The internal magnetic field of Mercury is anomalously weak compared with the fields of other solar system dynamos. Here we investigate the effect that magnetospheric currents may have on the internal dynamo process. Although strong dipolar dynamos are not markedly affected by such magnetospheric currents, a dynamo in a weak-dipole state can be stabilized in such a configuration by magnetospheric feedback. We suggest that Mercury's core dynamo was stabilized in a weak-field state early in Mercury's history, when the solar wind was much stronger than today, and has been maintained in that state to the present by magnetospheric feedback. A prediction of this scenario is that secular variation should occur more rapidly for Mercury's internal field than would be expected for some other models for the planet's weak field.
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 Mercury is the only inner planet other than Earth to possess a global magnetic field [Ness et al., 1974; Anderson et al., 2010], but the weakness of the field has posed a puzzle since its discovery. Combined magnetic field measurements during passages through the planet's magnetosphere by the Mariner 10 and MESSENGER spacecraft indicate that the dipole moment is about 250 nT RM3, where RM is Mercury's radius [Anderson et al., 2010]. In contrast, scaling laws for multipolar planetary dynamos, i.e., dynamos in a regime where multipolar components dominate the dipolar component, predict that the moment in a planet of Mercury's core size and internal composition should not be less than ∼1000 nT RM3 [Olson and Christensen, 2006]. Several classes of hypotheses have been developed to account for Mercury's weak field, including a thermoelectric dynamo generated by electrical currents along a topographically rough core-mantle boundary (CMB) [Stevenson, 1987]; remanent crustal magnetization coherent at long wavelengths [e.g., Aharonson et al., 2004]; a convective dynamo in which the ratio of inner and outer core dimensions localizes the convection [e.g., Heimpel et al., 2005]; a layered outer core with a non-convecting layer at the top [e.g., Christensen, 2006]; compositionally stratified outer core convection [Gómez-Pérez et al., 2010]; and double-layer convection leading to concentric dynamo-generating regions with opposing dipole moments [Vilim et al., 2009].
 Induction by external fields has long been recognized as a possible mechanism for sustaining sub-critical (i.e., non-convective) magnetic field generation in an electrically conductive core [Levy, 1979; Sarson et al., 1999; Sakuraba and Kono, 2000]. It has also been shown that external sources may modify the internal flow of self-sustained dynamos [Gómez-Pérez and Wicht, 2010]. Moreover, the combination of the weak internal dynamo field and comparatively strong solar wind at Mercury suggests that the magnetic field induced by magnetospheric currents may exert an important influence on the core dynamo [Glassmeier et al., 2007a; Heyner et al., 2010]. Chapman-Ferraro currents at Mercury's magnetopause induce a magnetic field at the CMB of as much as ∼60 nT, an important fraction of the estimated dipolar dynamo field at the CMB (∼500 nT) [Glassmeier et al., 2007b]. Because the external field is of magnetospheric origin, its direction and magnitude depend principally on two factors, the internal dipole moment and the solar wind pressure. These magnetospheric-induced fields always maintain a polarity that opposes that of the internal dipole [Grosser et al., 2004]. The strength of the Chapman-Ferraro currents, however, depends on the balance between the average solar wind pressure and the internal field, a balance that cannot be easily predicted.
 Here we explore the influence of magnetospheric fields on the dynamo-generated field by means of numerical models for Mercury's core dynamo. We implement numerically a feedback mechanism that is able to mantain a weak observable field, i.e., comparable to that observed at Mercury. Such feedback yields a weak field dominated by non-dipolar components and for which the dipole reverses frequently, although episodes of dipole polarity stability are also found. In the methodology section we describe the implementation of the dynamo and its feedback. Time-variable and time-averaged quantities describe the behavior of the magnetic field and are provided in the results section to facilitate a direct comparison of the effects of different feedback magnitudes. In the discussion section we analyze the implications of the feedback mechanism for Mercury, as well as for Earth, and we predict aspects of the observable field that can be contrasted with those of some other models proposed to explain the magnetic field of Mercury.
