1. Introduction
Robust eddy-driven zonal jets are ubiquitous in planetary atmospheres (Ingersoll 1990; Ingersoll et al. 2004; Vasavada and Showman 2005). Laboratory experiments, theoretical studies, and numerical simulations show that small-scale turbulence self-organizes into large-scale coherent structures, which are predominantly zonal and, furthermore, that the small-scale turbulence supports the jets against eddy mixing (Starr 1968; Huang and Robinson 1998; Read et al. 2007; Salyk et al. 2006). One of the simplest models, which is a test bed for theories regarding turbulence self-organization, is forced–dissipative barotropic turbulence on a beta plane.
An advantageous framework for understanding coherent zonal jet self-organization is the study of the statistical state dynamics (SSD) of the flow. SSD refers to the dynamics that governs the statistics of the flow rather than the dynamics of individual flow realizations. However, evolving the hierarchy of the flow statistics of a nonlinear dynamics soon becomes intractable; a turbulence closure is needed. Unlike the usual paradigm of homogeneous isotropic turbulence, when strong coherent flows coexist with the incoherent turbulent field, the SSD of the turbulent flow is well captured by a second-order closure (Farrell and Ioannou 2003, 2007, 2009; Tobias et al. 2011; Srinivasan and Young 2012; Bakas and Ioannou 2013a; Tobias and Marston 2013; Constantinou et al. 2014a,b; Thomas et al. 2014; Ait-Chaalal et al. 2016; Constantinou et al. 2016; Farrell et al. 2016; Farrell and Ioannou 2017; Fitzgerald and Farrell 2018a, 2019; Frishman and Herbert 2018; Bakas and Ioannou 2019b). Such a second-order closure comes in the literature under two names: stochastic structural stability theory (S3T; Farrell and Ioannou 2003) and cumulant expansion at second order (CE2; Marston et al. 2008). We refer to this second-order closure as S3T.
Using the S3T second-order closure it was first theoretically predicted that zonal jets in barotropic beta-plane turbulence emerge spontaneously out of a background of homogeneous turbulence through an instability of the SSD (Farrell and Ioannou 2007; Srinivasan and Young 2012). That is, S3T predicts that jet formation is a bifurcation phenomenon, similar to phase transitions, that appears as the turbulence intensity crosses a critical threshold. This prediction comes in contrast with the usual theories for zonal jet formation that involve anisotropic arrest of the inverse energy cascade at the Rhines scale (Rhines 1975; Vallis and Maltrud 1993). Jet emergence as a bifurcation was subsequently confirmed by comparison of the analytic predictions of the S3T closure with direct numerical simulations (Constantinou et al. 2014a; Bakas and Ioannou 2014). This flow-forming SSD instability is markedly different from hydrodynamic instability in which the perturbations grow in a fixed mean flow. In the flow-forming instability, both the coherent mean flow and the incoherent eddy field are allowed to change. The instability manifests as follows: a weak zonal flow that is inserted in an otherwise homogeneous turbulent field organizes the incoherent fluctuations to coherently reinforce the zonal flow. This instability has analytic expression only in the SSD, and we therefore refer to this new kind of instabilities as “SSD instabilities.” In particular, the flow-forming SSD instability of the homogeneous turbulent state to zonal jet mean-flow perturbations is also referred to as “zonostrophic instability” (Srinivasan and Young 2012).
Kraichnan (1976) suggested that the large-scale mean flow is supported by small-scale eddies. Indeed, when the large scales dominate the eddy field (i.e., when the large-scale shear time

Second-order closure captures the mean-flow dynamics despite differences in structure of eddy spectra. Shown here are the energy spectra for (left) a fully nonlinear simulation [see (1)] and (right) its quasilinear approximation (i.e., employing the second-order closure). Both simulations form four jets of similar strength. Setup as described in section 5 with
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1

