## 1. Introduction

Atmospheric turbulence is commonly observed to be organized into slowly varying large-scale structures such as zonal jets and coherent vortices. Prominent examples are the banded jets and the Great Red Spot in the Jovian atmosphere (Ingersoll 1990; Vasavada and Showman 2005). Laboratory experiments as well as direct numerical simulations of turbulent flows have shown that these coherent structures appear and persist for a very long time despite the presence of eddy mixing (Vallis and Maltrud 1993; Weeks et al. 1997; Read et al. 2004; Espa et al. 2010; Di Nitto et al. 2013).

A model that exhibits many aspects of turbulent self-organization into coherent structures yet is simple enough to extensively investigate is a barotropic flow on the surface of a rotating planet or on a beta plane with turbulence sustained by random stirring. Numerical simulations of this model have shown that robust zonal jets coexist with large-scale westward-propagating coherent waves (Sukoriansky et al. 2008; Galperin et al. 2010). These waves were found to either obey a Rossby wave dispersion or form nondispersive packets that are referred to as satellite modes (Danilov and Gurarie 2004) or zonons (Sukoriansky et al. 2008). In addition, the formation of these coherent structures was shown to be a bifurcation phenomenon. As the energy input of the stochastic forcing is increased, the flow bifurcates from a turbulent, spatially homogeneous state to a state in which zonal jets and/or nonzonal coherent structures emerge and are maintained by turbulence (Bakas and Ioannou 2013a; Constantinou et al. 2014). In this work, we address the eddy–mean flow dynamics underlying the emergence of both zonal and nonzonal structures.

Since organization of turbulence into coherent structures involves complex nonlinear interactions among a large number of degrees of freedom, which erratically contribute to the maintenance of the large-scale structure, an attractive approach is to study the statistical state dynamics (SSD) of the turbulent flow rather than single realizations of the turbulent field. Recently, the SSD of barotropic and baroclinic atmospheres has been studied by truncating the infinite hierarchy of cumulant equations to second order. Stochastic structural stability theory (S3T) is such a second-order Gaussian approximation of the full SSD, in which the third cumulant is parameterized as the sum of a known correlation function and a dissipation term (Farrell and Ioannou 2003). This is equivalent to a parameterization of the eddy–eddy nonlinearity as an exogenous random forcing with the required dissipation to remove the energy injected by the forcing. Such a representation is strongly supported by the results of previous studies. Linear inverse modeling studies showed that this parameterization is the best linear representation of the eddy–eddy nonlinear interactions in planetary turbulence (DelSole and Farrell 1996; DelSole 1996; DelSole and Hou 1999; DelSole 2004), while earlier studies have shown that the turbulent transport properties (heat and momentum fluxes) of the midlatitude transient climatology are accurately obtained as the stochastic response of the large-scale flow to stochastic forcing (Farrell and Ioannou 1994, 1995; Whitaker and Sardeshmukh 1998; Zhang and Held 1999). In addition, Bouchet et al. (2013) have shown that, in the limit of weak forcing and dissipation, the formal asymptotic expansion of the probability density function of the Euler equations around a mean flow that is assumed to only have a singular spectrum of modes comprises the second-order S3T closure with an additional stochastic term forcing the mean flow. Therefore, S3T formally describes the statistical equilibrium mean flow and the eddy statistics in this case, as the additional stochastic term only produces fluctuations around this statistical equilibrium. Similar to the S3T closure of the full SSD is the second-order cumulant expansion (CE2) closure in which the third-order cumulant is neglected without parameterization (Marston et al. 2008; Marston 2010, 2012; Tobias and Marston 2013). It has been shown that the predictions of S3T (or CE2) simulations are reflected in corresponding nonlinear simulations (O’Gorman and Schneider 2007; Srinivasan and Young 2012; Tobias and Marston 2013; Constantinou et al. 2014).

The second-order closure results in a nonlinear, autonomous dynamical system that governs the evolution of the mean flow and its consistent second-order perturbation statistics. Its fixed points define statistical equilibria, whose instability brings about structural reconfiguration of the mean flow and of the turbulent statistics. Previous studies employing S3T addressed the bifurcation from a homogeneous turbulent regime to a jet-forming regime in barotropic beta-plane turbulence and identified the emerging jet structures as linearly unstable modes of the homogeneous turbulent state equilibrium (Farrell and Ioannou 2003, 2007; Bakas and Ioannou 2011; Srinivasan and Young 2012; Parker and Krommes 2013, 2014). The S3T stability analysis of the homogeneous equilibrium was further advanced with the introduction of the continuum formulation by Srinivasan and Young (2012), who derived a compact analytic expression for the growth rate and frequency of the unstable structures. Interestingly, Carnevale and Martin (1982) using field theoretic techniques have arrived at the same stability equation for the statistical description of fluctuations about a homogeneous state.