 For the dynamo simulations in this paper, we use a pseudo-spectral algorithm to model the time-dependent flow within an electrically conductive liquid in a rotating spherical shell of internal and external radii ri and ro, respectively, that revolves with an angular velocity Ω = Ω z, where z is a unit vector aligned with the planet's spin axis [see Wicht, 2002; Gómez-Pérez and Wicht, 2010]. The system is controlled by several non-dimensional parameters. The Rayleigh number is given by Ra = α go ΔT L3(κ ν)−1, where α is the volumetric coefficient of thermal expansion, ΔT = T(ro) − T(ri) is the superadiabatic temperature difference between the top and bottom boundaries, L = ro − ri is the shell thickness, κ is the thermal diffusivity, and ν is the kinematic viscosity. The Ekman number is E = ν (Ω L2)−1; the Prandtl number is Pr = ν κ−1; and the magnetic Prandtl number is Pm = ν λ−1, where λ is the magnetic diffusivity.
 The field induced by magnetospheric currents is modeled as a uniform magnetic field that changes polarity in response to an axial-dipole polarity reversal. The magnetospheric-induced field (Bm) thus always opposes the internal axial dipole, i.e., Bm = −s(t) Bez, where s(t) equals 1 or −1 and indicates the dynamo axial dipole polarity, and Be is the magnitude of the induced field [Gómez-Pérez and Wicht, 2010]. The vector Bm neglects all contributions other than those acting on the internal axial dipole, as well as the modulation in Be expected from changes in surface current intensity and position of the magnetopause. The latter variation is likely to be important to the time evolution and stability of the feedback dynamo [Heyner et al., 2010].
 Because the size of a solid inner core in Mercury is unknown, we choose the Earth's radius ratio, ri/ro = 0.35, to allow for a direct comparison with previous numerical models. To assess the effects of feedback, we compare all results to an undisturbed solution, i.e., one lacking feedback or, equivalently, with Be = 0. Parameters for this control case result in a strong dipolar solution that is not expected to reverse spontaneously. These parameters include Ra = 9.65 × 106 = 13 Rac, where Rac is the Rayleigh number for the onset of convection [Al-Shamali et al., 2004]; E = 10−4; Pr = 1; and Pm = 2. The top and bottom boundaries are non-slip and constant in temperature. The mantle is assumed to be an insulator, and the inner core is taken to have the same electrical conductivity as the fluid core.
 We use non-dimensional quantities from the model output to characterize the magnetic field strength. The commonly used ratio of Lorentz to Coriolis forces (i.e., the Elsasser number) is defined by
where ρ is the bulk density of the fluid, the angled brackets indicate time average, and the square brackets indicate space average. Throughout this paper we use values of the magnetic induction normalized by (i.e., Λ = 〈[B2]〉).
 We evaluate the effect of magnetospheric feedback for two different sets of dynamo simulations. First, a self-sustained dynamo is allowed to develop without any external influence (i.e., Be = 0), resulting in a dipole-dominated, non-reversing, self-sustained dynamo. For time t > 0 we then set Be ≠ 0. We denote this set by its strong-field initial condition, or SIC. The second set uses the output of the SIC solution with Be = 0.5, which exhibits a weak dipolar field. Effectively, these simulations change from Be (t < 0) = 0.5 to a different value of Be for t > 0. Because the feedback modifies the flow field, simulations with a weakened-dipole initial condition, which we denote by WIC, achieve a different equilibrium than those with SIC.
 Volume- and surface-averaged Elsasser numbers are shown for SIC and WIC dynamos in Figure 1. In models with SIC, magnetospheric feedback does not affect dynamo strength (i.e., the solutions are similar to the control case) for Be = 0.1 or 0.25. In contrast, such feedback reduces the strength of the dynamo markedly for the cases Be = 0.5 and 1.0. A value of Be = 0.5 is equivalent in dimensional units to a field of ∼1.8 × 104 nT, an extremely high value to be generated by magnetospheric currents. We conclude that magnetospheric feedback is unlikely to be important in cases with strong dipolar dynamos.
 The situation is notably different for dynamos initialized with a weakened dipolar component (Figure 1). For sufficiently small magnetospheric fields (e.g., Be = 0.001), the dynamo returns to a strongly dipolar field in a time comparable to the magnetic diffusion time. The observable magnetic field, however, is controlled by the feedback for values of Be that are much lower than for the SIC models (e.g., Be = 0.01 and 0.1). For core properties matching those of Earth [Gubbins, 2007] but with Mercury's rotation rate, a value of Be = 0.01 is equivalent to Be ∼ 360 nT in dimensional units, about 6 times greater than the estimated magnitude of the magnetosphere-induced field at Mercury.