Second-order closure captures the mean-flow dynamics despite differences in structure of eddy spectra. Shown here are the energy spectra for (left) a fully nonlinear simulation [see (1)] and (right) its quasilinear approximation (i.e., employing the second-order closure). Both simulations form four jets of similar strength. Setup as described in section 5 with
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1
Second-order closure captures the mean-flow dynamics despite differences in structure of eddy spectra. Shown here are the energy spectra for (left) a fully nonlinear simulation [see (1)] and (right) its quasilinear approximation (i.e., employing the second-order closure). Both simulations form four jets of similar strength. Setup as described in section 5 with
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1
However, surprisingly, S3T remains accurate even at a perturbative level, that is, when the mean flows/jets are just emerging with
The dynamics that underlie the flow-forming SSD instability of the homogeneous state is well understood; Bakas and Ioannou (2013b) and Bakas et al. (2015) studied in detail this eddy–mean flow dynamics for barotropic flows and Fitzgerald and Farrell (2018b) for stratified flows. In these studies, the structures of the eddy field that produce upgradient momentum fluxes, and thus drive the instability, were determined in the appropriate limit
While the processes by which the flow-forming instability manifests are well understood, we lack comprehensive understanding of how this instability is equilibrated. For example, as the zonal jets grow, they often merge or branch to larger or smaller scales (Danilov and Gurarie 2004; Manfroi and Young 1999), multiple turbulence–jet equilibria exist (Farrell and Ioannou 2007; Parker and Krommes 2013; Constantinou et al. 2014a), and, also, transitions from various turbulent jet attractors may occur (Bouchet et al. 2019). Some outstanding questions include the following:
How is the equilibration of the flow-forming instability achieved and at which amplitude for the given parameters?
What are the eddy–mean flow dynamics involved in the equilibration process, and which eddies support the finite-amplitude jets?
What type of instabilities are involved in the observed jet variability phenomenology (jet merging and branching, multiple jet equilibria, transitions between various jet attractors), and what are the eddy–mean flow dynamics involved?
To tackle these questions, Parker and Krommes (2013) first pointed out the analogy of jet formation and pattern formation (Hoyle 2006; Cross and Greenside 2009). Exploiting this analogy Parker and Krommes (2014) were able to borrow tools and methods from pattern formation theory to elucidate the equilibration process. In particular, they demonstrated that, at small supercriticality, that is, when the turbulence intensity is just above the critical threshold for jet formation, the nonlinear evolution of the zonal jets follows Ginzburg–Landau (G–L) dynamics. In addition, Parker and Krommes (2014) examined the quantitative accuracy of the G–L approximation by comparison with turbulent jet equilibria obtained from the fully nonlinear S3T dynamics. Having established the validity of S3T dynamics even in the limit of very weak mean flows (as we have discussed above), it is natural to then proceed studying the G–L dynamics of this flow-forming instability and its associated equilibration process. The perturbative-level agreement of the S3T predictions with direct numerical simulations of the full nonlinear dynamics argues that the study of the equilibration of the flow-forming instability using the G–L dynamics is well founded.
In this work, we revisit the small-supercriticality regime of Parker and Krommes (2014). We thoroughly test the validity of the G–L approximation through a comparison with the fully nonlinear SSD closed at second order for a wide range of parameter values (section 5). Apart from the equilibrated flow-forming instability of the homogeneous turbulent state, which is governed by the G–L dynamics, we discover that an additional branch of jet equilibria exists for large values of
We investigate here the eddy–mean flow dynamics involved in the equilibration of the flow-forming instabilities, as well as those involved in the secondary side-band jet instabilities that occur (section 6). To do this, we derive the G–L equation in a physically intuitive way that allows for the comprehensive understanding of the nonlinear Landau term underlying the jet equilibration (section 4). Using methods similar to the ones developed by Bakas and Ioannou (2013b) and Bakas et al. (2015) we study the contribution of the forced eddies and their interactions in supporting the equilibrated finite-amplitude jets (section 6). Finally, to elucidate the equilibration of the new branch of jet equilibria that are not governed by the G–L dynamics, we develop an alternative reduced dynamical system that generalizes the G–L equation (section 6b). Using this reduced system we study the physical processes responsible for the equilibration of the new branch of jet equilibria.
2. Statistical state dynamics of barotropic β-plane turbulence in the S3T second-order closure

























Equation (1) is nondimensionalized using the forcing length-scale
The SSD of zonal jet formation in the S3T second-order closure comprise the dynamics of the first cumulant of the vorticity field
The overbars here denote zonal average, while dashes denote fluctuations about the mean. Thus,




















