Statistical State Dynamics: a new perspective on turbulence in shear flow

Brian F. FarrellDepartment of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138

Petros J. Ioannou∗

Department of Physics, National and Kapodistrian University of Athens, Athens, Greece

ABSTRACT

Traditionally, single realizations of the turbulent state have been the object of study in shear flowturbulence. When a statistical quantity was needed it was obtained from a spatial, temporal orensemble average of sample realizations of the turbulence. However, there are important advantagesto studying the dynamics of the statistical state (the SSD) directly. In highly chaotic systemsstatistical quantities are often the most useful and the advantage of obtaining these statistics directlyfrom a state variable is obvious. Moreover, quantities such as the probability density function (pdf)are often difficult to obtain accurately by sampling state trajectories even if the pdf is stationary.In the event that the pdf is time dependent, solving directly for the pdf as a state variable is theonly alternative. However, perhaps the greatest advantage of the SSD approach is conceptual:adopting this perspective reveals directly the essential cooperative mechanisms among the disparatespatial and temporal scales that underly the turbulent state. While these cooperative mechanismshave distinct manifestation in the dynamics of realizations of turbulence both these cooperativemechanisms and the phenomena associated with them are not amenable to analysis directly throughstudy of realizations as they are through the study of the associated SSD. In this review a selectionof example problems in the turbulence of planetary and laboratory flows is examined using recentlydeveloped SSD analysis methods in order to illustrate the utility of this approach to the study ofturbulence in shear flow.

1. Introduction

Adopting the perspective of statistical state dynamics(SSD) as an alternative to the traditional perspective af-forded by the dynamics of sample state realizations hasfacilitated a number of recent advances in understandingturbulence in shear flow. SSD reveals the operation in shearflow turbulence of previously obscured mechanisms, par-ticularly mechanisms arising from cooperative interactionamong disparate scales in the turbulence. These mecha-nisms provide physical explanation for specific phenomenaincluding formation of coherent structures in the turbulenceas well as more general and fundamental insights into themaintenance and equilibration of the turbulent state. More-over, these cooperative mechanisms and the phenomenaassociated with them are not amenable to analysis by thetraditional method of studying turbulence using samplestate dynamics. Another advantage of the SSD approachis that adopting the probability density function (pdf) asa state variable provides direct access to all the statisticsof the turbulence, at least within the limitations of theapproximations applied to the dynamics. Although theutility of obtaining the pdf directly as a state variable isobvious, the pdf is often difficult to obtain accurately by

sampling state trajectories even if the pdf is stationary. Inthe event that the pdf is time dependent, which is often thecase, solving directly for the pdf as a state variable is theonly alternative. While these are all important advantagesafforded by adopting the pdf as a state variable, the overar-ching advantage is that adopting the statistical state as thedynamical variable allows an understanding of turbulenceat a deeper level, which is the level in which the essentialcooperative mechanisms underlying the turbulent state aremanifest.

It is well accepted that complex spatially and temporallyvarying fields arising in physical systems characterized bychaos and involving interactions over an extensive range ofscales in space and time can be insightfully analyzed usingstatistical variables. In the study of turbulent systems,examining statistical measures for variables arising in theturbulence is a common practice; however, it is less commonto adopt statistical state variables for the dynamics of theturbulent system. The potential of SSD to provide insightinto the mechanisms underlying turbulence has been un-derexploited in part because obtaining the dynamics of thestatistical state has been assumed to be prohibitively diffi-cult in practice. An early attempt to use SSD in the study of

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turbulence was the formal expansion in cumulants by Hopf(Hopf 1952; Frisch 1995). However Hopf’s cumulant methodwas subsequently restricted in application, in large part dueto the difficulty of obtaining robust closure of the expan-sion. Another familiar example of a theoretical applicationof SSD to turbulence is provided by the Fokker-Planckequation which, while very useful conceptually, is generallyintractable for representing complex system dynamics ex-cept under very restrictive circumstances. Because of theperceived difficulty of implementing SSD to study systemsof the type typified by turbulent flows, the dynamics ofthese systems has been most often explored by simulatingsample state trajectories which are then analyzed to ob-tain an approximation to the assumed statistically steadyprobability density function of the turbulent state. Thisapproach fails to provide insight into phenomena that areassociated intrinsically with the dynamics of the statisticalstate rather than with the dynamics of sample realizations.The reason is that while the influence of multiscale coopera-tive phenomena on the statistical steady state of turbulenceis apparent from the statistics of sample realizations, the co-operative phenomena producing these statistical equilibriahave analytical expression only in the SSD of the associatedsystem. It follows that, in order to gain understanding ofturbulent equilibria, adopting the SSD perspective is essen-tial. But there is a more subtle insight into the dynamicsof turbulence afforded by the perspective of statistical statedynamics: while the statistical state of a turbulent systemmay asymptotically approach a fixed point, in which casethe mean statistics gathered from sample realizations forma valid representation of the stationary statistical state, thedynamics of the statistical state may instead be time depen-dent or even chaotic in which case the statistics obtainedfrom sample realizations would not in general correspondto a representation of the statistical state at any time. Thedynamics of such turbulent systems is accessible to analysisonly through its SSD.

Before introducing some illustrative examples of phe-nomena accessible to analysis through the use of SSD it isuseful to inquire further us to why this method has not beenmore widely exploited heretofore. In fact, as mentioned pre-viously, closure of cumulant expansions to obtain equilibriaof the SSD has been very extensively studied in associationwith isotropic homogeneous turbulence (Kraichnan 1959;Orszag 1977; Rose and Sulem 1978). However, the greatanalytical challenges posed by this approach and the limitedsuccess obtained using it served to redirect interest towarddimensional analysis and interpretation of simulations asa more promising approach to understanding the inertialsubrange. It turns out that the inertial subrange, whiledeceptively straightforward in dynamical expression, is in-trinsically and essentially nonlinear and as a result presentsgreat obstacles to analysis. However, the arguably more rel-evant forms of turbulence, at least in terms of applications

to meteorology, oceanography, astrophysics, MHD, and en-gineering fluid dynamics, is turbulence in shear flow at highReynolds number. At the core of these manifestations ofturbulence lies a linear dynamics which is revealed by lin-earization about the mean flow itself. This linearizationin turn uncovers an underlying simplicity of the dynamicsarising from the non-normality of the linear operator de-termining the set of dynamically relevant structures andtheir interaction with the mean shear. While not sufficientto eliminate the role of nonlinearity in the dynamics ofturbulence entirely, recognizing the centrality of this non-normality mediated interaction between perturbations andmean flows motivates a concept of central importance tounderstanding turbulence in shear flow which is the pivotalrole of quasi-linear interaction between a restricted set ofnon-normal structures and the mean flow. For our purposesa primary implication of this insight is that the SSD ofshear flow turbulence is amenable to study using closuresbased on this interaction.

Consider the large scale jets that are prominent fea-tures of planetary scale turbulence in geophysical flows ofwhich the banded winds of Jupiter and the Earth’s polarjets are familiar examples. These jets can be divided intotwo groups: forced jets and self-sustained jets. Jupiter’sjets are maintained by energy from turbulence excited byconvection arising from heat sources in the planet’s interior,and so the source of the turbulence from which the jets ofJupiter arise may be regarded as dynamically independentof the jet structure itself, such jets we refer to as forced. Incontrast, the Earth’s polar jets are maintained by turbu-lence arising from baroclinic growth processes drawing onthe potential energy associated with the meridional temper-ature gradient which is directly related to the jet structureby thermal wind balance. Because baroclinic growth mech-anisms depend strongly on jet structure it follows that themechanism producing the jet can not be separated dynami-cally from the mechanism producing the turbulence so thatthese problems must be solved together, such jets we referto as self-sustained. In the case of Jupiter’s jets the energysource is known from observations to be convection (Inger-soll 1990) leaving two fundamental problems presented bythe existence of these jets. The first is to explain how thejets arise from the turbulence and the second how they areequilibrated with the observed structure and maintainedwith this structure over time scales long compared to theexplicit time scales of the dynamics. Although these phe-nomena manifest prominently in simulations of sample staterealizations, neither is accessible to analysis using samplestate simulation while both have analytical expression andstraightforward solution when expressed using SSD (Farrelland Ioannou 2003, 2007). Moreover, in the course of solvingthe SSD problem for jet formation and equilibration a setof subsidiary results are obtained including identificationof the physical mechanism of jet formation (Bakas and

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Ioannou 2013b; Bakas et al. 2015), the structure of thefinite amplitude equilibrated jets, and prediction of the ex-istence of multiple equilibria jet states (Farrell and Ioannou2007, 2009a; Parker and Krommes 2013; Constantinou et al.2014a). But perhaps most significant is the insight obtainedfrom these SSD equilibria into the nature of turbulence: theturbulent state is revealed to be fundamentally determinedby cooperative interaction acting directly between the largeenergy bearing scales of the mean flow and the small scalesof the turbulence. This fundamental quasi-linearity, whichis revealed by SSD, is a general property of turbulence inshear flow recognition of which provides both conceptualclarity as well as analytical tractability to the turbulenceclosure problem.

