1. Introduction
Terrestrial and planetary circulations are described by nonlinear equations that support various types of waves in the linear limit. The real flows exhibit a complicated interplay between turbulence and waves. While in a certain range of scales, turbulent scrambling may overwhelm the wave behavior and lead to the disappearance of the dispersion relation (such as 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 and emergence of systems with strong wave–turbulence interaction. Systems that combine anisotropic turbulence and waves exhibit behavior very different from that of the classical isotropic and homogeneous turbulence (McIntyre 2001). In the context of large-scale atmospheric flows, the interaction between turbulence and waves yields dynamically rich macroturbulence (Held 1999; Schneider 2006), the notion that underlies diverse phenomena ranging from a local weather to a global climate. The subtlety of this interaction is further highlighted by the fact that, because of the planetary rotation and Taylor–Proudman theorem, large-scale flows are quasi-two-dimensionalized and may be conducive to the development of the inverse energy cascade (Read 2005). One of the parameters that characterize macroturbulence is the Rhines wavenumber, kR = (β/2U)1/2, or the Rhines scale, LR ∼ k−1R, where U is the rms fluid velocity, and β is the northward gradient of the Coriolis parameter (Rhines 1975; in the original paper, this wavenumber was denoted by kβ; here, kR is used instead, as the notation kβ is reserved for another wavenumber to appear later). At this scale, the inverse cascade supposedly becomes arrested, and the nonlinear turbulent behavior is replaced by excitation of linear Rossby waves. This scale has also been associated with flow reorganization into the bands of alternating zonal jets, the process known as zonation, and the width of the jets has often been scaled with k−1R. The Rhines scale plays a prominent role in many theories of large-scale atmospheric and oceanic circulation (see, e.g., James and Gray 1986; Vallis and Maltrud 1993; Held and Larichev 1996; Held 1999; Lapeyre and Held 2003; Schneider 2004; LaCasce and Pedlosky 2004; Vallis 2006) as well as in theories of planetary circulations (see, e.g., the review by Vasavada and Showman 2005) and, possibly, even stellar convection (see, e.g., Miesch 2003).
The meaning of the Rhines scale is not always clear, however. Even in the barotropic case, different regimes would arise for continuously forced and decaying flows, for flows with and without friction, and for flows in bounded and unbounded domains. As a result, the Rhines scale may be time-dependent, stationary, or entirely obscured by friction. In many studies, the scaling with k−1R is implied but the coefficient of proportionality varies in a wide range. There also exists a multitude of other scaling parameters that characterize various aspects of the flow and flow regimes. It is important, therefore, to understand the hierarchy of these parameters and the place of kR in this hierarchy. This is precisely the goal of the present study. This goal cannot be achieved without clarification 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 issues will be addressed theoretically; the theoretical results will be substantiated in numerical simulations.
The paper is organized in the following fashion. The next section presents the basics of the theory of two-dimensional (2D) turbulence with a β effect and accentuates its problems. Section 3 presents analysis of the interaction between inverse energy cascade of 2D turbulence and Rossby waves and discusses the notion of the cascade arrest. Section 4 elaborates this analysis and clarifies the meaning of the Rhines scale using numerical simulations of barotropic 2D turbulence on the surface of a rotating sphere. In addition, that section describes the hierarchy of scaling parameters and corresponding flow regimes as well as the interaction between turbulence and Rossby waves in different regimes. Finally, section 5 presents discussion and some conclusions.
2. Turbulence and Rossby waves: The basics
