1420 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2004 American Meteorological Society

A Local Model for Planetary Atmospheres Forced by Small-Scale Convection

K. SHAFER SMITH

Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

(Manuscript received 29 May 2003, in final form 14 January 2004)

ABSTRACT

An equivalent-barotropic fluid on the b plane, forced at small scales by random stirring and dissipated bylinear heat and vorticity drag, is considered as a local model for flow in the weather layer of internally forcedplanetary atmospheres. The combined presence of b, a finite deformation scale, and large-scale dissipationproduce novel dynamics with possible relevance to the giant gas planets, which are apparently driven by small-scale convective stirring. It is shown that in order for anisotropy to form, one must have b(el5)21/3 * 3.9, wheree is the (convectively driven) energy generation rate, l is the deformation wavenumber, and b is the Coriolisgradient. The critical value above is not equivalent to the barotropic stability criterion, and numerical simulationsdemonstrate that anisotropic flow with average zonal velocities that are supercritical with respect to the lattercan form. The formation of jets (a different matter) is not implied by the excess of zonal kinetic energy, andis instead sensitive to the relevant stability criterion for the flow geometry at hand. When b is sufficiently largethat anisotropy does form, the flow scale and rms zonal velocity are set by a combination of Rossby wavecascade inhibition, the total energy constraint imposed by the large-scale dissipation, and the partitioning betweenavailable potential and kinetic energies. The resulting theory demonstrates that a relatively narrow range ofparameters will allow for the formation of anisotropic flow with scale larger than the deformation scale. Thisis consistent with observations that indicate little separation between the jet scales and deformation scales onJupiter and Saturn.

1. Introduction

Explanations of the basic features of Jovian circula-tion have ranged from deep convective rolls to baro-clinic instability. Recently, based on analysis of datafrom the Galileo satellite, Gierasch et al. (2000) andIngersoll et al. (2000) have argued that small-scale shal-low convection from internally generated heat is wide-spread and likely represents the largest source of energyto the circulation. The organizing principle invoked isthat of an inverse cascade of energy in two-dimensionalbarotropic vorticity flow, halted at the Rhines scale, re-sulting in the large-scale zonal jets observed. The im-plied model is similar to that of Williams (1978), buthis model assumed forcing by baroclinic instability. Ap-plication of the Charney–Stern criterion to the observedjets on Jupiter and Saturn shows that they are baro-tropically unstable (Ingersoll and Pollard 1982), butnevertheless long-lived. Explanations of the stability ofthe jets range from a modified Charney–Stern criterionfor deep flows (Ingersoll and Pollard 1982), noting thatthe observed jets are stable with respect to Arnold’sSecond Stability criterion (Dowling 1994), and a recent

Corresponding author address: K. Shafer Smith, Center for At-mosphere Ocean Science, New York University, 251 Mercer Street,New York, NY 10012.E-mail: shafer@cims.nyu.edu

suggestion that a barotropic criterion modified to includefriction could be satisfied by Jovian jets (Galperin et al.2001). Numerical simulations of forced two-dimension-al b-plane turbulence typically produce steady multiple-jet solutions only when large-scale dissipation is alsopresent. In the absence of large-scale dissipation, jetscontinually merge until reaching the domain scale, bothon the torus and on the sphere (Cho and Polvani 1996;Chekhlov et al. 1996; Manfroi and Young 1999; Huanget al. 2001; Smith et al. 2002). Moreover, the resultingjets are barotropically stable (Vallis and Maltrud 1993;Huang et al. 2001). A second characteristic of the ob-served jets is that the jet scale is not much larger thanthe deformation scale (Allison 2000), but the dynamicalbasis for this observation has not yet been explained.

Both two-dimensional b-plane turbulence on the onehand, and equivalent-barotropic turbulence on the otherare relatively well-understood, while the combined ef-fects of a finite deformation scale and b, both of whichare nonnegligible in Jovian dynamics, have receivedonly recent attention (Cho and Polvani 1996; Kukharkinand Orszag 1996; Okuno and Masuda 2003). Both pa-rameters are fundamental to a wide range of phenomenain geophysical fluids. Moreover, the resulting dynamicsof the combined system are nontrivial. For example,while b foments the formation of multiple jets, Okunoand Masuda (2003) demonstrate that a finite radius of

15 JUNE 2004 1421S M I T H

deformation can suppress jet formation in freely evolv-ing flow. The additional effects of small-scale forcingand large-scale drag further complicate the problem, butnot so much as to make it intractable.

Kukharkin and Orszag (1996) demonstrate that add-ing a finite deformation scale to a stable-jet dominatedflow can destabilize jets and produce vortices of slightlylarger size than the deformation scale. Those authorsdid not produce a statistically steady state, however,since no large-scale dissipation was present, so little canbe said about the jet scale they observed. They alsointroduced a second b-induced nonlinearity to the equa-tions of motion, which, they argue, leads to a morerelevant model. This term leads, in their simulations, toa preponderance of anticyclonic vortices, but this is notrelevant to the balances that yield observed jet scales.

Huang et al. (2001) and Galperin et al. (2001) con-sider small-scale random forcing of two-dimensionalvorticity dynamics on the sphere, with b and a large-scale hypofriction (¹212 operator). However, no finitedeformation scale is included, and the hypofriction isknown to distort the cascade at long times and largescales (Borue 1994; Danilov and Gurarie 2001). Themodel is sufficiently complex to produce a spectrum,25ky

and Galperin et al. (2001) show that this is indeed thespectrum of the observed jets on Jupiter. The scale oftheir jets, however, was fit to the data by choosing anappropriate hypofriction coefficient.

