Emergence of large scale structure in planetary turbulence
aa r X i v : . [ phy s i c s . a o - ph ] M a r Emergence of large scale structurein planetary turbulence
Nikolaos A. Bakas ∗ and Petros J. Ioannou † Department of PhysicsNational and Kapodistrian University of AthensPanepistimiopolis, ZografosAthens 15784, Greece (Dated: August 15, 2018)Planetary and magnetohydrodynamic drift-wave turbulence is observed to self-organize into largescale structures such as zonal jets and coherent vortices. In this Letter we present a non-equilibriumstatistical theory, the Stochastic Structural Stability theory (SSST), that can make predictionsfor the formation and finite amplitude equilibration of non-zonal and zonal structures (lattice andstripe patterns) in homogeneous turbulence. This theory reveals that the emergence of large scalestructure is the result of an instability of the interaction between the coherent flow and the associatedturbulent field. Comparison of the theory with nonlinear simulations of a barotropic flow in a β -planechannel with turbulence sustained by isotropic random stirring, demonstrates that SSST predictsthe threshold parameters at which the coherent structures emerge as well as the characteristics of theemerging structures (scale, amplitude, phase speed). It is shown that non-zonal structures (latticestates or zonons) emerge at lower energy input rates of the stirring compared to zonal flows (stripestates) and their emergence affects the dynamics of jet formation. PACS numbers:
Turbulence in planetary atmospheres and in plasmaflows is commonly observed to be organized into largescale unidirectional (zonal) jets with long-lasting coher-ent eddies or vortices embedded in them [1, 2]. The jetscontrol the transports of heat and chemical species inplanetary atmospheres and separate the high tempera-ture plasma from the cold containment vessel wall in mag-netic plasma confinement devices. It is therefore impor-tant to understand the mechanisms for the emergence,equilibration and maintenance of these coherent struc-tures. In this Letter we present a theory that predictsthe regime changes occurring in the turbulent flow aswell as the amplitude, structure and propagation charac-teristics of both the zonal jets and the non-zonal coherentstructures that form in the flow. We then test this the-ory against non-linear simulations in a simple model offorced planetary and plasma turbulence.The simplest model that captures the turbulent dy-namics and its interaction with the zonal jets and the co-herent structures, is the stochastically forced barotropicvorticity equation on a plane tangent to the surface of arotating planet: ∂ t ζ + ψ x ζ y − ψ y ζ x + βψ x = − rζ − ν ∆ ζ + f. (1)The relative vorticity is ζ = ∆ ψ , ψ is the streamfunc-tion, ∆ = ∂ xx + ∂ yy is the horizontal Laplacian, x isin the zonal (east-west) direction and y is in the merid-ional (north-south) direction, β = 2Ω cos φ /R is thegradient of planetary vorticity, Ω the rotation rate ofthe planet, φ the latitude of the β -plane and R is the ∗ [email protected] † [email protected] radius of the planet. Equation (1) governs the dynam-ics of non-divergent motions at the midlatitudes of theplanet and is also the infinite effective Larmor radiuslimit of the Charney-Hasegawa-Mima equation that gov-erns drift-wave turbulence in plasmas. We are assum-ing linear damping with coefficient, r , representing theEkman drag induced by the horizontal boundaries andhyper-diffusion with coefficient ν that dissipates the en-ergy flowing into unresolved scales. The forcing term f is necessary to sustain turbulence, and may parameterizeprocesses that have not been included in the dynamics,such as the vorticity forcing from small scale convection.In many previous studies, this exogenous excitation istaken as a temporally delta correlated and spatially ho-mogeneous and isotropic random stirring. We will followthe same forcing protocol in this Letter and consider anisotropic ring forcing that is injecting energy at rate ǫ ina narrow ring of wavenumbers of width ∆ K f around thetotal wavenumber K f .We solve (1) in a doubly periodic domain of size 2 π × π .The calculations presented in this Letter are for β = 10, r = 0 . ν = 2 · − , K f = 8 and ∆ K f = 1, whichare reasonable planetary parameters. The results dis-cussed were verified to be insensitive to the forcing pro-tocol. To illustrate some of the characteristics of theturbulent flow and the emergence of coherent structures,we consider two indices. The first is the zonal mean flowindex [3] defined as the ratio of the energy of zonal jetsover the total energy, zmf = P l ˆ E ( k =0 ,l ) P kl ˆ E ( k,l ) , where ˆ E is thetime averaged energy power spectrum of the flow and k , l are the zonal and meridional wavenumbers respec-tively. The second is the non-zonal mean flow index de-fined as the ratio of the energy of the non-zonal modeswith scales lower than the scale of the forcing over the ǫ / ǫ c n z m f , z m f R β zmfnzmf FIG. 1. The zmf (red lines) and nzmf (blue lines) indicesas a function of energy input rate ǫ/ǫ c and the correspondingzonostrophy parameter R β for the non-linear (solid lines) andSSST (dashed lines) integrations. The critical value ǫ c (cor-responding to R β = 1 .
