Scaling laws for convection and jet speeds in the giant...

Submitted to Icarus

Scaling laws for convection and jet speeds in the giant planets

Adam P. Showman

University of Arizona, Tucson, AZ and Columbia University, New York, NY

Yohai Kaspi

California Institute of Technology

Glenn R. Flierl

Massachusetts Institute of Technology

abstract

Three dimensional studies of convection in deep spherical shells have been used to test the hypothesisthat the strong jet streams on Jupiter, Saturn, Uranus, and Neptune result from convection throughoutthe molecular envelopes. Due to computational limitations, these simulations must be performed atparameter settings far from Jovian values and generally adopt heat fluxes 5–10 orders of magnitudelarger than the planetary values. Several numerical investigations have identified trends for how themean jet speed varies with heat flux and viscosity, but no previous theories have been advanced toexplain these trends. Here, we show using simple arguments that if convective release of potentialenergy pumps the jets and viscosity damps them, the mean jet speeds split into two regimes. Whenthe convection is weakly nonlinear, the equilibrated jet speeds should scale approximately with F/ν,where F is the convective heat flux and ν is the viscosity. When the convection is strongly nonlinear,the jet speeds are faster and should scale approximately as (F/ν)1/2. We demonstrate how this regimeshift can naturally result from a shift in the behavior of the jet-pumping efficiency with heat flux andviscosity. Moreover, both Boussinesq and anelastic simulations hint at the existence of a third regimewhere, at sufficiently high heat fluxes or sufficiently small viscosities, the jet speed becomes independentof the viscosity. We show based on mixing-length estimates that if such a regime exists, mean jet speedsshould scale as heat flux to the 1/4 power. Our scalings provide a good match to the mean jet speedsobtained in previous Boussinesq and anelastic, three-dimensional simulations of convection within giantplanets over a broad range of parameters. When extrapolated to the real heat fluxes, these scalingssuggest that the mass-weighted jet speeds in the molecular envelopes of the giant planets are muchweaker—by an order of magnitude or more—than the speeds measured at cloud level.

——————–

1. Introduction

At the cloud levels near ∼1 bar pressure, numerouseast-west (zonal) jet streams dominate the meteorology

Corresponding author address:

A.P. Showman, Department of Planetary Sciences, Lunarand Planetary Laboratory, University of Arizona, Tucson,AZ 85721 ([email protected]). Currently on sabbat-ical at Columbia University, Department of Applied Physicsand Applied Mathematics, New York, NY 10027.

of the giant planets Jupiter, Saturn, Uranus, and Nep-tune, but the depth to which these jets extend into theinterior remains unknown. Endpoint theoretical scenar-ios range from weather-layer models where the jets areconfined to a layer several scale heights deep to modelswhere the jets extend throughout the molecular enve-lope (∼104 km thick) on cylinders parallel to the ro-tation axis (for a review see Vasavada and Showman2005). For Jupiter, in-situ observations by the Galileo

1

2 A.P. Showman et al., submitted to Icarus

probe at 7N latitude show that the equatorial jet ex-tends to at least ∼20 bars (∼150 km below the visibleclouds) (Atkinson et al. 1997), and indirect inferencessuggest that the jets at other latitudes extend to atleast ∼5–10 bars pressure (e.g., Dowling 1995; Legar-reta and Sanchez-Lavega 2008; Morales-Juberias andDowling 2005; Sanchez-Lavega et al. 2008). At Nep-tune, gravity data suggest that the fast jets are con-fined to the outermost few percent of the planet’s mass(Hubbard et al. 1991). Comparable data are currentlylacking for Jupiter and Saturn but will be obtainedby NASA’s Juno and Cassini missions, respectively, incoming years.

Three-dimensional (3D) numerical simulations of con-vection in rotating spherical shells have been performedby several groups to investigate the possibility that thejets on the giant planets result from convection in theinterior. Both free-slip and no-slip momentum bound-ary conditions at the inner and outer boundaries havebeen explored; of these, the free-slip case—which allowsthe development of strong zonal flows—is most relevantto giant planets. So far, such studies have neglectedthe high electrical conductivity and coupling to mag-netic fields expected to occur in the deep (∼> 1 Mbar)planetary interior.

In this line of inquiry, most studies to date make theBoussinesq assumption in which the basic-state density,thermal expansivity, and other background propertiesare assumed constant with planetary radius; the con-vection is driven by a constant temperature differenceimposed between the bottom hot boundary and top coldboundary (Aurnou et al. 2008; Aurnou and Heimpel2004; Aurnou and Olson 2001; Christensen 2001, 2002;Heimpel and Aurnou 2007; Heimpel et al. 2005). Thesestudies show that convection with free-slip boundariescan drive multiple jets with speeds greatly exceeding theconvective speeds. When the convection occurs withina thick shell, typically only ∼3–5 jets form (Aurnouand Olson 2001; Christensen 2001, 2002), whereas whenthe shell thickness is only ∼10% of the planetary ra-dius, then ∼15–20 jets can occur, similar to the numberobserved on Jupiter and Saturn (Heimpel and Aurnou2007; Heimpel et al. 2005).

In real giant planets, however, the density and ther-mal expansivity each vary by several orders of magni-tude from the cloud layer to the deep interior, and a newgeneration of convection models is emerging to accountfor this strong radial variation in basic-state proper-ties. Using the anelastic approximation, which accountsfor this layering, Evonuk and Glatzmaier (2006, 2007),Evonuk (2008) and Glatzmaier et al. (2009) present ide-alized two-dimensional (2D) simulations in the equa-torial plane exploring hypothetical basic-state density

profiles, with density varying by up to a factor of 55across the convection zone. In contrast, Jones andKuzanyan (2009) present 3D simulations using an ideal-ized basic-state density structure, with density varyingby a factor of up to 148, while Kaspi et al. (2009, 2010)present 3D simulations with a realistic Jovian interiorstructure, with density varying by nearly a factor of 104

from the deep interior to the 1-bar level. These anelas-tic studies likewise suggest that the jets could penetratedeeply through the molecular envelope.

A challenge with all of the above-described simu-lations is that, for computational reasons, they mustbe performed using heat fluxes and viscosities that dif-fer greatly from those on Jupiter, Saturn, Uranus, andNeptune. Thus, while simulations can produce jetswith speeds similar to the observed values of ∼100–200 m sec−1 for some combinations of parameters (e.g.,Aurnou et al. 2007, 2008; Aurnou and Olson 2001;Christensen 2001, 2002; Heimpel and Aurnou 2007;Heimpel et al. 2005), this does not imply that convec-tion in the interior of real giant planets would necessar-ily produce jets with such speeds. In fact, dependingon the parameter combinations, simulations with free-slip boundary conditions can produce jets that equi-librate to mean speeds1 ranging over three orders ofmagnitude, from less than 1 m sec−1 to 1000 m sec−1 ormore. Assessing the likely wind speeds in the molecu-lar envelopes of the real giant planets—and determiningwhether their observed jets can be pumped by convec-tion in their interiors—requires the development of atheory that can be extrapolated from the simulationregime to the planetary regime.

Currently, however, there is no published theory thatcan explain the jet speeds obtained in simulations withfree-slip boundaries nor their dependence on heat flux,viscosity, and other parameters. Several investigationshave presented scaling laws describing how the meanzonal-wind speeds vary with control parameters whenno-slip boundary conditions are used, as potentially rel-evant to Earth’s outer core (e.g., Aubert 2005; Aurnouet al. 2003). These are essentially theories for the mag-nitude of wind shear in the fluid interior for cases wherethe zonal velocity is pinned to zero at the boundaries.However, these scaling laws are not applicable to thegiant planets, where the fluid at the outer boundarycan move freely, lacks a frictional Ekman layer, and ex-hibits strong jets. Several attempts have also been madeto quantify how the convective velocities scale with pa-rameters, but—regardless of the boundary conditions—these relationships cannot be applied to the jet speeds

1Calculated in a suitably defined way, such as√

2K/M , whereK and M are the total kinetic energy and mass; this is essentiallythe mass-weighted characteristic wind speed of the fluid.

A.P. Showman et al., submitted to Icarus 3

because the convective and jet speeds can differ greatlyand their ratio may depend on the heat flux and otherparameters.

The goal of this paper, therefore, is to develop scal-ing laws for how the jet speeds depend on heat fluxand viscosity that explain the simulated behavior withinthe simulated regime and, ideally, allow an extrapola-tion to real planets. The simulations themselves makea number of simplifications (e.g., ignoring magnetohy-drodynamics in the deep interior at pressures exceeding∼1 Mbar), but in our view building a theory of thisidealized case is a prerequisite for understanding morerealistic systems.

We first quantify the degree of overforcing in cur-rent studies, since this issue has received little atten-tion in the literature (Section 2). Next, we quantifythe dependence of the convective speeds on planetaryparameters and compare them to results from an anelas-tic general circulation model from Kaspi et al. (2009)(Section 3). Armed with this information, we constructsimple scalings for the characteristic jet speeds in threeregimes. In the first two regimes, the viscosity is largeenough so that viscous damping of the jets providesthe dominant kinetic-energy loss mechanism. RegimeI is a strongly nonlinear regime where the jets domi-nate the total kinetic energy and the jet-pumping effi-ciency (i.e., the fraction of convectively released energyused to pump the jets) changes relatively slowly withthe parameters (Section 4a). Regime II, which occursclose to the critical Rayleigh number, is is a weaklynonlinear regime where the jets represent only a mod-est fraction of the total kinetic energy, the eastwardand outward convective velocity components are highlycorrelated, and the jet-pumping efficiency changes rel-atively rapidly with the parameters (Section 4b). Fi-nally, Christensen (2002) suggested that when the forc-ing is sufficiently strong or viscosity is sufficiently weak,the behavior enters an asymptotic regime where the jetspeeds become independent of the viscosity. We ex-plore this behavior—which we call Regime III—underthe assumption that the damping results from a turbu-lent eddy viscosity (Section 5). In Section 6, we combinethe three regimes and discuss extrapolations to Jupiter.In Section 7, we attempt to understand the thermal-wind shears that develop in the interior, evaluating inparticular the possibility that thermal-wind shears inoverforced simulations may be much larger than thoseon the giant planets. Section 8 concludes.

2. Degree of overforcing

For numerical reasons, current 3D simulations of con-vection in the giant planets must use viscosities manyorders of magnitude larger than the molecular viscosi-

ties. This results from the coarse grid resolution in themodels: to be numerically converged, such a modelmust have convective boundary layer and convectiveplume thicknesses of at least a gridpoint, and this re-quires very large viscosities. Given the enhanced damp-ing of the kinetic energy implied by this constraint, suchsimulations can achieve wind speeds similar to those onthe giant planets only by adopting heat fluxes orders ofmagnitude too large.

Moreover, even in the absence of such a viscousdamping, enhanced heat fluxes are computationallynecessary simply to achieve spin up within available in-tegration times. To illustrate, imagine a giant planetthat initially has no winds, and ask how long it wouldtake to spin the zonal jets up to full strength if thejets penetrate through the molecular region (down to∼1 Mbar pressure on Jupiter and Saturn). This cor-responds to a kinetic energy per area of ∼ u2∆p/g,where u is the characteristic jet speed (∼ 30 m sec−1

on Jupiter), g is gravity, and ∆p ∼ 1 Mbar is the pres-sure thickness across which the jets are assumed to ex-tend. On giant planets, the work to spin up the windscomes from the convection, driven by heat fluxes rang-ing from 1 W m−2 for Uranus and Neptune to 14 Wm−2

for Jupiter. Given these small available fluxes, the char-acteristic pumping time of the jets is ∼105 years. Thisis a lower limit, because it assumes that the efficiencyof converting the heat flux into kinetic energy is closeto 100%; in reality, the efficiency will be less, and muchof the power produced will be used to resist frictionallosses in the simulations. In contrast, published state-of-the-art high-resolution numerical simulations, if ex-pressed in dimensional units, typically integrate monthsto years (e.g., Aurnou et al. 2008; Aurnou and Ol-son 2001; Christensen 2001, 2002; Heimpel and Aurnou2007; Heimpel et al. 2005). Thus, the heat fluxes mustbe enhanced by at least 3–5 orders of magnitude simplyto allow the simulations to spin up within achievableintegration times.