The present work seeks to develop the turbulencephenomenology that applies when all of the major pa-rameters affecting the large-scale circulation are present.However, both in the theory derived and in the numer-ical simulations used to test that theory, we neglect theeffects of spherical geometry, dynamics precluded bythe quasigeostrophic approximation, vertical variationsin mean stratification and the explicit details of con-vection. Because we neglect spherical geometry fromthe outset, the present study does not address the globalcirculation problem in one pass. Instead, the familiarequivalent-barotropic model equation is taken to rep-resent local patches of the global circulation, and hencefixed values of Coriolis gradient b and deformationwavenumber l. By considering the latitudinal variationsof these parameters, the results can be extrapolated tothe globe. On can criticize the relevance of this approachto planetary circulation from the outset as being far toosimplified, but it nevertheless addresses the problem ofthe interaction of parameters of the large scale circu-lation (b, l, drag and energy generation) not yet ex-plored, and so the simplicity of the geometry might beconsidered a prerequisite for this study.

In section 2 we derive a criterion necessary for theformation of flow anisotropy, and two candidate theoriesfor the flow scale and rms zonal velocity. Limitationsto the theories are also discussed. In section 3 we de-scribe a large set of simulations performed to test theproposed theories. The simulations lead to a clear con-clusion about which theory best describes the results.

In section 4 we attempt to apply the theory to data forJupiter and Saturn. Finally, in section 5 we discuss im-plications of the results.

2. Theory

a. Model formulation

The barotropic vorticity equation is written

]q1 J(c, q) 5 F 1 D 2 rq, (2.1)

]t

where c is the barotropic streamfunction,

2 2q 5 ¹ c 2 l c 1 by (2.2)

is the potential vorticity, l is the deformation wave-number, and r is a potential vorticity drag (a singleinverse time scale that parameterizes both thermal andvorticity drag—lifting of this restriction will be consid-ered later in the text). We shall assume that F representsa random forcing (be it a parametrization of stirring byinteraction with baroclinic modes, or small-scale con-vection) confined to a scale , and D is an enstro-21k F

phy dissipation operator that acts only on wavenumbersk . kF.

The net energy generation rate, less that unavoidablyabsorbed by the enstrophy dissipation, is

g [ 2 c(F 1 D) da ,E7 8A

where ^ & denotes a time average, A is the domain area,and da is a differential area element. In this system, atscales smaller than the deformation scale, kinetic (K)and available potential (P) energy both cascade to largerscale. At scales larger than the deformation scale, Pcontinues to cascade toward larger scale, while K doesnot. (If a large-scale source of kinetic energy exists,kinetic energy will cascade toward smaller scale, untilit reaches the deformation scale). In numerical simu-lations, we shall equate the energy (K 1 P) generationrate with the upscale spectral transfer rate e 5 g, andhereafter refer only to e.

Multiplying (2.1) by 2c and integrating to form anenergy budget equation then reveals that the total sta-tistically equilibrated energy is exactly

eE 5 , (2.3)

2r

independent of b and l (see, e.g., Smith et al. 2002).While linear vorticity drag may have limited applica-bility to atmospheric dynamics, the dissipation of heatimplied by linear dissipation of potential vorticity in(2.1) is relatively physical. Furthermore, since availablepotential energy dominates kinetic energy at largescales, this part of the dissipation is the more crucial ofthe two for the present study.

1422 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. Schematic graphs of Rossby wave frequencies and turbulentstraining rates as functions of zonal wavenumber. Solid lines showRossby wave dispersion frequencies (2.7) with (a) zero and (b) non-zero deformation wavenumber l. Dashed lines show straining rates(2.8) of various flux strengths. Intersections imply that an upscalecascade of energy in zonal wavenumber will be inhibited by Rossbywaves. All three strain rates will yield anisotropic flow when dis-persion is given by curve (A). If the dispersion is given by (B), thencurve (a) will also lead to anisotropic flow, curve (b) will lead tomarginally anisotropic flow, and curve (c) will lead to isotropic flow.

A quadratic drag on vorticity is a more realistic pa-rameterization of atmospheric dissipation. While no ex-act theory exists for total energy with a combination oflinear heat dissipation and quadratic vorticity drag, itsvalue can be estimated, and the prior and followingarguments accordingly altered, but this avenue will notbe pursued here. See Grianik et al. (2004) for a treatmentof the effects of quadratic vorticity drag on the inversebarotropic cascade.

Smith et al. (2002) investigated the turbulence phe-nomenology of a similar system to that described here,but without b (as well as systems with b ± 0 and l 50), and found that at scales large compared to the de-formation scale, a linear drag on potential vorticity willact to yield a peak in the energy spectrum at

3 3 6 21 1/8k 5 (a r l e ) ,r (2.4)

where a [ 27C /44, and C is the Kolmogorov constant.Defining a length scale and time scale

21 2 21/3L 5 l , T 5 (l e) , (2.5)

we can rewrite (2.4) as3/8k /l 5 (ar) ,r (2.6)

where r 5 r(l2e)21/3. The addition of b to the dynamics,if sufficiently large, will lead to anisotropic flow. Thecritical value of b necessary for anisotropy to developis derived next.

b. Necessary conditions for anisotropic flow

Equivalent-barotropic Rossby waves have the dis-persion relation

kbv (k, l) 5 2 , (2.7)R 2 2 2k 1 l 1 l

[where k 5 (k, l) is the two-dimensional wavenumber],which vanishes at the largest scale, in distinction fromits barotropic counterpart, whose frequency is unbound-ed as k → 0, where k 5 | k | .

Assuming a constant stirring at small scales, and anensuing upscale cascade of energy (total energy for k. l and available potential energy for k , l), theturbulent straining rate in the Kolmogorov theory is

21 1/2 1/3 2/3t ( | k | ) 5 C e | k | . (2.8)

Anisotropy will begin to develop at scales larger thanthat at which the turbulent straining rate is equal to theRossby wave frequency (Maltrud and Vallis 1991). Onecan see immediately that the curves represented by (2.7)and (2.8) will not intersect unless b is large enough, ore or l is small enough (see Fig. 1).

For critical parameters, a single intersection between(2.7) and (2.8) will occur along the l 5 0 axis, and agraphical solution1 shows that kc 5 0.447l and

1 Let x [ (k/l)1/3 and a [ b/(C1/2e1/3l5/3), so that the relevant equa-tion for critical parameters is ax 5 x6 1 1. A single intersection canonly occur when a 5 1.57, and for this value of a, x 5 0.765.

