On the “arrest” of inverse energy cascade and the Rhines scale
Semion Sukoriansky,∗ Nadejda Dikovskaya, Department of Mechanical Engineering and Perlstone Center for Aeronautical Engineering Studies Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Boris Galperin College 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 quency domain demonstrates that Rossby waves co- exist with turbulence on all scales including those much smaller than LR thus indicating that the Rhines We revise the notion of the cascade “arrest” in a β- scale cannot be viewed as a crossover between turbu- plane turbulence in the context of continuously forced lence and Rossby wave ranges. flows using both theoretical analysis and numerical simulations. We demonstrate that the upscale en- ergy propagation cannot be stopped by a β-effect and can only be absorbed by friction. A fundamental 1. Introduction dimensional parameter in flows with a β-effect, the Rhines scale, LR, has traditionally been associated with the cascade “arrest” or with the scale separat- Terrestrial and planetary circulations are described ing turbulence and Rossby wave dominated spectral by nonlinear equations that support various types of ranges. We show that rather than being a measure waves in the linear limit. The real flows exhibit a of the inverse cascade arrest, LR is a characteris- complicated interplay between turbulence and waves. tic of different processes in different flow regimes. While in a certain range of scales, turbulent scram- In unsteady flows, LR can be identified with the bling may overwhelm the wave behavior and lead moving “energy front” propagating towards the de- to the disappearance of the dispersion relation (such creasing wavenumbers. When large-scale energy as in the small-scale range of stably stratified flows; sink is present, β-plane turbulence may attain sev- see e.g., Sukoriansky et al. (2005)), on other scales, eral steady-state regimes. Two of these regimes are the wave terms cause turbulence anisotropization and highlighted, friction-dominated and zonostrophic. In emergence of systems with strong wave-turbulence the former, LR does not have any particular signifi- interaction. Systems that combine anisotropic turbu- cance, while in the latter, the Rhines scale nearly co- lence and waves exhibit behavior very different from incides with the characteristic length associated with that of the classical isotropic and homogeneous tur- the large-scale friction. Spectral analysis in the fre- bulence (McIntyre 2001). In the context of large- scale atmospheric flows, the interaction between tur- ∗[email protected] bulence and waves yields dynamically rich macro-
1 turbulence (Held 1999; Schneider 2006), the notion cascade by the Rossby wave propagation which, thus, that underlies diverse phenomena ranging from a lo- becomes another focal point of this study. Both issues cal weather to a global climate. The subtlety of this will be addressed theoretically; the theoretical results interaction is further highlighted by the fact that due will be substantiated in numerical simulations. to the planetary rotation and Taylor-Proudman theo- rem, large-scale flows are quasi-two-dimensionalized The paper is organized in the following fashion. The and may be conducive to the development of the in- next section presents the basics of the theory of two- verse energy cascade (Read 2005). One of the param- dimensional (2D) turbulence with a β-effect and ac- eters that characterize macroturbulence is the Rhines centuates its problematics. Section 3 presents analy- 1/2 sis of the interaction between inverse energy cascade wavenumber, kR =(β/2U) , or the Rhines scale, −1 of 2D turbulence and Rossby waves and discusses the LR ∼ kR , where U is the r.m.s. fluid velocity and β is the northward gradient of the Coriolis parameter notion of the cascade “arrest.” Section 4 elaborates (Rhines 1975) (in the original paper, this wavenum- this analysis and clarifies the meaning of the Rhines scale using numerical simulations of barotropic 2D ber was denoted by kβ; here, kR is used instead, as turbulence on the surface of a rotating sphere. In ad- the notation kβ is reserved for another wavenumber to appear later). At this scale, the inverse cascade dition, that section describes the hierarchy of scaling supposedly becomes arrested, and the nonlinear tur- parameters and corresponding flow regimes as well as bulent behavior is replaced by excitation of linear the interaction between turbulence and Rossby waves Rossby waves. This scale has also been associated in different regimes. Finally, section 5 presents dis- with flow reorganization into the bands of alternating cussion and some conclusions. zonal jets, the process known as zonation, and the −1 width of the jets has often been scaled with kR . The Rhines scale plays a prominent role in many theories of large-scale atmospheric and oceanic circulation 2. Turbulence and Rossby waves: (see, e.g., James and Gray (1986); Vallis and Mal- The basics trud (1993); Held and Larichev (1996); Held (1999); Lapeyre and Held (2003); Schneider (2004); LaCasce and Pedlosky (2004); Vallis (2006)) as well as in the- The interaction between 2D turbulence and Rossby ories of planetary circulations (see, e.g., the review by waves has been considered in the pioneering study Vasavada and Showman (2005)) and, possibly, even by Rhines (1975). Starting with the barotropic vortic- stellar convection (see, e.g., Miesch (2003)). ity equation on a β-plane, he concentrated on mainly unforced barotropic flows caused by initially closely The meaning of the Rhines scale is not always clear, packed fields of eddies spreading from a state with a however. Even in the barotropic case, different δ-function like spectrum peaked at some wavenum- regimes would arise for continuously forced and de- ber k0. Among the major conclusions of the paper, caying flows, for flows with and without friction, and the following have the most relevance to the present for flows in bounded and unbounded domains. As study: a result, the Rhines scale may be time-dependent, stationary or entirely obscured by friction. In many −1 (1) at the wavenumber, denoted as kR, at which studies, the scaling with kR is implied but the coeffi- cient of proportionality varies in a wide range. There the root-mean-square