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Jupiter’s unearthly jets: a new turbulent model exhibiting

statistical steadiness without large-scale dissipation

Stephen I. Thomson, Michael E. McIntyreDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, UK

www.damtp.cam.ac.uk/user/sit23/, stephen.i.thomson@gmail.com; www.damtp.cam.ac.uk/user/mem2/, mem2@cam.ac.uk

Submitted to J. Atmos. Sci. 9 Dec 2014, revised 15 Jul & 5 Aug 2015

ABSTRACT

A longstanding mystery about Jupiter has been the straightness and steadiness of its weather-layerjets, quite unlike terrestrial strong jets with their characteristic unsteadiness and long-wavelengthmeandering. The problem is addressed in two steps. The first is to take seriously the classicDowling-Ingersoll (DI) 11

2-layer scenario and its supporting observational evidence, pointing toward

deep, massive, zonally-symmetric zonal jets in the underlying dry-convective layer. The second is toimprove the realism of the model stochastic forcing used to represent the effects of Jupiter’s moistconvection as far as possible within the 11

2-layer dynamics. The real moist convection should be

strongest in the belts where the interface to the deep flow is highest and coldest, and should generatecyclones as well as anticyclones with the anticyclones systematically stronger. The new modelforcing reflects these insights. Also, it acts quasifrictionally on large scales to produce statisticallysteady turbulent weather-layer regimes without any need for explicit large-scale dissipation, andwith jets that are approximately straight thanks to a guide-rail effect from the deep jets and toshear stability despite nonmonotonic potential-vorticity gradients, when the Rossby deformationlengthscale is not too large. Moderately strong forcing produces chaotic vortex dynamics andrealistic belt-zone contrasts in the model’s convective activity, through an eddy-induced sharpeningand strengthening of the weather-layer jets relative to the deep jets, tilting the interface betweenthem. Weak forcing, for which the only jet-sharpening mechanism is the passive (Kelvin) shearingof vortices (as in the “CE2” or “SSST” theories), produces unrealistic belt-zone contrasts.

1. Introduction

The aim of this work is to find the simplest stochasti-cally forced model of Jupiter’s visible weather layer thatreproduces the straightness and steadiness of the observedprograde jets, and the belt–zone contrasts in small-scaleconvective activity, under a forcing regime that is arguablycloser to the real planet’s than either (a) the forcing usedin orthodox beta-turbulence models, or (b) the purely an-ticyclonic forcing used in the recent work of Li et al. (2006)and Showman (2007). The new forcing is discussed below,after sketching the scientific background.

The weather layer is the cloudy, moist-convective layeroverlying a much deeper, hotter dry-convective layer. Suchvertical structure, though not directly observed, is to beexpected from the need to carry a substantial heat fluxfrom below and from the basic thermodynamics and esti-mated chemical composition of Jupiter’s atmosphere (e.g.,Sugiyama et al. 2006, and references therein).

Even at high latitudes, the observed weather-layer jetsare “straight” in the sense that apart from small-scale dis-turbances they closely follow parallels of latitude, especiallythe prograde jets, as clearly seen in the well-known an-

imated polar view from Cassini images.1 This extremestraightness contrasts with the meandering behavior foundin many single-layer model studies including the work of Liet al. (2006) and Showman (2007) and, for instance, Scottand Polvani (2007). Jupiter’s jets are also remarkably closeto being steady, as evidenced by the almost identical zonal-mean zonal wind profiles seen in 1979 and 2000, from cloudtracking in the Voyager 1 and Cassini images (Limaye 1986;Porco et al. 2003).

Because the weather layer appears turbulent and hasno solid lower boundary, we confine attention to modelsthat can reach statistically steady states without the large-scale friction used in the beta-turbulence models. We alsoavoid the use of large-scale Newtonian cooling as an eddy-damping mechanism. Real radiative heat transfer is notonly far more complicated, but also dependent on unknowndetails of the cloud structure within the weather layer andnear the interface with the dry-convective layer.

The extreme straightness and steadiness of Jupiter’sprograde jets make them strikingly dissimilar to the strong

1A movie is available from www.atm.damtp.cam.ac.uk/people//mem/nasa-cassini-index.html, by kind courtesy of NASA and theCassini mission (Porco et al. 2003, movie S1).

1

jets of the Earth’s atmosphere and oceans, with their con-spicuous, large-amplitude long wave meandering. By strongjets we mean the atmosphere’s tropopause and polar-nightjets, and the strongest ocean currents such as the GulfStream, the Kuroshio and the Agulhas. The cores of theseterrestrial strong jets are marked by concentrated isen-tropic gradients of Rossby–Ertel potential vorticity or gra-dients of ocean surface temperature, inversion of which im-plies sharp velocity profiles having width scales of the or-der of an appropriate Rossby deformation lengthscale LD.Such meandering strong jets are quite different from thestraighter but very weak “ghost” or “latent” jets in the Pa-cific ocean, visible only after much time-averaging (Maxi-menko et al. 2005, and references therein).

Jupiter’s jets are hardly weak. On the contrary, at leastsome of them are strong enough to look shear-unstable, bysome criteria, with nonmonotonic potential-vorticity gra-dients (e.g., Dowling and Ingersoll 1989; Read et al. 2006;Marcus and Shetty 2011). Here we argue that their straight-ness and steadiness may come from a different, strictly ex-traterrestrial combination of circumstances, which calls forsignificant modeling innovations.

We propose a new idealized model whose two mostcrucial aspects are as follows. The first is the stochasticforcing of turbulence by thunderstorms and other small-scale moist-convective elements injected into the weatherlayer from the underlying dry-convective layer.2 We as-sume that moist convection generates small cyclones andanticyclones, with a bias toward stronger anticylones. Sucha “potential-vorticity bias” or “PV bias” recognizes thatheat as well as mass is injected. This contrasts with themass-only, anticyclones-only scenarios of Li et al. (2006)and Showman (2007) on the one hand, and with the per-fectly unbiased small-scale forcing used in beta-turbulencemodels on the other. PV bias will enable us to dispensewith the large-scale friction required for statistical steadi-ness in beta-turbulence models.

The second aspect is the presence of zonally-symmetricdeep zonal jets in the underlying dry-convective layer. Theywill prove crucial to our model’s behavior. Here we followthe pioneering work of Dowling and Ingersoll (1989, here-after DI), who produced cloud-wind evidence pointing totwo remarkable and surprising conclusions. The first isthat the large-scale vortex dynamics, in latitudes around15◦–35◦ at least, is approximately the same as the dynam-ics of a potential-vorticity-conserving 112 -layer model, withthe upper layer representing the entire weather layer. DI’ssecond conclusion is that the cloud-wind data can be fitted

2We deliberately exclude other excitation mechanisms. In partic-ular, we exclude terrestrial-type baroclinic instabilities. These arearguably weak or absent because of the absence of a solid lower sur-face at the base of the weather layer, and because of the weak pole-to-equator temperature gradient — a weakness expected, in turn,from the well known “convective thermostat” argument (Ingersolland Porco 1978).

into this picture only if the underlying dry-convective layeris in large-scale relative motion. The simplest possibilityallowing a good fit is that the relative motion consists ofdeep zonally-symmetric zonal jets. Those deep jets musthave substantial velocities, comparable in order of magni-tude to jet velocities at cloud-top levels. To our knowledge,no subsequent cloud-wind study has overturned this secondconclusion. So we use a 112 -layer model with deep jets. Wetreat the deep jets as prescribed and steady, consistent withthe far greater depth and mass of the dry-convective layer.

The relevance of 112 -layer dynamics has recently gainedsupport from a different direction. Sugiyama et al. (2014)present results from a two-dimensional cloud resolving mo-del that includes the condensation and precipitation ofwater and other minor species. A model weather layeremerges whose stable stratification is sharply concentratednear the interface with the dry-convective layer, as a conse-quence of water-cloud behavior. This result suggests thatthe real weather layer could be surprisingly close to the112 -layer idealization with its perfectly sharp interface. Fur-ther such work is needed, if only because the real cloud-scale moist convection must itself be three-dimensional, assuggested by the morphology both of Jupiter’s folded fil-amentary regions and of terrestrial supercell or “tornadoalley” thunderstorms, with their intertwined patterns ofupdrafts, downdrafts, and precipitation.

DI’s evidence for deep jets remains important today be-cause, as yet, there are no other observational constraintson the existence or nonexistence of the deep jets, out-side the equatorial region. No such constraints are ex-pected until, hopefully, gravitational data come in fromthe Juno mission in 2016. Numerical studies of the dry-convective layer cannot address the question because theyneed to make speculative assumptions about conditionsat depth, including the effective bottom boundary condi-tions felt by Taylor-Proudman-constrained deep jets at lat-itudes within the associated tangent cylinder. Here thereis great uncertainty. There is a range of possible conditionswhose extremes are a slippery radiative layer well above themetallic-hydrogen transition, making deep jets easy to gen-erate, and a no-slip magnetohydrodynamic transition layerthat inhibits them (e.g., Guillot 2005; Jones and Kuzanyan2009; Liu et al. 2013; Gastine et al. 2014, and referencestherein). Jupiter’s prograde equatorial jet system needsseparate consideration, being almost certainly outside anyrelevant tangent cylinder or cylinders.

Zonal symmetry or straightness is plausible for any deep,dry-convective jets that may exist, in virtue of the scaleseparation between the jets themselves and the relativelytiny, Coriolis-constrained convective elements that excitethem, as seen in the numerical studies (e.g., Jones andKuzanyan 2009; Gastine et al. 2014, and references there-in).

Our own relatively modest aim, then, is to see whether,

2

on the basis of the DI scenario with prescribed deep, straightjets, and nonmonotonic upper PV gradients, an idealized112 -layer model can produce not only statistical steadinessin the absence of large scale dissipation but also realis-tic, quasi-zonal large scale weather-layer structures, withmoist-convective forcing strongest in the cyclonically-sheared“belts” and weakest in the anticyclonically-sheared “zones”.The folded filamentary regions and lightning observed onthe real planet, assumed to be symptomatic of moist con-vection, are concentrated in the belts (e.g., Porco et al.2003). In addition we aim to test the effectiveness, withinthe idealized model, of the beta-drift-mediated migrationof small anticyclones from belts into zones, following a sug-gestion by Ingersoll et al. (2000) that such migration mightbe significant.

We also look at the issue of shear instability and therelevance of Arnol’d’s second stability criterion for non-monotonic PV gradients (“A2 stability”), following sugges-tions by Dowling (1993) and Stamp and Dowling (1993).Jet straightness implies that the weather layer is eithermarginal or submarginal to shear instability; but the modelresults will force us to a stronger, and we believe novel,conclusion, differing from Dowling’s original suggestion,namely that the weather layer must be well below thethreshold for marginal instability. In the model, at least,and probably on the real planet also, slight submarginalityis insufficient to hold the jets realistically straight becauseit allows long-wave meandering to be too easily excited,even if not quite self-excited. Substantial submarginal-ity with nonmonotonic PV gradients restricts LD valuesto be substantially less than the jet spacing, as suggestedschematically in Fig. 1, at top right. A similar restrictionprobably holds on the real planet, constraining weather-layer depths and therefore abundances such as that of wa-ter.

