Zonal flow and Vortices From Numerical Models of Deep Convection in Giant Planets Kyoto, 7 November, 2013

Moritz Heimpel

University of Alberta, Department of Physics, Edmonton, Canada

Zonal flow and Vortices From Numerical Models of Deep Convection in Giant Planets

Kyoto, 7 November, 2013

Acknowledgments

• Thomas Gastine & Johannes Wicht (MPI for Solar System Research)

• Jonathan Aurnou (UCLA)

• Natalia Gomez Perez (Universidad de los Andes)

• Keith Cuff (University of Alberta)

• Compute Canada

• ^Westgrid

Jupiter: Zonal (East-West) flows and vortices

NASA Movie PIA02863

(Cassini flyby Oct 31-Nov 9, 2000)

Saturn: Zonal flows and vortices and storm (white spot)

NASA Movie PIA06082

(Cassini: February 6 and March 30, 2004)

Outline

Part I: Zonal flows on Jupiter, Saturn, and in Boussinesq and anelastic numerical models of rotating convection.

• Effect of spherical shell geometry

• The topographic beta-effect and Rhines scaling for incompressible flow

• The compressible beta-effect and Rhines scaling with strong density stratification

Part II: Storms and convective vortices on giant planets and anelastic models of rotating convection.

• Saturns Great White Spot of 2010/2011

• Modelling: The role of outer boundary stability conditions

• The role of viscous and thermal diffusion

Interior of Jupiter

The radius coordinate is not displayed here but scales linearly from the center to the pressure level of 1 bar. In both of the models the core consists of rocks, J

₄

/10

^!4

¼ ! 5 : 84, ¯ Y ¼ 27 : 5%, and T

₁

¼ 170 K. There is no degree of freedom in model J11b, and the only degree of freedom in model J11a, P

_m

, is set to 4 Mbar in order to give Z

_mol

> 1. The three most important differences are (1) the steep onset of dissociation and ionization of H atoms with the PPT in model J11b whereas LM-REOS yields a smooth increase of dissociation; (2) a high transition pressure of 4 Mbar in model J11a, well beyond the location of the PPT in SCvH-ppt; (3) a ra- tio of heavy elements Z

_mol

: Z

_met

of 4 : 3 in model J11b compared to 1 : 6 with LM-REOS. It is interesting to note that once neutral H and He

⁺

are formed in the SCvH-ppt model, they are almost immediately ionized further, whereas the fraction of ionized He of only 6% in the deep interior in model J11a is not resolved.

For model J11a, an ASCII data table containing the profiles of pressure, temperature, density, composition, and the figure func- tions along the radius is available in the electronic edition of the Astrophysical Journal. A shortened version of this data table is shown in Table 4, where the five rows, from top to bottom, present the 1 bar–level surface, the transition from the outer to the inner envelope, and the layer transition from the inner envelope to the core.

5. EVALUATION OF THE NEW RESULTS

In x 5 we discuss to what extent the new results for M

_c

, Z

_met

, Z

_mol

, and P

_m

obtained by using LM-REOS are in agreement with

experimental EOS data (e.g., principal Hugoniot curve), evolu- tion theory, element abundances, and H / He phase separation.

5.1. Core Mass M

_c

Saumon & Guillot (2004) found that H-EOS with a small maximum compression ratio of only 4 along the Hugoniot curve can yield small core masses lower than 3 M

_#

and comment that the apparent relation between the stiffness along the principal Hugoniot and the core mass may be not unique. In the same sense, our H-REOS reproduces the maximum compression ratio of 4.25 as derived from shock wave experiments, but the core masses range up to 7 M

_#

.

To understand the indirect effect of the compressibility ! of hydrogen on the core mass, we study in particular its impact in the deep interior ( !

_met

), around the layer boundary between the envelopes ( !

_m

), and in the outer molecular region ( !

_mol

). The core mass depends essentially and directly on the mass density

"

_met

in the deep interior: the higher "

_met

, the lower M

_c

. Clearly,

"

_met

can be enhanced either by !

