## 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 *even though* there could be differences in the eddy spectra and the concomitant eddy correlations; see, for example, Fig. 1.

However, surprisingly, S3T remains accurate *even at a perturbative level*, that is, when the mean flows/jets are just emerging with *even for very weak mean flows* should be attributed to the existence of the collective flow-forming instability, which seems to overpower the disruptive eddy–eddy nonlinear interactions.

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 *β* is the planetary vorticity gradient,

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

*β*plane with coordinates

^{1}

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, *C* is therefore homogeneous in *x*: *ξ*:

*C*through

*a*=

*b*implies that the function of

**x**

_{a}and

**x**

_{b}, for example, inside the square brackets on the right-hand side of (6), is transformed into a function of a single variable by setting

**x**

_{a}=

**x**

_{b}=

**x**. The total averaged energy density relaxes over the dissipation scale [which is of

*O*(1) in the nondimensional equations] to the energy supported under stochastic forcing and dissipation:Therefore, the total energy remains bounded under S3T dynamics (Bakas and Ioannou 2019a).

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

*n*is a real wavenumber that indicates the length scale of the jets. The corresponding eigenvalues

*σ*satisfy (see appendix A)where

*f*is the vorticity flux induced by the distortion of the eddy equilibrium field

*f*is given in (A9). This induced vorticity flux is referred to as the

*vorticity flux feedback on*

*σ*, and therefore, the emergent jets are not translating in the

*y*direction. The vorticity flux feedback at marginal stability,which is positive in this case, has the tendency to reinforce the preexisting jet perturbation

*ε*at which the homogeneous equilibrium becomes unstable to a jet with wavenumber

*n*is

*β*, there is a minimum energy input rate,above which the homogeneous state is unstable and jet formation occurs.

*n*. Angle

*ϑ*measures the inclination of the wave phase lines with respect to the

*y*axis. The precise expression for

*n*. In general, destabilizing vorticity fluxes are produced by waves with phase lines closely aligned to the

*y*axis (with small

Figure 3 shows the contribution *ϑ* for the most unstable jet *β*, as shown in Fig. 3a (Bakas et al. 2015). The contribution from all angles is small (of order *y* axis *β* can be understood by considering the evolution of wave groups in the sinusoidal flow and were studied in detail by Bakas and Ioannou (2013b).

## 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

*c*denotes that the derivatives are evaluated at the threshold point

*f*has a maximum at

*ε*–

*n*plane lie on the parabola:where

*σ*at supercriticality

*μ*. We find that jets with wavenumber

*μ*only jets withcan become unstable.

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 *μ* up to *β*. The parabolic approximation works best for intermediate *β* values, that is, for *β*. As it will be seen, this has implications on the validity of the weakly nonlinear dynamics derived next.

### b. G–L dynamics for weakly supercritical zonal jets

*μ*. Therefore, to obtain the dynamics that govern weakly supercritical zonal flows, we expand the mean flow

*C*of the S3T equations, (5), asGuided by (16) and (17), we have assumed that the zonal jet and its associated covariance evolve from the marginal values at the slow time scale

*μ*. At leading-order

*Y*in the amplitude

*A*, as well as that in

*A*can be obtained using this simplification, while the contribution to the asymptotic dynamics from the slow varying latitude

*Y*is the addition of a diffusion term with the diffusion coefficient

*Q*appears both at order

*must*be accompanied by a decrease in the eddy energy. This decrease is facilitated by a concomitant change of the eddy covariance at order

*y*and integrates to zero:At order

*defect*to counterbalance the energy growth of the mean flow. We refer to

*eddy energy correction term*. However, we note that the correction to the homogeneous part of the covariance does not only change the mean eddy energy but also other eddy characteristics, such as the mean eddy anisotropy, that also might play a role in the equilibration process.

*A*of the most unstable jet with wavenumber

*Y*and add the diffusion term

*φ*an undetermined phase that reflects the translational invariance of the system in

*y*. These equilibria are the possible finite-amplitude jets that emerge at low supercriticality. However, as will be shown in the next section, some of these equilibria are susceptible to a secondary SSD instability and evolve through jet merging or jet branching to the subset of the stable attracting states.