An additional issue arises in the case of self–sustainedjets that are maintained by baroclinic growth processes:establishment of the statistical mean turbulent state forsupercritical imposed meridional temperature gradients re-quires suppression of the unstable growth. It has long beenremarked that coincident with the equilibration of baroclinicinstability in supercritically forced baroclinic turbulenceis a characteristic organization of the flow into prominentlarge scale jets but this association remained an intriguingobservation because although fluctuating approximations toSSD equilibria occur in realizations, analytic expression ofthe mechanism of flow instability equilibration is in generalinaccessible within sample state dynamics. However, thisproblem can be directly solved using SSD: the equilibriumturbulent state is obtained in the form of fluctuation-freefixed points of the autonomous SSD. Moreover, the mecha-nism of equilibration is also identified as these fixed pointsare found to be associated with stabilization of the super-critical flow by the jets, in large part by confinement of theperturbation modes by the meridional jet structure to suffi-ciently small meridional scale that the instabilities are nolonger supported, a mechanism previously referred to as the“barotropic governor” (Ioannou and Lindzen 1986; James1987; Lindzen 1993; Farrell and Ioannou 2008, 2009c).

In the magnetic plasma confinement problem, which isof great importance to the quest for a practical fusion powersource, the formation of jets by cooperative interaction withdrift wave turbulence is fundamental to the effectivenessof the plasma confinement (Diamond et al. 2005). Thejet-mediated high confinement regime is another exampleof a phenomenon that arises from cooperative interactionacting directly between large jet and small perturbationscales in a turbulent flow that can only be studied directlyusing SSD. Drift wave turbulence in plasmas, governed bythe Charney-Hasegawa-Mima or Hasegawa-Wakatani equa-tions, parallels in dynamics the barotropic or baroclinicturbulence in the Earth’s atmosphere with the Lorenz forceplaying the role of the Coriolis force in the plasma case so itis not surprising that the same or very similar phenomenaoccur in these two systems. The time dependent statistical

state of drift wave turbulence has natural expression as thetrajectory of the statistical state evolving under its asso-ciated SSD (Farrell and Ioannou 2009b). The trajectoryof the statistical state of a turbulent system commonlyapproaches a fixed point corresponding to a statisticallysteady state but in the case of plasma turbulence the sta-tistical state instead often follows a limit cycle or even achaotic trajectory. These time-dependent states are distinctconceptually from the familiar limit cycles or chaotic trajec-tories of sample state realizations. Rather, these statisticalstate trajectories represent time-dependence or chaos ofthe cooperative dynamics of the turbulence which, whileapparent in observation of sample state temporal variabil-ity, has no counterpart in analysis based on sample statedynamics. An example of a time dependent statistical statetrajectory that is perhaps more familiar to the atmophericscience community is provided by the limit cycle behaviorof the Quasi-biennial Oscillation in the Earth’s equatorialstratosphere (Farrell and Ioannou 2003).

A manifestation of turbulence of great practical as wellas theoretical interest is that collectively referred to as wall-bounded shear flow turbulence, examples of which includepressure driven pipe and channel flows, flow between dif-ferentially moving plane surfaces, over airplane wings, andin the pressure forced shear flow of the convectively stableatmospheric boundary layers. These laminar shear flow ve-locity profiles have negative curvature and, consistent withthe prediction of Rayleigh’s theorem for their inviscid coun-terparts, these flows do not support inflectional instabilities.Two fundamental problems are posed by the turbulenceoccurring in wall-bounded shear flows: instigation of theturbulence, referred to as the bypass transition problem,and maintenance of the turbulent state once it has beenestablished. The second of these problems, maintenance ofturbulence in wall-bounded shear flow, is commonly associ-ated with what is referred to as the Self-Sustaining Process(SSP). Transition can occur either by a pathway intrinsicto the SSD of the background turbulence or alternativelytransition can be induced directly by imposition of a suf-ficiently large and properly configured state perturbation(Farrell and Ioannou 2012). The former bypass transitionmechanism has no analytical counterpart in sample statedynamics while the latter has been extensively studied usingsample state realizations as for example in Brandt et al.(2004). In contrast, the SSP mechanism maintaining theturbulent state is fundamentally a quasi-linear multiscaleinteraction amenable to analysis using SSD that can beidentified with a chaotic trajectory of the SSD (Farrell andIoannou 2012).

In this review one implementation of SSD, referred toas Stochastic Structural Stability Theory (S3T), will bedescribed. S3T is a second order cumulant expansion (CE2)closure that employs a stochastic parameterization to closethe expansion. S3T and alternative implementations of CE2

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have been used recently to study: barotropic turbulencein planetary atmospheres (Farrell and Ioannou 2003, 2007;Marston et al. 2008; Marston 2010; Srinivasan and Young2012; Marston 2012; Bakas and Ioannou 2013a; Parker andKrommes 2014; Constantinou et al. 2014a, 2016; Marstonet al. 2016; Woillez and Bouchet 2017), in equatorial dynam-ics (Farrell and Ioannou 2009a; Fitzgerald and Farrell 2017),in baroclinic turbulence (Farrell and Ioannou 2008, 2009c;Bernstein and Farrell 2010; Farrell and Ioannou 2017b), thegrowth of dry convective boundary layer in the atmosphere(Ait-Chaalal et al. 2016), turbulence in astrophysical flows(Tobias et al. 2011), drift-wave turbulence in plasmas (Far-rell and Ioannou 2009b; Parker and Krommes 2013), andthe turbulence of wall-bounded shear flows (Farrell andIoannou 2012; Constantinou et al. 2014b; Thomas et al.2014, 2015; Farrell et al. 2016, 2017b,a; Farrell and Ioannou2017a).

2. Implementation of SSD: S3T theory and analy-sis

S3T implements a closure at second order of the ex-pansion in cumulants of the system dynamics (Hopf 1952;Frisch 1995). The Hopf expansion equations govern thejoint evolution of the mean flow (first cumulant) and theensemble perturbation statistics (higher order cumulants).A second order closure is obtained by either a stochas-tic parameterization of the terms in the second cumulantequation that involve the third cumulant (Farrell and Ioan-nou 1993a,b; DelSole and Farrell 1996; DelSole 2004b) orsetting the third cumulant to zero (Marston et al. 2008).Restriction of the dynamics to the first two cumulants isequivalent to either neglecting or parameterizing by additivenoise the perturbation-perturbation interactions in the fullynon-linear dynamics, which removes the mechanism of thenonlinear perturbation cascade as well as nonlinear mixingfrom the dynamics. This closure results in a non-linear,autonomous dynamical system that governs the evolutionof the mean flow and its associated second order perturba-tion statistics. The S3T equations constitute a SSD which,when implemented as the dynamics of a zonal mean andthe covariance of perturbations from this zonal mean flow,governs the evolution of the statistical state representedby the zonal mean and a Gaussian approximation to theperturbation covariance that is consistent with it.

