## 1. Introduction

A regime in which jets, planetary-scale waves, and vortices coexist is commonly observed in the turbulence of planetary atmospheres, with the banded winds and embedded vortices of Jupiter and the Saturn North Polar vortex constituting familiar examples (Vasavada and Showman 2005; Sánchez-Lavega et al. 2014). Planetary-scale waves in the jet stream and vortices, such as cutoff lows, are also commonly observed in Earth’s atmosphere. Conservation of energy and enstrophy in undamped 2D turbulence implies continual transfer of energy to the largest available spatial scales (Fjørtoft 1953). This upscale transfer provides a conceptual basis for expecting the largest scales to become increasingly dominant as the energy of turbulence forced at smaller scale is continually transferred to the larger scales. However, the observed large-scale structure in planetary atmospheres is dominated not by incoherent large-scale turbulent motion, as would be expected to result from the incoherent phase relation of Fourier modes in a turbulent cascade, but rather by coherent zonal jets, vortices, and waves of highly specific form. Moreover, the scale of these coherent structures is distinct from the largest scale permitted in the flow. An early attempt to understand the formation of jets in planetary turbulence did not address the structure of the jet beyond attributing the jet scale to arrest of the incoherent upscale energy cascade at the length scale set by the value of the planetary vorticity gradient and a characteristic flow velocity (Rhines 1975). In Rhines’s interpretation, this is the scale at which the turbulent energy cascade is intercepted by the formation of propagating Rossby waves. While this result provides a conceptual basis for expecting zonal structures with spatial scale limited by the planetary vorticity gradient to form in beta-plane turbulence, the physical mechanism of formation, the precise morphology of the coherent structures, and their stability are not addressed by these general considerations.

Our goal in this work is to continue development of a general theory for the formation of finite-amplitude structures in planetary turbulence, specifically addressing the regime in which jets and planetary waves coexist. This theory identifies specific mechanisms responsible for formation and equilibration of coherent structures in planetary turbulence. A number of mechanisms have been previously advanced to account for jet, wave, and vortex formation. One such mechanism that addresses exclusively jet formation is vorticity mixing by breaking Rossby waves leading to homogenization of potential vorticity (PV) in localized regions (Baldwin et al. 2007; Dritschel and McIntyre 2008), resulting in the case of barotropic beta-plane turbulence in broad retrograde parabolic jets and relatively narrow prograde jets with associated staircase structure in the absolute vorticity. While PV staircases have been obtained in some numerical simulations of strong jets (Scott and Dritschel 2012), vorticity mixing in the case of weak to moderately strong jets is insufficient to produce a prominent staircase structure. Moreover, jets have been shown to form as a bifurcation from homogeneous turbulence, in which case the jet is perturbative in amplitude and wave breaking is not involved (Farrell and Ioannou 2003, 2007).

Equilibrium statistical mechanics has also been advanced to explain formation of coherent structures [e.g., by Miller (1990) and Robert and Sommeria (1991)]. The principle is that dissipationless turbulence tends to produce configurations that maximize entropy while conserving both energy and enstrophy. These maximum entropy configurations in beta-plane turbulence assume the coarse-grained structure of zonal jets (cf. Bouchet and Venaille 2012). However, the relevance of these results to the formation, equilibration, and maintenance of jets in strongly forced and dissipated planetary flows remains to be established.

Zonal jets and waves can also arise from modulational instability (Lorenz 1972; Gill 1974; Manfroi and Young 1999; Berloff et al. 2009; Connaughton et al. 2010). This instability produces spectrally nonlocal transfer to the unstable structure from forced waves and therefore presumes a continual source of waves with the required form. In baroclinic flows, baroclinic instability has been advanced as the source of these waves (Berloff et al. 2009). From the broader perspective of the statistical state dynamics theory used in this work, modulational instability is a special case of a stochastic structural stability theory (S3T) (Parker and Krommes 2014; Parker 2014; Bakas et al. 2015). However, modulational instability does not include the mechanisms for realistic equilibration of the instabilities at finite amplitude, although a Landau-type term has been used to produce equilibration of the modulational instability (cf. Manfroi and Young 1999).