velocity U is equal to the phase speed of Rossby waves with an average also exists a multitude of other scaling parameters 2 that characterize various aspects of the flow and flow orientation, cp = β/2k , there exists a subdivi- regimes. It is important, therefore, to understand the sion of the spectrum on turbulence (k>kR) and wave (k 2 3 1/5 (3) triad interactions of the modes with wavenum- kβ ∝ (β /) (the proportionality coefficient will bers close to kR require simultaneous resonance be established later) and the angular distribution re- in wavenumbers and frequencies; sembling a dumb-bell (Vallis and Maltrud 1993; Hol- loway 1986; Vallis 2006). The introduction of kβ (4) on the average, triad interactions transfer energy raises a slew of new questions. If there is an “arrest” to modes with smaller frequencies and smaller in the forced flows, which one of the two wavenum- wavenumbers causing the anisotropization of bers, kR and kβ, should be associated with the “ar- the flow field. As a result, the energy accumu- rest” scale? At which scale the anticipated in item lates in modes with small north-south wavenum- (2) Rossby wave excitation is triggered? Can Rossby bers that correspond to east-west currents, zonal waves coexist with turbulence on scales intermediate jets, giving rise to the process of zonation; −1 −1 between kR and kβ ? With regard to item (5) one (5) the width of the zonal jets scales with k−1; may ask which one of these two wavenumbers should R be used as a scaling parameter for the width of the (6) in a steadily forced flow, the spectrum is ex- zonal jets? Indeed, although the mechanism of zona- pected to develop a sharp peak at kR and rapidly tion has by now been solidly established, the scaling decrease for k>kR. of the jets’ width with the Rhines scale has not been conclusive as the coefficient of proportionality has so far been elusive. For instance, Hide (1966) used a The items (3) and (4) have been confirmed in nu- −1 merous theoretical, numerical and experimental stud- scale similar to kR for the width of the equatorial ies (see, e.g., Williams (1975, 1978); Holloway jets on Jupiter and Saturn while Williams (1978) and others applied it to the off-equatorial jets. Conclu- and Hendershott (1977); Panetta (1993); Vallis and −1 Maltrud (1993); Chekhlov et al. (1996); Cho and sive scaling with kR has materialized neither in the Polvani (1996); Nozawa and Yoden (1997); Huang Grenoble experiment (Read et al. 2004) nor in the and Robinson (1998); Huang et al. (2001); Read et al. recent analysis of the eddy-resolving simulations of (2004); Vasavada and Showman (2005); Galperin oceanic jets by Richards et al. (2006). et al. (2006)). A new light on zonation was shed by Chekhlov et al. (1996) have scrutinized the notion Balk (2005). Exploring a new invariant for inviscid of the “arrest” of the inverse energy cascade and as- 2D flows with Rossby waves (Balk 1991; Balk and serted that Rossby wave excitation cannot halt the van Heerden 2006), he showed that the transfer of en- cascade in principle. On the other hand, they have ergy from small to large scales takes place in such a confirmed that the β-term in the vorticity equation way that most of the energy is directed to the zonal impedes triad interactions hence reducing the effec- jets. The rest of the aforementioned issues, how- tiveness of nonlinear transfers and increasing their ever, remain controversial, particularly when com- characteristic time scale. Furthermore, Chekhlov pounded with other complicating factors such as the et al. (1996), Smith and Waleffe (1999) and Huang friction, continuous forcing, effects of the bound- et al. (2001) have shown that the spectrum of a aries, effects of stratification, etc. Let us consider, forced β-plane turbulence is strongly anisotropic. On for example, a realistic system in which the bottom large scales, the spectrum of the zonal modes is pro- friction acts like a large-scale drag. One can intro- 2 −5 portional to β ky . It is important to understand duce a wavenumber associated with that drag, kfr, the synergy between the slowing-up of the nonlin- as a wavenumber at which the characteristic friction ear interactions and spectrum steepening. The re- time is equal to that of the flow. If kfr >kR then duction in the effectiveness of the triad interactions the frictional processes would forestall the arrest and is due to the requirement of the simultaneous reso- the Rhines scale would require modification (James nance in wavenumbers and frequencies (Rhines 1975, and Gray 1986; Danilov and Gurarie 2002; Smith 1979; Holloway and Hendershott 1977; Carnevale et al. 2002). In flows with continuous small-scale and Martin 1982). The ensuing cascade anisotropiza- forcing at a rate , the equilibrium between the eddy tion facilitates strong energy flux to the zonal modes turnover time and the Rossby wave period selects an (Rhines 1975; Vallis and Maltrud 1993; Chekhlov anisotropic transition wavenumber with an amplitude 3 et al. 1996). Due to the frequency resonance imped- Considering a space of parameters that characterize iment, however, the zonal modes cannot effectively different regimes in barotropic, forced and dissipative transfer their energy to other modes. Since the to- β-plane turbulence, Galperin et al. (2006) have found tal energy flux must be conserved, the spectral en- that in a certain subspace, a flow attains a universal ergy density of the zonal modes has to increase. The regime with the zonal spectrum (1) and CZ ' 0.5. steepening of the zonal spectrum eventually reaches This subspace is wide enough to include important 2 −5 the β ky distribution at which the frequency reso- laboratory, terrestrial and planetary flows (Galperin nance condition is relaxed and the zonal modes can et al. 2006). again efficiently exchange energy with other modes. This exchange allows for the zonal modes to dispose Despite the progress made in understanding of var- 2 −5 ious aspects of a β-plane turbulence, there still re- of the energy excessive, on the average, of the β ky level and maintain a steady and statistically stable mains a great deal of confusion regarding the “arrest” 2 −5 of the inverse energy cascade and the Rhines scale. flow regime (Huang et al. 2001). The β ky spec- trum is somewhat reminiscent of the isotropic −5 dis- The fact that the interplay between turbulence and tribution discussed by Rhines (1975). He has put the Rossby waves has never been thoroughly investigated existence of such a sharp slope in doubt, however, only adds to this confusion. The next section gathers because it would imply a strong dependence on the some theoretical arguments that may help to clarify smallest wavenumbers with the highest energy levels this perplexity. In section 4 these arguments will be which would lead to a strong nonlocality negating the substantiated via numerical experimentation. dimensional arguments that facilitated the β2k−5 de- pendence in the first place. Sukoriansky et al. (2002) have elucidated that in the case of an anisotropic spectrum, the arguments of nonlocality may not be 3. Is the cascade really arrested? applicable hence one could expect that a flow regime with a steep zonal spectrum, The cascade “arrest” is often understood as a transi- tion of a flow character from strongly nonlinear and 2 −5 turbulent to weakly nonlinear and wave-dominated EZ (ky)=CZβ k , (1) y under the action of a nondissipative “extra-strain” in the Bradshaw (1973) terminology, a β-effect. This −1 may be established (here, CZ is an O(1) coefficient). transition presumably takes place at a scale kR . The stability of such a regime is stipulated by the Such an interpretation applies to unforced flows con- Rayleigh-Kuo criterion (Chekhlov et al. 1996; Huang sidered by Rhines (1979) in great detail (see also Ma- et al. 2001; McWilliams 2006). Furthermore, sim- jda and Wang (2006), chapters 10-13). Rhines (1979) ilarly to Lilly (1972) in the case of isotropic, non- showed that in unforced flows, both the nonlinear rotating flows, Sukoriansky et al. (2002) have argued term and a rate of the energy transfer to large scales that only the large-scale drag can balance the inverse decrease with time. This tendency can be traced to cascade and facilitate the establishment of a steady the impediment to triad interactions caused by the state in flows with a β-effect. They have reproduced frequency resonance condition discussed in the previ- the spectrum (1) in forced, dissipative simulations of ous section. A “soft” transition from a strongly non- 2D barotropic turbulence with a linear drag on the linear to a weakly nonlinear regime takes place at the surface of a rotating sphere and showed that in such crossover wavenumber, kR (Rhines 1979). This in- flows, kR is close to kfr. They have also found an terpretation of the cascade “arrest” is sometimes ap- evidence of the spectrum (1) on all four Solar giant plied to forced flows (James and Gray 1986). The ex- planets. Later, using the eddy-resolving simulations tension of the results valid for the unforced dynamics by Nakano and Hasumi (2005) of the North Pacific (=0) to flows with continuous forcing ( =06 ) may ocean, Galperin et al. (2004) have shown that this be problematic however. Already at the level of the spectrum may also be present in the barotropic mode dimensional analysis, the presence of the non-zero of the subsurface oceanic alternating zonal jets. and appearance of the non-trivial wavenumber kβ 4 point to significant differences between the two cases. In the remainder of this section, we shall elabo- Vallis and Maltrud (1993) have first recognized the rate various scaling relationships pertinent to forced importance of the transitional wavenumber, kβ.In anisotropic turbulence with a β-effect. These rela- their forced - dissipative simulations, Vallis and Mal- tionships will be used to further elucidate the absence trud (1993) observed a “piling up” of the computed of the cascade arrest. The flows will be assumed energy spectrum in the vicinity of kβ. This “pil- to contain a stationary random forcing maintaining ing up” took place in a short range of wavenumbers, an inverse energy cascade at a constant rate . This however, because the simulation parameters were set case has most relevance to the real world large-scale such that kβ was close to kR. The spectral steepen- terrestrial and planetary circulations (Galperin et al. ing was thought to be a result of the dumb-bell be- 2006). The importance of the parameter is under- coming a barrier to the energy flux to larger scales. scored by the conjecture that the diffusion coefficient Chekhlov et al. (1996) investigated a regime with of the poleward heat transport strongly depends on kβ kR. They have also observed the energy pil- it (Held and Larichev 1996; Held 1999; Lapeyre and ing up in the vicinity of kβ but they have attributed Held 2003). this phenomenon to a development of a new regime of circulation with the steep spectrum (1). Chekhlov Two classes of flows will be considered, those with- et al. (1996) have emphasized that although the to- out and with the action of the large-scale drag. The tal upscale energy flux in forced, undamped flows re- first class is rather unrealistic because, firstly, some mains constant, the rate of the “energy front” prop- kind of a drag is always present in all real flows, agation decreases due to the steepening of the zonal and, secondly, the large-scale energy condensation spectrum. The inverse cascade anisotropization com- would eventually distort flow configuration in long- bined with the zonal spectrum’s steepening facilitate term integrations (Smith and Yakhot 1993, 1994). the establishment of the zonostrophic regime in flows Such flows are important, however, for understand- with a large-scale drag (Galperin et al. 2006). ing of the dynamics of the transients and mechanisms that lead to the establishment of observable long-term Although the frequency resonance constraint im- patterns. These patterns are represented in the second pedes the triad interactions in both forced and un- class of flows. forced flows, it does not cause the “arrest” of the in- verse energy cascade which continues to pump en- Barotropic, small-scale forced, anisotropic 2D turbu- ergy to ever larger scales at a constant rate . The lence on a β-plane where x and y are directed east- frequency resonance impediment only facilitates the ward and northward, respectively, is described by the funneling of the energy flux into the zonal modes vorticity equation, leading to spectral anisotropization (Chekhlov et al. ∂ζ ∂ ∇−2ζ,ζ ∂ 1996) and