_met

or by Z

_met

. For example, the case of a smaller !

_met

leading to a smaller "

_met

occurs with LM- REOS. Furthermore, Z

_met

is diminished by both !

_m

, which re- duces the need for metals in order to adjust J

₂

, and the Z

_mol

chosen to reproduce J

₄

. Finally, Z

_mol

decreases with !

_mol

, for instance in the case of SCvH-ppt. Due to this propagation of effects, the behavior found in Saumon & Guillot (2004) may correspond to the coincidence of a small !

_m

and a large !

_met

between 5 and 15 Mbar (see Fig. 1 in Saumon & Guillot 2004). In agreement with Saumon & Guillot (2004) we conclude that the compress- ibility along the principal Hugoniot curve, which is restricted to densities below about 1 g cm

^!3

, does not determine the core mass alone. Experimental data for the hydrogen EOS off the principal Hugoniot curve, i.e., near the isentrope, are in this context urgently needed. For this, new experimental techniques such as reverber- ating shock waves or precompressed targets can be applied.

5.2. Abundance of Metals Z

_met

In our new Jupiter models with LM-REOS, Z

_met

is enriched over solar abundance by a factor of 5–10 and exceeds Z

_mol

by a factor of 4–30. This feature is consistent with the standard giant planet formation scenario, the core accretion model (Alibert et al.

2005; Pollack et al. 1996), where the planet grows first by ac- cretion of planetesimals onto a solid core embryo. If the core has grown such that surrounding nebula gas is attracted, an envelope forms and the planet grows by accretion of both gas and plan- etesimals, either sinking toward the core ( Pollack et al. 1996) or dissolving in the envelope. If the luminosity of the envelopes

Fig. 11.—Schemes of Jupiter models satisfying the same constraints; left, model J11a, ri g ht, model J11b. At the layer boundaries the values of pressure and temperature are given. The abundances of metals and of chemical species along the radius are indicated by gray scales. An arc segment corresponds to 100% in mass. For model J11a, an ASCII data table containing the profiles of pressure, temperature, density, composition, and the figure functions along the radius can be found in the electronic edition of the Astrophysical Journal.

TABLE 4

Jupiter Model J11a

m (M

_#

)

(1)

P

( Mbar) (2)

l ( ¯ R

_J

)

(3)

T ( K)

(4)

"

(g cm

^!3

) (5)

Y (%)

(6)

Z (%)

(7)

X

_H2

(8)

X

_He

(9)

X

_He^þ

(10)

s

₂

(11)

s

₄

(12)

s

₆

(13)

317.8336... 1.0000E ! 06 1.00000 1.7000E+02 1.6653E ! 04 23.307 2.072 1.0000 1.0000 0.0000 ! 4.501E ! 02 1.984E ! 03 ! 2.470E ! 04 .. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

223.6070... 3.9998E+00 0.72384 8.8682E+03 1.3239E+00 23.307 2.072 0.0095 1.0000 0.0000 ! 3.522E ! 02 1.062E ! 03 ! 1.742E ! 04 223.5810... 4.0013E+00 0.72380 8.8694E+03 1.5253E+00 24.476 16.616 0.0095 1.0000 0.0000 ! 3.522E ! 02 1.062E ! 03 ! 1.744E ! 04

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

2.75441... 3.8385E+01 0.08426 1.8571E+04 4.3293E+00 24.476 16.616 0.0000 0.9405 0.0574 ! 1.160E ! 02 1.063E ! 04 ! 7.278E ! 06 2.75308... 3.8393E+01 0.08426 1.8572E+04 1.8037E+01 0.000 0.000 0.0000 0.0000 0.0000 ! 1.160E ! 02 1.063E ! 04 ! 7.278E ! 06 Notes.—Table 4 is published in its entirety in the electronic edition of the Astrophysical Journal. A portion is shown here for guidance regarding its form and content.