The plan of the paper is as follows. Section 2 intro-duces the model. Section 3 introduces the PV-biased forc-ing and shows how it can act quasifrictionally. Section 4motivates our choice of parameters — emphasizing thosechoices, including submarginal LD, that lead to realistic,quasi-zonal weather-layer structures. Section 5 surveysthe main body of results. Section 6 discusses the chaoticvortex-interaction mechanisms that produce realistic struc-tures. We find that the migration mechanism is important.Section 7 shows that another mechanism — the Kelvinpassive-shearing mechanism (Thomson 1887), much dis-cussed in recent years under the headings “CE2”, “SSST”,and “zonostrophic instability” (e.g., Srinivasan and Young2014, and references therein) — has interesting effects butis unable to produce realistic weather-layer structures inour model. Section 8 presents some concluding remarksand suggestions for future work.

ζAAAK

Weatherlayer

Dry-convectivelayer

6-

z

y

LD

Fig. 1. Schematic of the model setup and motivation; see text. Thebar at top right indicates an LD value well below the threshold formarginal shear instability with nonmonotonic upper PV gradients.The notional cumulonimbus clouds, concentrated in the model belt,can be thought of as tending to generate vortex pairs with cyclonesbelow and anticyclones above. Such vortex pairs, called “hetons” or“heatons” in the oceanographic literature, can tilt and then propagatelike ordinary two-dimensional vortex pairs. Ordinary vortex pairs areall that can be accommodated in a 11

2-layer model. Ingersoll et al.

(2000) remind us that “both cyclonic and anticyclonic structures existwithin the belts” of the real planet, and succinctly summarize the casefor their being generated by moist convection.

2. Model formulation

We use a doubly-periodic, quasi-geostrophic, pseudo-spectral β-plane version of the 112 -layer model, with leapfrogtimestepping and a weak Robert filter. The model tries tomimic conditions in a band of northern-hemispheric lati-tudes containing two deep jets, one prograde and one retro-grade. The simplest way to achieve shear-stability proper-ties resembling those of a horizontally larger domain (Thom-son 2015) is to choose the model’s zonal (x) to latitudinal(y) aspect ratio to be 2:1. In most runs a 512×256 spatialgrid is used. Further detail is in Thomson (2015), and anannotated copy of the code is provided online through theauthors’ websites.

Figure 1 shows schematically a meridional slice throughthe model, with the upper or weather-layer jets shownstronger than the deep jets and the interface correspond-ingly tilted, as dictated by thermal-wind balance, with el-evation ζ say. The y-axis points northward and the x-axiseastward, out of the paper. The central raised, i.e., cold, in-terface is in a model belt, cyclonically sheared, with modelzones on either side and with the whole structure repeatedperiodically. The underlying dry-convective layer is mod-eled as adiabatic and infinitely deep, with constant po-tential temperature θ. The constant-θ interface with theweather layer is shown as the bottom of the shaded region.The interface is flexible and responsive to the dynamics.The shading is meant to indicate the stable stratificationsuggested by the work of Sugiyama et al. (2014), concen-trated near the base of the weather layer though less so inzones than in belts. Belt centre is where the interface ishighest, bringing it closest to the lifting condensation levelfor water. Such a configuration is consistent with thermal-wind balance and with the standard perception that theweather layer’s stable stratification — a positive vertical

3

gradient ∂θ/∂z — results from moist convection with theconvection strongest in the belts.

The model equations for q(x, y, t), the weather layer’slarge-scale quasi-geostrophic potential vorticity (PV), withforcing F (x, y, t) and small-scale dissipation D(x, y, t), are(

∂

∂t+ u · ∇

)q = F +D , (2.1)

q := ∇2ψ + βy − k2D(ψ − ψdeep

). (2.2)

Here ∇2 is the two-dimensional Laplacian in the xy plane,β is the local latitudinal gradient of the vertical compo-nent of the planetary vorticity, kD is the reciprocal of theRossby deformation length LD based on the weather-layer’smean depth and on g′, the reduced gravity at the interface;ψ(x, y, t) and u(x, y, t) are the geostrophic streamfunctionand velocity for the weather layer, such that u is horizon-tal with components u = (u, v) = (−∂ψ/∂y, ∂ψ/∂x), andψdeep is the geostrophic streamfunction for the prescribedsteady, zonally-symmetric zonal flow udeep = −∂ψdeep/∂yin the dry-convective layer. D is a quasi-hyperdiffusivedissipation in the form of a high-wavenumber spectral fil-ter, used only to maintain numerical stability. It will beignored in most of the theoretical discussion. We adopt thefilter described in appendix B of Smith et al. (2002). Themodel code evaluates ∇ψ and ∇q in spectral space beforeFFT-transforming to physical space and evaluating u · ∇qby pointwise multiplication, then transforming back.

Following Dowling (1993) and Stamp and Dowling (1993),we somewhat arbitrarily take the deep flow to have a sinu-soidal profile

udeep(y) = U0 + Umax sin( yL

), (2.3)

where U0 and Umax are constants. The lengthscale L is(2π)−1 times the domain’s y-period, the full wavelengthof the jet spacing, which we fix at 10,000km to representmid-latitude conditions.

The real deep-jet profiles may of course be different.However, they are not well known. DI’s analysis did, tobe sure, find rounded udeep(y) profiles, in striking con-trast with the sharper profiles found in some dry-convectivemodels. However, DI’s cloud-wind analysis may not havebeen accurate enough to fix udeep(y) with great precision.While cloud-wind analyses have greatly improved since then— see especially Asay-Davis et al. (2009) — we are notaware of any corresponding published estimates of udeep(y)profiles and their error bars.

With the exception of q, which contains the non-periodicterms βy − k2DU0y, all the model’s weather-layer fields areassumed to be doubly-periodic including the streamfunc-tion ψ and the zonal-mean gradient ∂q/∂y of q,

∂q

∂y= β − ∂2u

∂y2+ k2D (u− udeep) . (2.4)

The periodicity of ψ entails that∫ 2πL

0

udy = 0 , (2.5)

which implicitly assumes not only that we are in a par-ticular reference frame, but also that the domain-averagedangular momentum budget is steady.3 And without loss ofgenerality we may take the domain integrals of ψ and F tovanish: ∫∫

ψdxdy = 0 ,

∫∫F dxdy = 0 . (2.6)

The first of these follows from the freedom to add an ar-bitrary function of time t alone to the streamfunction ψ,with no effect on the quasi-geostrophic dynamics. Physi-cally, this says that a small variation in the total mass ofthe weather layer, due for instance to horizontally-uniformdiabatic processes, has no dynamical effect as long as themean weather-layer depth, hence LD value, can be consid-ered constant to leading order in Rossby number. From(2.2) and the first of (2.6) we then have

∂

∂t

∫∫q dxdy = 0 , (2.7)

which is consistent with (2.1) only if the second of (2.6) alsoholds. This can be seen by domain-integrating the fluxform of (2.1) and noting that v = ∂ψ/∂x = 0, and that∫∫Ddxdy = 0 in virtue of D’s restriction to the highest

wavenumbers. It is convenient to view (2.7) as a quasi-geostrophic counterpart to the “impermeability theorem”for the exact, Rossby–Ertel PV (e.g., Haynes and McIntyre1990).

From here on we ignore the small-scale dissipation D.The zonal-mean dynamics is then described by

∂q

∂t= − ∂(v′q′)

∂y+ F , (2.8)

where the primes denote departures from zonal averages( ). The model’s Taylor identity (e.g., Buhler 2014), whichallows the mean PV dynamics to be translated into meanmomentum dynamics, if desired,4 is

∂(u′v′)

∂y= − v′q′ . (2.9)

The Taylor identity is a consequence of (2.2) alone, as iseasily verified using ∂( )/∂x = 0, hence valid at all eddy

3In other words, any domain-averaged external zonal force is eithernegligible or balanced by a domain-averaged ageostrophic mean y-velocity.

4The mean momentum dynamics is given by the minus the indef-inite y-integral of (2.8), in which

∫dy(k2D∂ψ/∂t + F ) represents mi-

nus the Coriolis force from the ageostrophic mean y-velocity, whosey-derivative is related via mass conservation to layer-depth changesand to the forcing F conceived of as mass injection or withdrawal.

4

amplitudes and independent of forcing and dissipation. Bymultiplying (2.1) by −ψ, and continuing to ignore D, wefind the relevant energy equation to be

∂E

∂t= −

∫∫ψF dxdy (2.10)

where E is the kinetic plus available potential energy ofthe weather layer — the model’s only variable energy —divided by the mass per unit area. That is,

E :=

∫∫ (12 |∇ψ|

2 + 12k

2Dψ

2)dxdy . (2.11)

3. The forcing F

a. Impulsive injection of small vortices

The forcing F corresponds to repeated injections ofclose-spaced, east-west-oriented pairs of small vortices atrandom locations and in alternating order, cyclone-anticy-clone alternating with anticyclone-cyclone. In each pair,the cyclone is weaker than the anticyclone by a fractionalamount b, say, which we call the “fractional bias”, andwhich increases with vortex strength so as to express thenotion that the dry-convective layer supplies the weatherlayer with mass and heat but with relatively more mass inthe stronger convection events. Each vortex is impulsivelyinjected using a parabolic PV profile, Eq. (3.1) and inset toFig. 2 below. We acknowledge that this must be an exceed-ingly crude representation of vortex generation by the realthree-dimensional convection, whose structure and proper-ties are unknown and have yet to be plausibly modeled.However, a simplistic vortex-injection scheme may be thebest that can be done within the 112 -layer dynamics, and in-deed is a rather time-honored idea (e.g., Vallis et al. 1997,Section 3b; Li et al. 2006; Showman 2007; Humphreys andMarcus 2007, Section 5a and Fig. 6).

We use east-west-oriented pairs for two reasons. Oneis to make zonally averaged quantities such as q and v′q′

less noisy. The other, less obvious, reason is an interest inassessing whether the Kelvin passive-sheared-disturbancemechanism (also discussed under the headings “CE2” and“SSST” in the literature) has a significant role in any of theregimes we find. The Kelvin mechanism operates when theinjected vortices are so weak that they are passively shearedby the mean flow u(y), producing systematically slantedstructures and hence Reynolds stresses u′v′ potentially ableto cause jet self-sharpening by “zonostrophic instability”.

The Kelvin mechanism is entirely different from theinhomogeneous-PV-mixing mechanism that produces ter-restrial strong jets, through drastic piecewise rearrange-ment of a background PV gradient. It is also differentfrom the Rhines mechanism, in which the injected vor-tices are strong enough to undergo the usual vortex in-teractions, especially the vortex merging that produces anupscale energy cascade that is then arrested, or slowed,

by the Rossby-wave elasticity of an un-rearranged back-ground PV gradient. We are of course interested in whetherany of these mechanisms have significant roles. Regardingthe Kelvin mechanism, it is strongest when the forcing isanisotropic in the sense of east-west vortex-pair orientation(e.g., Shepherd 1985; Srinivasan and Young 2014). So ourchoice of east-west orientation will give the Kelvin mecha-nism its best chance.