reorganization of the flow field into a + + β ∇−2ζ = D + ξ, (2) lower-dimensional slow manifold decoupled from the ∂t ∂(x, y) ∂x fast Rossby waves (Smith and Lee 2005). This manifold manifests itself as a system of quasi-one- where ζ is the vorticity, ξ is the forcing, and D is the dimensional alternating zonal jets. The threshold of friction that includes a small-scale and a large-scale the spectral anisotropization is characterized by the components. In numerical simulations, the small- scale part of D is usually represented by a hypervis- wavenumber kβ while the width of the zonal jets is determined by the large-scale friction (Sukorian- cous term while the large-scale drag is linear. As was sky et al. 2002). It is important to emphasize that shown by Sukoriansky et al. (1999) and Sukoriansky forced β-plane turbulence never collapses to a lin- et al. (2002), a linear drag, being a physically plausi- ear state since the nonlinearity is the very factor that ble mechanism of the large-scale energy damping due sustains the slow manifold. It is clear, therefore, to the bottom friction, is a “soft” process which intro- that a β-effect cannot halt the inverse energy cas- duces only a small distortion to the inverse energy cade (Chekhlov et al. 1996; Sukoriansky et al. 2002; cascade that does not lead to a spurious accumulation Galperin et al. 2006). of energy in the lowest resolved modes. The con- stant β in (2) is the northward gradient of the Corio- 5 lis parameter, f, f = f0 + βy, where f0 is a refer- ence value of f. The forcing ξ, concentrated around some high wavenumber kξ and assumed to be ran- dom, zero-mean, Gaussian and white noise in time, supplies energy to the system at a constant rate. Part of this energy sustains the inverse cascade with the rate while the other part is lost on scales k>kξ due to the small-scale dissipation. On its left-hand side, Eq. (2) contains the nonlin- ear and the β terms which are dominant in differ- ent ranges of scales. Here, the balance between these terms will be analyzed using the notion of the characteristic time scale rather than the characteris- tic velocity as in Rhines (1975). The relatively high wave-number modes are nearly isotropic and obey the 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 spectrum E(k), U(k) ∝ [kE(k)]1/2, yielding 3 −1/2 τt(k)=[k E(k)] , where Figure 1: Normalized spectral energy transfer, T | |T | | 2/3 −5/3 E(k kc)/max E(k kc) , for kc =50. E(k)=CK k . (3) Here, C ' 6 is the Kolmogorov-Kraichnan con- K nonzonal, or the residual spectrum largely preserves stant. Equating τt and the Rossby wave period, 2 the shape given by the KBK law, Eq. (3). τRW (k)=−k /βkx, one can find a crossover wavenumber for the β-effect induced anisotropy The anisotropization of the inverse cascade has (Vallis and Maltrud 1993), been demonstrated using the results of direct nu- merical simulation (DNS) of a β-plane turbulence k ∝ (β3/)1/5. (4) β by Chekhlov et al. (1996) who considered the en- ergy transfer function, TE (k|kc), describing the k- Note that for a stationary , kβ is also stationary even if the flow itself is not in a steady state. Other charac- dependent spectral energy flow from all modes with teristic wavenumbers for both unsteady and steady- k>kc to a given mode k, k 6 anisotropic distribution of TE(k|kc) peaking at small integration time than similar simulations with no ro- 1 ky. tation; this tendency is reflected in the “− /4 law” (7). As a result of the spectral anisotropization, the zonal When a large-scale drag is present, an ensuing energy on the large scales exceeds its nonzonal coun- steady-state flow regime is determined by the ratios terpart, and the total energy, Etot(t), can be estimated between kfr, kβ, kd, and kξ (Galperin et al. 2006). based upon the zonal energy spectrum only, There exists a certain subspace of these parameters in which a flow undergoes zonation and develops a 2 kd U (t) 2 −4 universal stationary regime whose anisotropic spec- Etot(t) ' ' X EZ (ky) ∝ β km (t), (5) 2 trum in the inertial range (kfr 7 Rossby waves will also be presented. our units R =1, we shall not differentiate between the indeces and wavenumbers. In the unforced, nondissipative, linear limit, Eq. (9) 4. Turbulence and Rossby waves: gives rise to Rossby waves whose dispersion relation Results of numerical simula- is m ω = −2β , (11) tions m,n n(n +1) where β =Ω/R. We shall retain β in the forthcom- This section describes numerical experimentation ing equations in order to preserve the transparency with two-dimensional turbulent flows on the surface between the cases of a β-plane and rotating sphere. of a rotating sphere. The flow is governed by the 3 1/5 barotropic vorticity equation, The wavenumber nβ ∝ (β /) is the analogue of kβ. In the forced, dissipative regime, the balance ∂ζ = −J(ψ,ζ+ f)+ν∇2pζ − λζ + ξ, (9) between the small-scale forcing and large-scale dissi- ∂t pation gives rise to a steady state. The kinetic energy where ζ is the vorticity; ψ is the stream function de- spectrum can be calculated as fined as ∇2ψ = ζ; f = 2Ω sin θ is the Coriolis pa- n n(n +1) 2 Ω E(n)= X h|ψm| i, (12) rameter (or the “planetary vorticity”); is the angu- 4 n lar velocity of the sphere’s rotation; ν is the hyper- m=−n viscosity coefficient; p is the power of the hypervis- where the brackets indicate an ensemble or time av- cous operator (p was either 4 or 8 in this study); λ is erage (Boer 1983; Boer and Shepherd 1983). This the linear friction coefficient, and ξ is the small-scale spectrum can be represented as a sum of the zonal and forcing, respectively. In addition, J(ψ,ζ+ f) is the residual spectra, E(n)=EZ(n)+ER(n), where the Jacobian representing the nonlinear term, J(A, B)= zonal spectrum, EZ (n), corresponds to the addend 2 −1 (R cos θ) (AφBθ − AθBφ), and R is the sphere’s with m =0. In unsteady flows (Huang et al. 2001) radius. For convenience, the unit of length is set to and in the inertial range of the zonostrophic regime be the radius of the sphere. In these units, R =1and (Sukoriansky et al. 2002; Galperin et al. 2006) these will be omitted in the forthcoming derivations. The spectra are similar to those in β-plane turbulence, natural time scale is T =Ω−1 such that Ω=1.To 2 −5 explore the explicit dependency on Ω, in some of the EZ(n)=CZβ n ,CZ ∼ 0.5, (13a) 2/3 −5/3 