The study of planetary magnetic fields can provide important information on a planet's interior structure and composition. Here we use numerical dynamo models to demonstrate that it is possible to determine information on a fundamental core property: inner core size, from observations of intense, small-scale radial magnetic flux patches. The inner core size is discernable from the boundary between different types of intense magnetic flux patches inside and outside the inner core tangent cylinder. This type of study could help in determining the solid inner core size in Mercury, Ganymede, Jupiter and Saturn. For Mercury and Ganymede, the inner core size would then provide useful constraints on thermal evolution models for these bodies.
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 Determining the properties of a planet's deep interior is difficult because we cannot observe this region directly as we can the surface of a planet. On Earth, seismology has provided important information on the structure and composition of the deep interior including the core. For example the radii of the inner and outer cores were determined from seismic studies [Gutenberg, 1913; Lehmann, 1936]. In addition, properties of the Earth's outer core motions have been determined mostly from observations of the geomagnetic field secular variation (for a review see Bloxham and Jackson ). Unfortunately we are not, at present, able to conduct detailed seismic studies and observations of magnetic field secular variation for other planets to provide us with the same quality of information on their cores. We are however, capable of studying a planet's magnetic field morphology quite extensively with an orbiting spacecraft at relatively low altitude. This was demonstrated in convincing fashion by Mars Global Surveyor which mapped Mars' remanent crustal magnetic field to very high spherical harmonic degree and order [Acuna et al., 1999; Arkani-Hamed, 2004].
 In rapidly rotating fluids, a two-dimensionality is imposed on the fluid flow as a result of the Taylor-Proudman theorem [Proudman, 1916; Taylor, 1917]. In the spherical shell geometry of planetary cores (see Figure 1), this two-dimensionality results in the division of the fluid outer core into two dynamical regimes by the inner core tangent cylinder (an imaginary cylinder tangent to the inner core boundary and coaxial with the rotation axis). Numerical and experimental models have demonstrated that the region outside this cylinder has convective motions governed by columnar structures (motions with little variation in the direction of the rotation axis). Depending on the parameter regime, convection can occur in the form of steady cylindrical rolls [Busse, 1970] or highly variable (in existence, shape and time) columnar structures (examples of convection patterns in numerical dynamo models are given by Zhang and Schubert  and Kono and Roberts ). The two regions inside the tangent cylinder (one north and one south of the equatorial plane) are more constrained by the rapid rotation and therefore require more complex helical motions to transport heat or a compositionally buoyant element from the inner core boundary to the core-mantle boundary. These regions have also been studied experimentally, numerically and by observational analysis [e.g., Gubbins and Bloxham, 1987; Aurnou et al., 1998, 2003; Olson and Aurnou, 1999; Sreenivasan and Jones, 2005].
 Because magnetic fields in planetary cores are generated by interactions of fluid motions with existing magnetic fields, it seems likely that different motions and force balances inside and outside the tangent cylinder could lead to different magnetic field structures in these regions. This suggests that we may be able to use observed magnetic field structures to determine the division between these regions and hence, determine the size of the inner core in planets with active dynamos. Here we use numerical dynamo models to investigate whether any distinct magnetic field structures that are observable outside the core correlate with inner core size in an effort to guide future magnetic observations by planetary spacecraft missions.
2. Numerical Model
 Numerical dynamo models have shown that for planets with active dynamos, magnetic field morphology can provide information on certain planetary internal properties [e.g., Stanley and Bloxham, 2004; Stanley et al., 2005]. Although numerical models cannot at present work in an appropriate parameter regime for planetary cores, they can provide crucial insights into dynamo processes, and by appropriate interpretation of simulation results, we can use these models to obtain information on dynamo action in planetary cores. For a recent review on dynamo theory, numerical models and their application to the geodynamo, see Kono and Roberts .
 Here we examine the magnetic field morphology and its interactions with fluid motions in the Kuang and Bloxham [1997, 1999] numerical dynamo model using various values for the inner to outer core radius ratio (rio = rinner/router) and Rayleigh number (Ra). All other non-dimensional numbers are held fixed. The relevant equations, numerical methods and parameters are given by Kuang and Bloxham . Table 1 lists the relevant parameters for the models we analyze here.
The non-dimensional numbers varied in the dynamo models examined in this study: Ra = αTgohTro2/2Ωη is a modified Rayleigh number and rio is the inner to outer core radius ratio. Other non-dimensional parameters in the models are held fixed at E = ν/2Ωro2 = 2 × 10−5, Ro = η/2Ωro2 = 2 × 10−5, qk = κ/η = 1, Pr = ν/κ = 1 where E is Ekman number, ν is kinematic viscosity, Ω is rotation rate, ro is outer core radius, Ro is magnetic Rossby number, η is magnetic diffusivity, qk is magnetic Prandtl number, κ is thermal diffusivity, and Pr is Prandtl number. We use fixed heat flux boundary conditions on the temperature, finite electrically conducting boundary conditions on the magnetic field and impenetrable and stress-free boundary conditions on the velocity field. Our maximum spherical harmonic degree is 33 and order is 21 and we use similar hyperdiffusivities as Kuang and Bloxham . The hyperdiffusivities are used in order to work at more strongly supercritical Rayleigh numbers than possible without them. This results in damping the very small scale structures likely to be present in planetary cores. For possible dynamical effects of hyperdiffusivities on convection and dynamos, see Zhang and Jones ; Zhang et al.  and Grote et al. . The last column is the time average ratio of the octupole to dipole power in the fields where the power is defined in terms of the mean squared field intensity at the outer boundary of the dynamo source region.