3. The flow-forming instability and the underlying eddy–mean flow dynamics





























(a) Waves with small
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(a) Waves with small
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(a) Waves with small
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Figure 3 shows the contribution

The contribution
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The contribution
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The contribution
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4. The G–L dynamics governing the nonlinear evolution of the flow-forming instability
In this section we discuss how the equilibration of the zonal jet instabilities is achieved for energy input rates just above the critical threshold
To derive the asymptotic dynamics that govern the evolution of the jet amplitude we perform a multiple-scale perturbation analysis of the nonlinear dynamics near the marginal point. Before proceeding with the multiple-scale analysis we present an intuitive argument that suggests the appropriate slow time and slow meridional spatial scales.
a. The appropriate slow length scale and slow time scale






























Equations (16) and (17) establish the initial assertion: for
The validity of the approximate eigenvalue relation (17) as a function of supercriticality μ is shown in Figs. 4a and 4b. By comparing the exact growth rates as given by (10) and the growth rates obtained from the approximation (17), we see that the approximate eigenvalue dispersion may not be as accurate in three ways: predicting the maximum growth rate, predicting the wavenumber at which maximum growth occurs, and predicting the asymmetry of the exact growth rates about the maximal wavenumber. These three differences are indicated by the arrows in Figs. 4a and 4b and are quantified in Figs. 4c–e. Figure 4c compares the exact wavenumber of maximum growth

Validity of the approximate eigenvalue relation, (17). (a) Comparison of the growth rates for jet perturbations with wavenumber ν as predicted by the exact eigenvalue relation (10) (solid curve) and by the parabolic approximation, (17) (dashed curve), for supercriticality
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1

Validity of the approximate eigenvalue relation, (17). (a) Comparison of the growth rates for jet perturbations with wavenumber ν as predicted by the exact eigenvalue relation (10) (solid curve) and by the parabolic approximation, (17) (dashed curve), for supercriticality
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1
Validity of the approximate eigenvalue relation, (17). (a) Comparison of the growth rates for jet perturbations with wavenumber ν as predicted by the exact eigenvalue relation (10) (solid curve) and by the parabolic approximation, (17) (dashed curve), for supercriticality
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1
b. G–L dynamics for weakly supercritical zonal jets









































































5. Comparison of the predictions of G–L dynamics with S3T dynamics for the equilibrated jets









We also consider
Exact values of nondimensional planetary vorticity gradient


We calculate the finite-amplitude equilibrated jets of the nonlinear S3T dynamical system, (5), using Newton’s method with the initial guess provided by (29).2 All jet equilibria we compute in this section are hydrodynamically stable. At small supercriticalities, the jet amplitude is small, and the linear operator is dominated by dissipation. Thus, all instabilities we discuss here are SSD instabilities (see paragraph 3 in section 1).
a. Equilibration of the most unstable jet 

Consider first the most unstable jet perturbation with wavenumber

The amplitude
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The amplitude
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The amplitude
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Surprisingly, for

The bifurcation diagram for
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The bifurcation diagram for
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The bifurcation diagram for
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The jets on the lower and the upper stable branches are qualitatively different. Figures 6b–e compare the jet structure and spectra of two such equilibria in the case of
b. Equilibration of the side-band jets 

We now consider the jet equilibria that emerge from the equilibration of jet perturbations with wavenumbers close to











The amplitude
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The amplitude
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The amplitude
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Finally, we stress that the results in this section regarding the existence of the upper-branch equilibria as well as the accuracy of the G–L dynamics for the lower-branch equilibria are not quirks of the particular isotropic-forcing structure in (4). Similar qualitative behavior is found for forcing with anisotropic spectrum. Discussion regarding the effects of the structure of the forcing is found in appendix C.
6. The physical processes underlying the equilibration of the SSD instability of the homogeneous state
One of the main objectives of this paper is to study the processes that control the halting of the flow-forming instability both for the low-branch equilibria, which are governed by the G–L dynamics, and for the upper-branch equilibria (cf. Figs. 5 and 6).
a. Equilibration processes for the lower branch
For G–L dynamics, the equilibration of the instability for the most unstable jet perturbation with wavenumber