1/2 5 1/3b 5 1.5C (el ) .c (2.9)

Defining [ b(el5)21/3 and assuming C 5 6.0, this isb

b 5 3.9.c (2.10)

On a rotating planet, b decreases with latitude andl(} f ) increases with latitude, which implies, for a fixede, a distinct maximum latitude at which jets might beable to form. The critical b of (2.10) will later be shown,under certain circumstances, to yield flows that are su-percritical with respect to the inviscid Charney–Sterncriterion (CS).

c. CSM theory: Zonal flow scale via Charney–Sternmarginal criticality and total energy constraint

It is known that the jets that form in numerical modelsare marginally stable with respect to the CS. Despitethe inconsistency of this fact with observations of flowon the giant gas planets, we might expect that the cri-terion (along with some other constraints) could be usedto predict the scale and rms velocity of the supposedjets. In this subsection we derive zonal flow scales andvelocities based on this assumption, and term the re-sulting theory CSM (Charney–Stern marginal stability).

If b . bc, then energy will coalesce into zonal modes,and a cascade along the k 5 0 axis will ensue (Chekhlovet al. 1996; Marcus et al. 2000; Huang et al. 2001; Smithet al. 2002). If no drag is present, the cascade will pro-ceed all the way to the domain scale (Manfroi and Young1999; Huang et al. 2001). The theory presented in this

15 JUNE 2004 1423S M I T H

subsection assumes that stable jets will form at the zonalwavenumber l0 of marginal barotropic stability, deter-mined by the vanishing of the mean potential vorticitygradient, namely

2]q ] u25 b 2 1 l u 5 0.

2]y ]y

Assuming the mean zonal velocity is given by 5 u0usin(ly),2 we find the condition for marginal stability:

b2 2l 5 2 l . (2.11)0 u0

For a real solution to exist, one must have b/(u0l2) . 1.Determination of the zonal flow scale l0 requires ad-ditionally a prediction for the zonal velocity u0.

In order to estimate u0, we use the prediction for thetotal energy (2.3) and, separately, a scaling for the par-tition between available potential, meridional kinetic,and zonal kinetic energies. We make the following twoassumptions: 1) the ratio of the available potential (P)to kinetic (K) energy at a given wavenumber k scaleslike the ratio of that wavenumber to the deformationwavenumber l, that is, P/K . l2/ ; 2) since we are2l0

seeking a state in which we expect strong flow anisot-ropy to form, we assume the final circulation is primarilyzonal, so that u0 k ^y&, and the energy is coalescedaround the meridional wavenumber l0. So, expressingthe total energy as the sum of kinetic and availablepotential parts, we can rewrite (2.3) as

2 2l u e01 1 . . (2.12)21 2l 2 2r0

Equations (2.11) and (2.12) can be solved simulta-neously,3 resulting in the solution

12 2 1/2l . [(1 1 4b r) 2 1], (2.13)0 2

where l0 5 l0/l. The rms zonal velocity is

u0 2 1/2 21. 2[(1 1 4b r) 1 1] , (2.14)b

where u0 5 u0(el21)21/3. The solutions are exact—theapproximate equality symbols in (2.13) and (2.14) resultfrom the fact that (2.12) is an approximation. Note that

2 In making the assumption of a sinusoidal velocity profile, wehave assumed all the zonal energy has accumulated in one wave-number. This is of course not accurate: observed jets are asymmetric.Generally, eastward jets are strong and sharp while westward jets arewide and weak, implying a broadband spectrum of energy along thel axis in wavenumber space. More complex estimates for the zonalvelocity profile would likely introduce nondimensional form factorsinto the above calculation, but would not change the basic scalingdependencies.

3 Letting x 5 /l 2 . 0 and y 5 b/(u 0l 2) . 0 [and noting that2l0

b2r/(l 4e) 5 2 r] gives the pair of equations x 5 y 2 1 andb1/x 5 y 2 /( 2 r) 2 1. Elimination of x results in a cubic equation inby, which can be readily solved.

u0/ 5 u0l2/b, and that the right-hand side of (2.14)bnever exceeds unity.

Note also that we have used an inviscid stability cri-terion. Niino (1982) suggests a modified stability cri-terion that includes a dependence on the drag r. Thiscriterion may provide a more relevant assessment forthe current model, but is difficult to apply to the casesconsidered here, and will not be pursued. In any case,the effects of drag are taken into account in the follow-ing theory.

d. SI theory: Zonal flow scale via spectral integrationand total energy constraint

The problem with the above theory is that it restrictsthe energy in zonal modes to a scale and magnitude thatsatisfy CS stability. Because we want a scaling that canpotentially apply to observed jets, we separate the issuesof jet stability and zonal energy. After all, there is noth-ing to preclude energy coalescing on the zonal axis (k5 0 in spectral space) without forming stable jets. Ar-guing strictly from cascade dynamics, and knowing thatenergy cascades to a scale that is a combined functionof b and drag (Huang et al. 2001; Smith et al. 2002),we should be able to derive a scale for the energy peakon the zonal axis (l0). In this section we proceed assuggested, and term the resulting theory for zonal scaleand energy as spectral integration (SI).

In regimes of strong anisotropy, the kinetic energyspectrum is dominated by spectrally steep contributionsalong the zonal axis (see, e.g., Chekhlov et al. 1996),roughly obeying the form

2 25K (k) 5 C b k ,b (2.15)

where K is the spectrum of kinetic energy, defined suchthat

`

K 5 K (k) dk.E0

Using a similar relation for the spectrum A of availablepotential energy, we know that

2 22A(k) 5 l k K ,

and so the total energy spectrum for anisotropic flowsis (see, e.g., Merilees and Warn 1972)

2 25 2 22E(k) 5 C b k (1 1 l k ).b (2.16)

We can estimate the zonal flow scale by integrating(2.16) from l0 to `, neglecting the 25/3 contributionbelow the anisotropy boundary, and setting the resultequal to e/2r,

` 24 2 26l l l e0 02E (k) dk 5 C b 1 5 .E b 1 24 6 2rl0

1424 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

The solution4 is given by the single positive real rootof

1/6 1/3g g

l 5 1 1 1 20 1 2 1 2[ !3 3

1/21/3g

1 1 2 1 2 . (2.17)1 2 ]! 3

The parameter g 5 Cb2 r/2 is positive, and the dis-b

criminant of the square root changes sign at g 5 3.However, when the discriminant becomes negative, thetwo factors within parentheses are complex conjugatesof one another, and so their sum is real. The positive,real root of the equation is continuous across the criticalparameter.