We now review the derivation of the S3T system startingfrom the discretized Navier-Stokes equations which can beassumed to take the generic form:

dxidt

=∑j,k

aijkxjxk −∑j

bijxj + fi . (1)

The flow variable xi could be the velocity component atthe i-th location of the flow and the discrete set of equa-tions (1) could arise from discretization of the continuous

fluid equations on a spatial grid. The linear term∑j bijxj

represents dissipation with bij a positive definite matrix.Any externally imposed body force is specified by fi. In fluidsystems

∑i,j,k aijkxixjxk vanishes identically implying in

the absence of dissipation and forcing that E = 12

∑i x

2i is

conserved.Consider now the averaging operator ( · ). This averaging

operator could be the zonal mean in a planetary flow, butfor now it is left unspecified but it will be assumed that itsatisfies the Reynolds postulates that require that(a): λφ(t) + µψ(t) = λφ(t) + µψ(t) for all real numbers λ,µ, and

(b): φ(t)ψ(t) = φ(t) ψ(t).It is also assumed that the averaged quantities are at leastas differentiable and integrable as the original unaveragedfields and that the averaging operator commutes with timetranslations, so that(

∂φ

∂t

)=∂φ

∂t,

(∫dt φ

)=

∫dt φ . (2)

Note that the running time average:

φ(t) ≡ 1

2T

∫ t+T

t−Tφ(τ)dτ , (3)

commutes with time translations, satisfies the linearityassumption (a), but does not define an averaging operatoras it does not satisfy condition (b). These postulates implythe dependent variable xi can be decomposed into a mean,and a perturbation part: xi = Xi + x′i, where Xi ≡ xi, andx′i = 0 and most importantly that

xixj = XiXj +Xix′j +Xjx′i + x′ix′j = XiXj + x′ix

′j ,

and that also

dXi

dt=dXi

dt,dx′idt

=dx′idt

= 0 . (4)

In the development that follows we assume that upondecomposing the external forcing fi = Fi + f ′i that theFi ≡ fi are deterministic while the f ′i are stochastic. Takingthe average of (1) we obtain that the mean and perturbationvariables evolve according to:

dXi

dt−∑j,k

aijkXjXk +∑j

bijXj =∑j,k

aijkx′jx′k + Fi,

(5a)

dx′idt

=∑j

Aij(X)x′j + f ′iNL

+ f ′i , (5b)

whereAij(X) =

∑k

(aikj + aijk)Xk − bij , (6)

4

andf ′iNL

=∑j,k

aijk

(x′jx′k − x′jx′k

). (7)

The term∑j Aij(X)x′j is bilinear in X and x′ and repre-

sents the influence of the mean flow on the perturbationdynamics while the quadratically nonlinear term f ′

NLrep-

resents the perturbation-perturbation interactions whichare responsible for the turbulent cascade in the pertur-bation variables. The term FRi ≡

∑j,k aijkx

′jx′k, in (5a)

is the perturbation Reynolds stress divergence and repre-sents the influence of the perturbations on the mean flow.Equations (5) determine the evolution of the mean flowand the perturbation variables under the full non-lineardynamics (1) and will be referred to as the NL equations.

The quasi-linear (QL) approximation to NL results whenthe perturbation-perturbation interactions (given by term

f ′NL

in (5b)) are either neglected entirely or replaced by astochastic parameterization while the influence of pertur-bations on the mean is retained fully by incorporating theterm FR in the nonlinear mean equation (5a). The QLapproximation of (5) under the stochastic parameterization

f ′iNL

+ f ′i =√ε∑j fijdBtj is:

dXi

dt−∑j,k

aijkXjXk +∑j

bijXj =∑j,k

aijkx′jx′k + Fi,

(8a)

dx′i =

n∑j=1

Aij(X)x′jdt+√ε fij dBtj . (8b)

The noise terms, dBtj , are independent delta correlatedinfinitesimal increments of a one-dimensional Brownianmotion at time t (cf. Øksendal (2000)) satisfying:

〈dBti〉 = 0 , 〈dBtidBsj〉 = δijδ(t− s) dt , (9)

in which 〈 · 〉 denotes the ensemble average over realizationsof the noise. Equations (8) will be referred to as the QLequations. In the absence of forcing and dissipation theQL equations conserve the energy EQL = 1

2

∑i

(X2i + x′2i

)and, in the presence of bounded deterministic forcing, Fi,and dissipation, realizations of the dynamics (8b) have allmoments finite at all times. In general the energy conservedin NL differs from that conserved in QL and their differenceis E−EQL =

∑iXix

′i. This cross term vanishes identically

if summation over i provides equivalent action as the chosenaveraging operator, otherwise the equality of the energyinvariants is true only on average. For example if theaveraging operator is the zonal mean and the index refersto the value of the variable on a spatial grid the crossterm vanishes and then E = EQL. We will require thatthe averaging operators have the property that the QLinvariants (energy, enstrophy, etc) are the same as thecorresponding NL invariants.

Consider N realizations of the perturbation dynam-ics (8b) evolving under excitation by statistically indepen-dent realizations of the forcing but all evolving under theinfluence of a common mean flow X according to

dx′ri =

n∑j=1

Aij(X)x′rj dt+√ε fijdB

rtj , (r = 1, . . . , N).

(10)Denote with superscript r the r-th realization so that xrj(t)corresponds to the forcing dBrtj . Assume further that themean flow X is evolving under the influence of the averageFR over these N realizations, so that:

dXi

dt−∑j,k

aijkXjXk +∑j

bijXj =∑j,k

aijkCNjk + Fi ,

(11)

with

CNij =1

N

N∑r=1

x′ri x′rj , (12)

the N -ensemble averaged perturbation covariance matrix.To motivate this ensemble consider it to correspond to thephysical situation in which the averaging operator is thezonal mean and assume that over a latitude circle the zonaldecorrelation scale is such that the latitude circle may beconsidered to be populated by N independent perturbationstructures, all of which contribute additively to the Reynoldsstresses that collectively maintains the zonal mean flow.

An explicit equation for the evolution of CNij is obtainedusing the Ito lemma and (10):

dCNij =1

N

N∑r=1

(dx′ri x

′rj + x′ri dx

′rj

)+ ε

n∑k=1

fikfTkjdt

=

n∑k=1

(Aik(X)CNkj + CNikA

Tkj(X) + εfikf

Tkj

)dt+

+

√ε

N

N∑r=1

n∑k=1

(fikx

′rj + fjkx

′ri

)dBrtk . (13)

The stochastic equation (13) should be understood in theIto sense, so that the variables x′ and the noise dBt areuncorrelated in time and the ensemble mean of each of(fikx

′rj + fjkx

′ri

)dBrtk vanishes at all times. The corre-

sponding differential equation in the physically relevantStratonovich interpretation is obtained by removing fromequation (13) the term ε

∑nk=1 fikf

Tkj dt. However, both

interpretations produce identical covariance evolutions be-cause in the Stratonovich interpretation the mean of

√ε(fikx

′rj + fjkx

′ri )dBrtk

is nonzero and equal exactly to the term ε∑nk=1 fikf

Tkjdt

that was removed from the Ito equation. This results

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because the noise in (10) enters additively. The noiseterm in (13) can be further reduced using the Ito isom-etry (cf. Øksendal (2000)) according to which any noiseof the form

∑mk=1 gk(x′1, . . . , x

′n) dBtk can be replaced by

the single noise process√∑m

k=1 g2k(x′1, . . . , x

′n) dBt, in the

sense that both processes have the same probability distri-bution function. Applying the Ito isometry to the noiseterms in (13) we obtain

√ε

N

N∑r=1

n∑k=1

(fikx

′rj + fjkx

′ri

)dBrtk =

=

√ε

N

√√√√ n∑k=1

fikfikCNjj + fjkfjkCNii + 2fikfjkCNij dBt ,

and (13) becomes:

dCNij =

n∑k=1

(Aik(X)CNkj + CNikA

Tkj(X) + εfikf

Tkj

)dt+

+

n∑k=1

√ε

NRik(CN ) dBtkj , (14)

where dBtij is an n× n matrix of infinitesimal incrementsof Brownian motion, and the elements of Rij are:

Rij(CN ) =

√QiiCNjj +QjjCNii + 2QijCNij . (15)

with Qij =∑nk=1 fikf

Tkj . Equations (11) and (14) which

govern the evolution of the mean flow interacting with Nindependent perturbation realizations will be referred to asthe ensemble quasi-linear equations (EQL).