Most convection papers cast the equations in nondi-mensional form, rendering unclear the actual degree towhich these simulations are overforced. One can quan-tify the degree of overforcing as follows. In these simula-tions, the specified control parameters are the Rayleighnumber Ra, the Ekman number E, and the Prandtlnumber Pr, defined as follows:

Ra =αg∆TD3

κν, E =

ν

ΩD2, P r =

ν

κ(1)

where α, g, κ, and ν are the thermal expansivity, grav-ity, thermal diffusivity, and kinematic viscosity. D isthe thickness of the convecting layer, ∆T is the im-posed temperature drop across this layer, and Ω is the


Fig. 1. Illustration of the parameter space associatedwith convection in rotating spherical shells. The modified-flux Rayleigh number Ra∗

F (abscissa) and Ekman number E(ordinate) can be viewed as non-dimensionalized heat fluxand viscosity, respectively. The Prandtl number, giving theratio of viscosity to thermal diffusivity, constitutes a thirddimension. Published simulations access a small region ofparameter space with values of Ra∗

F and E typically 10−6 orgreater, but the actual values for Jupiter are ∼10−14 or less.It is unknown whether convection at Jupiter-like values ofRa∗

F and E would produce Jupiter-like wind speeds.

planetary rotation rate.

The heat flux, which is an output, is typically castas a Nusselt number Nu, which gives the ratio of thetotal (convective plus conductive) heat flux to a refer-ence conductive flux that would be conducted across asystem with the same temperature drop and thickness:

Nu =Ftot

Fcond

=DFtot

κρcp∆T, (2)

where Ftot is the total flux, the thermal conductivityis κρcp, and Fcond = κρcp∆T/D. Away from the criti-cal Rayleigh number, the total heat flux approximatelyequals the convective heat flux.

So how may we determine the dimensional heat fluxfor a nondimensional simulation with specified Ra, E,Pr, and Nu? Using (1) and (2) shows that

Ftot =Ω3D2ρcp

αg

E3RaNu

Pr2. (3)

Because convection is driven by buoyancy associatedwith a heat flux, an alternate Rayleigh number, definedin terms of the heat flux rather than a temperaturecontrast, is useful:

RaF =αgFtotD

4

ρcpνκ2. (4)

Christensen (2002) argued that, in the limit of smalldiffusivities, the fluid behavior should become indepen-dent of the diffusivities, and he defined a modified heat-flux Rayleigh number that is independent of these diffu-sivities. This can be constructed by multiplying Eq. (4)by E3Pr−2, giving2

Ra∗

F =αgFtot

ρcpΩ3D2. (5)

In the limit of small diffusivities, one thus expects thatthe behavior should scale with Ra∗

F rather than Ra.Casting Eq. (5) as an expression for heat flux, we have

Ftot =Ω3D2ρcp

αgRa∗

F . (6)

For a giant planet, all the dimensional parame-ters in Eq. (6) are known. Consider Jovian valuesof Ω = 1.74 × 10−4 sec−1, D ≈ 2 × 104 km, cp =1.3 × 104 J kg−1 K−1, and g = 26 m sec−2. For inter-preting Boussinesq simulations in the context of giantplanets, an ambiguity is that the thermal expansivityand density vary greatly from atmospheric values atthe top to liquid values in the interior. Using atmo-spheric values corresponding to a pressure of ∼100 bars,where the temperature is ∼600 K, we have ρ ≈ 4 kg m−3

and α ≈ 1.6 × 10−3 K−1. Considering the simulationin Christensen (2001), where Pr = 1, E = 3 × 10−5,Ra = 108, and Nu ≈ 10, we then have heat fluxes of∼105 W m−2. However, given that the vast bulk of theinterior behaves as a liquid, much more appropriate val-ues are ρ ∼ 103 kg m−3 and α ∼ 10−5 K−1, relevant tothe bulk of Jupiter’s interior. These values imply fluxesof ∼109–1010 W m−2. Similar estimates imply that thesimulations of Heimpel and Aurnou (2007) and Aurnouet al. (2008) adopt a heat flux of ∼108 W m−2. Fortheir simulations of Uranus and Neptune, Aurnou et al.(2007) used heat-flux scaling laws to estimate a dimen-sional heat flux of ∼10 Wm−2; however, inserting theirparameter values into Eq. (6) instead indicates a heatflux of ∼107 W m−2.

Given the great variation of density and thermal ex-pansivity with radius on a giant planet, future studieswill increasingly adopt models that incorporate the fullradial variation of these parameters. This will requireextending the definition of Rayleigh number. We adoptthe following definition, in which a mass-weighted av-erage is performed over the radially varying quantities

2Christensen (2002) made alternate definitions of Ra∗

F and Nuthat included a non-dimensional factor of η (the ratio of the inner-to-outer radius adopted in the simulations) in the numerator ofEqs. 2 and 4. We forgo this factor here.


(Kaspi et al. 2009):

Ra∗

F =1

Ω3D2〈αgFtot

ρcp〉, (7)

where the mass-weighted average is defined as 〈...〉 =∫

(...)ρr2 dr/∫

ρr2 dr.

Anelastic simulations that adopt realistic radial pro-files for the density and thermal expansivity lack theambiguity described above. Kaspi et al. (2009) forcethe flow not through a constant-temperature boundarycondition but by introducing internal heating and cool-ing. Integrating their heating profile over the depthof the planet gives an effective heat flux with a max-imum of ∼109 W m−2 in the deep interior with valuesdecaying to zero in the atmosphere. Similarly, Jonesand Kuzanyan (2009) estimate a dimensional heat fluxof ∼109 W m−2 for their simulations, broadly consistentwith the estimates made here.

The implication is that the simulations in Chris-tensen (2001, 2002), Aurnou and Olson (2001), Heimpeland Aurnou (2007), Aurnou et al. (2008), Kaspi et al.(2009) and related studies are overforced by ∼6–10 or-ders of magnitude. The overforcing compensates forthe numerical need to use high diffusivities and achievea steady state over reasonable time scales.

3. Convective velocities and buoyancies

In a giant planet, convection in the interior trans-ports the interior heat flux. To order-of-magnitude, onethus expects that

F ∼ ρcpwδT, (8)

where F is the convected heat flux, w is the characteris-tic magnitude of the vertical convective velocity, and δTis the characteristic magnitude of the temperature dif-ference between a convective plume and the surroundingfluid.

In a rapidly rotating, low-viscosity fluid, the scalingfor convective velocities is typically written (e.g., Boub-nov and Golitsyn 1990; Fernando et al. 1991; Golitsyn1980, 1981; Stevenson 1979)3

w ∼

(

αgF

ρcpΩ

)1/2

. (9)

It is known that Eq. (9) provides a reasonable repre-sentation of convective velocities for laboratory exper-iments in rotating tanks (e.g., Boubnov and Golitsyn

3Equation (9) can be heuristically derived by assuming thatthe convective buoyancy force per mass, gα δT , is balanced bythe vertical Coriolis force due to the turbulent eddy motions,Ωu′, where u′ is the horizontal eddy velocity. Assuming that theturbulent eddy motions are approximately isotropic (u′ ∼ w) andinvoking Eq. (8) then leads to Eq. (9).

Table 1. Key to Symbols

Anelastic simulations using Kaspi et al. (2009) model

Symbol Ra∗

F E Prup-pointing triangle varies 4× 10−4 10right-pointing triangle varies 8× 10−4 10down-pointing triangle varies 9× 10−4 10left-pointing triangle varies 6× 10−4 10circle varies (1.5–15)× 10−4 10diamond varies 4 × 10−4 0.8–12

Boussinesq simulations from Christensen (2002)

Symbol Ra∗

F E Prtriangle varies 3 × 10−4 1diamond varies 10−4 1circle varies 3 × 10−5 1square varies 1 × 10−5 1

1990; Fernando et al. 1991; Golitsyn 1981). Several au-thors have also suggested that Eq. (9) is relevant forthe dynamo-generating regions of planetary interiors,where a three-way balance between buoyancy, Coriolis,and Lorentz forces is often assumed (e.g., Starchenkoand Jones 2002; Stevenson 2003). To our knowledge,however, no broad assessment of its accuracy has beenmade for convection in the molecular interior of a gi-ant planet. To do so, we performed three-dimensionalnumerical simulations of convection in Jupiter’s inte-rior using an anelastic general circulation model basedon the MITgcm (Kaspi et al. 2009). The radial depen-dence of basic-state density and thermal expansivity aretaken from the SCVH equation of state (Saumon et al.1995). Simulations with a broad range of parameterswere explored (Table 1).

Our simulations show that Eq. (9) provides a goodapproximation over a wide range of heat fluxes and rota-tion rates. Figure 2 shows the mass-weighted mean con-vective velocities from the Kaspi et al. (2009) anelasticsimulations versus modified-flux Rayleigh number andcompares it to Eq. (9). For the vast majority of simula-tions, the simulations differ less than a factor of ∼3 fromthe scaling, indicating that Eq. (9) is quite accurate.This analysis complements Kaspi et al. (2009)’s findingthat, within a specific simulation, Eq. (9) captures theapproximate radial dependence of the convective veloc-ity across the interior, over which α/ρ varies by a factorof ∼ 105 (Fig. 2, bottom). Together, these results sug-gest that Eq. (9) is valid not only for the mass-weightedmean velocities but locally within the fluid.

Laboratory experiments suggest that a prefactor shouldexist on the right side of Eq. (9) with a value close to2 (Boubnov and Golitsyn 1990; Fernando et al. 1991;Golitsyn 1981); however, our simulations are better


0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70w′

u′

(

u′w′

)1

2

(

αgF

ρCpΩ

)1

2

r

RJ

m s

−1

b

10−6

10−5

10−4

10−3

10−2

101

102

m s

−1

(

αgF

ρCpΩ

)1

2

Ra∗

F

w

a

Fig. 2. Top: Mass-weighted convective vertical windspeed (calculated as the deviation of vertical wind speedfrom its zonal mean) versus modified heat-flux Rayleighnumber for a range of anelastic simulations. Dashed line isthe scaling of Eq. (9). See Table 1 for definitions of the sym-bols. Bottom: Radial variation of the horizontally averagedvelocity components for a particular simulation. Depictsdeviation of vertical velocity from its zonal mean (dashed),deviation of the zonal velocity from its zonal mean (dashed-dotted), the square root of their zonally averaged correlation(u′w′)1/2 (dotted), and the scaling from Eq. (9) evaluatedusing the radially varying thermal expansivity α, basic-statedensity ρ, heat flux F , and gravity. For more details on theexperiments see Kaspi (2008, chapter 6.3).

matched when no prefactor is used.

We can nondimensionalize the convective velocitiesusing a convective Rossby number, Roconv ≡ w/(ΩD).The convective flux, F , equals Ftot minus the conduc-tive flux. The conductive flux is only important near thecritical Rayleigh number, and in this limit it becomesFcond from Section 2. Making the approximation thatthe conductive flux equals Fcond, Eq. (9) nondimension-alizes to yield

Roconv ∼ γ(Ra∗

F − Ra∗critF )1/2, (10)

where Ra∗critF is the critical modified flux Rayleigh num-

ber for the onset of convection. Thus, at the criti-cal Rayleigh number, the expression properly predictszero convective velocities, and at greatly supercriticalRayleigh numbers it predicts Roconv ∼ γ(Ra∗

F )1/2. Wehave introduced a dimensionless prefactor γ for gener-ality, but consistent with Fig. 2, we will assume γ = 1when performing numerical estimates.