64) is the energy input rate at whichthe SSST predicts structural instability of the homogeneousturbulent state. Zonal jets emerge here for ǫ > ǫ nl , with ǫ nl = 15 . ǫ c (corresponding to R β = 1 . k l (a) −5 0 5−505 123456 k l (b) −5 0 5−505 246810 FIG. 2. Time averaged energy power spectra, log ( ˆ E ( k, l )),obtained from non-linear simulation of Eq. (1) at (a) ǫ/ǫ c =2 . ǫ/ǫ c = 30. In (a) the flow is dominated by a( | k | , | l | ) = (1 ,
5) non-zonal coherent structure. In (b) the flowis dominated by a coherent zonal flow at ( k, | l | ) = (0 , total energy: nzmf = (cid:16)P kl : K
5) that correspond to coherentstructures propagating westward (cf. Fig. 3(a),(b)) withapproximately the Rossby wave phase speed for this wave.However, at larger ǫ the propagation speed of these struc- x y (a) x t (b) x y (c) t y (d) FIG. 3. (a) Snapshot of the streamfunction ψ ( x, y, t ) and(b) Hovm¨oller diagram of ψ ( x, y = π/ , t ) obtained from non-linear simulation for ǫ/ǫ c = 2 .
6. The thick lines in (b) cor-respond to the phase speed obtained from the stability equa-tion (6). (c) Snapshot of the streamfunction ψ ( x, y, t ) and(d) Hovm¨oller diagram of the x -averaged ψ ( y, t ), which is ob-tained from non-linear simulations at ǫ/ǫ c = 30. tures departs from that of Rossby waves. The presenceand properties of such non-linear waves in similar sim-ulations, were also reported recently [6]. For ǫ > ǫ nl indicated in Fig. 1 (corresponding to R β = 1 . k, | l | ) = (0 ,
3) inthe spectrum of Fig. 2(b) correspond to coherent zonaljets (cf. Fig. 3(c),(d)). From Fig. 1 we see that whilethe jets contain over half of the total energy, substantialpower remains in non-zonal structures. Previous studies,refer to the coherent non-zonal structures in this regimein which strong zonal jets dominate the flow (referredto as zonostrophic regime [7]) as satellite modes [8] orzonons [4, 6].The emergence of jets has been described in previousstudies in terms of an anisotropic inverse energy cas-cade [9–11], or in terms of inhomogeneous mixing of vor-ticity [5, 12], or in terms of a direct transfer of energyfrom small scale waves into the zonal jets, through eithernon-linear interactions between finite amplitude Rossbywaves [13, 14], or through shear straining of the smallscale waves by the jet [15]. However, the mechanismfor the emergence of the non-zonal structures remainselusive. Statistical equilibrium theory applied in the ab-sence of forcing and dissipation, has been able to pre-dict both jets and coherent vortices as maximum entropystructures [16] and a recent study has shown correspon-dence of the theoretical results with non-linear simula-tions in the limit of weak forcing and dissipation [17].Nevertheless, the relevance of these results in planetaryand plasma flows that are strongly forced and dissipatedand are therefore out of equilibrium remains to be shown.In this Letter we present results of an alternative, non-equilibrium statistical theory, that is termed as Stochas-tic Structural Stability Theory (SSST) [18, 19] or Sec-ond Order Cumulant Expansion theory (CE2) [3, 20, 21].While recent studies have demonstrated that SSST canpredict the structure of zonal flows in turbulent fluids[3, 22, 23], the results