The interaction between 2D turbulence and Rossby waves has been considered in the pioneering study by Rhines (1975). Starting with the barotropic vorticity equation on a β plane, he concentrated on mainly unforced barotropic flows caused by initially closely packed fields of eddies spreading from a state with a δ function–like spectrum peaked at some wavenumber k0. Among the major conclusions of the paper, the following have the most relevance to the present study:
At the wavenumber, denoted as kR, at which the root-mean-square velocity U is equal to the phase speed of Rossby waves with an average orientation, cp = β/2k2, there exists a subdivision of the spectrum on turbulence (k > kR) and wave (k < kR) modes.
The expansion of the flow field to kR triggers wave propagation and slowing down, or the arrest, of the cascade to smaller wavenumbers.
Triad interactions of the modes with wavenumbers close to kR require simultaneous resonance in wavenumbers and frequencies.
On average, triad interactions transfer energy to modes with smaller frequencies and smaller wavenumbers causing the anisotropization of the flow field. As a result, the energy accumulates in modes with small north–south wavenumbers that correspond to east–west currents, zonal jets, giving rise to the process of zonation.
The width of the zonal jets scales with k−1R.
In a steadily forced flow, the spectrum is expected to develop a sharp peak at kR and rapidly decrease for k > kR.
Items 3 and 4 have been confirmed in numerous theoretical, numerical, and experimental studies (see, e.g., Williams 1975, 1978; Holloway and Hendershott 1977; Panetta 1993; Vallis and Maltrud 1993; Chekhlov et al. 1996; Cho and Polvani 1996; Nozawa and Yoden 1997; Huang and Robinson 1998; Huang et al. 2001; Read et al. 2004; Vasavada and Showman 2005; Galperin et al. 2006). A new light on zonation was shed by Balk (2005). Exploring a new invariant for inviscid 2D flows with Rossby waves (Balk 1991; Balk and van Heerden 2006), he showed that the transfer of energy from small to large scales takes place in such a way that most of the energy is directed to the zonal jets. The rest of the aforementioned issues, however, remain controversial, particularly when compounded with other complicating factors such as the friction, continuous forcing, effects of the boundaries, effects of stratification, etc. Let us consider, for example, a realistic system in which the bottom friction acts like a large-scale drag. One can introduce a wavenumber associated with that drag, kfr, as a wavenumber at which the characteristic friction time is equal to that of the flow. If kfr > kR then the frictional processes would forestall the arrest and the Rhines scale would require modification (James and Gray 1986; Danilov and Gurarie 2002; Smith et al. 2002). In flows with continuous small-scale forcing at a rate ϵ, the equilibrium between the eddy turnover time and the Rossby wave period selects an anisotropic transition wavenumber with an amplitude kβ ∝ (β3/ϵ)1/5 (the proportionality coefficient will be established later); its angular distribution resembles a dumbbell (Vallis and Maltrud 1993; Holloway 1986; Vallis 2006). The introduction of kβ raises a slew of new questions. If there is an arrest in the forced flows, which one of the two wavenumbers, kR and kβ, should be associated with the arrest scale? At which scale is the anticipated in item 2 Rossby wave excitation triggered? Can Rossby waves coexist with turbulence on scales intermediate between k−1R and k−1β? With regard to item 5, one may ask which one of these two wavenumbers should be used as a scaling parameter for the width of the zonal jets? Indeed, although the mechanism of zonation has by now been solidly established, the scaling 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 scale similar to k−1R for the width of the equatorial jets on Jupiter and Saturn while Williams (1978) and others applied it to the off-equatorial jets. Conclusive scaling with k−1R has materialized neither in the Grenoble experiment (Read et al. 2004) nor in the recent analysis of the eddy-resolving simulations of oceanic jets by Richards et al. (2006).


