Emergence of non-zonal coherent structures. Nikolaos A. Bakas Department of Physics, University of Ioannina, Ioannina, Greece

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1 Emergence of non-zonal coherent structures Nikolaos A. Bakas Department of Phsics, Universit of Ioannina, Ioannina, Greece Petros J. Ioannou Department of Phsics, National and Kapodistrian Universit of Athens, Athens, Greece arxiv:5.58v [phsics.ao-ph] Jan 5 Planetar turbulence is observed to self-organize into large-scale structures such as zonal jets and coherent vortices. One of the simplest models that retains the relevant dnamics of turbulent self-organization is a barotropic flow in a beta-plane channel with turbulence sustained b random stirring. Non-linear integrations of this model show that as the energ input rate of the forcing is increased, the homogeneit of the flow is first broken b the emergence of non-zonal, coherent, westward propagating structures and at larger energ input rates b the emergence of zonal jets. The emergence of both non-zonal coherent structures and zonal jets is studied using a statistical theor, Stochastic Structural Stabilit Theor S3T). S3T directl models a second-order approimation to the statistical mean turbulent state and allows the identification of statistical turbulent equilibria and stud of their stabilit. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analtic epressions for the zonal and non-zonal large-scale coherent flows that emerge as a result of structural instabilit are obtained and the equilibration of the incipient instabilities is studied through numerical integrations of the S3T dnamical sstem. The dnamics underling the formation of zonal jets are also investigated. It is shown that zonal jets form from the upgradient momentum flues that result from the shearing of the eddies b the emerging infinitesimal large-scale flow. Finall, numerical simulations of the nonlinear equations confirm the characteristics scale, amplitude and phase speed) of the structures predicted b S3T, even in highl non-linear parameter regimes such as the regime of zonostrophic turbulence. Atmospheric and oceanic turbulence is commonl observed to be organized into spatiall and temporall coherent structures such as zonal jets and coherent vortices. A simple model that retains the relevant dnamics, is a barotropic flow on a β-plane with turbulence sustained b random stirring. Numerical simulations of the stochasticall forced barotropic vorticit equation on the surface of a rotating sphere or on a β-plane, have shown the coeistence of robust zonal jets and of large-scale westward propagating coherent structures that are referred to as satellite modes Danilov and Gurarie ) or zonons Sukariansk et al. 8; Galperin et al. ). Emergence of these coherent structures in barotropic turbulence has also another feature. As the energ input of the stochastic forcing is increased or dissipation is decreased, there is a sudden onset of coherent zonal flows Srinivasan and Young ; Constantinou et al. ) and non-zonal coherent structures Bakas and Ioannou ). This argues that the emergence of coherent structures in a homogeneous background of turbulence is a bifurcation phenomenon. An advantageous method to stud such a phenomenon, is to adopt the perspective of statistical state dnamics of the flow, rather than look into the dnamics of sample realizations of direct numerical simulations. This amounts to stud the dnamics and stabilit of the statistical equilibria arising in the turbulent flow, which are fied points of the equations governing the evolution of the flow statistics. This approach is followed in the Stochastic Structural Stabilit Theor S3T) Farrell and Ioannou 3) or Second Order Cumulant Epansion theor CE) Marston et al. 8). This theor is based on two building blocks. The first is to do a Renolds decomposition of the dnamical variables

2 into the sum of a mean value that represents the coherent flow and fluctuations that represent the turbulent eddies and then form the cumulants containing the information on the mean values first cumulant) and on the edd statistics higher order cumulants). The second building block is to truncate the equations governing the evolution of the cumulants at second order b either parameterizing the terms involving the third cumulant Farrell and Ioannou 993a,b,c; DelSole and Farrell 99; DelSole ) or setting the third cumulant to zero Marston et al. 8; Tobias et al. ; Srinivasan and Young ). Restriction of the dnamics to the first two cumulants is equivalent to neglecting the edd-edd interactions in the full non-linear dnamics and retaining onl the interaction between the eddies with the instantaneous mean flow. While such a second order closure might seem crude at first sight, there is strong evidence to support it Bouchet et al. 3). Previous studies emploing S3T have alread addressed the bifurcation from a homogeneous turbulent regime to a jet forming regime in barotropic β-plane turbulence and identified the emerging jet structures both numericall Farrell and Ioannou 7) and analticall Bakas and Ioannou ; Srinivasan and Young ) as linearl unstable modes to the homogeneous turbulent state equilibrium. It was also shown that the resulting dnamical sstem for the evolution of the first two cumulants linearized around the homogeneous equilibrium possesses the mathematical structure of the dnamical sstem of pattern formation Parker and Krommes 3). Comparison of the results of the stabilit analsis with direct numerical simulations have shown that the structure of zonal flows that emerge in the non-linear simulations can be predicted b S3T Srinivasan and Young ; Constantinou et al. ). However, these research efforts, have assumed that the ensemble average emploed in S3T is equivalent to a zonal average, a simplification that treats the non-zonal structures as incoherent and cannot address their emergence and effect on the jet dnamics. In addition, the edd-mean flow dnamics underling the S3T instabilit even in the jet formation case, that involve onl the interactions of small scale waves with the large-scale coherent structures are not clear. So the goals in this article are the following. The first goal is to adopt a more general interpretation of the ensemble average, in order to address the emergence of coherent non-zonal structures. We adopt the more general interpretation that the ensemble