On the relationship between zonal jets and dynamo action in giant planets



[1] Jupiter and Saturn exhibit similar large-scale dynamical features. Each planet has a prograde equatorial jet and a deeply seated dipolar magnetic field. Compared to Jupiter, Saturn's jet is broader and faster, while its magnetic field is weaker and more axially symmetric. The Sun also has prograde equatorial flow and a large-scale axial magnetic field. While the depth of the Sun's differential rotation is well constrained by helioseismology, the depth to which the zonal winds penetrate in the giant planets is not known and has been a subject of debate. Although magnetic braking has been invoked as the mechanism to slow the winds at depth, such a mechanism has not previously been demonstrated. Here we present the first self-consistent numerical planetary dynamo models in which slow convection in the interior dynamo source region coexists with strong zonal flow near the outer surface. The models include radially variable electrical conductivity and show that prograde zonal flow penetrates to a depth where Lorentz forces balance the Reynolds stress, which drives the equatorial jet. Our results imply that major differences between the surface zonal flows of Jupiter and Saturn arise from the different depths and conditions of a transition layer analogous to the solar tachocline. This transition layer, the planetary tachocline, separates the high velocity, semiconducting molecular envelope from the slow moving liquid metal interior dynamo.

1. Introduction

[2] The dynamics of giant planets and stars are controlled, in large part, by the interaction of rotation, convection and the global magnetic field, which is generated in the region of the deep interior where fluid motions and electrical conductivity are sufficient to maintain a dynamo process. Shock experiments have yielded estimates of the hydrogen metallization pressure at 140 GPa, where hydrogen attains σ ∼ 2 × 105 S/m, the minimum electrical conductivity for a metal [Nellis et al., 1996]. This marks the transition from the molecular envelope to the liquid metal interior, and corresponds to estimated radii of 0.84RJ and 0.63RS for Jupiter and Saturn respectively [Helled et al., 2009; Liu et al., 2008]. Recent quantum mechanical ab-initio models, which calculate the dissociation of molecular H2 to atomic H, result in lower metallization pressures of ∼50 GPa, corresponding to 0.9RJ and 0.7RS [Holst et al., 2008; Nettelmann et al., 2008]. Within the semiconducting molecular envelope, the electrical conductivity decreases exponentially with radius to about 30 GPa, and then superexponentially toward the shallow atmosphere [Nellis et al., 1996].

[3] The dynamos of the giant planets reside mainly in their convective liquid metal H-He cores [Jones et al., 2010; Stevenson, 1982]. The dynamo source region, which likely includes part of the deep molecular envelope, is subject to strong magnetic Lorentz forces that effectively damp large-scale zonal flows [Kirk and Stevenson, 1987; Liu et al., 2008; Stanley and Glatzmaier, 2010]. A typical fluid velocity in deep interiors of the giant planets may be ∼10−3 m/s–about an order of magnitude faster than flow driving the geodynamo in Earth's liquid iron core, and five orders of magnitude slower than the shallow winds of Jupiter and Saturn. The Rossby number Ro = V/(ΩR), scales the flow velocity V to the rotation rate Ω for bodies of radius R. In the giant planets low Ro is consistent with small scale (latitudinally confined) meridional circulation [Liu and Schneider, 2010], and stable (non-reversing), dipolar magnetic fields [Olson and Christensen, 2006].

[4] In contrast to the giant planets, the solar dynamo is seated in the tachocline and outer convection zone. The Sun's frequently reversing, multipolar magnetic field is consistent, from the perspective of planetary dynamo models, with relatively high Ro in the dynamo source region [Olson and Christensen, 2006]. Furthermore, observed global meridional flow seems to be linked to the quasiperiodic solar dynamo reversal frequency [Dikpati and Charbonneau, 1999; Hathaway et al., 1996]. Despite these major differences between the dynamics of the Sun and the giant planets, a comparison of flow structures shows striking similarities, which suggests the universal nature of the interaction of rotation and convection. Strong prograde zonal flow in the equatorial region is observed at the surface of each of the three bodies. Values of Ro for the equatorial flow of the Sun, Jupiter, and Saturn are roughly 0.2, 0.008, and 0.04 respectively. Helioseismology has shown that the Sun's differentially rotating surface zonal flow is underlain by flow structures that roughly conserve angular velocity down to the solar tachocline − a thin shear layer at about 0.7 R that is thought to effectively separate the fast zonal flow in the convection zone from the stably stratified radiative interior, which rotates nearly as a rigid body [Schou et al., 1998].