Considering a space of parameters that characterize different regimes in barotropic, forced and dissipative β-plane turbulence, Galperin et al. (2006) have found that in a certain subspace, a flow attains a universal regime with the zonal spectrum (1) and CZ ≃ 0.5. This subspace is wide enough to include important laboratory, terrestrial, and planetary flows (Galperin et al. 2006).
Despite the progress made in understanding of various aspects of β-plane turbulence, there still remains a great deal of confusion regarding the arrest of the inverse energy cascade and the Rhines scale. The fact that the interplay between turbulence and Rossby waves has never been thoroughly investigated only adds to this confusion. The next section gathers some theoretical arguments that may help to clarify this perplexity. In section 4, these arguments will be substantiated via numerical experimentation.
3. Is the cascade really arrested?
The cascade arrest is often understood as a transition of a flow character from strongly nonlinear and turbulent to weakly nonlinear and wave-dominated under the action of a nondissipative extra strain in the Bradshaw (1973) terminology, a β effect. This transition presumably takes place at a scale k−1R. Such an interpretation applies to unforced flows considered by Rhines (1979) in great detail (see also Majda and Wang 2006, chapters 10–13). Rhines (1979) showed that in unforced flows, both the nonlinear term and a rate of the energy transfer to large scales decrease with time. This tendency can be traced to the impediment to triad interactions caused by the frequency resonance condition discussed in the previous section. A soft transition from a strongly nonlinear to a weakly nonlinear regime takes place at the crossover wavenumber, kR (Rhines 1979). This interpretation of the cascade arrest is sometimes applied to forced flows (James and Gray 1986). The extension of the results valid for the unforced dynamics (ϵ = 0) to flows with continuous forcing (ϵ ≠ 0) may be problematic however. Already at the level of the dimensional analysis, the presence of the nonzero ϵ and appearance of the nontrivial wavenumber kβ point to significant differences between the two cases. Vallis and Maltrud (1993) have first recognized the importance of the transitional wavenumber, kβ. In their forced dissipative simulations, Vallis and Maltrud (1993) observed a “piling up” of the computed energy spectrum in the vicinity of kβ. This piling up took place in a short range of wavenumbers, however, because the simulation parameters were set such that kβ was close to kR. The spectral steepening was thought to be a result of the dumbbell becoming a barrier to the energy flux to larger scales. Chekhlov et al. (1996) investigated a regime with kβ ≫ kR. They have also observed the energy piling up in the vicinity of kβ but they have attributed this phenomenon to a development of a new regime of circulation with the steep spectrum (1). Chekhlov et al. (1996) have emphasized that although the total upscale energy flux in forced, undamped flows remains constant, the rate of the energy front propagation decreases due to the steepening of the zonal spectrum. The inverse cascade anisotropization combined with the zonal spectrum’s steepening facilitates the establishment of the zonostrophic regime in flows with a large-scale drag (Galperin et al. 2006).
Although the frequency resonance constraint impedes the triad interactions in both forced and unforced flows, it does not cause the arrest of the inverse energy cascade, which continues to pump energy to ever-larger scales at a constant rate ϵ. The frequency resonance impediment only facilitates the funneling of the energy flux into the zonal modes leading to spectral anisotropization (Chekhlov et al. 1996) and reorganization of the flow field into a lower-dimensional slow manifold decoupled from the fast Rossby waves (Smith and Lee 2005). This manifold manifests itself as a system of quasi-one-dimensional alternating zonal jets. The threshold of the spectral anisotropization is characterized by the wavenumber kβ while the width of the zonal jets is determined by the large-scale friction (Sukoriansky et al. 2002). It is important to emphasize that forced β-plane turbulence never collapses to a linear state since the nonlinearity is the very factor that sustains the slow manifold. It is clear, therefore, that a β effect cannot halt the inverse energy cascade (Chekhlov et al. 1996; Sukoriansky et al. 2002; Galperin et al. 2006).
In the remainder of this section, we shall elaborate various scaling relationships pertinent to forced anisotropic turbulence with a β effect. These relationships will be used to further elucidate the absence of the cascade arrest. The flows will be assumed to contain a stationary random forcing maintaining an inverse energy cascade at a constant rate ϵ. This case has most relevance to the real world large-scale terrestrial and planetary circulations (Galperin et al. 2006). The importance of the parameter ϵ is underscored by the conjecture that the diffusion coefficient of the poleward heat transport strongly depends on it (Held and Larichev 1996; Held 1999; Lapeyre and Held 2003).
Two classes of flows will be considered, those without and with the action of the large-scale drag. The first class is rather unrealistic because, first, some kind of a drag is always present in all real flows, and, second, the large-scale energy condensation would eventually distort flow configuration in long-term integrations (Smith and Yakhot 1993, 1994). Such flows are important, however, for understanding of the dynamics of the transients and mechanisms that lead to the establishment of observable long-term patterns. These patterns are represented in the second class of flows.






