. Generally, the magnetic Prandtl number found in planetary interiors is rather small. Therefore, it is reasonable to suggest that at small magnetic Prandtl number, dynamo action in spherical shells powered by internal heating tends to produce non-dipole dominant magnetic fields. [6] In Ganymede s core, when convection is driven by FeS flotation or inner core growth, buoyancy sources are located at the bottom of the convecting region, suggesting a dynamo similar to heated from below dynamos. Similarly, convection driven by Fe snow resembles internally heated convection, since Fe snow establishes a stable compositional gradient with its maximum attained at the core-mantle boundary, the same place where the temperature gradient in internally heated convection peaks. Therefore, it is expected that Ganymede s present dipole dominant magnetic field is not generated by the Fe snow process. [7] To advance our knowledge of the magnetic field generation process in Ganymede s interior, dynamo models are constructed in which deep inner core growth, Fe-snow and FeS flotation are adopted to drive convection. We explore the possible connections between the morphology of Ganymede s observed magnetic field and the convection driving modes: Fe or FeS precipitation. 2. Model Description [8] A series of simulations of the magnetic field generation process in Ganymede s core is performed. Considering that pure thermally driven convection is inefficient in powering a dynamo [Gubbins et al., 2003] and the heat flux out of Ganymede can be easily conducted by an adiabat [Hauck et al., 2006], we take compositional convection as the primary dynamo energy source. Compositional convection can be driven by Fe snow, FeS flotation or inner core growth. The latter two mechanisms are similar in that light materials are formed at the bottom boundary, then buoyantly rise, mixing the convecting region and generating convective motions. Therefore, FeS flotation is used as a proxy for the latter two convection driving mechanisms. [9] We consider a rotating, Boussinesq fluid spherical shell where convection and dynamo action occur. The shell has inner radius r i and outer radius r o, and the radius ratio is denoted by c = r i /r o. Although it is not clear whether there is a solid inner core in Ganymede, it is still appropriate to use a spherical shell to represent the dynamo source region, since in Ganymede s core the effectively convecting region has the shape of a spherical shell. For high enough bulk core sulfur contents (>6 wt %S), Fe snow precipitation can occur at a shallow level (near the CMB). The Fe-snow migrates deeper into the core because of its higher density. When we consider convection driven by Fe snow, the convecting region is chosen as follows: the top boundary is set to be where the Fe snow precipitation first takes place, while the bottom boundary is placed where the net buoyancy of Fe solids is zero. Because of the relatively higher density of Fe solids, they keep falling in the core till they remelt at a very deep level, thus the existence of the bottom boundary is ensured. Similarly, when the bulk core is on the FeS-rich side of the eutectic (>22.5 wt %S), solidification of FeS is possible. FeS solids could form either at mid-core depths or deep near the center of the core, then float up and remelt before reaching the CMB. For convection driven by FeS flotation, we choose the convecting region as follows: the bottom boundary is set to be where the FeS solidification first takes place, while the top boundary is located where the net buoyancy of FeS solids is zero. It is apparent that the top boundary does exist and should be at some level near the CMB. It is important to note that the convecting regions we choose are not the entire fluid core but only a part of it. This is a first approximation, since we only take into account convective motions produced by sinking Fe solids or floating FeS solids. Our model is different from the Mercury dynamo model presented by Vilim et al. [2010]. They consider deep iron snow cases in which Fe snow precipitation occurs at rather deep levels and two sources of compositional convection are taken into account, one from the Fe solids falling toward the planetary center, and another from the liberated sulfur floating toward the CMB. However, such 2of7