Like Eq. (10), many of our subsequent scalings willdepend on the difference between the actual and criticalRayleigh numbers. For notational brevity, we thereforedefine this difference as

Ra∆F ≡ Ra∗

F − Ra∗critF . (11)

Equations (8–9) imply that the fractional density as-sociated with convective plumes should scale as

αδT ∼

(

FΩα

ρcpg

)1/2

. (12)

Aubert et al. (2001) proposed an alternate scalingfor the convective velocities in the limit of negligibleviscosities, Roconv ∼ (Ra∗

F )2/5. They performed rotat-ing laboratory experiments in liquid gallium and water,and showed that this expression provides a reasonablefit to the convective velocities inferred for their exper-iments. In the numerical simulations of Christensen(2002) and Christensen and Aubert (2006), the poloidalcomponent of the velocity field (which includes the con-vective velocities) depends on the Ekman number, butat small Ekman number seems to be converging towardan asymptotic dependence that is reasonably well rep-resented by this 2/5 scaling (with a prefactor of 0.5). Inour case, Eqs. (9)–(10) provides a slightly better fit tothe convective velocities, although the 2/5 scaling (witha prefactor) is also adequate. Conversely, overplottingEqs. (9)–(10) against Christensen (2002)’s simulationdata shows that they match essentially as well as the2/5 scaling.


4. Scaling for the jet speeds: Viscous regimes

a. Regime I: Strongly nonlinear viscous regime

Published simulations show that the mean jet speedsdepend significantly on the control parameters Ra∗

F

and E. Convection can produce jets with Jupiter-likewind speeds for some combinations of these parameters,but other combinations produce flows with wind speedsfar from the jovian regime, with domain-averaged jetspeeds ranging across three orders of magnitude evenwithin the limited parameter space explored in currentinvestigations (Fig. 1). The processes that determinehow mean jet speeds depend on Ra∗

F and E are not un-derstood, however. Although several investigations rel-evant to giant planets have identified trends for the Ra∗

F

and E-dependence of the jet speeds (Christensen 2002;Kaspi et al. 2009), no theories have yet been proposedto explain these empirical trends. Moreover, extrapo-lating the simulations into the jovian parameter regimecan only be performed once a theory for the Ra∗

F andE-dependence of the mean jet speeds has been devel-oped. In the absence of such a theory, it is unknownwhether convection at Jupiter-like values of Ra∗

F and Ewould generate Jupiter-like jet speeds.

Figure 3 shows how the mean jet speeds U (or equiv-alently, the mean Rossby number Ro ≡ U/(ΩD), whereD is the thickness of the simulation domain) dependon the control parameters Ra∗

F and E in the simula-tions from Christensen (2002) (top panel) and Kaspiet al. (2009) (bottom panel).4 At present, these arethe only published investigations with free-slip bound-ary conditions that present sufficient numerical data toaccurately characterize the dependence of domain-meanwind speeds over a broad range of Ra∗

F and E. In bothcases, larger Ra∗

F and smaller E promote faster windspeeds. This makes sense qualitatively because, for agiven rotation rate and planetary size, larger Ra∗

F im-plies larger heat flux (i.e., stronger forcing of the flow),while smaller E implies smaller viscosity (i.e., weakerdamping of the flow). Kaspi et al. (2009) found thatconstant-wind-speed contours exhibit a slope (in thelog(Ra∗

F )–log(E) diagram) of approximately 5/4. Thedata from Christensen (2002) plotted in Fig. 3 like-wise indicate that the wind-speed contours in his caseexhibit a mean slope of approximately 1. For bothsets of simulations, Fig. 3 also shows that the contourstend to be widely spaced toward the right and tightlyspaced toward the left, with an approximate transitionat Ro ≈ 0.02. This suggests a transition between tworegimes.

4For Jupiter’s radius and rotation rate, the Rossby numbersshown in Fig. 3 correspond to domain-averaged jet speeds rangingfrom ∼0.8 to 1000m sec−1.

0.00

1372

90.

0032

171

0.00

6092

90.

0115

390.

0176

640.

0270

390.

0413

91

0.06

3359

0.09

6988

log 1

0(E

)

−6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5−5

−4.5

−4

−3.5

0.0003

0.001

0.003

0.01

0.03

0.1

0.00

1372

90.

0032

171

0.01

1539

0.01

7664

0.02

7039

0.04

1391

0.06

3359

log10

(Ra∗

F )

log 1

0(E

)

−6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5−4

−3.5

−3

−2.5

0.0003

0.001

0.003

0.01

0.03

0.1

Fig. 3. Dependence of mass-weighted, domain-averagedjet speed on parameters from published simulations. Depictsthe Rossby number (color and contours) as a function ofmodified flux Rayleigh number (abscissa) and Ekman num-ber (ordinate) for Boussinesq simulations from Christensen(2002) (top) and Kaspi et al. (2009) (bottom). Note thatcontours are evenly spaced in log(Ro) and that all figures inthis paper use identical contour values and colorbars, facil-itating intercomparison. In both panels, the plusses denotethe locations of individual simulations. The range of Ekmannumbers plotted is different in the two cases; anelastic casesare computationally more demanding and hence were runat larger Ekman numbers than the Boussinesq cases. Small-scale structure, particularly in the bottom plot, is probablynot real. The robust feature is the overall trend of posi-tive slopes of constant-Ro contours, with mean slopes closeto one, and Ro values ranging from ∼0.1 on the right to∼0.0001 on the left. Note that, in both sets of simulations,the contours are widely spaced at Rossby numbers exceed-ing ∼0.02 (right side of plot) but become tightly spaced atRossby numbers less than ∼0.02 (left side of plot), suggest-ing a regime transition.


Here, we construct a simple scaling theory based onenergetic arguments to attempt an explanation for thejet speeds in the regime of fast jets (Ro ∼> 0.02 in Fig. 3,

corresponding to jet speeds exceeding ∼ 100 m sec−1),which we call Regime I.

First consider the forcing. A convecting fluid parceltraveling at a vertical speed w releases potential energyper mass at a rate P ∼ gwδρ/ρ, where ρ is densityand δρ is the magnitude of the density contrast be-tween plumes and the background environment. Usingthe equation of state δρ = αρδT , where ρ represents abackground density and δT is the magnitude of the tem-perature contrast between plumes and the backgroundenvironment, we can write P ∼ gwα δT . This potentialenergy is primarily converted into convective kinetic en-ergy. Some fraction ǫ of this energy is used to pump thezonal jets; we provisionally assume ǫ is constant but re-turn to this assumption in Section 4c. Integrating overthe planetary mass, the total power (in W) pumped intothe jets is then approximately

Ptot ∼ 4πǫ

∫

αδTgwρr2 dr, (13)

where the mass of an infinitesimal spherical shell of ra-dial thickness dr is 4πρr2 dr. Both δT and w are a priori

unknown, but they are related by the fact that convec-tion transports the planet’s interior heat flux (Eq. 8),which implies that w δT ∼ F/(ρcp). Thus, the poweravailable to pump the jets can be written

Ptot ∼ 4πǫ

∫

αgF

cpr2 dr. (14)

Energy loss occurs through friction, which we assumeacts as a diffusive damping of the winds with a viscosityν (for the simulations, this would be the model viscositythat enters the definition of the Rayleigh and Ekmannumber; see Eq. 1). The power per mass dissipatedby viscosity is approximately ν∇2u2 ∼ νk2u2, wherek is the wavenumber of the structures dominating thedissipation, that is, the dominant wavenumber of ∇2u2.This implies a total rate of kinetic energy loss given by

Kloss ∼ 4π

∫

νk2u2ρr2 dr. (15)

Now we equate energy gain (Eq. 14) and loss (Eq. 15)to obtain an expression for mean jet speed. In the caseof a Boussinesq fluid, we note that α and ρ are constantsand that F , g, cp, and u are approximately constant(within a factor of ∼3) and can be pulled out of theintegral, yielding

u ∼ k−1

(

αgǫF

ρcpν

)1/2

(16)

The case of an anelastic fluid is complicated by the largevariation of α and ρ with radius, and even u may varysignificantly from the top to the bottom of the domain(Kaspi et al. 2009). However, in the Kaspi et al. (2009)simulations, the mean jet speeds are relatively constant(within a factor of ∼ 3) throughout most of the domain;only the top few % of the fluid mass experiences signifi-cant wind shear along the Taylor columns. To obtain acrude expression for the mass-weighted mean jet speedin the molecular region, we therefore assume that u isconstant, allowing us to write:

u ∼ k−1( ǫ

ν

)1/2

[∫ αgFρcp

ρr2 dr∫

ρr2 dr

]1/2

. (17)

Let us nondimensionalize these expressions so thatthey may be compared with the simulation resultsshown in Fig. 3. For both cases, we use Ro ≡ u/(ΩD)and the definition of E given in Eq. (1). Using the ap-propriate definition of the modified flux Rayleigh num-ber yields the same nondimensional expression for boththe Boussinesq and anelastic cases. Again approximat-ing the convective flux F as the total flux minus Fcond,we obtain

Ro ≈ǫ1/2

kD

(

Ra∆F

E

)1/2

, (18)

where Ra∆F ≡ Ra∗

F − Ra∗critF (see Eq. 11). In Eq. (18),

Ra∗

F is given by Eq. (5) for the Boussinesq case and byEq. (7) for the anelastic case. Note that, in the highlysupercritical regime in which we expect Eq. (18) to bevalid, the critial Rayleigh number is generally negligible,so that Ra∆

F approximately equals Ra∗

F .

Thus, this simple theory predicts that the character-istic jet speed is proportional to (F/ν)1/2, or equiv-alently that the characteristic Rossby number is ap-proximately proportional to (Ra∗

F /E)1/2. Increases ordecreases in Ra∗

F and E by the same factor leave Rounchanged; in other words, this theory predicts thatcontant-Ro contours should have a slope of one in theRa∗

F –E plane. This explains the fact that the empiri-cally determined slopes in the simulations are close toone (see Fig. 3). In our theory, this behavior occursbecause the forcing depends on the flux in the sameway as damping depends on viscosity (namely, linearly).Thus, equal increases (or decreases) in the flux and theviscosity cause alterations to the forcing and dampingthat cancel out, leading to a mean jet speed that isunchanged.

To further compare the theory to the simulations,Fig. 4 plots our solution (Eq. 18) as contours of Rossbynumber versus Ra∗

F and E. A choice of kD is neces-sary to evaluate Eq. (18). Kaspi et al. (2009)’s sim-ulations show that, although the jets themselves are

A.P. Showman et al., submitted to Icarus 90.

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846

log 1

0(E

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log10

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F )

−6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5−5

−4.5

−4

−3.5

0.0003

0.001

0.003

0.01

0.03

0.1

Fig. 4. Prediction of the scaling law for Regime I givenin Eq. (18), in which the jet-pumping efficiency ǫ is as-sumed constant. Depicts contours of Rossby number versusmodified flux Rayleigh number (abscissa) and Ekman num-ber (ordinate). Adopts kD = 5π and ǫ = 0.3 (see text).Note the similarity of the slopes and values of the contourswith the simulation results plotted in Fig. 3, particularly forRossby numbers greater than ∼0.02.

broad (with widths comparable to the domain thick-ness), the curvature of the wind ∇2u, and thus the dis-sipation, exhibits a dominant wavelength of about D/5(see their Fig. 10d). High-wavenumber structure in theflow shear also occurs in Christensen (2002)’s simula-tions. To account for this high-wavenumber structure,we adopt kD ≈ 5π, and we calculate Fig. 4 using thisvalue and ǫ = 0.3.

Figure 4 reiterates the approximate agreements inthe slopes of constant-Ro contours (compare Fig. 4 withFig. 3). Moreover, the Ro values themselves also agreesurprisingly well, particularly in the regime of fast zonaljets where Ro ∼> 0.02. Throughout this part of the pa-rameter space, the predicted Rossby number (Fig. 4)matches those obtained in the simulations (Fig. 3) towithin a factor of ∼2 at a given Ra∗

F and E, althoughthe discrepancy approaches an order of magnitude to-ward the left side of the plot (at Ro ∼< 0.02) where thesimulated Rossby numbers seem to undergo a regimeshift that is not captured by Eq. (18). Such a shiftcould result for example from a dependence of ǫ or k onRa∗

F and E, a possibility not considered up to now. Wereturn to this point in the next two subsections.

b. Regime II: Weakly nonlinear viscous regime

For sufficiently small values of Ra∗

F , both the Boussi-nesq and anelastic simulations shown in Fig. 3 exhibita regime shift where the dependence of Rossby number

on Ra∗

F and E becomes steeper than the (Ra∗

F /E)1/2

dependence discussed in the previous subsection. AsFig. 3 shows, at Rossby numbers smaller than ∼0.01–0.02, the constant-Ro contours becomes closely spaced,with a spacing indicating that a scaling (Ra∗

F /E)ξ withξ ≈ 1 might provide an approximate fit. There is also anindication that the slopes of the constant-Ro contoursdecrease toward the left side of the plot, particularly forChristensen’s simulations.