Given knowledge of the zonal flow scale, the rmszonal velocity can be found approximately (neglectingmeridional velocities) by integrating the kinetic energyspectrum:

`2 2u C bb0 24. K (k) dk 5 l .E 02 4l0

In terms of the nondimensional parameters already in-troduced,

1/2u Cb0 225 l (2.18)01 2b 2

with l0 given by (2.17). Equations (2.17) and (2.18) shouldbe compared to (2.13) and (2.14). Based on estimates fromChekhlov et al. (1996), the constant Cb is about 0.2–0.4,so that Cb/2 ; 0.1–0.2, and the factor cannot be neglected.So in this case, g is similar to the factor 2r that appearsbin (2.13) and (2.14) of the CSM theory, but differs by thenondimensional prefactor Cb/2.

In the small- and large-g limits, expressions (2.17)and (2.18) take on simpler forms. Specifically,

1/6(2g/3) , g K 1l 5 and (2.19)0 1/45(4g/3) , g k 1,

21/32g, g K 11/2 1 23u Cb0 5 (2.20)21/21 2b 2 4g, g k 1.1 23

The error incurred by using the small-g limit of (2.19)when g 5 1 is about 20%, and about 10% for the large-g limit. An order of magnitude increase or decrease ing reduces the error to less than 1% for each estimate,respectively.

4 Write the resulting equation as x3 2 gx 2 2g/3 5 0, where x 5(l0/l)2 and g 5 Cbb2r/(2el4).

e. Limitations imposed by drag

The above solutions assume that energy can cascadefreely along the meridional wavenumber axis until it ishalted by the combined b–r scale in (2.13) or (2.17),but if surface drag is large enough, the cascade willnever reach scales where b becomes important. In theabsence of b, drag itself can halt the cascade isotropi-cally at the scale given by (2.6). On the other hand, forlarge enough b, the flow will be anisotropic with, per-haps, a zonal flow scale determined by one of the abovetheories. For CSM theory we can find the critical valueof drag rc such that, for r . rc, no anisotropy will formand the cascade will peak isotropically at kr. To do this,we set kr in (2.6) to l0 in (2.13). For SI theory, we canproceed similarly with the approximations in (2.19).

For CSM theory the physical solution5 consists of twopositive, real roots, for all b $ a1/2(27/4)1/6 . 2.6, whichis smaller than the critical value of b required for anisotro-py to form, given by (2.10). With b 5 bc . 3.9, valuesof the drag r that are either too small or too large willlead to isotropic flow. For higher values of b (e.g., b 55bc), the intersections move to extremely small and largevalues of g, respectively, and so the zonal flow scale pre-dicted by (2.13) is always smaller than the drag-inducedscale (2.6) for physically reasonable values of g. Anisotro-py is thus always expected for moderately supercrit-ical .b

For SI theory, we compare the approximations in(2.19) to (2.6). Setting the low-g limit expression forl0 in (2.19) equal to the drag scale kr in (2.6), we finda critical drag

1/54 8C bb 8/5r 5 . (0.01)b .c 91 281a

Choosing b 5 bc, the minimum value that will lead toanisotropy, gives us rc . 0.1 and g 5 0.15, validatinga posteriori our small-g assumption. Using the high-glimit leads to a similar value. In simulations reportedin the next section, drag never exceeds r 5 0.01, or atenth of the critical value at minimum b.

If drag or b is so large that the cascade peak scale isat or smaller than the deformation scale (an unphysicallimit), then the drag scale (2.6) is no longer valid, sincethat scale is derived under the assumption that k K l.Rather, in the opposite limit k k l, we revert to two-dimensional vorticity dynamics and the appropriate iso-tropic halting scale is

3/2k 5 (a r) .r 2 (2.21)

However, this limit has no relevance to the current prob-lem, since we are interested in scales k ; l0, and allthe physical systems to which the theory developed heremight apply have jets at scales on order of or largecompared to the deformation scale.

5 Letting x 5 (ar)1/4, solutions are the positive, real roots of x3 2( 2/a)x 1 1 5 0.b

15 JUNE 2004 1425S M I T H

TABLE 1. Nondimensional parameters and their dimensional rep-resentations. The time and space scales used in nondimensionalizationare given by (2.5).

Nondimensional parameter Dimensional form

ulbrg

u 21b

u(el21)21/3

ll21

b(el5)21/3

r(el2)21/3

(Cb/2)b2re21l24

ul2b21

TABLE 2. Model parameter values and steady-state results for the numerical simulations reported.

Run l b r b r g 0l u0l2/b |y |/|u| Jets?