 The ratio between inertial and Coriolis forces has been recognized to be indicative of the temporal stability and magnetic field geometry of the solution [Olson and Christensen, 2006]. We define the local Rossby number , where the Rossby number is Ro = urms (Ω L)−1, and
is the dominant harmonic degree l for fluid flow with velocity field u. The solutions presented here may be grouped by Rol; see Table 1. All simulations in which the solution is controlled by magnetospheric feedback satisfy Rol ∼ 0.8. In contrast, the observable field is independent of feedback for simulations where Rol ∼ 0.6. For WIC models, a relatively weak level of magnetospheric feedback (e.g., Be = 0.01) is able to control the internal flow, allowing for the development of a high value of the fluid velocity (i.e., a high value of the magnetic Reynolds number Rm = urmsL λ−1).
Table 1. Parameters for and Time- and Space-Averaged Results From Dynamo Simulationsa
Be is the magnitude of the imposed field, Λe = Be2 the Elsasser number associated with the imposed field, Rm the magnetic Reynolds number, Λ the time- and volume-averaged Elsasser number per unit volume, Rol the local Rossby number, Λcmb the time- and surface-averaged Elsasser number at the CMB per unit area, Λcmb,l=1 the time- and surface-averaged Elsasser number of the dipole component (l = 1) at the CMB per unit area, fr the number of reversals per unit time (magnetic diffusion time), and q the ratio of the imposed field magnitude (Be) to the surface-averaged dipole magnitude at the CMB (√Λcmb,l=1). The first to fifth rows correspond to SIC simulations, and the sixth to eighth rows show results for WIC simulations.
218 ± 13
7.60 ± 0.72
0.056 ± 0.005
5.75 ± 0.65
2.3129 ± 0.2505
222 ± 13
7.05 ± 0.71
0.058 ± 0.005
5.42 ± 0.68
2.2839 ± 0.2921
6 × 10−2
229 ± 12
6.12 ± 0.57
0.061 ± 0.005
4.86 ± 0.58
2.2148 ± 0.2296
2 × 10−1
295 ± 16
1.02 ± 0.66
0.081 ± 0.005
1.08 ± 0.63
0.7736 ± 0.3137
295 ± 16
1.03 ± 0.68
0.081 ± 0.005
3.23 ± 0.66
2.9352 ± 0.3722
296 ± 14
0.94 ± 0.15
0.081 ± 0.004
0.29 ± 0.06
0.0323 ± 0.0068
295 ± 14
0.94 ± 0.16
0.081 ± 0.004
0.29 ± 0.09
0.0166 ± 0.0221
243 ± 36
5.37 ± 3.12
0.065 ± 0.013
3.87 ± 2.50
1.5012 ± 1.0520
 A good way to represent the magnitude of feedback is to compare the magnetospheric field and the dipolar poloidal component of the dynamo field at the CMB. We introduce q = (Λe/Λcmb,l=1)1/2, where Λcmb,l=1 is the Elsasser number of the dipolar component of the magnetic field at the CMB; see Table 1. Estimated values for planets are qM = 0.16 for Mercury and qE = 0.0005 for Earth [Glassmeier et al., 2007b]. In WIC models, we find that magnetic fields are controlled by feedback for values of q ≥ 0.078 (which is less than qM and thus predicts feedback control at Mercury) and independent of feedback for q ≤ 0.001 (which is greater than qE and thus predicts a negligible effect of feedback at Earth). These model results are thus consistent with the importance of a feedback dynamo at Mercury and a lack of magnetospheric influence on Earth's dynamo.
 The dipole component of the solutions stabilized in a weak-field state by magnetospheric feedback varies rapidly in time. The rate of reversals per unit time, fr, is shown in Table 1. The dipole polarity and the contribution from the dipole to the total field as functions of time are shown in Figure 2a for three simulations, Be = 0, Be = 0.001 with WIC, and Be = 0.01 with WIC. Solutions independent of feedback (e.g., Be = 0 and Be = 0.001) result in a non-reversing, dipole-dominated magnetic field. In contrast, the solutions controlled by magnetospheric feedback, e.g., Be = 0.01 with WIC, have a substantial multipolar magnetic field contribution and exhibit dipole polarity reversals with an average period of 3 kyr, although there are intervals of stable polarity up to ∼100 kyr in duration (for comparison, the magnetic diffusion time in Mercury is τλ ∼ 50 kyr).
 A weak dipolar field component at the CMB does not necessarily imply that the observable field must be multipolar. The radial component of the magnetic field at spacecraft altitude (∼200 km from the planetary surface) is shown in map view in Figures 2b and 2c at t = 3τλ for the simulations in Figure 2a. The observable field from the feedback-controlled model, from which a representative snapshot of a stable dipole episode is shown in Figure 2d, exhibits a weak magnitude and a dipolar/quadrupolar morphology consistent with measurements of the magnetic field of Mercury [Uno et al., 2009; Anderson et al., 2010].
