On the “arrest” of inverse energy cascade and the Rhinesscale

Semion Sukoriansky∗, Nadejda Dikovskaya,Department of Mechanical Engineering

and Perlstone Center for Aeronautical Engineering StudiesBen-Gurion University of the Negev, Beer-Sheva 84105, Israel

Boris GalperinCollege of Marine Science, University of South Florida

140 7th Avenue South, St. Petersburg,FL 33701, USA

Submitted to the Journal of the Atmospheric Sciences on December 17, 2006

Abstract

We revise the notion of the cascade “arrest” in aβ-plane turbulence in the context of continuously forcedflows using both theoretical analysis and numericalsimulations. We demonstrate that the upscale en-ergy propagation cannot be stopped by aβ-effect andcan only be absorbed by friction. A fundamentaldimensional parameter in flows with aβ-effect, theRhines scale,LR, has traditionally been associatedwith the cascade “arrest” or with the scale separat-ing turbulence and Rossby wave dominated spectralranges. We show that rather than being a measureof the inverse cascade arrest,LR is a characteris-tic of different processes in different flow regimes.In unsteady flows,LR can be identified with themoving “energy front” propagating towards the de-creasing wavenumbers. When large-scale energysink is present,β-plane turbulence may attain sev-eral steady-state regimes. Two of these regimes arehighlighted, friction-dominated and zonostrophic. Inthe former,LR does not have any particular signifi-cance, while in the latter, the Rhines scale nearly co-incides with the characteristic length associated withthe large-scale friction. Spectral analysis in the fre-

∗semion@bgu.ac.il

quency domain demonstrates that Rossby waves co-exist with turbulence on all scales including thosemuch smaller thanLR thus indicating that the Rhinesscale cannot be viewed as a crossover between turbu-lence and Rossby wave ranges.

1. Introduction

Terrestrial and planetary circulations are describedby nonlinear equations that support various types ofwaves in the linear limit. The real flows exhibit acomplicated interplay between turbulence and waves.While in a certain range of scales, turbulent scram-bling may overwhelm the wave behavior and leadto the disappearance of the dispersion relation (suchas in the small-scale range of stably stratified flows;see e.g., Sukoriansky et al. (2005)), on other scales,the wave terms cause turbulence anisotropization andemergence of systems with strong wave-turbulenceinteraction. Systems that combine anisotropic turbu-lence and waves exhibit behavior very different fromthat of the classical isotropic and homogeneous tur-bulence (McIntyre 2001). In the context of large-scale atmospheric flows, the interaction between tur-bulence and waves yields dynamically rich macro-

1

turbulence (Held 1999; Schneider 2006), the notionthat underlies diverse phenomena ranging from a lo-cal weather to a global climate. The subtlety of thisinteraction is further highlighted by the fact that dueto the planetary rotation and Taylor-Proudman theo-rem, large-scale flows are quasi-two-dimensionalizedand may be conducive to the development of the in-verse energy cascade (Read 2005). One of the param-eters that characterize macroturbulence is the Rhineswavenumber,kR = (β/2U )1/2, or the Rhines scale,LR ∼ k−1

R , whereU is the r.m.s. fluid velocity andβ is the northward gradient of the Coriolis parameter(Rhines 1975) (in the original paper, this wavenum-ber was denoted bykβ; here,kR is used instead, asthe notationkβ is reserved for another wavenumberto appear later). At this scale, the inverse cascadesupposedly becomes arrested, and the nonlinear tur-bulent behavior is replaced by excitation of linearRossby waves. This scale has also been associatedwith flow reorganization into the bands of alternatingzonal jets, the process known as zonation, and thewidth of the jets has often been scaled withk−1

R . TheRhines scale plays a prominent role in many theoriesof large-scale atmospheric and oceanic circulation(see, e.g., James and Gray (1986); Vallis and Mal-trud (1993); Held and Larichev (1996); Held (1999);Lapeyre and Held (2003); Schneider (2004); LaCasceand Pedlosky (2004); Vallis (2006)) as well as in the-ories of planetary circulations (see, e.g., the review byVasavada and Showman (2005)) and, possibly, evenstellar convection (see, e.g., Miesch (2003)).

The meaning of the Rhines scale is not always clear,however. Even in the barotropic case, differentregimes would arise for continuously forced and de-caying flows, for flows with and without friction, andfor flows in bounded and unbounded domains. Asa result, the Rhines scale may be time-dependent,stationary or entirely obscured by friction. In manystudies, the scaling withk−1

R is implied but the coeffi-cient of proportionality varies in a wide range. Therealso exists a multitude of other scaling parametersthat characterize various aspects of the flow and flowregimes. It is important, therefore, to understand thehierarchy of these parameters and the place ofkR inthis hierarchy. This is precisely the goal of the presentstudy. This goal cannot be achieved without clarifica-tion of the notion of the “arrest” of the inverse energy

cascade by the Rossby wave propagation which, thus,becomes another focal point of this study. Both issueswill be addressed theoretically; the theoretical resultswill be substantiated in numerical simulations.

The paper is organized in the following fashion. Thenext section presents the basics of the theory of two-dimensional (2D) turbulence with aβ-effect and ac-centuates its problematics. Section 3 presents analy-sis of the interaction between inverse energy cascadeof 2D turbulence and Rossby waves and discusses thenotion of the cascade “arrest.” Section 4 elaboratesthis analysis and clarifies the meaning of the Rhinesscale using numerical simulations of barotropic 2Dturbulence on the surface of a rotating sphere. In ad-dition, that section describes the hierarchy of scalingparameters and corresponding flow regimes as well asthe interaction between turbulence and Rossby wavesin different regimes. Finally, section 5 presents dis-cussion and some conclusions.

2. Turbulence and Rossby waves:The basics

The interaction between 2D turbulence and Rossbywaves has been considered in the pioneering studyby Rhines (1975). Starting with the barotropic vortic-ity equation on aβ-plane, he concentrated on mainlyunforced barotropic flows caused by initially closelypacked fields of eddies spreading from a state with aδ-function like spectrum peaked at some wavenum-ber k0. Among the major conclusions of the paper,the following have the most relevance to the presentstudy:

(1) at the wavenumber, denoted askR, at whichthe root-mean-square velocityU is equal to thephase speed of Rossby waves with an averageorientation,cp = β/2k2, there exists a subdivi-sion of the spectrum on turbulence (k > kR) andwave (k < kR) modes;

(2) the expansion of the flow field tokR triggerswave propagation and slowing down, or the “ar-rest” of the cascade to smaller wavenumbers;

2

(3) triad interactions of the modes with wavenum-bers close tokR require simultaneous resonancein wavenumbers and frequencies;

(4) on the average, triad interactions transfer energyto modes with smaller frequencies and smallerwavenumbers causing the anisotropization ofthe flow field. As a result, the energy accumu-lates in modes with small north-south wavenum-bers that correspond to east-west currents, zonaljets, giving rise to the process of zonation;

(5) the width of the zonal jets scales withk−1R ;

(6) in a steadily forced flow, the spectrum is ex-pected to develop a sharp peak atkR and rapidlydecrease fork > kR.

The items (3) and (4) have been confirmed in nu-merous theoretical, numerical and experimental stud-ies (see, e.g., Williams (1975, 1978); Hollowayand Hendershott (1977); Panetta (1993); Vallis andMaltrud (1993); Chekhlov et al. (1996); Cho andPolvani (1996); Nozawa and Yoden (1997); Huangand Robinson (1998); Huang et al. (2001); Read et al.(2004); Vasavada and Showman (2005); Galperinet al. (2006)). A new light on zonation was shed byBalk (2005). Exploring a new invariant for inviscid2D flows with Rossby waves (Balk 1991; Balk andvan Heerden 2006), he showed that the transfer of en-ergy from small to large scales takes place in such away that most of the energy is directed to the zonaljets. The rest of the aforementioned issues, how-ever, remain controversial, particularly when com-pounded with other complicating factors such as thefriction, continuous forcing, effects of the bound-aries, effects of stratification, etc. Let us consider,for example, a realistic system in which the bottomfriction acts like a large-scale drag. One can intro-duce a wavenumber associated with that drag,kfr,as a wavenumber at which the characteristic frictiontime is equal to that of the flow. Ifkfr > kR thenthe frictional processes would forestall the arrest andthe Rhines scale would require modification (Jamesand Gray 1986; Danilov and Gurarie 2002; Smithet al. 2002). In flows with continuous small-scaleforcing at a rateε, the equilibrium between the eddyturnover time and the Rossby wave period selects ananisotropic transition wavenumber with an amplitude

kβ ∝ (β3/ε)1/5 (the proportionality coefficient willbe established later) and the angular distribution re-sembling a dumb-bell (Vallis and Maltrud 1993; Hol-loway 1986; Vallis 2006). The introduction ofkβ

raises a slew of new questions. If there is an “arrest”in the forced flows, which one of the two wavenum-bers,kR andkβ, should be associated with the “ar-rest” scale? At which scale the anticipated in item(2) Rossby wave excitation is triggered? Can Rossbywaves coexist with turbulence on scales intermediatebetweenk−1

R andk−1β ? With regard to item (5) one

may ask which one of these two wavenumbers shouldbe used as a scaling parameter for the width of thezonal jets? Indeed, although the mechanism of zona-tion has by now been solidly established, the scalingof the jets’ width with the Rhines scale has not beenconclusive as the coefficient of proportionality has sofar been elusive. For instance, Hide (1966) used ascale similar tok−1

R for the width of the equatorialjets on Jupiter and Saturn while Williams (1978) andothers applied it to the off-equatorial jets. Conclu-sive scaling withk−1

R has materialized neither in theGrenoble experiment (Read et al. 2004) nor in therecent analysis of the eddy-resolving simulations ofoceanic jets by Richards et al. (2006).