[5] The shallow azimuthal (zonal) flow velocity of Jupiter and Saturn, based on cloud motions [Porco et al., 2003; Sanchez-Lavega et al., 2000; Vasavada and Showman, 2005], is shown in Figure 1. A purely observational basis for estimating the depth to which the zonal winds penetrate in the giant planets is not currently available. However, the upcoming Juno mission to Jupiter could provide data sufficient to detect the gravitational signal of deep zonal winds [Kaspi et al., 2010]. Using analytical models based on radial conductivity distributions, Liu et al. [2008] compared the planetary luminosity to ohmic heating associated with deep zonal flows, and estimated that the minimum depths to which fast zonal flow can penetrate correspond to 0.96RJ for Jupiter and 0.86RS for Saturn. They then argued that, at those depths, forces are insufficient to truncate the cylindrical projections of high latitude zonal jets. This led them to conclude that fast zonal flows at such depths are unlikely, and that the transition to fast zonal flow occurs only at much shallower depths. However, the assumptions concerning magnetic field and flow morphology underlying those conclusions have been challenged [Glatzmaier, 2008] (see auxiliary material).

Figure 1.

Observed surface wind profiles of Jupiter and Saturn. The Rossby number is Ro = Vϕ/(ΩR), where Ω is the planetary rotation rate and Vϕ is the azimuthal (zonal) velocity, inferred by tracking of clouds at the planetary radius R. Jupiter data (black curve) and southern hemisphere Saturn data (red curve) are from the Cassini mission [Vasavada and Showman, 2005]. Voyager 1 and 2 Saturn data are shown by the blue curve [Sanchez-Lavega et al., 2000].

[6] Models of convection and flow in rotating spheres have postulated that zonal flows extend into the deep interior [Busse, 1976, 1983; Yano et al., 2003]. More recent numerical models showed that deeply driven and seated zonal flows are consistent with the latitudinally flat thermal emission profiles of Jupiter and Saturn [Aurnou et al., 2008], and that the scaling of equatorial and high latitude jet widths can be explained by a tangent cylinder (TC), the imaginary axial cylinder of radius where the equatorial jet is truncated [Heimpel and Aurnou, 2007; Heimpel et al., 2005]. The jet structure of those incompressible (Boussinesq approximation) models has been verified by models of compressible rotating convection, using the anelastic approximation [Jones and Kuzanyan, 2009]. In this paper we use dynamo models with radially variable electrical conductivity to study the depth of fast zonal flow.

2. Methods and Scaling

[7] Our numerical model is based on a legacy pseudo-spectral dynamo code that solves the coupled non-dimensional equations for conservation of mass, momentum, and energy for a Boussinesq, conducting fluid in a spherical shell, bounded on the inside by a rigid conducting sphere (inner core), which is allowed to change its rotation due to viscous and magnetic torques [Christensen et al., 1998; Glatzmaier, 1984; Glatzmaier and Roberts, 1995; Wicht, 2002]. Here we include recently implemented radially variable electrical conductivity in the fluid layer. Details of the numerical method and definitions of non-dimensional input parameters are given by Gómez-Pérez et al. [2010]. The electrical conductivity, which is the inverse of the magnetic diffusivity λ = μ−1σ−1, where the magnetic permeability μ is essentially that of free space, is constant in the rigid inner sphere of radius ri = 0.35. For each case the magnetic Prandtl number at the inner sphere boundary is Pm = v/λ(ri) = 3. However, the three model cases have different radial electrical conductivity profiles. The conductivity is continuous across the inner boundary and varies slowly in the deep fluid region. Starting at a specified radius rm, the electrical conductivity decreases exponentially, by three orders of magnitude, to a minimum value at the outer boundary (ro = 1). The three cases have rm = 0.7, 0.8 and 0.9 (see Figure 2h). Other input parameters are, for the three cases, identical: the Prandtl number Pr = 1, Ekman number E = 1 × 10−5, and the Rayleigh number Ra = 1.3 × 108. Velocity boundary conditions are no-slip at the inner core boundary and free-slip at the outer boundary. Uniform temperature boundary conditions are imposed at both boundaries and numerical runs were initiated with a small temperature perturbation. Time stepping was carried out for greater than 1000 rotations, which was sufficient to achieve a quasi-steady state in the kinetic and magnetic energy time series. The spatial resolution of these full-sphere calculations is defined by the maximum spherical harmonic degree (lmax = 200) and by the grid, which has 17 radial levels in the inner core and 65 radial levels in the fluid outer core. No hyperdiffusivity was used for these runs.

Figure 2.