The impulsive vortex injections, corresponding theoret-ically to temporal delta functions in the forcing functionF , are actually spread over time intervals 2∆t to avoid ex-citing the leapfrog computational mode, where ∆t is thetimestep. This is still fast enough for advection to be neg-ligible, implying that the injections are instantaneous togood approximation. The parabolic profile of the resultingchange ∆q(r) in the PV field is given by

∆q(r) = qpk

(1− r2

r20

)(r 6 r0) , (3.1)

the peak vortex strength qpk being positive for a cycloneand negative for its accompanying anticyclone. The rela-tive radius r := |x−xc| with x = (x, y) denoting horizontalposition and xc = (xc, yc) the position of the vortex cen-ter. The radius r0 is taken as small as we dare, consistentwith reasonable resolution and realistic-looking vortex in-teractions (Thomson 2015, §§2.1.1, 3.5.2). In most casesr0 = 4∆x where ∆x is the grid size.

b. The complementary forcing

Thanks to the peculiarities of quasi-geostrophic dynam-ics and to our model choices we need the forcing to satisfy(2.6b). The model code does this automatically, by as-signing zero values to all spectral components having totalwavenumber zero. The most convenient way to see what itmeans, however, is to think of each injected vortex as sat-isfying (2.6b) individually. When, for instance, a smallanticyclone is injected, it is accompanied by a domain-wide cyclonic “complementary forcing”, in the form of asmall, spatially-constant contribution, additional to (3.1)and spread over the entire domain, such that

∫∫F dxdy

is zero. That is not to say that the dynamical responseto a single injection is domain-wide. Rather, the comple-mentary forcing is no more than a convenient bookkeepingdevice to guarantee that the forcing is consistent, at alltimes, with our choice of model setup including the choice(2.6) and its consequence (2.7).

Consider for instance a localized mass injection. Thedynamical response is formation of an anticyclone, namelya negative anomaly in the q field together with the asso-ciated mass and velocity fields obtainable by PV inversion(e.g., Hoskins et al. 1985, and references therein). Thosefields describe an outward mass shift and anticyclonically-circulating winds, the whole structure extending outwards

5

and decaying exponentially on the lengthscale LD. Thecomplementary forcing is quite different. It can be picturedas a uniform, domain-wide withdrawal of a compensatingamount of mass that is small, of the order of the Rossbynumber, and has no dynamical effect whatever. The com-plementary forcing is an artificial device to keep the mass ofthe model weather layer exactly constant. Of course withanticyclonic bias a domain-wide mass withdrawal wouldhave to occur in reality, presuming statistical steadiness,and would involve cloud physics and radiative heat transfer(e.g., Li et al. 2006). However, that aspect of the problemis invisible to the quasi-geostrophic dynamics.5

c. The vortex-injection scheme

We have explored many vortex-injection schemes, withmany choices of the way in which injected vortex strengths|qpk| and fractional bias b are made to increase with the in-terface elevation or coldness ζ. The simplest choices, withstrengths increasing monotonically with ζ, produced run-away situations with vortices far stronger than the realplanet’s mean shears and observed vortices, incompatiblewith our aim of finding flow regimes that are both realisticand statistically steady.

After much experimentation, the following vortex-injec-tion scheme proved successful, one aspect of which is that|qpk| is never allowed to exceed a set value qmax > 0. Thesensitivity to interface elevation or coldness is set by a pa-rameter ψlim > 0, in terms of which the definition of ζ willbe written as

ζ(x, y, t) =ψdeep(y)− ψ(x, y, t)

ψlim, (3.2)

whereψdeep(y) := LUmax cos

( yL

), (3.3)

corresponding to the y-oscillatory or jet-like part of thedeep flow (2.3).6 Injections are done one pair at a time,with the intervening time intervals selected at random froma specified range [4∆t, tmax], with uniform probability. Theminimum value 4∆t ensures that injection events do notoverlap in time. The maximum value tmax is usually cho-sen to be much larger, such that 1

2 tmax, close to the averagetime interval, is of the same order as the background shear-ing time L/Umax. We interpret these temporally sparse in-

5A reviewer’s comment prompts us to remark that spatially uni-form layer-mass changes are invisible to quasi-geostrophic channeldynamics also. The first of (2.6), and the remarks below (2.6), stillapply.

6Because of its double periodicity, our idealized model has noway of representing large-scale gradients in ψdeep except insofar asdψdeep/dy = −udeep enters the background PV gradient (2.4). Ofcourse the model also ignores the real planet’s other large-scale gradi-ents, and associated mean meridional circulations, for instance large-scale gradients in temperature, in composition including hydrogenortho–para fraction (e.g., Read et al. 2006, Fig. 10), and in the Cori-olis parameter and LD.

1�

1/2

1/4

1/8

-

6

r0 r0

ζFig. 2. Functions used in the vortex-injection scheme. The insetat top left shows the ∆q(r) profile for an injected cyclone of radiusr0 � LD, discussed in Eq. (3.1)ff. The top, light-colored curve inthe main figure shows the function ρ(ζ) defined in (3.8), equivalently|qpkA|/qmax when the choice (3.9) is made. The remaining curvesshow qpkC/qmax for different nonzero biases, according to (3.6)–(3.9)and the text below (3.6). From bottom up we have b = 1 (qpkC = 0 forall ζ) and then qpkC/qmax for bmax = 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.Note, however, that (3.9) is used subject to the further conditionsdescribed in (3.10)ff.

jections as idealizing the intermittency of real convection,probably governed by slow but chaotic dry-convective dy-namics along with time-variable structure near the inter-face (e.g., Showman and de Pater 2005; Sugiyama et al.2014).

For each injection event a location x = (x, y) is chosenat random and a close-spaced but non-overlapping pair ofvortices, each of radius r0 as specified in (3.1), is injectedat the pair of neighboring points

(xc, yc) = (x± 12s, y) . (3.4)

where the separation s is fixed at

s = 2r0 + ∆x . (3.5)

We denote the respective strengths by qpk = qpkC > 0 forthe cyclone and qpk = qpkA < 0 for the anticyclone, withmagnitudes always in the ratio B 6 1 where

B =

∣∣∣∣qpkCqpkA

∣∣∣∣ = (1− b) . (3.6)

The fractional bias b is either 1, to give anticyclones only,as in Li et al. (2006) and in Showman (2007), or

b = b(ζ) = bmax ρ(ζ) (3.7)

in all other cases, where bmax < 1 is a positive constantand where ρ(ζ) is the three-piece ramp function defined by

ρ(ζ) :=

0 (ζ 6 − 12 ),

12 + ζ (− 1

2 6 ζ 6 12 ),

1 (ζ > 12 ) .

(3.8)

6

This function is plotted as the top, light-colored curve inthe main part of Fig. 2.

It remains to choose how |qpkA| varies. The simplestchoice would be

|qpkA| = qmax ρ(ζ) , (3.9)

so that |qpkA|/qmax is given by the light-colored curve inFig. 2. Then qpkC/qmax is given by the curves underneath,for values of bmax < 1, plotted using (3.9) with (3.6)–(3.8).The label 1 marks the null curve qpkC/qmax = 0 for theanticyclones-only case b = 1. However, the choice (3.9)still produces runaway situations incompatible with statis-tical steadiness, except when qmax is made too small toproduce significant small-scale vortex activity. For largerqmax values, enough to produce such activity, the typicalbehavior is the growth and unbounded strengthening of alarge cyclone. The large cyclone’s cold-interface footprintin ζ, still larger in area, induces strong local injections fromwhich the small injected cyclones tend to migrate inwardand the small anticyclones outward to give a cumulative,and apparently unbounded, increase in the large cyclone’ssize and strength. (Notice by the way that this mechanismis quite different from the classic vortex merging or upscaleenergy cascade. The possibility of an unbounded increasein vortex strength is another peculiarity of quasi-geostro-phic theory, predicting its own breakdown as Rossby num-bers increase.)

Large, strong cyclones have correspondingly large q′

values, motivating our final choice, which is to use (3.6)–(3.9) — remembering that qpkC = B |qpkA| — wheneverthe local q′ value satisfies

max(B |qpkA|+ q′, |qpkA| − q′

)6 qmax (3.10)

whereas, if (3.9) gives a |qpkA| value that makes the left-hand side of (3.10) greater than qmax, then |qpkA| is reducedjust enough to achieve equality, i.e., reduced just enough tosatisfy (3.10) with B unchanged. The second argument ofthe max function in (3.10) covers the possibility that stronganticyclones with large negative q′ might occur, though itis the first argument that prevails in all the cases we haveseen.

The limitations thus placed on the strongest vorticesinjected are interpreted here as reflecting not only the lim-itations of quasi-geostrophic theory, but also the unknownlimitations of the real, three-dimensional moist convectionas a mechanism for generating coherent vortices on thelarger scales represented by our model. On smaller scalesone must expect three-dimensionally turbulent vorticityfields with still stronger peak magnitudes — as terrestrialtornadoes remind us — though, with no solid lower sur-face, the details are bound to be different. For one thing,net mass injection rates are bound to be modified by suchphenomena as evaporation-cooled, precipitation-weighted

thunderstorm downdrafts, also called microbursts, contri-buting negatively. The concluding remarks in Section 8 willsuggest a possible way of replacing (3.10) by something lessartificial, albeit paid for by further expanding the model’sparameter space.

d. Quasifrictional effects

As mentioned earlier, the bias b has quasifrictional ef-fects. These are most obvious in the zonal-mean dynamicsdescribed by (2.8), under the constraints (2.5)–(2.7). Be-cause of thermal-wind balance and the positive slope ofthe ramp function ρ(ζ), the sign of F (y, t) tends on aver-age to be anticyclonic in belts and cyclonic in zones when-ever the upper or weather-layer jets are stronger than thedeep jets (3.3), the case sketched in Fig. 1. The converseholds in the opposite case. So F tends on average to re-duce differences between shears in the upper jets and in thedeep jets. There is a corresponding quasifrictional effect onlarge cyclones. By contrast, fluctuations such as those giv-ing rise to the eddy-flux term −∂(v′q′)/∂y in (2.8) can actin the opposite sense, in some cases giving rise to realisticinterface-temperature structures in the manner sketched inFig. 1.

We find that the quasifrictional effects can be under-stood alternatively from environment-dependent negativecontributions to the right hand side of the energy equation(2.10), competing with the positive, environment-indepen-dent “self-energy” inherent in each injection. This con-trasts with the standard, perfectly unbiased forcing used inbeta-turbulence theory (e.g., Srinivasan and Young 2012),which is designed such that the self-energy is the only con-tribution, allowing one to prescribe a fixed, positive energyinput rate ε, which along with spectral narrowness is thenormal prelude to using Kolmogorovian arguments. How-ever, it would then be necessary to introduce a separatelarge-scale dissipation term, as would be necessary also ifcyclonic bias, b < 0, were to be used in our scheme. (Notsurprisingly, taking b < 0 has antifrictional effects. Whenwe tried it, the most conspicuous result was self-excitationof unrealistic long-wave undulations.)