 When using an inner to outer core radius ratio appropriate for studying the Earth's dynamo (rio ≈ 0.35), this numerical model is capable of reproducing many salient features of the Earth's magnetic field such as the field's axial-dipole dominance, secular variation, intensity, strong high latitude flux patches and reversals. All the models we discuss here produce large-scale fields dominated by their axial dipole components; the same class of fields found in Earth, Jupiter, Saturn, Ganymede and possibly Mercury. We have chosen our Rayleigh numbers such that the models all produce dynamos with magnetic Reynolds numbers of O(100) in an effort to compare models that produce a similar vigor of magnetic field generation.
Figure 2 shows the radial component of the magnetic field at the core-mantle boundary for the models in Table 1 (not all the models are shown in the interest of space, but they all have similar characteristics to those shown). The large-scale magnetic fields in all the models are similar in that they are dominated by their axial-dipole components, however there is a marked difference between the models in their smaller-scale fields. Although all the models contain intense small scale flux patches, the regions where they form and their configuration differ.
 The most striking feature is that intense reversed flux spots (spots with opposite polarity to the dipole polarity in the hemisphere where they occur) only appear in regions outside the tangent cylinder. This is evident in Figure 2 where there is a good correlation between the maximum latitudinal extent of the reversed flux spots and the circle denoting the intersection of the inner core tangent cylinder with the core mantle boundary. Since this circle occurs at different latitudes for different rio, the extent of the reversed flux spots varies between the models with different rio. Some of the models also produce intense flux spots inside the tangent cylinder, but they are only of normal polarity (i.e. the same direction as the dipole in that hemisphere).
 The timestep shown in Figure 2a (rio = 0.35) does not have reversed flux extending to near the tangent cylinder boundary (it is present in equatorial regions). Intense reversed flux spots occur less frequently in time at higher latitudes in our very thick shell models then in our thinner shell models. We have chosen to included a more representative timestep in Figure 2a, rather than a more special case with a high latitude reversed flux spot. We will demonstrate in the next section that there is complimentary evidence that would aid in determining the inner core size for thick shell models. Whether reversed flux spots occur less frequently in thick shell planetary dynamos, or numerical models miss an important dynamical effect that results in more frequent high latitude reversed flux spots is currently unclear, and will be addressed in future work. The Earth is an important test case for a thick shell dynamo model and we will discuss the presence of high latitude intense reversed flux in Earth observations in the next section.
 The reversed flux spots found outside the tangent cylinder in our models can be generated by two mechanisms. The first is the interaction of convective radial motions on toroidal magnetic field to create poloidal field which then diffuses across the core-mantle boundary. This mechanism has been proposed for the generation of Earth's mid-latitude flux spot pairs [Bloxham, 1986] and has been observed in previous dynamo models [Christensen et al., 1998; Olson et al., 1999]. The second is magnetic field instabilities which result from strong toroidal fields. In regions where the toroidal field reverses zonal direction, the strong oppositely directed field lines can re-connect; this diffusive instability results in an upwelling motion as Lorentz forces drive flow horizontally away from the point of reconnection. The upwelling then expels field from the dynamo region, resulting in a pair of radial flux spots. This mechanism can produce equatorially antisymmetric spots near the equator where the toroidal field changes sign and also at higher latitudes in some of our thin shell models (large rio) which have multiple toroidal field nodal lines. In regions where the toroidal field is strongest, magnetic field gradient instabilities can also result in flux expulsion and the formation of pairs of spots.
 Inside the tangent cylinder, we either find no intense small-scale flux spots (in models where convection has not initiated inside the tangent cylinder) or we find only strong normal polarity flux spots (in models where convection has initiated inside the tangent cylinder, such as in Figure 2e). These strong spots generally occur closer to the inner core tangent cylinder than to the pole and are accompanied by weaker fields near the poles. They may be the result of wave motions inside the tangent cylinder or downwellings near the tangent cylinder that act to converge magnetic field lines into local areas. It appears that magnetic field instabilities are suppressed inside the tangent cylinder in our models. There are several possible explanations for this and combinations of them may be important in any specific model: (1) In some models, differential rotation is weak inside the tangent cylinder resulting in weaker toroidal fields and hence the lack of strong toroidal fields initiating magnetic instabilities. (2) For diffusive instabilities, after reconnection has occurred, the resulting upwelling is strongly inhibited inside the tangent cylinder due to the rotational constraint on motions in the direction of the rotation axis (i.e. Taylor's constraint). (3) The role of the large scale poloidal field may also be important and there is a difference in the relative direction of the poloidal field lines to the upwelling direction inside the tangent cylinder (they are both approximately parallel to the rotation axis) whereas outside the tangent cylinder, the upwelling direction is mostly in the cylindrically radial direction which is perpendicular to the large vertical component of the dipolar field lines. These mechanisms will be examined more closely in future studies.