(a) The amplitude
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(a) The amplitude
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(a) The amplitude
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For
For large β,

(a) The contribution
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(a) The contribution
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(a) The contribution
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Figure 8b shows the contribution of the two processes in
b. Equilibration processes for the upper-branch jets
We have seen in the discussion surrounding Fig. 6 that the














Figure 10 shows the mean-flow growth rates [e.g.,

(left) The mean-flow growth rates (a)
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(left) The mean-flow growth rates (a)
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(left) The mean-flow growth rates (a)
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There exist, however, two additional points of intersection, both of which are accessible to the flow through paths in the
The qualitative agreement between the approximate dynamics of the reduced dynamical system (40) and the nonlinear S3T dynamics reveal that it could be a useful tool for exploring the phase space of the S3T system. For example, the bifurcation structure of Fig. 6 could be obtained by plotting the growth rates obtained using the adiabatic approximation. Figure 11 shows the curves of zero tendencies for various values of the supercriticality. For low subcritical values

The locus of zero mean-flow tendencies in
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The locus of zero mean-flow tendencies in
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The locus of zero mean-flow tendencies in
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For low supercriticality,
7. Eckhaus instability of the side-band jets
In this section we study the stability of the side-band jet equilibria. As noted by Parker and Krommes (2014), these harmonic jet equilibria are susceptible to Eckhaus instability, a well-known result for harmonic equilibria of the G–L equation (Hoyle 2006). Here, we present the main results of the Eckhaus instability and compare them with fully nonlinear S3T dynamics.
a. An intuitive view of the Eckhaus instability















Solid curve shows a sinusoidal equilibrium jet
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Solid curve shows a sinusoidal equilibrium jet
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Solid curve shows a sinusoidal equilibrium jet
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To summarize, because of the diffusive nature of the vorticity flux feedback, there is a tendency to go toward
b. A formal view of the Eckhaus instability






















(a) The most unstable wavenumber for the Eckhaus instability
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(a) The most unstable wavenumber for the Eckhaus instability
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(a) The most unstable wavenumber for the Eckhaus instability
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c. Comparison with S3T dynamics
We first compare the stability analysis for the harmonic jets derived in the weakly nonlinear limit of G–L dynamics to nonlinear dynamics in the S3T system. Note that the growth rate of the Eckhaus instability is much less than the corresponding growth rate of the flow-forming instability of the homogeneous state of a jet for almost all wavenumbers ν. Figure 13b compares the growth rate


























The equilibration of the Eckhaus instability under S3T dynamics. (a) The evolution of the mean flow
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1

The equilibration of the Eckhaus instability under S3T dynamics. (a) The evolution of the mean flow
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The equilibration of the Eckhaus instability under S3T dynamics. (a) The evolution of the mean flow
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Figure 15 shows the comparison of the growth rates for the other unstable side-band jet equilibria illustrated in Fig. 7. We see once more that for

Growth rate for the Eckhaus instability of the finite-amplitude jets. Shown is the growth rate as a function of supercriticality μ for (a) β1, (b) β6, and (c) β192, obtained from the stability analysis for the equilibrium jets with wavenumbers
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1