Simulations by Chekhlov et al. (1996), Smith and Waleffe (1999), and Huang et al. (2001) of the evolving turbulence with a β effect have shown that as the energy propagates to wavenumbers smaller than kβ, the nature of the flow and its spectral characteristics change markedly. At a wavenumber close to kβ, the zonal spectrum, EZ(ky), undergoes reorganization and attains a steep distribution (1) while the nonzonal, or the residual spectrum largely preserves the shape given by the KBK law, Eq. (3).
The anisotropization of the inverse cascade has been demonstrated using the results of direct numerical simulation (DNS) of a β-plane turbulence by Chekhlov et al. (1996) who considered the energy transfer function,
The spectral energy transfer has also been calculated by Read et al. (2007) using the data from the Grenoble experiment (Jets special collection). In qualitative similarity to Fig. 1, they obtained an anisotropic distribution of








When a large-scale drag is present, an ensuing steady-state flow regime is determined by the ratios between kfr, kβ, kd, and kξ (Galperin et al. 2006). There exists a certain subspace of these parameters in which a flow undergoes zonation and develops a universal stationary regime whose anisotropic spectrum in the β-dominated range kfr < k < kβ approaches a distribution given by Eqs. (1) and (3). The steep zonal spectrum is better pronounced for kfr ≪ kβ; it also extends to the region k > kβ to its intersection with the KBK, isotropic, modal k−8/3 spectrum as explained in Sukoriansky et al. (2002). The parameter subspace of this regime is delineated by three conditions: 1) the β-dominated range is sufficiently wide; 2) the forcing operates on scales not impacted upon by the β effect; and 3) the frictional wavenumber is large enough [kfr ≳ 4(2π/L), L being the system size] to avoid the large-scale energy condensation (Smith and Yakhot 1993, 1994). For convenience, Galperin et al. (2006) have coined this regime zonostrophic. Sukoriansky et al. (2002) have shown that for this regime, the friction wavenumber, kfr, is the final destination of the moving energy front given by Eq. (7), and thus kR ∼ kfr. The latter relationship will be confirmed in the next section. Similarly to the unsteady regime, kR is not a scale of inverse cascade termination by a β effect; the advance of the energy front can only be stopped by the large-scale drag. Outside the zonostrophic regime, the flow depends on some additional parameters and kR ceases to be the final destination of the moving energy front; the latter is still associated with kfr, however.
From the previous studies and the scaling arguments in this section, we draw two main conclusions: 1) in unsteady flows, the marching in time of the moving energy front slows down from the −3/2 to −1/4 law upon crossing of kβ and the front’s location can be related to kR; 2) in steady flows, kR can be identified with kfr in the zonostrophic regime. These conclusions will be confirmed in numerical simulations of barotropic 2D turbulence on the surface of a rotating sphere described in the next section. In addition, using the frequency analysis in Fourier space, an analysis of the interplay between turbulence and Rossby waves will also be presented.
4. Turbulence and Rossby waves: Results of numerical simulations














The values of the parameters used in unsteady simulations are summarized in Table 1. In total, four numerical experiments were performed. Although Ω and ϵ varied in a wide range in these experiments, their values were set such as to ensure nβ < nξ. For statistical analysis of the results, ensemble averaging over 80 to 110 independent realizations was employed.
At first, the energy evolution was investigated. It is well known that without rotation, the total energy of the flow increases linearly with time, Etot(t) = ϵt. We wanted to verify that, first, the linear trend is preserved in the case of nonzero rotation and, second, the zonal and nonzonal total energy components, EZtot(t) and ERtot(t) respectively, would follow a similar trend [here, Etot(t) = EZtot(t) + ERtot(t); ERtot(t) contains both the eddy and wave energies]. Figure 2 shows the evolution of the total energy and its components for experiment 1 in a simulation of the duration of 10 000 planetary days (for experiment 1, 1 day = 2πT). Initially, the zonal energy is very small and the entire energy is mostly concentrated in the nonzonal component. That component 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, after initial restructuring, both EZtot(t) and ERtot(t) grow linearly in time for as long as the flow evolution remains unobstructed by the action of a large-scale drag or domain boundaries.
We have conducted an additional simulation featuring β = 0 and the same ϵ as the one used in experiment 1. The evolution of the total energy in that simulation was indistinguishable from the case with β ≠ 0. The only difference was that in the case β = 0, the energy front required much shorter time to reach the largest scales of the system. The total energy at that moment is marked by a triangle in Fig. 2. Clearly, the shortening of the evolution time is a result of the fast front propagation described by Eq. (8). For β ≠ 0, the system can accumulate significantly higher energy; according to Eq. (5), this energy scales with β2 and is independent of ϵ. This increased energetic capacity of the system is a direct result of the spectral anisotropization and steepening of the zonal spectrum according to Eq. (1).
The behavior of the total energy components exhibited in Fig. 2 requires some clarification. Further insight comes from the consideration of the evolution of the total and residual energy spectra for the same experiment 1 shown in Fig. 3. Driven by the inverse cascade, the spectrum expands toward the small wavenumber end. Until the transitional wavenumber nβ is reached, the energy front, nm(t), marked by the black dots in Fig. 3a, can be identified as a wavenumber with the highest energy. For n < nm, the spectral energy density rapidly decreases. As in classical 2D turbulence, the ensemble-averaged spectra at the wavenumbers swept by nm remain in a quasi–steady state. The approximate observance of the classical isotropic KBK distribution (13b) is an indication that a β effect does not yet have a strong influence on these scales. This behavior changes when nm becomes smaller than nβ. The spectrum begins to steepen eventually attaining the level dictated by Eq. (13a). As discussed earlier, the saturation of the spectrum at this level is a slow process during which the energy transfer continues not only to the mode nm but also to the modes with n < nm. As a result, along with the slope (13a), the energy spectrum also forms a plateau for n ≲ nm. Let us emphasize that the moving energy front nm is not only a transitional wavenumber between the −5 slope and a plateau but it also corresponds to the number of the zonal jets (Chekhlov et al. 1996).
By comparing Figs. 2 and 3, we establish that the aforementioned change in the behaviors of EZtot(t) and ERtot(t) is concurrent with the restructuring of the spectra from isotropic, KBK to strongly anisotropic distributions upon crossing nβ. Although at large times, ERtot is relatively small, the nonzonal modes play a crucial role in maintaining the zonal flows as they effectively preserve the upscale energy cascade.
Note that the evolution of the nonzonal spectrum ER(n) shown in Fig. 3b provides practically no information on a flow field transformation under the action of a β effect. Indeed, ER(n) retains an approximate KBK distribution, Eq. (13b), with nearly the same value of the coefficient CK for both nβ < nm and nβ > nm. Only the zonal spectrum, EZ(n), reflects the anisotropization by attaining a steeper slope.