Comparisons of the jet structure predicted by S3T with direct numerical simulations have shown that the structure of zonal flows that emerge in the nonlinear simulations can be predicted by S3T (Srinivasan and Young 2012; Tobias and Marston 2013; Constantinou et al. 2014). However, Srinivasan and Young (2012) found quantitative differences between the predictions of S3T as seen in the bifurcation diagram for the emergence of jets and the corresponding diagram obtained from the nonlinear simulations, calling into question the validity of the S3T (or CE2) approximations when the mean flow is very weak. Constantinou et al. (2014) demonstrated that this discrepancy was due to the prior emergence of nonzonal coherent structures that modified the background equilibrium spectrum and showed that S3T predictions were accurate when this modification in the background spectrum was accounted for.

The nonzonal structures were treated in these studies as incoherent because of the assumption that the ensemble average over the forcing realizations is equivalent to a zonal average and therefore their emergence and effect on the jet dynamics could not be directly addressed. By making the alternative interpretation of the ensemble mean as a Reynolds average over the fast turbulent motions that was introduced in earlier studies of atmospheric blocking (Bernstein 2009; Bernstein and Farrell 2010), Bakas and Ioannou (2013a, 2014) addressed the emergence of the nonzonal coherent structures in barotropic beta-plane turbulence in terms of the parameters *β* is the gradient of the planetary vorticity, *ε* is the energy input rate of the forcing, and

Typical parameter values for geophysical flows. The typical forcing length scale *L _{f}* is taken as the deformation radius in each geophysical setting.

The S3T dynamics that underlie the formation of large-scale structure cannot depend on turbulent anisotropic inverse cascade processes because local-in-wavenumber-space eddy–eddy interactions are absent in S3T. In S3T, large-scale structure emerges from a cooperative instability arising from the nonlocal in wavenumber space interaction between the large-scale mean flow and the forced, small-scale turbulent eddies. The eddy–mean flow dynamics of this cooperative instability has been investigated by Bakas and Ioannou (2013b) for the case of zonal jet emergence in the limit of

This paper is organized as follows: In section 2 we derive the S3T system for a barotropic flow and the resulting eigenvalue problem addressing the stability of the homogeneous statistical equilibrium. In section 3 we transform the eigenvalue problem to a rotated frame of reference so that the formation of zonal jets and nonzonal structures can be studied under a uniform framework. In section 4 we identify the eddy–mean flow dynamics underlying the S3T instability for isotropic stochastic forcing and in section 5 we study the effect of the forcing anisotropy to the S3T instability. The results are summarized in sections 6 and 7.

## 2. Formulation of S3T dynamics and emergence of nonzonal coherent structures

*x*and

*y*Cartesian coordinates along the zonal and the meridional direction respectively and with planetary vorticity gradient,

*ψ*, as

*J*is the two-dimensional Jacobian,

*r*, which typically models Ekman drag in planetary atmospheres. Turbulence is maintained by the external stochastic forcing

*ξ*, which models exogenous processes, such as turbulent convection or energy injected by baroclinic instability. We assume that

*ε*. We nondimensionalize (1) using the dissipation time-scale

To construct the S3T dynamical system in the continuous formulation of Srinivasan and Young (2012), we proceed as follows:

- The averaged fields, denoted with uppercase letters, are calculated by taking a time average, denoted with
, over an intermediate time scale, larger than the time scale of the turbulent motions but smaller than the time scale of the large-scale motions. Deviations from the mean (eddies) are denoted with dashes and lowercase letters. For example, the vorticity field is split as , where . The equations for the mean and the eddies that derive from (1) arewhereis the linear perturbation operator about the instantaneous mean flow and . Neglecting the nonlinear term in (2b) we obtain the quasi-linear system: - The quasi-linear system (4) under the ergodic assumption that the time average over the intermediate time scale is equal to an ensemble average produces the S3T system:where
*C*is the ensemble-mean eddy-vorticity spatial covariance between pointsand ; *Q*is the spatial covariance of the delta-correlated and spatially homogeneous forcing, defined byandis the ensemble-mean vorticity forcing of the large scales by the eddy field, given byThe subscript *a*(*b*) in the operators indicates that the coefficients of the operator are evaluated on*a*(*b*) and that the operator acts only on the variable( ). The subscript indicates that any function of and is evaluated at the same point, .