Column headings from left to right: mass coordinate, pressure, level coordinate (radius; see eq. [8]), temperature, mass density, mass fraction of helium, mass fraction of metals (H

₂

O), particle fraction of H

₂

molecules with respect to the H subsystem, particle fraction of neutral He (col. [9]) and of singly ionized He (col. [10]) with respect to the He subsystem, and cols. (11)–(13): the dimensionless figure functions s

₂

; s

₄

; s

₆

.

NETTELMANN ET AL.

1226 Vol. 683

Nettelmann et al., (2008):

Ab-initio equation of state

Constraints on deep-seated zonal winds inside Jupiter and Saturn 655

Fig. 1. Electrical conductivity and magnetic diffusivity distributions inside giant planets: (a) Jupiter; (b) Saturn. Values of conductivity and magnetic diffusivity are plotted in the left and right panels, respectively. Solid lines depict mean value; dashed lines bound the range of uncertainties.

tribution, which we have neglected, from impurities x in addition to that from hydrogen:

n_e = ^nH2 exp

!

−_2K^Eg

BT

"

+ #

x

n_x exp

!

−_2K^Ex

BT

"

, (2)

where n_x and E_x express the number density of the electrons and the energy gap due to an impurity. Alkali metals are sources of small band gap impurities. They may also contribute to the radiative opacity thus insuring adiabaticity (Guillot et al., 2004;

Guillot, 2005). The mixing ratio of an alkali metal in the interior of a giant planet is presumably similar to that determined from its cosmic abundance. With these abundances, a band gap of a few electron volts would lead to a conductivity of 10⁻⁶ ∼ ^{10−4 S m−1}

at T ∼ 1000 K, significantly above the value due to hydrogen in the outer shells of giant planets.

In magnetohydrodynamics it is conventional to characterize the electrical conductivity σ in terms of the magnetic diffusiv- ity λ = (µ₀σ)⁻¹, where µ₀ is the magnetic permeability. Fig. 1 shows that the electrical conductivity of hydrogen decreases expo- nentially outward from the metallic conducting region. Therefore, the magnetic diffusivity increases exponentially outward. The scale height of magnetic diffusivity

H_λ(r) = λ(r)

dλ(r)/dr (3)

is shown in Fig. 2.

3. Ohmic dissipation based on inward extrapolation of the external magnetic field

We approximate the planet’s magnetic field as axisymmet- ric; Jupiter’s dipole tilt is about 10^◦ and Saturn’s less than 0.1^◦ (Connerney, 1993). Then we evaluate the azimuthal component of the magnetic field produced by differential rotation acting on the poloidal components. The maximum penetration depth is that of the level above which the associated Ohmic dissipation matches the planet’s net luminosity, L^.

To proceed, we need to know the poloidal magnetic field above the maximum penetration depth. Here we assume that it can be determined by inward extrapolation of the planet’s external mag- netic field. This assumption is appropriate provided the magnetic Reynolds number based on the convective velocity field, R_m^c , re- mains small down to the maximum penetration depth, which our estimates suggest it does.³

Lack of accurate magnetic field measurements at high latitudes close to Jupiter and Saturn makes the inward extrapolation of their external magnetic fields somewhat uncertain. Thus we cannot ex- clude the possibility that where R^c_m ≫ 1 the magnetic field might be closely aligned with the rotation axis. This possibility is exam- ined in Section 4.

3.1. Derivation of Ohmic dissipation

The time evolution of the magnetic field satisfies

∂B

∂t = ∇ × (U× ^B) − ∇ × $

λ(r)∇ × ^B%

, (4)

where U and B denote velocity and magnetic field. We work in spherical coordinates and set U = ^Uφeφ = ^{r sin}θ Ωe_φ. The genera- tion of toroidal field from poloidal field is described by

∂B_φ

∂t = ^rsinθ

!∂Ω

∂r B_r + ¹_r ∂Ω

∂θ ^Bθ

"

+ ¹_r ∂

∂r

! λ ∂

∂r(r B_φ)

"