Equation (21) was verified in a series of simulations of the zonostrophic (filled dots in Fig. 5) and marginally zonostrophic (unfilled triangles adjacent to the zonostrophic region) regimes with widely varying values of λ (5 × 10−4 ≤ λ ≤ 2.5 × 10−3), ϵ (3.3 × 10−9 ≤ ϵ ≤ 9 × 10−8), and Ω (0.3 ≤ Ω ≤ 2). Note that due to the steep zonal spectrum, zonostrophic flows have low variability and require long-term integrations to assemble records sufficient for statistical analysis. Galperin and Sukoriansky (2005) have estimated that a record length of up to 100τ after attaining the steady state, τ = (2λ)−1 being the time scale associated with the large-scale drag, would be adequate for the spectral analysis. Accordingly, our steady-state simulations were of duration of about 60 to 100τ. The results of these simulations are summarized in Fig. 6. Note that the value of the coefficient in the correlation between nfr and nR in Eq. (21) had to be adjusted to 1.2. As one can see, the linear relationship between nR and nfr holds faithfully. One concludes, therefore, that, first, similarly to the unsteady case, a β effect does not halt the inverse energy cascade which is now damped by friction, and, second, in the zonostrophic regime, the Rhines scale is, in fact, a scale that characterizes the effect of the large-scale drag.




Equation (23) has been validated in numerical simulations with large values of the drag coefficient, λ, which was chosen such that nβ/nR ≲ 1.5. These simulations are shown as unfilled squares in Fig. 5. Figure 7 shows a good agreement between the actual values of nfr and those calculated from Eq. (23) with the coefficient adjusted to 10.
There still remains a question about the proper scaling of the width of the zonal jets which in some studies was found to be close to n−1R. When the jets have approximately homogeneous spatial distribution, their width can be estimated from the number of the jets, njet. It is of interest, therefore, to compare njet with nfr and nR.
Figure 8, derived from simulations of the zonostrophic regime, indicates that njet ≃ nR. The spread of the data points is caused by the discreteness of njet. In the range 4 ≤ njet ≤ 10, for example, unavoidable unity deviations appear like a 10%–25% noise. In fact, given this uncertainty, nR provides a faithful estimation of the number of the zonal jets. Using the correlation between nfr and nR shown in Fig. 6, one concludes that both n−1fr and n−1R are appropriate scaling parameters for the jets’ width in the zonostrophic regime. We have also attempted to scale njet with nR calculated using the nonzonal velocities only. In that case, the correlation between njet and nR was considerably worse than that in Fig. 8 and, therefore, it is not shown here.