These stronger dependences of Rossby number onRa∗

F suggest a regime shift where different processes setthe characteristic Rossby number at low Rossby num-ber than at high Rossby number. In the fast-zonal-windviscous regime explored in the previous subsection, theRayleigh number is strongly supercritical, and the con-vection is chaotic and nonlinear. At lower Rayleighnumbers, when the Rayleigh number is only modestlygreater than the critical value for convection, the con-vection is laminar and more spatially organized. Here,we explore how this transition in convective behaviormight lead to the regime shift in the jet speeds seen inChristensen (2002)’s simulations.

To do so, we consider an alternate approach basedthe zonal momentum balance. Consider a cylindri-cal coordinate system whose axis aligns with the ro-tation axis, with coordinates (s, λ, z) corresponding tocylindrical radius (i.e., distance from the rotation axis),longitude, and distance above or below the equatorialplane, respectively. The zonal-mean zonal momentumequation then reads (cf Kaspi et al. 2009)

∂u

∂t+ 2Ωvs +

1

ρ∇ · (ρ uv) +

1

ρ∇ · (ρ u′v′) = ν∇2u (19)

where u is the zonal wind, vs is the cylindrically radialwind component (that is, the velocity component awayfrom the rotation axis), overbars denote zonal means,primes deviations therefrom, and ρ is the mean density(constant in the Boussinesq case and a specified func-tion of radius in the anelastic case). On the left side,the second term is the Coriolis acceleration. The thirdterm represents advection due to the mean-meridionalflow, and the fourth term represents the accelerationcaused by eddies (i.e., Reynolds stress convergences).Generally speaking, for the geostrophic jovian regime,the Reynolds stress term dominates over the advectionterm (e.g. Kaspi et al. 2009).

We now average the equation in z (i.e. along the di-rection of Taylor columns). The Coriolis accelerationscancel out, because mass continuity prohibits any net,column-averaged motion toward or away from the rota-tion axis. Coupled with the free-slip boundary condi-tions and symmetry of the zonal-wind structure aboutthe equatorial plane, the z-averaging also removes thez-components of the divergence and Laplacian terms.


Neglecting the mean-flow advection terms, the column-averaged, steady-state balance becomes

1

s2ρ

∂

∂s(s2u′v′sρ) ≈ ν

∂

∂s

[

1

s

∂

∂s(su)

]

(20)

which states that, in steady state, viscous drag associ-ated with shear of the mean zonal wind ∂u/∂s balancesjet acceleration caused by eddies. Balances analogousto this expression have been written, for example, byBusse and Hood (1982), Busse (1983a), Busse (1983b),and Cardin and Olson (1994).5

We here use this balance to achieve a simple expres-sion for the jet speeds. Approximate the left-hand sideas k u′v′s, where k is the wavenumber (in cylindrical ra-dius) over which u′v′s varies significantly. Furthermore,suppose that both u′ and v′s scale like our expected con-vective velocity scale, so that u′v′s ∼ Cw2, where w isthe convective velocity (e.g., from Eq. 9) and C is acorrelation coefficient equal to one when u′ and v′s areperfectly correlated (i.e. when eastward u′ always oc-curs with outward v′s and vice versa) and equal to zerowhen u′ and v′s exhibit no correlation. We approximatethe right-hand side as νk2u. The momentum balancethen becomes

u ∼ Cw2

νk(21)

Nondimensionalizing, the Rossby number associatedwith the jets is

Ro ∼C

kD

Ro2conv

E(22)

An equation analogous to this was derived by Chris-tensen (2002, his Eq. 3.12). Inserting our expression forthe convective velocities (Eq. 10) yields

Ro ∼Cγ2

kD

Ra∆F

E. (23)

Thus, this theory predicts that, if the degree of cor-relation between u′ and v′s is independent of the controlparameters (i.e., if C is constant), then the domain-mean jet speed scales as the convective flux over the vis-cosity, or equivalently the mean Rossby number scalesas (Ra∗

F −Ra∗critF )/E. Away from the critical Rayleigh

number, increases or decreases in Ra∗

F and E by thesame factor leave Ro unchanged; therefore, as with thetheory presented in the previous subsection, Eq. (23)predicts that constant-Ro contours should have slopesclose to one in the logarithmic Ra∗

F –E plane. Near thecritical Rayleigh number, however, the predicted slopes

5An analogous balance was considered by Aubert et al. (2001)and Aubert (2005) but for the no-slip boundary condition whereboundary-layer friction plays the dominant role in the damping.

differ from one. If the critical modified flux Rayleighnumber depends on E to a positive power less thanone, then the constant-Ro contours increase in slope,becoming more vertical; if, however, Ra∗crit

F depends onE to a power greater than one, then the constant-Rocontours decrease in slope, becoming more horizontal.

To evaluate Eq. (23) quantitatively, we require anexpression for Ra∗crit

F . By performing many simula-tions, Christensen (2002) determined empirically thecritical Rayleigh number for each Ekman number heexplored.6 Cardin and Olson (1994) performed a linearinstability analysis of rotating convection in a spheri-cal shell and found that, at E ≪ 1, the critical valueof the ordinary Rayleigh number is a constant timesE−17/15Pr4/3(1 + Pr)−4/3. Using the relationships be-tween Ra and Ra∗

F (see Section 2), and noting that theNusselt number equals one at the onset of convection,shows that the critical modified flux Rayleigh numbershould then scale as E28/15Pr−2/3(1 + Pr)−4/3. Mostof Christensen’s simulations adopt a Prandtl number ofone and we find that his empirically determined criticalRayleigh numbers are well matched by the expression

Ra∗critF ≈ 20E28/15. (24)

Given this dependence, contours of constant Rossbynumber in the logarithmic Ra∗

F –E plane should de-crease in slope as one approaches the critical Rayleighnumber (i.e., as one moves toward lower Ra∗

F values).This is qualitatively consistent with the behavior exhib-ited by Christensen’s simulations (see Fig. 3).

c. Combining Regimes I and II

We have discussed two distinct viscous regimes: (i)a regime of fast zonal winds and strongly supercriticalconvection where the simulated jet speeds seem well ex-plained by the assumption that ǫ is constant (Regime I),and (ii) a regime of slow zonal winds and weakly su-percritical convection where the jet speeds are approxi-mately explained by the assumption that the correlationbetween the u′ and v′s velocity components is strong androughly constant (Regime II). These led to two distinctscalings for the jet speeds: Eqs. (18) and (23) for thetwo regimes, respectively. Two issues now arise: First,how do we combine these regimes? Second, why shouldenergetics and momentum considerations lead to dif-

6Christensen found that the critical values of a modifiedRayleigh number Ra∗ ≡ RaE2Pr−1 are 0.001005, 0.002413,0.006510, and 0.016790 for Ekman numbers of 10−5, 3 × 10−5,10−4, and 3 × 10−4, respectively. This is equivalent to mod-ified flux critical Rayleigh numbers, Ra∗crit

F , of 1.005 × 10−8,

7.2 × 10−8, 6.5 × 10−7, and 5.04 × 10−6 for those same Ekmannumbers, respectively. Equation (24) gives values that agree withthese to within ∼10%.


Fig. 5. Top: Snapshots in the equatorial plane of twoanelastic, 3D simulations from Kaspi (2008) illustrating howthe correlation between the convective velocity componentsdepends on the supercriticality. Each slice shows half ofthe equatorial plane of one simulation (note however thatthe simulations each span 360 of longitude). Color depictsstreamfunction and arrows denote velocity component in theequatorial plane. The left simulation is weakly supercriti-cal; the strong correlation between outward and eastwardvelocity components is obvious. The right simulation isstrongly supercritical; the correlation between the outwardand eastward velocity components is weaker because of thecomplex, turbulent convective structure. Bottom: Depictscorrelation coefficient C versus Ra∗

F for a range of anelasticsimulations7. As expected qualitatively from the top panels,C decreases with Ra∗

F .

ferent scalings? Both approaches should, in principle,yield the same answer within any given regime.

The resolution to both issues lies in the dependenceof the correlation coefficient C and the convective jet-

pumping efficiency ǫ on the control parameters. Con-sider, for example, the correlation coefficient. When theRayleigh number is weakly supercritical and the flow isgeostrophic, the convection forms broad, laminar con-vection rolls, which in a sphere become tilted in theprograde direction (Busse 1983a, b, 2002; Busse andHood 1982; Christensen 2002; Kaspi 2008; Sun et al.1993; Tilgner and Busse 1997, and others). This leadsto highly correlated velocity components u′ and v′s, asillustrated in Fig. 5 (top): ascending fluid parcels moveprograde while descending parcels move retrograde. Asa result, at weakly supercritical Rayleigh numbers, thecorrelation coefficient C should be close to one, at leastoutside the tangent cylinder. On the other hand, whenthe convection is strongly supercritical, the convectivestructure is chaotically time-dependent and spatiallycomplex; the velocity components u′ and v′s are posi-tively correlated in some regions but negatively corre-lated in others. This is illustrated in Fig. 5 (bottom).The net correlation is usually still positive, but thelarge degree of cancellation implies that, at strongly su-percritical Rayleigh numbers, the correlation coefficientshould drop significantly below one (Christensen 2002).As a result, one expects the correlation coefficient C tovary slowly with Ra∗

F near the critical Rayleigh num-ber but to decrease more rapidly with increasing Ra∗

F

at sufficiently supercritical Rayleigh numbers. This be-havior can be seen in correlation coefficients calculatedfor a range of simulations versus Ra∗

F (Fig. 5, bottom).

Next consider the jet-pumping efficiency ǫ. Thescaling (18), wherein the Rossby number scales with(Ra∗

F /E)1/2, assumes that ǫ is constant. While this as-sumption seems to work well at sufficiently supercriticalRayleigh numbers, it must fail in the weakly supercrit-ical regime where u′ and v′s are highly correlated. Thisis because, in this weakly nonlinear regime, the meanjet speeds scale quadratically with the convective veloc-ities, and the power exerted to pump the jets scales asthe Reynolds stresses times the jet speeds, namely asthe fourth power of the convective velocities. The de-pendence of this quantity on the heat flux differs fromthe dependence of the convective potential-energy re-lease on the heat flux. Since ǫ is the ratio of thesequantities, ǫ cannot be constant in this regime.

A regime shift analogous to that seen for the jetspeeds in Fig. 3—where Rossby numbers scale approxi-mately with Ra∗

F /E at low Rossby number and approx-imately as (Ra∗

F /E)1/2 high Rossby number—can occurif the correlation coefficient depends weakly on Ra∗

F atlow Ra∗

F yet strongly on Ra∗

F at high Ra∗

F . Likewise, itcan occur if the jet-pumping efficiency depends stronglyon Ra∗

F at low Ra∗

F yet weakly at high Ra∗

F .

To quantify these arguments, let us relate ǫ and C to


each other. By definition, the efficiency ǫ is the ratio ofwork done to pump the jets to the work made availableby convection:

ǫ ∼u 1

ρ∇ · (ρu′v′)

gwα δT, (25)

where it is understood that both the numerator andthe denominator represent global averages. Expressingthe numerator as Ckw2, the denominator as gαF/ρcp

(using Eq. 8), and non-dimensionalizing, we obtain

ǫ ∼RoCkDRo2

conv

Ra∆F

. (26)

Using Eq. (22) for the Rossby number yields the expres-sion

ǫ ∼ C2 Ro4conv

ERa∆F

. (27)

If we adopt Roconv ≈ γ(Ra∆F )1/2 for concreteness, we

obtain finally

ǫ ∼ C2γ4 Ra∆F

E. (28)

Therefore, ǫ and C cannot simultaneously be constantwhen Ra∗

F or E are varied. Holding one constant re-quires the other to become a function of Ra∗

F and E.