Results from the numerical model. (a–c) Snapshots of the magnetic field scaled by the upper color bar, are shown in model units B/equation image (see also equation (4)). Azimuthal averages of the radial component equation imager are shown in Figures 2a and 2b for cases rm = 0.7 and rm = 0.9 respectively. In Figure 2c, Br is shown near the outer surface for rm = 0.8 in Mollweide projection. Radii rm and ri = 0.35 are indicated in Figure 2a and 2b, and in Figure 2c, where the intersections of the tangent cylinders for those radii with the outer boundary are shown. (d and e) Meridional snapshots of ϕ - averaged azimuthal velocity equation image, in units of Ro for rm = 0.7 and rm = 0.9, each scaled by the color bar to the right. (f) Vϕ near the outer surface for the rm = 0.8 case. The velocities for Figure 2f are obtained by using the color scale for e multiplied by 10. (g) The three velocity profiles correspond to (h) the three electrical conductivity profiles. (i) The major force balances: Rm, ARs, A, and Λ (see text), averaged over time and ϕ for the rm = 0.8 case. We refer to the radii bounded by the dashed lines in Figure 2i as the planetary tachocline (see also Figure 3).

[8] To explain the relationship between the magnetic and flow fields in our models, and to estimate the depth to which fast zonal flow may exist in giant planets, we estimate balances between rotational (Coriolis) and magnetic (Lorentz) forces, and magnetic diffusion, which drive and resist large scale flows in spherical shells. We neglect buoyancy for this discussion because it acts radially, and we are mainly interested in horizontal motions. The magnetic Reynolds number

equation image

is best understood as the ratio of inertially driven magnetic induction to magnetic diffusion timescales. Thus magnetic field induction overtakes diffusion for Rm > 1, and Rm scales the local magnetic field generation. Via the omega effect, azimuthal differential rotation can act upon the poloidal component of the magnetic field Bp to amplify the azimuthal component BϕRmBp [Roberts, 2007]. Whereas the characteristic length L of planetary dynamos is the radial extent of entire metallic region, the characteristic length of the semiconducting molecular region is the scale height of the magnetic diffusivity L = Lλ = λ(dλ/dr)−1 [Liu et al., 2008]. For our models the scale height in the region of nearly constant conductivity (r < rm) is more than one order of magnitude greater than that in the variable conductivity region (r > rm). For example, in the case of rm = 0.8, we have L = rmri = 0.45 inside rm, and L = Lλ = 0.031 outside rm. The Alfvén number

equation image

where B is the magnetic intensity and ρ is the density, represents the balance between Lorentz and inertial forces. The locally generated magnetic field increases with depth as the magnetic diffusivity decreases, but saturates as the flow velocity decreases, due to increasing Lorentz forces. The classical saturation limit for slowly rotating systems is reached where A ∼ 1 the condition for kinetic and magnetic energy density equipartition [Galloway et al., 1977]. The Reynolds stress is proportional to the correlation between the zonal velocity Vϕ and the non-zonal (radial or latitudinal) convective velocity VC. We refer to the ratio of the Lorentz force to the Reynolds stress as the Reynolds stress Alfvén number:

equation image

where the overbar indicates an average over the azimuthal direction ϕ. The azimuthal balance represented by ARs suggests that we can replace B by BϕRmBp. However, such a representation may oversimplify magnetic field amplification, which occurs via nonlinear feedback between turbulent flow and the toriodal and poloidal magnetic field components. In planets the relationship between those components is not observable. However, equation (3) is appropriate for our numerical model since we obtain the magnetic field components directly. Since the Reynolds stress provides the torque that drives the zonal flow of giant planets and stars [Brummell et al., 1998], and the Lorentz force opposes differential rotation, we expect that the maximum depth of a rotationally driven equatorial jet will occur where ARs ∼ 1. The Elsasser number,

equation image

where Ω is the planetary rotation rate, is the ratio of Lorentz to Coriolis forces. Numerical models typically have Λ of order unity (magnetostrophic balance). For Λ ≪ 1, a strong rotational constraint limits helicity and poloidal field generation, while for Λ ≫ 1 the organization of axial convection cells by the Coriolis force deteriorates, and the magnetic field tends to become smaller-scale and multipolar. Thus, the magnetostrophic balance represents an optimum state for the generation of a large scale external magnetic field.

3. Results and Implications for Giant Planets

[9] All three cases produce dynamos of similar strength, with an RMS value of the Elsasser number ΛRMS ∼ 0.04 at the outer boundary. Convection drives strong magnetic fields in the high conductivity fluid region, decreasing outward across rm into the low-conductivity region. The field near the outer boundary is nearly an axial dipole. Fast equatorial zonal flow develops outside the region of high conductivity in each case. This is shown Figures 2d2g. The equatorial jets for the cases rm = 0.7 and rm = 0.9 are seen to occupy the volume outside the tangent cylinder defined by the radial conductivity structure (see Figures 2d and 2e). For the equatorial jets in each case, the radius where Vϕ crosses zero corresponds roughly to the radius where ARs = 1. Since our model conserves angular momentum and Vϕ is relative to the velocity of initial (non-convective) solid body rotation, this result implies that the depth of the equatorial jet is set by the balance between the Reynolds stress, which drives the jet, and the Lorentz force, which damps it. For the case with rm = 0.8, saturation of Rm, A and ARs occurs between the roll-of radius of the conductivity profile rm = 0.8 and rm = 0.76, where Λ = 1 (see Figures 2h and 2i).