Another approach to understanding the jet–wave coexistence regime is based on the idea that jets and waves interact in a cooperative manner. Such a dynamic is suggested, for example, by observations of a prominent wavenumber-5 disturbance in the Southern Hemisphere (Salby 1982). Using a zonally symmetric two-layer baroclinic model, Cai and Mak (1990) demonstrated that storm-track organization by a propagating planetary-scale wave resulted in modulation in the distribution of synoptic-scale transients configured on average to maintain the organizing planetary-scale wave. The symbiotic forcing by synoptic-scale transients that, on average, maintains planetary-scale waves was traced to barotropic interactions in the studies of Robinson (1991) and Qin and Robinson (1992). While diagnostics of simulations such as these are suggestive, comprehensive analysis of the essentially statistical mechanism of the symbiotic regime requires obtaining solutions of the statistical state dynamics underlying it, and indeed the present work identifies an underlying statistical mechanism by which transients are systematically organized by a planetary-scale wave so as to, on average, support that planetary-scale wave in a spectrally nonlocal manner.

S3T provides a statistical state dynamics (SSD)-based theory accounting for the formation, equilibration, and stability of coherent structures in turbulent flows. The underlying mechanism of jet and wave formation revealed by S3T is the spectrally nonlocal interaction between the large-scale structure and the small-scale turbulence (Farrell and Ioannou 2003). S3T is a nonequilibrium statistical theory based on a closure comprising the nonlinear dynamics of the coherent large-scale structure together with the consistent second-order fluxes arising from the incoherent eddies. The S3T system is a cumulant expansion of the turbulence dynamics closed at second order (cf. Marston et al. 2008), which has been shown to become asymptotically exact for large-scale jet dynamics in turbulent flows in the limit of zero forcing and dissipation and infinite separation between the time scales of evolution of the large-scale jets and the eddies (Bouchet et al. 2013; Tangarife 2015). S3T has been employed to understand the emergence and equilibration of zonal jets in planetary turbulence in barotropic flows on a beta plane and on the sphere (Farrell and Ioannou 2003, 2007, 2009a; Marston et al. 2008; Srinivasan and Young 2012; Marston 2012; Constantinou et al. 2014; Bakas and Ioannou 2013b; Tobias and Marston 2013; Parker and Krommes 2014), in baroclinic two-layer turbulence (Farrell and Ioannou 2008, 2009c), in the formation of dry convective boundary layers (Ait-Chaalal et al. 2016), and in drift–wave turbulence in plasmas (Farrell and Ioannou 2009b; Parker and Krommes 2013). It has been used in order to study the emergence and equilibration of finite-amplitude propagating nonzonal structures in barotropic flows (Bakas and Ioannou 2013a, 2014; Bakas et al. 2015) and the dynamics of blocking in two-layer baroclinic atmospheres (Bernstein and Farrell 2010). It has also been used to study the role of coherent structures in the dynamics of the 3D turbulence of wall-bounded shear flows (Farrell and Ioannou 2012; Thomas et al. 2014, 2015; Farrell et al. 2015, manuscript submitted to *J. Fluid Mech.*).

In certain cases, a barotropic S3T homogeneous turbulent equilibrium undergoes a bifurcation in which nonzonal coherent structures emerge as a function of turbulence intensity prior to the emergence of zonal jets, and when zonal jets emerge a new type of jet–wave equilibrium forms (Bakas and Ioannou 2014). In this paper, we use S3T to further examine the dynamics of the jet–wave coexistence regime in barotropic beta-plane turbulence. To probe the jet–wave–turbulence dynamics in more depth, a separation is made between the coherent jets and large-scale waves and the smaller-scale motions, which are considered to constitute the incoherent turbulent component of the flow. This separation is accomplished using a dynamically consistent projection in Fourier space. By this means, we show that jet states maintained by turbulence may be unstable to emergence of nonzonal traveling waves and trace these unstable eigenmodes to what would, in the absence of turbulent fluxes, have been damped wave modes of the mean jet. Thus, we show that the cooperative dynamics between large-scale coherent and small-scale incoherent motion is able to transform damped modes into unstable modes by altering the mode structure, allowing it to tap the energy of the mean jet.