Let us now make the simplest possible assumption re-garding this regime shift—we postulate a regime withC ≈ constant = 1 at low Rayleigh number and aregime with ǫ ≈ constant ≡ ǫmax at high Rayleighnumber. Quantitatively, there is no rigorous expecta-tion that C or ǫ need be constant, nor (as describedpreviously) is this assumption actually necessary for aregime shift to occur. Nevertheless, C and ǫ have upperlimits of 1, so if some process causes them to increasewith increasing (or decreasing) Ra∗

F , they might nat-urally plateau—at least over some range of parameterspace—upon approaching their upper limits. Still, thisassumption should be viewed as simply a crude startingpoint for comparison with the simulations; future theo-retical work on what sets the Ra∗

F - and E-dependenceof the correlation coefficient and jet-pumping efficiencyis warranted.

Given these postulates, Eq. (28) implies that

C =

1, Ra∆F ∼< ǫmaxγ

−4E;

ǫ1/2maxγ−2

(

ERa∆

F

)1/2

, Ra∆F ∼> ǫmaxγ

−4E(29)

and

ǫ =

γ4 Ra∆

F

E , Ra∆F ∼< ǫmaxγ

−4E;

ǫmax, Ra∆F ∼> ǫmaxγ

−4E(30)

To smoothly transition between the regimes, we adoptthe expression

C ∼

[

1 +γ2

ǫ1/2max

(

Ra∆F

E

)1/2]−1

(31)

which is merely the inverse of one over the value of Cfrom the first regime plus one over the value of C fromthe second regime. This gives behavior equivalent toEq. (29) in the two limits with a smooth transition inbetween. Given this expression, ǫ can then be evaluatedfrom Eq. (28), yielding

ǫ ∼ γ4

[

1 +γ2

ǫ1/2max

(

Ra∆F

E

)1/2]−2

Ra∆F

E. (32)

Figure 6 plots ǫ and C versus Ra∗

F from availabledata and compares them to Eqs. (31)–(32). In particu-lar, the blue triangles denote values of C and ǫ thatwe rigorously computed7 from the three-dimensionalconvective velocity and entropy fields for a sequence ofanelastic simulations with E = 4 × 10−4. Additionally,the red circles denote values of C rigorously calculatedby Christensen (2002) for a sequence of his simulationsat E = 10−5 (see his Table 3; note that no data on ǫ areavailable for Christensen’s simulations.) Blue and redsolid curves denote the predictions of Eqs. (31)–(32) forE = 4×10−4 and 10−5, respectively, and can be directlycompared to the symbols of the same color.

Key points are as follows. First, the data indicatethat, for both sets of simulations, C decreases withRa∗

F , while ǫ increases with Ra∗

F . Second, and moreimportantly, the slopes of the trends change across therange of Ra∗

F explored. The trends of C in both setsof simulations depend relatively weakly on Ra∗

F towardthe left but seem to steepen toward the right. For ex-ample, Christensen’s data indicate that a factor-of-twochange in Ra∗

F cause a 1.28-fold change in C at low Ra∗

F

but a 2.7-fold change in C at high Ra∗

F . Conversely,the data suggest that ǫ depends on Ra∗

F strongly atsmall Ra∗

F but more weakly at larger Ra∗

F . As Eq. (28)shows, these trends must go hand-in-hand; consistencybetween the two quantities requires the trend in C to

7 The correlation coefficient and jet-pumping efficiency werecalculated from the anelastic simulations as

C =[u′v′s]

[(u′2 v′2)1/2]

and

ǫ =[u 1

ρ∇ · (ρu′

v′)]

[gwαδT ]

where [...] =∫

(...)ρs ds/∫

ρsds indicates a mass-weighted aver-age over the equatorial plane.


Fig. 6. Top: The correlation coefficient C, the ratio be-tween the Reynolds stress u′v′

s and the product of the root-mean-square velocity magnitudes. Bottom: ǫ, the fractionof potential energy released by convection that is used topump the jets. Blue triangles depict values we calculatedrigorously from anelastic simulations at E = 4×10−4, whilered squares denote values Christensen (2002) calculated rig-orously from Boussinesq simulations at E = 10−5. In bothcases, values represent flow behavior outside the tangentcylinder. Curves plot Eqs. (31)–(32) at an Ekman num-ber of 4×10−4 (solid blue) and 10−5 (dashed red) assumingγ = 1 and ǫmax = 0.3, with Ra∗crit

F evaluated using 20E28/15

for the Boussinesq simulations and 0.4E28/15 for the anelas-tic simulations (the difference resulting from the differingPrandtl numbers; see Table 1).

steepen if that in ǫ flattens. Moreover, for both C andǫ, these changes in the slopes with Ra∗

F are preciselythe ingredients that can generate a regime shift whereinRossby numbers depend strongly on Ra∗

F at low Ra∗

F

(as postulated for Regime II in Section 4b) but moreweakly on Ra∗

F at high Ra∗

F (as postulated for RegimeI in Section 4a). Third, the data suggest that, at con-stant Ra∗

F , the correlation coefficient increases with in-creasing E.

The theoretical curves for C and ǫ match the datasurprisingly well (Fig. 6). All the qualitative trendsdiscussed above for the data are predicted by the the-ory: C decreases and ǫ increases with Ra∗

F ; the slope ofC(Ra∗

F ) steepens toward the right while that of ǫ(Ra∗

F )flattens toward the right; and at constant Ra∗

F the val-ues of C increase with E. The theory also predicts thatat constant Ra∗

F , the values of ǫ should decrease withincreasing E, which can be tested with future simu-lations. Moreover, for the parameters chosen (γ = 1and ǫmax = 0.3), the numerical values of the theoreticalǫ and C curves agree relatively well with the simula-tion data, particularly for the anelastic simulations atE = 4 × 10−4. On the other hand, the theoretical cor-relation coefficients at E = 10−5 overpredict those forthe corresponding Boussinesq simulations by up to afactor of several, particularly at the higher-Ra∗

F values.Of course, the technique adopted to smooth betweenthe two regimes (cf Eqs. 31–32) will affect the degree towhich the curves agree with the data, so the details ofthe comparisons should be considered tentative.

What are the implications of this discussion for thejet speeds? Inserting our expression for ǫ into Eq. (18)or our expression for C into Eq. (23) leads to a single,self-consistent expression for the Rossby number span-ning both viscous regimes. For simplicity, we adoptEqs. (29)–(30) rather than their smoothed counterparts,leading to the expression

Ro =1

kDmin

[

γ2 Ra∗

F − Ra∗critF

E, ǫ1/2

max

(

Ra∗

F − Ra∗critF

E

)1/2]

.

(33)which is equivalent to the minimum of Eqs. (18) and(23). Thus, arguments based on energetics and mo-mentum arguments are now self-consistent: both ap-proaches predict that the Rossby number scales as(Ra∗

F − Ra∗critF )/E at weakly supercritical Rayleigh

numbers but as (Ra∗

F − Ra∗critF /E)1/2 at strongly su-

percritical Rayleigh numbers.

Figure 7 plots the jet speed predicted by Eq. (33)as contours of Rossby number versus Ra∗

F and E, witha contour spacing that is even in log(Ro) and contourvalues identical to those in Figs. 3 and 4. Regime I,in which the jet-pumping efficiency ǫ is constant, lies

14 A.P. Showman et al., submitted to Icarus0.

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0176

640.

0270

390.

0413

910.

0633

590.

0969

880.

1484

6

log 1

0(E

)

log10

(Ra∗

F )

−6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5−5

−4.5

−4

−3.5

0.0003

0.001

0.003

0.01

0.03

0.1

Fig. 7. Prediction of our scaling law, Eq. (33), that com-bines Regimes I and II. Depicts contours of Rossby numberversus modified flux Rayleigh number (abscisssa) and Ek-man number (ordinate). Adopts kD = 5π, γ = 1, andǫmax = 0.3 (see text). Note the similarity of the slopes andvalues of the contours with the simulation results plotted inFig. 3. Contours are evenly spaced in log(Ro), with contourvalues identical to those in Figs. 3 and 4. For reference, thedashed line depicts the critical modified flux Rayleigh num-ber given by Eq. (24). Above the dashed line, convectiondoes not occur.

toward the lower right, while Regime II, in which thecorrelation coefficient C is constant, lies toward theupper left. The transition between the regimes man-ifests visually as a jump in the spacing of the constant-Ro contours and occurs at a Rossby number of ∼0.02for the parameters chosen, roughly consistent with theregime transition in the simulated data (see Fig. 3).Within Regime II, the slopes of the contours decreaseas the Rossby number is decreased and the modifiedflux Rayleigh number approaches the critical value. Asimilar decrease in the slope of the contours at smallRossby number is evident in Christensen (2002)’s dataplotted in Fig. 3. Moreover, throughout most of theparameter space accessed by the simulations, the pre-dicted Rossby number (Fig. 7) matches those obtainedin the simulations (Fig. 3) within a factor of ∼2.

5. Scaling for the jet speeds in Regime III: the

asymptotic regime

The simulated behavior in Fig. 3 is reasonably wellexplained by the theory presented in Section 4, whereRossby numbers scale approximately with (Ra∗

F−Ra∗critF )/E

at low Rayleigh numbers (Regime II) and as (Ra∗

F /E)1/2

at high Rayleigh numbers (Regime I). However, Chris-tensen (2002) argued that at the highest Ra∗

F he ex-plored for each value of E (or equivalently the lowest Evalues explored for each value of Ra∗

F ) his simulationsapproached an asymptotic regime where the mean equi-librated jet speed became independent of the values ofthe diffusivities and depend only weakly on the heatflux. This behavior manifests in Fig. 3(top) as botha widening of the constant-Ro contours and significantincrease in their slope beyond one toward the far rightedge of the plot, neither of which is predicted by thescalings in Section 4. Based on an empirical fit to hissimulation results, Christensen (2002) proposed that inthis asymptotic limit—which we call Regime III—theRossby number associated with the characteristic jetspeeds scales as8

Ro = 0.53(Ra∗

F )1/5. (34)

At present, however, a physical basis for this em-pirical fit is lacking. To see whether Eq. (34) can betheoretically explained, we present here a scaling anal-ysis based on mixing-length theory. We compare it toavailable simulations and evaluate its implications forthe Jovian parameter regime.