average is a Renolds average over the fast turbulent motions Bernstein 9; Bernstein and Farrell ). With this definition of the ensemble mean, we obtain the statistical dnamics of the interaction of the coarse-grained ensemble average field, which can be zonal or non-zonal coherent structures represented b their vorticit, with the fine-grained incoherent field represented b the vorticit second cumulant and we revisit the structural stabilit of the homogeneous equilibrium under this assumption. The second goal is to stud in detail the edd-mean flow dnamics underling the S3T instabilit focusing on the eample of jet formation. And the third goal is to compare the characteristics of the structures that emerge in S3T against non-linear simulations, even in highl non-linear regimes that at first glance present a challenging test for the restricted dnamics of S3T.. Formulation of Stochastic Structural Stabilit Theor under a generalized average Consider a nondivergent barotropic flow on a β-plane with cartesian coordinates =, ). The velocit field, u = u, v), is given b u, v) = ψ, ψ), where ψ is the streamfunction. Relative vorticit ζ,, t) = ψ, evolves according to the non-linear NL) equation: t + u ) ζ + βv = rζ ν ζ + εf e, ) where = + is the horizontal Laplacian, β is the gradient of planetar vorticit, r is the coefficient of linear dissipation that tpicall parameterizes Ekman drag in planetar atmospheres and ν is the coefficient of hper-diffusion that dissipates enstroph flowing into unresolved scales. The eogenous forcing term f e, parameterizes processes such as small scale convection or baroclinic instabilit, that are missing from the barotropic dnamics and is necessar to sustain turbulence. We assume that f e is a temporall delta correlated and spatiall homogeneous random stirring injecting energ at a rate ε and having a two-point, two-time correlation function of the form: f e,, t )f e,, t ) = δt t )Ξ,,, ), ) where the brackets denote an ensemble average over the different realizations of the forcing. S3T describes the statistical dnamics of the first two same time cumulants of ). The equations governing the evolution of the first two cumulants are obtained as follows. We decompose the vorticit field into the averaged field, Z = T [ζ], defined as a time average over an intermediate time scale and deviations from the mean or eddies, ζ = ζ Z. The intermediate time scale is larger than the time scale of the turbulent motions but smaller than the time scale of the large scale motions. With this decomposition we rewrite ) as: t + U ) Z + βv = T [u ζ ] rz ν Z, 3) where U = [U, V ] = [ Ψ, Ψ] and u = [u, v ] = [ ψ, ψ ] are the mean and the edd velocit fields respectivel. The mean vorticit is therefore forced b the divergence of the mean vorticit flues. The edd vorticit

3 ζ evolves according to: t + U ) ζ + β + Z)v + u Z = = rζ ν ζ + f e + T [u ζ ] u ζ }{{}, ) f nl where f nl is the term involving the non-linear interactions among the turbulent eddies. A closed sstem for the statistical state dnamics is obtained b first neglecting the edd-edd term f nl to obtain the quasi-linear sstem, t + U ) Z + βv = T [u ζ ] rz ν Z, 5) t + U ) ζ + β + Z)v + u Z = = rζ ν ζ + εf e, ) In order to obtain the statistical dnamics of the quasilinear sstem 5)-) we adopt the general interpretation that the ensemble average over the forcing realizations is equal to the time average over the intermediate time scale Bernstein 9; Bernstein and Farrell ). Under this assumption, the slowl varing mean flow Z is also the first cumulant of the vorticit Z = ζ, where the brackets denote the ensemble average. The time mean of the vorticit flu is equal to the ensemble mean of the flu T [u ζ ] = u ζ. The flues can be related to the second cumulant C,, t) ζ )ζ ), which is the correlation function of the edd vorticit between the two points i = i, i ), i =,. We hereafter indicate the dnamic variables that are functions of points i = i, i ) with the subscript i, that is ζ i ζ i ). B making the identification that the flues at point are equal to the value of the two variable function u ζ evaluated at the same point = =, we write the flues as: u ζ = u ζ =. 7) Epressing the velocities in terms of the vorticit [u, v ] = [, ]ζ, where is the integral operator that inverts vorticit into the streamfunction field, we obtain the vorticit flues as a function of the second cumulant, in the following manner: u ζ = [ u ζ =, v ζ ] = [ = ζ ζ, ] = ζ ζ = [ = C), ] = C). 8) = Consequentl, the first cumulant evolves according to: t Z + UZ + V β + Z ) + rz + ν Z = = C) = C) =. 9) Multipling A5) for t ζ b ζ and A5) for t ζ b ζ, adding the two equations and taking the ensemble average ields the equation for the second cumulant C: t C A + A )C = ε f e ζ + f e ζ, ) where A i = U i i β + i Z) i i + i Z i i r ν i, ) governs the dnamics of linear perturbations about the instantaneous mean flow U. The right hand side of ) is the correlation of the eternal forcing with vorticit, which for delta correlated stochastic forcing is independent of the state of the flow and is equal at all times to the prescribed forcing covariance: ε f e ζ + f e ζ = ε f e f e = εξ. Therefore The second cumulant evolves then according to: t C = A + A )C + εξ, ) and forms with Eq. 9) the closed autonomous sstem of S3T theor that determines the statistical dnamics of the flow approimated at second order. The S3T sstem has bounded solutions cf. Appendi A) and the fied points Z E and C E, if the eist, define statistical equilibria of the coherent structures with vorticit, Z E, in the presence of an edd field with second order cumulant or covariance, C E. The structural stabilit of these statistical equilibria addresses the parameters in the phsical sstem which can lead to abrupt reorganization of the turbulent flow. Specificall, when an equilibrium of the S3T equations becomes unstable as a phsical parameter changes, the turbulent flow bifurcates to a different attractor. In this work, we show that coherent structures emerge as unstable modes of the S3T sstem and equilibrate at finite amplitude. The predictions of S3T regarding the emergence and characteristics of the coherent structures are then compared to the non-linear simulations of the stochasticall forced barotropic flow.. S3T instabilit and emergence of finite amplitude large-scale structure The homogeneous equilibrium with no mean flow Z E =, C E = Ξ r, 3) is a fied point of the S3T sstem 9) and ) in the absence of hperdiffusion cf. Appendi B). The linear stabilit of the homogeneous equilibrium can be addressed b performing an eigenanalsis of the S3T sstem linearized about this equilibrium. The eigenfunctions in this case have the plane wave form δz = Z nm e in+im e σt, δc = C nm, ỹ)e in+im e σt, ) where =, = + )/, ỹ =, = + )/, n and m are the and wavenumbers of the eigenfunction and σ = σ r + iσ i is the eigenvalue with σ r = Reσ), σ i = Imσ) being the growth rate and frequenc of the mode respectivel. The eigenvalue σ satisfies the 3