B1 20 800 0.02 5.40 2.70 3 1023 8.00 3 1023 0.504 1.23 0.892 NoB2B3B4B5B6B7B8B9B10

1.50 3 103

2.00 3 103

2.50 3 103

3.00 3 103

4.00 3 103

8.00 3 103

1.50 3 104

2.00 3 104

4.00 3 104

10.213.617.020.427.154.3

102136271

0.02800.05000.07800.1130.2000.8002.825.00

20.0

0.5300.5790.6260.6840.7201.021.391.551.77

0.7640.6190.5290.4680.3880.2400.1390.1080.0559

0.5210.4450.4000.3630.2900.1680.1400.1040.0726

YesYesYesYesYesYesYesYesYes

L1L2L3L4L5

510304060

3.00 3 103 0.0200 20564.610.46.413.26

6.80 3 1023

4.30 3 1023

2.10 3 1023

1.70 3 1023

1.30 3 1023

28.81.800.0220

7.00 3 1023

1.40 3 1023

1.941.150.4790.4040.272

0.07540.2591.2231.7643.274

0.1610.1930.4940.7400.940

YesYesYes

?No

RA1RA2RA3RA4

20 3.00 3 103 0.01000.05000.1000.200

20.4 1.40 3 1023

6.80 3 1023

0.01360.0271

0.05630.2810.5631.13

0.5570.7460.9171.21

0.5510.3520.2960.240

0.3450.4260.4700.555

YesYesYesYes

RB1RB2RB3

20 8.00 3 103 0.01000.05000.100

54.3 1.40 3 1023

6.80 3 1023

0.0136

0.4002.004.00

0.9571.331.55

0.3190.1750.130

0.1340.2300.313

YesYesYes

LG 80 6.00 3 1023 0.0100 4.04 5.39 3 1024 8.80 3 1024 0.176 2.03 0.821 ?

NV 20 3.00 3 103 0.0200* 20.4 2.70 3 1023 0.113 0.553 0.470 0.356 Yes

* Drag was applied only to the vortex-stretching for run NV.

3. Numerical experiments

a. The numerical model

The simulations reported below were performed usinga two-dimensional dealiased spectral model with 2562

equivalent horizontal gridpoints (kmax 5 127)—addi-tionally, one simulation using 5122 equivalent horizon-tal gridpoints was performed to demonstrate the ro-bustness of the results with increasing resolution. Themodel uses a leapfrog time step to advance the solution,and a weak Robert filter to suppress the computationalmode. The flow is forced with an isotropic random Mar-kovian forcing at high wavenumber, typically kF 5 80(kF 5 160 for the high-resolution run). Enstrophy isdissipated with a highly scale-selective exponential cut-off filter with explicitly vanishing dissipation below acutoff wavenumber kcut . For simulations presented here,kcut 5 kF 1 5. Traditional hyperviscosity, by contrast,dissipates at all scales. Even above the cutoff wave-

number, the dissipation level does not become signifi-cant until near the maximum wavenumber in the com-putational domain. Thus, while we focus here on main-taining as wide an inverse cascade range as possible,we have allowed a reasonable direct cascade for wave-numbers k . kF. The small-scale dissipation, despiteits presence only at k . kF, removes some of the inputenergy. The model calculates this loss at each time stepand uses it to set the magnitude of the forcing such thatthe net upscale energy transfer rate, after losses to theenstrophy filter, is fixed at e 5 1. Appendix B of Smithet al. (2002) details the forcing function and enstrophyfilter, with the exception that in the present paper, theforcing is altered (as explained in the previous sentence)to normalize the forcing completely, including for lossesof energy to the enstrophy filter.

b. Summary of simulations and comparison totheories

We describe a set of simulations devised to test therelations derived in section 2. The results are describedin terms of the nondimensional parameters defined insection 2, all of which are summarized in Table 1. Fourseries of simulations, along with a set of special or ex-treme cases, are reported and their parameters are de-scribed in Table 2. The simulation series vary b, de-formation wavenumber (l), and drag (r) with other pa-rameters fixed. The fourth series varies drag at a dif-ferent value of b than the first drag series. The four

1426 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 2. Comparison of kinetic energy spectra for simulations B5(solid), NV (dashed), and HR (dashed–dotted). Simulation HR usesthe same parameters as B5 but with twice the numerical resolutionand forcing dissipation cutoff scales half as large as for B5. Simu-lation NV uses the same parameters as B5, but with no drag appliedto the vorticity.

FIG. 3. Nondimensional energy peak wavenumber l0 5 l0l21 vsg 5 (Cb/2)b2re21l24. The simulated peak meridional wavenumberl0 is calculated as the centroid of a slice of the time-averaged kineticenergy spectrum along the y axis, which is a very steep spectrum(26) and so gives a good statistical indication of the spectral peak.The CSM theory prediction (2.13) is plotted as a dashed line, andthe SI theory prediction (2.17) is plotted as a dashed–dotted line. Theparameter value Cb 5 0.2 is used for all calculations.

FIG. 4. Nondimensional jet velocity u021 5 u0l2b21 vs g (definedb

in the caption of Fig. 3). The nondimensional jet velocity must beless than unity to satisfy the inviscid CS condition [see (2.11)]. Thesimulated jet velocity u0 is calculated as the square root of twice thetotal time-averaged kinetic energy, and hence assumes that the me-ridional velocity is negligible. The CSM theory prediction (2.14) isplotted as a dashed line and the SI theory prediction (2.18) is plottedas a dashed–dotted line. The deviations at small g are due to theweak anisotropy of the flow, which is not accounted for by eithertheory. The parameter value Cb 5 0.2 is used for all calculations.

series will be referred to as the B series, L series, RAseries, and RB series, respectively. The special casesconsist of one run with a set of parameters chosen togive a very small value of g (called LG), one simulationin which the linear drag is applied only to the vortexstretching term (i.e., thermal drag without vorticity drag:NV for ‘‘no vorticity drag’’), and one simulation per-formed at twice the resolution (HR for ‘‘high resolu-tion’’). The latter two special cases were performed us-ing parameters identical to the simulation labeled B5,the canonical simulation. Smaller values than r 5 0.01were not performed because the number of time stepsrequired to reach a statistically steady state grows in-versely with the drag, and the lowest drag simulationsperformed required millions of time steps in some cases.

The parameters for simulation LG were chosen toinvestigate a lower, but still numerically realizable valueof g at which anisotropy might still form. The limita-tions are that b is large enough for anisotropy to form;drag is small enough that it does not control the inversecascade; and that drag is not too small, so that numericalequilibration is attainable in reasonable time. The valuesof the parameters used are listed in Table 2. Note that

is only slightly larger than c 5 3.9, and so the flow˜ ˜b bshould be barely supercritical for anisotropy formation.