experiments T was kept fixed while Ω varied. ER(n)=CK n ,CK ∼ 5to6. (13b) On the surface of a unit sphere, the stream function The zonal and residual spectra intersect at the transi- can be represented via the spherical harmonics de- tional wavenumber, composition, 3/10 3 1/5 3 1/5 CZ β β nβ = ' 0.5 . N n CK m m Ψ(µ, φ, t)=X X ψn (t)Yn (µ, φ), (10) (14) n=1 m=−n To investigate the energy front propagation in un- m where Yn (µ, φ) are the spherical harmonics (the as- steady flows, a series of long-term simulations was sociated Legendre polynomials); µ = sin θ; φ is the performed using Eq. (9) decomposed in spheri- longitude; θ is the latitude; n and m are the total and cal harmonics according to Eq. (10). A Gaussian zonal wavenumbers, respectively, and N is the total grid was employed with resolutions of 400×200 and truncation wavenumber. Conventionally, the indeces 720×360 nodes and 2/3 dealiasing rule (rhomboidal n, m and N are nondimensional. However, when ap- truncations R133 and R240, respectively). The hy- pear in equations below, the wavenumbers n and m perviscosity coefficient, ν, was chosen such as to ef- have the dimension of the inverse length. Since in fectively suppress the enstrophy range. The Gaussian 8 Table 1: Parameters of the four runs shown in Fig. 4. Experiment Ω nβ 111.44 × 10−8 17.6 212.58 × 10−9 24.8 3 2.5 1.42 × 10−8 30.6 431.44 × 10−8 34.1 −3 random forcing was distributed amongst all modes 10 nξ =83, 84, 85 and nξ =99, 100, 101 for R133 and R240, respectively; it had a constant variance and was E E tot tot uncorrelated in time and between the modes. −4 10 The values of the parameters used in unsteady sim- ulations are summarized in Table 1. In total, four R numerical experiments were performed. Although Ω −5 E 10 tot and varied in a wide range in these experiments, their values were set such as to ensure n −7 At first, the energy evolution was investigated. It is 10 2 3 4 well known that without rotation, the total energy of 10 10 10 the flow increases linearly with time, Etot(t)=t. t/(2π) We wanted to verify that, firstly, the linear trend is preserved in the case of nonzero rotation and, sec- ondly, the zonal and nonzonal total energy compo- Figure 2: Evolution of the total energy, Etot(t) (thick Z R solid line), and its zonal (dashed line) and non- nents, Etot(t) and Etot(t), respectively, would fol- Z Z zonal (dashed-dotted line) components, Etot(t) and low a similar trend [here, Etot(t)=Etot(t)+ R R R Etot(t), respectively, in Experiment 1. A triangle on Etot(t); Etot(t) contains both the eddy and wave en- ergies]. Figure 2 shows the evolution of the total en- Etot(t) shows the time required for the energy front ergy and its components for Experiment 1 in a sim- to reach the largest scales in the system in the case ulation of the duration of 10,000 planetary days (for when β =0but is the same as in Experiment 1. Experiment 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 preserves an approximately constant value and then returns to a linear growth while remaining considerably smaller than Etot(t). At those later times, the total energy is mostly contained in the zonal component. Generally, Z R after initial restructuring, both Etot(t) and Etot(t) grow linearly in time for as long as the flow evolution remains unobstructed by the action of a large-scale 9 drag or domain boundaries. We have conducted an additional simulation featur- ing β=0 and the same as the one used in Experiment 1. The evolution of the total energy in that simulation was indistinguishable from the case with β =06 . The only difference was that in the case β =0, the “en- ergy front” required much shorter time to reach the −4 10 largest scales of the system. The total energy at that 8.9 17.8 C β2n−5 a moment is marked by a triangle in Fig. 2. Clearly, the E(n) Z shortening of the evolution time is a result of the fast −6 3.6 front propagation described by Eq. (8). For β =06 , 10 the system can accumulate significantly higher en- 2.1 ergy; according to Eq. (5), this energy scales with 1.2 2 β and is independent of . This increased energetic −8 10 0.5 capacity of the system is a direct result of the spec- n tral anisotropization and steepening of the zonal spec- β 0 1 2 trum according to Eq. (1). 10 10 n 10 −4 The behavior of the total energy components exhib- 10 ited in Fig. 2 requires some clarification. Further C ε2/3n−5/3 b E (n) K insight comes from the consideration of the evolution R of the zonal and residual energy spectra for the same −6 Experiment 1 shown in Fig. 3. Driven by the inverse 10 cascade, the spectrum expands towards the small wavenumber end. Until the transitional wavenum- ber n is reached, the “energy front”, n (t), marked −8 β m 10 by the black dots in Fig. 3a, can be identified as a nξ wavenumber with the highest energy. For n 10 −1/2 −3/2 the Rhines scale: 80 34ε t 4 n m 1 2 3 2U 1/2 8t 1/4 n−1(t)= ' , (15) 40 R β β2 20 −1/4 which shows that nR is time-dependent, nR ∝ t . If the evolution of the moving “energy front,” nm(t), n =1.7n is plotted against n (t), then one expects to detect 10 m R R two different power laws for the dependence of nm on nR. At the early stages, while nm(t) >nβ and the effect of the β-term is not yet felt, nm would evolve 3 5 10 20 according to the “− /2” law, Eq. (8). Upon reach- n =(8εt/β2)−1/4 R ing nβ, as follows from Eqs. (7) and (15), nm and nR become proportional to each other. As estimated from the data, the numerical coefficient between nm 3 Figure 4: The moving “energy front,” nm, in un- and nR is approximately equal to 1.7. The “− /2” 1 steady simulations. The transition from the “−3/2“ and “− /4” evolution laws and the transition between 1 to the “− /4” evolution law in the vicinity of the tran- them are confirmed in Fig. 4 for the four experiments sitional wavenumbers nβ located at the intersections summarized in Table 1. The initial evolution is in- of the respective dashed lines is clearly visible for deed according to the “−3/2” law which changes to 1 all four simulations. The figure demonstrates the ab- the “− /4” law when nm approaches nβ. Clearly, the sence of the halting wavenumber for the inverse en- change in the exponent of the evolution law is con- ergy cascade; the ”energy front“ can penetrate to the sistent with the change in the spectral slope shown in