3.1. Case Study: Earth
 Earth is the one planet whose inner core size we know. We therefore examine if our method of using the extent of reversed flux spots to determine inner core size would give an accurate result for the Earth. A recent map of the radial magnetic field at the core surface constructed from a combination of Oersted and Magsat satellite data is shown by Jackson [2003, Figure 1]. Although the inner core tangent cylinder is not shown, it happens to have a similar latitudinal extent as Antarctica, whose outline is shown. There are two reversed flux spots in the southern hemisphere (one beneath South America and one beneath Africa) whose latitudinal extents correlate well with the inner core tangent cylinder boundary. It appears that using our method would obtain a relatively good result for the Earth's inner core size. Because the growth of these reversed flux patches has reduced the Earth's dipole moment, it has been suggested that they may be linked to the beginning of a field reversal [Constable, 1992; Gubbins and Coe, 1993; Hulot et al., 2002]. From our models, we would suggest it is also possible that these spots are just a result of magnetic field instabilities and not necessarily related to the beginning of a reversal. Although some other observational models have reversed flux inside the tangent cylinder [e.g., Hulot et al., 2002], it is weaker than the strong flux spots outside the tangent cylinder and has been attributed to a polar vortex resulting from a thermal wind inside the tangent cylinder [Olson and Aurnou, 1999]. Our results are therefore still valid when we use these Earth models: the inner core can still be inferred from the extent of strong reversed flux.
3.2. Other Useful Magnetic Features
 The motivation of this study was to guide future spacecraft magnetic observations interested in determining a planet's inner core size. Although our models suggest the smaller scale intense reversed flux spots are the most obvious feature that correlates with inner core size, from a practical standpoint, they are also the most difficult to observe. One needs good spatial resolution as well as good timing (since the spots are variable) in order to use them as a proxy for inner core size. We therefore examine whether other, more easily observed magnetic field structures could complement reversed flux spots in determining this property.
 Another feature that is observed in the magnetic field are concentrations of normal-polarity flux at high-latitude occurring in pairs arranged symmetrically about the equator. The latitude at which they occur in the Earth is just outside the tangent cylinder [Gubbins and Bloxham, 1987]. These flux spot pairs have been interpreted as due to concentrations of field lines associated with downwellings in convection rolls outside the core. We see this form of flux spot in our numerical models as well, although they seem to be more regular in the small inner core (i.e. thick shell) models. Because they are also small scale features, they may be as difficult to observe as reversed flux spots, but they may be less time variable and hence, more likely to be observed. As can be seen in Figure 2a, these symmetric flux spot pairs would be crucial in the specific timestep shown for determining the inner core size since there are no high-latitude reversed spots in this image.
 Another property we investigate is the power in the large scale fields. We find that inner core size is not an obvious function (e.g. monotonic) of the surface magnetic power in the large scale (dipole, quadrupole, octupole) components. Our results agree with those of Heimpel et al.  who examine the power spectra for dynamos in various shell geometries at near critical Rayleigh numbers. However, one noticeable effect is that models with a small (Earth-like) inner core have a low octupole contribution, whereas those with a large inner core have a highly variable and frequently high octupole contribution (ratios of the octupole to dipole power are given in Table 1). This strong octupole power is directly related to the reversed flux patches outside the tangent cylinder since, for a relatively thin shell, the tangent cylinder intersection with the core surface occurs very near the nodal lines of an axisymmetric octupolar field. Our models therefore suggest that a strong octupole power relative to the dipole power could be a good indicator of a large inner core size (thin fluid shell). Since the octupole component of the field would require less spatial resolution to measure by an orbiting spacecraft than small-scale flux spots, this might be an important component of the field to examine since it could indicate whether the fluid core has a thick or thin shell geometry.
 By using just the small scale reversed magnetic flux spots, one could discern the intersection of the tangent cylinder with a planet's surface and hence obtain a fairly good estimate of a planet's inner core size. In specific situations, supporting evidence could also be obtained from other magnetic structures (antisymmetric higher latitude flux spot pairs or power in the octupole component of the field) for a more confident determination of the inner core size. Since the inner core size is directly related to the thermal evolution of the planet, knowing it provides useful constraints for thermal evolution models (e.g. how much sulfur or radioactive elements need to be present in a planetary core [Stevenson et al., 1983; Schubert et al., 1988; Hauck et al., 2006]). The models examined here all have large scale fields dominated by axial dipoles and so our method can only comment on planets with active dynamos whose large scale fields are dominated by axial dipoles. It therefore has potential usefulness for missions to Mercury, Ganymede, Jupiter and Saturn.