In this work, we also extend the S3T stability analysis of homogeneous equilibria (Farrell and Ioannou 2003, 2007; Srinivasan and Young 2012; Bakas and Ioannou 2013a,b, 2014; Bakas et al. 2015) to the S3T stability of jet equilibria. We present new methods for the calculation of the S3T stability of jet equilibria, extending the work of Farrell and Ioannou (2003) and Parker and Krommes (2014), which was limited to the study of the S3T stability of jets only with respect to zonal perturbations, to the S3T stability of jets to nonzonal perturbations.

## 2. Formulation of S3T dynamics for barotropic beta-plane turbulence

*x*is the zonal direction and

*y*is the meridional direction, and with the flow confined to a periodic channel of size

*ψ*as

*x*–

*y*plane. The component of vorticity normal to the plane of motion is

*r*and viscous dissipation with coefficient

*ν*. The stochastic excitation maintaining the turbulence

Equation (1) is nondimensionalized using length scale *L* and time scale *T*. The double-periodic domain becomes 2π × 2π, and the nondimensional variables in (1) are

We review now the formulation of the S3T approximation to the SSD of (1). The S3T dynamics was introduced in the matrix formulation by Farrell and Ioannou (2003). Marston et al. (2008) showed that S3T comprises a canonical second-order closure of the exact statistical state dynamics and derived it alternatively using the Hopf formulation. Srinivasan and Young (2012) obtained a continuous formulation that facilitates analytical explorations of S3T stability of turbulent statistical equilibria.

*ψ*). The example, for example,

*f*and

*g*,

*t*and the coordinates of the two points

*C*. For example, the eddy vorticity flux divergence source term,

*C*as follows:in which

*t*, in (8) is to be considered a function of two independent spatial variables and

*t*by setting

*C*:Both terms

^{1}and consequently (11) simplifies to the time-dependent Lyapunov equation:

Using parameterization (13) to account for both the eddy–eddy nonlinearity,

Because the scale separation assumed in (16) is only approximately satisfied in many cases of interest, an alternative formulation of S3T will now be obtained in which separation into two independent interacting components of different scales is implemented [a similar formulation was independently derived by Marston et al. (2016)]. This formulation makes more precise the dynamics of the coherent jet and wave interacting with the incoherent turbulence regime in S3T.

*I*the identity. Similarly, vorticity and velocity fields are decomposed into

## 3. Specification of the parameters used in this work

Assume that the large-scale phase coherent motions occupy zonal wavenumbers

^{2}With this normalization, the rate of energy injection by the stochastic forcing in (1), (16), (21), (22), and (23) is

*ε*and is independent of the state of the system, because

*ξ*has been assumed temporally delta correlated.

We choose *e*-folding time for linear damping of 40 days. The diffusion coefficient *e*-folding time for scales on the order of

## 4. S3T jet equilibria

*ε*. For all values of

*ε*and all homogeneous stochastic forcings, there exist equilibria that are homogeneous (both in

*x*and

*y*) withwhere

*ε*exceeds a critical value. For values of

*ε*exceeding this critical value, zonal jets arise from a supercritical bifurcation (Farrell and Ioannou 2003, 2007; Srinivasan and Young 2012; Parker and Krommes 2013, 2014; Constantinou et al. 2014). These jets are constrained by the periodic domain of our simulations to take discrete values of meridional wavenumber

*ε*–

*n*

_{y}plane separating the region in which only stable homogeneous turbulence equilibria exist from the region in which stable or unstable jet equilibria exist is shown, for the chosen parameters, in Fig. 1. This marginal curve was calculated using the eigenvalue relation for inhomogeneous perturbations to the homogeneous S3T equilibrium in the presence of diffusive dissipation, in the manner of Srinivasan and Young (2012) and Bakas and Ioannou (2014), with the wavenumber