The S3T system (5) is a closure of the statistical dynamics of (2) at second order. Being autonomous, it may posses statistical equilibria

*Q*, there is always the homogeneous S3T equilibriumwith no mean flow and a homogeneous eddy field—that is, with a translationally invariant covariance

*ε*exceeds the critical value

*β*and on the structure of

*Q*. In this work we focus our analysis close to the instability threshold

^{1}wherewith

*μ*modulates the anisotropy of the spectrum of the forcing and takes values

*μ*are shown in Fig. 2. When

## 3. Emergence of nonzonal structures as zonal flows in a rotated frame

*f*determining the amplitude and relative phase of the vorticity flux feedback induced by the mean-flow eigenfunction

*σ*. When the real part of

*f*is positive, the induced vorticity flux is upgradient, and when it is negative, the flux is downgradient. It is shown in appendix A thatwith

*σ*of (15) satisfy the equationFor

^{2}that the eigenvalue corresponding to unstable zonal jet eigenfunctions have

^{3}

*ϑ*between their phase lines and the

*β*by writing

*β*.

In the following sections we will determine the contribution of the various waves to the vorticity flux feedback and identify the angle *ϑ* that produces the most significant contribution to this feedback. We will also calculate *n* for

## 4. Eddy–mean flow dynamics leading to formation of zonal and nonzonal structures for isotropic forcing

### a. Induced vorticity fluxes for

*β*:with

*β*and is shown in Fig. 4a. For

*φ*. Therefore all zonal and nonzonal eigenfunctions with the same wavenumber

*n*grow at the same rate. Upgradient fluxes (

*n*are induced by waves with phase lines inclined at angles satisfying

*y*axis surrender instantaneously momentum to the mean flow and reinforce it, whereas pairs with

*K*positive and proportional to the anisotropy factor

*μ*[cf. appendix B and Bakas and Ioannou (2013b)].

*β*to the vorticity flux feedback, we plot

*n*and wave angle

*ϑ*. We choose to scale

*β*tempers both the upgradient (for roughly

The asymptotic analysis of Bakas and Ioannou (2013b), which is formally valid for

Consider now the case of a nonzonal perturbation (Fig. 4c). We observe that the angles for which the waves have significant positive or negative contributions to the vorticity flux feedback are roughly the same as in the case of zonal jets. In addition, the vorticity flux feedback factor decreases with the angle *φ* of the nonzonal perturbations [cf. (25)]. As a result, zonal jet perturbations always produce larger vorticity fluxes compared to nonzonal perturbations and are therefore the most unstable in the limit

### b. Induced vorticity fluxes for

Consider first the emergence of nonzonal structures in the limit *f _{r}* for the case of nonzonal structures at

Resonant triads do not occur for all mean-flow perturbations *ϑ*. The resonant and nonresonant contribution for a typical case in region B is shown in Fig. 7c. Note that it is the resonant contributions that determine the vorticity flux feedback. However, they produce a negative vorticity flux feedback (a downgradient tendency), which is stabilizing, a result that holds for all

*j*indicates the value of the functions at the

*j*th of the

*M*roots of

*β*. It is

*j*th resonant contribution to the total vorticity flux feedback depends only on the sign of

*n*. For

*n*(corresponding to region D), the vorticity flux feedback is negative and

An interesting exception to the results discussed above occurs for the important case of zonal jet perturbations (*y* axis (remember that, for smaller *β*, the region that produces destabilizing fluxes extends up to *β*,

To summarize, although zonal jets and most nonzonal perturbations induce fluxes that decay as *β*, resonant and near-resonant interactions arrest the decay rate of certain nonzonal perturbations by a factor of *n* and *φ* (cf. Fig. 5), the vorticity flux feedback is negative for

### c. Induced vorticity fluxes for

We have seen that in the singular case of isotropic forcing the only process available for the emergence of mean flows is the fourth-order antidiffusive vorticity flux feedback induced by the variation of the group velocity of the forced eddies due to the mean flow shear. For *β* increases, this growth is reduced since the waves interact with an effective integral shear within their propagation extent, which is weak and, eventually, as we have seen in the previous section, for *β*. This occurs for *β* maximizes the S3T instability for all forcing spectra.