+ λ

r²

∂

∂θ

! ₁

sinθ

∂

∂θ (sinθB_φ)

"

. (5)

We seek a steady-state solution noting that B_φ scales propor- tional to λ⁻¹, and H_λ is much smaller than the length scale for the meridional variation of U_φ and B. Thus we neglect r⁻¹∂/∂θ

with respect to ∂/∂r, which is equivalent to assuming that j_r ≪ ^jθ.4

3 An axisymmetric poloidal field is invariant under differential rotation.

4 A toy problem illustrating the effects that H_λ ≪ ^Hρ has on j is presented in Appendix A.

Liu et al., 1998:

Electrical conductivity in

the molecular envelope

E ⇥u

⇥t + (u · ⌅)u ⌅²u

⇥

+ 2ˆz ⇤ u = ⌅P + RaE P r

g

g_oˆrT + 1

P m (⌅ ⇤ B) ⇤ B

⌅ · u = 0 ⌅ · B = 0

⇥T

⇥t + u · ⌅T = 1

P r⌅²T

⇥B

⇥t = ⌅ ⇤ (u ⇤ B) 1

P m⌅ ⇤ ( (⌅ ⇤ B))

MHD rotating convection dynamo conservation equations (Boussinesq)

Rayleigh Ekman

Prandtl Magnetic Prandtl

E =

D² P r = ⇥

Dimensionless parameters

Ra = g_o T D³

⇥⇤

Numerical Model

The pseudo-spectral dynamo code MAGIC solves the Boussinesq dynamo equations (Glatzmaier and Roberts, 1995; Wicht, 2002). The

implementation of radially variable magnetic diffusivity, (λ = λ(r))

(Gomez-Perez, et al., 2010) allows the simultaneous simulation of the internal dynamo and the strong equatorial jet.

P r = ⇥

o

m

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 (l_max= 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.

[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 bal- ances 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 inter- ested in horizontal motions. The magnetic Reynolds number

Rm ¼VL

! ð1Þ

is best understood as the ratio of inertially driven magnetic induction to magnetic diffusion timescales. Thus magnetic field induction overtakes diffusion forRm> 1, andRmscales the local magnetic field generation. Via the omega effect, azimuthal differential rotation can act upon the poloidal component of the magnetic fieldB_pto amplify the azimuthal component B_" ∼ RmB_p [Roberts, 2007]. Whereas the char- acteristic length L of planetary dynamos is the radial extent of entire metallic region, the characteristic length of the 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/pffiffiffiffiffiffiffiffiffiffiffiffi#$!W (see also equation (4)). Azimuthal averages of the radial component B_r are shown in Figures 2a and 2b for casesr_m = 0.7 andr_m= 0.9 respectively. In Figure 2c,B_r is shown near the outer surface forr_m= 0.8 in Mollweide projection. Radiir_m andr_i = 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 " ‐ averaged azimuthal velocityV_", in units ofRofor r_m= 0.7 andr_m= 0.9, each scaled by the color bar to the right. (f) V_"near the outer surface for ther_m= 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, A_Rs, A, and L (see text), averaged over time and 8 for the r_m = 0.8 case. We refer to the radii bounded by the dashed lines in Figure 2i as the planetary tachocline (see also Figure 3).

HEIMPEL AND GÓMEZ PÉREZ: ZONAL JETS AND DYNAMO ACTION L14201 L14201

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Variable conductivity σ(r) dynamo:

High σ inside r

m

, expon. decreasing σ outside r

m

.

E = 10

^-5

, Ra = 1.3 x 10

⁸

, Pr = 1, Pm = 3, No hyperdiffusion.

Heimpel and Gómez Pérez, 2011

semiconducting molecular region is the scale height of the magnetic diffusivity L = L_l = l(dl/dr)⁻¹ [Liu et al., 2008].