smallest wavenumbers available in the system. Fig. 3a. The transition between the two regimes is a complicated and non-universal process further exac- erbated by the discrete character of the front propaga- By comparing Figs. 2 and 3 we establish that the tion. Furthermore, the location of nm in the transition Z area may be somewhat ambiguous as evident from aforementioned change in the behaviors of Etot(t) R Fig. 3a. Nevertheless, the transition wavenumbers and Etot(t) is concurrent with the restructuring of the spectra from isotropic, KBK to strongly anisotropic are quite close to the values of nβ in Table 1 for all four runs. Figure 4 clearly demonstrates the absence distributions upon crossing nβ. Although at large R of inverse cascade termination. Although a β-effect times, Etot is relatively small, the nonzonal modes play a crucial role in maintaining the zonal flows as is strong in the range nm 11 ters, for instance, n n fr = f β . (16) nR nR The functional dependency varies according to a flow regime. Using an extensive series of long-term steady-state simulations of 2D turbulence with a β- effect, we have derived a detailed classification of possible regimes in the space of parameters nβ and 15 nR (Galperin et al. 2006). These regimes are delin- eated by the dashed lines in Fig. 5. Two of these n R regimes, zonostrophic and friction-dominated, are of particular interest to the present study. The former is confined to the subspace outlined by the chain in- 10 equality nξ & 4nβ & 8nR & 30. (17) This inequality instructs us that (1) the forcing acts 5 on scales only weakly impacted by a β-effect; (2) there exists a meaningful zonostrophic inertial range whose width is defined by the ratio nβ /nR, and (3) there exists a sufficient number of the lowest modes to resolve the large-scale friction processes and avoid 0 0 5 10 15 20 25 the large-scale condensation. nβ The friction-dominated or, briefly, friction regime oc- cupies the subspace to the left of the top dashed line Figure 5: Possible flow regimes in 2D turbulence in Fig. 5 and is delineated by the inequality with a β-effect in the space of parameters nR and n β . 1.5. (18) nβ. The subspace of the friction-dominated regimes nR (unfilled squares) occupies the area to the left of the upper dashed line while the subspace of the zonos- In the friction regime, the zonostrophic inertial range trophic regime (filled dots ) outlined by the chain in- is practically non-existent; only the classical KBK in- equality (17) is confined to the area between the mid- ertial range may survive for n>nfr. dle, bottom and vertical dashed lines for the R133 experiments. For higher resolution, the vertical line Let us consider now how Eq. (16) changes in dif- ferent regimes. In the zonostrophic regime, when moves to larger values of nβ. nβ/nR 1, the dependence on this parameter ceases (practically, this dependence becomes weak already at nβ /nR & 2) and one obtains nfr/nR ' const. Let us calculate the numerical value of the proportionality constant. Since denotes the net part of the energy input that goes to the inverse cas- cade, the spectrally-integrated energy balance equa- tion reads E = . (19) tot 2λ Using this relationship, we obtain the Rhines 12 20 40 n n fr fr 15 30 n =1.2n 10 fr R 20 5 10 0 0 0 10 20 30 40 0 5 10 15 5 2 1/4 10(n /n ) n n =(β λ/4ε) R β R R Figure 7: The correlation between the actual and the- Figure 6: The friction scale, nfr, determined as the intersection of the plateau and the −5 ranges oretical values of nfr in simulations of the large-scale drag dominated 2D turbulence with a β-effect. in the energy spectrum, vs. the Rhines scale, nR, in simulations of the zonostrophic (filled dots) and marginally-zonostrophic (empty dots) regimes with a linear large-scale drag. marginally-zonostrophic (unfield triangles adjacent to the zonostrophic region) regimes with widely vary- ing values of λ (5 × 10−4 ≤ λ ≤ 2.5 × 10−3), wavenumber in the form (3.3 × 10−9 ≤ ≤ 9 × 10−8), and Ω(0.3 ≤ Ω ≤ 2). 1/4 Note that due to the steep zonal spectrum, zonos- λβ2 n = . (20) trophic flows have low variability and require long- R 4 term integrations to assemble records sufficient for On the other hand, since the energy spectrum is com- statistical analysis. Galperin and Sukoriansky (2005) prised of a plateau for n 13 and Gurarie (2002); Smith et al. (2002); James and 12 n Gray (1986)). The functional relationship between jet nfr/nR and nβ /nR can then be established using the assumption that the energy spectrum is isotropic and described by the KBK distribution, Eq. (3). The total 8 kinetic energy in this case is given by the expression ' 3 2/3 −2/3 Etot 2 CK nfr , where nfr is the wavenum- ber with the maximum energy. Using the energy bal- ance equation, Eq. (19), we obtain 4 3/2 3 1/2 nfr =(3CK) (λ /) . (22) Using Eq. (22) and the definitions of nβ, Eq. (14), 0 and nR, Eq. (20), we further find 0 4 8 12 n 5 R 3/2 nR nfr = (12CZ) nR nβ 5 Figure 8: The correlation between the number of the nR ' 14.7 nR. (23) zonal jets, njet, estimated from the velocity profile in nβ the physical space, and nR obtained from numerical This relationship is very different from the one de- simulations of the zonostrophic regime. rived for the zonostrophic regime, Eq. (21), because now nfr depends on and is independent of β such that the proportionality between nfr and nR is lost. Summarizing, we note that the dependence of nfr on jets, njet. It is of interest, therefore, to compare njet nR given by Eq. (16) has two limiting cases pertain- with nfr and nR. ing to the zonostrophic and friction regimes. In the Figure 8, derived from simulations of the zonos- former, nfr and nR are related linearly and are gener- ally quite close while in the latter they diverge from trophic regime, indicates that njet ' nR. The spread each other by a stiff power law. In the intermedi- of the data points is caused by the discreteness of njet. In the range 4 ≤ njet ≤ 10, for example, unavoidable ate cases, the relationship between nfr on nR can be expected to be nonlinear and complicated which ex- unity deviations appear like a 10-25% noise. In fact, plains why the scaling of the zonal jets’ width with given this uncertainty, nR provides a faithful estima- −1 tion of the number of the zonal jets. Using the corre- nR has been so evasive in diverse flows. lation between nfr and nR shown in Fig. 6, one con- −1 −1 Equation (23) has been validated in numerical sim- cludes that both nfr and nR are appropriate scal- ulations with large values of the drag coefficient, λ, ing parameters for the jets’ width in the zonostrophic which was chosen such that nβ/nR . 