*y*with period

## 5. S3T stability of the jet equilibria

*x*perturbations has been investigated previously by Farrell and Ioannou (2003, 2007) for periodic domains and by Parker and Krommes (2014) and Parker (2014) for infinite domains. A comprehensive methodology for determining the stability of jet equilibria to zonal and nonzonal perturbations was developed by Constantinou (2015). Recalling these results, perturbations

^{3}with

*R*as in (8),

*x*) direction, the mean-flow eigenfunctions are harmonic functions in

*x*, and also, because the equilibrium mean flow and covariance are periodic in

*y*with period

*α*[i.e.,

*y*

*α*in

*y*(Cross and Greenside 2009; Parker and Krommes 2014). Therefore, the eigenfunctions take the following form:with

*y*with period

*α*and

*α*. We have chosen

^{4}The zonal wavenumber

*x*, and the Bloch wavenumber

*y*(Constantinou 2015). The eigenvalue

*σ*determines the S3T stability of the jet as a function of

*x*with phase velocity

*x*and

*y*and correspond to a wave. These perturbations, when unstable, can form nonzonal large-scale structures that coexist with the mean flow, as in Bakas and Ioannou (2014). For jets with meridional periodicity

The maximum growth rate *ε* are jets with wavenumber *ε* marginally exceeds the critical ^{5} to

The maximum growth rate of the jet equilibria to

## 6. The mechanism destabilizing S3T jets to nonzonal perturbations

We now examine the stability properties of the

Because the jet

Although it is not formed as a result of a traditional hydrodynamic instability, this S3T instability is very close in structure to the least-stable eigenfunction of

For the S3T unstable eigenfunction shown in Fig. 4d, the growth rate

This same mechanism is responsible for the S3T destabilization of the

## 7. Equilibration of the S3T instabilities of the equilibrium jet

We next examine equilibration of the

Consider the energetics of these large scales consisting of the

The instantaneous rates of change of the energy of the

### a. Case 1: instability at

Consider first the equilibration of the

### b. Case 2: instability at

The equilibration of the jet at

## 8. Discussion

### a. Correspondence between the S3T dynamics (16) and the projected S3T dynamics (23)

Stability of a two-jet state to jet and wave perturbations in the projected S3T formulation (23) is shown in Fig. 2. For parameters for which the base state becomes unstable to nonzonal large-scale perturbations, this base state transitions to a new equilibrium in which the jet coexists with a coherent wave. The stability calculation, its energetics, and equilibration process are studied in the framework of the projected S3T equations in (23), which allows a clear separation between the contribution of the coherent jet interaction and that of the incoherent eddies to the instability and equilibration processes. This stability analysis using the projected S3T system produces essentially the same results as were obtained using the S3T system (16) (cf. Fig. 2 with Figs. 10a,b and Figs. 3e,f with Figs. 10c,d). The equilibrated states produced by these two S3T systems are also very similar (cf. Fig. 11).

### b. Reflection of ideal S3T dynamics in QL simulations

The ideal S3T equilibrium jet and the jet–wave states that we have obtained are imperfectly reflected in single realizations of the flow because fluctuations may obscure the underlying S3T equilibrium (cf. Farrell and Ioannou 2003, 2016). The infinite ensemble ideal incorporated in the S3T dynamics can be approached in the QL [governed by (22)] by introducing in the equation for the coherent flow an ensemble-mean Reynolds stress obtained from a number of independent integrations of the QL eddy equations with different forcing realizations.

Consider, for example, the jet–wave S3T regime at

Both S3T and ensemble simulations isolate and clearly reveal the mechanism by which a portion of the incoherent turbulence is systematically organized by large-scale waves to enhance the organizing wave. However, as in simulation studies revealing this mechanism at work in baroclinic turbulence (Cai and Mak 1990; Robinson 1991), the large-scale wave retains a substantial incoherent component in individual realizations. This is expected in the strongly turbulent atmosphere, considering that even stationary waves at planetary scale, which are strongly forced by topography, are revealed clearly only in seasonal average ensembles.