For our models the scale height in the region of nearly constant conductivity (r < r_m) is more than one order of magnitude greater than that in the variable conductivity region (r > r_m). For example, in the case of r_m = 0.8, we have L = r_m − r_i = 0.45 inside r_m, and L = L_l = 0.031 outside r_m. The Alfvén number

A ¼

ffiffiffiffiffiffiffiffiffiffiffi B²

!"V² s

; ð2Þ

where B is the magnetic intensity and r 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 V_# and the non‐zonal (radial or latitu- dinal) convective velocity V_C. We refer to the ratio of the Lorentz force to the Reynolds stress as the Reynolds stress Alfvén number:

A_Rs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B²

!"V_CV_# s

; ð3Þ

where the overbar indicates an average over the azimuthal direction #. The azimuthal balance represented by A_Rs

suggests that we can replace B by B_# ∼ RmB_p. 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 com- ponents is not observable. However, equation (3) is appro- priate 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 dif- ferential rotation, we expect that the maximum depth of a rotationally driven equatorial jet will occur where A_Rs ∼ 1.

The Elsasser number,

L ¼ B²

!"$W; ð4Þ

where W is the planetary rotation rate, is the ratio of Lorentz to Coriolis forces. Numerical models typically have L of order unity (magnetostrophic balance). For L $ 1, a strong rotational constraint limits helicity and poloidal field gener- ation, while for L % 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 LRMS ∼ 0.04 at the outer boundary. Convection drives strong magnetic fields in the high conductivity fluid region, decreasing outward across r_m 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 2d–2g. The equatorial jets for the cases r_m = 0.7 and r_m = 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 V_# crosses zero corresponds roughly to the radius where A_Rs = 1. Since our model conserves angular momentum and V_# 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 r_m = 0.8, saturation of Rm, A and A_Rs occurs between the roll‐of radius of the conductivity profile r_m = 0.8 and r_m = 0.76, where L = 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 L = 1 classically represents a fully developed dynamo, it is natural to use this condition to define the bottom of the planetary tachocline (PT). A sche- matic 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 R_S (where Rm ∼ 1) to r = 0.77 (where A ∼ L ∼ 1), which is close to the result of our r_m = 0.8 numerical model. Similarly, for Jupiter the estimated PT range is 0.96 R_J – 0.93 R_J. None of our models Figure 3. Schematic of proposed dynamical structure of the

Jovian planets. The estimated radii for Rm ∼ 1, A_Rs ∼ 1, and L ∼ A ∼ 1 are, for Saturn 0.88 R_S, 0.83 R_S, and 0.77 R_S, respectively, and for Jupiter 0.96 R_J, 0.955 R_J, and 0.93 R_J, respectively.

HEIMPEL AND GÓMEZ PÉREZ: ZONAL JETS AND DYNAMO ACTION L14201 L14201

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Schematic diagram of

possible giant planet interior dynamical structure.

Reynolds stress Alfvén number,

semiconducting molecular region is the scale height of the magnetic diffusivity L = L

_l

= l (d l /dr)

⁻¹

[Liu et al., 2008].

For our models the scale height in the region of nearly constant conductivity (r < r

_m

) is more than one order of magnitude greater than that in the variable conductivity region (r > r

_m

). For example, in the case of r

_m

= 0.8, we have L = r

_m

− r

_i

= 0.45 inside r

_m

, and L = L

_l

= 0.031 outside r

_m

. The Alfvén number

A ¼

ffiffiffiffiffiffiffiffiffiffiffi B

²

!"V

²

s

; ð 2 Þ

where B is the magnetic intensity and r 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 V

_#

and the non‐zonal (radial or latitu- dinal) convective velocity V

_C

. We refer to the ratio of the Lorentz force to the Reynolds stress as the Reynolds stress Alfvén number:

A

_Rs

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B

²

!"V

_C

V

_#

s

; ð 3 Þ

where the overbar indicates an average over the azimuthal direction #. The azimuthal balance represented by A

_Rs

suggests that we can replace B by B

_#

∼ RmB

_p

. 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 com- ponents is not observable. However, equation (3) is appro- priate 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 dif- ferential rotation, we expect that the maximum depth of a rotationally driven equatorial jet will occur where A

_Rs

∼ 1.