When F and other quasifrictional effects with b > 0 arestrong enough to produce realistic, statistically steady flowregimes, we find that upper-jet profiles tend to be pulledfairly close to deep-jet profiles. This tendency shows up ro-bustly in test runs initialized with upper jets both weakerand stronger than the deep jets. In most cases, therefore,we use a standard initialization in which the upper-jet pro-files are the same as the deep-jet profiles (3.3), makingζ = 0 to start with, and the average forcing spatially uni-form. We then observe how the upper profiles, ζ, and theimplied forcing all change in response to the eddy flux v′q′

in (2.8).

7

4. Parameter choices

a. Sensitivity

It turns out that the interesting cases, statistically stea-dy with realistic, quasi-zonal ζ or interface-temperaturestructure, occupy only a small region within the model’svast parameter space. Not surprisingly, the behavior is sen-sitive to qmax and LD values, which govern the strength andnature of the model’s vortex activity all the way from caseswith no such activity — having only the Kelvin (CE2/SSST)passive-shearing mechanism — up to cases with vortex ac-tivity so violent as to disrupt the quasi-zonal ζ structurealtogether. It turns out that the Kelvin mechanism is un-able to produce realistic ζ structure. See Section 7 below.The most interesting cases, our main focus, turn out tobe those exhibiting chaotic vortex interactions just strongenough to make an impact on the y-profiles of v′q′ in (2.8).

A big surprise, though, was that the behavior is verysensitive to the choice of bmax, with the most interestingcases clustered around small values . 10−1. This was es-pecially surprising in view of the past work of Li et al.(2006) and Showman (2007) using purely anticyclonic forc-ing b = 1. The different behavior seems related in part tothe absence of deep jets in their studies but presence inours, in combination with the quasifrictional F effect.

b. LD values and A2 stability

In considering choices of LD, and remembering its lat-itude dependence, we would like to respect observationalas well as theoretical constraints. However, observationalconstraints from the comet-impact waves are controversialand unclear.7 Also, observational constraints from the DIwork and its successors apply mainly to the lower latitudesof the Great Red Spot and other large anticyclonic Ovals,roughly 15◦–35◦. The original DI work suggested LD val-ues at, say, 35◦S, that were not strongly constrained butwere estimated as roughly in the range 1500–2250km. Themore recent work of Shetty and Marcus (2010), based on amuch more sophisticated cloud-wind methodology, appearsto constrain LD values more tightly, for instance producingvalues close to 1900km at latitudes around 33.5◦S from ananalysis of the flow around a large anticyclone, Oval BA.However, unlike DI, who used 112 -layer primitive-equationdynamics, Shetty and Marcus (2010) assumed that quasi-geostrophic 112 -layer dynamics applies accurately to the realplanet. In any case, it is likely that all these estimates ap-ply to the locally deeper weather layer expected near large

7One reason is that even if the comet-impact waves were gravitywaves guided by the weather layer they would have had a differentstructure in the underlying dry-convective layer, more like surface-gravity-wave structure than that of 11

2-layer dynamics. Another rea-

son is the case made by Walterscheid (2000) that the observed comet-impact waves were in any case more concentrated in Jupiter’s strato-sphere.

anticyclones, suggesting somewhat smaller LD values fur-ther eastward or westward, as well as further poleward invirtue of the increasing Coriolis parameter.

Our approach will be to reserve judgement on theseissues, and simply to find a range of LD values for whichthe idealized model behavior looks realistic.

As indicated near the end of Section 1, we need tokeep the model’s jets straight by excluding long-wave shearinstability. With nonmonotonic upper PV gradients ourmodel has a shear-instability threshold strongly influencedby the value of LD, as does, almost certainly, the real planetas well. Linear theory shows that, when the threshold isslightly exceeded, the instability first kicks in as a long-wave undulation, phase-coherent between adjacent jets, afact that we have cross-checked in test runs with the un-forced model showing, in addition, that the undulationequilibrates nonlinearly to a moderately small amplitude(Thomson 2015, §4.1) without otherwise disturbing thePV distribution, to any significant extent. Such a phase-coherent long-wave undulation would be conspicuous onthe real planet but is not observed. To get straight jets wemust stay below the shear-instability threshold.

In our model, for upper-jet profiles kept close to thedeep-jet profiles (3.3) by the quasifrictional F effect, theupper PV gradients are indeed nonmonotonic, and stronglyso if we take plausible values of Umax and β−k2DU0. RecallEqs. (2.3)–(2.4), noting that the y-oscillatory part of the k2Dterm in (2.4) is small when the upper and lower jet profilesare close, that is, when u(y) is close to udeep(y)− U0.

If for instance we take Umax ' 30ms−1 and β − k2DU0

anywhere between the value zero suggested by DI’s resultsand the value of β itself at the equator, ' 5×10−12 s−1m−1,then we get strongly nonmonotonic ∂q/∂y essentially be-cause, with L = (2π)−1 × 10, 000km = 1592km, we have∂2u/∂y2 values in the range ±Umax/L

2 = ±12 × 10−12

s−1m−1, whose magnitude is well in excess of β at anylatitude.

The model’s jet flow is then shear-unstable for suffi-ciently large LD/L but stabilized when LD/L is taken be-low the threshold already mentioned, despite the stronglynonmonotonic ∂q/∂y. The existence of that threshold wasrecognized by Ingersoll and Cuong (1981) and, as pointedout by Dowling (1993), is related to the “A2 stabilization”described by Arnol’d’s second stability theorem. It arisesbecause reducing LD/L reduces the intrinsic phase speedsand lateral reach of even the longest, hence fastest possible,pair of counterpropagating Rossby waves, each wave propa-gating upstream on adjacent prograde and retrograde jets.These reductions suppress the instability by “destroyingthe ability of the two Rossby waves to keep in step” (McIn-tyre and Shepherd 1987, p. 543; see also Hoskins et al.1985, Fig. 18ff., and references therein) — i.e., to phase-lock, with each wave holding its own against the mean flow.That is why the first wavelength to go unstable for slightly

8

supermarginal LD/L is the longest available wavelength,with zonal wavenumber kmin, say.

For unbounded or doubly-periodic domains the A2 the-orem says that the flow is shear-stable if a constant c canbe found such that

k2D + k2min >∂q/∂y

u− c, (4.1)

where as before k2D := L−2D . For the sinusoidal profilesof our standard initialization, and for β − k2DU0 = 0 assuggested by DI, it happens that (4.1), with c = 0, isa necessary as well as a sufficient condition for stability(Stamp and Dowling 1993). The right-hand side of (4.1) isthen just L−2, independent of Umax, and the threshold isprecisely at k2D = L−2 − k2min.

In our model, with its 2:1 aspect ratio, we have k2min =L−2/4. Therefore, LD . (4/3)1/2L = 1838km should beenough to exclude long-wave shear instability, as long asthe upper profiles u(y, t) stay close to the deep profilesudeep(y). It is arguable, however, that since the much largerdomain of the real planet should correspond to k2min �L−2, it might be more appropriate to take LD . L = 1592km. Having regard to these considerations, we decided touse LD values 1500km or less in most of our model runs.In any case it turned out that for larger LD the typicalresult was unrealistically strong, or even violent, long-wavedisturbances.

In sections 5 and 6 we describe and illustrate the model’sbehavior for LD = 1200km and 1500km and for forcingsjust strong enough to produce chaotic vortex interactions.In such cases the model robustly approaches stable, statis-tically steady states with fairly straight jets, and realisticζ structures, over significant ranges of qmax and bmax andwith nonmonotonic upper PV gradients ∂q/∂y.

c. Other parameters including β0 and q∗max

We fix Umax = 35ms−1 as a compromise between lowand midlatitude values, and choose two values of

β0 := β − k2DU0 , (4.2)

namely zero and 4.03× 10−12 s−1m−1. Both choices make∂q/∂y strongly nonmonotonic. The value zero requires pro-grade U0, roughly consistent with DI’s results; see theirFig. 4b and its idealization in Stamp and Dowling (1993).Prograde U0 is in any case expected in latitudes outsidea tangent cylinder of the dry-convective layer (e.g., Jonesand Kuzanyan 2009). The value 4.03×10−12 s−1m−1 is thevalue of β itself at latitude 35◦. For convenience we referto these two cases β0 = 0 and β0 = 4.03 × 10−12 s−1m−1

as “pure-DI” and “midlatitude” respectively, remembering,however, that nothing is known about actual U0 values atthe higher latitudes.

Because the quasifrictional F effect tends to pull ourmodel’s upper jets more or less close to its deep jets, the

strongest upper mean shears ∂u/∂y tend to have ordersof magnitude similar to that of the strongest deep shearUmax/L = 2.199× 10−5 s−1. So a convenient dimensionlessmeasure of qmax is

q∗max :=qmax

Umax/L> 0 . (4.3)

The parameter q∗max governs the likely fate of vortices in-jected into background shear of order Umax/L. We takevalues ranging from q∗max = 0.5 up to q∗max = 32. At thelow end of the range, practically all the injected vorticesare shredded, i.e., sheared passively and destroyed. (Thereis still, of course, a quasifrictional F effect.) In the high-est part of the range, say 16 . q∗max . 32, the strongestinjected vortices all survive even in adverse shear, e.g., an-ticyclones in cyclonic shear. For intermediate values onetypically sees survival in favorable shear only. To distin-guish the three types of behavior we call the injections“weak”, “strong”, and “semi-strong” respectively.

The behavior is roughly consistent with the classic studyof Kida (1981) on single vortices in shear. In place of theparabolic profile ∆q(r) defined by (3.1) above, Kida’s anal-ysis assumes that LD = ∞ and uses a top hat or “vor-tex patch” profile. It should be qualitatively relevant forsmall vortex radius r0 � LD. Kida’s condition for an an-ticyclone of strength qpatch to survive in cyclonic shearS is q∗patch > 6.72 where q∗patch := |qpatch/S |, as can be

shown straightforwardly from his equations.8 Model testruns with single-vortex injections and vortex-pair injec-tions (Thomson 2015, Figs. B10, B11 respectively) behaveas expected from Kida’s analysis. For instance, a singleparabolic anticyclone of peak strength |qpkA| injected intocyclonic shear S is destroyed when |qpkA/S | . 7. It sur-vives almost intact when |qpkA/S | & 16, then behavingalmost like Kida’s patch with the same Kelvin circulation,i.e., with the same value of 2π

∫ r00

∆q rdr so that |qpatch| =12 |qpkA|. For intermediate values 8 . |qpkA/S | . 16 a smallcore survives while the outskirts are eroded away.