4 non-dimensional equation: ε πr 3 L d kd l Ñ / K )ˆΞ k, l) f m k ñ l) [ñ m k + l ] +) + m ñ ) k + l+ i β k K s k + ñ) K ) + σ + ) K K = s = σ + )Ñ iñ β, 5) where L f is a characteristic length scale, σ = σ/r and ñ, m) = L f n, m) are the non-dimensional eigenvalue and wavenumbers respectivel, ε = ε/r 3 L f ) is the non-dimensional energ injection rate of the forcing, β = βl f /r is the nondimensional planetar vorticit gradient, ˆΞk, l) = π Ξ, ỹ)e ik ilỹ d dỹ, ) is the Fourier transform of the forcing covariance, K = k + l, K s = k+ñ) + l+ m), Ñ = ñ + m, k + = k+ñ/ and l + = l + m/ cf. Appendi B). For a forcing with the mirror smmetr ˆΞk, l) = ˆΞk, l) in wavenumber space and for ñ, the eigenvalues satisf the relations: σ ñ, m) = σ ñ, m), and σ ñ, m) = σ ñ, m), 7) impling that the growth rates depend on ñ and m. As a result, the plane wave δz = cosn + m) and an arra of localized vortices δz = cosn) cosm), have the same growth rate, despite their different structure. For zonall smmetric perturbations with ñ =, onl the second relation in 7) holds and 5) reduces to the eigenvalue relation derived b Srinivasan and Young ) for the emergence of jets in a barotropic β-plane. We consider the case of a ring forcing that injects energ at rate ε at the total wavenumber K f : ˆΞk, l) = K f δ k + l K f ), 8) and obtain the eigenvalues σ b numericall solving 5). For small values of the energ input rate, σ r < for all wavenumbers and the homogeneous equilibrium is stable. At a critical ε c the homogeneous flow becomes S3T unstable and eponentiall growing coherent structures emerge. The critical value, ε c, is calculated b first determining the energ input rate ε t ñ, m) that renders wavenumbers ñ, m) neutral σ rñ, m) = ), and then b finding the minimum energ input rate over all wavenumbers: ε c = min ñ, m) ε t. The critical energ input rate ε c as a function of β is shown in figure. In addition, the corresponding critical zonostroph parameter R β =.7 ε c β ) / which was used in previous studies to characterize the emergence and structure of zonal jets in planetar turbulence Galperin et al. ), is shown as a function of β in figure. The absolute minimum energ input rate required is ε c = 7 and occurs at β min = 3.5, while the minimum zonostroph parameter required for the emergence of coherent flows is R β =.8 and occurs for β. For β β min, the structures that first become marginall stable are zonal jets with n = ). The critical input rate increases as ε c β for β and the homogeneous equilibrium is structurall stable for all ecitation amplitudes when β =. However, the structural stabilit for β = is an artifact of the assumed isotrop of the ecitation and the assumption of a barotropic flow. In the presence of even the slightest anisotrop Bakas and Ioannou, 3b), or in the case of a stratified flow Parker and Krommes 5), zonal jets are S3T unstable and are epected to emerge even in the absence of β. For β > β min, the marginall stable structures are non-zonal and ε c grows as ε c β / for β. Since the critical forcing for the emergence of zonal jets also shown in figure ), increases as ε c β for β, for large values of β non-zonal structures first emerge and onl at significantl higher ε zonal jets are epected to appear. Investigation of these results with other forcing distributions revealed that the results for β are independent of the structure of the forcing Bakas et al. 5). The parameter regime of S3T instabilit is now related to the results of previous studies and to geophsical flows. Previous studies have identified a parameter regime which is distinguished b robust, slowl varing zonal jets as well as propagating, non-dispersive, non-zonal coherent structures Galperin et al. ). This regime that is termed as zonostrophic, is in a region in parameter space in which the zonostroph parameter is large R β.5) and the scale k β =.5β 3 /ε) /5 in which anisotropization of the turbulent spectrum occurs is sufficientl larger than the forcing scale k β /K f /). This regime is shown in figure to be highl supercritical for all β. In addition, Bakas and Ioannou ) calculated indicative order of magnitude values of β and ε for the Earth s atmosphere and ocean as well as for the Jovian atmosphere. From these values we calculated the relevant zonostroph parameter R β and indicated the three geophsical flows in figure. We can see that all three cases are supercritical: the Jovian atmosphere is highl supercritical and is well within the zonostrophic regime, while the Earth s atmosphere and ocean are slightl supercritical at least within the contet of the simplified barotropic model). We now eamine the growth rate and dispersion properties of the unstable modes for ε > ε c and consider first the case β =, with ε = ε c. The growth rate of the maimall growing eigenvalue, σ r, and its associated frequenc of the mode, σ i, are plotted in figure 3a) as a function of ñ and m. We observe that the region in wavenumber space defined roughl b < ñ < /, and / < m < is unstable, with the maimum growth rate occurring for zonal structures ñ = ) with m.8. The frequenc