While the eddy–mean flow interaction of both zonal and nonzonal perturbations is dominated by the same dynamics when *β* (*β* increases, the resonant contributions start playing an important role for nonzonal perturbations as there is enhanced contribution to the vorticity flux feedback in the vicinity of the

## 5. Effect of anisotropic forcing on S3T instability

*μ*(cf. Fig. 2). The maximum value of

*n*depends in this case linearly on

*μ*(cf. appendix B),For nonzonal perturbations, the main contribution comes from forced waves satisfying the resonant condition

*μ*[cf. (A7c)]. The anisotropicity affects only the magnitude of

*μ*. However, the effect on the maximum vorticity feedback is weak, as the spectral selectivity of the resonances renders the characteristics of the most unstable nonzonal structure independent of the spectrum of the forcing. That is,

*μ*(cf. Figs. 11b,c).

*μ*:This shows that there can be upgradient vorticity fluxes leading to S3T instability for

*y*direction (meridional jets) (cf. Fig. 11c).

It is worth noting that Srinivasan and Young (2014) also find that that the eddy momentum fluxes are proportional to *μ* when a constant shear flow is stochastically forced with power spectrum (11). This result is intriguing as the two studies address two different physical regimes. This work treats the limit appropriate for emerging structures in which the shear time scale is far larger than the dissipation time scale with the fluxes determined by the instantaneous response of the eddies on the shear. Srinivasan and Young (2014) study the opposite limit in which the mean-flow shear is finite and the shear time scale is much shorter than the dissipation time scale with the fluxes determined by the integrated influence of the shear on the eddies over their whole life cycle, which may include complex effects such as reflection and absorption at critical levels.

In summary,

- the S3T instability of the homogeneous state is a monotonically increasing function of
*μ*for all*β*, - the forced waves that contribute most to the instability are structures with small
*γ*(i.e., waves with phase lines nearly aligned with the*y*axis, as in Fig. 2a), and - the anisotropy of the excitation affects prominently the S3T stability of the homogeneous state only for
.

## 6. Discussion

In this work we addressed the dynamics underlying the onset of the S3T instability leading to the formation of large-scale structure but not the nonlinear development and equilibration of the instability. The emergent structure may be susceptible to either hydrodynamic or structural secondary instabilities as it reaches finite amplitude [cf. Farrell and Ioannou 2003, 2007; Parker and Krommes (2014) for zonal jets and Bakas and Ioannou (2014) for nonzonal flows]. For example, the most unstable jet structure for marginally unstable parameters is at the scale of the forcing *β*, maximum instability occurs at a scale slightly larger than ^{4} while for larger *β*, maximum instability occurs at about

*β*plane in the absence of forcing and dissipation. Parker and Krommes (2015) have recently shown that in the inviscid limit the modulational instability of a Rossby wave

## 7. Conclusions

The mechanism for formation of coherent structures in a barotropic beta plane under a spatially homogeneous and temporally delta-correlated stochastic forcing was examined in the framework of stochastic structural stability theory (S3T). In this framework, a second-order closure for the dynamics of the flow statistics is obtained by ignoring or parameterizing the eddy–eddy nonlinearity. The resulting deterministic system for the joint evolution of the coherent flow and of the second-order turbulent eddy covariance admits statistical equilibria.

For a spatially homogeneous forcing covariance, a homogeneous state with no mean coherent structures is such an equilibrium solution of the S3T dynamical system. When a critical energy input rate of the forcing is exceeded, this homogeneous equilibrium is unstable and propagating nonzonal coherent structures and/or stationary zonal jets emerge in agreement with direct numerical simulations. To identify the processes that lead to the formation of coherent structures, the vorticity fluxes induced by a plane wave-mean flow, which is the eigenfunction of the linearized S3T system around the homogeneous equilibrium, were calculated close to the bifurcation point and closed form asymptotic expressions for these fluxes were obtained. Upgradient fluxes in this limit are consistent with S3T instability and coherent structure formation.

The induced fluxes were calculated in a rotated frame of reference, in which the plane wave-mean flow corresponds to a zonal jet evolving in a beta plane with a nonmeridional planetary vorticity gradient. This was done because in this rotated frame of reference the intuition gained by previous studies for the eddy–mean flow dynamics underlying zonal jet formation can be utilized to clarify the dynamics underlying nonzonal wave formation, or formation of zonal jets when the effect of topography is equivalent to a nonmeridional planetary vorticity gradient.