The parameter ψlim in (3.2) has to be chosen empiri-cally. We want the resulting ζ fields to range over valueswithin, or slightly exceeding, the range − 1

2 6 ζ 6 + 12 that

8The number 6.72 can be verified from the first line of Kida’sEq. (3.4), by setting s = 1 and plotting the right-hand side overan interval r ∈ (0, 1). In Kida’s notation r = 1 corresponds to acircular vortex and r = 0 to a vortex shredded by the shear into aninfinitely thin filament. Taking s = 1 picks out the case of an initiallycircular vortex. The vortex is shredded if for all r the right-handside of Kida’s (3.4) stays between ±1, while if it dips below −1 thevortex survives. For adverse pure shear such a dip occurs whenever,in Kida’s notation, ω/2e = −ω/2γ > 6.7215, where ω = qpatch < 0and 2γ = S > 0, the shear. For favorable pure shear there is nosharp threshold, but for instance shear with 0 < −ω/2e = ω/2γ <0.5 distorts an initially circular vortex beyond aspect ratio 25. Itsdestruction is then a practical certainty for finite LD, or for almostany background differing from Kida’s strictly steady, strictly constantpure shear of infinite spatial extent.

9

Fig. 3. Zonal-mean zonal velocity profiles u(y) / m s−1 for the pure-DI case with LD = 1200 km and q∗max = 16, all shown at time t = 120Earth years. The inner, dashed curve is udeep(y) − U0. The heavysolid curve is the upper profile u(y) for the anticyclones-only run,b = 1 for all ζ (run DI-12-16-1). The lighter solid curves show u(y),in order of increasing peak |u|, respectively for bmax = 1/4, 1/8, 1/16,1/32, 1/64, and 0 (runs DI-12-16-4, DI-12-16-8,... DI-12-16-∞).

corresponds to the sloping part of the ramp function ρ(ζ).A satisfactory choice is found to be ψlim = Λq∗max whereΛ = 4.47 × 106 m2s−1. This is used in all the runs men-tioned here, all the way from q∗max = 0.5 to q∗max = 32. Theprecise value of Λ is not critical. Any neighboring value willproduce similar results.

Table 1 and its caption summarize the parameter choicesfor the most important runs. In the first column, DI means“pure DI” and ML “midlatitude”, corresponding to theβ0 values shown in column 2. See text following (4.2).The numbers within each label, DI-12-16-1 etc., are short-hand for LD, q∗max, and bias respectively, as shown also incolumns 3–5. Thus for instance DI-12-16-1 labels a pure-DI run in which LD = 1200km, q∗max = 16, and bias b = 1,i.e., anticylones-only forcing. When the final number ex-ceeds 1, as in runs DI-12-16-4 to DI-12-16-∞, it is thereciprocal of bmax in (3.7). In the column marked “Sta-tistically steady?”, the blank entries “ – ” signify runs notyet steady but likely to have become so, in our judgement,had the run been continued for long enough.

5. Main results

a. Pure-DI with LD =1200 km, q∗max = 16, and varying bias

We focus at first on the pure-DI case with LD = 1200km and q∗max = 16, then comment briefly on similaritiesand differences for LD = 1500km and for midlatitude cases.Further details are given in Thomson (2015). It is forLD = 1200km, well below the A2 stability threshold, thatwe obtain the widest ranges of q∗max and bmax over whichmodel flows are realistic and statistically steady. Broadlyspeaking, the range of q∗max values that produce such flows

Fig. 4. Domain-averaged total energy per unit mass, in Jkg−1 orm2s−2, against time in Earth years for the same set of pure-DI runsas in Fig. 3, with LD = 1200 km and q∗max = 16. The lowest, heavysolid curve is for the anticyclones-only run, b = 1 for all ζ. Thelighter solid curves, reaching successively higher energies, correspondrespectively to bmax = 1/4, 1/8, 1/16, 1/32, 1/64, and 0.

are found to be in or near the semi-strong regime.Figures 3–7 show results for the first seven runs in Ta-

ble 1, in which bias is varied in the pure-DI case withLD = 1200km and q∗max = 16.

In Fig. 3 the inner, dashed curve is the deep-jet velocityprofile udeep(y) − U0. The solid curves are upper-jet pro-files u(y) for the different biases, after 120yr (Earth years)of integration from the standard initialization. The modelbelt lies approximately in the y-interval between 2500kmand 7500km, where the mean shears are cyclonic, corre-sponding to the central portion of Fig. 3, and of Fig. 1also. The model zone is in the periphery and its peri-odic extension. The upper-jet profiles u(y) begin with theanticyclones-only run, DI-12-16-1, which has b = 1 forall ζ. This is the first solid curve, heavier than the restand only slightly different from the dashed curve. Thelighter solid curves, peaking at successively higher valuesof |u|, correspond to runs DI-12-16-4 to DI-12-16-∞, i.e.,to bmax = 1/4, 1/8, 1/16, 1/32, 1/64, and 0 respectively.We also ran bmax = 1/2; the profile, not shown, hardlydiffers from the dashed curve and the heavy, b = 1 profile.

Evidently the actual mean shears in the model belt areeither close to, or somewhat greater in magnitude than,the nominal value Umax/L in (4.3). Many of the injec-tions in these runs are semi-strong, depending on injectionlocations. A small minority can be strong. The varyingbehavior of the injections is further discussed below, inconnection with an illustrative movie.

As anticipated, reducing the bias reduces the quasifric-tional F effect, allowing stronger upper jets. In these pure-DI runs there is no dynamical difference between progradeand retrograde jets, which on average are sharpened andstrengthened by the same amounts.

10

Run β0/ s−1m−1 LD/ km q∗max Bias trun/Years Statistically FiguresEq.(4.2)ff. Eq.(2.2)ff. Eq.(4.3) Eq.(3.6)ff. steady?

DI-12-16-1 0 1200 16 1 125 Yes 3, 4, 6, 7DI-12-16-4 0 1200 16 1/4 125 Yes 3, 4, 6, 7DI-12-16-8 0 1200 16 1/8 250 Yes 3, 4, 6, 7DI-12-16-16 0 1200 16 1/16 710 Yes 3, 4, 5, 6, 7DI-12-16-32 0 1200 16 1/32 710 Yes 3, 4, 6, 7DI-12-16-64 0 1200 16 1/64 125 No 3, 4, 6, 7DI-12-16-∞ 0 1200 16 0 125 No 3, 4, 6, 7ML-12-16-1 4.03× 10−12 1200 16 1 125 Yes 8, 9ML-12-16-4 4.03× 10−12 1200 16 1/4 125 Yes 8, 9ML-12-16-8 4.03× 10−12 1200 16 1/8 250 Yes 8, 9ML-12-16-16 4.03× 10−12 1200 16 1/16 290 Yes 8, 9ML-12-16-32 4.03× 10−12 1200 16 1/32 240 – 8, 9ML-12-16-64 4.03× 10−12 1200 16 1/64 125 No 8, 9ML-12-16-∞ 4.03× 10−12 1200 16 0 125 No 8, 9DI-15-16-16 0 1500 16 1/16 295 Yes Thomson (2015)ML-15-16-16 4.03× 10−12 1500 16 1/16 60 – Thomson (2015)

DI-15-8-8 0 1500 8 1/8 60 – Thomson (2015)DI-15-8-16 0 1500 8 1/16 320 Yes Thomson (2015)ML-15-8-8 4.03× 10−12 1500 8 1/8 60 – Thomson (2015)ML-15-8-16 4.03× 10−12 1500 8 1/16 315 Yes Thomson (2015)DI-12-8-16 0 1200 8 1/16 125 – Thomson (2015)ML-12-8-16 4.03× 10−12 1200 8 1/16 110 – Thomson (2015)DI-12-32-16 0 1200 32 1/16 330 Yes Thomson (2015)DI-12-1-16 0 1200 1 1/16 350 – Thomson (2015)DI-12-1-64 0 1200 1 1/64 300 Yes 10, 11, 12, 13

DI-12-0.5-64 0 1200 0.5 1/64 300 Yes 10, 11, 12, 13

Table 1. Parameter values chosen. In column 5, the bias values are b = 1 or bmax < 1; see text surrounding Eq.(3.7). The trun columnshows the length of each run, rounded down to the nearest 5 Earth years. Statistical steadiness is assessed at t = trun. The blank entries “ – ”signify runs not yet steady but judged likely to become so. The parameter ψlim defined in (3.2) is set equal to Λq∗max where Λ = 4.47 × 106

m2s−1; Λ is always held fixed. Also, tmax, defined below Eq. (3.3), is 24.06 hours in all runs except the last two, DI-12-1-64 and DI-12-0.5-64,for which tmax = 0.2406 hours. Other parameters held fixed are Umax = 35 m s−1, L = 1591.55 km, r0 = 156.25 km, ∆x = ∆y = 39.0625 km,∆t = 50s, Umax/L = 2.199× 10−5 s−1, and the injected-vortex separation s = 2r0 + ∆x = 351.56 km, Eq. (3.5). However, ∆x, ∆y and ∆t arehalved in numerical resolution tests (e.g., Thomson 2015, §3.5.2).

The runs with bmax ranging from 1/4 to 1/16, and therun with b = 1, are all close to statistical steadiness, consis-tent with the flattening-out of the corresponding curves inFig. 4. These give domain-averaged total energy in Jkg−1

against time t, with bias decreasing upward from curve tocurve. Total energies are dominated by 1

2 |u|2+ 1

2k2D|ψ|2, the

kinetic plus available potential energy of the zonal-meanflow, contributing in roughly equal proportions. Domain-averaged eddy energies, not shown, are relatively small butalso flatten out, for the runs in question. The run withbmax = 1/32 corresponds to the topmost of the three en-ergy curves that reach 250yr. It is evolving toward statisti-cal steadiness but does not come close to it until somethinglike 500yr of integration. The run with b = 1, included forcomparison and contrast with Li et al. (2006) and Show-man (2007), is statistically steady apart from a decadal-timescale vacillation (heavy curve at bottom of Fig. 4).

However, in that run the upper jets are hardly strongerthan the deep jets, as seen in Fig. 3, and the ζ structure iscorrespondingly unrealistic.

b. Details and movie for a realistic example

We focus on run DI-12-16-16. Fig. 5 shows snapshotsof ζ in contours and q in grayscale for that run, at timet = 120yr. A corresponding q-field movie is provided inthe online supplemental material, in grayscale and colorversions. The bars on the right show LD. Solid contoursin Fig. 5a show positive ζ, a cold, elevated interface thatincreases moist-convective activity. The heavy solid con-tour marks the value ζ = + 1

2 at which the ramp functionρ(ζ) saturates. Dashed contours show negative ζ, a warm,depressed interface that reduces moist-convective activity.The structure of this ζ field is sufficiently zonal to countas realistic, by our criterion that the model should reflect

11

Fig. 5. Snapshots of the ζ and q fields at time t = 120 Earth yearsfor run DI-12-16-16. In the top panel (a), dashed contours shownegative ζ and solid contours positive ζ, with contour interval 0.1 inthe dimensionless units of Fig. 2. The heavy solid contour marks thevalue ζ = +0.5 at which the ramp function ρ(ζ) saturates. In thebottom panel (b), which is the first frame of the supplemental movie,the grayscale is in units of Umax/L = 2.199 × 10−5 s−1, like q∗max.The strongest vortices slightly exceed the grayscale range, with thelarge cyclone in mid-belt, y ' 5000 km, peaking at q = 17.8Umax/Land the largest anticyclone peaking at q = −18.4Umax/L, to the farsouth-south-west near x ' 4500 km. There is one other out-of-rangevortex, the small cyclone north-north-west of the large cyclone, nearx ' y ' 6400 km, which peaks at q = 17.7Umax/L.

the real planet’s preference for stronger convection in beltsthan in zones.