5 a). σ jets r σ non zon r σ r jets σ r non zon m β β.. β ñ ǫc 3 3 β Figure : The critical energ input rate ε c for structural instabilit thick solid line) and the critical energ input rate for structural instabilit of zonal jets solid line) as a function of β. The behavior of these critical values for β and β is indicated with the dashed asmptotes. In the light gra region onl non-zonal coherent structures emerge, while in the dark gra region both zonal jets and non-zonal coherent structures emerge. The thin dotted vertical line β = β min separates the unstable region: for β < β min zonal structures grow the most, whereas for β > β min non-zonal structures grow the most. Rβ zonostrophic regime atmosphere.5 3 β Jupiter ocean Figure : The critical zonostroph parameter R β =.7 ε c β ) / for structural instabilit thick line) and the corresponding critical parameter for structural instabilit of zonal jets thin line) as a function of β. The shaded region denotes the zonostrophic regime for which both the inequalities R β.5 and k β /K f / are satisfied. The stars denote the position of the Earth s atmosphere and ocean as well as the Jovian atmosphere in the R β, β parameter space. β / m b) ñ. Figure 3: Dispersion relation of the unstable modes for β = panel a) and β = panel b). The contours show the growth rate σ r and the shading shows the frequenc σ i of the unstable modes. For β O), stationar zonal jets are more unstable and for β, westward propagating non-zonal structures are more unstable. For both panels, the energ input rate is ε = ε c. of the unstable modes is zero for zonal jet perturbations ñ = ) and non-negative for all other wavenumbers ñ ). Using the smmetries 7), this implies that real unstable mean flow perturbations δz propagate in the retrograde direction if ñ and are stationar when ñ =. As ε increases the instabilit region epands and roughl covers the sector / < Ñ <, with zonal structures having a larger growth rate compared to non-zonal structures, a result that holds for an ε when β < β min. For β > β min the non-zonal structures have alwas larger growth rate. This is illustrated in figure 3b), showing the growth rates and frequencies of the unstable modes for β =. For larger β values there is a tendenc for the frequenc of the unstable modes to conform to the corresponding Rossb wave frequenc σ R = βñ ñ + m, 9) a tendenc that does not occur for smaller β. A comparison between the frequenc of the unstable modes and the Rossb wave frequenc is shown in figure in a plot of σ i / σ R. For slightl supercritical ε, the ratio is close to one and the unstable modes satisf the Rossb wave dispersion relation. At higher supercriticalities though, σ i departs from the Rossb wave frequenc b as much as % for the case of ε = 5 ε c shown in figure b))

6 m m.8.. a) ñ.5.5 b) ñ Figure : Ratio of the frequenc of the unstable modes σ i over the corresponding frequenc of a Rossb wave with the same wavenumbers σ R at a) ε = ε c and b) ε = 5 ε c when β =. Values of one denote an eact match with the Rossb wave frequenc. 3. Analsis of the edd-mean flow dnamics underling jet formation In this section, we investigate the edd-mean flow dnamics leading to jet formation. These dnamics should have the propert of directl channeling energ from the turbulent motions to the coherent flow without the presence of a turbulent cascade. Previous studies have identified such mechanisms for the maintenance of zonal jets. Huang and Robinson 998) showed that shear straining of the turbulent field b the jet produced upgradient momentum flues that maintained the jet against dissipation. A simple case that clearl illustrates the phsical picture for the mechanism of shear straining is to consider the evolution of eddies in a planar, inviscid constant shear flow. The eddies are sheared b the mean flow into thinner elliptical shapes, while their vorticit is conserved. For an elongated edd this implies that the edd velocities decrease and the edd energ is transferred to the mean flow through upgradient momentum flues. This mechanism can operate when the time required for the eddies to shear over is much shorter than the dissipation time scale. The reason is that in this limit even the eddies with streamfunctions leaning against the shear that initiall widen significantl gaining momentum, have the necessar time to shear over, elongate and surrender their momentum to the mean flow. Given that for an emerging jet the characteristic shear time scale is necessaril infinitel longer than the dissipation time scale, it needs to be shown that shear straining can produce upgradient momentum flues in this case as well. In addition, previous studies have shown that shearing of isotropic eddies on an infinite domain does not produce an net momentum flues Shepherd 985; Farrell 987; Hollowa ) and should have no effect on the S3T instabilit Srinivasan and Young ). Therefore another mechanism should be responsible for producing the upgradient flues in the case of an isotropic forcing. In order to investigate the edd-mean flow dnamics underling the S3T instabilit, we calculate the vorticit flu divergence that is induced when the statistical equilibrium 3) is perturbed b an infinitesimal coherent structure δz. For an S3T unstable structure, the induced flu divergence tends to enhance the coherent structure δz producing the positive feedback required for instabilit. So the goal of this section is to illuminate the edd-mean flow dnamics leading to this positive feedback and to understand qualitativel wh the homogeneous equilibrium is more stable for small and large values of β. For zonal mean flows 9), ) are simplified to: t U = u v ru = C) = ru, ) and where t C = A + A )C + Ξ, ) A i = U i i β i i U) i i r, ) respectivel. As a result the zonal mean flow is driven b the momentum flu divergence of the eddies. The perturbation in vorticit covariance δc that is induced b the mean flow perturbation δu can be estimated immediatel b assuming that the sstem )-) is ver close to the stabilit boundar, so that the growth rate is small. In this case the mean flow evolves slow enough that it remains in equilibrium with the edd covariance, that is dδc/dt. Bakas and Ioannou 3b) showed that the ensemble mean momentum flu induced b an infinitesimal sinusoidal mean flow perturbation δu = ɛ sinm), where ɛ i.e the eigenfunction of B)), is equal in this case to the integral over time and over all zonal wavenumbers of the responses to all point ecitations in the direction: δ u v = π u v t)dtdξdk, 3) where u v t) is the momentum flu at time t produced b: Gk, ξ) = Bk)h ξ)e ik+il ξ). ) The Green s function G has the form of a wavepacket with an amplitude Bk) and a carrier wave with wavenumbers k, l ) that is modulated in the direction b the wavepacket envelope h). The characteristics of the amplitude, the wavenumber and the envelope depend on the forcing characteristics, but in an case the calculation of the ensemble mean momentum flues is reduced to calculating the momentum flues over the life ccle of wavepackets that are initiall at different latitudes and then adding their relative contributions.