In the limit of a weak planetary vorticity gradient (

In the limit of strong planetary vorticity gradient *β*. Zonal jets continue to induce upgradient vorticity fluxes through wave shearing, which decrease as

Finally, the relation of the S3T instability and modulational instability of finite-amplitude Rossby waves was discussed. Parker and Krommes (2015) showed that the growth rates obtained when three Rossby waves interact with the primary finite-amplitude Rossby wave match exactly in the inviscid limit that the growth rates obtained by the S3T stability analysis for the homogeneous equilibrium with the vorticity covariance produced by the primary Rossby wave. It was shown in this work that this agreement can be found for more general cases (e.g., when the covariance is produced by any linear combination of Rossby waves with the same total wavenumber). Such an agreement occurs because retaining only the interaction between four waves in modulational instability is equivalent to neglecting the eddy–eddy nonlinearity in S3T. The equivalence of the dynamics underlying modulational and S3T instability in this case shows that S3T stability analysis generalizes modulational instability analysis to stochastically forced dissipative flows. However, contrary to modulational instability, the underlying S3T dynamics can capture both the emergence of large-scale structure and its equilibration. In addition, the dynamics underlying modulational instability can be interpreted under the alternative eddy–mean flow view adopted in this work.

Nikolaos Bakas is supported by the AXA Research Fund and Navid Constantinou acknowledges the support of the Alexander S. Onassis Public Benefit Foundation. Nikolaos Bakas would also like to thank Izumi Saito for fruitful discussions on the results of this work.

# APPENDIX A

## Eddy-Vorticity Flux Response to a Mean-Flow Perturbation

*σ*has the spatial structureThe power spectrum of the homogeneous part of the covariance eigenfunction,

*f*:with

*σ*, which can be shown to be exactly the stability equation obtained by Bakas and Ioannou (2014). The stability equation can be written in terms of the real and imaginary part of

*σ*asThe real part of the vorticity flux feedback

*ε*increases as

*β*and therefore the frequency of the marginally unstable waves is approximately equal to the Rossby phase frequency.

*n*and nonzonality parameter

*φ*is upgradient, and the marginal energy input rate is

Note that *φ* is shifted by

# APPENDIX B

## Asymptotic Expressions for the Induced Vorticity Flux Feedback

In this appendix we calculate in closed form asymptotic expressions for the vorticity flux feedback induced by a mean-flow perturbation in the form of a zonal jet in the rotated frame of reference with wavenumber *n* for

### a. Case

*n*satisfying

*β*. Since

*ϑ*with respect to the

*n*when

*β*all waves with

*n*for

*μ*. The maximum feedback factor is in this caseand is achieved for mean flows with

*β*:producing upgradient fluxes for

### b. Case

*ϑ*, then the feedback factor is

*ϑ*, then as we will show in this appendix,

*β*in cases in which

*I*, we split the range of integration to a small range close to the roots of

*M*is the total number of the roots of

*j*denotes the value at

*d*is of

*η*is shown as a function of

*κ*(which is a rough measure of the distance between the roots) in Fig. B2. We observe that the maximum value is attained at

Consider now nonzonal perturbations *n* (region B in Fig. 7a), and for any given *φ*, *n* (regions A and C in Fig. 7a), ^{B1}

The maximum response, which is *β*, the location and the size of the region of maximum response depends on *β* through the dependence of *κ* on *β*. However, as *β* increases this dependence is weak and as *β*, making it exceedingly hard to locate for large *β*, and the asymptotic approach of

# APPENDIX C

## Formal Equivalence between S3T Instability of a Homogeneous Equilibrium with Modulational Instability of a Corresponding Basic Flow

In this appendix we demonstrate the formal equivalence between the modulational instability (MI) of any solution of the barotropic equation, which may be in general time dependent but has stationary power spectrum, with the S3T instability of the homogeneous state with the same power spectrum.

**n**, which is orthogonal to

**n**and

**n**, and the waves

**p**(as in MI studies), (C6) is referred to as the four-mode truncation or “4MT” system.

*m*th Bessel function of the first kind, this can be shown to be the nondispersive structurepropagating westward with velocity

*β*, where

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^{1}

The power spectrum of a spatially homogeneous covariance is the Fourier transform of the covariance

^{2}

We have been unable to find a counterexample to these assertions or to prove them when

^{3}

While the phase speed of the marginally unstable nonzonal structures almost matches the corresponding Rossby phase speed for *β*, we have found that the results presented in this work are not sensitive to the value of the frequency.

^{4}

The value of *n* that produces maximum instability has a nonuniform limit as

^{B1}

It can be shown that fluxes from the resonant contributions for