A dynamo model for axisymmetrizing Saturn s magnetic field

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1 Click Here for Full Article GEOPHYSICAL RESEARCH LETTERS, VOL. 37,, doi: /2009gl041752, 2010 A dynamo model for axisymmetrizing Saturn s magnetic field S. Stanley 1 Received 11 November 2009; revised 1 February 2010; accepted 3 February 2010; published 3 March [1] Magnetic field measurements demonstrate that Saturn s internally generated magnetic field has an extremely small dipole tilt. The nearly perfect axisymmetry of Saturn s dipole is troubling because of Cowling s theorem which states that an axisymmetric magnetic field cannot be maintained by a dynamo. A possible mechanism to axisymmetrize the observed field involves differential rotation in a stably stratified electrically conducting layer surrounding the dynamo. Here we use numerical dynamo models to study the axisymmetrizing effects of stably stratified layers surrounding the dynamo. We find that a thin stably stratified layer which undergoes differential rotation due to thermal winds as a result of pole to equator temperature differences can produce a more axisymmetrized field. Surprisingly, we find that the direction of the zonal flows and their equatorial symmetry is a crucial factor for magnetic field axisymmetry since some zonal flows act to destabilize the dynamo producing non axisymmetric fields. Citation: Stanley, S. (2010), A dynamo model for axisymmetrizing Saturn s magnetic field, Geophys. Res. Lett., 37,, doi: /2009gl Introduction 1 Department of Physics, University of Toronto, Toronto, Ontario, Canada. Copyright 2010 by the American Geophysical Union /10/2009GL041752$05.00 [2] The Cassini mission has confirmed the previous results of the Pioneer 11 and Voyager I and II missions that Saturn possesses a dynamo generated magnetic field [Acuna and Ness, 1980; Smith et al., 1980; Ness et al., 1981, 1982; Dougherty et al., 2005]. The field is axially dipolar dominated as are the fields of Earth, Jupiter, Ganymede and most likely Mercury. However, Saturn is unique in the level of axisymmetry observed. According to observations, Saturn s dipole tilt is less than 1 from its rotation axis [Smith et al., 1980; Connerney et al., 1982, 1984; Giampieri and Dougherty, 2004]. In contrast, the other axial dipole dominated fields have tilts of about 10 [Connerney, 1993; Stevenson, 2009]. [3] Saturn s dynamo is generated in its hydrogen shell in regions where compression results in a significant electrical conductivity [Stevenson, 1982a; Guillot, 2005]. Theoretical studies and shock pressure measurements show that the transition from molecular to metallic hydrogen occurs at about GPa which corresponds to a depth of about 0.5 Saturn radii (R s = km) [Stevenson and Salpeter, 1977; Nellis et al., 1996; Hubbard et al., 2002]. [4] Saturn s extreme axisymmetry appears problematic because of Cowling s Theorem which states that a dynamo cannot generate a purely axisymmetric field [Cowling, 1933]. However, Cowling s Theorem applies to the magnetic field in the dynamo generation region whereas the axisymmetric magnetic field observations are made outside the planet. If a mechanism can be found that takes a nonaxisymmetric field inside the dynamo region and axisymmetrizes the resulting observed field outside the dynamo region, then there is no conflict with Cowling s Theorem. [5] Stevenson [1982b] proposed a mechanism for this axisymmetrization. In the molecular metallic hydrogen transition region, helium may become immiscible in metallic hydrogen and therefore rain out of the metallic hydrogen. This will produce a compositionally stably stratified layer at the top of the metallic conducting region [Stevenson, 1977; Fortney and Hubbard, 2003; Stixrude and Jeanloz, 2008]. Stevenson [1982b] proposed that thermal winds (axial gradients in zonal wind velocity [e.g., Aubert, 2005]) could be driven in this stable layer due to latitudinal temperature differences at the top of the layer produced, for example, by solar insolation. He argued that these thermal winds would act to axisymmetrize the observed magnetic field by shearing out the non axisymmetric field components at the top of the dynamo region. It is also possible that natural convection patterns in Saturn s molecular atmosphere could result in latitudinal temperature variations at its base (i.e. the top of the stable layer), similar in equatorial symmetry to those produced by solar insolation [Aurnou et al., 2008]. This helium rainout layer could also explain several other Saturn anomalies such as the helium deficiency is Saturn s atmosphere (by the sequestering