Growth rate for the Eckhaus instability of the finite-amplitude jets. Shown is the growth rate as a function of supercriticality μ for (a) β1, (b) β6, and (c) β192, obtained from the stability analysis for the equilibrium jets with wavenumbers
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1
Growth rate for the Eckhaus instability of the finite-amplitude jets. Shown is the growth rate as a function of supercriticality μ for (a) β1, (b) β6, and (c) β192, obtained from the stability analysis for the equilibrium jets with wavenumbers
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1
8. Conclusions
We examined the dynamics that underlies the formation and support of zonal jets at finite amplitude in forced–dissipative barotropic beta-plane turbulence using the statistical state dynamics of the turbulent flow closed at second order. Within this framework, jet formation is shown to arise as a flow-forming instability (or zonostrophic instability) of the homogeneous statistical equilibrium turbulent state when the nondimensional parameter
When supercriticality
According to G–L dynamics, the harmonic unstable modes of the homogeneous equilibrium state equilibrate at finite amplitude. The predicted amplitude of the jet that results from the equilibration of the most unstable mode with wavenumber
The amplitudes of the jets that emerge from the side-band jet instabilities of the most unstable mode of the flow-forming instability (i.e., the jets that emerge at scales
The physical and dynamical processes underlying the equilibration of the flow-forming instability were then examined using three methods. The first was the decomposition of the nonlinear term in the G–L equation governing the equilibration of the instability in two terms. One involves the change in the homogeneous part of the eddy covariance that is required by total energy conservation. The other involves the vorticity flux feedback resulting from the interaction of the most unstable jet with wavenumber
For the G–L branch, the central physical process responsible for the equilibration is the reduction in the upgradient vorticity flux that occurs through the change in the homogeneous part of the eddy covariance. For low values of β, the instability is quickly quenched, and the jets equilibrate at low amplitude. The reason is that the contribution of the eddies that induce upgradient fluxes and drive the instability is weakened as the jets emerge, while simultaneously, the contribution of the eddies that induce downgradient fluxes is increased. As a result, the jets equilibrate at a small amplitude and are supported by the same eddies that drive the instability.
For large values of β, both the upgradient and the downgradient contributions are almost equally weakened, thus leading to a slow decay of the growth rate and to an equilibrated jet with a much larger amplitude. Because the equilibrium amplitude is large, the stabilizing fluxes that are multiplied by the square of the jet amplitude in the G–L equation are dominant, and therefore, at equilibrium, the jet is supported by the eddies that were initially hindering its growth (these eddies have phase lines that form small angles with the meridional but different than zero).
For the new branch of jet equilibria the main physical process responsible for the equilibration is the interaction of the
Finally, the stability of the equilibrated side-band unstable jet perturbations was examined. For an infinite domain, zonal jets with scales close to the scale
The predictions for the stability boundary and the growth rate of the Eckhaus instability were then compared to the stability analysis of the jet equilibria using the fully nonlinear S3T system and the methods developed in Constantinou (2015). For
We note that the comparison of the G–L dynamics with nonlinear S3T integrations, as well as investigation of the equilibration process with an anisotropic ring forcing, showed that the results in this study are not sensitive to the forcing structure.
A question that rises naturally is whether the results discussed here are relevant for strong turbulent jets. Strong turbulent jets also undergo bifurcations as the turbulence intensity increases. There are, however, qualitative differences compared to weak jets: strong jets always merge to larger scales, while weak jets can either merge or branch to reach a scale close to
Acknowledgments
The authors thank Jeffrey B. Parker for helpful comments on the first version of the manuscript. N.A.B. was supported by the AXA Research Fund. N.C.C. was partially supported by the NOAA Climate and Global Change Postdoctoral Fellowship Program, administered by UCAR’s Cooperative Programs for the Advancement of Earth System Sciences and also by the National Science Foundation under Award OCE-1357047.
APPENDIX A
S3T Formulation and Eigenvalue Relation of the Flow-Forming Instability
In this appendix we derive the eigenvalue relation of the flow-forming instability. The eigenvalue relation was first derived by Srinivasan and Young (2012). Here, we repeat the derivation mainly to introduce some notation and terminology that will prove to be helpful in understanding the nonlinear equilibration of the flow-forming instability.










































APPENDIX B
Ginzburg–Landau Equation for the Weakly Nonlinear Evolution of a Zonal Jet Perturbation about the Homogeneous State




























































































































The contribution of the two feedbacks
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The contribution of the two feedbacks
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The contribution of the two feedbacks
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APPENDIX C
Nonisotropic Ring Forcing











We first note that we obtain similar results to the isotropic-forcing case regarding the comparison of the G–L predictions to the fully nonlinear dynamics (not shown). That is, both the existence of the upper-branch equilibria and the relative quantitative success of the G–L dynamics (after the proposed modifications) in predicting the amplitude and instability of the equilibrated jets are insensitive to forcing structure.
Regarding the physical processes underlying the equilibration of the jets, we show in Fig. C1a the amplitude

As in Fig. 8, but for anisotropic forcing (C1) with
Citation: Journal of the Atmospheric Sciences 76, 3; 10.1175/JAS-D-18-0148.1