We hypothesize that, if the viscosity is sufficientlysmall, the damping will be determined not by the molec-

8Because he included an inner-to-outer radius ratio in his def-inition of Ra∗

F , Christensen (2002) quotes the relationship witha pre-factor of 0.65 rather than 0.53.


ular viscosity but by an eddy viscosity associated withturbulent diffusion. The concept of an eddy viscosityis relevant only when the convection is sufficiently su-percritical (so that the eddies behave in a nonlinear,turbulent manner), leading us to adopt the approachof Section 4a. To obtain an expression for mean windspeed, we again equate forcing (Eq. 14) and damping(Eq. 15) but use the eddy viscosity, νeddy, in place of νin Eq. (15). We adopt a standard mixing-length formu-lation for the eddy viscosity, νeddy ≈ wk−1

mix, where w isthe convective velocity (given by Eqs. 9–10) and kmix

is the wavenumber associated with the mixing length(e.g., a typical distance traversed by coherent convec-tive plumes). In the case of a Boussinesq fluid, where αand ρ are constant and F , g, and cp are approximatelyconstant, this yields

u ∼ǫ1/2Ω1/4k

1/2mix

kγ1/2

(

αgF

ρcp

)1/4

(35)

where ǫ is assumed approximately constant as in Section4a. For an anelastic fluid, where α and ρ vary greatlywith radius, we follow the approach used in Section 4a,yielding

u ∼ǫ1/2Ω1/4

kγ1/2

∫

αgFcp

r2 dr

∫

(

αgFcp

)1/2

k−1mixρ

1/2r2 dr

1/2

. (36)

Let us nondimensionalize these expressions so thatthey can be compared with existing Boussinesq andanelastic numerical results and the empirical asymp-totic scaling (Eq. 34) proposed by Christensen (2002).Defining the Rossby number as Ro ≡ u/(ΩD) and as-suming that kmix is constant and that Ra∗

F ≫ Ra∗critF ,

Eqs. (35)–(36) can be nondimensionalized to yield

Ro =ǫ1/2Γ(kmixD)1/2

kDγ1/2(Ra∗

F )1/4 (37)

where Ra∗

F is given by Eq. (5) for the Boussinesq caseand Eq. (7) for the anelastic case. Γ is a dimensionlessconstant that depends on the planet’s basic-state radialdensity structure and is given by

Γ =

(∫

ρr2 dr)1/2(

∫

αgfcp

r2 dr)1/2

∫

(

αgfcp

)1/2

ρ1/2r2 dr

1/2

(38)

where f(r) is an order-unity dimensionless functionthat describes the radial dependence of the heat flux,F = F0f(r), and F0 is a constant (independent of ra-dius) that gives the characteristic value of the heat flux(e.g., halfway through the layer). Note that, for the

10−6

10−5

10−4

10−3

10−2

10−3

10−2

10−1

Ros

sby

num

ber

Ra∗

F

χ (Ra∗

F )1/4

b

10−3

10−2

10−1

Ros

sby

num

ber

χ (Ra∗

F )1/4

0.53 (Ra∗

F )1/5

a

Fig. 8. Comparison of our asymptotic scaling for the jetspeeds, Eq. (37) (solid curves) with Christensen’s empiricalasymptotic fit from Eq. (34) (dashed curve) and simulationresults from Christensen (2002) (panel a) and Kaspi et al.(2009) (panel b). Solid curves are evaluated using ǫ = 0.3,γ = 1, and χ = 1.


Boussinesq case, Γ ≡ 1. Interestingly, for the Kaspiet al. (2009) model, Γ = 1.02 despite the large variationof α and ρ with radius.

This simple theory therefore predicts that, in theasymptotic regime, the characteristic jet speed scalesas F 1/4, or equivalently the characteristic Rossby num-ber scales as (Ra∗

F )1/4, with no explicit dependence onthe molecular or numerical viscosity. Our exponentof 1/4 compares favorably to—but is modestly steeperthan—Christensen’s empirically determined exponentof 1/5. Notably, our theoretical exponent is significantlysmaller than that for Regimes I and II (Section 4),where, at constant E, jet speeds scale with F 1/2 or Fdepending on assumptions. In our asymptotic theory,the weak dependence on Ra∗

F occurs because the eddyviscosity decreases with decreasing heat flux (unlike thebehavior in the viscous regime, where the viscosity is afixed parameter). As a result, a weakly forced flow hasextremely weak damping, whereas a strongly forced flowhas extremely strong damping; this leads to an equili-brated jet speed that depends only weakly on heat flux.Specifically, the kinetic-energy forcing scales with F ,and because the convective velocities scale with F 1/2

(Eq. 9), the eddy viscosity and therefore the kinetic-energy damping also scale with F 1/2. This leads to anequilibrated kinetic energy scaling with F 1/2 and a jetspeed scaling with F 1/4.

Figure 8 explicitly compares the theoretical scal-ing (Eq. 37) with the empirical scaling (Eq. 34) andthe simulation results of Christensen (2002) and Kaspiet al. (2009). The theoretical scaling is shown fora value of the prefactor ǫ1/2(kmixD)1/2γ−1/2(kD)−1,which we call χ, equal to 1. For this value, the mag-nitudes of the jet speeds predicted by our Eq. (37)match Christensen’s fit within a factor of ∼2 (com-pare solid and dashed lines in Fig. 8a). For the val-ues of kD ≈ 5π adopted previously, this would requirekmix ≈ (5π)2/D, implying a rather short mixing length2πk−1

mix of ∼0.03D. This seems at least qualitativelyconsistent with the fact that the vorticity structure ofconvective plumes exhibits little coherence in cylindricalradius s (see Christensen 2002). However, it is unclearwhether the value of kD ≈ 5π relevant for the viscousregime is also appropriate to the asymptotic regime.

Overall, the close agreement between the heat-fluxdependences of our asymptotic scaling and Christensen’sempirical fit is encouraging and argues that the basicidea encapsulated by our scaling—that the damping de-pends on F to a power modestly less than that of theforcing—is correct.

At high Rayleigh numbers, both the Boussinesq andanelastic simulations (Fig. 8a and b, respectively) con-verge toward a trend similar to the shallow slope of the

predicted asymptotic scaling. However, the simulationsdiverge from the asymptotic scalings at low Rayleighnumbers as the effects of the model diffusivities becomestrong. Christensen’s fit, and the scaling presented inthis section, are intended to describe the asymptotic

behavior in the limit of negligible diffusivities (for dis-cussion see Christensen 2002).

When extrapolated to Jovian values of Ra∗

F , thisasymptotic regime predicts that the mass-weighted meanvelocities in the molecular envelopes of Jupiter and Sat-urn are small, ∼0.1–1 m sec−1. We consider this is-sue further, and combine the viscous and asymptoticregimes into a single scaling, in the next subsection.

6. Combining the three scalings

We have derived scalings for the jet speeds in threeregimes: two regimes in which the numerical viscositydominates the damping (Regimes I and II in Sections 4aand 4b, respectively) and another in which the viscos-ity is determined by turbulence i.e., an eddy viscosity(Regime III in Section 5). Here, we combine the scal-ings.

Our basic approach in deriving the scalings was tobalance forcing against damping; the damping repre-sents a numerical (or molecular) viscosity in Regime Ibut an eddy viscosity in Regime III. Here, we supposethat both molecular and eddy viscosities operate simul-taneously, and that whichever viscosity is larger willdominate. Because larger viscosity implies smaller equi-librated jet speeds, we therefore expect that the smaller

of the two Rossby numbers will dominate. Combiningthe two viscous regimes with the asymptotic regime, wecan thus roughly say

Ro =1

kDmin

[

γ2 Ra∗

F − Ra∗critF

E,

ǫ1/2max

(

Ra∗

F − Ra∗critF

E

)1/2

, ǫ1/2maxΓ

(kmixD)1/2

γ1/2(Ra∗

F )1/4

]

(39)

where we have assumed that k is constant and ǫmax

has the same values in the second and third terms onthe right side. Neglecting Ra∗crit

F in the expressionfor Regime I, the transition between the asymptoticRegime III and the viscous Regime I occurs for Ekmannumbers

Etr ≈γ(Ra∗

F )1/2

Γ2kmixD. (40)

At a given Ra∗

F , the viscous regime occurs for largeEkman numbers while the asymptotic regime occursfor small Ekman numbers, with a transition near Etr.Analogously, at constant E, the viscous regime domi-nates at small Ra∗

F and the asymptotic regime occurs at

A.P. Showman et al., submitted to Icarus 170.

0005

8592

0.00

1372

9

0.00

3217

1

0.00

6092

9

0.01

1539

0.01

7664

0.02

7039

0.04

1391

0.06

3359

0.09

6988

0.14

846

log10

(Ra∗

F )

log 1

0(E

)

−6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3 −2.5−6

−5.5

−5

−4.5

−4

−3.5

0.0003

0.001

0.003

0.01

0.03

0.1

Fig. 9. Predicted jet speeds for a scaling law that com-bines Regimes I, II, and III, given in Eq. (39), for the caseǫmax = 0.3, kD = 5π, and γ = 1. The value of kmix

is chosen so that the prefactor in the asymptotic regime,χ ≡ ǫ

1/2max(kmixD)1/2/(γ1/2kD), equals one. Depicts con-

tours of Rossby number versus modified flux Rayleigh num-ber (abscissa) and Ekman number (ordinate). In the vis-cous regimes, contours tilt upward to the right, whereas inthe asymptotic regime they are vertical. For reference, thedashed line depicts the critical modified flux Rayleigh num-ber given by Eq. (24). Above the dashed line, convectiondoes not occur.

large Ra∗

F , with a transition near modified flux Rayleighnumbers of Γ4γ−2(kmixD)2E2.

To illustrate the combined scaling, Fig. 9 depictsEq. (39) as contours of Rossby number versus Ra∗

F andE for the specific case ǫ = 0.3, γ = 1, and kD = 5π(the same values as in all previous figures). The valueof kmix is chosen so that the prefactor in the asymptotic

regime ǫ1/2max(kmixD)1/2(γ1/2kD)−1 ≡ χ = 1. The vis-

cous regimes lie in the upper left and exhibit slopedcontours. The asymptotic regime lies at the lowerright and exhibits vertical contours, consistent with theexpectation that Rossby number is independent of Ethere. As expected from Eq. (40), the transition be-tween Regimes I and III slopes gradually upward tothe right; the asymptotic regime extends to larger Ewhen Ra∗

F is larger. For the value of kmix adoptedin Fig. 9, Christensen’s simulations lie predominantlywithin the viscous regimes; however, the predicted tran-

sition to the asymptotic regime occurs close to Chris-tensen’s highest-Ra∗

F cases for each value of E. If Fig. 9is correct, it would suggest that the asymptotic regimewould become clear at Ekman numbers an order of mag-nitude smaller than Christensen explored for each valueof Ra∗

F . Note, however, that in reality the transition be-tween the regimes will probably occur smoothly acrossa broad strip straddling Etr(Ra∗

F ); if so, it might benecessary to reach Ekman numbers as low as 10−7 tosee the asymptotic regime clearly.

Now we examine Fig. 10a, which plots our combinedscaling (Eq. 39) versus Ra∗

F for several values of E andcompares them to Christensen (2002)’s Boussinesq sim-ulations. Curves and simulation data are shown forE = 10−5 (black), 3×10−5 (magenta), 10−4 (blue), and3 × 10−4 (red). The three regimes are clearly visible inthe scalings and agree favorably with the simulationsthroughout the plotted range. The gradual increase inthe slopes in Regime II near the left edge—which canbe seen in both the scalings and the simulations, espe-cially at Ekman numbers of 3×10−5 and 10−4—resultsfrom the fact that Ra∗

F approaches its critical value(evaluated for the scalings using Eq. 24). Althoughthe match is not perfect, the scalings not only approx-imately reproduce the asymptotic behavior suggestedby Christensen but also exhibit the trend (suggestedby Christensen’s simulations) where the transition tothe asymptotic regime ocurs at larger Ra∗

F for largerE. The values of Ra∗

F at these transitions are similarin the scalings as in Christensen’s simulations to withinan order of magnitude for most E values. Moreover, theactual values of Rossby number predicted by the scal-ing at a given Ra∗

F and E match those of Christensen’ssimulations to within a factor of ∼2.