3 Figure 1. The solid curve represents the compositional gradient for the cases of inner core growth and FeS flotation; it decreases as r increases. The dashed curve is the compositional gradient for the Fe snow case; it is approximately proportional to r. two-layer convection leads to a rather weak surface magnetic field, which is not consistent with the observed magnetic field of Ganymede. In consequence, we only consider one convection layer, in which falling Fe solids or floating FeS solids are the convection driving source. [10] In the case of Ganymede s core, the buoyancy caused by differences in composition is significant in driving core convection. The sum of pressure and gravity forces can be written as f ¼ rp þ Cg; where P is the modified pressure which has absorbed centrifugal force and C is a quantity called the co-density [Braginsky and Roberts, 1995] defined by C ¼ a S S a j j: Here S and j are, respectively, the specific entropy and the mass fraction of light material (negative values indicate heavy materials, such as Fe), and a S and a j are entropic and compositional expansion coefficients. Since thermal convection likely plays a less important role in powering Ganymede s dynamo [Gubbins et al., 2003; Hauck et al., 2006], here we only consider compositionally driven convection which is thermodynamically more efficient than the thermal component of convection; the term a S S is discarded as the result of this assumption [Vilim et al., 2010]. [11] The mass fraction j satisfies the compositional transport equation [Kono and Roberts, 2001] j t þ u rj ¼ kj r 2 j þ s j ; where u is the fluid velocity, and k j is the chemical diffusivity. The source term s j represents the secular change in the bulk composition of the convecting layer on a much ð1þ ð2þ ð3þ longer timescale than the convection timescale, because a continuous net inward transfer of Fe (Fe snow) or net outward transfer of FeS (FeS flotation) will modify the mean composition of the convecting region on a long timescale. j can be written as j + j, where j is the basic state mass fraction and j is the deviation from it. When there is no convection, the basic state mass fraction j satisfies k j r 2 j þ s j ¼ 0: For Fe snow, we require that j/ r =0atr = r i, since the net buoyancy of Fe solids is zero there. For FeS flotation, we require that j/ r =0atr = r o, since the net buoyancy of FeS solids is zero there. Therefore the compositional gradient profile can be determined as ð4þ j sj ¼ r 3k j r r3 o r 2 ; for FeS flotation; ð5þ j sj ¼ r 3k j r r3 i r 2 ; for Fe snow: ð6þ As shown in Figure 1, the compositional gradient profile maintained by FeS flotation decreases rapidly as r increases. This property is similar to that of the temperature gradient profile established by a fixed temperature contrast between inner and outer boundaries. The compositional gradient profile maintained by Fe snow is nearly proportional to r, resembling the temperature gradient profile caused by internal heating or secular cooling. [12] Length, time, mass fraction and magnetic induction are scaled by d = r o r i, d 2 /n, Dj and (rmhw) 1/2, respectively, where d is the shell thickness, h is the magnetic diffusivity, m is the magnetic permeability, r is the density, W is 3of7

4 Table 1. Results of the Dynamo Computations E Ra Pm E mag Rm B dip f dip P/T Boundary Type FeS Flotation fixed co-density fixed co-density fixed co-density fixed co-density fixed co-density flux fixed co-density flux fixed co-density fixed co-density Fe Snow fixed co-density fixed co-density fixed co-density fixed co-density fixed co-density flux fixed co-density flux fixed co-density fixed co-density the rotation rate about the z-axis and Dj is the basic state mass fraction contrast between the inner and outer boundaries. The dimensionless governing equations for convection and dynamo action in the shell can then be written E u t þ ðu rþu r2 u þ 2^z u ¼ rpþra r j r o þ 1 ð Pm rb ÞB; ð7þ ru ¼ 0; j t þ u rj ¼ u rj þ 1 Pr r2 j ; B t ¼rðuBÞ 1 Pm rrb; rb ¼ 0; subject to boundary conditions ð8þ ð9þ ð10þ ð11þ u ¼ 0atr¼1= ð1 cþ; c= ð1 cþ; ð12þ j ¼ 0atr¼1= ð1 cþ; c= ð1 cþ; ð13þ B ¼ 0 at infinity;rb ¼ 0 when r > 1= ð1 cþ and r < c= ð1 cþ ð14þ Non-dimensional input parameters are defined as the radius ratio c = r i /r o, the Prandtl number Pr = n/k, the magnetic Prandtl number Pm = n/h, the Ekman number E = n/(wd 2 ) and the Rayleigh number a j g o Djd/nW. Both constant codensity and constant co-density flux boundary conditions are applied to the inner and outer boundaries. There are no significant differences in the models for either co-density boundary condition, since the compositional gradient profile is already determined for both the Fe snow and FeS flotation cases [Vilim et al., 2010]. The implementation of the boundary conditions for magnetic induction B is presented in Chan et al. [2007] and Zhan et al. [2011]. The equations are solved via a finite element method, which is quite different from normally used spectral methods. Our code is a modified version of that presented by Chan et al. [2007], and has been validated against the benchmark of Christensen et al. [2001]. 