[10] Here we estimate the radii at which important force balances occur in Jupiter and Saturn. We base these estimates on our model results and on the conductivity profiles of Liu et al. [2008], which are based on the conductivity models of Nellis et al. [1996]. Since Λ = 1 classically represents a fully developed dynamo, it is natural to use this condition to define the bottom of the planetary tachocline (PT). A schematic representation of our estimates, applied to the giant planets, is shown in Figure 3. We estimate that Saturn's PT ranges in radial extent from r = 0.88 RS (where Rm ∼ 1) to r = 0.77 (where A ∼ Λ ∼ 1), which is close to the result of our rm = 0.8 numerical model. Similarly, for Jupiter the estimated PT range is 0.96 RJ – 0.93 RJ. None of our models adequately represent Jupiter in terms of depths of the layers. Nevertheless, comparing the velocity profiles of the rm = 0.9 and rm = 0.7 cases, we find that both the breadth of the surface equatorial jet, as well as its characteristic velocity scales linearly with the depth of the low-conductivity region. This is consistent with observations of the zonal winds of Jupiter and Saturn (see Figure 1), and the present understanding of the conductivity structure of the giant planets.

Figure 3.

Schematic of proposed dynamical structure of the Jovian planets. The estimated radii for Rm ∼ 1, ARs ∼ 1, and Λ ∼ A ∼ 1 are, for Saturn 0.88 RS, 0.83 RS, and 0.77 RS, respectively, and for Jupiter 0.96 RJ, 0.955 RJ, and 0.93 RJ, respectively.

[11] Although our numerical models reproduce the large-scale dynamical structure of the Jupiter and Saturn, there are significant differences between the input, as well as output, of our numerical models and the estimated parameters for the giant planets. For the dipolar dynamo calculations presented here, we limited Ro of the equatorial jet to about 1/5 that of the maximum surface wind speeds of Jupiter and Saturn. Higher Ro calculations (unpublished) also yield prograde equatorial jets, but multipolar dynamos, which result from higher Ro in the dynamo region. The relationship between high Ro and multipolar magnetic fields is consistent with dynamo scaling, and has been observed in previous models [Gómez-Pérez and Heimpel, 2007; Olson and Christensen, 2006]. In addition our mean model convective velocities are about one order of magnitude lower than the zonal velocities. For planets, with very low diffusion, the ratio of zonal and convective velocities is several orders of magnitude. Future work should produce stronger (and more realistic) velocity gradients, which are likely required for dipolar interior dynamos to coexist with higher Ro equatorial jets.

[12] For 3D numerical deep convection and dynamo models computational resolution requires the use of thermal, magnetic and viscous diffusion parameters several orders of magnitude greater than those in the planets. With our model Ekman number E = νΩ−1L−2 = 10−5, the Jovian values Ω = 1.8 × 10−4 s−1 and L = 6 × 107 m yield a scaled kinematic viscosity of ν = 6 × 106 m s−1, roughly six orders of magnitude greater than estimates of turbulent diffusivities of the giant planets [Starchenko and Jones, 2002]. These high numerical diffusion values, which are typical of planetary deep convection and dynamo models, mean that to drive numerical flows with velocities of planetary scale, heat flow in numerical models must be greatly over-forced to obtain kinematic conditions that are representative of planets [Jones and Kuzanyan, 2009; Showman et al., 2011]. However, the high model heat flow values may not be problematic in terms of using models to interpret large-scale planetary flow fields [Jones and Kuzanyan, 2009]. Clearly, the viscous and ohmic dissipation in our numerical models are also very high. However, we find that losses from the integrated ohmic and viscous dissipation are substantially less than the mean model heat flow. This is consistent with a previous study that scaled a set of numerical models to the geodynamo [Christensen and Tilgner, 2004]. Since our models do not yield strong, nor steady high-latitude jets we cannot conclude anything about ohmic dissipation associated with truncation of the cylindrical projection of fast high latitude flows. However, we find that the existence of the equatorial jet does not yield particularly high ohmic losses associated with the velocity gradient near the tangent cylinder (see auxiliary material).

[13] While Mercury, Earth, Uranus and Neptune also have internal dynamos, we have focused here on Jupiter and Saturn, which, like the Sun, are composed mostly of hydrogen and helium. In addition, our results may be applicable to a multitude of exoplanets [Baraffe et al., 2010; Showman et al., 2010]. Since most of the discovered exoplanets are the size of Jupiter or larger, studying these bodies broadens our understanding of the relationships between the dynamics of giant planets and the Sun and stars.


[14] Computational resources were provided by WestGrid and Compute Canada. Funding was provided by NSERC.

[15] The Editor thanks Adam Showman and an anonymous reviewer for their assistance in evaluating this paper.