presented in this Letter, demon-strate that an extended version of SSST can predict theemergence of both zonal and non-zonal coherent struc-tures in planetary turbulence and can capture their finiteamplitude manifestations. The emergence of non-zonaland zonal structures described above is similar to forma-tion of the lattice and stripe patterns in homogeneousthermal non-equilibrium systems [24]. The analogy be-tween the formation of stripes and zonal jets has beenrecently emphasized using SSST dynamics [25]. In thisletter we formulate the SSST dynamics that can producelattice states in fully turbulent flows.SSST describes the statistical dynamics of the first twoequal time cumulants of Eq. (1). The first cumulant is Z ( x , t ) ≡ h ζ i (the brackets denote an ensemble average)and the second cumulant C ( x , x , t ) ≡ h ζ ′ ζ ′ i is a func-tion of the vorticity deviation ζ ′ i = ζ i − Z i at the twopoints x i = ( x i , y i ) ( i = 1 ,
2) at time t . It can be shownfrom (1) that the equations for the evolution of the twocumulants are: ∂ t Z + U Z x + V ( β + Z y ) + rZ + ν ∆ Z == ∂ x (cid:0) ∂ y ∆ − C (cid:1) x = x − ∂ y (cid:0) ∂ x ∆ − C (cid:1) x = x , (2) ∂ t C = ( A + A ) C + Ξ . (3)The linear operator A i = − U i ∂ x i − V i ∂ y i − ( β + Z y i ) ∂ x i ∆ − i ++ Z x i ∂ y i ∆ − i − r − ν ∆ i , (4)acts at the points x i = ( x i , y i ) and governs the dynam-ics of linear perturbations about the instantaneous meanflow U = [ U, V ] = [ − ∂ y h ψ i , ∂ x h ψ i ]. In (3), Ξ containsthe covariance of the external forcing and terms relatedto third order cumulants. A second order closure is ob-tained if the third order cumulant is ignored and Ξ is set to be the spatial covariance of the stochastic forcing f .In most earlier studies of SSST or CE2, the ensemble av-erage was assumed to represent a zonal average. In thisLetter, we adopt the more general interpretation that theensemble average represents a Reynolds average with theensemble mean representing coarse-graining. This inter-pretation has been adopted in the SSST study of turbu-lence in baroclinic flows [26, 27]. With this interpretationof the ensemble mean, the SSST system (2)-(3) providesthe statistical dynamics of the interaction of the ensembleaverage field, which can be a zonal or a non-zonal coher-ent structure, with the fine-grained field, represented inthe theory through its covariance C . The SSST systemdefines an autonomous dynamics and its fixed points de-fine a new type of turbulent statistical equilibria. Whilethese equilibria formally exist only in the infinite ensem-ble limit, it has been shown that their characteristicsmanifest even in single realizations of the turbulent sys-tem. The structural stability of these turbulent equilib-ria can be addressed in SSST by studying their stability.Specifically, when an equilibrium of the SSST equationsbecomes unstable, the turbulent flow bifurcates to a dif-ferent attractor. This theory therefore predicts param-eters of the physical system which can lead to abruptreorganization of the turbulent flow.The SSST equations (2)-(3), admit for ν = 0 the simpleequilibrium U E = V E = 0 , C E = Ξ2 r , (5)that has zero large scale flow and a homogeneouseddy field with spatial covariance