Chekhlov et al. (1996) have scrutinized the notionof the “arrest” of the inverse energy cascade and as-serted that Rossby wave excitation cannot halt thecascade in principle. On the other hand, they haveconfirmed that theβ-term in the vorticity equationimpedes triad interactions hence reducing the effec-tiveness of nonlinear transfers and increasing theircharacteristic time scale. Furthermore, Chekhlovet al. (1996), Smith and Waleffe (1999) and Huanget al. (2001) have shown that the spectrum of aforcedβ-plane turbulence is strongly anisotropic. Onlarge scales, the spectrum of the zonal modes is pro-portional to β2k−5

y . It is important to understandthe synergy between the slowing-up of the nonlin-ear interactions and spectrum steepening. The re-duction in the effectiveness of the triad interactionsis due to the requirement of the simultaneous reso-nance in wavenumbers and frequencies (Rhines 1975,1979; Holloway and Hendershott 1977; Carnevaleand Martin 1982). The ensuing cascade anisotropiza-tion facilitates strong energy flux to the zonal modes(Rhines 1975; Vallis and Maltrud 1993; Chekhlov

3

et al. 1996). Due to the frequency resonance imped-iment, however, the zonal modes cannot effectivelytransfer their energy to other modes. Since the to-tal energy flux must be conserved, the spectral en-ergy density of the zonal modes has to increase. Thesteepening of the zonal spectrum eventually reachestheβ2k−5

y distribution at which the frequency reso-nance condition is relaxed and the zonal modes canagain efficiently exchange energy with other modes.This exchange allows for the zonal modes to disposeof the energy excessive, on the average, of theβ2k−5

y

level and maintain a steady and statistically stableflow regime (Huang et al. 2001). Theβ2k−5

y spec-trum is somewhat reminiscent of the isotropic−5 dis-tribution discussed by Rhines (1975). He has put theexistence of such a sharp slope in doubt, however,because it would imply a strong dependence on thesmallest wavenumbers with the highest energy levelswhich would lead to a strong nonlocality negating thedimensional arguments that facilitated theβ2k−5 de-pendence in the first place. Sukoriansky et al. (2002)have elucidated that in the case of an anisotropicspectrum, the arguments of nonlocality may not beapplicable hence one could expect that a flow regimewith a steepzonal spectrum,

EZ(ky) = CZβ2k−5

y , (1)

may be established (here,CZ is anO(1) coefficient).The stability of such a regime is stipulated by theRayleigh-Kuo criterion (Chekhlov et al. 1996; Huanget al. 2001; McWilliams 2006). Furthermore, sim-ilarly to Lilly (1972) in the case of isotropic, non-rotating flows, Sukoriansky et al. (2002) have arguedthat only the large-scale drag can balance the inversecascade and facilitate the establishment of a steadystate in flows with aβ-effect. They have reproducedthe spectrum (1) in forced, dissipative simulations of2D barotropic turbulence with a linear drag on thesurface of a rotating sphere and showed that in suchflows, kR is close tokfr. They have also found anevidence of the spectrum (1) on all four Solar giantplanets. Later, using the eddy-resolving simulationsby Nakano and Hasumi (2005) of the North Pacificocean, Galperin et al. (2004) have shown that thisspectrum may also be present in the barotropic modeof the subsurface oceanic alternating zonal jets.

Considering a space of parameters that characterizedifferent regimes in barotropic, forced and dissipativeβ-plane turbulence, Galperin et al. (2006) have foundthat in a certain subspace, a flow attains a universalregime with the zonal spectrum (1) andCZ ' 0.5.This subspace is wide enough to include importantlaboratory, terrestrial and planetary flows (Galperinet al. 2006).

Despite the progress made in understanding of var-ious aspects of aβ-plane turbulence, there still re-mains a great deal of confusion regarding the “arrest”of the inverse energy cascade and the Rhines scale.The fact that the interplay between turbulence andRossby waves has never been thoroughly investigatedonly adds to this confusion. The next section gatherssome theoretical arguments that may help to clarifythis perplexity. In section 4 these arguments will besubstantiated via numerical experimentation.

3. Is the cascade really arrested?

The cascade “arrest” is often understood as a transi-tion of a flow character from strongly nonlinear andturbulent to weakly nonlinear and wave-dominatedunder the action of a nondissipative “extra-strain” inthe Bradshaw (1973) terminology, aβ-effect. Thistransition presumably takes place at a scalek−1

R .Such an interpretation applies to unforced flows con-sidered by Rhines (1979) in great detail (see also Ma-jda and Wang (2006), chapters 10-13). Rhines (1979)showed that in unforced flows, both the nonlinearterm and a rate of the energy transfer to large scalesdecrease with time. This tendency can be traced tothe impediment to triad interactions caused by thefrequency resonance condition discussed in the previ-ous section. A “soft” transition from a strongly non-linear to a weakly nonlinear regime takes place at thecrossover wavenumber,kR (Rhines 1979). This in-terpretation of the cascade “arrest” is sometimes ap-plied to forced flows (James and Gray 1986). The ex-tension of the results valid for the unforced dynamics(ε=0) to flows with continuous forcing (ε 6= 0) maybe problematic however. Already at the level of thedimensional analysis, the presence of the non-zeroε and appearance of the non-trivial wavenumberkβ

4

point to significant differences between the two cases.Vallis and Maltrud (1993) have first recognized theimportance of the transitional wavenumber,kβ. Intheir forced - dissipative simulations, Vallis and Mal-trud (1993) observed a “piling up” of the computedenergy spectrum in the vicinity ofkβ. This “pil-ing up” took place in a short range of wavenumbers,however, because the simulation parameters were setsuch thatkβ was close tokR. The spectral steepen-ing was thought to be a result of the dumb-bell be-coming a barrier to the energy flux to larger scales.Chekhlov et al. (1996) investigated a regime withkβ kR. They have also observed the energy pil-ing up in the vicinity ofkβ but they have attributedthis phenomenon to a development of a new regimeof circulation with the steep spectrum (1). Chekhlovet al. (1996) have emphasized that although the to-tal upscale energy flux in forced, undamped flows re-mains constant, the rate of the “energy front” prop-agation decreases due to the steepening of the zonalspectrum. The inverse cascade anisotropization com-bined with the zonal spectrum’s steepening facilitatethe establishment of the zonostrophic regime in flowswith a large-scale drag (Galperin et al. 2006).

Although the frequency resonance constraint im-pedes the triad interactions in both forced and un-forced flows, it does not cause the “arrest” of the in-verse energy cascade which continues to pump en-ergy to ever larger scales at a constant rateε. Thefrequency resonance impediment only facilitates thefunneling of the energy flux into the zonal modesleading to spectral anisotropization (Chekhlov et al.1996) and reorganization of the flow field into alower-dimensional slow manifold decoupled from thefast Rossby waves (Smith and Lee 2005). Thismanifold manifests itself as a system of quasi-one-dimensional alternating zonal jets. The threshold ofthe spectral anisotropization is characterized by thewavenumberkβ while the width of the zonal jetsis determined by the large-scale friction (Sukorian-sky et al. 2002). It is important to emphasize thatforced β-plane turbulence never collapses to a lin-ear state since the nonlinearity is the very factor thatsustains the slow manifold. It is clear, therefore,that a β-effect cannot halt the inverse energy cas-cade (Chekhlov et al. 1996; Sukoriansky et al. 2002;Galperin et al. 2006).

In the remainder of this section, we shall elabo-rate various scaling relationships pertinent to forcedanisotropic turbulence with aβ-effect. These rela-tionships will be used to further elucidate the absenceof the cascade arrest. The flows will be assumedto contain a stationary random forcing maintainingan inverse energy cascade at a constant rateε. Thiscase has most relevance to the real world large-scaleterrestrial and planetary circulations (Galperin et al.2006). The importance of the parameterε is under-scored by the conjecture that the diffusion coefficientof the poleward heat transport strongly depends onit (Held and Larichev 1996; Held 1999; Lapeyre andHeld 2003).