Figure 10b compares our combined scaling againstsimulations performed using the Kaspi et al. (2009)model. Magenta squares, blue upward-pointing trian-gles, and red right-pointing triangles present the Rossbynumbers from anelastic simulations performed respec-tively at E = 1.5 × 10−4, 4 × 10−4, and 8 × 10−4 usingthe Jovian radial structure as in Kaspi et al. (2009),while the black squares present Rossby numbers from aBoussinesq simulation at E = 3× 10−5 performed withthe same model. Scaling predictions are overplotted inthe corresponding colors (black, magenta, blue, and redfor Ekman numbers of 3 × 10−5, 1.5 × 10−4, 4 × 10−4,and 8 × 10−4, respectively). In these runs, the Prandtlnumber equals 10, leading to a smaller critical Rayleighnumber than given in Eq. (24); as a result, the plottedranges are sufficiently supercritical that the scaling pre-dictions do not exhibit a gradual change in slope nearthe left edge of the plot (in contrast to the behavior inpanel a). Again, the scalings agree well with the simu-lations; the discrepancy is everywhere less than a factor


10−4

10−3

10−2

10−1

Ros

sby

Num

ber

a

E = 1×10−5

E = 3×10−5

E = 1×10−4

E = 3×10−4

10−7

10−6

10−5

10−4

10−3

10−2

10−3

10−2

10−1

Ros

sby

Num

ber

Ra∗

F

b

E = 3×10−5

E = 1.5×10−4

E = 4×10−4

E = 8×10−4

Fig. 10. Predicted jet speeds for a scaling law that com-bines Regimes I, II, and III, given in Eq. (39), for the caseǫmax = 0.3, kD = 5π, and χ = 1. Depicts Rossby numberversus modified flux Rayleigh number for several values ofE for our combined scaling (curves) and from simulations(symbols). Panel a: Christensen (2002)’s Boussinesq sim-ulations, at Ekman numbers of 1 × 10−5, 3 × 10−5, 10−4,and 3 × 10−4, with scaling predictions evaluated at thosesame values (plotted in the corresponding colors). Panelb: Anelastic simulations at Ekman numbers of 1.5 × 10−4,4 × 10−4, and 8 × 10−4, along with a Boussinesq simula-tion at 3 × 10−5, performed using the Kaspi et al. (2009)model. Curves give scaling predictions evaluated at thosesame values (plotted in corresponding colors).

of 2. This agreement is encouraging considering that,together, Fig. 10a and b present results spanning factorsof 200,000 in Ra∗

F , 1000 in Ro, and 80 in E.

Interestingly, Fig. 9 suggests that Regime I disap-pears for sufficiently small Ekman and modified fluxRayleigh numbers (E ≤ 10−7 and Ra∗

F ≤ 10−8 for theparameters chosen). If so, then at very small Ekmannumbers, Regime II abuts directly against Regime III.No numerical simulations have yet been performed atthese small parameter values, however, and how thistransition occurs remains an open question. Equa-tion (39) may need modification in this low-Ra∗

F andlow-E portion of the parameter space pending furthernumerical and theoretical work.

Next, we extrapolate the scalings to Jupiter, wherethe Ekman and modified flux Rayleigh numbers are∼10−15 and ∼10−13–10−14, respectively. Despite theuncertainties, Eq. (39) would suggest that Jupiter liesin the asymptotic regime. For the Jovian values ofRa∗

F and E, the scalings predict Rossby numbers of∼3× 10−5–6× 10−4 and the implied zonal wind speedsin the interior are ∼0.1–1 m sec−1. Although numeri-cal simulations that are overforced by orders of magni-tude can easily produce Jupiter-like jet speeds (Chris-tensen 2001, 2002; Heimpel and Aurnou 2007), bothChristensen’s asymptotic fit and our mixing-length scal-ings therefore suggest that, given the weakly forcedconditions of Jupiter’s interior, the zonal-jet speeds inthe interior are much slower than the observed cloud-level values (which reach ∼150 m sec−1 for Jupiter and400 m sec−1 for Neptune).

7. Thermal wind

Several authors have suggested that the molecular in-teriors of giant planets should exhibit zonal winds thatare constant on columns parallel to the rotation axis(the z axis), following the expectations of the Taylor-Proudman theorem (Busse 1976; Ingersoll and Pollard1982). This Taylor-Proudman behavior is expectedwhen a geostrophic, low-viscosity fluid has a barotropicstructure where isobars and iso-density contours align(i.e., where ∇ρ×∇p = 0), as might be expected if con-vective mixing homogenizes the interior entropy.

However, convective plumes lead to internal densityperturbations that (while small) are non-zero, so thejets should exhibit some degree of thermal-wind shearin the z direction even though the flow is geostrophi-cally balanced at large scales. Boussinesq simulationstend to exhibit modest shear in z (Aurnou and Ol-son 2001; Christensen 2001, 2002; Heimpel and Aurnou2007; Heimpel et al. 2005, and others), but recent 3Danelastic simulations exhibit significant variation of thezonal wind along the z axis (Kaspi et al. 2009). This


comparison hints that compressibility could allow shearto develop in the z direction. On the other hand, largerheat fluxes lead to larger convective density perturba-tions, which could allow greater wind shear to developin overforced simulations than in the real planetary in-teriors. Sorting out the relative roles of overforcing andcompressibility in enabling shear along the jets is there-fore necessary to understand whether Jupiter can ex-hibit significant shear with z along its jets. Ultimately,we wish to understand the magnitude of wind shear seenin both Boussinesq and anelastic simulations, how suchwind shear should depend on the heat flux and com-pressibility, and the extent to which real giant planetsshould exhibit (or lack) significant wind shear in z alongthe jets. Here we address these questions using simplescaling arguments.

To obtain a generalized version of the thermal-windequation valid for deep atmospheres, we start with thevorticity balance for a geostrophic fluid where frictionforces are weak (Pedlosky 1987, p. 43):

2(Ω · ∇)v − 2Ω∇ · v = −∇ρ ×∇p

ρ2, (41)

where Ω is the planetary rotation vector, v is the 3Dvelocity vector, ρ is density, and p pressure. Castingthis in a cylindrical coordinate system, with z alignedalong Ω, and taking the longitudinal component of thisequation yields

2Ω∂u

∂z= −

∇ρ ×∇p

ρ2· λ, (42)

where λ is the unit vector in the longitudinal directionand u is the zonal wind.

Let’s now express the baroclinic term in terms ofdensity gradients along isobars. Consider an infinites-imal patch of fluid, and suppose there is an angle δbetween the vectors ∇p and ∇ρ. There is thus alsoan angle δ between isobars and iso-density surfaces.From geometry, the partial derivative of density alongthe isobar is (∂ρ/∂l)p = |∇ρ| sin δ, where l is the co-ordinate along the isobar along which density varies(see Fig. 11). Since the magnitude of ∇ρ × ∇p is just|∇ρ ×∇p| = |∇p| |∇ρ| sin δ, we can say that

|∇ρ ×∇p| = |∇p|

(

∂ρ

∂l

)

p

. (43)

We wish to apply this equation to the non-hydrostatic,convecting interior of a giant planet. But the non-hydrostatic component of the pressure gradient is ex-tremely small compared to the hydrostatic componentfor this case, so to good approximation we can say∇p ≈ ∇phydrostatic = −ρg. (This would not be valid

Fig. 11. Geometry relating total density gradient to den-sity gradient along isobars. Consider an infinitesimal patchof fluid with isobars and iso-density surfaces misaligned, asshown. The angle between these surfaces is δ. Consider alsotwo iso-density surfaces, with densities ρ1 and ρ2 and a den-sity difference of ∆ρ ≡ ρ1 − ρ2, that are spaced a distance∆b apart. The magnitude of the total density gradient isthus |∇ρ| = ∆ρ/∆b. The magnitude of the partial deriva-tive of density along the isobar is just (∂ρ/∂l)p = ∆ρ/∆l,where ∆l is the separation of the iso-density surfaces asprojected onto the isobar. Since ∆b/∆l = sin δ, we have(∂ρ/∂l)p = |∇ρ| sin δ.


if convection produced fractional density perturbationsof order unity, but in reality the convective fractionaldensisty pertubations are ∼ 10−4 or less for the interiorof a giant planet.) Thus,

|∇ρ ×∇p| ≈ −ρg

(

∂ρ

∂l

)

p

. (44)

Inserting this expression into Eq. (42) yields

2Ω∂u

∂z≈

g

ρ

(

∂ρ

∂l

)

p

(45)

to good approximation.9 This expression implies thatthe zonal-wind shear of the jets in z is

∂u

∂z≈

gkjet

2Ω

(

δρ

ρ

)

p

, (46)

where kjet is the meridional wavenumber of the jets and(δρ/ρ)p is the fractional density difference (along iso-bars) across a jet.

Internal density contrasts are caused by convection.To estimate the wind shear of the jets in z, we assumethat the convectively induced interior density contrastsbecome organized on the jet scale, so that (δρ/ρ)p inEq. (46) can be equated with our convective estimateof δρ/ρ from Eq. (12). Integrating, we obtain

∆u ≈kjet

2

∫(

Fαg

ρcpΩ

)1/2

dz, (47)

where ∆u is the characteristic, order-of-magnitude dif-ference in zonal wind from top to bottom of a constant-s column. In the Boussinesq case, the integrand is ap-proximately constant, in which case ∆u ≈ kjetD(Fαg/Ωρcp)

1/2,where we have dropped the factor of 2 because a typi-cal column is somewhat longer than D. In the anelasticcase, α/ρ varies by orders of magnitude along a Taylorcolumn and the integral must be evaluated.

Let us nondimensionalize these expressions in termsof a differential Rossby number, ∆Ro ≡ ∆u/(ΩD),which represents the characteristic difference in localRossby number from top to bottom of a column atconstant s. (This can be viewed as simply a nondi-mensional measure of the column-integrated zonal-windshear.) Doing so leads to the same expression for theBoussinesq and anelastic cases, namely,

∆Ro ≈ Γ2Dkjet(Ra∗

F )1/2 (48)

9Equation (45) is equivalent to Kaspi et al. (2009, Eq. 38) butis more general because it allows the possibility of compositionalas well as thermal density gradients on isobars.

where for the Boussinesq case Ra∗

F is given by Eq. (5)and Γ2 ≡ 1 while for the anelastic case Ra∗

F is given byEq. (7) and Γ2 is given by the dimensionless expression

Γ2 =

∫

(

fαgρcp

)1/2

dz(∫

ρr2 dr)1/2

D(

∫

αgfcp

r2 dr)1/2

(49)

where as before f(r) is a dimensionless, order-unityfunction that gives the radial dependence of the heatflux (see Section 5). For purposes of evaluating Γ2,we simply evaluate the first integral in the numeratorwith respect to r rather than z, leading to the valueΓ2 ≈1 for the interior structure adopted in Kaspi et al.(2009). Motivated by the possibility of zonal flow inEarth’s outer core, Aurnou et al. (2003) derived a re-lationship analogous to Eq. (48) for the special case ofa Boussinesq fluid; Eq. (48) generalizes this to includecases where α and ρ vary radially.

An alternate way to represent the wind shear is toestimate the ratio between the characteristic zonal windat the top and bottom, or more precisely the ratio ofcharacteristic zonal wind at the intersection of a sur-face of constant cylindrical radius (constant s) with theoutermost boundary to that at the innermost bound-ary or equatorial plane, as appropriate. The estimatesof mass-weighted mean zonal wind (or Ro) presented inSections 4 and 5 provide a rough measure of the zonalwind at the bottom. The characteristic dimensionlesszonal wind at the top is then Ro + ∆Ro. Their ratio is

utop

ubottom

∼ 1 +∆Ro

Ro. (50)

The plus sign corresponds to the case where the char-acteristic Rossby number is greater at the top than thebottom, as seen in the anelastic simulations (Kaspi et al.2009). If we adopt our viscous scaling for Ro fromRegime I, which provides an adequate representationof the mass-weighted mean winds in the regime of fastzonal winds simulated by Christensen (2002) and Kaspiet al. (2009), we obtain

utop

ubottom

∼ 1 +Γ2D

2kkjetE1/2

ǫ1/2. (51)

Note that, unlike the expression for ∆Ro, this expres-sion for utop/ubottom is independent of Ra∗

F .

Thus, this simple theory predicts that the charac-teristic change in zonal wind along a column is pro-portional to F 1/2, or equivalently that the change incharacteristic Rossby number along a column is pro-portional to (Ra∗

F )1/2. This prediction results directlyfrom the expectation that the internal density contrastsassociated with convection are proportional to F 1/2 via


Eq. (12): flows with greater convective heat fluxes ex-hibit greater convective density contrasts and hence candevelop greater thermal-wind shear in the z direction ifthose density contrasts become organized on the largescale. On the other hand, because the mass-weightedcharacteristic zonal wind itself is approximately propor-tional to F 1/2 in Regime I (Section 4a), this theory alsopredicts that the ratio of zonal wind at the top andbottom of a column is approximately independent ofRa∗

F in Regime I. This behavior would not necessar-ily continue into asymptotic Regime III, however: sincepredicted mass-weighted jet speeds scale with F 1/4 inthat regime, the fractional changes in jet speed along zwould be expected to scale with (Ra∗

F )1/4 there.