3. Results [13] Table 1 gives details on the 16 models we have integrated for this study. In all models the radius ratio c is fixed at 0.4. The parameter range is E =10 4 to 10 3 for the Ekman number, Pm =2 5for the magnetic Prandtl number, Ra varies up to 10 times the critical value and the Prandtl number Pr is set to 1 in all simulations. It should be kept in mind that due to computational limitations, and just like all accessible numerical dynamo simulations, our simulations operate in a parameter regime still very far from that of Ganymede s core. [14] We define several outputs, which are all averaged over times much longer than core flow R timescales, the 1 magnetic energy density E mag = ð2v s PmEÞ V s B 2 dv, the magnetic Reynolds number Rm = dn/h where v represents the rms velocity, the rms intensity of the dipole field on the outer boundary B dip, the ratio f dip of the dipole strength on the outer boundary to the rms amplitude of the magnetic field on the outer boundary, and the ratio of the magnetic energy of the poloidal component to that of toroidal component P/T. Run times are more than 10 magnetic diffusion time units, ensuring that solutions are fully developed and reach a statistically steady state that is not affected by initial conditions. [15] Three obvious results can be concluded from Table 1. First, all the P/T values are very close to one, so the poloidal and toroidal components are comparable to each other for every dynamo, suggesting that the a 2 process is responsible for magnetic field generation. Second, from the values of f dip, we know that no dipole dominant dynamos are found in 4of7

5 Figure 2. (a) The radial component of the magnetic field at the outer boundary r = r o for the FeS flotation model. (b) The radial component of the magnetic field at the outer boundary for the Fe snow model. (c) Magnetic power spectrum at r = r o for the FeS flotation model, normalized by the total power at r = r o. (d) Magnetic power spectrum at r = r o for the Fe snow model, normalized by the total power at r = r o. (e) The z- component of (ru) at the equatorial plane for the FeS flotation model. (f) The z- component of (r u) at the equatorial plane for the Fe snow model. The models in this figure are for E =3 10 4, Pm =3,Ra = 550. Fe snow driven models while all FeS flotation driven models give dipole dominant dynamos. Third, for all models with Fe snow, the magnetic energy is rather small, in spite of very large Rm, indicating that Fe snow is not an efficient way to power a dynamo, similar to internal heating. [16] To redimensionalize our results, we use W = s 1 as Ganymede s angular rate of rotation, r = 5990 kg m 3 as the mean core density and assume a magnetic diffusivity of 2 m 2 s 1 (the same as that used by Hauck et al. [2006]). From B dip values listed in Table 1, for FeS flotation cases, the rescaled rms strength of the dipole field on the CMB falls in the range nt. The dipole field geometrically attenuates as (R c /R G ) 3, where R G = 2631 km is the radius of Ganymede, R c is the radius of its core and we take R c = 0.25R G [Hauck et al., 2006], so the surface value of the dynamo generated magnetic field is in the range nt. This is in good agreement with the observed value of Ganymede s equatorial surface field of 750 nt [Kivelson et al., 2002]. For Fe snow cases, the surface value of the magnetic field varies from 24 nt to 72 nt, a rather weak field compared to observation. [17] Although the a 2 mechanism is responsible for magnetic field generation in both the Fe snow and FeS flotation models, their magnetic fields have significantly different patterns. Figure 2 shows maps of the radial component of the magnetic field at the outer boundary. At r = r o, the magnetic 5of7