Two classes of flows will be considered, those with-out and with the action of the large-scale drag. Thefirst class is rather unrealistic because, firstly, somekind of a drag is always present in all real flows,and, secondly, the large-scale energy condensationwould eventually distort flow configuration in long-term integrations (Smith and Yakhot 1993, 1994).Such flows are important, however, for understand-ing of the dynamics of the transients and mechanismsthat lead to the establishment of observable long-termpatterns. These patterns are represented in the secondclass of flows.

Barotropic, small-scale forced, anisotropic 2D turbu-lence on aβ-plane wherex andy are directed east-ward and northward, respectively, is described by thevorticity equation,

∂ζ

∂t+∂

(∇−2ζ, ζ

)

∂(x, y)+ β

∂

∂x

(∇−2ζ

)= D + ξ, (2)

whereζ is the vorticity,ξ is the forcing, andD is thefriction that includes a small-scale and a large-scalecomponents. In numerical simulations, the small-scale part ofD is usually represented by a hypervis-cous term while the large-scale drag is linear. As wasshown by Sukoriansky et al. (1999) and Sukorianskyet al. (2002), a linear drag, being a physically plausi-ble mechanism of the large-scale energy damping dueto the bottom friction, is a “soft” process which intro-duces only a small distortion to the inverse energycascade that does not lead to a spurious accumulationof energy in the lowest resolved modes. The con-stantβ in (2) is the northward gradient of the Corio-

5

lis parameter,f , f = f0 + βy, wheref0 is a refer-ence value off . The forcingξ, concentrated aroundsome high wavenumberkξ and assumed to be ran-dom, zero-mean, Gaussian and white noise in time,supplies energy to the system at a constant rate. Partof this energy sustains the inverse cascade with therateεwhile the other part is lost on scalesk > kξ dueto the small-scale dissipation.

On its left-hand side, Eq. (2) contains the nonlin-ear and theβ terms which are dominant in differ-ent ranges of scales. Here, the balance betweenthese terms will be analyzed using the notion of thecharacteristic time scale rather than the characteris-tic velocity as in Rhines (1975). The relatively highwave-number modes are nearly isotropic and obeythe classical KBK theory of 2D turbulence in the en-ergy range. For those modes, a conventionally de-fined eddy turnover time,τt(k), is a ratio of the lo-cal length scale,k−1, and the local velocity scale,U (k), which can be estimated from the KBK en-ergy spectrumE(k), U (k) ∝ [kE(k)]1/2, yieldingτt(k) = [k3E(k)]−1/2, where

E(k) = CKε2/3k−5/3. (3)

Here,CK ' 6 is the Kolmogorov-Kraichnan con-stant. Equatingτt and the Rossby wave period,τRW (k) = −k2/βkx, one can find a crossoverwavenumber for theβ-effect induced anisotropy(Vallis and Maltrud 1993),

kβ ∝ (β3/ε)1/5. (4)

Note that for a stationaryε, kβ is also stationary evenif the flow itself is not in a steady state. Other charac-teristic wavenumbers for both unsteady and steady-state flows arekξ andkd, where the latter is associ-ated with the small-scale dissipation. For steady-stateflows, this group should also include the large-scalefriction wavenumber,kfr.

Simulations by Chekhlov et al. (1996), Smith andWaleffe (1999) and Huang et al. (2001) of the evolv-ing turbulence with aβ-effect have shown that asthe energy propagates to wavenumbers smaller thankβ, the nature of the flow and its spectral character-istics change markedly. At a wavenumber close tokβ, the zonal spectrum,EZ(ky), undergoes reorga-nization and attains a steep distribution (1) while the

Figure 1: Normalized spectral energy transfer,TE(k|kc)/max|TE(k|kc)|, for kc = 50.

nonzonal, or the residual spectrum largely preservesthe shape given by the KBK law, Eq. (3).

The anisotropization of the inverse cascade hasbeen demonstrated using the results of direct nu-merical simulation (DNS) of aβ-plane turbulenceby Chekhlov et al. (1996) who considered the en-ergy transfer function,TE(k|kc), describing thek-dependent spectral energy flow from all modes withk > kc to a given modek, k < kc. The modekc is set arbitrarily but such thatkc < kξ. Figure1 showsTE(k|kc) for a simulation withkc = 50(which was smaller thankβ = 79). The hole in themiddle of the Fig. 1 indicates that the “energy front”had not yet reached those wavenumbers. There is astriking difference between energy transfer to zonalmodeskx → 0 and the rest of the modes. One cansee that most of the energy flux is directed towardsthe zonal modes; the trend that dramatically increaseswith decreasingk. The spectral energy transfer hasalso been calculated by Read and co-workers usingthe data from the Grenoble experiment (this issue).In qualitative similarity to Fig. 1, they obtained an

6

anisotropic distribution ofTE(k|kc) peaking at smallky.

As a result of the spectral anisotropization, the zonalenergy on the large scales exceeds its nonzonal coun-terpart, and the total energy,Etot(t), can be estimatedbased upon the zonal energy spectrum only,

Etot(t) 'U2(t)

2'

kd∑

km

EZ(ky) ∝ β2k−4m (t), (5)

whereU (t) is the rms of the zonal velocity, andkm(t)is the smallest wavenumber with the spectrum (1) attime t. As follows from (5),

km(t) ∝ [β/2U (t)]1/2 = kR(t), (6)

and so one concludes that in unsteady flows with thezonal spectrum (1), the Rhines wavenumber,kR, istime-dependent and provides the location of the mov-ing “energy front.” Clearly, Eq. (6) does not implythe arrest of the inverse cascade; in an unboundeddomain, after sufficiently long time,kR can attain anarbitrarily small value. The unsteady flow under con-sideration highlights the difference betweenkR andkβ; not onlykR < kβ (and it may be thatkR kβ),but alsokR is time-dependent whilekβ is stationary.

Combining (5) with the linear trendEtot(t) = εt, oneobtains

km(t) ∝ kR(t) ∝ β1/2(εt)−1/4. (7)

A similar estimate of the evolution of the moving“energy front” for the classical KBK turbulence withno rotation yields the “−3/2” law (Rose and Sulem1978),

k0m(t) ∝ ε−1/2t−3/2. (8)

Comparing the “−1/4” and “−3/2” evolution laws,Eqs. (7) and (8), respectively, one concludes thatrather thanhalting the inverse cascade, aβ-effectonly slows down the up-scale march of the energyfront. The deceleration of the energy front propaga-tion is caused by the prolongation of the time stretchnecessary to saturate the zonal modes to their energylevels specified by the spectrum (1) [instead of (3)]at the same rate of the energy transfer,ε. As a con-sequence, numerical simulations on aβ-plane or onthe surface of a rotating sphere require much longer

integration time than similar simulations with no ro-tation; this tendency is reflected in the “−1/4 law” (7).

When a large-scale drag is present, an ensuingsteady-state flow regime is determined by the ratiosbetweenkfr, kβ, kd, andkξ (Galperin et al. 2006).There exists a certain subspace of these parametersin which a flow undergoes zonation and develops auniversal stationary regime whose anisotropic spec-trum in the inertial range (kfr < k < kβ) approachesa distribution given by Eqs. (1) and (3). The sharpzonal spectrum is better pronounced forkfr kβ;it also extends to the regionk > kβ to its intersec-tion with the KBK, isotropic, modalk−8/3 spectrumas explained in Sukoriansky et al. (2002). The pa-rameter subspace of this regime is delineated by threeconditions: (1) the inertial range is sufficiently large;(2) the forcing operates on scales not impacted uponby theβ-effect; and (3) the frictional wavenumber islarge enough [kfr & 4(2π/L), L being the systemsize] to avoid the large-scale energy condensation(Smith and Yakhot 1993, 1994). For convenience,Galperin et al. (2006) have coined this regime zonos-trophic. Sukoriansky et al. (2002) have shown thatfor this regime, the friction wavenumber,kfr, is the“final destination” of the moving energy front givenby Eq. (7), and thuskR ∼ kfr. The latter relation-ship will be confirmed in the next section. Similarlyto the unsteady regime,kR is not a scale of inversecascade termination by aβ-effect; the advance of theenergy front can only be stopped by the large-scaledrag. Outside the zonostrophic regime, the flow de-pends on some additional parameters andkR ceasesto be the “final destination” of the moving energyfront; the latter is still associated withkfr, however.

From the previous studies and the scaling argumentsin this section, we draw two main conclusions: (1) inunsteady flows, the marching in time of the moving“energy front” slows down from the “−3/2” to “−1/4”law upon crossing ofkβ and the front’s location canbe related tokR; (2) in steady flows,kR can be iden-tified with kfr in the zonostrophic regime. These con-clusions will be confirmed in numerical simulationsof barotropic 2D turbulence on the surface of a ro-tating sphere described in the next section. In addi-tion, using the frequency analysis in Fourier space,an analysis of the interplay between turbulence and

7

Rossby waves will also be presented.