7 As the wavepacket propagates in the latitudinal direction, its meridional wavenumber and frequenc are going to change due to shearing b the mean flow and due to the change of the mean vorticit gradient β U. The resulting time variable momentum flu u v t) can be calculated using ra tracing. According to standard ra tracing arguments, the wave action is conserved along a ra in the absence of dissipation) leading to the momentum flu: u v t) = B A M t)e rt h ηt)), 5) where A M t) = kl t /k + l t ) is the momentum flu of the carrier wave that determines the amplitude of the flues of the wavepacket and l t, ηt) are the time dependent meridional wavenumber and position of the wavepacket respectivel Andrews et al. 987). Because of the small amplitude of the mean flow perturbation δu, the wavenumber and position of the packet var slowl on a time scale Oɛt) compared to the dissipation time scale /r and the dominant contribution to the time integral in 3) comes from small times. We can therefore seek asmptotic solutions of the form l t = l + ɛl +, ηt) = ξ + c t + ɛη t) +, ) where c = βkl /k + l ) is the group velocit in the absence of a mean flow and calculate the integral of u v t) over time from the leading order terms. Substituting ) in 5) we obtain: u v t) = B A M )e rt h ξ c t) }{{} u v R ) ɛ B dam l t)e rt h ξ c t) dl t l } {{} u v S ɛ B A M )η t)e rt d d h ξ c t). }{{} u v β 7) The first term, u v R, arises from the momentum flu produced b a wavepacket in the absence of a mean flow. Because A M ) = kl /k + l) is odd with respect to wavenumbers, this term does not contribute to the ensemble averaged momentum flu when integrated over all wavenumbers and will be hereafter ignored. The second term, u v S, arises from the small change in the amplitude of the flu A M over a dissipation time scale. The third term, u v β, arises from the small change in the position of the packet η compared to a propagating packet in the absence of a mean flow. To summarize, the infinitesimal mean flow refracts the wavepacket due to shearing b the mean flow and due to the change of the mean vorticit gradient and slightl changes the amplitude of the flues as well as slightl speeds up or slows down the wavepacket. The sum of these two effects will produce the induced momentum flues. a. The limit of small scale wavepackets with a short propagation range In order to clearl illustrate the behavior of the edd flues, we consider the limit of β = βl f /r, where L f is the scale of the wavepackets and in addition we assume that the scale of the mean flow, /m, is much larger than the scale of the wavepackets ml f. In this limit, the wavepackets are dissipated before propagating far from the initial position and the effect of the change in the mean vorticit gradient is higher order. As a result, Bakas and Ioannou 3b) show that l and η decrease monotonicall with time with rates independent of δu and proportional to the shear δu ξ) at the initial position ξ: ) dam l = δu ξ)kt, η = βδu ξ) kt. 8) dl t l That is, the amplitude of the flu A M and the group velocit of the packets change onl due to the shearing of the phase lines of the carrier wave according to the local shear. Consider in this limit the first term, u v S, arising from the small amplitude change. Since the wavepacket is dissipated before it propagates awa, we can ignore to first order propagation: ) u v S = ɛ B dam l t)e rt h ξ c t) dl t l ) B dam δu ξ)kt e rt h ξ), 9) dl t l so that the packet grows/decas in situ. Since the wave packet is rapidl dissipated, the integrated momentum flu over its life time will be given to a good approimation b the instantaneous change in the flu that is proportional to da M /dl t ) l. Figure 5 illustrates the amplitude of the momentum flu as a function of the angle θ t = arctanl t /k) of the phase lines of the carrier wave of the packet with the -ais. It is shown that the momentum flu of a wavepacket with θ < π/ that is with phase lines close to the meridional direction) ecited in regions II or III, will increase within the dissipation time scale. Compared to an unsheared wavepacket, this process leads to upgradient momentum flu. The opposite occurs for waves ecited in regions I and IV with θ > π/) that produce downgradient flu, as their momentum flu decreases. We now consider the second term, u v β arising from the effect of propagation on the momentum flu. The group occurring over the dissipation time scale /r that is incremental in shear time units 7

8 .5. I II.. a).3. uv. AM III IV θt = arctanlt/k) uvβ b) 3 3 Figure 5: Amplitude of the momentum flues, A M t), of wavepackets as a function of the angle θ t = arctanl t/k) between the phase lines of the central wave and the -ais. The vertical lines separate the regions with θ t < π/ II and III) and θ t > π/ I and IV). velocit is given b c g = βa M in this case and as a result a wavepacket starting in region III, will propagate towards the north c.f. figure 5). Because shearing slows down the waves in region III η da M /dl t )), the wavepacket will flu its momentum from southern latitudes compared to when it moved in the absence of the shear flow. This is shown in figure a) illustrating the distribution of momentum flu of an unsheared and a sheared perturbation whose amplitudes are constant. Figure b) plots this difference, u v β, and shows that the flu is downgradient in this case. The same happens for waves ecited in region II, while the waves ecited in regions I and IV produce upgradient flu. The net momentum flues produced b an ensemble of wavepackets, will therefore depend on the spectral characteristics of the forcing that determine the regions I-IV), in which the forcing has significant power. Bakas and Ioannou 3b) show that for the isotropic forcing 8): δ u v = = π u v Sdξdk + u π v βdξdk 3 ε β r d 3 δu 3πKf d 3. 