5. Discussion and conclusions
One of the main results of the presented here theoretical analysis and numerical simulations is the conclusion that a nondissipative extra strain in the vorticity equation, a β term, can cause no arrest of the inverse energy cascade in 2D turbulence. The effect of the β term on energy transfer manifests in cascade anisotropization and formation of quasi-1D structures, zonal jets. This result can be likened to that in 3D flows with pure rotation where nondissipative extra strains are brought about by the Coriolis force. Again, turbulent cascade in this case is not arrested but anisotropized leading to self-organization of the flow field into quasi-2D, large-scale columnar vortices aligned with the rotation axis (Smith and Lee 2005). Even in the case when extra strains do enter the kinetic energy equation as sink terms, they cannot totally suppress turbulent cascade. For example, in the case of 3D flows with stable stratification, there also exists strong anisotropization. Along the direction of the extra force (i.e., the gravity), turbulent exchange diminishes, but in the normal planes, turbulence is not suppressed and may even be enhanced (Sukoriansky et al. 2005; Smith and Waleffe 2002). In all these cases, the extra strains support various types of linear waves, which can coexist with turbulence in wide ranges of scales.
Our simulations revealed neither a sharp separation between the regions of turbulence and wave domination nor a large-scale threshold of the Rossby wave propagation. The opposite is true; similarly to the internal and inertial waves, Rossby waves can coexist with turbulence in a wide range of scales. Rather than being a scale of the cascade arrest, the Rhines scale may characterize many different phenomena. In the present study, this conclusion is illustrated by two examples. In unsteady flows, LR appears as a scale of the largest energy containing structures, while in a steady-state zonostrophic regime, the Rhines scale coincides 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 depends on the flow regime. It is not surprising, therefore, that LR has been used as a scaling parameter characterizing various aspects of flows with a β effect. As an example, one may recall that the scaling with LR has been applied both to the equatorial (Hide 1966) and off-equatorial (Williams 1978) Jovian jets although their widths are quite different. Generally, the Rhines scale is a basic dimensional parameter in flows not necessarily related to geostrophic turbulence where a β effect is a salient feature (see, e.g., Pedlosky 1998; Nof et al. 2004).




For the Grenoble experiment, using the parameters reported in Read et al. (2007), one obtains Rβ ∈ (0.5, 2.3); that is, the experimental flow was marginally zonostrophic. It is not surprising, therefore, that the flow field obtained in some experiments exhibited zonation, spectral anisotropization, and build up of the zonal spectrum described by Eq. (13a).
For the oceanic flows, the magnitude of ϵ can be estimated between 10−9 and 10−11 m2 s−3, the latter value follows, for example, from the realistic, eddy-resolving simulations by Nakano and Hasumi (2005). With such values of ϵ and U of the order of 10 cm s−1, one finds that Rβ ∈ (1, 2.8). Similarly to the Grenoble experiment, the oceanic flows are on the verge between the zonostrophic and friction-dominated regimes. Visually, oceanic flows are quite erratic as would be expected in the transitional regime. However, averaging in time reveals zonation (see, e.g., Maximenko et al. 2005; Ollitrault et al. 2006) and spectral anisotropization (Zang and Wunsch 2001) suggesting that Rβ is rather close to 2. In addition, some numerical models show the build up of the zonal spectrum according to Eq. (13a) (Galperin et al. 2004). Even though the zonostrophic inertial range in the ocean is small and the barotropic currents are relatively weak, by virtue of the zonation that penetrates through considerable depth these currents can still play an important role in the dynamic and transport processes. Studies of these processes are now becoming an area of an active research (see, e.g., Galperin et al. 2004; Smith 2005; Richards et al. 2006; Ollitrault et al. 2006; Nadiga 2006; Eden 2006).
For the solar giant planets, ϵ can be estimated at 10−8 m2 s−3 (Galperin et al. 2006) yielding Rβ ∼ 10, 25, 30, and 40 for Jupiter, Saturn, Uranus, and Neptune, respectively. The large values of Rβ on all four solar giant planets indicate that their atmospheric circulations feature well established zonostrophic regimes. Indeed, the energy spectra of the zonal flows on these planets are consistent with Eq. (13a) in both the slope and the magnitude (Galperin et al. 2001; Sukoriansky et al. 2002).
The present results emphasize the importance of the barotropic mode in flows with a small Burger number. In the case of the zonostrophic regime with a profound inertial range (or large Rβ), the barotropic mode contains most of the kinetic energy and hence governs the large-scale circulation and its energetics. As shown in Sukoriansky et al. (2002), the total kinetic energy of a circulation in this regime is determined by Ω, R, and nfr only and is independent of the external forcing or the potential energy conversion rate unless these dependencies are implicitly embedded in the friction wavenumber nfr the physical mechanism of which is not always well understood and is a subject of an ongoing research (see, e.g., Müller et al. 2005).
Acknowledgments
We are grateful to Drs P. L. Read, M. E. McIntyre, W. R. Young, A. Showman, H.-P. Huang, and N. Maximenko for numerous discussions and comments during the course of this research. Thoughtful comments from anonymous reviewers helped us to improve and clarify the manuscript. Partial support of this study by the ARO Grant W911NF-05-1-0055 and the Israel Science Foundation Grant 134/03 is greatly appreciated.
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Normalized spectral energy transfer,
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