4. Turbulence and Rossby waves:Results of numerical simula-tions

This section describes numerical experimentationwith two-dimensional turbulent flows on the surfaceof a rotating sphere. The flow is governed by thebarotropic vorticity equation,

∂ζ

∂t= −J(ψ, ζ + f) + ν∇2pζ − λζ + ξ, (9)

whereζ is the vorticity;ψ is the stream function de-fined as∇2ψ = ζ; f = 2Ω sin θ is the Coriolis pa-rameter (or the “planetary vorticity”);Ω is the angu-lar velocity of the sphere’s rotation;ν is the hyper-viscosity coefficient;p is the power of the hypervis-cous operator (p was either 4 or 8 in this study);λ isthe linear friction coefficient, andξ is the small-scaleforcing, respectively. In addition,J(ψ, ζ + f) is theJacobian representing the nonlinear term,J(A,B) =(R2 cos θ)−1(AφBθ −AθBφ), andR is the sphere’sradius. For convenience, the unit of length is set tobe the radius of the sphere. In these units,R = 1 andwill be omitted in the forthcoming derivations. Thenatural time scale isT = Ω−1 such thatΩ = 1. Toexplore the explicit dependency onΩ, in some of theexperimentsT was kept fixed whileΩ varied.

On the surface of a unit sphere, the stream functioncan be represented via the spherical harmonics de-composition,

Ψ(µ, φ, t) =N∑

n=1

n∑

m=−n

ψmn (t)Y m

n (µ, φ), (10)

whereY mn (µ, φ) are the spherical harmonics (the as-

sociated Legendre polynomials);µ = sin θ; φ is thelongitude;θ is the latitude;n andm are the total andzonal wavenumbers, respectively, andN is the totaltruncation wavenumber. Conventionally, the indecesn,m andN are nondimensional. However, when ap-pear in equations below, thewavenumbers n andmhave the dimension of the inverse length. Since in

our unitsR = 1, we shall not differentiate betweenthe indeces and wavenumbers.

In the unforced, nondissipative, linear limit, Eq. (9)gives rise to Rossby waves whose dispersion relationis

ωm,n = −2βm

n(n+ 1), (11)

whereβ = Ω/R. We shall retainβ in the forthcom-ing equations in order to preserve the transparencybetween the cases of aβ-plane and rotating sphere.

The wavenumbernβ ∝ (β3/ε)1/5 is the analogueof kβ. In the forced, dissipative regime, the balancebetween the small-scale forcing and large-scale dissi-pation gives rise to a steady state. The kinetic energyspectrum can be calculated as

E(n) =n(n + 1)

4

n∑

m=−n

〈|ψmn |2〉, (12)

where the brackets indicate an ensemble or time av-erage (Boer 1983; Boer and Shepherd 1983). Thisspectrum can be represented as a sum of the zonal andresidual spectra,E(n) = EZ(n)+ER(n), where thezonal spectrum,EZ(n), corresponds to the addendwith m = 0. In unsteady flows (Huang et al. 2001)and in the inertial range of the zonostrophic regime(Sukoriansky et al. 2002; Galperin et al. 2006) thesespectra are similar to those inβ-plane turbulence,

EZ(n) = CZβ2n−5, CZ ∼ 0.5, (13a)

ER(n) = CKε2/3n−5/3, CK ∼ 5 to 6. (13b)

The zonal and residual spectra intersect at the transi-tional wavenumber,

nβ =(CZ

CK

)3/10 (β3

ε

)1/5

' 0.5(β3

ε

)1/5

.

(14)

To investigate the energy front propagation in un-steady flows, a series of long-term simulations wasperformed using Eq. (9) decomposed in spheri-cal harmonics according to Eq. (10). A Gaussiangrid was employed with resolutions of 400×200 and720×360 nodes and2/3 dealiasing rule (rhomboidaltruncations R133 and R240, respectively). The hy-perviscosity coefficient,ν, was chosen such as to ef-fectively suppress the enstrophy range. The Gaussian

8

Table 1: Parameters of the four runs shown in Fig. 4.

Experiment Ω ε nβ

1 1 1.44× 10−8 17.6

2 1 2.58× 10−9 24.8

3 2.5 1.42× 10−8 30.6

4 3 1.44× 10−8 34.1

random forcing was distributed amongst all modesnξ = 83, 84, 85 andnξ = 99, 100, 101 for R133 andR240, respectively; it had a constant variance and wasuncorrelated in time and between the modes.

The values of the parameters used in unsteady sim-ulations are summarized in Table 1. In total, fournumerical experiments were performed. AlthoughΩand ε varied in a wide range in these experiments,their values were set such as to ensurenβ < nξ.For statistical analysis of the results, ensemble av-eraging over 80 to 110 independent realizations wasemployed.

At first, the energy evolution was investigated. It iswell known that without rotation, the total energy ofthe flow increases linearly with time,Etot(t) = εt.We wanted to verify that, firstly, the linear trend ispreserved in the case of nonzero rotation and, sec-ondly, the zonal and nonzonal total energy compo-nents,EZ

tot(t) andERtot(t), respectively, would fol-

low a similar trend [here,Etot(t) = EZtot(t) +

ERtot(t); ER

tot(t) contains both the eddy and wave en-ergies]. Figure 2 shows the evolution of the total en-ergy and its components for Experiment 1 in a sim-ulation of the duration of 10,000 planetary days (forExperiment 1, 1day =2πT ). Initially, the zonal en-ergy is very small and the entire energy is mostly con-centrated in the nonzonal component. That compo-nent at first grows linearly, then temporarily preservesan approximately constant value and then returns toa linear growth while remaining considerably smallerthanEtot(t). At those later times, the total energy ismostly contained in the zonal component. Generally,after initial restructuring, bothEZ

tot(t) andERtot(t)

grow linearly in time for as long as the flow evolutionremains unobstructed by the action of a large-scale

102

103

104

10−7

10−6

10−5

10−4

10−3

t/(2π)

Etot

EtotZ

EtotR

Etot

Figure 2: Evolution of the total energy,Etot(t) (thicksolid line), and its zonal (dashed line) and non-zonal (dashed-dotted line) components,EZ

tot(t) andER

tot(t), respectively, in Experiment 1. A triangle onEtot(t) shows the time required for the energy frontto reach the largest scales in the system in the casewhenβ = 0 but ε is the same as in Experiment 1.

9

drag or domain boundaries.

We have conducted an additional simulation featur-ingβ=0 and the sameε as the one used in Experiment1. The evolution of the total energy in that simulationwas indistinguishable from the case withβ 6= 0. Theonly difference was that in the caseβ = 0, the “en-ergy front” required much shorter time to reach thelargest scales of the system. The total energy at thatmoment is marked by a triangle in Fig. 2. Clearly, theshortening of the evolution time is a result of the fastfront propagation described by Eq. (8). Forβ 6= 0,the system can accumulate significantly higher en-ergy; according to Eq. (5), this energy scales withβ2 and is independent ofε. This increased energeticcapacity of the system is a direct result of the spec-tral anisotropization and steepening of the zonal spec-trum according to Eq. (1).

The behavior of the total energy components exhib-ited in Fig. 2 requires some clarification. Furtherinsight comes from the consideration of the evolutionof the zonal and residual energy spectra for the sameExperiment 1 shown in Fig. 3. Driven by the inversecascade, the spectrum expands towards the smallwavenumber end. Until the transitional wavenum-bernβ is reached, the “energy front”,nm(t), markedby the black dots in Fig. 3a, can be identified as awavenumber with the highest energy. Forn < nm,the spectral energy density rapidly decreases. As inclassical 2D turbulence, the ensemble-averaged spec-tra at the wavenumbers swept bynm remain in aquasi-steady state. The approximate observance ofthe classical isotropic KBK distribution (13b) is anindication that aβ-effect does not yet have a stronginfluence on these scales. This behavior changeswhennm becomes smaller thannβ. The spectrum be-gins to steepen up eventually attaining the level dic-tated by Eq. (13a). As discussed earlier, the satu-ration of the spectrum at this level is a slow processduring which the energy transfer continues not onlyto the modenm but also to the modes withn < nm.As a result, along with the slope (13a), the energyspectrum also forms a plateau forn . nm. Let usemphasize that the moving “energy front”nm is notonly a transitional wavenumber between the−5 slopeand a plateau but it also corresponds to the number ofthe zonal jets (Chekhlov et al. 1996).

100

101

102

10−8

10−6

10−4

100

101

102

10−8

10−6

10−4

ER(n)

n

CKε2/3 n−5/3

nβ

CZβ2n−58.9 17.8

3.6

2.11.2

0.5

a

b

E(n)

n

nξ

Figure 3: Total (a) and nonzonal (b) energy spectraat different times in Experiment 1 of unsteady sim-ulations (λ=0). The spectra are marked by the to-tal energy (times105) accumulated in the flow fieldfrom the beginning of simulations. Black dots showthe location of the “energy front”nm. The transitionfrom the−5/3 to−5 slope aroundnβ is clearly visiblein plate a. Before the transition, the spectral energydensity rapidly decreases forn < nm. After the tran-sition, the energy accumulates at the modesn < nm

facilitating the emergence of the plateau.