3) The first integral is zero, because the gain in momentum occurring for θ < π/ waves ecited in regions II, III) is full compensated b the loss in momentum for θ > π/ waves ecited in regions I, IV) since for the isotropic forcing all possible wave orientations are equall ecited. The net momentum flues are therefore produced b the u v β term and are upgradient, because the loss in momentum occurring for θ < π/, is over compensated b the gain in momentum for θ > π/. The momentum flues are also Figure : a) Comparison of the momentum flues of an unsheared wavepacket ecited in regions II thick solid line) and III solid line) to the momentum flues of a sheared wavepacket shown b the corresponding dashed lines, when onl the change in propagation is taken into account. A snapshot of the flues at t =./r is shown. The planetar vorticit gradient is β =., the wavepacket has initial vorticit h) = e, k + l =, θ = π/ and B =. b) The difference in momentum flues between a sheared and an unsheared wavepacket calculated over their life ccle. proportional to the third derivative of δu ielding a hperdiffusive momentum flu divergence that tends to reenforce the mean flow and is therefore destabilizing. These destabilizing flues are proportional to β and as a result, the energ input rate required to form zonal jets increases as / β in this limit. It is worth noting that the first term integrates to zero onl for the special case of the isotropic forcing, as even the slightest anisotrop ields a non-zero contribution from u v S. For eample consider the forcing covariance Ξ,,, ) = cos k )) e ) /δ that mimics the forcing of the barotropic flow b the most unstable baroclinic wave, which has zero meridional wavenumber. In this case the forcing that is centered at l = in wavenumber space, injects significant power in a band of waves in regions II and III and therefore u v S ields upgradient flues. b. The effects of the change in the mean vorticit gradient and the finite propagation range In order to take into account the effect of the change in the vorticit gradient, we retain higher order terms with respect to ml f in l and η. In this case it can be shown that l decreases with time at a rate proportional to U ξ) + U ξ) Bakas and Ioannou 3b). Since the local shear and the local change in the vorticit gradient have different signs, the wavepacket is sheared less and as a result we epect reduced momentum flues compared to the limit discussed in section.. Indeed, for the isotropic 8

9 t forcing: δ u v 3 ε β r 3πK f d 3 δu d 3 K f ) d 5 δu d 5. 3) That is, the change in the mean vorticit gradient has a stabilizing effect. We finall rela the assumption that β. In this case, l and η are affected b an integral shear and mean vorticit gradient over the region of propagation. For larger β, the wavepacket will encounter regions of both positive and negative shear and as a result, the momentum flues that are qualitativel proportional to the integral shear over the propagation region will be reduced. In the limit β, the region of propagation is the whole sinusoidal flow with consecutive regions of positive and negative shear and the integral shear along with the flues will asmptoticall tend to zero. As a result, the energ input rate required for structural instabilit of zonal jets increases with β in this limit.. Equilibration of the S3T instabilities We now investigate the equilibration of the instabilities b studing the S3T sstem 9), ) discretized in a doubl periodic channel of size π π. We approimate the monochromatic forcing 8), b considering the narrow band forcing ˆΞk, l) = K { f, for k + l K f K f K f, for k + l, 3) K f > K f where k, l assume integer values, that injects energ at rate ε in a narrow ring in wavenumber space with radius K f and width K f. We consider the set of parameter values β =, r =., ν =.9, K f = and K f =, for which β =. The integration is therefore in the parameter region of figure in which the non-zonal structures are more unstable than the zonal jets. The growth rates of the coherent structures for integer values of the wavenumbers, n and m are calculated from the discrete version of equation 5) obtained b substituting the integrals with sums over integer values of the wavenumbers Bakas and Ioannou 3a). We first consider the supercritical energ input rate ε = ε c. For these parameters onl non-zonal modes are unstable, with the perturbation with n, m) =, 5) growing the most. At t =, we introduce a small random perturbation, whose streamfunction is shown in figure 7a). After a few e-folding times, a harmonic structure of the form Z = cos) cos5) dominates the large-scale flow. The energ of this large scale structure shown in figure 7b), increases rapidl and eventuall saturates. At this point the large-scale flow gets attracted to a traveling wave finite amplitude equilibrium structure cf. figure 7c)) close in 5 a) c) E 3 b) 5 t 3 d) Figure 7: Equilibration of the S3T instabilities. a) Streamfunction of the initial perturbation. b) Energ evolution of the initial perturbation shown in panel a) as obtained from the integration of the S3T equations 9) and ) dashed line) and from the integration of the ensemble quasilinear EQL) sstem )-3) with N ens = solid line) and N ens = dash-dotted line) ensemble members that is discussed in section. c) Snapshot of the streamfunction Ψ eq of the traveling wave structure and d) Hovmöller diagram of Ψ eq, = π/, t) for the finite equilibrated traveling wave. The thick dashed line shows the phase speed obtained from the stabilit equation 5). The energ input rate is ε = ε c and β =. form to the harmonic Z = cos) cos5) that propagates westward. This is illustrated in the Hovmöller diagram of ψ, = π/, t) shown in 7d). The sloping dashed line in the diagram corresponds to the phase speed of the traveling wave, which is found to be approimatel the phase speed of the unstable n, m) =, 5) eigenmode. Consider now the energ input rate ε = ε c. While the maimum growth rate still occurs for the n, m ) =, 5) non-zonal structure, zonal jet perturbations are unstable as well. If the S3T dnamics are restricted to account onl for the interaction between zonal flows and turbulence b emploing a zonal mean rather than an ensemble mean, an infinitesimal jet perturbation will grow and equilibrate at finite amplitude. To illustrate this we integrate the S3T dnamical sstem )-) restricted to zonal flow coherent structures. The energ of the small zonal jet perturbation δz =. cos) is shown in figure 8 to grow and saturate at a constant value and the streamfunction of the equilibrated jet is shown in the left inset in figure 8. However, in the contet of the generalized S3T analsis that takes into account the dnamics of the interaction between coherent non-zonal structures and jets, we find that these S3T jet equilibria can be saddles: stable to zonal jet perturbations but unstable to non-zonal perturbations. To show this, we consider the evolution of the same jet perturbation δz =. cos) under the generalized S3T dnamics 9), ) and find that the flow follows the zonall 9