### c. Reflection of ideal S3T dynamics in nonlinear simulations

Consider now the reflection of the S3T jet–wave regime in nonlinear (NL) and ensemble NL simulations. Ensemble simulations of the NL system (21) were performed by introducing in the mean equations in (21a), the ensemble average of

When all waves with

It could be maintained that because isotropic ring forcing suppresses eddy–eddy interactions, the agreement between S3T and NL should be expected (cf. Bakas et al. 2015, their appendix C). This property follows from the fact that a barotropic fluid excited in an infinite channel with an isotropic ring forcing with spectrum *e*-folding times of 4 and 60 days, respectively. For these parameters, S3T theory predicts that the 5 jet equilibrium at

## 9. Conclusions

Large-scale coherent structures, such as jets, meandering jets, and waves embedded in jets are characteristic features of turbulence in planetary atmospheres. While conservation of energy and enstrophy in inviscid 2D turbulence predicts spectral evolution leading to concentration of energy at large scales, these considerations cannot predict the phase of the spectral components and therefore cannot address the central question of the organization of the energy into specific structures such as jets and the coherent component of planetary-scale waves. To study structure formation, additional aspects of the dynamics beyond conservation principles must be incorporated in the analysis. For this purpose, SSD models have been developed and used to study the formation of coherent structure in planetary-scale turbulence. In this work, an SSD model was formulated for the purpose of studying the regime of coexisting jets and waves. In this model, a separation in zonal Fourier modes is made by projection in order to separate a coherent structure equation, in which only the gravest zonal harmonics are retained, from the remaining spectrum, which is assumed to be incoherent and gives rise to the ensemble-mean second-order statistics associated with the incoherent turbulence. This second-order SSD model is closed by a stochastic forcing parameterization that accounts for both the neglected nonlinear dynamics of the small scales as well as the forcing maintaining the turbulence. The equation for the large scales retains its nonlinearity and its interaction through Reynolds stress with the eddies.

In this model, jets form as instabilities and equilibrate nonlinearly at finite amplitude. A stable mode of the Rossby wave spectrum associated with these jets is destabilized for sufficiently strong forcing by interaction with eddy Reynolds stresses. This destabilization is found to have, in some cases, the remarkable property of resulting from destabilization of the retrograde Rossby wave to mean jet interaction by structural modification of this damped mode arising from its interaction with the incoherent turbulence, thereby transforming it into an unstable mode of the mean jet. In other cases, comparable contributions are found from direct forcing by the Reynolds stresses, as in S3T instability with projections at

## Acknowledgments

We thank Nikolaos Bakas for useful discussions on the energetics of the equations in spectral space. We also thank the anonymous reviewers for their comments that led to improvement of the paper. N.C.C. would like to thank Prof. Georgios Georgiou for his hospitality and support during the summer of 2015 at the University of Cyprus. B.F.F. was supported by NSF AGS-1246929. N.C.C was partially supported by the NOAA Climate and Global Change Postdoctoral Fellowship Program, administered by UCAR’s Visiting Scientist Programs.

## REFERENCES

Ait-Chaalal, F., , T. Schneider, , B. Meyer, , and J. B. Marston, 2016: Cumulant expansions for atmospheric flows.

,*New J. Phys.***18**, 025019, doi:10.1088/1367-2630/18/2/025019.Bakas, N. A., , and P. J. Ioannou, 2013a: Emergence of large scale structure in barotropic

*β*-plane turbulence.,*Phys. Rev. Lett.***110**, 224501, doi:10.1103/PhysRevLett.110.224501.Bakas, N. A., , and P. J. Ioannou, 2013b: On the mechanism underlying the spontaneous emergence of barotropic zonal jets.

,*J. Atmos. Sci.***70**, 2251–2271, doi:10.1175/JAS-D-12-0102.1.Bakas, N. A., , and P. J. Ioannou, 2014: A theory for the emergence of coherent structures in beta-plane turbulence.