of helium in the rainout layer) and Saturn s excess luminosity (by providing an additional heat source due to gravitational heating in the dissociation process) [Stevenson, 1982a]. [6] Stevenson [1982b] carried out a cartesian thin layer analytic study to determine the level of axisymmetrization of such a layer and determined that axisymmetrization should occur if the magnetic Reynolds number of the dynamo action is relatively large. In addition, kinematic dynamo studies have examined the feasibility of a stably stratified layer in axisymmetrizing a magnetic field [Love, 2000; Schubert et al., 2004]. These models prescribe a velocity field and examine the evolution of the resulting magnetic field. They have shown that stably stratified layers surrounding dynamo regions can affect the symmetry of the produced magnetic field. However, these models do not always produce axisymmetric fields. The morphology of the resulting field depends on both the imposed flows in the stably stratified layer and the geometry of the field in the dynamo generation region. Since these models prescribe the velocity fields, they don t examine the interactions between the stable layer and underlying dynamo region, nor the dynamic effects of the magnetic fields, but they do provide a numerically expedient method to examine dynamo processes. [7] Recent 3 D dynamic dynamo models have also studied this process. Christensen and Wicht [2008] incorporate a very thick stably stratified layer surrounding a dynamo and 1of5

2 Table 1. Model Parameters and Resulting Dipole Tilts a Model r so Ra TBC (l, m) Average Dipole Tilt ( ) Dipole Tilt in Stable Epoch ( ) (2,0) (2,0) (3,0) (2,0) (2,0) a Ra = ag o hr o 2 /2Wh is a modified Rayleigh number, a is buoyancy expansion coefficient, g o is gravitational acceleration at the outer boundary, r o is radius of the outer boundary, h is buoyancy flux at the inner boundary, fixed at 0.35r o, W is angular rotation rate and h is magnetic diffusivity. r so is the ratio of the stable layer radius to the outer boundary radius. TBC (thermal boundary condition) is the spherical harmonic mode of the imposed laterally variable heat flux pattern, l is spherical harmonic degree and m is spherical harmonic order. The rms of the lateral variations is approximately 3 times the average superadiabatic heat flux. Other non dimensional parameters in the models are fixed at E = n/2wr o 2 =2 10 5, Ro = h/2wr o 2 =2 10 5, q k = /h =1,Pr = n/ =1whereE is Ekman number, v is kinematic viscosity, Ro is magnetic Rossby number, q k is magnetic Prandtl number, is thermal diffusivity, and Pr is the Prandtl number. We use fixed heat flux boundary conditions on the temperature, finite electrically conducting boundary conditions on the magnetic field and impenetrable and stressfree boundary conditions on the velocity field. Our maximum spherical harmonic degree l is 33 and order m is 21 and we use similar hyperdiffusivities as (Kuang and Bloxham, 1999). The average dipole tilt is calculated over several magnetic diffusion times which may include periods of reversals. The dipole tilt in stable epoch is the average dipole tilt during a dipole diffusion time when the field is not reversing nor experiencing large excursions (tilts >10 degrees). show that in certain parameter regimes, a more axisymmetric field can result. Although the thickness of the stable layer is not well constrained in Saturn, a very thick stable layer was not envisioned in the original helium rainout mechanism. Stanley and Mohammadi [2008] investigate the effects of thin stably stratified layers surrounding the dynamo. They demonstrate that a thin stably stratified layer, by itself, does not act to axisymmetrize the field. In contrast, because of interactions between the stable and unstable layers, a thermal wind is generated that can disrupt the dynamo sometimes resulting in more non axisymmetric, non dipolar fields. However, in these models, the poleto equator temperature variations envisioned by Stevenson to drive thermal winds in the layer were not included. [8] Here we use a 3 D dynamic dynamo model to investigate the effect of thermal winds resulting from temperature variations on the outer boundary of a thin stably stratified layer surrounding the dynamo region. Because our model is dynamic, the stable and unstable layers can interact and the fluid flows are influenced by Lorentz forces. Details of the numerical models are given in section 2, results in section 3 and discussions in section 4. their application to the geodynamo, see Kono and Roberts [2002]. [10] We make an outer portion of the dynamo region stable to