The stochastic term in the EQL vanishes as the numberof realizations increases and in the limit N → ∞ we ob-tain the autonomous and deterministic system of StochasticStructural Stability theory (S3T) for the mean X and the as-sociated perturbation covariance matrix Cij = limN→∞ CNij :

dXi

dt−∑j,k

aijkXjXk +∑j

bijXj =∑j,k

aijkCjk + Fi ,

(16a)

dCijdt

=

n∑k=1

(Aik(X)Ckj + CikA

Tkj(X)

)+ εQij . (16b)

3. Remarks on the S3T system

i. We have followed a physically based derivation ofthe S3T equations as in Farrell and Ioannou (2003).These equations can also be obtained using Hopf’sfunctional method (Hopf 1952; Frisch 1995; Marstonet al. 2008).

ii. The choice of averaging operator as well as the choiceof the stochastic parameterization for the perturbation-perturbation interaction and the external perturba-tion forcing used in the S3T model must be consistentwith the dynamics of the turbulent flow being studied.For example, using the zonal mean as an averagingoperator is appropriate when studying jet formation.

iii. In the case of homogenous isotropic turbulence thestochastic excitation must be very carefully fashionedin order to obtain approximately valid statistics usinga stochastic closure (Kraichnan 1971) while in shearflow the form of the stochastic excitation is not crucial.The reason is that in shear flow the operator Aijis non-normal and a restricted set of perturbationsparticipate strongly in the interaction with the meanflow. As a result the statistical state of the turbulenceis primarily determined by the quasilinear interactionbetween the mean and these perturbations rather thanby nonlinear interaction among the perturbations.

iv. The S3T dynamics exploits the idealization of aninfinite ensemble of perturbations interacting withthe mean. It follows that S3T becomes increasinglyaccurate as the number of effectively independentperturbations contributing to influence the mean in-creases.

v. It is often the case that the zonal mean is an attractivechoice for the averaging operator. A stable fixed pointof the associated S3T system then corresponds to astatistical turbulent state comprising a mean zonaljet and fluctuations about it with covariance Cij sothat the pdf of the perturbation field is Gaussian withdistribution:

p(x′1, . . . , x′n) =

exp[− 1

2

∑i,j(C

−1)ijx′ix′j

]√

(2π)n det(C).

vi. S3T theory can also be applied to problems in whicha temporal rather than a spatial mean is appropriate.The interpretation of the ensemble mean is then asa Reynolds average over an intermediate time scale,in which interpretation the perturbations are highfrequency motions while the mean constitutes theslowly varying flow components (Bernstein and Farrell2010; Bakas and Ioannou 2013a; Constantinou et al.2016).

vii. Often the attractor of the S3T dynamics is a fixedpoint representing a regime with stable statistics.However, the attractor of the S3T dynamics neednot be a fixed point and in many cases a stable peri-odic orbit emerges as the attracting solution. In manyturbulent systems large scale observables exhibit slow

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and nearly periodic fluctuation despite short timescales for the underlying dynamics and the lack ofexternal forcing to account for the long time scale(i.e. the Quasi Biennial Oscillation in the Earth’satmosphere, the solar cycle). S3T provides a mech-anism for such phenomena as reflections of a limitcycle attractor of the ideal S3T dynamics (Farrell andIoannou 2003).

viii. S3T provides an analytical mechanism for investigat-ing the sensitivity of the statistical mean state ofa turbulent system (the climate as represented bythe model) to perturbations of system parameters.Small changes in system parameters typically causecorrespondingly small and linearly related variationin the statistical state from which the sensitivity ofthe model climate can be inferred.

ix. However, at the bifurcation points of the SSD smallchanges of the system parameters produce large changesin the statistical state of the turbulence. For example,S3T dynamics predicts bifurcation from the statisti-cal homogeneous regime in which there are no zonalflows to a statistical regime with zonal flows when aparameter changes or predicts the transition from aregime characterized by two jets to a regime with asingle jet.

x. The EQL equations (11) and (14) contain informa-tion about the fluctuations remaining in the ensembledynamics when the number of ensemble members isfinite. These fluctuation statistics can be used todetermine the statistics of noise induced transitionsbetween ideal S3T equilibria.

xi. The close correspondence of S3T and NL simulationssuggests that turbulence in shear flow can be es-sentially understood as determined by quasi-linearinteraction occurring directly between a spatial ortemporal mean flow and perturbations. This resultprovides a profound simplification of the dynamicsof turbulence, identifies the mechanism determiningthe statistical mean state in a turbulent flow, andshows that the role of nonlinearity in the dynamicsof turbulence is highly restricted.

xii. The S3T system has bounded solutions and if desta-bilized typically equilibrates to a fixed point whichcan be identified with statistically stable states of tur-bulence (Farrell and Ioannou 2003). Moreover, theseequilibria closely resemble observed statistical states.For example S3T applied to an unstable baroclinicflow takes the form of baroclinic adjustment that isobserved to occur in observations and in simulations(Stone and Nemet 1996; Schneider and Walker 2006;Farrell and Ioannou 2008, 2009c).

4. Applying S3T to study SSD equilibria and theirstability

S3T dynamics comprises interaction between the meanflow, X, and the turbulent Reynolds stress obtained fromthe associated second order covariance of the perturbationfield, C. A fixed point of this system, when stable, cor-responds to a stationary statistical mean turbulent state.When rendered unstable by change of a system parame-ter, these equilibria predict structural reorganization ofthe whole turbulent field leading to establishment of anew statistical mean state. These bifurcations correspondto a new type of instability in turbulent flows associatedwith statistical mean state reorganization. Although suchreorganizations have been commonly observed there hasnot heretofore been a theoretical method for analyzing orpredicting them.

S3T dynamics comprises interaction between the meanflow, X, and the turbulent Reynolds stress obtained fromthe associated second order covariance of the perturbationfield, C. A fixed point of this system, when stable, cor-responds to a stationary statistical mean turbulent state.When rendered unstable by change of a system parame-ter, these equilibria predict structural reorganization ofthe whole turbulent field leading to establishment of anew statistical mean state. These bifurcations correspondto a new type of instability in turbulent flows associatedwith statistical mean state reorganization. Although suchreorganizations have been commonly observed there hasnot heretofore been a theoretical method for analyzing orpredicting them.

We consider first stability of an equilibrium probabilitydistribution function in the context of S3T dynamics. TheS3T equilibrium is determined jointly by an equilibriummean flow Xe and a perturbation covariance, Ce, thattogether constitute a fixed point of the S3T equations (16a)and (16b):∑

j,k

aijkXejX

ek −

∑j

bijXej +

∑j,k

aijkCejk + Fi = 0 ,

(17a)∑k

(Aik(Xe)Cekj + CeikA

Tkj(X

e))

+ εQij = 0 . (17b)

The linear stability of a fixed point statistical equilib-rium of the S3T system (Xe, Ceij) is determined from theassociated perturbation equations

d δXi

dt=∑k

Aik(Xe)δXk +∑j,k

aijkδCjk , (18a)

d δCijdt

=∑k

(δAikC

ekj + CeikδA

Tkj+

+Aik(Xe)δCkj + δCikATkj(X

e)), (18b)

7

withδAij =

∑k

(aikj + aijk) δXk . (19)

The asymptotic stability of such a fixed point is deter-

mined by assuming solutions of the form (δXi, δCij)eσt

with δAij = δAijeσt and by obtaining the eigenvalues, σ,

and the eigenfunctions of the system:

σδXi =∑k

Aik(Xe)δXk +∑j,k

aijk δCjk , (20a)

σδCij =∑k

[δAikC

ekj + Ceik δA

T

kj+

+Aik(Xe)δCkj + δCikATkj(X

e)], (20b)

If the attractor of the S3T is a limit cycle then (16) hastime varying periodic solutions (Xp

i (t), Cpij(t)) with periodT . The S3T stability of this periodically varying statisticalstate is determined by obtaining the eigenvalues of thepropagator of the time dependent version of (18) over theperiod T .

If the attractor of the S3T is a limit cycle then (16) hastime varying periodic solutions (Xp

i (t), Cpij(t)) with periodT . The S3T stability of this periodically varying statisticalstate is determined by obtaining the eigenvalues of thepropagator of the time dependent version of (18) over theperiod T .