In order to ensure that our results are not dependenton resolution, simulation HR was performed using iden-tical parameters to simulation B5, but with twice theresolution and half the forcing scale. A comparison ofthe resulting kinetic energy spectra is shown in Fig. 2(along with a comparison to simulation NV, discussedlater). The total energy in the two simulations is within5%, as are the derived zonal flow scale and rms zonalvelocity. We conclude that the results are not dependenton resolution.

Figures 3 and 4 show the compiled nondimensionalenergy peak scales and rms zonal velocities for all thesimulations performed, plotted as functions of g. A val-ue of Cb 5 0.2 was used in calculating g for the sim-ulations performed. The peak meridional wavenumberl0 is calculated as the centroid of a slice of the time-averaged kinetic energy spectrum along the y axis,

15 JUNE 2004 1427S M I T H

which is a very steep spectrum (26) and so gives agood statistical indication of the spectral peak. The rmszonal velocity u0 is calculated as the square root of twicethe total time-averaged kinetic energy, and hence as-sumes that the meridional velocity is negligible. Thevalues plotted, as well as the anisotropy (defined as theratio of rms meridional to zonal velocities) for each runare also tabulated in Table 2. The theories from sections2c and 2d are plotted along with the model output. CSMtheory is plotted as a function of g by substituting 2g/Cb

for 2 r in (2.13) and (2.14).bThe model output for the peak meridional energy

scale in Fig. 3 is clustered along the prediction of SItheory for the whole range of g considered, with thegreatest variations for simulations LG, B10, and L1.Error bars generated from the model data would besmaller than the size of the symbols in the plot, and soare not displayed. In any case, both ‘‘theories’’ are farfrom exact, and so exact correspondence is not expected.A major error, for example, is that for runs with close tobthe critical value, the flow is not highly anisotropic, andso meridional velocities cannot be neglected. Consid-ering Fig. 4, the deviations from data at small g are dueto the weak anisotropy of the flow, which is not ac-counted for by either theory. The close correspondenceof the L series simulations with SI theory at small g issurprising. The trends of the output are neverthelesssufficiently consistent with the scale predictions for SItheory to rule out CSM theory as a predictor for theflow scale.

Note that while g varies over 4 decades, the ratio ofzonal flow scale to deformation scale varies only be-tween about 3.0 and 0.5, the former occurring for verysmall g. Apparently, it is very difficult to form zonalflow scales, much less stable jets, that are much largerthan the deformation scale. This is consistent with theobserved near coincidence of the two scales on Jupiterand Saturn.

c. Jet stability versus flow anisotropy

The model data for the rms zonal velocity in Fig. 4is plotted as u0/ 5 u0l2/b, which is just the super-bcriticality for CSM theory (the CS criterion). Note thatthe curve for CSM theory asymptotes to unity at smallg, consistent with the constraint (2.11) used to deriveit. The model output, however, clearly does not obeythis constraint, and again is more consistent with the SItheory.

The physical space zonal velocity fields for L seriesruns and for run LG are shown in Fig. 5 and those forthe first six B series runs (B1–B6) in Fig. 6. The purposeof these figures is to assess the flows for the presenceof stable jet structures, so no scale is given on the plots.Simulations L3, L4, L5, LG, B1, and marginally B2have rms zonal velocities exceeding the CS limit (seeFig. 4). Nevertheless, only those with & 4 (B1, L5,bLG) show almost no sign of jets. Simulations B2, L4,

and L3 all show jet structure, despite that they violatethe CS criterion, but the jets are very wiggly and brokenin some places. Indications for the presence of jets aretabulated in the final column of Table 2.

One can infer from the results that it is appropriateto separate the issues of jet stability and flow anisotropy.The latter is controlled by the steady-state spectral dis-tribution of turbulent energy, while the former surelydepends on the relevant stability criterion for the flowsat hand. For example, Dowling (1994) argues that jetson Jupiter are critically stable with respect to Arnold’sSecond Stability criterion. Ingersoll and Pollard (1982),by contrast, claim that a Charney–Stern criterion mod-ified for deep flows characterizes Jovian jets. More re-cently Galperin et al. (2001) pointed out that any sta-bility criterion should include drag. Most basically,McIntyre and Shepherd (1987) show that even peri-odicity in the meridional (y) direction, or the lack there-of, can affect the relevant stability criteria. In any case,none of these approaches determine the energy level ofanisotropic flow, but rather only the ability for stablejets to form given the generation of anisotropic energy.

d. Thermal drag versus vorticity drag

The restriction that the drag on the system be appliedto both the stretching and vorticity, and be equal, is toorestrictive to be directly applicable to physical systems.However, in systems with large-scale (compared to thedeformation scale) jets, the available potential energywill dominate, and so the thermal drag should accom-plish most of the dissipation. Simulation NV was per-formed as a test for this limit. The parameters used forthis simulation are identical to simulation B5 except thatno drag is applied to the vorticity (the extreme case).Explicitly, decomposing the drag term from (2.1),

2 2rq 5 r ¹ c 2 r l c,z c

we have set rz 5 0 in simulation NV, while in all othersimulations we have set rz 5 rc 5 r. The results for thescale and velocity of zonal flow are displayed with adiamond in Figs. 3 and 4. The zonal velocity is nearlyidentical for the two runs, while the zonal scale is largerin simulation NV than in B5. While the kinetic energyis nearly the same in the two runs, the available potentialenergy in NV is 35% higher. One might have expectedthe reverse to occur, given that vorticity drag dampskinetic energy while thermal drag damps potential en-ergy. Instead, the lack of vorticity drag has allowed theinverse cascade to proceed to slightly larger scale, andthis change in scale is reflected more dramatically inthe potential energy spectrum, which is amplified atlarge scale and reduced at small scale by the factorl2/k2. A comparison of the kinetic energy spectra forsimulations B5 and NV is shown in Fig. 2. The changein zonal scale in the absence of vorticity drag is a weak-ness of the theory.

1428 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 5. Instantaneous zonal velocity fields for L series and LG simulations. These figuresdemonstrate the presence and absence of jets in various simulations (no scale is necessary). Seetext and Table 2.