convection in order to mimic the effect of the helium rainout layer in Saturn. In order to drive zonal flows in this layer, akin to the mechanism suggested by Stevenson [1982b], we impose latitudinally variable thermal boundary conditions at the top of the stable layer. This boundary would correspond to the boundary between the helium rainout layer and the surrounding molecular hydrogenhelium atmosphere in Saturn. The radius of the stable layer is constant in all the models as the outer 10 percent of the dynamo region. This would correspond to a layer of 3000 km thickness in Saturn. Our goal is to investigate the effects of the resulting thermal winds in the stably stratified layer surrounding the dynamo on the axisymmetry of the observed field. For simplification, we use a Boussinesq model and do not model the gradual transition in electrical conductivity between the molecular and metallic regions. [11] The stability of the layer is maintained through the background co density state, similar to the implementation of stable layers in other dynamo studies [e.g., Stanley and Bloxham, 2004; Stanley and Mohammadi, 2008; Christensen and Wicht, 2008]. The level of stability is unconstrained in Saturn, and we have therefore chosen a value such that very little convection penetrates the stable layer. We have found that a non dimensional background co density gradient: dc o /dr = 5 is adequate. The relevant equations, numerical methods and parameters are given by Kuang and Bloxham [1999]. Table 1 lists the parameters used in our models. We compare models with stable layers and different patterns of latitudinally variable thermal boundary conditions to models with (1) a stable layer but no variable thermal boundary condition (model 4), (2) no stable layer but a Saturn like thermal boundary condition (model 3), and (3) standard models with no stable layer nor variable thermal boundary condition (models 1 and 2). A Saturn like model implementing the thermal signatures proposed by 2. Numerical Model [9] We use the Kuang and Bloxham [1997, 1999] numerical dynamo model which has been shown to reproduce 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. For a recent review on dynamo theory, numerical models and Figure 1. Schematic view of dynamo region geometry and thermal boundary conditions. The solid inner region is shown in black, the electrically conducting metallic hydrogen region in orange, and the stably stratified helium rainout layer in blue. The rotation axis is depicted as the black vertical line and the temperature variations on the outer boundary are as indicated. 2of5

3 Figure 2. Filled contours of the radial component of the non dimensional magnetic field. The fields are plotted (left) at the top of the dynamo region (at r = 0.5R S ) and (right) at the surface (at r = R S ) for models (a) 1, (b) 6, (c) 5 and (d) 7. Stevenson [1982b] (warm equator, cold poles) is represented by models 7 and 8 and a schematic of the model set up is shown in Figure 1. We also investigate other patterns including models with equatorial regions cooler than polar regions (model 5) and models with an octupole pattern (model 6) in order to determine the influence of the equatorial symmetry and direction of zonal flows on the magnetic fields. 3. Results [12] The average dipole tilts for the models are given in Table 1. Only the models with Saturn like patterns and stable layers (models 7 & 8) produce fields with smaller dipole tilts than the standard models (1 & 2). It appears that the equatorial symmetry of the thermal boundary condition and the sign of the pattern are important in determining whether an axisymmetric field results. Table 1 also demonstrates that model 3, with no stable layer but Saturn like thermal boundary conditions, and model 4, with a stable layer but homogenous thermal boundaries, do not produce a more axisymmetric field (the latter is demonstrated by Stanley and Mohammadi [2008]). For models that experience reversals or large excursions, the average dipole tilt will increase due to the time spent in these epochs. We therefore also calculate the average dipole tilt in a stable epoch where the field experiences no reversals or large excursions. As Table 1 shows, even during these stable epochs, only the Saturn like models produce more axisymmetric fields than the standard models. [13] In Figure 2 we plot the radial magnetic field at the top of the stable layer (outer boundary of the model = 0.5R s ) and at the surface (1R s ) for various models in Table 1. Only the Saturn like model (Figure 2d) has a more axisymmetric field than the standard model (Figure 2a). Although some non axisymmetric small scale features are visible in the 3of5