,*J. Fluid Mech.***740**, 312–341, doi:10.1017/jfm.2013.663.Bakas, N. A., , N. C. Constantinou, , and P. J. Ioannou, 2015: S3T stability of the homogeneous state of barotropic beta-plane turbulence.

,*J. Atmos. Sci.***72**, 1689–1712, doi:10.1175/JAS-D-14-0213.1.Baldwin, M. P., , P. B. Rhines, , H.-P. Huang, , and M. E. McIntyre, 2007: The jet-stream conundrum.

,*Science***315**, 467–468, doi:10.1126/science.1131375.Berloff, P., , I. Kamenkovich, , and J. Pedlosky, 2009: A mechanism of formation of multiple zonal jets in the oceans.

,*J. Fluid Mech.***628**, 395–425, doi:10.1017/S0022112009006375.Bernstein, J., , and B. F. Farrell, 2010: Low-frequency variability in a turbulent baroclinic jet: Eddy–mean flow interactions in a two-level model.

,*J. Atmos. Sci.***67**, 452–467, doi:10.1175/2009JAS3170.1.Bouchet, F., , and A. Venaille, 2012: Statistical mechanics of two-dimensional and geophysical flows.

,*Phys. Rep.***515**, 227–295, doi:10.1016/j.physrep.2012.02.001.Bouchet, F., , C. Nardini, , and T. Tangarife, 2013: Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier–Stokes equations.

,*J. Stat. Phys.***153**, 572–625, doi:10.1007/s10955-013-0828-3.Cai, M., , and M. Mak, 1990: Symbiotic relation between planetary and synoptic-scale waves.

,*J. Atmos. Sci.***47**, 2953–2968, doi:10.1175/1520-0469(1990)047<2953:SRBPAS>2.0.CO;2.Connaughton, C. P., , B. T. Nadiga, , S. V. Nazarenko, , and B. E. Quinn, 2010: Modulational instability of Rossby and drift waves and generation of zonal jets.

,*J. Fluid Mech.***654**, 207–231, doi:10.1017/S0022112010000510.Constantinou, N. C., 2015: Formation of large-scale structures by turbulence in rotating planets. Ph.D. thesis, National and Kapodistrian University of Athens, 204 pp. [Available online at http://www.didaktorika.gr/eadd/handle/10442/35501?locale=en.]

Constantinou, N. C., , B. F. Farrell, , and P. J. Ioannou, 2014: Emergence and equilibration of jets in beta-plane turbulence: Applications of stochastic structural stability theory.

,*J. Atmos. Sci.***71**, 1818–1842, doi:10.1175/JAS-D-13-076.1.Cross, M., , and H. Greenside, 2009:

*Pattern Formation and Dynamics in Nonequilibrium Systems.*Cambridge University Press, 552 pp.Dritschel, D. G., , and M. E. McIntyre, 2008: Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers.

,*J. Atmos. Sci.***65**, 855–874, doi:10.1175/2007JAS2227.1.Farrell, B. F., , and P. J. Ioannou, 2002: Perturbation growth and structure in uncertain flows. Part II.

,*J. Atmos. Sci.***59**, 2647–2664, doi:10.1175/1520-0469(2002)059<2647:PGASIU>2.0.CO;2.Farrell, B. F., , and P. J. Ioannou, 2003: Structural stability of turbulent jets.

,*J. Atmos. Sci.***60**, 2101–2118, doi:10.1175/1520-0469(2003)060<2101:SSOTJ>2.0.CO;2.Farrell, B. F., , and P. J. Ioannou, 2007: Structure and spacing of jets in barotropic turbulence.

,*J. Atmos. Sci.***64**, 3652–3665, doi:10.1175/JAS4016.1.Farrell, B. F., , and P. J. Ioannou, 2008: Formation of jets by baroclinic turbulence.