dictated directly fromthe forcing. We now investigate the SSST stabilityof this equilibrium as a function of the energy inputrate, ǫ , and relate the outcome of this stability analy-sis to the results in the non-linear simulations of (1).The stability of the homogeneous equilibrium (5) is as-sessed by introducing small perturbations of the form[ δZ, δC ] = [ δZ nm , δC nm ] e in ( x + x ) / im ( y + y ) / e σt tothe SSST equations (2)-(3) linearized about the equi-librium (5) and calculating the eigenvalue σ . When ℜ ( σ ) >
0, the structure with ( x, y ) wavenumbers ( n, m )is unstable and will emerge. It can be shown that σ sat-isfies the non-dimensional equation˜ ǫK f π ∆ K f X k,l ( ˜ m ˜ k − ˜ n ˜ l ) h ˜ n ˜ m (˜ k − ˜ l ) + ( ˜ m − ˜ n )˜ k + ˜ l + i (1 − ˜ N / ˜ K )2 i ˜ k + (˜ k + ˜ n + ˜ l + ˜ m ) − i ˜ n (cid:16) ˜ K + ˜ K s (cid:17) / σ + 2) ˜ K ˜ K s = (˜ σ + 1) ˜ N − i ˜ n. (6)In this equation (˜ n, ˜ m, ˜ k, ˜ l ) = ( n, m, k, l ) r/β , ˜ σ = σ/r are the non-dimensional wavenumbers and growth raterespectively, ˜ K = ˜ k + ˜ l , ˜ K s = (˜ k + ˜ n ) + (˜ l + ˜ m ) ,˜ N = ˜ n + ˜ m , ˜ k + = ˜ k + ˜ n/ l + = ˜ l + ˜ m/ k, l ) satisfying | ˜ K − ( K f r/β ) | < ∆ K f r/β [28]. The non-dimensionalenergy input rate ˜ ǫ = ǫβ /r , which is the bifurcatingparameter in this Letter, is related to the zonostrophy | n | | m | (a) | n | | m | (b) FIG. 4. The growth rate, ℜ ( σ ) as a function of the integervalued wavenumbers | n | , | m | of the coherent structures for (a) ǫ/ǫ c = 2 . ǫ/ǫ c = 30 (only positive values of ℜ ( σ )are shown). Unstable waves with | n | > ℑ ( σ ) > parameter through R β = 0 . ǫ / . For n = 0 the stabilityequation (6) reduces to the equation that determines theemergence of zonal flows [3, 29].For small values of the energy input rate ˜ ǫ , ℜ ( σ ) < ǫ c the homo-geneous flow becomes SSST unstable, symmetry break-ing occurs, and coherent structures emerge. The critical˜ ǫ c is defined as min ( n,m ) ˜ ǫ t , where ˜ ǫ t is the energy inputrate that renders wavenumbers ( n, m ) neutral ( ℜ ( σ ) = 0).The critical ˜ ǫ c depends in general on the forcing charac-teristics and for the ring forcing at K f = 8, ˜ ǫ c = 2 . · or R β = 1 .
64 [30]. The growth rates as a function of theinteger valued wavenumbers, ( n, m ), of the structure areshown in Fig. 4. For ǫ/ǫ c = 2 .
6, the structure with thelargest growth rate, is non-zonal with ( | n | , | m | ) = (1 , ℑ ( σ ) = 0 .
4, implying retrograde propagationof the eigenstructure. Note also that for this energy in-put rate, the zonal flows are SSST stable and jets arenot expected to form. For ǫ/ǫ c = 30, both zonal jets andnon-zonal structures are unstable, but the zonal jets havesmaller growth rates compared to the non-zonal struc-tures [31]. The zonal jets are stationary ( ℑ ( σ ) = 0), incontrast to the non-zonal coherent structures that alwayspropagate in the retrograde direction. Numerical integra-tion of the SSST system (2)-(3), shows that for ǫ > ǫ c the unstable structures typically equilibrate at finite am-plitude after an initial period of exponential growth. Asa result (2)-(3) admit in general multiple equilibria. Fig-ure 5(a) shows the equilibrium structure with the largestdomain of attraction, when ǫ/ǫ c = 2 .