Normalized spectral energy transfer,
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
Normalized spectral energy transfer,
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

Evolution of the total energy, Etot(t) (thick solid line), and its zonal (dashed line) and nonzonal (dashed–dotted line) components, EZtot(t) and ERtot(t), respectively, in experiment 1. A triangle on Etot(t) shows the time required for the energy front to reach the largest scales in the system in the case when β = 0 but ϵ is the same as in experiment 1.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

Evolution of the total energy, Etot(t) (thick solid line), and its zonal (dashed line) and nonzonal (dashed–dotted line) components, EZtot(t) and ERtot(t), respectively, in experiment 1. A triangle on Etot(t) shows the time required for the energy front to reach the largest scales in the system in the case when β = 0 but ϵ is the same as in experiment 1.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
Evolution of the total energy, Etot(t) (thick solid line), and its zonal (dashed line) and nonzonal (dashed–dotted line) components, EZtot(t) and ERtot(t), respectively, in experiment 1. A triangle on Etot(t) shows the time required for the energy front to reach the largest scales in the system in the case when β = 0 but ϵ is the same as in experiment 1.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

(a) Total and (b) nonzonal energy spectra at different times in experiment 1 of unsteady simulations (λ = 0). The spectra are marked by the total energy (× 105) accumulated in the flow field from the beginning of simulations. Black dots show the location of the energy front nm. The transition from the −5/3 to −5 slope around nβ is clearly visible in (a). Before the transition, the spectral energy density rapidly decreases for n < nm. After the transition, the energy accumulates at the modes n < nm facilitating the emergence of the plateau.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

(a) Total and (b) nonzonal energy spectra at different times in experiment 1 of unsteady simulations (λ = 0). The spectra are marked by the total energy (× 105) accumulated in the flow field from the beginning of simulations. Black dots show the location of the energy front nm. The transition from the −5/3 to −5 slope around nβ is clearly visible in (a). Before the transition, the spectral energy density rapidly decreases for n < nm. After the transition, the energy accumulates at the modes n < nm facilitating the emergence of the plateau.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
(a) Total and (b) nonzonal energy spectra at different times in experiment 1 of unsteady simulations (λ = 0). The spectra are marked by the total energy (× 105) accumulated in the flow field from the beginning of simulations. Black dots show the location of the energy front nm. The transition from the −5/3 to −5 slope around nβ is clearly visible in (a). Before the transition, the spectral energy density rapidly decreases for n < nm. After the transition, the energy accumulates at the modes n < nm facilitating the emergence of the plateau.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The moving energy front, nm, in unsteady simulations. The transition from the −3/2 to the −1/4 evolution law in the vicinity of the transitional wavenumbers nβ located at the intersections of the respective dashed lines is clearly visible for all four simulations. The figure demonstrates the absence of the halting wavenumber for the inverse energy cascade; the energy front can penetrate to the smallest wavenumbers available in the system.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The moving energy front, nm, in unsteady simulations. The transition from the −3/2 to the −1/4 evolution law in the vicinity of the transitional wavenumbers nβ located at the intersections of the respective dashed lines is clearly visible for all four simulations. The figure demonstrates the absence of the halting wavenumber for the inverse energy cascade; the energy front can penetrate to the smallest wavenumbers available in the system.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
The moving energy front, nm, in unsteady simulations. The transition from the −3/2 to the −1/4 evolution law in the vicinity of the transitional wavenumbers nβ located at the intersections of the respective dashed lines is clearly visible for all four simulations. The figure demonstrates the absence of the halting wavenumber for the inverse energy cascade; the energy front can penetrate to the smallest wavenumbers available in the system.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