,*J. Atmos. Sci.***65**, 3353–3375, doi:10.1175/2008JAS2611.1.Farrell, B. F., , and P. J. Ioannou, 2009a: Emergence of jets from turbulence in the shallow-water equations on an equatorial beta plane.

,*J. Atmos. Sci.***66**, 3197–3207, doi:10.1175/2009JAS2941.1.Farrell, B. F., , and P. J. Ioannou, 2009b: A stochastic structural stability theory model of the drift wave-zonal flow system.

,*Phys. Plasmas***16**, 112903, doi:10.1063/1.3258666.Farrell, B. F., , and P. J. Ioannou, 2009c: A theory of baroclinic turbulence.

,*J. Atmos. Sci.***66**, 2444–2454, doi:10.1175/2009JAS2989.1.Farrell, B. F., , and P. J. Ioannou, 2012: Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow.

,*J. Fluid Mech.***708**, 149–196, doi:10.1017/jfm.2012.300.Farrell, B. F., , and P. J. Ioannou, 2016: Statistical state dynamics: A new perspective on turbulence in shear flow.

*Zonal Jets*, B. Galperin and P. L. Read, Eds., Cambridge University Press, in press. [Available online at http://arxiv.org/abs/1412.8290.]Fjørtoft, R., 1953: On the changes in the spectral distribution of kinetic energy for two-dimensional, nondivergent flow.

,*Tellus***5A**, 225–230, doi:10.1111/j.2153-3490.1953.tb01051.x.Frisch, U., 1995:

*Turbulence: The Legacy of A. N. Kolmogorov.*Cambridge University Press, 312 pp.Gill, A. E., 1974: The stability of planetary waves on an infinite beta-plane.

,*Geophys. Astrophys. Fluid Dyn.***6**, 29–47, doi:10.1080/03091927409365786.Kasahara, A., 1980: Effect of zonal flows on the free oscillations of a barotropic atmosphere.

,*J. Atmos. Sci.***37**, 917–929, doi:10.1175/1520-0469(1980)037<0917:EOZFOT>2.0.CO;2.Lorenz, E. N., 1972: Barotropic instability of Rossby wave motion.

,*J. Atmos. Sci.***29**, 258–269, doi:10.1175/1520-0469(1972)029<0258:BIORWM>2.0.CO;2.Manfroi, A. J., , and W. R. Young, 1999: Slow evolution of zonal jets on the beta plane.

,*J. Atmos. Sci.***56**, 784–800, doi:10.1175/1520-0469(1999)056<0784:SEOZJO>2.0.CO;2.Marston, J. B., 2012: Planetary atmospheres as nonequilibrium condensed matter.

,*Annu. Rev. Condens. Matter Phys.***3**, 285–310, doi:10.1146/annurev-conmatphys-020911-125114.Marston, J. B., , E. Conover, , and T. Schneider, 2008: Statistics of an unstable barotropic jet from a cumulant expansion.

,*J. Atmos. Sci.***65**, 1955–1966, doi:10.1175/2007JAS2510.1.Marston, J. B., , G. P. Chini, , and S. M. Tobias, 2016: The generalized quasilinear approximation: Application to zonal jets. 5 pp. [Available online at http://arxiv.org/abs/1601.06720.]

Miller, J., 1990: Statistical mechanics of Euler equations in two dimensions.

,*Phys. Rev. Lett.***65**, 2137–2140, doi:10.1103/PhysRevLett.65.2137.Parker, J. B., 2014: Zonal flows and turbulence in fluids and plasmas. Ph.D. thesis, Princeton University, 157 pp. [Available online at http://arks.princeton.edu/ark:/88435/dsp01h989r543m.]

Parker, J. B., , and J. A. Krommes, 2013: Zonal flow as pattern formation.

,*Phys. Plasmas***20**, 100703, doi:10.1063/1.4828717.Parker, J. B., , and J. A. Krommes, 2014: Generation of zonal flows through symmetry breaking of statistical homogeneity.