1.5. These regime. We have also attempted to scale njet with nR simulations are shown as unfilled squares in Fig. 5. calculated using the nonzonal velocities only. In that Figure 7 shows a good agreement between the actual case, the correlation between njet and nR was con- values of nfr and those calculated from Eq. (23) with siderably worse than that in Fig. 8 and, therefore, it the coefficient adjusted to 10. is not shown here. There still remains a question about the proper scal- We now want to analyze the interplay between tur- ing of the width of the zonal jets which in some stud- bulence and Rossby waves in order to determine −1 ies was found to be close to nR . Note that the scaling whether nR (or nβ) separate turbulence and Rossby of the jets’ width only makes sense when the jets have wave regions or turbulence and waves coexist in some approximately homogeneous spatial distribution and range of scales. To answer this question, it is instruc- their width can be estimated from the number of the tive to analyze the Fourier-transform of the two-time 14 n 5 10 20 40 evident from Fig. 9, Rossby waves are present at n n/n β 0.41 0.82 1.64 3.28 far exceeding both nR and nβ indicating that there is m/n no separation between Rossby waves and turbulence; 1 5 5 0.2 0.2 both processes coexist on virtually all scales. Figure −0.4 0 −0.2 0 −0.2 0 −0.2 0 0.2 5 10 depicts a more complicated wave-turbulence in- 0.4 0.2 5 0.2 teraction. As in Fig. 9, the spectral spikes at small −0.4 0 −0.2 0 −0.2 0 −0.2 0 0.2 n are sharp but now they begin to show signs of a 0.6 0.05 2 5 0.2 strong wave-turbulence interaction. At larger n, due −0.4 0 −0.2 0 −0.2 0 −0.2 0 0.2 to a stronger turbulence, the spikes broaden and be- 0.02 1 5 0.2 0.8 come displaced from the values stipulated by the lin- ,m,n) ω −0.4 0 −0.2 0 −0.2 0 −0.2 0 0.2 ear dispersion relationship. This behavior is indica- U( 0.01 0.5 0.2 1.0 5 tive of a complex nature of turbulence - wave inter- −0.4 0 −0.2 0 −0.2 0 −0.2 0 0.2 action in the case of the zonostrophic regime which ω Ω / may manifest itself in the development of the Rossby wave frequency shift or in the appearance of non- linear waves different from Rossby waves. Simi- Figure 9: The velocity autocorrelation function, larly to Fig. 9, Rossby waves and turbulence coexist U(ω,m,n) × 107, for the friction dominated regime; for wavenumbers exceeding both n and n . Note n ' 9, n ' 12. The triangles correspond to the R β R β that the wave-turbulence interaction is stronger and linear dispersion relation (11). more subtle in the zonostrophic than in the friction- dominated regime; a more detailed investigation of velocity autocorrelation function given by this interaction will be given elsewhere. An important result from this study is that there exist neither a sharp n(n +1) m 2 transition between linear and nonlinear regimes nor U(ω,m,n)= h|ψn (ω)| i, (24) 4 a distinct boundary between turbulence and Rossby m wave spectral ranges. This result reinforces our ear- where ψn (ω) is a time Fourier transformed spectral m lier conclusion that neither nβ nor nR can be associ- coefficient ψn (t) defined in Eq. (10). In 2D turbu- lence without Rossby waves, U(ω,m,n) would be ated with the turbulence - Rossby wave transition. expected to have a symmetric bell shape around the zero frequency. When the waves are present, they cause U(ω,m,n) to develop spikes at frequencies 5. Discussion and conclusions corresponding to the dispersion relation. The corre- lation function U(ω,m,n) is therefore a convenient tool to establish and/or diagnose the dispersion rela- One of the main results of the presented here theo- tions in data. A similar approach based upon the anal- retical analysis and numerical simulations is the con- ysis of the autocorrelation function of the sea surface clusion that a non-dissipative extra-strain in the vor- elevation obtained from the satellite altimetry has ticity equation, a β-term, can cause no arrest of the been used by other researchers (see, e.g., Glazman inverse energy cascade in 2D turbulence. The effect and Cheng (1999); Glazman and Weichman (2005)) of the β-term on energy transfer manifests in cascade to diagnose the baroclinic Rossby waves in the ocean. anisotropization and formation of quasi-1D struc- The results of our analysis are shown in Figs. 9 tures, zonal jets. This result can be likened to that and 10 for the friction dominated and zonostrophic in 3D flows with pure rotation where non-dissipative regimes, respectively. In the former case, for small extra-strains are brought about by the Coriolis force. wavenumbers, Rossby waves produce sharp spikes Again, turbulent cascade in this case is not arrested almost exactly corresponding to the linear dispersion but anisotropized leading to self-organization of the relation (11). Similar spikes also appear at higher flow field into quasi-2D, large-scale columnar vor- wavenumbers although they are not as sharp because tices aligned with the rotation axis (Smith and Lee they are progressively broadened by turbulence. As 2005). Even in the case when extra-strains do enter 15 n 5 10 20 40 the Rhines and friction scales is not universal and de- n/n β 0.36 0.72 1.45 2.90 pends on the flow regime. It is not surprising, there- m/n 50 0.2 4 0.4 fore, that LR has been used as a scaling parame- 0.2 ter characterizing various aspects of flows with a β- −0.4 0 −0.4 0 −0.5 0 0.5 −1 0 1 4 0.1 4 effect. As an example, one may recall that the scaling 0.4 0.2 with LR has been applied both to the equatorial (Hide −0.4 0 −0.4 0 −0.5 0 0.5 −1 0 1 4 0.1 2 0.2 1966) and off-equatorial (Williams 1978) Jovian jets 0.6 although their widths are quite different. Generally, −0.4 0 −0.4 0 −0.5 0 0.5 −1 0 1 0.1 the Rhines scale is a basic dimensional parameter in 4 2 0.2 0.8 flows where a β-effect is a salient feature not nec- ,m,n) −0.4 0 −0.4 0 −0.5 0 0.5 −1 0 1 ω 4 0.1 4 essarily related to geostrophic turbulence (see, e.g., U( 1.0 0.2 Pedlosky (1998); Nof et al. (2004)). It