4. Application to Jupiter and Saturn

The theories presented can be applied directly to lim-ited aspects of the observed zonal wind data for thegiant gas planets. However, their simplicity and limi-tations must be taken into account, and resulting pre-

dictions restricted to their regimes of validity. The the-oretical framework presented considers only barotropicvorticity dynamics, and will not be relevant too closeto the equator. Second, apart from the overall forcinglevel, details of the convective forcing are not repre-sented. Third, the homogeneous nature of the theories

15 JUNE 2004 1429S M I T H

FIG. 6. Instantaneous zonal velocity fields for first six B series simulations. Comments incaption to Fig. 5 apply.

precludes the prediction of jet directions and asym-metries. Nevertheless, both theories can be testedagainst the observed scale and speed of the zonally av-eraged profiles of those jets beyond the equatorial zone.Equation (2.9) also predicts latitudes above which jetsshould not form.

In order to apply SI theory to planetary data, five

physical parameters must be specified: the rotation rateV, the planetary radius a, the deformation wavenumberl, the convective energy generation rate e, and the dragr. Here, the theory will be applied to the zonal windson Jupiter and Saturn, the known planetary parametervalues for which are given in Table 3. The planet radiiand rotation rates are well known, and Allison (2000)

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TABLE 3. Planetary data for Jupiter and Saturn. Estimates of gravitywave speed are from Allison (2000) and estimates of generation anddrag for Jupiter are from Galperin et al. (2001).

Parameter Jupiter Saturn

Planet radius (a)Planet rotation rate (V)Gravity wave speed (NH)Eddy generation (e)Drag (r)

7.15 3 107 m1.76 3 1024 s21

;700 m s21

;1027 m2 s23

;3 3 10212 s21

6.03 3 107 m1.64 3 1024 s21

;1500 m s21

??

FIG. 7. (a) Eastward jet velocity maxima and (b) meridional spectraljet scale vs absolute planetographic latitude for Jupiter. The solidlines are estimates from SI theory [(2.18) and (2.17)] using valuesparameters given in Table 2, and the dashed lines are the estimatesfrom CSM theory [(2.14) and (2.13)] using the same parameter val-ues. The data points and error bars are from Garcıa-Melendo andSanchez-Lavega (2001). Squares denote Northern Hemisphere points,and circles denote Southern Hemisphere points. See text for expla-nation of scale analysis.

provides estimates of the product of stratification andscale height from gravity wave speed observations.From these, one can estimate the deformation wave-number as

4pVl(f) . sin(f), (4.1)

NH

where f is the latitude. The Coriolis gradient is

2Vb(f) 5 cos(f), (4.2)

a

Galperin et al. (2001), using data from Gierasch et al.(2000), given an estimate of the energy flux e for Jupiterfrom convective stirring, and also estimate the drag timescale (the low end of their estimate is quoted in Table3). No estimates of the latter two values are known forSaturn.

Figure 7a plots the maxima of Jovian eastward jetvelocities and velocity errors versus absolute planeto-graphic latitude from Garcıa-Melendo and Sanchez-Lavega (2001) data, shown as squares for NorthernHemisphere jets and circles for Southern Hemispherejets. The solid line is the prediction from SI theory(2.18) using the parameter estimates described above,and the dashed line is the equivalent prediction usingCSM (2.14). As it was with the simulation data, thenondimensional value Cb 5 0.2 is used for the SI theoryprediction. (Latitudinal estimates for the Jovian valuesof g and that result from the parameter estimates arebshown as solid lines in Fig. 9.) Apart from the anom-alously sharp and intense jet at 238N, SI theory lieswithin the scatter of the data, while CSM greatly un-derpredicts jet speeds at all latitudes.

Figure 7b plots the the meridional scale of the zonallyaveraged zonal wind for Jupiter against the predictionsof SI theory (2.17) (solid line) and CSM theory (2.13)(dashed line). The scale analysis is based on the fullprofile of Jovian zonally averaged zonal wind fromGarcıa-Melendo and Sanchez-Lavega (2001), and is ex-plained in the appendix. The results for the scale anal-ysis show a more striking departure between theoriesthan for jet speeds. The SI theory roughly fits the data,and when the variation of deformation scale and b aretaken into account, SI theory predicts an increase in jetscale toward the equator, but nearly constant scales atmidlatitudes, as observed. CSM theory, by contrast, isabsolutely inconsistent with the data. The CSM predic-

tion (dashed) increases to tenfold larger jets than ob-served at midlatitudes (exceeding the plot axes).

Figure 8 is similar to Fig. 7, but uses the data ofSanchez-Lavega and Rojas (2000) for Saturn’s zonallyaveraged zonal wind. The theoretical estimates for theSaturnian flow, however, require parameter values forwhich no published estimates exist, as mentioned above.Without further guidance, drag is estimated to be the sameas that for Jupiter, while the convective generation rate eis adjusted to fit the data. A value of e ; 1026 m2 s23 isfound to best fit the jet speed maxima data with SItheory. The requisite latitudinal estimates for the valuesof g and for Saturn based on this value are shown asbthe dashed lines in Fig. 9. The SI and CSM curves in

15 JUNE 2004 1431S M I T H

FIG. 8. Same as Fig. 7, except that no independent estimates forthe drag and convective generation rate on Saturn exist. The drag isestimated to be the same as that for Jupiter, and the convective forcingrate e is chosen to give the best fit to the data. The value found is10 times the value estimated for Jupiter. See Fig. 9 for the resultingestimates of g and b. The data points and error bars are from Sanchez-Lavega and Rojas (2000).

FIG. 9. Estimated values for and g as functions of latitudebfor Jupiter ( solid ) and Saturn (dashed). Horizontal lines denotecritical values for ( . 3.9 implies anisotropic flow) and g (g˜ ˜b b, 3 3 10 22 implies flow will be supercritical with respect to theinviscid Charney–Stern criterion).

Fig. 8 for both jet speed (Fig. 8a) and scale (Fig. 8b)use the resulting values of g.