The q snapshot in Fig. 5b is dominated by small vor-tices, produced by injections followed by migration — es-pecially of small anticyclones from the belt into the zone— as well as by occasional vortex merging and other inter-actions. Cyclones are shown dark and anticyclones light.The small vortices move around chaotically, under theirneighbors’ influence and that of the background shear. Yetvortex merging and upscale energy cascading are inhibitedto a surprising extent. This lack of upscale cascading willbe discussed further in Section 6.

Conspicuous in Fig. 5b is a single, relatively large cy-clone near x ' 7500km and y ' 5000km. The strengthof this cyclone fluctuates but is statistically steady. Thesnapshot shows it slightly larger and stronger than average.

The strength is governed by reinforcement through vortexmerging and internal migration on the one hand (Section 6below), competing with attrition by erosion and quasifric-tional effects on the other. The cyclone is strong enough toproduce a conspicuous footprint in the ζ field, superposedon the quasi-zonal structure (Fig. 5a). The footprint takesthe form of a cold patch or elevated area extending outwardfrom the core of the cyclone on the lengthscale LD.

The velocity field of the large cyclone modifies the back-ground shear and strain quite substantially, such that someof the nearby injection events are strong rather than semi-strong. An example can be seen in the movie, startingwest-south-west of the large cyclone at trel = 0.408, wheretime trel runs from 0 to 1 in units of movie duration, justunder an Earth month. The injected anticyclone survivesas it travels around the large cyclone, protected by thecyclone’s anticyclonic angular shear, then slowly migratesinto the model zone to the north, across the retrogradejet. The accompanying cyclone, caught in the same anti-cyclonic angular-shear environment, suffers partial erosionalmost immediately after injection. It has a much shorterlifetime and ends up completely shredded, at trel ' 0.58,after one more partial erosion event.

Another clear example of migration from belt to zone,this time southward, occurs between trel ' 0.65 and trel '0.9. A recently-injected anticyclone partly merges witha pre-existing anticyclone, near x ' 15000km and y '3000km, and then migrates from belt to zone across thesouthern, prograde jet.

The snapshot in Fig. 5b is taken at the start of themovie, trel = 0. At that instant, there has just been an in-jection almost due west of the large cyclone, near x ' 5000km. That injection proves to be semi-strong. Its anticy-clone, seen on the left in Fig. 5b, is shredded immediately.However, its cyclone is also shredded shortly afterward,again by the anticyclonic angular-shear environment. Dur-ing the cyclone’s short lifetime (trel . 0.13), it migratesinward through a small radial distance, as can be checkedby comparing its positions south and then north of thelarge cyclone, at trel = 0.040 and 0.098 respectively. Suchevents are frequent and are clearly part of what builds thelarge cyclone, whose typical q-structure is sombrero-like, astrong core surrounded by a weaker, fluctuating cyclonicPV anomaly, easier to see in the color movie than in thegrayscale snapshot. That structure is alternately built upand eroded by a chaotic sequence of vortex interactions andinjections.

Also notable in the movie is an injection making arare direct hit on the inner core of the large cyclone, attrel = 0.044. Thanks to the condition (3.10) this injectionbehaves as a semi-strong injection. In this case the injectedanticyclone is shredded into a tight spiral and acts quasi-frictionally. By contrast, the injected cyclone stays almostcompletely intact, and migrates through a small radial dis-

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Fig. 6. Zonal-mean PV profiles q(y) for the same set of pure-DI runsas in Fig. 3, with LD = 1200 km and q∗max = 16. The PV is in unitsof Umax/L = 2.199 × 10−5 s−1 and is time-averaged from t = 115yr to t = 125yr to reduce fluctuations. The heavy curve is the q(y)profile for the anticyclones-only run, b = 1 for all ζ, and the lightercurves with increasingly large peak |q| values correspond respectivelyto bmax = 1/4, 1/8, 1/16, 1/32, 1/64, and 0. The initial profile, notshown, is sinusoidal with amplitude 1 unit, its central peak only justbeyond the flat part of the b = 1 heavy curve.

tance to the center. The net effect is a slight reductionin the overall size and strength of the large cyclone, fromabove average to below average.

c. Varying bias again

Most of the statistically steady runs produce a single,relatively large cyclone in a similar way. Its average size in-creases as bmax, and hence quasifriction, are reduced. RunsDI-12-16-64 and DI-12-16-∞, with bmax = 1/64 and 0,were terminated at t = 125yr because by then they haddeveloped single cyclones large enough to produce an unre-alistic, grossly nonzonal, footprint-dominated ζ structure.

The sharpened peaks of the jet profiles for bmax =1/16 in Fig. 3 correspond to sharp steps in the q field,as seen in Fig. 5b as sharp transitions between light grayand darker gray. These PV steps, embedded as they arein relatively uniform surroundings, resemble the cores ofterrestrial strong jets apart from their relatively limitedmeandering, which is much more Jupiter-like. The forma-tion of such steps from an initially smooth q field points toPV mixing across belts and zones as a contributing mech-anism. A role for PV mixing is consistent with the chaoticappearance of the small-scale vortex interactions.

The PV steps persist into the two regimes with thesmallest bmax values 1/64 and 0 and the largest cyclones.However, the PV steps are no longer reflected in the cor-responding u(y) profiles in Fig 3, the outermost two pro-files. Being Eulerian means, they are more rounded simplybecause the large cyclones make the jets meander morestrongly. A Lagrangian mean, not shown, would follow the

Fig. 7. Relative moist-convective activity A(y)/Amax for the sameset of pure-DI runs as in Fig. 3, except that bmax = 1/64 and bmax = 0are omitted. Here A(y) is defined in Eq. (5.1), and Amax is the largestvalue in the set shown. The heavy curve is for the anticyclones-onlyrun b = 1. The lighter curves are for bmax = 1/4, 1/8, 1/16, and1/32, peaking successively further to the right. The small wigglesarise from the statistical fluctuations in the vortex-injection scheme,showing up despite time-averaging from 60 to 120yr.

meandering and still reveal sharpened jet profiles — indeedeven sharper than the sharpest in Fig. 3.

Figure 6 shows the Eulerian-mean q profiles for thesame set of pure-DI runs, at time t ' 120yr (see caption).For bmax > 1/16, the profiles reflect the same inhomo-geneous-PV-mixing structure, though the large cyclone inFig. 5b creates a noticeable blip near y ' 5000km, in theq profile for bmax = 1/16. Similar blips, corresponding tolarger cyclones, become strong and then dominant as bmax

is reduced to 1/32, 1/64, and 0; and the large cyclones arestill growing in those runs. It is interesting to see whatlook like hints of PV mixing even for b = 1, the heavycurve, although the departure from the initial, sinusoidalq(y) profile is then very weak (see figure caption), and un-able to produce a realistic ζ field.

Figure 7 gives an alternative view of the model’s belt–zone structure for bmax > 1/32. It shows in arbitrary unitsa positive-definite measure A of average injection strength,defined by

A(y) = |∆q(r)|xt

(5.1)

where the averaging is over many injections, both zonallyand in time. In Fig. 7 the time-averaging is from 60yr to120yr. Here ∆q(r) represents the cores of injected anticy-clones only, as defined in (3.1), i.e., omitting their cycloniccompanions and complementary forcings. Thus, A = 0would signal a complete absence of injections. IncreasingA-values signal increasingly strong injections, on average,regardless of bias. The wiggles in the curves arise from thestatistical fluctuations in the vortex-injection scheme.

The belt-to-zone variation in injection strength seen inFig. 7 is consistent with the realistic ζ structures found for

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Fig. 8. Zonal-mean zonal velocity profiles u(y) / m s−1 for the mid-latitude case with LD = 1200 km and q∗max = 16, at time t = 120Earth years. Note that all the retrograde jet profiles are rounded.Biases vary as in Fig. 3. These are the runs labeled ML-12-16-1,ML-12-16-4,... ML-12-16-∞.

moderately small values of bmax, with the strongest injec-tions concentrated in mid-belt. The unrealistic ζ structurefor b = 1 produces, as expected, relatively little belt-to-zone variation in injection strength (heavy curve). Thebelt-to-zone variation increases as bmax decreases toward1/32, but then decreases again (not shown) because of thedominance of the large cyclone, within whose footprint thecondition (3.10) weakens the injections.

d. Other cases and parameter values

Results for q∗max = 8 (not shown; see Thomson 2015)show similar behavior, though the tendency to form largecyclones is weaker, and there are cases in which the largestcyclones come in pairs. Most injections are then weak orsemi-strong. For q∗max = 32, by contrast, many injectionsare strong, resulting in relatively violent vortex activity. Along run DI-12-32-16 has been carried out for q∗max = 32and bmax = 1/16. It shows unrealistic, strongly nonzonalζ structure, briefly described at the end of Section 6. Forq∗max = 1 or less, practically all injections are weak. The re-sponse is then governed mainly by the Kelvin and F mech-anisms. See Section 7.

For the “midlatitude” case, with β0 = β − k2DU0 =4.03 × 10−12 s−1m−1, the value of β itself at latitude 35◦,the results (Thomson 2015) are broadly similar except thatjet-sharpening is more effective for the prograde than forthe retrograde jets. The u and q profiles for LD = 1200km and q∗max = 16 at time t ' 120yr are shown in Figs. 8and 9 (see captions). Again, only the runs with b = 1 orbmax > 1/16 are close to statistical steadiness at t ' 120yr.

Notice from Fig. 9 that the q profiles are still stronglynonmonotonic. The ζ and q fields for bmax = 1/16 are

Fig. 9. Zonal-mean PV profiles q(y) for the same set of midlatituderuns as in Fig. 8, with LD = 1200 km and q∗max = 16. The PVis in units of Umax/L = 2.199× 10−5 s−1 and is time-averaged fromt = 115yr to t = 125yr to reduce fluctuations. Biases vary as in Fig. 6.The background PV gradient 4.03×10−12 s−1m−1 makes each profileshear over, with total displacement 1.83 units (4.03× 10−12 s−1m−1

× 107 m ÷ 2.199× 10−5 s−1 = 1.83).

similar to those in Fig. 5, except that the PV step near y =7500km is distinctly weaker and the large cyclone distinctlystronger. Concomitantly, the ζ field is somewhat less zonal,with a stronger and larger cyclonic footprint.