To test these scalings, we analyzed the three-dimensionalnumerical simulations from Kaspi et al. (2009) and sup-plemented these with additional simulations performedusing the same model. Both Boussinesq and and anelas-tic simulations were considered. Figure 12 shows thecharacteristic values of ∆Ro and utop/ubottom calcu-lated from these numerical simulations. The Boussinesqcases (filled symbols) generally have ∆Ro ∼ 10−4–10−2

and ratios utop/ubottom of 1 to 2. This implies onlymodest (factor of ∼< 2) variation of zonal wind in z,consistent with the coherent nature of the jets across thedomain in the Boussinesq cases (see Kaspi et al. 2009,Fig. 4, right panel). On the other hand, the anelas-tic simulations (open symbols) have ∆Ro ∼ 0.01–0.1 ormore, with ratios utop/ubottom ∼5–20, consistent withthe significant wind shear along columns in the anelasticcases (see Kaspi et al. 2009, Fig. 4, left panel). Includedin Fig. 12 are the scaling predictions (Eqs. 48 and 51).

Interestingly, the scalings seem to provide an approx-imately valid upper limit for the values of ∆Ro andutop/ubottom from the simulations. This suggests thatthe basic idea encapsulated here—that the convectivedensity contrasts set the large-scale density contrasts(on isobars) and hence determine the allowable thermal-wind shear in the interior—is correct.

On the other hand, the scalings do not seem to of-fer an explanation for the difference in wind shear seenin Boussinesq and anelastic simulations. The prefactorΓ2—which reflects the radial variation of ρ, α, and otherparameters—is nearly 1 not only in the Boussinesq casebut in the anelastic case too, at least for the verticalstructure adopted in the Kaspi et al. (2009) simulations.As a result, the scalings (48) and (51) predict the samewind shear as a function of Ra∗

F for both cases, contraryto the simulation results. Conceivably, the convectivedensity perturbations could develop greater large-scaleorganization in the anelastic cases than the Boussinesqcases, promoting greater wind shear in the anelasticcases. This issue warrants future investigation.

10−6

10−5

10−4

10−3

10−2

100

101

Ro t/R

o b

RaF*

b

10−4

10−2

100

∆ R

o a

Fig. 12. Two measures of the zonal-wind shear in z (i.e.parallel to the rotation axis) as a function of Ra∗

F for aseries of anelastic simulations and theory. a and b depictcharacteristic difference in the Rossby number between thetop and bottom and the ratio of characteristic zonal windat the top and bottom, respectively. Curves denote our pre-dicted scalings of Eq. (48) (top) and Eq. (51) (bottom) usingkD = kjetD = 5π. Red solid symbols are Boussinesq sim-ulations; open symbols are anelastic simulations with thefull Jovian radial profiles of basic-state density, thermal ex-pansivity, and heat flux as described in Kaspi et al. (2009).In b, curves are depicted for four of the same E values asthe anelastic simulations (4× 10−4, 6× 10−4, 8× 10−4, and9 × 10−4 in blue, black, red, and green, respectively) andevaluated using ǫ = 1.


8. Conclusions

Over the past two decades, several groups have per-formed three-dimensional numerical simulations of con-vection in spherical shells to test the hypothesis thatconvection in the molecular interiors pumps the fastzonal jet streams observed on Jupiter, Saturn, Uranus,and Neptune. These studies show that the zonal jetspeeds can range over more than three orders of mag-nitude depending on the heat flux, viscosity, and otherparameters (e.g., Christensen 2002; Kaspi et al. 2009).Here, we presented simple theoretical arguments to ex-plain the characteristic jet speeds—and their depen-dence on heat flux and viscosity—seen in these studies.

Our main findings are as follows:

• We demonstrated that the characteristic convec-tive velocities in the Kaspi et al. (2009) simula-tions scale well with (αgF/ρcpΩ)1/2. While sucha rotating scaling has long been proposed and isconsistent with rotating laboratory experiments(e.g., Fernando et al. 1991), this to our knowledgeis the first demonstration of its applicability tothe interiors of giant planets over a wide range ofheat fluxes and rotation rates.

• Next, we attempted to explain the jet speeds ob-tained by Christensen (2002) and Kaspi et al.(2009) in three-dimensional numerical simulationsof convection with free-slip boundaries. At weaklysupercritical Rayleigh numbers, linear theory (e.g.,Busse 1983a; Busse and Hood 1982) and nu-merical simulations (e.g., Christensen 2002) showthat the eastward and outward convective veloc-ity components are highly correlated outside thetangent cylinder. We showed that if the degreeof correlation C between the eastward and out-ward convective velocity components is constant(as expected at very low supercriticalities) andif the jets are damped by a numerical viscosity,then the mean jet speeds should scale as the con-vective heat flux over the viscosity (Regime II).This scaling explains the jet speeds obtained inboth Boussinesq and anelastic simulations at lowRayleigh numbers, where the jet speeds are rela-tively weak.

• On the other hand, at higher Rayleigh num-ber, the eastward and outward convective veloc-ity components become increasingly decorrelated.We showed that, if the fraction of convective en-ergy release used to pump the jets (ǫ) is a con-stant in this regime, and if the jets are dampedby a numerical viscosity, then the mean jet speedsshould scale as (F/ν)1/2. This scaling (Regime I)

explains quite well the mean jet speeds obtainedin the simulations from Christensen (2002) andKaspi et al. (2009) in cases when the jets domi-nate the total kinetic energy, as occurs at highlysupercritical Rayleigh numbers.

• The relationship between the correlation coeffi-cient C and the jet-pumping efficiency ǫ showshow the transition between these two regimescan naturally occur. A nearly constant correla-tion coefficient implies that the jet-pumping ef-ficiency must increase strongly with Ra∗

F . Whenthe jet-pumping efficiency finally plateaus near itsmaximum value of 1, then the correlation coeffi-cient must begin to decrease strongly with fur-ther increases in Ra∗

F (see Eq. 28). Thus, a nat-ural regime transition occurs between Regime IIat low Rayleigh numbers and Regime I at highRayleigh numbers. Moreover, we rigorously cal-culated C and ǫ from the anelastic simulationsand showed that, as expected, C decreases andǫ increases with Ra∗

F . A simple, heuristic the-ory that transitions smoothly between constantcorrelation coefficient at low Ra∗

F and a constantjet-pumping efficiency at high Ra∗

F explains thequalitative features of the correlation coefficientand jet-pumping efficiency as calculated from thesimulations.

• Both the Boussinesq (Christensen 2002) and anelas-tic simulations hint at the existence of a thirdregime where, at sufficiently large heat flux and/orsufficiently small viscosity, the characteristic jetspeeds become independent of the viscosity. Weconstructed a simple mixing-length scaling for thebehavior in this regime under the assumption thatthe jet-pumping efficiency is constant and thedamping results from an eddy viscosity (ratherthan the molecular or numerical viscosity). Thisscaling suggests that the mean jet speed in themolecular interior should scale with the convec-tive heat flux to the 1/4 power. Given the sim-plicity of our model, this agrees favorably with theasymptotic fit suggested by Christensen (2002),which proposes that jet speeds should scale withthe convective heat flux to the 1/5 power. Bothscalings suggest that the mean wind speeds in themolecular interior should be significantly weakerthan the jet speeds measured at the cloud level.

• A scaling that combines these three regimes canexplain (to within a factor of ∼2) the mass-weighted mean jet speeds obtained in both theBoussinesq and anelastic simulations ranging acrossa factor of 105 in Ra∗

F , 102 in Ekman number,


and 103 in Rossby number. Importantly, whenthe modified flux Rayleigh number is defined in anappropriate mass-weighted way, the same scalings

apply to both Boussinesq and anelastic cases.

• Finally, we showed that, under the assumptionthat the convective density perturbations becomeorganized on the large scale, the variation in zonalwind in z (i.e., in the direction parallel to the rota-tion axis) in the convective interior should scale asF 1/2. In practice, we found that the scaling servesas an upper limit. At present, the scalings do notexplain why the anelastic simulations exhibit suchdifferent wind shear than the Boussinesq cases.

It is important to emphasize that the scaling theo-ries presented in Sections 4–6 apply only to the mass-weighted mean wind speeds. Although we attemptedto understand the wind shear along columns in Sec-tion 7, our theory does not provide information on thedetailed three-dimensional structure of the jets, such astheir variation in latitude.

Several challenges remain for the future. First, wedid not consider the effect of the Prandtl number onthe jet speeds, mostly because a careful characteriza-tion of the Ra∗

F - and E-dependence of the jet speedsover a wide range has only been performed for a verylimited number of Pr values (1 for Christensen (2002)and 10 for Kaspi et al. (2009)). However, several stud-ies show that Pr does affect the jet speeds (e.g., Aubertet al. 2001; Christensen 2002; Kaspi 2008), althoughone would expect this dependence to disappear in theasymptotic limit. Extending our scalings to include thePr dependence is an important goal for future work.

Second, the simulations we set out to explain—thoseof Christensen (2002) and Kaspi et al. (2009)—assumethe convection occurs in a thick shell, for which the re-gion outside the tangent cylinder dominates the totalmass, volume, and kinetic energy. As shown by sev-eral authors, however, the dynamical mechanisms anddetails of jet pumping differ for the fluid inside andoutside the tangent cylinder (e.g., Heimpel and Au-rnou 2007). Thus, the scaling properties could differfor simulations in a thin shell (Heimpel and Aurnou2007; Heimpel et al. 2005), for which the high-latituderegions inside the tangent cylinder will more stronglyinfluence mass-weighted mean flow properties.

Third, we assumed that the wavenumber k—whichrepresents the length scale for the variation of zonal-wind shear and Reynolds stress with cylindrical radiuss—was a constant with the Rayleigh and Ekman num-bers. To zeroth order, this seems a reasonable approx-imation, because simulations suggest that the widthsof the jets outside the tangent cylinder are set by the

adopted shell thickness and not by the Ekman andRayleigh numbers.10 Nevertheless, analytic solutions inthe linear limit suggest that the zonal wavelengths nearthe critical Rayleigh number should scale as E1/3 (e.g.,Busse 1970; Roberts 1968), and scaling arguments inthe context of convection with no-slip boundaries sug-gest that the sizes of vortex rolls scale as E1/5 (Aubertet al. 2001). Thus, the wavenumber representing thes-variation of the zonal-jet-shear and Reynolds stressescould potentially have a weak dependence on E. Al-though our assumption of constant k seems to workreasonably well over the simulated range, it would beworth relaxing this assumption in the future.

Our scaling arguments—and the simulations they at-tempt to explain—assume the fluid is electrically neu-tral. However, the interiors of Jupiter and Saturn be-come metallic at pressures exceeding ∼1 Mbar (Guillotet al. 2004), and the resulting magnetohydrodynamic ef-fects could influence the jet dynamics even in the molec-ular region (e.g., Glatzmaier 2008; Kirk and Stevenson1987; Liu et al. 2008). For example, Liu et al. (2008)argued that Ohmic dissipation would limit strong jetsto regions above 96% (86%) of the radius on Jupiter(Saturn). They further argued that, if the interioris isentropic and exhibits columnar Taylor-Proudmanbehavior, the winds would be weak not only in theelectrically conducting region but throughout the con-vection zone. This result is consistent with ours al-though it invokes different physics. Moreover, the sim-ulations we investigated do not include solar forcingor latent heating. These effects may induce significantinternal density perturbations—and therefore thermal-wind shear—within a few scale heights of the observ-able cloud deck (e.g., Kaspi and Flierl 2007; Lian andShowman 2008, 2009; Schneider and Liu 2009; Williams2003). The development of convective models that in-clude these effects is a major goal for the future.

Acknowledgments. This work was supported byNSF grant AST-0708698 and NASA grant NNX07AF35Gto APS, NSF grant AST-0708106 to GRF, and a NOAAGlobal and Climate Change Postdoctoral Fellowshipadministered by the University Corporation for Atmo-spheric Research.

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