5. Remarks on S3T instability

i. Consider an equilibrium mean flow Xe; if ε vanishesidentically then Ce also vanishes and S3T stabilitytheory collapses to the familiar hydrodynamic insta-bility of this mean flow, which is governed by thestability of A(Xe). Consequently, S3T instability of(Xe, 0) implies the hydrodynamic instability of Xe.However, if ε does not vanish identically then the in-stability of the equilibrium state (Xe, Ce) introducesa new type of instability, which is an instability ofthe collective interaction between the ensemble meanstatistics of the perturbations and the mean flow. Itis an instability of the SSD and can be formulatedonly within this framework. Eigenanalysis of the S3Tstability equations (18) provides a full spectrum ofeigenfunctions comprising mean flows and associatedcovariances that can be ranked according to theirgrowth rate. These eigenfunctions underly the behav-ior of QL and NL simulations. An example of this canbe seen in the EQL system, governed by (11) and (14),which provides noisy reflections of the S3T equilibriaand their stability. Specifically: the response of anEQL simulation of a stable S3T equilibrium manifestsstructures reflecting stochastic excitations of the sta-ble S3T eigenfunctions by the fluctuations in EQL(Constantinou et al. 2014a).

ii. Hydrodynamic stability is determined by eigenanal-ysis of the n × n matrix A(Xe). However, in orderto determine the S3T stability of (Xe, Ce) the eigen-values of the (n2 + n)× (n2 + n) system of equations(18) must be found and special algorithms have beendeveloped for this calculation (Farrell and Ioannou2003; Constantinou et al. 2014a).

iii. If (Xe, Ce) is a fixed point of the S3T system, thenXe is necessarily hydrodynamically stable, i.e. A(Xe)is stable. This follows because the equilibrium co-variance Ce that solves (17b) is determined from thelimit

Ceij = ε limt→∞

∫ t

0

∑k,l

eA(Xe)(t−s)ik Qkl e

AT (Xe)(t−s)lj ds ,

(21)which does not exist if A(Xe) is neutrally stable orunstable and consequently in either case Xe is nota realizable S3T equilibrium. This argument gener-alizes to S3T periodic orbits: if S3T has a periodicsolution, (Xp(t), Cp(t)) of period T , then the pertur-bation operators A(Xp(t)) must be Floquet stable, i.e.the propagator over a period T of A(Xp(t)) has eigen-values, λ, with |λ| < 1, so that the time dependentmean states Xp(t) are hydrodynamically stable.

iv. While S3T stable solutions are necessarily also hydro-dynamically stable, the converse is not true: hydrody-namic stability does not imply S3T stability. We willgive examples below of hydrodynamically stable flowsthat are S3T unstable. This is important because itcan lead to transitions between turbulent regimes thatare the result of cooperative S3T instability ratherthan instability of the associated laminar flow.

6. Applying S3T to study the SSD of beta-planeturbulence

Consider a barotropic midlatitude beta-plane modelfor the dynamics of jet formation and maintenance in theEarth’s upper troposphere or in Jupiter’s atmosphere atcloud level. For simplicity assume a doubly periodic channelwith x and y Cartesian coordinates along the zonal andthe meridional direction respectively. The nondivergentzonal and meridional velocity fields are expressed in termsof a streamfunction, ψ, as u = −∂yψ and v = ∂xψ. Theplanetary vorticity is 2Ω+βy, with Ω the planetary rotationrate and β the planetary vorticity gradient evaluated at thelatitude of the midpoint of the channel. The relative vortic-ity is q = ∆ψ where ∆ ≡ ∂2xx + ∂2yy is the Laplacian. TheNL dynamics of this system is governed by the barotropicvorticity equation:

∂tq + u ∂xq + v ∂yq + βv = D +√ε f . (22)

8

0 2 4 6−2

−1

0

1

2

3

4x 10

−3

y0 2 4 6

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

y

vq for β=0 vq for β=0

vq for β=10

vq for β=10

δ U

Figure 1: Mean flow acceleration resulting from a smallmean flow perturbation imposed on a background of homo-geneous turbulence. In the absence of a perturbation jet thevorticity flux v′q′ = 0. The turbulence is distorted by the jetperturbation inducing vorticity fluxes that tend to amplifythe imposed jet perturbation. In the left panel is shown aGaussian jet perturbation together with the accelerations thatit induces for β = 0, 10. The mean flow acceleration is not thesame function as the δU that was introduced. Right panel:when the δU is sinusoidal the mean flow acceleration has thesame form resulting in exponential growth that leads to theemergence of large scale jets. Calculations were performed ina doubly periodic square beta plane box of length 2π. Thecoefficient of linear damping is r = 0.1.

The term D represents linear dissipation with the zonalcomponent of the flow (corresponding to zonal wavenumberk = 0) dissipated at the rate rm while the non-zonal com-ponents are dissipated at rate r > rm (cf. Constantinouet al. (2014a)). This dissipation specification allows use ofRayleigh damping while still modeling the physical effect ofsmaller damping rate at the large jet scale than at the muchsmaller perturbation scale. Periodic boundary conditionsare imposed in x and y with periodicity 2πL. Distances arenondimensionalized by L = 5000 km and time by T = L/U ,where U = 40 m s−1, so that the time unit is T = 1.5 dayand β = 10 corresponds to a midlatitude value. Turbu-lence is maintained by stochastic forcing with spatial andtemporal structure, f , and variance ε.

Choosing as the averaging operator the zonal mean, i.e.

φ(y, t) =∫ 2π

0φ(x, y, t) dx/2π and decomposing the fields in

zonal mean components and perturbations we obtain thediscretized barotropic QL system:

dU

dt= v′q′ − rm U , (23a)

dqkdt

= Ak(U)qk +√εFkdBtk , (23b)

where U is the mean flow state and the subscript k =1, . . . , Nk, in (27b) indicates the zonal wavenumber and qkthe Fourier coefficient of the perturbation vorticity that

has been expanded as q′ = <(∑Nk

k=1 qkeikx)

. The Nk wave

numbers include only the zonal wavenumbers that are ex-cited by the stochastic forcing because the k 6= 0 Fouriercomponents do not directly interact in (27b) and thereforethe perturbation response is limited to the wavenumbers di-rectly excited by the stochastic forcing. The linear operator

of the perturbation dynamics (27b) is given by:

Ak(U) = −ikU − ik(β −D2U)∆−1k − r , (24)

with ∆k = D2 − k2, ∆−1k its inverse, and D2 = ∂yy. Thecontinuous operators are discretized and approximated bymatrices. The perturbation velocity appearing in (27a) is

given by v′ = <(∑Nk

k=1 ik∆−1k qkeikx)

, and the meridional

vorticity flux accelerating the mean flow is:

v′q′ =

Nk∑k=1

k

2diag

[=(∆−1k Ck

)], (25)

with = denoting the imaginary part, Ck = qkq†k the single

ensemble member covariance, † the Hermitian transpose,and diag the diagonal elements of a matrix. The forcingstructure is chosen to be non-isotropic with matrix elements:

Fkij = ck

[e−(yi−yj)

2/(2s2) + e−(yi−2π−yj)2/(2s2)+

+ e−(yi+2π−yj)2/(2s2)], (26)

with s = 0.2/√

2 and the normalization constants chosenso that energy is injected at each zonal wavenumber k atunit rate. The delta correlation in time of the excitationensures that this energy injection rate is the same in theQL and NL simulations and is independent of the stateof the system. This forcing is chosen to model forcing ofthe barotropic jet in the upper atmosphere by baroclinicinstability. For more details cf. Constantinou et al. (2014a).

Choosing as the averaging operator the zonal mean, i.e.

φ(y, t) =∫ 2π

0φ(x, y, t) dx/2π and decomposing the fields in

zonal mean components and perturbations we obtain thediscretized barotropic QL system:

dU

dt= v′q′ − rm U , (27a)

dqkdt

= Ak(U)qk +√εFkdBtk , (27b)

where U is the mean flow state and the subscript k =1, . . . , Nk, in (27b) indicates the zonal wavenumber and qkthe Fourier coefficient of the perturbation vorticity that

has been expanded as q′ = <(∑Nk

k=1 qkeikx)

. The Nk wave

numbers include only the zonal wavenumbers that are ex-cited by the stochastic forcing because the k 6= 0 Fouriercomponents do not directly interact in (27b) and thereforethe perturbation response is limited to the wavenumbers di-rectly excited by the stochastic forcing. The linear operatorof the perturbation dynamics (27b) is given by:

Ak(U) = −ikU − ik(β −D2U)∆−1k − r , (28)

with ∆k = D2 − k2, ∆−1k its inverse, and D2 = ∂yy. Thecontinuous operators are discretized and approximated by

9

matrices. The perturbation velocity appearing in (27a) is

given by v′ = <(∑Nk

k=1 ik∆−1k qkeikx)

, and the meridional

vorticity flux accelerating the mean flow is:

v′q′ =

Nk∑k=1

k

2diag

[=(∆−1k Ck

)], (29)

with = denoting the imaginary part, Ck = qkq†k the single

ensemble member covariance, † the Hermitian transpose,and diag the diagonal elements of a matrix. The forcingstructure is chosen to be non-isotropic with matrix elements:

Fkij = ck

[e−(yi−yj)

2/(2s2) + e−(yi−2π−yj)2/(2s2)+

+ e−(yi+2π−yj)2/(2s2)], (30)

with s = 0.2/√

2 and the normalization constants chosenso that energy is injected at each zonal wavenumber k atunit rate. The delta correlation in time of the excitationensures that this energy injection rate is the same in theQL and NL simulations and is independent of the stateof the system. This forcing is chosen to model forcing ofthe barotropic jet in the upper atmosphere by baroclinicinstability. For more details cf. Constantinou et al. (2014a).