10

5 10 20

10

20

40

80

1 2

nR

=(8εt/β2)−1/4

nm

34ε−1/2t−3/2

3

nm

=1.7nR

4

Figure 4: The moving “energy front,”nm, in un-steady simulations. The transition from the “−3/2“to the “−1/4” evolution law in the vicinity of the tran-sitional wavenumbersnβ located at the intersectionsof the respective dashed lines is clearly visible forall four simulations. The figure demonstrates the ab-sence of the halting wavenumber for the inverse en-ergy cascade; the ”energy front“ can penetrate to thesmallest wavenumbers available in the system.

By comparing Figs. 2 and 3 we establish that theaforementioned change in the behaviors ofEZ

tot(t)andER

tot(t) is concurrent with the restructuring of thespectra from isotropic, KBK to strongly anisotropicdistributions upon crossingnβ. Although at largetimes,ER

tot is relatively small, the nonzonal modesplay a crucial role in maintaining the zonal flows asthey effectively preserve the upscale energy cascade.

Note that the evolution of the nonzonal spectrumER(n) shown in Fig. 3b provides practically no in-formation on a flow field transformation under the ac-tion of aβ-effect. Indeed,ER(n) retains an approx-imate KBK distribution, Eq. (13b), with nearly thesame value of the coefficientCK for bothnβ < nm

and nβ > nm. Only the zonal spectrum,EZ(n),reflects the anisotropization by attaining a steeperslope.

For unsteady flows, the linear growth of the total en-ergy,Etot = εt, yields the following expression for

the Rhines scale:

n−1R (t) =

(2Uβ

)1/2

'(

8εtβ2

)1/4

, (15)

which shows thatnR is time-dependent,nR ∝ t−1/4.If the evolution of the moving “energy front,”nm(t),is plotted againstnR(t), then one expects to detecttwo different power laws for the dependence ofnm

onnR. At the early stages, whilenm(t) > nβ and theeffect of theβ-term is not yet felt,nm would evolveaccording to the “−3/2” law, Eq. (8). Upon reach-ing nβ, as follows from Eqs. (7) and (15),nm andnR become proportional to each other. As estimatedfrom the data, the numerical coefficient betweennm

andnR is approximately equal to 1.7. The “−3/2”and “−1/4” evolution laws and the transition betweenthem are confirmed in Fig. 4 for the four experimentssummarized in Table 1. The initial evolution is in-deed according to the “−3/2” law which changes tothe “−1/4” law whennm approachesnβ. Clearly, thechange in the exponent of the evolution law is con-sistent with the change in the spectral slope shown inFig. 3a. The transition between the two regimes is acomplicated and non-universal process further exac-erbated by the discrete character of the front propaga-tion. Furthermore, the location ofnm in the transitionarea may be somewhat ambiguous as evident fromFig. 3a. Nevertheless, the transition wavenumbersare quite close to the values ofnβ in Table 1 for allfour runs. Figure 4 clearly demonstrates the absenceof inverse cascade termination. Although aβ-effectis strong in the rangenm < nβ, the moving energyfront, nm, always reaches the lowest wavenumbersavailable in the system.

Consider now a steady-state flow with a linear large-scale drag. By its very nature, such a flow is freeof moving fronts; the march of the moving energyfront had been damped by the drag. The final, sta-tionary destination of the moving front,nm, will nowbe identified with the friction wavenumbernfr whosemagnitude depends on the drag coefficient,λ, and,possibly, also onβ andε such that generally, one canwrite nfr = f(λ, β, ε). According to the Bucking-ham’sΠ-theorem, this system is fully characterizedby two functionally related non-dimensional parame-

11

0 5 10 15 20 250

5

10

15

nR

nβ

Figure 5: Possible flow regimes in 2D turbulencewith a β-effect in the space of parametersnR andnβ. The subspace of the friction-dominated regimes(unfilled squares) occupies the area to the left of theupper dashed line while the subspace of the zonos-trophic regime (filled dots ) outlined by the chain in-equality (17) is confined to the area between the mid-dle, bottom and vertical dashed lines for the R133experiments. For higher resolution, the vertical linemoves to larger values ofnβ.

ters, for instance,

nfr

nR= f

(nβ

nR

). (16)

The functional dependency varies according to aflow regime. Using an extensive series of long-termsteady-state simulations of 2D turbulence with aβ-effect, we have derived a detailed classification ofpossible regimes in the space of parametersnβ andnR (Galperin et al. 2006). These regimes are delin-eated by the dashed lines in Fig. 5. Two of theseregimes, zonostrophic and friction-dominated, are ofparticular interest to the present study. The formeris confined to the subspace outlined by the chain in-equality

nξ & 4nβ & 8nR & 30. (17)

This inequality instructs us that (1) the forcing actson scales only weakly impacted by aβ-effect; (2)there exists a meaningful zonostrophic inertial rangewhose width is defined by the rationβ/nR, and (3)there exists a sufficient number of the lowest modesto resolve the large-scale friction processes and avoidthe large-scale condensation.

The friction-dominated or, briefly, friction regime oc-cupies the subspace to the left of the top dashed linein Fig. 5 and is delineated by the inequality

nβ

nR. 1.5. (18)

In the friction regime, the zonostrophic inertial rangeis practically non-existent; only the classical KBK in-ertial range may survive forn > nfr.

Let us consider now how Eq. (16) changes in dif-ferent regimes. In the zonostrophic regime, whennβ/nR 1, the dependence on this parameterceases (practically, this dependence becomes weakalready atnβ/nR & 2) and one obtainsnfr/nR 'const. Let us calculate the numerical value of theproportionality constant. Sinceε denotes the netpart of the energy input that goes to the inverse cas-cade, the spectrally-integrated energy balance equa-tion reads

Etot =ε

2λ. (19)

Using this relationship, we obtain the Rhines

12

0 5 10 150

5

10

15

20

nfr

nR

=(β2λ/4ε)1/4

nfr

=1.2nR

Figure 6: The friction scale,nfr, determined asthe intersection of the plateau and the−5 rangesin the energy spectrum, vs. the Rhines scale,nR,in simulations of the zonostrophic (filled dots) andmarginally-zonostrophic (empty dots) regimes with alinear large-scale drag.

wavenumber in the form

nR =(λβ2

4ε

)1/4

. (20)

On the other hand, since the energy spectrum is com-prised of a plateau forn < nfr and the inertialpart well approximated by Eq. (13a) forn > nfr,the total kinetic energy can be calculated by inte-grating this composite spectrum with respect ton.The integration yields an approximate relationship,Etot ' (5/4)CZ(Ω/R)2n−4

fr (Galperin et al. 2001).Using this expression and Eq. (20), one can relatenfr

to nR,

nfr = (10CZ)1/4

(λβ2

4ε

)1/4

' 1.5nR. (21)

The coefficient5/4 in the expression forEtot abovewas stipulated by the assumption that the energyspectrum atn < nfr is flat. This assumption is onlyan approximation; in real flows, this coefficient mayneed to be adjusted.

Equation (21) was verified in a series of simula-tions of the zonostrophic (filled dots in Fig. 5) and

0 10 20 30 400

10

20

30

40n

fr

10(nR

/nβ)5nR

Figure 7: The correlation between the actual and the-oretical values ofnfr in simulations of the large-scaledrag dominated 2D turbulence with aβ-effect.

marginally-zonostrophic (unfield triangles adjacentto the zonostrophic region) regimes with widely vary-ing values ofλ (5 × 10−4 ≤ λ ≤ 2.5 × 10−3), ε(3.3×10−9 ≤ ε ≤ 9×10−8), andΩ (0.3 ≤ Ω ≤ 2).Note that due to the steep zonal spectrum, zonos-trophic flows have low variability and require long-term integrations to assemble records sufficient forstatistical analysis. Galperin and Sukoriansky (2005)have estimated that a record length of up to 100τafter attaining the steady state,τ = (2λ)−1 beingthe time scale associated with the large-scale drag,would be adequate for the spectral analysis. Accord-ingly, our steady-state simulations were of durationof about 60 to 100τ . The results of these simula-tions are summarized in Fig. 6. Note that the valueof the coefficient in the correlation betweennfr andnR in Eq. (21) had to be adjusted to 1.2. As one cansee, the linear relationship betweennR andnfr holdsfaithfully. One concludes, therefore, that, firstly, sim-ilarly to the unsteady case, aβ-effect does not halt theinverse energy cascade which is now damped by fric-tion, and, secondly, in the zonostrophic regime, theRhines scale is, in fact, a scale that characterizes theeffect of the large-scale drag.