In Fig. 9, note that . 3.9 for latitudes up to aboutb558 on Jupiter, and latitudes up to about 708 on Saturn.Actual jets do exist at latitudes higher than these predictedcritical latitudes, but do fail to form at polar latitudes.Also note that g is very small in the midlatitudes on bothplanets, implying CS supercritical zonal energies andscales (this is implied for all g & 3 3 1022), consistentwith observations.

5. Discussion

We have investigated a geometrically simple but para-metrically complex representation of forced and dissi-

pated turbulence as a local model for flow in rotating,stratified atmospheres driven by small-scale convectivestirring. The new and interesting aspect of the presentwork is the simultaneous inclusion of b, a finite defor-mation scale and large-scale drag in a model stirred bysmall-scale random forcing. The linear drag is arguablythe weakest aspect of the investigation, but provides aconstraint on the overall flow energy, removing a majoruncertainty.

The first result is that, in order for the flow to becomeanisotropic, b must exceed a critical value that is afunction of the local deformation scale and the eddyenergy generation rate (2.10). This condition is notequivalent to a stability criterion, but rather determineswhether large-scale meridional flow is inhibited. Whenconditions are favorable for the production of anisot-ropy, the energy constraint imposed by the drag, in con-junction with the Rhines–Chekhlov spectrum (2.15) andthe partition between kinetic and potential energy, pre-dicts the steady-state zonal flow scale and rms velocitywith some accuracy. This theory was referred to as SItheory and was compared to an alternate candidate thatused the inviscid Charney–Stern criterion as a predictorfor jet scale. A wide range of numerical simulationswere performed and demonstrate that the former theoryis consistent with the results of the simulations whilethe latter is not. The implication is that zonal energylevels and scales that would be supercritical to the CScondition, were they to form into jets, can be generated.

The formation of stable jets is another matter, andlikely depends on the geometry of the flow. Anisotropyis shown to form without stable jets in inviscid CS su-percritical parameter regions, though additionally wefind that semistable supercritical jets can be formed evenwith our simple doubly periodic spectral model. In any

1432 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

case, it is clear that marginal barotropic stability doesnot predict the presence, scale, or strength of turbulentanisotropic flow.

Despite the simplicity of the theory, we find that, asa local model, its predictions bear quantitative resem-blance to the observed jet scales and strengths on Jupiter.The SI theory predicts that meridional jet scale shouldbe roughly constant in midlatitudes but increase in scaletoward the equator, and that jets should cease to formtoward the poles, all as observed. The prediction for thelatitudinal distribution of jet speeds and scales on Jupiteris moreover consistent with the data in some detail.Application of the theory to Saturn required choosinga convective forcing rate 10 times that on Jupiter to bestfit the data. Varying this single parameter, however,yields predictions for the latitudinal distribution of jetspeeds and scales that are consistent with Saturnian datain detail, notably predicting the observed twofold largerjet scales and fourfold faster jet speeds observed thererelative to Jupiter. It will be interesting to see whetherfuture observations yield measurements consistent withthis prediction for convective forcing on Saturn. Thecombination of model simplicity and parameter data in-completeness makes this the roughest of comparisons.Nevertheless, the presented theory provides a relativelysimple explanation for the most basic features of theobserved midlatitude zonal wind profiles on the gasplanets.

Acknowledgments. The author thanks the three re-viewers of this paper for their careful analysis and con-structive comments, all of which led to an improvedpaper. Some of the simulations reported here were com-puted on the supercomputing facilities at NOAA’s Geo-physical Fluid Dynamics Laboratory.

APPENDIX

Analysis of Meridional Scale Zonal Wind Data

In order to extract the scale of the jets as a functionof latitude, as shown in Figs. 7b and 8b, a sliding win-dowed (or ‘‘short time’’) fast Fourier transform (FFT)is calculated at each latitude, starting and ending at lat-itudes where the data begins and ends, less one half thewidth of the window. The scales are then calculated as2p/kc, where kc is the centroid wavenumber of eachFFT density spectrum. Error bars are taken as one halfthe second moment of each spectrum. Points are plottedfor each half window width, so that windowed dataoverlaps. Northern and Southern Hemisphere data isstaggered to allow for clarity in the plot.

Window widths are chosen large enough to resolvethe full widths of jets (outside the Tropics) but smallenough to allow for a sufficient range of scale calcu-lations to demonstrate the latitudinal variation of theresults. A range of window widths can be used, but mustbe chosen to balance spectral resolution against spatial

resolution (the larger the window, the better the FFT,but the narrower the range of latitudes for which scalescan be calculated). Within the usable and meaningfulrange of possible window widths, the results do not varysignificantly.

For the Jovian Northern Hemisphere, the Garcıa-Me-lendo and Sanchez-Lavega (2001) data consists of 256velocity values at evenly spaced latitudes ranging from0.38 to 76.78. An 188 window width was used for theNorthern Hemisphere data and scales are plotted fromoverlapping windowed regions whose centers are spaced98 apart.

Jovian Southern Hemisphere data from Garcıa-Me-lendo and Sanchez-Lavega (2001) consists of 226 ve-locity values at evenly spaced latitudes ranging from20.38 to 267.88. The same window width and spacingof plotted scales as for the Northern Hemisphere datais used, but they are offset by 38 to allow separationfrom the Northern Hemisphere scale points.

The same analysis is made of the Saturnian data foundin Sanchez-Lavega and Rojas (2000). The SaturnianNorthern Hemisphere data consists of 155 velocity val-ues at nearly evenly spaced latitudes ranging from 08to 80.78. A larger window width of 41.78 was necessaryboth to resolve the larger jet widths, and to give suf-ficient resolution to the FFTs, given the smaller numberof data points. Scale points are plotted at separations of118.

The Saturnian Southern Hemisphere data consists of102 velocity values that are nearly evenly spaced forlatitudes ranging from 210.78 to 270.98. Data existsfor the equatorward latitudes 218, 21.58, and 22.18,but data for latitudes between 22.18 and 210.78 is un-available because they are obscured by Saturn’s rings.Window widths of 62.78 were necessary to produce use-ful results. The scale points are plotted at a separationof 118.

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