The barotropic S3T system is:

dU

dt=

Nk∑k=1

−k2=(diag

(∆−1k Ck

))− rmU , (31a)

dCkdt

= Ak(U)Ck + CkA†k(U) + εQk , (31b)

with Ck =⟨qkq†k

⟩and Qk = FkF

†k. The imaginary part

in (31a) requires that we add to the system an equationfor the conjugate of the covariance. This is necessary fortreating the S3T equations as a dynamical system andfor analyzing the stability of S3T equilibria. Alternativelywe can treat the real and imaginary part of the perturba-tion covariance as separate variables to obtain a real S3Tdynamical system as in (16).

Under the assumption that the stochastic forcing in theperiodic channel is homogeneous (31a) and (31b) admitthe homogeneous equilibrium

Ue = 0 , Cek =ε

2rQk , (32)

as shown in Appendix A. The assumption of homogeneityof the excitation is crucial for obtaining a covariance (32)and associated turbulent equilibrium with no flow, whichrequires that the excitation does not lead to a momen-tum flux convergence. If the excitation were confined toa latitude band, the homogeneous state could not be anS3T equilibrium. In that case, Rossby waves that originatefrom the region of excitation would dissipate in the far-field

0 1 2 3 4 5 6 7 8 9 10 11 12

100

101

102

103

n

ǫ/ǫc

Figure 2: S3T stability diagram showing the predicted zonalflow equilibria as a function of the number of jets, n, and themarginal fractional amplitude of excitation ε/εc in the doublyperiodic channel (dashed curve). The amplitude, εc, is theminimal excitation amplitude, obtained from S3T stabilityanalysis, that renders the homogeneous state unstable. Forparameter values below the dashed curve the homogeneousstate is S3T stable and no jets are predicted to emerge. Abovethe dashed curve the flow is S3T unstable and new statisticalsteady states emerge characterized by the number of finite am-plitude jets across the channel, n. S3T stable finite amplitudeequilibrium jets are indicated with a full circle. Note that forgiven excitation amplitude there exist multiple S3T stableequilibria characterized by different numbers of jets. UnstableS3T equilibria determined with Newton iterations are indi-cated with open circles. Near the curve of marginal stabilitythe S3T unstable modes other than the most unstable oneat wavenumber 6 do not continue to stable finite amplitudestructures with the same wavenumber as the instability. Thiscan be understood as a manifestation of the universal Eckhausinstability as discussed in section 5.2.4 (Parker and Krommes2015) and in Parker and Krommes (2013). As the excitationamplitude increases jet bifurcations occur resulting in the suc-cessive establishment of stable S3T equilibria with smaller n.Zonal wavenumbers k = 1, . . . , 14 are forced, β = 10, r = 0.1,rm = 0.01. (Adapted from Constantinou (2015).)

producing momentum flux convergence into the excitationregion and acceleration of the mean flow there. The analysisthat follows assumes that the forcing is homogeneous sothat the homogeneous state (32) is an S3T equilibrium forall parameter values. The question is whether this homoge-neous equilibrium is S3T stable. If it becomes S3T unstableat certain parameter values this would be an example of aflow that is hydrodynamically stable but S3T unstable.

a. Formation and structural stability of beta-plane jets

To motivate our intuition for jet formation by coopera-tive interaction across spatial scales in beta-plane turbulenceconsider an infinitesimal perturbation of the homogeneousturbulence in the form of a small amplitude jet, δU , as forexample the one shown in Fig. 1a, and calculate the vortic-ity fluxes induced by this perturbation mean flow assuming

10

that the turbulence adjusts to δU so that it satisfies theequilibrium form of (31b) with zonal mean flow perturba-tion δU . The modification produced in the turbulence fieldby this mean flow perturbation produces the steady statecovariance satisfying the Lyapunov equation:

Ak(δU)Ck + CkA†k(δU) = −εQk , (33)

for k = 1, . . . , Nk. By solving (33) we find that introducingthe infinitesimal jet δU breaks the homogeneity of the tur-bulence resulting in an ensemble mean acceleration whichcan be calculated from (29). This induced acceleration,which is shown in Fig. 1a, is upgradient and tends to re-inforce the mean flow perturbation that induced it, andthis occurs even in the absence of β (it vanishes for β = 0only if the forcing covariance is isotropic). Repeated exper-imentation shows that this positive feedback occurs for anymean flow perturbation under any broadband excitation(isotropic or non-isotropic) as long as there is power at suf-ficiently high zonal wavenumbers1. This universal propertyof reinforcement of preexisting mean flow perturbations,revealed through S3T analysis, underlies the ubiquitousphenomenon of emergence of large scale structure in tur-bulent flows and explains why homogeneous equilibriumstates are unstable to jet perturbations2. It is clear that theacceleration induced by the test perturbation δU shown inFig. 1 is not identical to δU and therefore is not an eigen-function but if there exists a mean flow perturbation thatinduces accelerations with the same form as the imposedmean flow perturbation then this perturbation would be aneigenfunction of the S3T stability equations and it wouldgrow or decay exponentially without change of form. S3Tprovides the framework for determining systematically thefull spectrum of such eigenfunctions enabling full descrip-tion of the evolution of a perturbation of small amplitudenear an S3T equilibrium state. In fact, homogeneity in yof the mean state assures that the mean flow component ofthe eigenfunctions of the S3T equilibrium (32) are harmonicso that modulo phase δUn = sin(ny), where n indicatesthe number of jets associated with this eigenfunction. InFig. 1b is shown a verification that a single harmonic inducesmean flow acceleration of the same form and is thereforean eigenfunction of the S3T stability equations (18).

Consider the stability of the homogeneous equilibriumstate (32) as a function of the excitation amplitude ε. Forε = 0 the equilibrium Ue = 0 and Cek = 0 is S3T stable.For ε > 0 the universal process of reinforcement of animposed jet occurs and therefore if rm = 0 the homogeneousS3T equilibrium would immediately become unstable. For

1This implies that numerical simulations must be adequately re-solved for jet formation to occur.

2The dynamics leading to this behavior in barotropic flows isdiscussed in Bakas and Ioannou (2013b); Bakas et al. (2015). Thecounterpart of this process for three dimensional flows is discussed inFarrell and Ioannou (2012); Farrell et al. (2017a,b).

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ǫ/ǫc

Em/E

NL

QL

S3T

Figure 3: The ratio of the energy of the zonal mean flow,Em, to the total energy E, in S3T (solid and diamonds), QL(dashed and dots), and NL (dash-dot and circles) as a functionof forcing amplitude ε/εc. S3T predicts that the homogeneousflow becomes unstable at ε/εc = 1 with a symmetry breakingbifurcation giving rise to finite amplitude zonal jets. Thisprediction of S3T is reflected accurately in the sample integra-tions of both QL and NL. The agreement between NL and S3Treveals that zonal jet formation is a bifurcation phenomenonand the fact that S3T predicts both the inception of the in-stability and the finite equilibration of the emergent flowsdemonstrates that the essential dynamics of the formationand nonlinear equilibration is captured by QL/S3T. Otherparameters as in Fig. 2. (Adapted from Constantinou et al.(2014a).)

rm > 0, instability occurs for ε > εc, where εc is theforcing amplitude that renders the S3T stability equationneutrally stable. The normalized critical forcing amplitudefor the S3T eigenfunctions with meridional wavenumbern = 1, . . . , 11 is shown in Fig. 2. S3T predicts that forthese parameters the maximum S3T instability occurs formean flow perturbations δU = sin(ny) with n = 6 and thusS3T predicts that the breakdown of the homogeneous stateoccurs with the emergence of 6 jets in this channel.

b. Structure of jets in beta-plane turbulence

For parameter values exceeding those required for in-ception of the S3T instability the S3T attractor comprisesstatistical steady states with finite amplitude jets that canbe characterized by their zonal mean flow index Em/E,where Em is the kinetic energy density of the mean flowand E to the total kinetic energy density of the flow, as inSrinivasan and Young (2012). A plot of Em/E as a functionof forcing amplitude as predicted by S3T and as observedin QL and NL, is shown in Fig. 3 and the correspondingmeridional structures of the jet equilibria for various pa-rameter values are shown in Fig. 4. Jets correspondingto S3T equilibria must be hydrodynamically stable, whichgenerally requires with the mean vorticity gradient β −Ueyyof the equilibrium jets not change sign as the coefficient of