,*New J. Phys.***16**, 035006, doi:10.1088/1367-2630/16/3/035006.Qin, J., , and W. A. Robinson, 1992: Barotropic dynamics of interactions between synoptic and low-frequency eddies.

,*J. Atmos. Sci.***49**, 71–79, doi:10.1175/1520-0469(1992)049<0071:BDOIBS>2.0.CO;2.Rhines, P. B., 1975: Waves and turbulence on a beta-plane.

,*J. Fluid Mech.***69**, 417–433, doi:10.1017/S0022112075001504.Robert, R., , and J. Sommeria, 1991: Statistical equilibrium states for two-dimensional flows.

,*J. Fluid Mech.***229**, 291–310, doi:10.1017/S0022112091003038.Robinson, W. A., 1991: The dynamics of low-frequency variability in a simple model of the global atmosphere.

,*J. Atmos. Sci.***48**, 429–441, doi:10.1175/1520-0469(1991)048<0429:TDOLFV>2.0.CO;2.Salby, M. L., 1982: A ubiquitous wavenumber-5 anomaly in the Southern Hemisphere during FGGE.

,*Mon. Wea. Rev.***110**, 1712–1721, doi:10.1175/1520-0493(1982)110<1712:AUWAIT>2.0.CO;2.Sánchez-Lavega, A., and et al. , 2014: The long-term steady motion of Saturn’s hexagon and the stability of its enclosed jet stream under seasonal changes.

,*Geophys. Res. Lett.***41**, 1425–1431, doi:10.1002/2013GL059078.Scott, R. K., , and D. G. Dritschel, 2012: The structure of zonal jets in geostrophic turbulence.

,*J. Fluid Mech.***711**, 576–598, doi:10.1017/jfm.2012.410.Srinivasan, K., , and W. R. Young, 2012: Zonostrophic instability.

,*J. Atmos. Sci.***69**, 1633–1656, doi:10.1175/JAS-D-11-0200.1.Tangarife, T., 2015: Kinetic theory and large deviations for the dynamics of geophysical flows. Ph.D. thesis, Ècole Normale Supérieure de Lyon, 186 pp. [Available online at http://www.theses.fr/2015ENSL1037.]

Thomas, V., , B. K. Lieu, , M. R. Jovanović, , B. F. Farrell, , P. J. Ioannou, , and D. F. Gayme, 2014: Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow.

,*Phys. Fluids***26**, 105112, doi:10.1063/1.4898159.Thomas, V., , B. F. Farrell, , P. J. Ioannou, , and D. F. Gayme, 2015: A minimal model of self-sustaining turbulence.

,*Phys. Fluids***27**, 105104, doi:10.1063/1.4931776.Tobias, S. M., , and J. B. Marston, 2013: Direct statistical simulation of out-of-equilibrium jets.

,*Phys. Rev. Lett.***110**, 104502, doi:10.1103/PhysRevLett.110.104502.Vasavada, A. R., , and A. P. Showman, 2005: Jovian atmospheric dynamics: An update after

*Galileo*and*Cassini*.,*Rep. Prog. Phys.***68**, 1935–1996, doi:10.1088/0034-4885/68/8/R06.

^{1}

Assumption (13) implies identity (14) even when

^{2}

A stochastic term *ξ* has units *Q* has dimensions

^{3}

These perturbations equations are valid for equilibria inhomogeneous in both *x* and *y* directions. In the case of perturbations that are homogeneous in *x*, the projection operators are redundant.

^{4}

The covariance eigenfunction does not need to be symmetric or Hermitian in its matrix representation, but both symmetric and asymmetric parts have the same growth rate. For a discussion of the properties of covariance eigenvalue problems see Farrell and Ioannou (2002).

^{5}

The periodic boundary conditions always allow the existence of a jet eigenfunction with zero growth and with structure that of the *y* derivative of the equilibrium jet and covariance. This eigenfunction leads to a translation of the equilibrium jet and its associated covariance in the *y* direction. The existence of this neutral eigenfunction can be a verified by taking the *y* derivative of (28). We do not include this obvious neutral eigenfunction in the stability analysis.