In the friction regime, the drag may be so strong thatits impact extends to the scales where the flow is un-affected byβ (this aspect was considered by Danilov

13

and Gurarie (2002); Smith et al. (2002); James andGray (1986)). The functional relationship betweennfr/nR andnβ/nR can then be established using theassumption that the energy spectrum is isotropic anddescribed by the KBK distribution, Eq. (3). The totalkinetic energy in this case is given by the expressionEtot ' 3

2CKε2/3n

−2/3fr , wherenfr is the wavenum-

ber with the maximum energy. Using the energy bal-ance equation, Eq. (19), we obtain

nfr = (3CK)3/2(λ3/ε)1/2. (22)

Using Eq. (22) and the definitions ofnβ, Eq. (14),andnR, Eq. (20), we further find

nfr = (12CZ)3/2

(nR

nβ

)5

nR

' 14.7(nR

nβ

)5

nR. (23)

This relationship is very different from the one de-rived for the zonostrophic regime, Eq. (21), becausenow nfr depends onε and is independent ofβ suchthat the proportionality betweennfr andnR is lost.Summarizing, we note that the dependence ofnfr onnR given by Eq. (16) has two limiting cases pertain-ing to the zonostrophic and friction regimes. In theformer,nfr andnR are related linearly and are gener-ally quite close while in the latter they diverge fromeach other by a stiff power law. In the intermedi-ate cases, the relationship betweennfr onnR can beexpected to be nonlinear and complicated which ex-plains why the scaling of the zonal jets’ width withn−1

R has been so evasive in diverse flows.

Equation (23) has been validated in numerical sim-ulations with large values of the drag coefficient,λ,which was chosen such thatnβ/nR . 1.5. Thesesimulations are shown as unfilled squares in Fig. 5.Figure 7 shows a good agreement between the actualvalues ofnfr and those calculated from Eq. (23) withthe coefficient adjusted to 10.

There still remains a question about the proper scal-ing of the width of the zonal jets which in some stud-ies was found to be close ton−1

R . Note that the scalingof the jets’ width only makes sense when the jets haveapproximately homogeneous spatial distribution andtheir width can be estimated from the number of the

0 4 8 120

4

8

12n

jet

nR

Figure 8: The correlation between the number of thezonal jets,njet, estimated from the velocity profile inthe physical space, andnR obtained from numericalsimulations of the zonostrophic regime.

jets,njet. It is of interest, therefore, to comparenjet

with nfr andnR.

Figure 8, derived from simulations of the zonos-trophic regime, indicates thatnjet ' nR. The spreadof the data points is caused by the discreteness ofnjet.In the range4 ≤ njet ≤ 10, for example, unavoidableunity deviations appear like a 10-25% noise. In fact,given this uncertainty,nR provides a faithful estima-tion of the number of the zonal jets. Using the corre-lation betweennfr andnR shown in Fig. 6, one con-cludes that bothn−1

fr andn−1R are appropriate scal-

ing parameters for the jets’ width in the zonostrophicregime. We have also attempted to scalenjet with nR

calculated using the nonzonal velocities only. In thatcase, the correlation betweennjet andnR was con-siderably worse than that in Fig. 8 and, therefore, itis not shown here.

We now want to analyze the interplay between tur-bulence and Rossby waves in order to determinewhethernR (or nβ) separate turbulence and Rossbywave regions or turbulence and waves coexist in somerange of scales. To answer this question, it is instruc-tive to analyze the Fourier-transform of the two-time

14

−0.4

0.2−0.2

5

−0.2

5

0.20−0.2

0.20−0.2

−0.2

5

−0.2

1

−0.4

1

−0.2

5

0.20−0.2

−0.2

5

−0.4

0.05

−0.2

2

−0.2

5

0.20−0.2

−0.4

0.01

−0.2

0.5

−0.2

5

0.20−0.2

−0.4

0.02

n/nβ

m/n 0.41 0.82 1.64 3.28

0.2

0.4

0.6

0.8

1.0

n 5 10 20 40

U(ω

,m,n

)

ω/Ω

0.2

0.2

0.2

0.2

0.2

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

Figure 9: The velocity autocorrelation function,U (ω,m, n)× 107, for the friction dominated regime;nR ' 9, nβ ' 12. The triangles correspond to thelinear dispersion relation (11).

velocity autocorrelation function given by

U (ω,m, n) =n(n+ 1)

4〈|ψm

n (ω)|2〉, (24)

whereψmn (ω) is a time Fourier transformed spectral

coefficientψmn (t) defined in Eq. (10). In 2D turbu-

lence without Rossby waves,U (ω,m, n) would beexpected to have a symmetric bell shape around thezero frequency. When the waves are present, theycauseU (ω,m, n) to develop spikes at frequenciescorresponding to the dispersion relation. The corre-lation functionU (ω,m, n) is therefore a convenienttool to establish and/or diagnose the dispersion rela-tions in data. A similar approach based upon the anal-ysis of the autocorrelation function of the sea surfaceelevation obtained from the satellite altimetry hasbeen used by other researchers (see, e.g., Glazmanand Cheng (1999); Glazman and Weichman (2005))to diagnose the baroclinic Rossby waves in the ocean.The results of our analysis are shown in Figs. 9and 10 for the friction dominated and zonostrophicregimes, respectively. In the former case, for smallwavenumbers, Rossby waves produce sharp spikesalmost exactly corresponding to the linear dispersionrelation (11). Similar spikes also appear at higherwavenumbers although they are not as sharp becausethey are progressively broadened by turbulence. As

evident from Fig. 9, Rossby waves are present atnfar exceeding bothnR andnβ indicating that there isno separation between Rossby waves and turbulence;both processes coexist on virtually all scales. Figure10 depicts a more complicated wave-turbulence in-teraction. As in Fig. 9, the spectral spikes at smalln are sharp but now they begin to show signs of astrong wave-turbulence interaction. At largern, dueto a stronger turbulence, the spikes broaden and be-come displaced from the values stipulated by the lin-ear dispersion relationship. This behavior is indica-tive of a complex nature of turbulence - wave inter-action in the case of the zonostrophic regime whichmay manifest itself in the development of the Rossbywave frequency shift or in the appearance of non-linear waves different from Rossby waves. Simi-larly to Fig. 9, Rossby waves and turbulence coexistfor wavenumbers exceeding bothnR andnβ. Notethat the wave-turbulence interaction is stronger andmore subtle in the zonostrophic than in the friction-dominated regime; a more detailed investigation ofthis interaction will be given elsewhere. An importantresult from this study is that there exist neither a sharptransition between linear and nonlinear regimes nora distinct boundary between turbulence and Rossbywave spectral ranges. This result reinforces our ear-lier conclusion that neithernβ nornR can be associ-ated with the turbulence - Rossby wave transition.

5. Discussion and conclusions

One of the main results of the presented here theo-retical analysis and numerical simulations is the con-clusion that a non-dissipative extra-strain in the vor-ticity equation, aβ-term, can cause no arrest of theinverse energy cascade in 2D turbulence. The effectof theβ-term on energy transfer manifests in cascadeanisotropization and formation of quasi-1D struc-tures, zonal jets. This result can be likened to thatin 3D flows with pure rotation where non-dissipativeextra-strains are brought about by the Coriolis force.Again, turbulent cascade in this case is not arrestedbut anisotropized leading to self-organization of theflow field into quasi-2D, large-scale columnar vor-tices aligned with the rotation axis (Smith and Lee2005). Even in the case when extra-strains do enter

15

−0.4

50

−0.4

4

−0.4

0.1

0.50−0.5 10 −1

0.50−0.5

10 −1

10 −1 −0.4

0.1

−0.4

4

−0.4

4

−0.4

4−0.4

0.1

−0.4

0.10.50−0.5

0.50−0.5 10 −1

−0.4

0.2

0.50−0.5 10 −1

m/n

n/nβ

0.2

0.4

0.6

0.8

1.0

0.36 0.72 1.45 2.90

n 5 10 20 40

U(ω

,m,n

)

ω/Ω

0.2

0.2

0.2

0.2

0.4 4

4

2

2

4

0 0

0 0

0 0

0 0

0 0

Figure 10: The velocity autocorrelation function,U (ω,m, n), for the zonostrophic regime;nR ' 5,nβ ' 14. The scaling coefficients forU are105 forn = 5 and 10 and108 for n = 20 and 40. The trian-gles correspond to the linear dispersion relation (11).

the kinetic energy equation as sink terms, they can-not totally suppress turbulent cascade. For example,in the case of 3D flows with stable stratification, therealso exists strong anisotropization. Along the direc-tion of the extra-force (i.e., the gravity), turbulent ex-change diminishes, but in the normal planes, turbu-lence is not suppressed and may even be enhanced(Sukoriansky et al. 2005; Smith and Waleffe 2002).In all these cases, the extra-strains support varioustypes of linear waves which can coexist with turbu-lence in wide ranges of scales.