11

0 2 4 60

10

20

β − U ey y

0 2 4 6

−0.2

0

0.2

ǫ=

2ǫc

U e

100

101

100

E (k = 0, l )

0 2 4 6

0

20

40

0 2 4 6

−1

0

1

ǫ=

150ǫc

100

101

100

0 2 4 6

0

50

100

0 2 4 6

−2

0

2

ǫ=

500ǫc

100

101

100

0 2 4 6

0

50

100

y0 2 4 6

−10

0

10

ǫ=

10000ǫc

y10

0

101

100

l

Figure 4: Left: The equilibrium mean flow Ue for excitationamplitudes ε/εc = 2, 8, 150, 500, 104. Center: the correspond-ing mean vorticity gradient β − Ueyy. Right: The energyspectrum of the zonal mean flow. For the highly supercriticaljets the energy spectrum approaches the approximate l−5

dependence on meridional wavenumber l found in NL simula-tions. We argue that in the inviscid limit this slope shouldapproach l−4 as the prograde jet becomes increasingly sharp.Other parameters as in Fig. 2. (Adapted from Constantinou(2015).)

the linear damping becomes vanishingly small. Consistentwith being constrained by this criterion, the equilibratedjets at high supercriticality shown in Fig. 4 assume thecharacteristic shape of sharply pointed prograde jets, withvery large negative curvature, and smooth retrograde jetswhich satisfy β ≈ Ueyy. This consideration leads to theprediction that the mean spacing of the jets at high su-percriticality is approximately given by

√|Umin|/β, with

Umin the peak retrograde mean zonal velocity. At low su-percriticality jet amplitude is too low for instability to be afactor in constraining jet structure with typical planetaryvalues of β and the jets equilibrate with nearly the structureof their associated eigenmode, as discussed in Farrell andIoannou (2007). While the spacing of highly supercriticaljets could be interpreted as corresponding to Rhines scaling,the mechanism that produces this scaling is associated withthe modal stability boundary of the finite amplitude jetand is unrelated to the traditional interpretation of Rhines’scaling in terms of arrest of turbulent cascades.

S3T also predicts that the Fourier energy spectrum ofthe mean zonal flow of the highly supercritical jets hasapproximate l−5 dependence on meridional wavenumber, l,

y

NL U (y , t ) , ǫ/ǫ c =20

0 500 1000 1500 20000

2

4

6

y

QL U (y , t ) , ǫ/ǫ c =20

0 500 1000 1500 20000

2

4

6

t

y

S3T U (y , t ) , ǫ/ǫ c =20

0 500 1000 1500 20000

2

4

6

−0.4

−0.2

0

0.2

0.4

−0.4

−0.2

0

0.2

0.4

−0.4

−0.2

0

0.2

0.4

Figure 5: Hovmoller diagrams of jet emergence in NL, QLand S3T simulations for excitation ε/εc = 20 as in Fig. 3.Shown for the NL, QL and S3T simulations are U(y, t). In allsimulations the jet structure that first emerges is the n = 6maximally growing jet structure predicted by S3T stabilityanalysis. After a series of mergers S3T is attracted to astatistical steady state with a n = 4 jet. The whole process ofjet mergers and S3T equilibration is accurately reflected in theQL and NL simulations. This figure shows that S3T predictsthe structure, growth and equilibration of strongly forced jetsin both the QL and NL simulations. Other parameters as inFig. 2.

in accord with highly resolved nonlinear simulations (Sukar-iansky et al. 2002; Danilov and Gurarie 2004; Galperin et al.2004) as well as observations (Galperin et al. 2014). Thefinite amplitude jets obtained as fixed point equilibria inS3T are characterized by near discontinuity in the shear atthe maxima of the prograde jets. This discontinuity wouldbe consistent with a zonal energy spectrum proportionalto l−4. However, this discontinuity can not materialize inthe presence of diffusion and it is smoothed resulting in theapproximate l−5 dependence seen in simulations. Note thatS3T not only predicts the energy spectrum of the zonal jetsbut in addition the structure and therefore the phase ofthe spectral components. The occurrence of this power lawis theoretically anticipated by S3T because the upgradientfluxes act as a negative diffusion on the mean flow and asa result tend to produce constant shear equilibria at eachflank of the jet resulting in the near discontinuity observedin the derivative of the prograde jet (Farrell and Ioannou2007).

12

−0.5 −0.25 0 0.25 0.50

2

4

6

U

y

ǫ/ǫc = 20

Figure 6: The S3T equilibrium jet (solid-blue) for ε/εc = 20,and its reflection in a NL simulation (dash-dot-black) anda QL simulation (dashed - red). The jets in NL and QLundergo small fluctuations as is evident in Fig. 5 so shown isthe average jet structure over 500 units of time. This figureshows that S3T predicts the structure of strongly forced jetsin both the QL and NL simulations. Other parameters as inFig. 2.

c. Comments on statistical equilibria in beta-plane turbulence

turbulence are predictions of S3T that could not resultfrom analysis within the NL and QL systems as in both NLand QL turbulent states with stable statistics do not exist asfixed point equilibria so that a meaningful structural stabil-ity analysis of statistically stable equilibria states can not beperformed. However, as revealed in Fig. 3, Fig. 5 and Fig. 6,reflections of the bifurcation structure and of the finite am-plitude equilibria predicted by the S3T closure dynamicscan be clearly seen both in QL and NL sample simulations.The reflection in NL of the bifurcation structure predictedby S3T for ε ≈ εc is particularly significant because it showsthat the S3T perturbation instability faithfully reflects thephysics underlying jet formation in turbulence even as itmanifests for infinitesimal mean flows. For a more detaileddiscussion cf. Constantinou et al. (2014a).

7. Applying S3T to study SSD equilibria in baro-clinic turbulence

A 2D barotropic fluid lacks a source term to main-tain vorticity against dissipation and therefore it cannotself-sustain turbulence. Perhaps the simplest model thatself-sustains turbulence is the baroclinic two-layer model,which is essentially two barotropic fluids sharing a horizon-tal boundary. We use this model to study the statisticalstate dynamics of baroclinic turbulence. The perturbation-perturbation nonlinearity has been shown to be accuratelyparameterized in S3T studies of baroclinic turbulence byusing a state independent stochastic excitation as a closure

(DelSole and Farrell 1996; DelSole and Hou 1999; DelSole2004b). The dynamics of baroclinic turbulence has alsobeen studied using S3T with a state dependent closure withsimilar results (Farrell and Ioannou 2009c).

Consider a two layer fluid with quasi-geostrophic dy-namics. The layers are of equal depth h/2 and bounded byhorizontal rigid walls at the bottom and the top. Variablesin the top layer are denoted with subscript 1, and in thebottom layer with subscript 2. The top layer density is ρ1and the bottom ρ2 with ρ2 > ρ1 and the potential vortic-ity of the layer is qi = ∆ψi + βy + (−1)i2λ2(ψ1 − ψ2)/2(i = 1, 2), with λ2 = 2f20 /(g

′h), where g′ = g(%2 − %1)/%1 isthe reduced gravity, and f0 the Coriolis parameter so that,in terms of the Rossby radius of deformation Ld =

√g′h/f0,

λ =√

2/Ld. The flow is relaxed to a constant temperaturegradient in the meridional direction (y) which through thethermal wind relation induces, in the absence of turbulenceand at equilibrium, a meridionally independent mean shearHT in the zonal (x) mean velocity. This shear is takenwithout loss of generality to result from relaxation to zerovelocity in the bottom layer (2) and a constant mean flowwith stream function −HT y in the top layer (1). Relaxationto this velocity structure is induced by Newtonian coolingand the bottom layer is dissipated by Ekman damping. Thequasi-geostrophic dynamics governing this system is givenby:

∂tq1 + J(ψ1, q1) = 2λ2rTψ1 − ψ2 +HT y

2, (34a)

∂tq2 + J(ψ2, q2) = −2λ2rTψ1 − ψ2 +HT y

2− r∆ψ2 ,

(34b)

in which the advection of potential vorticity is expressedusing the Jacobian J(ψ, q) = ∂xψ∂yq − ∂yψ∂xq, rT is thecoefficient of the Newtonian cooling and r is the coefficientof Ekman damping. Equations (34) have been made non-dimensional with length scale Ld = 1000 km and timescale 1 day. With this scaling λ =