The pattern of results for LD = 1500km is again broadlysimilar, except that realistic quasi-zonal structure is moreeasily disrupted. Vortex interactions reach across some-what greater distances, and the whole system is somewhatcloser to the A2-marginal threshold. This makes the jetsmore liable to large-scale meandering. The most realisticζ fields are obtained for a narrower range of q∗max values,closer to 8 than 16. For further detail see Thomson (2015).

e. Regarding statistical steadiness

As an extreme test of statistical steadiness, we ran ourmain case DI-12-16-16 out to t = 710yr and compareddetails at later times with the 120-yr results shown above.All the mean profiles remain nearly indistinguishable fromthose in Figs. 3, 6, and 7. The q fields are qualitativelyindistinguishable from Fig. 5b. In particular, the largestcyclone and anticyclone have similar sizes. By way of il-lustration a snapshot of the q field at 600yr is shown inThomson (2015), Fig. 4.10.

As a further check, we produced a time series of domain-maximum cyclone strength over the whole time intervalfrom t = 0 to 710yr. The time series showed fairly strongfluctuations, mostly in the range 15–20 in units of Umax/L.Many of these fluctuations are fleeting and are, we think,due to transient Gibbs fringes produced by the high-wave-number filter, during interactions involving the strongestsmall cyclones. The main point, however, is that even this

14

sensitive time series looks statistically steady from t ' 100yr onward, all the way out to t = 710yr.

6. Mechanisms in play

Repeated vortex merging, and the resulting upscale en-ergy cascade, are often taken for granted as central to alltwo-dimensionally turbulent flows. It therefore came as asurprise to us to discover the relative unimportance of thatmechanism in our most realistic cases, in which the stochas-tic forcing is nevertheless strong enough to produce chaoticvortex interactions. One reason is the sparseness of our vor-tex injection scheme, idealizing the intermittency of realJovian moist convection as in Li et al. (2006). This con-trasts with the extremely dense forcing — dense both spa-tially and temporally — used in orthodox beta-turbulencestudies. They use a spatially dense forcing of a specialkind, in order to achieve spectral narrowness (e.g., Srini-vasan and Young 2014, Eq. (20) and Fig. 1f).

Sparse forcing need not, by itself, lead to sparse vortexfields, in a model with numerical dissipation small enoughto allow long vortex lifetimes. For isolated strong vorticesof standard size r0 = 4∆x, injected into favorable shear,and with peak strength 16 times the shear, we find thatlifetimes under numerical dissipation alone are typically ofthe order of years, albeit variable because they depend onthe Robert filter and on bulk advection speeds across thegrid (Thomson 2015, §2.1.1).

For comparison, average injection rates are of the orderof 2 vortex pairs per day in all the cases just described; andso the modest number of small vortices seen in snapshotslike Fig. 5b can be explained only if vortex lifetimes aremore like months than years. Vortex lifetimes are thereforelimited not by numerical dissipation but by the chaoticvortex interactions themselves, as illustrated by the erosionand shredding events seen in the q-field movie discussed inSection 5b. Vortex-merging events do occur, as noted inthat discussion, but require extremely close encounters andare much rarer than vortex erosion events.9

The longest-lived small vortices are the anticyclones inthe zone, shown white in Fig. 5b. Of these, the weakestcome from local injections, corresponding to low values ofthe ramp function ρ(ζ), and the strongest from migrationevents like the two described in Section 5b. Such migra-tion, of small but relatively strong anticyclones from beltto zone, can be attributed to a combination of chaotic,quasi-random walking away from strong-injection sites on

9A reviewer asks for more evidence against vortex merging andupscale energy cascading. Our best answer, beyond saying that thesupplemental movie described in Section 5b shows typical behavior— as also seen in the output from other runs — is to call attention tothe extremely long run described in Section 5e. Even after 710yr ofintegration, vortex merging has made no net headway against vortexerosion and shredding; and the distribution of vortex sizes remainsqualitatively indistinguishable from that in Fig. 5b, with all but oneof the vortices remaining small in comparison with LD.

the one hand, and the so-called “beta-drift” or “beta-gyre”mechanism on the other.

As is well known, and as we have verified by experi-mentation with our model, a single vortex injected into abackground PV gradient will immediately advect the back-ground gradient to produce a pair of opposite-signed PVanomalies on either side, traditionally called “beta-gyres”,whose induced velocity field advects the original vortex to-ward background values closer to its own PV values. Thismigration mechanism weakens as the anomalies wrap upinto a spiral pattern around the original vortex. Neverthe-less, the mechanism appears to have a role in helping ananticyclone to cross a jet, from belt to zone either north-ward or southward. Such an anticyclone typically carrieswith it a wrapped-up cyclonic fringe, which is subsequentlyeroded away.

The range of anticyclone sizes and strengths illustratedin Fig. 5b, up to the largest anticyclone, which is at the bot-tom of the figure near x ∼ 5000km, shows that there musthave been occasional merging events during the preceding120yr. The size of this largest anticyclone can be comparedwith that of the injected anticyclone near x ∼ y ∼ 5000km, As already emphasized, however, most vortex encoun-ters produce erosion rather than merging. The outcomeis statistical steadiness with no further net growth of thelargest anticyclone, even over 710yr, though it remains anopen question whether further such growth occurs in real-istic parameter ranges as yet unexplored, for instance withlarger r0 or injections less sparse.

The inward migration of injected cyclones within a largercyclone plays a role in the buildup and persistence of thelarge cyclones we observe. For instance the example in themovie between trel ' 0.04 and 0.10 (Section 5b) does, onclose inspection, show a local beta-drift mechanism in oper-ation, the neighboring PV contours being weakly twisted inthe sense required, as is especially clear around trel ' 0.10.The corresponding mechanism for large anticyclones seemstoo weak to compete with erosion, in the regimes exploredso far that have realistic ζ structure.

For brevity, no profiles of v′q′ and ∂(v′q′)/∂y are shownhere. Their qualitative characteristics are, however, sim-ple, and easy to see for realistic, statistically steady stateslike that of Fig. 5. For then, broadly speaking, ζ and bare maximal in mid-belt and minimal in mid-zone, hav-ing regard to (3.7) and to quasi-zonal ζ fields like thatin Fig. 5a. In a statistically steady state the right-handside of (2.8) must vanish, after sufficient time-averaging.Thus averaged, ∂(v′q′)/∂y must therefore have the samey-profile as F . Apart from a sign reversal and an additiveconstant, such F profiles are shaped like smoothed versionsof the profiles of A in Fig. 7, in realistic cases, although −Ftends to be more strongly peaked in mid-belt because ofthe dependence (3.7) of b upon ζ. The peak in −F comesfrom injections that are both stronger, as indicated by the

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Fig. 10. Zonal-mean zonal velocity profiles u(y) / m s−1 for a pair ofKelvin-dominated, pure-DI runs DI-12-0.5-64 and DI-12-1-64 withLD = 1200 km and q∗max = 0.5 (darker solid curve) and q∗max = 1.0(lighter solid curve), with bmax = 1/64 and with injection rates t−1

max

increased by a factor 100, as specified in the caption to Table 1.Although it makes little difference to these profiles, they have beentime-averaged from t = 108 to 202 Earth years for consistency withthe profiles of ζ, q, and A shown below, some of which are moresubject to fluctuations within a statistically steady state. The dashedcurve is the deep-jet velocity profile udeep(y)− U0 as before.

mid-belt peak in A(y), and more strongly biased because bitself peaks in mid-belt. The additive constant is required

in order to make∫ 2πL

0F dy = 0, for consistency with (2.6b).

Periodicity and the Taylor identity (2.9) require, more-

over, that∫ 2πL

0v′q′ dy = 0. The v′q′ profile therefore has

to be qualitatively like an additive constant plus the peri-odic part of the indefinite integral of −A(y). Again aftersufficient time-averaging this is a smooth, quasi-sinusoidalcurve going positive to the south and negative to the northof mid-belt, which is consistent with the migration of stronganticyclones from belt to zone already noted. Both v′q′ andu′v′ are upgradient at nearly all latitudes, in pure-DI cases.Explicit diagnostics of the model output confirm this qual-itative picture (Thomson 2015, Figs. 4.12, 4.13). There isno PV-mixing signature in the statistically steady state es-sentially because, as illustrated in Fig. 5b, the mixing hasalready taken place.

The upgradient migration of small anticyclones frombelt to zone produces a clustering of anticyclones, on thelargest possible meridional scale, into regions whose back-ground is already anticyclonic. Energetically, this of courseimplies a direct energy transfer from the smallest to thelargest scales, as distinct from a Kolmogorovian turbulentenergy cascade via intermediate scales. The direct trans-fer is balanced by the quasifrictional F effect, taking thelarge-scale energy back out of the system.

The run DI-12-32-16 (for details again see Thomson2015) develops not only a large cyclone but also a large an-ticyclone, perhaps reminiscent of the real planet’s Ovals, al-

Fig. 11. Zonal-mean interface-elevation profiles ζ(y) for the samepair of Kelvin-dominated runs, with q∗max = 0.5 (darker curve) andq∗max = 1.0 (lighter curve), both time-averaged from t = 108 to 202Earth years as before. Without the time averaging, the q∗max = 1.0curve would be less symmetric and would fluctate noticeably, becauseof vacillations mentioned in the text.

though more symmetrically located within the model zone.There are two more caveats about this run. First and mostimportant, the accompanying ζ structure is footprint-dom-inated, and not quasi-zonal as in Fig. 5a. So we count it asunrealistic. The model’s large anticyclone depends less onbelt-to-zone migration than on strong injections directlyinto the zone.

Second, the two large vortices and their periodic imagesform a vortex street, more precisely a doubly-periodic vor-tex lattice, constrained by the 2:1 geometry of the modeldomain. Without extending our study to a much larger do-main we cannot, therefore, claim to be capturing possiblevortex-street properties in any natural way.

7. The Kelvin mechanism

The Kelvin/CE2/SSST passive-shearing mechanism hasgained increased attention recently (e.g., Srinivasan andYoung 2014, and references therein). It is one of three verydifferent mechanisms for creating and sharpening jets, theother two being the Rhines and PV-mixing mechanisms al-ready mentioned. The Kelvin mechanism is simple to un-derstand, especially when the weak forcing is anisotropicin the same sense as that of our injected vortex pairs, withtheir east-west orientation. Such pairs are immediatelysheared into phase-tilted structures producing upgradientReynolds stresses u′v′. The Taylor identity (2.9) deter-mines the accompanying v′q′ field. That field describes aneddy PV flux that is upgradient at some latitudes y anddowngradient at others and, as indicated by the y deriva-tive in (2.9), involves subtle phase relations sensitive to they-gradients of disturbance amplitude and shearing rates.