4 Figure 3. Filled contours of (top) the non dimensional axisymmetric thermal winds and (bottom) differential rotation in meridional slices for models (a) 1, (b) 6, (c) 5 and (d) 7. In Figure 3 (top) we plot du /dz where u is the axisymmetric zonal flow and z is the coordinate in the direction of the rotation axis. In Figure 3 (bottom) we plot u /s where s is cylindrical radius. In Figures 3b 3d, the radius of the stably stratified layer is shown in black. Saturn like model in the 0.5R s plot, they have very little power in the surface field and the large scale components are extremely axisymmetric. A model with a different symmetry of vhf pattern is shown in Figure 2b and a model with the same mode but opposite sign to the Saturn model is shown in Figure 2c. Both of these models produce nondipolar fields and possess less axisymmetry than the Saturnlike and standard models. [14] In our models, latitudinal temperature variations result in thermal winds that drive zonal flows in both the stable and unstable layers (shown in Figure 3). In the standard model, these thermal winds are due to the pole to equator temperature perturbations that occur naturally as the result of convection. In models with stable layers and laterally homogeneous thermal boundary conditions, the temperature variations at the top of the stable layer drives thermal winds in the stable layer. The rotational constraint results in the interaction of the stable and unstable layer and can disrupt the natural zonal flows in the convection region. This can destabilize the dynamo resulting in more nonaxisymmetric fields as occurs in models 3 6. Only the Saturn like model (Figure 3d) produces a temperature perturbation of the same sign and geometry as those due to natural convection in the convective interior. The thermal winds in these models therefore do not act to disrupt the dynamo source region. Rather, they produce a shearing effect in the stable layer that axisymmetrizes the field. The importance of the direction of the zonal flows (resulting from the sign of the temperature perturbation at the outer boundary) is evident by examining model 5 (Figure 3c). The colder equatorial regions and warmer poles (the opposite of the expected Saturn pattern) does not produce favorable thermal winds and hence destabilizes the dynamo. Similarly, thermal winds of opposite equatorial symmetry to those produced by convection (such as model 6, Figure 3b) can destabilize the dynamo. 4. Discussions [15] We have shown that latitudinally varying thermal boundary conditions at the top of a stable layer surrounding a dynamo region can significantly effect the morphology of the resulting field. Not only are boundary conditions that are similar to those proposed to be Saturn like (hotter equator, colder poles) capable of producing axisymmetrized fields, they are the only pattern that produces axisymmetrized fields. Merely having the presence of zonal flows in the stable layer does not guarantee axisymmetrization. [16] Although our models are simplified in comparison to Saturn s interior, the mechanism demonstrated here: interaction of the stable and unstable layers due to the influences of rapid rotation, should still hold for the planet. In future 4of5

5 work we will investigate the effects of varying the magnitude of stratification and thermal variation and coupling between the molecular and metallic regions since we currently impose a stress free boundary condition such that the base of the molecular atmospheric layers do not couple to the stable layer. These studies should provide important constraints on the interior of Saturn and perhaps aid in determining basic properties of Saturn such as its bulk rotation rate [Giampieri and Dougherty, 2004]. [17] Acknowledgments. S. Stanley is partially funded by the National Science and Research Council of Canada (NSERC). Simulations were performed on supercomputing resources partially funded by the Canadian Foundation for Innovation and the Ontario Research Fund. References Acuna, M., and N. Ness (1980), Magnetic field of Saturn Pioneer 11 observations, Science, 207, Aubert, J. (2005), Steady zonal flows in spherical shell dynamos, J. Fluid Mech., 542, Aurnou, J., M. Heimpel, L. Allen, E. 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Jeanloz (2008), Fluid helium at conditions of giant planetary interiors, Proc.Natl.Acad.Sci.U.S.A., 105, 11,071 11,075, doi: /pnas S. Stanley, Department of Physics, University of Toronto, 60 St. George. St., Toronto, ON M5S 1A7, Canada. (stanley@physics.utoronto.ca) 5of5