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

^{2}grid resolution and

*A*on the same domain. We approximate the delta function in the ring forcing, (4), aswith

We also consider *ν* is realizable within our domain *only* if its dimensional wavenumber

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 *β*. We see that, for

Surprisingly, for *μ* (see Figs. 5c,d). Specifically, there exists a branch of stable equilibria apart from the jets connected to the homogeneous equilibrium (cf. triangles in Figs. 5c,d vs the circles). For *ε* is varied. The two stable branches are connected with a branch of unstable equilibria (open circles) that were also found using Newton’s method.

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 *β*. For example, for

*β*. While the functional dependence of the equilibrated amplitude on

*μ*is qualitatively captured by (28) (dashed lines), there are significant quantitative differences, especially for

*β*, the amplitude of the jets close to the bifurcation point is accurately predicted, and for the intermediate value of

*β*shown in Fig. 7c the additional upper branch of equilibria is found and this branch has the same characteristics as the upper branch of

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 *β*; Fig. 8a shows the amplitude of the most unstable jet, *β*. For

*β*can be understood by considering the contribution of the various wave components to

*β*.

For *ϑ* are counteracted by *ϑ* are enhanced by

For large *β*, *β*, note that, as *β* increases, (i) the heights of the dipole peaks grow linearly with *β*, (ii) the widths of the dipole peaks decrease as *β* as

*β*. For small values of

*β*, waves with angles

Figure 8b shows the contribution of the two processes in *β*. We observe that the main contribution to the coefficient *β*. Only for ^{3} The same results also hold for the case of the anisotropic forcing (see Fig. C1). Therefore, we conclude that, for most values of *β*, the mean flow is stabilized by the change in the homogeneous part of the covariance because of conservation of the total energy that leads to a concomitant reduction of the upgradient fluxes. For

### b. Equilibration processes for the upper-branch jets

We have seen in the discussion surrounding Fig. 6 that the

*C*, compute the vorticity fluxes, and decompose them into their Fourier components:with

*m*positive integer. Then, from the mean flow equation, (5a), we obtain that the mean-flow components satisfy

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

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

*μ*. Similarly, the second term on the right-hand side of (42b) is proportional to the vorticity flux feedback from the interaction of

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

*ν*. (Similarly, for an equilibrium jet with

*ν*.)

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*in polar form aswhere

*R*is the amplitude and

*ν*. Note that the equilibria with wavenumbers

### 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

*q*is

*β*, the parabolic profile is not so accurate, and therefore, the criterion developed fails. For example, for both

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

## 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 *subcritical values* of the flow-forming instability of the homogeneous state (i.e., for

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 *β*, large quantitative discrepancies occur with a few exceptions, but the qualitative picture of the dynamics with branching (merging) into the stable jet equilibrium remains.

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.

*C*by the mean flow

*k*, (A9) becomeswith

## APPENDIX B

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

*μ*:along with the linear and nonlinear operators

*y*and

*Y*, respectively.

*μ*. As discussed in section 4, we further assume that the amplitude

*A*, as well as

*Y*. This way, operators

*Y*. In this case, the order-

*c*on terms

*j*.

*C*, and they are responsible for the equilibration of the flow-forming SSD instability.

*β*,

*ϑ*with the

*y*axis, we substitute the ring forcing power spectrum, (4). After expressing the integrand in polar coordinates

*k*, we obtainwhere

## APPENDIX C

### Nonisotropic Ring Forcing

*γ*determines the degree of anisotropy of the forcing. The isotropic case of (4) is recovered for

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 *β*. For *β* for *β*, the equilibration is dominated by the interaction of the most unstable jet with the homogeneous covariance correction.

Last, we note that for anisotropic forcing, a similar qualitative decomposition of

## APPENDIX D

### Eckhaus Stability of G–L Dynamics

*A*in polar form, (53); we then substitute into (27) and separate real and imaginary parts to obtain

*λ*areInstability occurs when

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^{1}

In numerical simulations, we approximate the delta function in (4) as a Gaussian with narrow width—see section 5 for more details.

^{2}

For details regarding Newton’s root-finding algorithm for system (5), the reader is referred to appendix I in the thesis of Constantinou (2015).

^{3}

Further analysis on the relative contributions of the forced eddies for the two distinct processes can be found in appendix B.