10 3. E Ez t 8 t Figure 8: Energ evolution of an initial jet perturbation δz =. cos) for the zonall restricted S3T dnamics )-) dashed line) and the generalized S3T dnamics 9), ) thin line). The insets show a snapshot of the mean flow streamfunction at t = 7 left) and the streamfunction of the equilibrated structure at t = 35 right) under the generalized S3T dnamics. The parameters are ε = ε c and β =. restricted S3T dnamics and equilibrates to the same finite amplitude zonal jet cf. figure 8). At this point we insert a small random perturbation to the equilibrated flow. Soon after, non-zonal undulations grow and the flow transitions to the stable Z = cos) cos5) traveling wave state that is also shown in figure 8. As a result, the finite equilibrium zonal jet structure is S3T unstable to coherent non-zonal perturbations and is not epected to appear in non-linear simulations despite the fact that the zero flow equilibrium is unstable to zonal jet perturbations. We will elaborate more on this issue in the net section. Finall, consider the case ε = 3 ε c. At this energ input rate, the finite amplitude non-zonal traveling wave equilibria become S3T unstable. To show this, we consider the nonzonal traveling wave equilibrium obtained b the evolution of the small non-zonal perturbation δz =. cos) cos5) to the homogeneous state that is shown at the left inset in figure 9 and impose a small random zonal perturbation. The evolution of the zonal energ E z = /)U, where the overbar denotes a zonal average, is shown in figure 9. After an initial transition period, the zonal perturbations grow eponentiall and the flow transitions to the jet equilibrium state shown at the right inset in figure 9. Note however, that the jet equilibrium structure is not zonall smmetric. This is a new tpe of S3T equilibrium: it is a mi between a zonal jet and a non-zonal traveling wave with the same meridional scale. These mied equilibria appear to be the attractors for larger energ input rates as well. This is illustrated in figure showing the structure of the mied equilibrium at ε = 5ε c. The equilibrium structure consists of a large amplitude zonall smmetric jet with larger scale Figure 9: Zonal energ evolution of a random zonal perturbation imposed on the non-zonal traveling wave equilibrium shown in the left inset. The streamfunction of the equilibrated structure is shown in the right inset. The energ input rate is ε = 3 ε c and β =. compared to the mied state in figure 9. Embedded in it are non-zonal vortices with the same meridional scale and with about % the energ of the zonal jet. These vortices that are shown in figure b) to have approimatel the compact support structure Ψ = cos) cos) propagate westward as shown in the Hovmöller diagram in figure c). 5. Comparison to ensemble mean quasi-linear and non-linear simulations a. Comparison to an ensemble of quasi-linear simulations Within the contet of the second order cumulant closure, the S3T formulation allows the identification of statistical turbulent equilibria in the infinite ensemble limit, in which the fluctuations induced b the stochastic forcing are averaged to zero. However, these S3T equilibria and their stabilit properties are manifest even in single realizations of the turbulent sstem. For eample, previous studies using S3T obtained zonal jet equilibria in barotropic, shallow water and baroclinic flows in close correspondence with observed jets in planetar flows Farrell and Ioannou 7, 8, 9b,a). In addition, previous studies of S3T dnamics restricted to the interaction between zonal flows and turbulence in a β-plane channel showed that when the energ input rate is such that the zero mean flow equilibrium is unstable, zonal jets also appear in the non-linear simulations with the structure scale and amplitude) predicted b S3T Srinivasan and Young ; Constantinou et al. ). A ver useful intermediate model that retains the wavemean flow dnamics of the S3T sstem while relaing the infinite ensemble approimation is the quasi-linear sstem 5)-). Under the ergodic assumption, this can be inter-

11 t 3 5 b) c) a) nzmf, zmf Rβ nzmf NL nzmf EQL ε/εc zmf EQL zmf NL Figure : Mied zonal jet-traveling wave S3T equilibrium for ε = 5 ε c and β =. a) Snapshot of the streamfunction Ψ eq of the equilibrium state. b) Contour plot of the non-zonal component Ψ eq Ψ eq of the equilibrium structure, where the overline denotes a zonal average. c) Hovmöller diagram of Ψ eq, = π/, t) for the equilibrated structure. preted as an ensemble of quasi-linear equations EQL) in which the ensemble mean can be calculated from a finite number of ensemble members. Its integration is done as follows. A pseudo-spectral code with a 8 8 resolution and a fourth order Runge-Kutta scheme for time stepping is used to integrate 5)-) forward. At each time step, N ens separate integrations of ) are performed with the eddies evolving according to the instantaneous flow. Then the ensemble average vorticit flu divergence is calculated as the average over the N ens simulations and 5) is stepped forward in time according to those flues. The EQL sstem reaches a statistical equilibrium at time scales of the order of t eq O/r) and the integration was carried on until t = t eq in order to collect accurate statistics. We choose the same parameter values as in the S3T integrations in section 5 β =, r =., ν =.9, K f = and K f = ). For these parameters β = ), S3T predicts the emergence of propagating non-zonal structures when the energ input rate eceeds the critical threshold ε c, and the emergence of mied zonal jet-traveling wave states when the finite amplitude traveling wave states become structurall unstable to zonal jet perturbations. In order to eamine whether the same bifurcations occur in the EQL sstem, we consider two indices that measure the power concentrated at scales larger than the scales forced. The first is the zonal mean flow inde defined as in Srinivasan and Young ), as the ratio of the energ of zonal jets with scales larger than the scale of the forcing over the total energ zmf = l:l<k f K f Êk =, l) kl Êk, l), 