To suppress small-scale vortex activity and to allow

16

Fig. 12. Zonal-mean PV profiles q(y) for the same pair of Kelvin-dominated runs, with q∗max = 0.5 (darker solid curve) and q∗max = 1.0(lighter solid curve), both time-averaged from t = 108 to 202 Earthyears as before; see text. The dashed, sinusoidal curve is the initialPV profile.

the Kelvin mechanism to dominate, we must take q∗max

small enough to ensure that injections are almost alwaysweak. Figures 10–12 show statistically steady u, ζ, andq profiles from a pair of pure-DI runs with q∗max = 0.5and 1 (darker and lighter curves respectively), the runs la-beled DI-12-0.5-64 and DI-12-1-64. As before, we takeLD = 1200km. The Kelvin mechanism is so weak that, inorder to see it working and to reach statistical steadiness,we had to reduce bmax to 1/64 and to increase the averageinjection rate by two orders of magnitude, i.e., we had toreduce tmax by a factor 100, as specified in the caption toTable 1.

The jets are indeed sharpened and the jet-core q profilessteepened, in both cases, creating in turn a ζ structure thatis interesting but unrealistic. As Fig. 11 shows, the centralpart of the belt is relatively warm, ζ negative, with onlythe edges cold, ζ positive. The corresponding A profilesfrom (5.1) are given in Fig. 13, showing an inhibition ofconvective activity in mid-belt. No such inhibition is seenon the real planet.

The unrealistic ζ structure arises from the way theKelvin mechanism works in a model with no artificial Ray-leigh friction, corresponding to the low-friction limit foundby Srinivasan and Young (2014), their µ→ 0. Each shearedvortex-pair structure survives as long as it can, throughnearly the whole range of phase-tilt angles. It is only whenthe orientation has become nearly zonal that the structureis destroyed by the model’s high-wavenumber filter. Thusits lifetime is inversely proportional to the local backgroundshear ∂u/∂y. The Reynolds stress u′v′, time-averaged overmany injections, is also, therefore, inversely proportionalto ∂u/∂y, as in Eq. (44) of Srinivasan and Young (2014).So as long as the u profile remains close to its initiallysinusoidal shape while the ζ field remains flat, keeping in-

Fig. 13. Relative moist-convective activity profiles A(y)/Amax from(5.1) for the same pair of Kelvin-dominated runs, with q∗max = 0.5(darker curve) and q∗max = 1.0 (lighter curve), both time-averagedfrom t = 108 to 202 Earth years as before. In the central part ofthe belt convective activity is inhibited, in both runs, for the reasonsexplained in the text.

jection strengths uniform everywhere, the profile of −u′v′has a smooth, positive-valued U shape within the belt, witha broad minimum in mid-belt and a steep increase towardeach jet extremum. In the zone, with the sign of ∂u/∂yreversed, it is +u′v′ rather than −u′v′ that is positive andU-shaped; and there is a very steep transition at each jetextremum producing a sharp, narrow peak in the zonalforce −∂(u′v′)/∂y, positive at the prograde jet and nega-tive at the retrograde.

The jet-sharpening is therefore strongly localized, witha small y-scale � LD. It begins with narrow peaks grow-ing at the extrema of the otherwise-sinusoidal u profile,with |∂u/∂y| reduced everywhere else. The PV profile de-velops correspondingly sharp steps, cut into the sides ofits initially sinusoidal shape. That is, there is localizedjet-sharpening but — in striking contrast with Fig. 3 —weakening rather than strengthening of u at most otherlatitudes y. The thermal-wind tilt of the interface is there-fore, at most other latitudes, opposite to what it was in thecases discussed in Section 5, except within narrow regionsnear the jet peaks. That is the essential reason why thebelt develops a warm, negative-ζ central region with cold,positive-ζ regions only in the outer parts of the belt.

The change in the y-profile of ζ and hence of injectionstrengths then reacts back on the u′v′ profiles, but in arather smooth way that leaves the qualitative pattern un-changed. Indeed, the back-reaction acts as a positive feed-back that reinforces the pattern, because the warm beltcenter weakens the injections there and thus deepens thecentral minimum in the U-shaped profile of −u′v′. That iswhy the lighter curve in Fig. 10, corresponding to the lessweakly forced run, q∗max = 1, shows a u profile more con-spicuously weakened across most of the belt. The resulting

17

thermal-wind tilt further reinforces the central warmth ofthe belt.

The stronger feedback for q∗max = 1 appears to be re-sponsible for the central dip in the q profile seen in thelighter curve in Fig. 12. The central dip gives rise to a weaklong-wave shear instability (though stronger when LD =1500km), because we then have nonmonotonic ∂q/∂y on asmaller y-scale, putting pairs of counterpropagating Rossbywaves within reach of each other. This long-wave insta-bility produces vacillations in the form of weak travelingundulations with zonal wavenumber 1. The vacillationshardly affect the u and q profiles, but show up more clearlyin a time sequence of ζ profiles. The corresponding ζ profilein Fig 11 (lighter curve) has been time-averaged to reducethe effects of these vacillations.

For q∗max = 0.5 we see a weak and entirely different,zonally symmetric, mode of instability that causes spon-taneous y-symmetry breaking in the central region of thebelt, 4000km . y . 6000km. For instance the u profilegiven by the darker solid curve in Fig. 10 shows a tinydeparture from antisymmetry about mid-belt. The darkercurves in Figs. 11 and 13 are more conspicuously asymmet-ric in the central region. Considering a u profile consistingof a constant cyclonic shear plus a small wavy perturba-tion, we see that such a perturbation is zonostrophicallystable — because the abovementioned inverse proportion-ality reduces |u′v′| wherever |∂u/∂y| increases — but “ther-mostrophically unstable” via the feedback from ζ, whichevidently has the opposite effect on |u′v′| and predominatesin this case.

8. Concluding remarks

In view of the Kelvin regime’s lack of realism we re-turn to the model’s realistic, statistically steady, pure-DIregimes that have been our main focus (Sections 5–6). Inthose regimes, not only is u′v′ persistently upgradient, butalso v′q′, at nearly all latitudes y, after sufficient time av-eraging. The small-scale vortex activity produces a persis-tent migration of small anticyclones from belts to zones.It is this upgradient migration that strengthens the up-per jets, as distinct from the inhomogeneous PV mixingthat sharpens them. The importance of migration on thereal planet was suggested by Ingersoll et al. (2000). Inthe model it is mediated by quasi-random walking awayfrom strong-injection sites, via chaotic vortex interactions,in combination with the beta-drift mechanism (Section 6above).

Both mechanisms are entirely different from the Kelvinjet-sharpening mechanism because the latter involves novortex activity, as already emphasized, but only passiveshearing of injected vortex pairs by the background zonalflow u(y). Passive shearing of small-scale anomalies is alsowhat seems to produce the upgradient u′v′ in the real

planet’s cloud-top winds (e.g., Salyk et al. 2006). Yet, inour model at least, as shown in Section 7, the Kelvin mech-anism cannot produce a realistic ζ structure. It thereforecannot, in this model, produce a realistic belt-to-zone con-trast in moist-convective activity.

We suggest therefore that the cloud-top u′v′ on the realplanet must be a relatively shallow phenomenon, whosevertical scale is much smaller than the depth of the weatherlayer. It is most likely, we suggest, to result not from theshearing of tall, columnar vortices resembling the injectedvortices in our model but, rather, from the shearing ofthe real weather-layer’s small scale, baroclinic, fully three-dimensional fluid motions. Such motions, including shallowvortices and the real filamentary moist convection are, ofcourse, outside the scope of any 112 -layer model and notsimply related to PV fields like that of Fig. 5b above.

As is well known, the same conclusions are suggestedby the implausibly large kinetic-energy conversion ratesobtained when the cloud-top u′v′ field is assumed to ex-tend downward, along with ∂u/∂y, throughout the en-tire weather layer. When one vertically integrates cloud-top conversion rates u′v′ ∂u/∂y, whose global average ∼10−4 Wkg−1, then global integration gives numbers “inthe range 4–8% of the total thermal energy emitted byJupiter” (Salyk et al. 2006, and references therein). Suchlarge conversion rates are overwhelmingly improbable, ina low-Mach-number fluid system such as Jupiter’s weatherlayer.

Consistent with these considerations, the model’s flowregimes with realistic ζ structures have conversion ratesu′v′ ∂u/∂y that are still positive, but about two orders ofmagnitude smaller. For instance, in the case examined inSection 5b we find u′v′ ∂u/∂y values that fluctuate arounda time-mean close to 1×10−6 Wkg−1 (Thomson 2015, Fig.4.13). Such values are much more plausible, for the wholeweather layer, than the observed cloud-top values ∼ 10−4

Wkg−1.As well as the model’s success in producing flow re-

gimes with realistically straight weather-layer jets and re-alistic, internally generated belt–zone contrasts in moist-convective activity, we note again the implied restrictionon LD values. Such a restriction holds in the model andalso, very probably, on the real planet. In the model, wefound that realistic behavior requires LD . 1500km at 35◦

latitude. It therefore seems probable that values such asthe 5000km used for all the midlatitude flow fields pre-sented in Li et al. (2006), for instance, are unrealisticallylarge.

Perhaps the weakest aspect of the current model isthe artificial condition (3.10) that we adopted in order toavoid strong-cyclone runaway. As is well known, the realplanet’s large cyclones can be intensely convective, presum-ably because they have cold, high-ζ footprints. No 112 -layervortex-injection scheme can come close to representing the

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three-dimensional reality. An attractive compromise, anda possible way of dispensing with (3.10), might be to in-troduce an eddy viscosity whose value intensifies wheneverand wherever the model’s convective activity intensifies.This might capture some of the dissipative effects of thereal, three-dimensionally turbulent moist convection, whilestill avoiding the use of Rayleigh friction or other such ar-tifice.

A localized, convection-dependent eddy viscosity wouldhave the advantage of, probably, allowing realistic statisti-cally steady states with a simpler vortex-injection scheme,such as that described by Eqs. (3.2)–(3.9) alone. It couldautomatically expand the core sizes, and dilute the peakstrengths, of the strongest injected vortices and thus pre-vent strong-cyclone runaway. As an added bonus it mighteven produce realistic cases in which large anticyclonesform (cf. end of Section 6). Because the real planet’s largeanticyclones are not ubiquitous, there may be a certaindelicacy about the conditions that allow them to form.

Such questions must await future studies. These couldinclude studies using general circulation models in whichthe lower boundary conditions, about which there is somuch uncertainty, are replaced by conditions correspondingto a flexible interface above prescribed deep jets.

Acknowledgements: Emma Boland and Andy Thomp-son kindly helped us with the model code. We also thankthem for useful comments and and/or encouragement alongwith James Cho, Tim Dowling, David Dritschel, LeighFletcher, Boris Galperin, Thomas Gastine, Gary Glatz-maier, Peter Haynes, Xianglei Huang, Andy Ingersoll, ChrisJones, Kirill Kuzanyan, Junjun Liu, Inna Polichtchouk, Pe-ter Read, Tapio Schneider, Richard Scott, Sushil Shetty,Adam Showman, Steve Tobias, Richard Wood, Bill Young,and Roland Young. The paper’s reviewers suggested manyuseful clarifications, for which we are grateful. We alsothank the UK Science and Technology Facilities Councilfor financial support in the form of a research studentship.

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