is important, −0.4 0 −0.4 0 −0.5 0 0.5 −1 0 1 ω/Ω therefore, to understand what processes are charac- terized by LR and determine an appropriate veloc- ity scale U. In situations where the spectrum is un- Figure 10: The velocity autocorrelation function, known and the flow is affected by multiple factors, U(ω,m,n), for the zonostrophic regime; nR ' 5, the identification of the dominant processes and the 5 nβ ' 14. The scaling coefficients for U are 10 for correct choice of U are essential (see, e.g., Barry et al. n =5and 10 and 108 for n =20and 40. The trian- (2002)). gles correspond to the linear dispersion relation (11). As was shown in section 4, a nondimensional pa- rameter Rβ = nβ/nR plays a key role in quasi-2D, steady state turbulent flows with a β-effect. By virtue the kinetic energy equation as sink terms, they can- of determining the type of a flow regime, this pa- not totally suppress turbulent cascade. For example, rameter reveals an important information on various in the case of 3D flows with stable stratification, there aspects of large-scale circulations. Recasting Rβ in also exists strong anisotropization. Along the direc- terms of the basic flow parameters, we obtain tion of the extra-force (i.e., the gravity), turbulent ex- 3/10 5 1/10 change diminishes, but in the normal planes, turbu- 1/2 CZ βU Rβ =2 2 lence is not suppressed and may even be enhanced CK (Sukoriansky et al. 2005; Smith and Waleffe 2002). βU5 1/10 In all these cases, the extra-strains support various ' 0.7 . 2 (25) types of linear waves which can coexist with turbu- lence in wide ranges of scales. This number can also be defined in terms of the ex- Our simulations revealed neither a separation be- ternal flow parameters, λ, β and , tween the regions of turbulence and wave domina- 3/10 2 1/20 tion nor a large-scale threshold of the Rossby wave 1/2 CZ β Rβ =2 5 propagation. The opposite is true; similarly to the CK λ internal and inertial waves, Rossby waves can coex- β2 1/20 ' 0.7 . (26) ist with turbulence in a wide range of scales. Rather λ5 than being a scale of the cascade arrest, the Rhines scale may characterize many different phenomena. In The zonostrophic regime is characterized by Rβ > 2 the present study, this conclusion is illustrated by two (in addition to the smallness of the Burger number, examples. In unsteady flows, LR appears as a scale Bu; see e.g. Galperin et al. (2006)) while for the of the largest energy containing structures, while in friction-dominated regime, Rβ . 1.5. The range a steady state zonostrophic regime, the Rhines scale 1.5 . Rβ . 2 corresponds to a transitional regime coincides with the scale of the large-scale friction. whose properties are not universal. It is instructive to Furthermore, it is shown that the relationship between calculate the values of Rβ for different environments. 16 To illustrate the significance of Rβ, we shall consider planets are consistent with Eq, (13a) in both the slope a wide variety of environments with small Bu rang- and the magnitude (Galperin et al. 2001; Sukoriansky ing from the small-scale turntable used in the Greno- et al. 2002). ble experiment (Read et al. 2004) (see also Read et al., this issue) and to the terrestrial oceans to the solar An interesting implication can be drawn regarding the giant planets’ weather layers. “barotropic governor” (James and Gray 1986; James 1987) the essence of which is the reduction of the For the Grenoble experiment, using the parameters potential-kinetic energy conversion rate of the baro- reported in Read et al. (this issue), one obtains Rβ ∈ clinic instability due to the increase of the horizontal (0.5, 2.3), i.e., the experimental flow was marginally barotropic shear resulting from the inverse cascade in zonostrophic. It is not surprising, therefore, that the the barotropic mode. In light of the present results, flow field obtained in some experiments exhibited it is tempting to think of the barotropic mode as a zonation, spectral anisotropization, and build up of governor in a rather broader sense than just a regu- the zonal spectrum described by Eq. (13a). lator of the baroclinic instability. In the case of the zonostrophic regime with a profound inertial range For the oceanic flows, the magnitude of can be es- (or large R ), the barotropic mode contains most of −9 −11 2 −3 β timated between 10 and 10 m s , the lat- the kinetic energy and hence truly governs the large- ter value follows, for example, from the realistic, scale circulation and its energetics. As shown in eddy resolving simulations by Nakano and Hasumi Sukoriansky et al. (2002), the total kinetic energy of (2005). With such values of and U of the or- a circulation in this regime is determined by Ω, R and der of 10 cm s−1, one finds that R ∈ (1, 2.8). β nfr only and is independent of the external forcing or Similarly to the Grenoble experiment, the oceanic the energy conversion rate unless these dependencies flows are on the verge between the zonostrophic and are implicitly embedded in the friction wavenumber friction-dominated regimes. Visually, oceanic flows nfr the physical mechanism of which is not always are quite erratic as would be expected in the transi- well understood and is a subject of an ongoing re- tional regime. However, averaging in time reveals search (see, e.g., M¨ulleret al. (2005)). zonation (see, e.g., Maximenko et al. (2005); Ol- litrault et al. (2006)) and spectral anisotropization (Zang and Wunsch 2001) suggesting that Rβ is rather Acknowledgement We are grateful to Drs P.L. close to 2. In addition, some numerical models Read, M.E. McIntyre, W.R. Young, A. Showman, show the build up of the zonal spectrum according H.-P. Huang and N. Maximenko for numerous dis- to Eq. (13a) (Galperin et al. 2004). Even though the cussions and comments during the course of this zonostrophic inertial range in the ocean is small and research. Thoughtful comments from anonymous the barotropic currents are relatively weak, by virtue reviewers helped us to improve and clarify the of the zonation that penetrates through considerable manuscript. Partial support of this study by the depth these currents can still play an important role ARO grant W911NF-05-1-0055 and the Israel Sci- in the dynamic and transport processes. 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