Our simulations revealed neither a separation be-tween the regions of turbulence and wave domina-tion nor a large-scale threshold of the Rossby wavepropagation. The opposite is true; similarly to theinternal and inertial waves, Rossby waves can coex-ist with turbulence in a wide range of scales. Ratherthan being a scale of the cascade arrest, the Rhinesscale may characterize many different phenomena. Inthe present study, this conclusion is illustrated by twoexamples. In unsteady flows,LR appears as a scaleof the largest energy containing structures, while ina steady state zonostrophic regime, the Rhines scalecoincides with the scale of the large-scale friction.Furthermore, it is shown that the relationship between

the Rhines and friction scales is not universal and de-pends on the flow regime. It is not surprising, there-fore, thatLR has been used as a scaling parame-ter characterizing various aspects of flows with aβ-effect. As an example, one may recall that the scalingwithLR has been applied both to the equatorial (Hide1966) and off-equatorial (Williams 1978) Jovian jetsalthough their widths are quite different. Generally,the Rhines scale is a basic dimensional parameter inflows where aβ-effect is a salient feature not nec-essarily related to geostrophic turbulence (see, e.g.,Pedlosky (1998); Nof et al. (2004)). It is important,therefore, to understand what processes are charac-terized byLR and determine an appropriate veloc-ity scaleU . In situations where the spectrum is un-known and the flow is affected by multiple factors,the identification of the dominant processes and thecorrect choice ofU are essential (see, e.g., Barry et al.(2002)).

As was shown in section 4, a nondimensional pa-rameterRβ = nβ/nR plays a key role in quasi-2D,steady state turbulent flows with aβ-effect. By virtueof determining the type of a flow regime, this pa-rameter reveals an important information on variousaspects of large-scale circulations. RecastingRβ interms of the basic flow parameters, we obtain

Rβ = 21/2

(CZ

CK

)3/10 (βU5

ε2

)1/10

' 0.7(βU5

ε2

)1/10

. (25)

This number can also be defined in terms of the ex-ternal flow parameters,λ, β andε,

Rβ = 21/2

(CZ

CK

)3/10 (β2ε

λ5

)1/20

' 0.7(β2ε

λ5

)1/20

. (26)

The zonostrophic regime is characterized byRβ > 2(in addition to the smallness of the Burger number,Bu; see e.g. Galperin et al. (2006)) while for thefriction-dominated regime,Rβ . 1.5. The range1.5 . Rβ . 2 corresponds to a transitional regimewhose properties are not universal. It is instructive tocalculate the values ofRβ for different environments.

16

To illustrate the significance ofRβ, we shall considera wide variety of environments with smallBu rang-ing from the small-scale turntable used in the Greno-ble experiment (Read et al. 2004) (see also Read etal., this issue) and to the terrestrial oceans to the solargiant planets’ weather layers.

For the Grenoble experiment, using the parametersreported in Read et al. (this issue), one obtainsRβ ∈(0.5, 2.3), i.e., the experimental flow was marginallyzonostrophic. It is not surprising, therefore, that theflow field obtained in some experiments exhibitedzonation, spectral anisotropization, and build up ofthe zonal spectrum described by Eq. (13a).

For the oceanic flows, the magnitude ofε can be es-timated between10−9 and 10−11 m2 s−3, the lat-ter value follows, for example, from the realistic,eddy resolving simulations by Nakano and Hasumi(2005). With such values ofε and U of the or-der of 10 cm s−1, one finds thatRβ ∈ (1, 2.8).Similarly to the Grenoble experiment, the oceanicflows are on the verge between the zonostrophic andfriction-dominated regimes. Visually, oceanic flowsare quite erratic as would be expected in the transi-tional regime. However, averaging in time revealszonation (see, e.g., Maximenko et al. (2005); Ol-litrault et al. (2006)) and spectral anisotropization(Zang and Wunsch 2001) suggesting thatRβ is ratherclose to 2. In addition, some numerical modelsshow the build up of the zonal spectrum accordingto Eq. (13a) (Galperin et al. 2004). Even though thezonostrophic inertial range in the ocean is small andthe barotropic currents are relatively weak, by virtueof the zonation that penetrates through considerabledepth these currents can still play an important rolein the dynamic and transport processes. Studies ofthese processes are now becoming an area of an ac-tive research (see, e.g., Galperin et al. (2004); Smith(2005); Richards et al. (2006); Ollitrault et al. (2006);Nadiga (2006)).

For the solar giant planets,ε can be estimated at10−8

m2 s−3 (Galperin et al. 2006) yieldingRβ ∼ 10, 25,30 and 40 for Jupiter, Saturn, Uranus and Neptune,respectively. The large values ofRβ on all four solargiant planets indicate that their atmospheric circula-tions feature well established zonostrophic regimes.Indeed, the energy spectra of the zonal flows on these

planets are consistent with Eq, (13a) in both the slopeand the magnitude (Galperin et al. 2001; Sukorianskyet al. 2002).

An interesting implication can be drawn regarding the“barotropic governor” (James and Gray 1986; James1987) the essence of which is the reduction of thepotential-kinetic energy conversion rate of the baro-clinic instability due to the increase of the horizontalbarotropic shear resulting from the inverse cascade inthe barotropic mode. In light of the present results,it is tempting to think of the barotropic mode as agovernor in a rather broader sense than just a regu-lator of the baroclinic instability. In the case of thezonostrophic regime with a profound inertial range(or largeRβ), the barotropic mode contains most ofthe kinetic energy and hence truly governs the large-scale circulation and its energetics. As shown inSukoriansky et al. (2002), the total kinetic energy ofa circulation in this regime is determined byΩ,R andnfr only and is independent of the external forcing orthe energy conversion rate unless these dependenciesare implicitly embedded in the friction wavenumbernfr the physical mechanism of which is not alwayswell understood and is a subject of an ongoing re-search (see, e.g., Muller et al. (2005)).

Acknowledgement We are grateful to Drs P.L.Read, M.E. McIntyre, W.R. Young, A. Showman,H.-P. Huang and N. Maximenko for numerous dis-cussions and comments during the course of thisresearch. Thoughtful comments from anonymousreviewers helped us to improve and clarify themanuscript. Partial support of this study by theARO grant W911NF-05-1-0055 and the Israel Sci-ence Foundation grant No. 134/03 is greatly appreci-ated.

References

Balk, A., 1991: A new invariant for Rossby wave sys-tems.Phys. Lett. A, 155, 20–24.

— 2005: Angular distribution of Rossby wave en-ergy.Phys. Lett. A, 345, 154–160.

Balk, A. and F. van Heerden, 2006: Conservation

17

style of the extra invariant for Rossby waves.Phys-ica D, 223, 109–120.

Barry, L., G. Craig, and J. Thuburn, 2002: Polewardheat transport by the atmospheric heat engine.Na-ture, 415, 774–777.

Boer, G., 1983: Homogeneous and isotropic turbu-lence on the sphere.J. Atmos. Sci., 40, 154–163.

Boer, G. and T. Shepherd, 1983: Large-scale two-dimensional turbulence in the atmosphere.J. At-mos. Sci., 40, 164–184.

Bradshaw, P., 1973: Effects of streamline curvatureon turbulent flow.AGARDograph, 169.

Carnevale, G. and P. Martin, 1982: Field theoreti-cal techniques in statistical fluid dynamics: Withapplication to nonlinear wave dynamics.Geophys.Astrophys. Fluid Dynamics, 20, 131–164.

Chekhlov, A., S. Orszag, S. Sukoriansky, B. Galperin,and I. Staroselsky, 1996: The effect of small-scale forcing on large-scale structures in two-dimensional flows.Physica D, 98, 321–334.

Cho, J.-K. and L. Polvani, 1996: The emergence ofjets and vortices in freely evolving, shallow-waterturbulence on a sphere.Phys. Fluids, 8, 1531–1552.

Danilov, S. and D. Gurarie, 2002: Rhines scale andspectra of theβ-plane turbulence with bottom drag.Phys. Rev. E, 65, 067301.

Galperin, B., H. Nakano, H.-P. Huang, andS. Sukoriansky, 2004: The ubiquitous zonaljets in the atmospheres of giant planets andEarth’s oceans.Geophys. Res. Lett., 31, L13303,doi:10.1029/2004GL019691.

Galperin, B. and S. Sukoriansky, 2005: Energy spec-tra and zonal flows on theβ-plane, on a rotatingsphere, and on giant planets.Marine Turbulence:Theories, Observations, and Models, H. Baumert,J. Simpson, and J. S¨undermann, eds., CambridgeUniversity Press, 472–493.

Galperin, B., S. Sukoriansky, N. Dikovskaya,P. Read, Y. Yamazaki, and R. Wordsworth, 2006:Anisotropic turbulence and zonal jets in rotating

flows with aβ-effect. Nonlinear Proc. Geophys.,13, 83–98.

Galperin, B., S. Sukoriansky, and H.-P. Huang, 2001:Universaln−5 spectrum of zonal flows on giantplanets.Phys. Fluids, 13, 1545–1548.

Glazman, R. and B. Cheng, 1999: Altimeter obser-vations of baroclinic oceanic inertia-gravity waveturbulence.Proc. R. Soc. London, 455, 91–123.