33) Figure : The zmf and nzmf indices defined in 33) and 35) respectivel, as a function of energ input rate ε/ε c and the zonostroph parameter R β for the non-linear NL) integrations and an ensemble of quasi-linear EQL) integrations dashed line) with N ens = ensemble members as described in section. The critical value ε c = 8. is the energ input rate at which the S3T predicts structural instabilit of the homogeneous turbulent state. Zonal jets emerge for ε > ε nl, with ε nl = 5ε c. The parameters are β =, r =., ν =.9 and the forcing is an isotropic ring in wavenumber space with radius K f = and width K f =. where Êk, l) = T T ˆζ k + l ) + Ẑ k + l dt 3) is the time averaged total energ power spectrum of the flow at wavenumbers k, l). The second is the non-zonal mean flow inde defined as the ratio of the energ of the non-zonal modes with scales larger than the scale of the forcing over the total energ: nzmf = kl:k<k f K f Êk, l) kl Êk, l) zmf. 35) If the structures that emerge are coherent, then these indices quantif their amplitude. Figure shows both indices as a function of the energ input rate ε and as a function of the corresponding values of the zonostroph inde R β for EQL simulations with N ens = members. The rapid increase of the nzmf inde for ε > ε c corresponding to R β >.55), illustrates that this regime transition in the flow predicted b S3T with the emergence of non-zonal structures manifests in the quasi-linear dnamics as well. We now consider the case ε = ε c in detail in which the traveling wave structure Z = cos) cos5) is maintained in the S3T integrations. We observe, that the S3T equilibria manifest in the EQL simulations with the addition of some thermal noise due to the stochasticit of the forcing that is retained in this sstem. This is illustrated in figure 7b showing the

12 l t l t!*# ) ' % $ % & $ %+%, %+%%& % %+%%& %+%,!"# ) ' % $ % & $ %+% %+%' % %+%' %+% %+%) 5 5 a) 5 5 k b) Figure : Snapshot of the mean streamfunction Ψ at statistical equilibrium obtained from the ensemble mean quasilinear simulations with N ens = members for ε = ε c panel a) and ε = 5ε c panel b). The parameters are as in figure. energ growth of the coherent structure for N ens = and N ens =. The energ of the coherent structure in the EQL integrations fluctuates around the values predicted b the S3T sstem with the fluctuations decreasing as / N ens. However, even with onl ensemble members we get an estimate that is ver close to the theoretical estimate of the infinite ensemble members obtained from the S3T integration. The structure of the traveling wave equilibrium in the quasi-linear simulations shown in figure a) and its phase speed not shown) are also in ver good agreement with the corresponding structure and phase speed obtained from the S3T integration. The second transition in which zonal jets emerge is more intriguing. While the homogeneous equilibrium is structurall unstable to zonal jets when ε sz = 5.ε c, the finite amplitude zonal jet equilibria are structurall unstable and the flow stas on the attractor of the non-zonal traveling wave equilibria cf. figure 8). When ε > ε nl, the non-zonal traveling wave equilibria become S3T unstable while at these parameter values the S3T sstem has mied zonal jet-traveling wave equilibria which are stable cf. figure ). The rapid increase in the zmf inde with the concomitant rapid decrease in the nzmf inde shown in figure, illustrates that this regime transition manifests in the EQL sstem as well with similar mied zonal-traveling wave states appearing. The structure of the mied zonal jet-traveling wave equilibrium for ε = 5ε c is shown in b) and similar to the S3T equilibrium in figure, it consists mainl of zonal jets and the compact support vortices Z cos) cos) embedded in the jets. We therefore conclude that the EQL sstem accuratel captures the characteristics of the emerging structures. b. Comparison to non-linear simulations In order to compare the predictions of S3T to the nonlinear simulations, we solve ) with the narrow band forcing 3) on a doubl periodic channel of size π π using the same pseudospectral code as in the EQL simulations and the same parameter values. Figure shows the nzmf and zmf indices as a function of the energ input rate ε for the 5 5 c) 5 5 k d) Figure 3: Time averaged energ power spectra, logêk, l)), obtained from the non-linear NL) simulation of ) at ε/ε c = panel a) and ε/ε c = 5 panel c). Hovmöller diagram of ψ, = π/, t) at ε/ε c = panel b) and ε/ε c = 5 panel d). The thick dashed lines correspond to the phase speed obtained from the eigenvalue relation 5). NL simulations. The rapid increase in the nzmf inde for ε > ε c shows that the non-linear dnamics share the same bifurcation structure as the S3T statistical dnamics. In addition, the stable S3T equilibria are in principle viable repositories of energ in the turbulent flow and the nonlinear sstem is epected to visit their attractors for finite time intervals. Indeed for ε = ε c, the pronounced peak at k, l ) =, 5) of the time averaged power spectrum shown in figure 3a) illustrates that the traveling wave equilibrium with k, l ) =, 5) that emerges in the S3T integrations, is the dominant structure in the NL simulations. Comparison of the energ spectra obtained from the EQL and the NL simulations not shown), reveals that the amplitude of this structure in the quasi-linear and in the non-linear dnamics almost matches. Remarkabl, the phase speed of the S3T traveling wave matches with the corresponding phase speed of the k, l ) =, 5) structure observed in the NL simulations, as can be seen in the Hovmöller diagram in figure 3b). Such an agreement in the characteristics of the emerging structures between the EQL and NL simulations occurs for a wide range of energ input rates as can be seen b comparing the nzmf indices in figure. As a result, S3T predicts the dominant non-zonal propagating structures in the non-linear simulations, as well as their amplitude and phase speed. We now focus on the second regime transition with the emergence of zonal jets. The increase in the zmf inde in the NL simulations for ε > ε nl that is shown in figure, indicates the emergence of jets roughl at the bifurcation point of the S3T and EQL simulations. However, the energ input rate threshold for the emergence of jets is larger in the