The Elsasser number,

L ¼ B

²

!"$W ; ð 4 Þ

where W is the planetary rotation rate, is the ratio of Lorentz to Coriolis forces. Numerical models typically have L of order unity (magnetostrophic balance). For L $ 1, a strong rotational constraint limits helicity and poloidal field gener- ation, while for L % 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 L

_RMS

∼ 0.04 at the outer boundary. Convection drives strong magnetic fields in the high conductivity fluid region, decreasing outward across r

_m

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 2d–2g. The equatorial jets for the cases r

_m

= 0.7 and r

_m

= 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 V

_#

crosses zero corresponds roughly to the radius where A

_Rs

= 1. Since our model conserves angular momentum and V

_#

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 r

_m

= 0.8, saturation of Rm, A and A

_Rs

occurs between the roll‐of radius of the conductivity profile r

_m

= 0.8 and r

_m

= 0.76, where L = 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 L = 1 classically represents a fully developed dynamo, it is natural to use this condition to define the bottom of the planetary tachocline (PT). A sche- matic 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 R

_S

(where Rm ∼ 1) to r = 0.77 (where A ∼ L ∼ 1), which is close to the result of our r

_m

= 0.8 numerical model. Similarly, for Jupiter the estimated PT range is 0.96 R

_J

– 0.93 R

_J

. None of our models Figure 3. Schematic of proposed dynamical structure of the

Jovian planets. The estimated radii for Rm ∼ 1, A

_Rs

∼ 1, and L ∼ A ∼ 1 are, for Saturn 0.88 R

_S

, 0.83 R

_S

, and 0.77 R

_S

, respectively, and for Jupiter 0.96 R

_J

, 0.955 R

_J

, and 0.93 R

_J

, respectively.

HEIMPEL AND GÓMEZ PÉREZ: ZONAL JETS AND DYNAMO ACTION L14201 L14201

4 of 6 represents the balance

between Lorentz force, which damps fast zonal flow, and

Reynolds stress, which drives it.

A

Rs

~ 1

estimates the maximum depth of fast zonal flow.

Estimated radii for ARs ∼ 1 are, for Saturn 0.83 RS, and for Jupiter 0.95 RJ.

Heimpel, Aurnou, & Wicht., 2005

Part I: Zonal flow in spherical shells: Reynolds stress spins columnar convection (Busse columns) into zonal flows.

Busse, 1976

Jupiter Saturn Numerical model

Heimpel, Aurnou & Wicht, 2005

Jupiter & Saturn winds, and numerical model

Model: E=3x10^-6, Ra=5.6 x 10⁸, Pr=0.1, ri/ro=0.9, 8-fold symmetry Boussinesq approximation.

Constant top-bottom temperature difference drives convection.

Free-slip top and bottom boundary conditions

Conservation of potential vorticity Topographic effect

R.L. Stewart, 2004

For inviscid flow:

Π = ( ω

_z

+ f)/H .

The motion of fluid columns conserves

Π

Leads to a topographic parameter

β = - 2 Ω (dH/dx)/H

ω

₂

>ω

₁

ω

₁

Topographic β-effect effect applied to spherical geometry

β = - 2 Ω (dh/ds)/h

Rhines scale for 3D sphere or spherical shell

k ≃ ^(|β|/2U)

^1/2

^s

k ≃ sinθ (|β|/2U)

^1/2

g

= 2⇡

r

_o

k = 2⇡

r

_o

sin ✓

s U h

⌦ | dh/ds |

Spherical Shell Rhines Scale

Outside TC Inside TC

(Notes: 1. Scaling discontinuity across TC; 2. λ depends on U)

Jet width scaling

Rhines scale for Boussinesq model and Jupiter

Heimpel & Aurnou (2007)

L

_R

U

Jupiter: Best jet widths fit for ri/ro = 0.86

Turbulent convection in rapidly rotating spherical shells 549

Fig. 6. Velocities and jet boundaries for (a) Jupiter and (b) the numerical model. It is noted that the horizontal scale is different in (a) and (b). Observed profiles of the absolute value of the surface velocity |u| are shown as black curves. The actual velocity profiles (showing the retrograde jets in gray) are plotted for reference.