Glazman, R. and P. Weichman, 2005: Meridionalcomponent of oceanic Rossby wave propagation.Dyn. Atmos. Oceans, 38, 173–193.

Held, I., 1999: The macroturbulence of the tropo-sphere.Tellus A, 51, 59–70.

Held, I. and V. Larichev, 1996: A scaling theory forhorizontally homogeneous, baroclinically unstableflow on a beta plane.J. Atmos. Sci., 53, 946–952.

Hide, R., 1966: On the circulation of the atmospheresof Jupiter and Saturn.Planet. Space. Sci., 14, 669–675.

Holloway, G., 1986: Considerations on the theory oftemperature spectra in stably stratified turbulence.J. Phys. Oceanogr., 16, 2179–2183.

Holloway, G. and M. Hendershott, 1977: Stochas-tic closure for nonlinear Rossby waves.J. FluidMech., 82, 747–765.

Huang, H.-P., B. Galperin, and S. Sukoriansky, 2001:Anisotropic spectra in two-dimensional turbulenceon the surface of a rotating sphere.Phys. Fluids,13, 225–240.

Huang, H.-P. and W. Robinson, 1998: Two-dimensional turbulence and persistent zonal jets ina global barotropic model.J. Atmos. Sci., 55, 611–632.

James, I., 1987: Suppression of baroclinic instabilityin horizontally sheared flows.J. Atmos. Sci., 44,3710–3720.

James, I. and L. Gray, 1986: Concerning the effectof surface drag on the circulation of a baroclinicplanetary atmosphere.Quart. J. R. Met. Soc., 112,1231–1250.

18

LaCasce, J. and J. Pedlosky, 2004: The instability ofRossby basin modes and the oceanic eddy field.J.Phys. Oceanogr., 34, 2027–2041.

Lapeyre, G. and I. Held, 2003: Diffusivity, kinetic en-ergy dissipation, and closure theories for the pole-ward eddy heat flux.J. Atmos. Sci., 60, 2907–2916.

Lilly, D. K., 1972: Numerical simulation study oftwo-dimensional turbulence.Geophys. Fluid Dyn.,3, 289–319.

Majda, A. and X. Wang, 2006:Nonlinear Dynam-ics and Statistical Theories for Basic GeophysicalFlows. Cambridge University Press.

Maximenko, N., B. Bang, and H. Sasaki, 2005: Ob-servational evidence of alternating zonal jets inthe world ocean.Geophys. Res. Lett., 32, L12607,doi:10.1029/2005GL022728.

McIntyre, M., 2001: Balance, potential-vorticityinversion, Lighthill radiation, and the slowquasimanifold.Proc. IUTAM/IUGG/Royal IrishAcademy Symposium on Advances in Mathemati-cal Modeling of Atmosphere and Ocean Dynam-ics held at the Univ. of Limerick, Ireland, 2-7 July2000, P. Hodnett, ed., Kluwer Academic Publish-ers, 45–68.

McWilliams, J., 2006:Fundamentals of GeophysicalFluid Dynamics. Cambridge University Press.

Miesch, M., 2003: Numerical modeling of the so-lar tachocline. II. Forced turbulence with imposedshear.Astrophys. J., 586, 663–684.

Muller, P., J. McWilliams, and M. Molemaker, 2005:Routes to dissipation in the ocean: the two-dimensional/three-dimensional turbulence conun-drum. Marine Turbulence: Theories, Observa-tions, and Models, H. Baumert, J. Simpson, andJ. Sundermann, eds., Cambridge University Press,397–405.

Nadiga, B., 2006: On zonal jets inoceans. Geophys. Res. Lett., 33, L10601,doi:10.1029/2006GL025865.

Nakano, H. and H. Hasumi, 2005: A series of zonaljets embedded in the broad zonal flows in the Pa-cific obtained in eddy-permitting ocean general cir-culation models.J. Phys. Oceanogr., 35, 474–488.

Nof, D., S. Van Gorder, and T. Pichevin, 2004: Adifferent outflow length scale?J. Phys. Oceanogr.,34, 793–804.

Nozawa, T. and S. Yoden, 1997: Formation of zonalband structure in forced two-dimensional turbu-lence on a rotating sphere.Phys. Fluids, 9, 2081–2093.

Ollitrault, M., M. Lankhorst, D. Fratantoni,P. Richardson, and W. Zenk, 2006: Zonalintermediate currents in the equatorial At-lantic Ocean.Geophys. Res. Lett., 33, L05605,doi:10.1029/2005GL025368.

Panetta, R., 1993: Zonal jets in wide baroclinicallyunstable regions: persistence and scale selection.J. Atmos. Sci., 50, 2073–2106.

Pedlosky, J., 1998: Ocean Circulation Theory.Springer, second edition.

Read, P., 2005: From mixing to geostrophy:geostrophic turbulence in atmospheres, oceans,and the laboratory.Marine Turbulence: Theories,Observations, and Models, H. Baumert, J. Simp-son, and J. S¨undermann, eds., Cambridge Univer-sity Press, 406–422.

Read, P., Y. Yamazaki, S. Lewis, P. Williams,K. Miki-Yamazaki, J. Sommeria, H. Didelle,and A. Fincham, 2004: Jupiter’s and Sat-urn’s convectively driven banded jets in thelaboratory. Geophys. Res. Lett., 31, L22701,doi:10.1029/2004GL020106.

Rhines, P., 1975: Waves and turbulence on a beta-plane.J. Fluid Mech., 69, 417–443.

— 1979: Geostrophic Turbulence.Annu. Rev. FluidMech., 11, 401–411.

Richards, K., N. Maximenko, F. Bryan, andH. Sasaki, 2006: Zonal jets in the Pa-cific Ocean. Geophys. Res. Lett., 33, L03605,doi:10.1029/2005GL024645.

Rose, H. and P. Sulem, 1978: Fully developed turbu-lence and statistical mechanics.J. de Phys. (Paris),39, 441–484.

19

Schneider, T., 2004: The tropopause and the ther-mal stratification in the extratropics of a dry atmo-sphere.J. Atmos. Sci., 61, 1317–1340.

— 2006: The general circulation of the atmosphere.Annu. Rev. Earth Planet. Sci., 34, 655–688.

Smith, K., 2005: Tracer transport along and acrosscoherent jets in two-dimensional turbulent flow.J.Fluid Mech., 544, 133–142.

Smith, K., G. Boccaletti, C. Henning, I. Marinov,C. Tam, I. Held, and G. Vallis, 2002: Turbulent dif-fusion in the geostrophic inverse cascade.J. FluidMech., 469, 13–48.

Smith, L. and Y. Lee, 2005: On near resonancesand symmetry breaking in forced rotating flowsat moderate Rossby number.J. Fluid Mech., 535,111–142.

Smith, L. and F. Waleffe, 1999: Transfer of energyto two-dimensional large scales in forced, rotat-ing three-dimensional turbulence.Phys. Fluids, 11,1608–1622.

— 2002: Generation of slow large scales in forcedrotating stratified turbulence.J. Fluid Mech., 451,145–168.

Smith, L. and V. Yakhot, 1993: Bose condensationand small-scale structure generation in a randomforce driven 2D turbulence.Phys. Rev. Lett., 71,352–355.

— 1994: Finite-size effects in forced two-dimensional turbulence.J. Fluid Mech., 274, 115–138.

Sukoriansky, S., B. Galperin, and A. Chekhlov,1999: Large scale drag representation in simula-tions of two-dimensional turbulence.Phys. Fluids,11, 3043–3053.

Sukoriansky, S., B. Galperin, and N. Dikovskaya,2002: Universal spectrum of two-dimensional tur-bulence on a rotating sphere and some basic fea-tures of atmospheric circulation on giant planets.Phys. Rev. Lett., 89, 124501.

Sukoriansky, S., B. Galperin, and I. Staroselsky,2005: A quasinormal scale elimination model of

turbulent flows with stable stratification.Phys. Flu-ids, 17, 085107.

Vallis, G., 2006:Atmospheric and Oceanic Fluid Dy-namics. Cambridge University Press.

Vallis, G. and M. Maltrud, 1993: Generation of meanflows and jets on a beta plane and over topography.J. Phys. Oceanogr., 23, 1346–1362.

Vasavada, A. and A. Showman, 2005: Jovian at-mospheric dynamics: an update after Galileoand Cassini.Rep. Prog. Phys., 68, 1935–1996,doi:10.1088/0034–4885/68/8/R06.

Williams, G., 1975: Jupiter’s atmospheric circula-tion. Nature, 257, 778.

— 1978: Planetary circulations: 1. Barotropic rep-resentation of Jovian and terrestrial turbulence.J.Atmos. Sci., 35, 1399–1426.

Zang, X. and C. Wunsch, 2001: Spectral descrip-tion of low-frequency oceanic variability.J. Phys.Oceanogr., 31, 3073–3095.

20