6. This structurecoincides in this case with the finite amplitude equilib-rium of the fastest growing ( | n | , | m | ) = (1 ,
5) eigenfunc-tion and propagates as illustrated in Fig. 5(b) in the ret-rograde direction with a speed approximately equal tothe phase speed of this unstable eigenstructure. A proxyfor the amplitude of these equilibrated structures are thezmf and nzmf indices calculated for the SSST integra-tions that are shown in Fig. 1. As the energy input rateincreases, the non-zonal structures equilibrate at largeramplitudes. However, for ǫ > ǫ nl , the equilibria with thelargest domain of attraction are zonal jets and the flowis dominated by these structures (cf. Fig. 1). x y (a) x t (b) FIG. 5. (a) Snapshot of the streamfunction ψ ( x, y, t ) and(b) Hovm¨oller diagram of ψ ( x, y = π/ , t ) obtained from theSSST integrations for ǫ/ǫ c = 2 .
6. The thick dashed lines in(b) show the phase speed obtained from the stability equation(6).
The results of the SSST analysis are now compared tonon-linear simulations. First of all, the SSST stabilityanalysis accurately predicts the critical ǫ c for the emer-gence of non-zonal structures in the non-linear simula-tions as shown in Fig. 1. The finite amplitude equilibriaobtained when ǫ > ǫ c also correspond to the dominantstructures in the non-linear simulations. For ǫ/ǫ c = 2 . | n | , | m | ) = (1 , ,
5) waves observed in the non-linearsimulations and the amplitude of these structures as illus-trated by the nzmf index are approximately equal to thephase speed and amplitude of the corresponding SSSTequilibrium structure (cf. Figs. 1, 3 and 5). For ǫ > ǫ nl ,in both nonlinear and SSST simulations zonal jets emergeand the power of the non-zonal structures is substantiallyreduced. Comparison of the number of jets and their am-plitude between the SSST and the nonlinear simulationsalso shows good agreement (not shown). This demon-strates that the SSST system can predict the amplitudeand characteristics of both the non-zonal and the zonalstructures that emerge in the turbulent flow.While the regime transition that occurs at ǫ c is pre-dicted by the stability equation (6), the second transition,which is associated with the emergence of zonal flows andoccurs at ǫ nl , is more intriguing. The stability equation(6) predicts that the zonal structures become unstable at ǫ sz = 4 ǫ c < ǫ nl . In previous studies of SSST dynamicsrestricted to the interaction between zonal flows and tur-bulence, these initially unstable structures were found toequilibrate at finite amplitude [3, 22]. Preliminary calcu-lations show that within the context of this generalizedSSST analysis that takes into account the dynamics ofnon-zonal structures as well, these equilibria are found tobe saddles that are stable to zonal but unstable to non-zonal perturbations. The threshold for the emergence ofjets in the SSST integrations and in the nonlinear simu-lations is therefore determined as the energy input rateat which an SSST stable, finite amplitude zonal jet equi-librium exists. A method to correctly obtain the criticalinput rate ǫ nl has been recently developed [22]. It startsby recognizing that for ǫ c < ǫ < ǫ sz , the spectrum in theturbulent flow is modified ( C E = Ξ / r ) and is given bythe covariance ˜ C E associated with the finite amplitudeequilibria similar to the ones shown in Fig. 5. If this mod-ification is taken into account, then the stability analysisaround the equilibrium [ U E = 0 , ˜ C E ] correctly predicts ǫ nl .In summary, we presented a theory that showsthat large scale structure in barotropic planetary anddrift-wave turbulence arises through systematic self-organization of the turbulent Reynolds stresses, throughnon-local interactions and in the absence of cascades.The theory allowed the