The jet boundaries are plotted as light solid lines of latitude. The dashed lines of latitude are at the peaks of the jets nearest the equator. The jet width associated with those peaks is twice the latitude distance from the dashed line to the next jet boundary at higher latitude. The scaling velocityU (θ) (red for Jupiter and blue for the numerical model) is calculated from(23), which is based on the assumption of constant Rhines length in each of the three regions; north, south and equatorial.

(For interpretation of the references to color in this figure ledend, the reader is referred to the web version of this article.)

the velocity scale U is taken to be ⟨|u|⟩^J, which represents lati- tudinal mean of the azimuthally averaged zonal surface velocity for each jet.

Figs. 7–10 show the applicability of Rhines scaling to the zonal flow on Jupiter and for our numerical simulations. Fig. 7 compares the observed and simulated jet widths against the jet widths predicted by Rhines scaling. For the simulations, the correspondence between measured and predicted jet widths is seen to be close outside the tangent cylinder, although, for the χ = 0.9 case the highest latitude simulated jets are signifi- cantly narrower than predicted by Rhines scaling. For both the χ = 0.85 and χ = 0.9 cases the simulated equatorial jets are much broader than predicted. For Jupiter the predicted jet width values are calculated using χ = 0.9 in (9) and (13). Although there is clearly some misfit between the measured and predicted jet widths, it is remarkable that Jupiter’s measured jet widths correspond well to the values calculated using Rhines scaling, except at the highest latitudes, where, similar to the simulated cases, Rhines scaling overestimates the jet widths.

Fig. 8 shows the measured jet widths for Jupiter against the predicted jet widths for several values of the radius ratio χ. Fig. 9 shows the magnitude of the difference between the mea- sured and predicted jovian jet widths (the misfit) with the same set of χ—values as inFig. 7. These two figures show that, while it is plausible to fit Jupiter’s jet widths with those predicted by Rhines scaling for any spherical shell layer depth (or ra- dius ratio χ), the best fits occur for radius ratios between about

0.85 and 0.95. This is graphically quantified in Fig. 10, which shows the misfit between measured and predicted values, av- eraged over all measured jovian jets, as a function of χ. Here we see that, compared to what Rhines scaling predicts for full sphere (χ = 0) or an extremely shallow layer (χ ≃1) the mini- mum misfit (at χ = 0.86) is improved by roughly 33%.

3.4. A geometrical jet scaling function

The geostrophic Rhines wavelength, in radians of latitude on the surface of a rapidly rotating spherical shell, is speci- fied by (9) and (13) outside and inside the tangent cylinder, respectively. In this formulation λ_g depends on the spherical geometry, the velocity scale U, the angular rotation rate Ω and the characteristic length-scales of the system, r_o and D. We may isolate the purely geometrical part of the Rhines scaling relationship by writing

(14) ξ = λ_g

2π

!r_oΩ

U

which we refer to as the jet scaling parameter. Combining (9), (13) and (14) we see that, outside the tangent cylinder,

(15) ξ(θ) =

! 1

cosθ ; |θ| ! cos⁻¹χ,

Turbulent convection in rapidly rotating spherical shells 549

Fig. 6. Velocities and jet boundaries for (a) Jupiter and (b) the numerical model. It is noted that the horizontal scale is different in (a) and (b). Observed profiles of the absolute value of the surface velocity |u| are shown as black curves. The actual velocity profiles (showing the retrograde jets in gray) are plotted for reference.

The jet boundaries are plotted as light solid lines of latitude. The dashed lines of latitude are at the peaks of the jets nearest the equator. The jet width associated with those peaks is twice the latitude distance from the dashed line to the next jet boundary at higher latitude. The scaling velocityU (θ) (red for Jupiter and blue for the numerical model) is calculated from (23), which is based on the assumption of constant Rhines length in each of the three regions; north, south and equatorial.