Possible flow regimes in 2D turbulence with a β effect in the space of parameters nR and nβ. The subspace of the friction-dominated regimes (unfilled squares) occupies the area to the left of the upper dashed line while the subspace of the zonostrophic regime (filled dots) outlined by the chain inequality (17) is confined to the area between the middle, bottom, and vertical dashed lines for the R133 experiments. For higher resolution, the vertical line moves to larger values of nβ.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

Possible flow regimes in 2D turbulence with a β effect in the space of parameters nR and nβ. The subspace of the friction-dominated regimes (unfilled squares) occupies the area to the left of the upper dashed line while the subspace of the zonostrophic regime (filled dots) outlined by the chain inequality (17) is confined to the area between the middle, bottom, and vertical dashed lines for the R133 experiments. For higher resolution, the vertical line moves to larger values of nβ.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
Possible flow regimes in 2D turbulence with a β effect in the space of parameters nR and nβ. The subspace of the friction-dominated regimes (unfilled squares) occupies the area to the left of the upper dashed line while the subspace of the zonostrophic regime (filled dots) outlined by the chain inequality (17) is confined to the area between the middle, bottom, and vertical dashed lines for the R133 experiments. For higher resolution, the vertical line moves to larger values of nβ.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The friction scale, nfr, determined as the intersection of the plateau and the −5 ranges 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.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The friction scale, nfr, determined as the intersection of the plateau and the −5 ranges 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.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
The friction scale, nfr, determined as the intersection of the plateau and the −5 ranges 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.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The correlation between the actual and theoretical values of nfr calculated using (23) in simulations of the large-scale drag dominated 2D turbulence with a β effect.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The correlation between the actual and theoretical values of nfr calculated using (23) in simulations of the large-scale drag dominated 2D turbulence with a β effect.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
The correlation between the actual and theoretical values of nfr calculated using (23) in simulations of the large-scale drag dominated 2D turbulence with a β effect.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The correlation between the number of the zonal jets, njet, estimated from the velocity profile in the physical space, and nR obtained from numerical simulations of the zonostrophic regime.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The correlation between the number of the zonal jets, njet, estimated from the velocity profile in the physical space, and nR obtained from numerical simulations of the zonostrophic regime.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
The correlation between the number of the zonal jets, njet, estimated from the velocity profile in the physical space, and nR obtained from numerical simulations of the zonostrophic regime.
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The velocity autocorrelation function, U(ω, m, n) × 107, for the friction dominated regime; nR ≃ 9, nβ ≃ 12. The triangles correspond to the linear dispersion relation (11).
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The velocity autocorrelation function, U(ω, m, n) × 107, for the friction dominated regime; nR ≃ 9, nβ ≃ 12. The triangles correspond to the linear dispersion relation (11).
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
The velocity autocorrelation function, U(ω, m, n) × 107, for the friction dominated regime; nR ≃ 9, nβ ≃ 12. The triangles correspond to the linear dispersion relation (11).
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The velocity autocorrelation function, U(ω, m, n), for the zonostrophic regime; nR ≃ 5, nβ ≃ 14. The scaling coefficients for U are 105 for n = 5 and 10 and 108 for n = 20 and 40. The triangles correspond to the linear dispersion relation (11).
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1

The velocity autocorrelation function, U(ω, m, n), for the zonostrophic regime; nR ≃ 5, nβ ≃ 14. The scaling coefficients for U are 105 for n = 5 and 10 and 108 for n = 20 and 40. The triangles correspond to the linear dispersion relation (11).
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1
The velocity autocorrelation function, U(ω, m, n), for the zonostrophic regime; nR ≃ 5, nβ ≃ 14. The scaling coefficients for U are 105 for n = 5 and 10 and 108 for n = 20 and 40. The triangles correspond to the linear dispersion relation (11).
Citation: Journal of the Atmospheric Sciences 64, 9; 10.1175/JAS4013.1