determination of conditions forthe emergence of non-zonal coherent structures in ho-mogeneously forced flows and we have demonstrated,through comparison with nonlinear simulations, that itpredicts both the emergence and the finite amplitudeequilibration of structure. An advance made in this Let-ter is the development of the theoretical framework thataccounts for the emergence of non-zonal states in homo- geneous turbulence. These non-zonal (or lattice) stateswere found to propagate westward, and their relation towestward propagating vortex rings in the ocean and co-herent vortices in planetary atmospheres will be the sub-ject of future research. The homogeneous turbulent flowwas found to be more unstable to non-zonal structure.We think that the finite amplitude non-zonal states aresusceptible to secondary instability at higher supercriti-cality, and as a result the prevalent structures in plane-tary flows are zonal jets. ACKNOWLEDGMENTS
This research was supported by the EU FP-7 underthe PIRG03-GA-2008-230958 Marie Curie Grant. Theauthors acknowledge the hospitality of the Aspen Cen-ter for Physics supported by the NSF (under grant No.1066293), where part of this work was done. The authorswould also like to thank Navid Constantinou and BrianFarrell for fruitful discussions. [1] A. R. Vasavada and A. P. Showman, Rep. Prog. Phys. , 1935 (2005).[2] P. H. Diamond, S. I. Itoh, K. Itoh, and T. S. Hahm,Plasma Phys. Control. Fusion , R35 (2005).[3] K. Srinivasan and W. R. Young, J. Atmos. Sci. , 1633(2012).[4] B. H. Galperin, S. Sukoriansky, and N. Dikovskaya,Ocean Dyn. , 427 (2010).[5] R. K. Scott and D. G. Dritchel, J. Fluid Mech. , 576(2012).[6] S. Sukariansky, N. Dikovskaya, and B. Galperin, Phys.Rev. Lett. , 178501 (2008).[7] B. H. Galperin, S. Sukoriansky, N. Dikovskaya, P. Read,Y. Yamazaki, and R. Wordsworth, Nonlinear Proc.Geoph. , 83 (2006).[8] S. Danilov and D. Gurarie, Phys. of Fluids , 2592(2004).[9] P. B. Rhines, J. Fluid Mech. , 417 (1975).[10] G. K. Vallis and M. E. Maltrud, J. Phys. Oceanogr. ,1346 (1993).[11] S. Nazarenko and B. Quinn, Phys. Rev. Letters ,118501 (2009).[12] D. G. Dritchel and M. E. McIntyre, J. Atmos. Sci. ,855 (2008).[13] A. E. Gill, Geophys. Fluid Dyn. , 29 (1974).[14] C. Connaughton, B. Nadiga, S. Nazarenko, andB. Quinn, J. Fluid Mech. , 207 (2010).[15] W. A. Robinson, J. Atmos. Sci. , 2109 (2006).[16] F. Bouchet and A. Venaille, Phys. Rep. , 227 (2012).[17] F. Bouchet and E. Simonnet, Phys. Rev. Lett. ,094504 (2009). [18] B. F. Farrell and P. J. Ioannou, J. Atmos. Sci. , 2101(2003).[19] B. F. Farrell and P. J. Ioannou, J. Atmos. Sci. , 3652(2007).[20] J. B. Marston, E. Conover, and T. Schneider, J. Atmos.Sci. , 1955 (2008).[21] J. B. Marston, Annu. Rev. Condens. Matter Phys. , 285(2012).[22] N. C. Constantinou, P. J. Ioannou, and B. F. Far-rell, J. Atmos. Sci. (2013), (sub judice, arXiv:1208.5665[physics.flu-dyn]).[23] S. M. Tobias and J. B. Marston, Phys. Rev. Lett. (2013),(to appear, arXiv:1209.3862 [physics.flu-dyn]).[24] M. Cross and H. Greenside, Pattern Formation and Dy-namics in Nonequilibrium Systems (Cambridge Univer-sity Press, 2009) p. 489.[25] J. B. Parker and J. A. Krommes, (2013),arXiv:1301.5059v1 [physics.ao-ph].[26] J. Bernstein,
Dynamics of turbulent jets in the at-mosphere and ocean , Ph.D. thesis, Harvard University(2009).[27] J. Bernstein and B. F. Farrell, J. Atmos. Sci. , 452(2010).[28] Hyperdiffusion can be readily included in (6) in order toobtain correspondence with the nonlinear simulations.[29] N. A. Bakas and P. J. Ioannou, J. Fluid Mech. , 332(2011).[30] It can be shown that ˜ ǫ c is a rapidly decreasing functionof K f and the critical value for instability asymptoticallyoccurs at lim K f →∞ ˜ ǫ c = 23 or R β = 0 . β > β min . For the given isotropicforcing, β min = 4 . rK ff