(For interpretation of the references to color in this figure ledend, the reader is referred to the web version of this article.)

the velocity scale U is taken to be ⟨|u|⟩^J, which represents lati- tudinal mean of the azimuthally averaged zonal surface velocity for each jet.

Figs. 7–10 show the applicability of Rhines scaling to the zonal flow on Jupiter and for our numerical simulations. Fig. 7 compares the observed and simulated jet widths against the jet widths predicted by Rhines scaling. For the simulations, the correspondence between measured and predicted jet widths is seen to be close outside the tangent cylinder, although, for the χ = 0.9 case the highest latitude simulated jets are signifi- cantly narrower than predicted by Rhines scaling. For both the χ = 0.85 and χ = 0.9 cases the simulated equatorial jets are much broader than predicted. For Jupiter the predicted jet width values are calculated using χ = 0.9 in (9) and (13). Although there is clearly some misfit between the measured and predicted jet widths, it is remarkable that Jupiter’s measured jet widths correspond well to the values calculated using Rhines scaling, except at the highest latitudes, where, similar to the simulated cases, Rhines scaling overestimates the jet widths.

Fig. 8 shows the measured jet widths for Jupiter against the predicted jet widths for several values of the radius ratio χ. Fig. 9 shows the magnitude of the difference between the mea- sured and predicted jovian jet widths (the misfit) with the same set of χ—values as inFig. 7. These two figures show that, while it is plausible to fit Jupiter’s jet widths with those predicted by Rhines scaling for any spherical shell layer depth (or ra- dius ratio χ), the best fits occur for radius ratios between about

0.85 and 0.95. This is graphically quantified in Fig. 10, which shows the misfit between measured and predicted values, av- eraged over all measured jovian jets, as a function of χ. Here we see that, compared to what Rhines scaling predicts for full sphere (χ = 0) or an extremely shallow layer (χ ≃1) the mini- mum misfit (at χ = 0.86) is improved by roughly 33%.

3.4. A geometrical jet scaling function

The geostrophic Rhines wavelength, in radians of latitude on the surface of a rapidly rotating spherical shell, is speci- fied by (9) and (13) outside and inside the tangent cylinder, respectively. In this formulation λ_g depends on the spherical geometry, the velocity scale U, the angular rotation rate Ω and the characteristic length-scales of the system, r_o and D. We may isolate the purely geometrical part of the Rhines scaling relationship by writing

(14) ξ = λ_g

2π

!r_oΩ

U

which we refer to as the jet scaling parameter. Combining (9), (13) and (14) we see that, outside the tangent cylinder,

(15) ξ(θ) =

! 1

cosθ ; |θ| ! cos⁻¹χ,

Outer BC’s: Free slip, Interior - Solar heat flux = 4/3 - cos(latitude).

Inner BC’s: No slip, zero heat flux. Internal heating.

Ra = 2.2 x 10

⁹

, E = 3 x 10

^-6

, Pr = 0.1, lmax = 240, light hyperdiffusion Full sphere, radius ratio r

i/

r

o

= 0.85

Outer boundary prograde velocity

Outer boundary temperature

Azimuthally averaged prograde

velocity

Meridional Temperature

slice

Outer BCs: Solar heat flux cos(lat), Free slip.

Inner BCs: Zero heat flux, No slip Ra = 2.2 x 10

⁹

, E = 3 x 10

^-6

, Pr = 0.1

Radius ratio r

i/

r

o

= 0.85

−80 −60 −40 −20 0 20 40 60 80

−0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

−80 −60 −40 −20 0 20 40 60 80

−100 0 100 200 300 400 500

Wind speed (m/s)

Cassini

Voyager 1 & 2

−0.01 0 0.01 0.02 0.03 0.04 0.05

Rossby number

Latitude

Saturn

_Numerical

model