6 Figure 3. The ratio of the dipole strength on the outer boundary to the rms amplitude of the magnetic field on the outer boundary, f dip, plotted as a function of time for an Fe snow model, with E =3 10 4, Ra = 550 and Pm =3. field produced by the FeS flotation model is characterized by strong flux spots concentrated at midlatitudes. Also, flux spots with opposite signs tend to be symmetric about the equator. For the Fe snow model, the magnetic field is much smaller in scale, and we can see that flux spots of both signs on the outer boundary are irregularly distributed in both hemispheres. [18] Magnetic power spectra at r = r o with respect to spherical harmonic degree l are plotted for both models in Figure 2, clearly demonstrating the dipole nature of the FeS flotation model and the non-dipole nature of the Fe snow model. The magnetic power spectra are calculated via the approach given by Stanley and Bloxham [2006]. In Figure 2c for the Fe flotation model, the magnetic power spectrum attains its maximum at l = 1 and decays rapidly as l increases. The dominance of the dipole is evident. In Figure 2d for the Fe snow model, the magnetic spectrum shows a broad maximum at degrees 2 to 6 and a slow decaying trend at higher degrees. The contribution from the dipole component is rather small. [19] The different magnetic field patterns are mainly the result of different convection structures. In each model, convection is quasi-columnar outside the tangent cylinder. However, with Fe snow the number of convection cells is larger by almost an order of magnitude, and the kinetic energy is higher. Contours of the z- component of vorticity given in Figures 2e and 2f show that when convection is driven by FeS flotation, convection cells tend to concentrate toward the tangent cylinder, whereas for Fe snow, they are more evenly distributed in the shell. These differences can be explained by the radial dependence of the compositional gradient. In the Fe flotation case, the compositional gradient is steepest in the vicinity of the inner boundary, and convection should be most vigorous there, while in the Fe snow case, the gradient varies more smoothly in the radial direction, and its maximum is at the outer boundary. [20] We also found that when simulating an Fe snow driven dynamo, the dipole could be dominant from the beginning for a quite long time. Figure 3 illustrates the variation of f dip with time for an Fe snow model. It is clearly shown that the dipole is dominant from the beginning for as long as 2 magnetic diffusion time units, then it takes another 2 magnetic diffusion time units for the dipole to become much weaker than other multipole components. Dynamo simulations are usually run only for several magnetic diffusion time units. As demonstrated here, if the run-time is not long enough, Fe snow dynamos can be incorrectly characterized as dipole dominant. 4. Discussion [21] We have shown that different driving mechanisms can significantly affect the morphology of the generated field. Convection driven by FeS flotation results in dipole dominant dynamos. No dynamo powered by the Fe snow process shows dipole dominance. According to Ganymede s present magnetic field morphology, we suggest that its core convection is mainly driven by FeS flotation or inner core growth. The convective state in Ganymede s core is determined by the bulk core sulfur content and the mantle temperature. Buoyancy can be released by Fe snow and inner core growth or by Fe snow and FeS flotation. Here we do not rule out the possibility that Fe snow exists in Ganymede s core, but if it does, it makes a secondary contribution to powering the dynamo while FeS flotation or inner core growth is the primary driver. [22] Although our models are simplified in comparison to Ganymede s interior, we are able to include the basic physics, such as fast rotation and compositional convection. In future work we will incorporate the effects of tidal heating and coupling between Ganymede s core and mantle into our model. These studies provide important constraints on the interior of Ganymede and help us to understand its thermal evolution. [23] Acknowledgments. We thank the Shanghai Supercomputer center (SSC) and NASA Ames for providing computing resources. X. Zhan and G. Schubert acknowledge support from NASA NNX09AB57G. G. Schubert also acknowledges support from NSF AST References Aubert, J., S. Labrosse, and C. Poitou (2009), Modelling the palaeo-evolution of the geodynamo, Geophys. J. Int, 179, , doi: /j x x. Bland, M. T., A. P. Showman, and G. Tobie (2008), The production of Ganymede s magnetic field, Icarus, 198, , doi: /j. icarus Braginsky, S. I., and P. H. 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