 As suggested by kinematic simulations [Glassmeier et al., 2007a; Heyner et al., 2010], magnetospheric feedback may be important for Mercury's core dynamo. We have found that a weakened-dipole dynamo may be substantially modified by feedback for a sufficiently strong magnetospheric field. In the absence of such feedback, or when the magnetospheric-generated field is too small, a weak-field dynamo in our simulations returns to a strongly dipolar state. There is a threshold level to the magnetospheric-generated field, however, above which a weakened internal dynamo cannot overcome control by magnetospheric feedback and return to a strong-field state. For the conditions adopted for the WIC simulations in this paper, that limit lies in the range 0.01 > Be > 0.001. The critical value of Be for effective feedback, however, is likely sensitive to core convection (i.e., Ra, E, Pr, Pm) and to the dynamo–solar wind interaction (i.e., solar wind pressure and magnetopause position) and thus may have a different value from the models in this paper under different overall conditions.
 The models of this paper therefore suggest a scenario to account for Mercury's weak internal field. If at some point in the history of Mercury's dynamo the dipole field was particularly weak, then magnetospheric feedback could have been responsible for stabilizing the planet's weak field until the present. Such a weakened dipole may have been caused by any of several mechanisms. There may have been a non-dipolar seed field at the onset of the dynamo, the result, say, of a weak and highly variable interplanetary magnetic field. Alternatively, a weakened-dipole magnetic field could have occurred naturally during the life of the dynamo, as a result of core geometry [e.g., Heimpel et al., 2005] or large convective forcing [Kutzner and Christensen, 2002]. Because solar wind pressure in the early solar system (ρsw,e) was up to 1,000 times larger than at present (ρsw) [Wood et al., 2002], magnetospheric-generated fields at Mercury would have been stronger early in the history of the global dynamo and would have been even more effective at maintaining Mercury's weak-field state than today.
 Although magnetospheric feedback is not expected to be a major influence on modern dynamos for solar system planets other than Mercury, such feedback may have been more widespread in the early solar system given the stronger solar wind from the young Sun [Wood et al., 2002]. Planets in the early solar system would have been subjected to a solar wind pressure comparable to or larger than that at Mercury's present orbital distance (RMo) as far from the Sun as (ρsw,e/ρsw)1/2RMo ∼ 13 AU. For this reason, early dynamos in other terrestrial planets might have been affected by magnetospheric feedback. A much greater thickness of non-conducting mantle, however, would have lessened the influence of close-in magnetospheric currents, particularly in the largest terrestrial planets.
 If magnetospheric feedback is important in Mercury at present, then secular variation in the field should occur more rapidly than would be expected from some other dynamo models, e.g., a deep-seated dynamo [Christensen, 2006]. Identification of secular variation in the internal field can be assessed with spacecraft measurements, but it will be difficult to do so because of the high variability of external magnetic fields of magnetospheric origin [e.g., Uno et al., 2009]. An accurate assessment of magnetospheric currents and stable solar wind conditions will be necessary to estimate the internal field and to explore its time variability. Extended temporal baselines for magnetic field measurements, e.g., including observations from both the MESSENGER and BepiColombo [Benkhoff et al., 2010] missions, will probably be crucial for such exploration.
 1. Magnetospheric feedback is capable of controlling fluid flow in the outer core for dynamos with weakened-dipole initial conditions.
 2. Such magnetospheric feedback is quantitatively important for Mercury.
 3. Magnetospheric feedback would have exerted a stronger influence early in Mercury's history and may have been of broader importance for early solar system dynamos. Thick non-conductive silicate mantles, however, may have shielded a dynamo even from magnetospheric currents substantially stronger than at present in the larger terrestrial planets.
 4. Under control by magnetospheric feedback, secular variation of the dynamo-generated field at Mercury may be more rapid than for some dynamo models that don't incorporate such feedback. However, separating internal and external fields in magnetic field measurements made from orbit will be challenging and may limit the ability to observe such variability in Mercury. Long temporal baselines that span multiple spacecraft missions will be helpful in exploring the nature and rate of Mercury's secular variations.
 We thank M. H. Heimpel and G. Anglada-Escudé for constructive comments and discussions. This work has been supported by the NASA MESSENGER project and the NASA Planetary Geology and Geophysics Program, under contract NASW-00002 and grant NNX07AP50G, respectively.