√2, the velocity unit is

11.5 ms−1 and a typical midlatitude value of β = 1.4. Theseequations can be expressed alternatively in terms of thebarotropic ψ = (ψ1 +ψ2)/2 and baroclinic θ = (ψ1 −ψ2)/2streamfunctions as:

∂t∆ψ + J(ψ,∆ψ) + J(θ,∆θ) + βψx = −r2

∆(ψ − θ)

(35a)

∂t∆λθ + J(ψ,∆λθ) + J(θ,∆ψ) + βθx =

=r

2∆(ψ − θ) + 2λ2rT

(θ +

HT

2y

)(35b)

where ∆λ ≡ ∆− 2λ2.The barotropic and baroclinic streamfunctions are de-

composed into a zonal mean (denoted with capitals) anddeviations from the zonal mean (referred to as perturba-

13

tions) as:

ψ = Ψ + ψ′ , θ = Θ + θ′ , (36)

and denote with U = −∂yΨ the barotropic zonal meanflow and with H = −∂yΘ the baroclinic zonal mean flow.With this decomposition we obtain the equations for theevolution of the barotropic and baroclinic zonal mean flows:

∂tU = ψ′xψ′yy + θ′xθ

′yy −

r

2(U −H)− rmU + νD2U

(37a)

∂tD2λH = D2

(ψ′xD

2λθ′ + θ′xψ

′yy

)+r

2D2(U −H)+

+ 2λ2rT

(H − HT

2

)− rmD2

λH + νD2D2H ,

(37b)

with D2 ≡ ∂y, D2λ ≡ ∂y − 2λ2, subscripts x and y denoting

differentiation and the overline denoting zonal averaging.The corresponding barotropic and baroclinic components ofthe zonal mean flow deviations are U = −Ψy and H = −Θy.The equations for the evolution of the perturbations are:

∂t∆ψ′ + U∂x∆ψ′ +H∂x∆θ′ + (β −D2U)∂xψ

′−

−D2H∂xθ′ = −r

2∆(ψ′ − θ′)− rp∆ψ′ + ν∆∆ψ′

− J(ψ′,∆ψ′)′ − J(θ′,∆θ′)′, (38a)

∂t∆λθ′ +H∂x∆ψ′ + U∂x∆λθ

′ + (β −D2U)∂xθ′−

−D2λH∂xψ

′ =r

2∆(ψ′ − θ′) + 2λ2rT θ

′ − rp∆λθ′+

+ ν∆∆θ′ − J(ψ′,∆λθ′)′ − J(θ′,∆ψ′)′ ,

(38b)

with the prime Jacobians denoting the perturbation-perturbationinteractions,

J(A,B)′ = J(A,B)− J(A,B) . (39)

We have allowed in (37) and (38) for linear dissipationof the mean at rate rm and of the perturbations at raterp. This choice in decay rates reflects the different ratesof dissipation of the mean flow and the perturbation field,which in natural flows is concentrated at smaller scales. Wehave also included diffusive dissipation of the velocity fieldwith coefficient ν. Equations (37) and (38) comprise theNL system that governs the two layer baroclinic flow. Weimpose periodic boundary conditions at the channel wallson Ψ, Θ, ψ′, θ′ as in Haidvogel and Held (1980); Panetta(1993). These boundary conditions can be verified to requirethat the zonally and meridionally averaged velocity at alltimes remains equal to that of the radiative equilibriumflow, which in turn implies that the temperature differencebetween the channel walls, and therefore the channel meancriticality, remains fixed.

The corresponding QL system is obtained by substitut-ing for the perturbation-perturbation interaction a stateindependent and temporally delta correlated stochastic exci-tation together with sufficient added diffusive dissipation toobtain an approximately energy conserving closure (DelSoleand Farrell 1996; DelSole and Hou 1999; DelSole 2004b).Under these assumptions the QL perturbation equationsfor the Fourier components of the barotropic and baroclinicstreamfunction are:

dψkdt

= Aψψk ψk + Aψθk θk +√ε∆−1k Fk ξ

ψ(t) , (40a)

dθkdt

= Aθψk ψk + Aθθk θk +√ε∆−1kλFk ξ

θ(t) , (40b)

in which we have assumed that the barotropic and baroclinicstreamfunctions are excited respectively by

√ε∆−1k Fkξψ

and√ε∆−1kλFkξ

θ. We also assume that ξψ and ξθ are inde-pendent temporally delta correlated stochastic processes ofunit variance and that the perturbation fields have beenexpanded as ψ′ = <

(∑k ψke

ikx)

and θ′ = <(∑

k θkeikx).

The operators Ak linearized about the mean zonal flowU = [U,H]T are

Ak(U) =

(Aψθk AψθkAθψk Aθθk

)(41)

with :

Aψψk = ∆−1k[−ikU∆k − ik

(β −D2U

)]− r

2− rp + ν∆k,

(42a)

Aψθk = ∆−1k[−ikH∆k + ikD2H

]+r

2, (42b)

Aθψk = ∆−1kλ

[−ikH∆k + ikD2

λH +r

2∆k

], (42c)

Aθθk = ∆−1kλ[−ikU∆kλ − ik

(β −D2U

)−

− r

2∆k + 2rTλ

2 + ν∆k∆k

]− rp ,

(42d)

and ∆k ≡ D2− k2, ∆kλ ≡ ∆k− 2λ2. Continuous operatorsare discretized and the dynamical operators approximatedby finite dimensional matrices. The states ψk and θk arerepresented by a column vector with entries the complexvalue of the barotropic and baroclinic streamfunction atthe collocation points in (y).

The corresponding S3T system is obtained by formingfrom the QL equations (40) the Lyapunov equation forthe evolution for the zonally averaged covariance of theperturbation field which takes the form:

dCkdt

= Ak(U)Ck + Ck A†k(U) + εQk , (43)

with the covariance for the wavenumber k zonal Fouriercomponent defined as:

Ck =

(Cψψk CψθkCψθ†k Cθθk

), (44)

14

where Cψψk = 〈ψkψ†k〉, Cψθk = 〈ψkθ†k〉, Cθθk = 〈θkθ†k〉 with〈 • 〉 denoting ensemble averaging, which under the ergodicassumption is equal to the zonal average. The covarianceof the stochastic excitation

Qk =

(∆−1k FkF

†k∆−1†k 0

0 ∆−1kλFθkF

θ†k ∆−1†kλ

), (45)

has been normalized so that for each k a unit of energy perunit time is injected by the excitation and therefore theamplitude of the excitation is controlled by the parameterε.

The S3T equations for the mean flow (37) in terms ofthe perturbation covariances are:

dU

dt=∑k

k

2diag

[=(D2Cψψk +D2Cθθk

)]−

− r

2(U −H)− rmU + νD2U, (46a)

dD2λH

dt= D2

∑k

k

2diag

[=(D2λC

ψθ†k +D2Cψθk

)]+

+r

2D2(U −H) + 2λ2rT

(H − HT

2

)−

− rmD2λH + νD2D2H . (46b)

As is the case in the previous barotropic examples, forhomogeneous forcing there also exists a meridionally andzonally homogeneous turbulent S3T equilibrium state con-sisting of an equilibrium zonal mean flow equal to thethermal wind balanced radiative equilibrium flow Ue =[HT /2, HT /2] corresponding to layer velocities Ue1 = HT

and Ue2 = 0 and a perturbation field with covariances, Cekwhich satisfy the corresponding steady state Lyapunov equa-tions (43) (for the explicit expression of the equilibriumcovariance see DelSole and Farrell (1995)). However, unlikein the previous case of the barotropic beta-plane example,this homogeneous equilibrium state is realizable only forparameter values for which Ue is hydrodynamically (baro-clinically) stable. In the absence of dissipation instabilityoccurs for this constant flow when HT > β/λ2 or, in termsof the criticality parameter ξ = max(U1 − U2)λ2/β, whenξ > 1. This dissipationless criticality parameter is custom-arily used despite the fact that ξ > 1 implies instabilityonly when the meridionally uniform flow is disipationless.In the presence of dissipation the homogeneous S3T equilib-rium Ue = [HT /2, HT /2] is realizable when ξ < ξc, whereξc is the critical value for instability for the parametersof the flow being studied. Although with ξ < ξc the ho-mogeneous equilibrium state is stable to the traditionalbaroclinic modal instability, it becomes S3T unstable atsome critical forcing amplitude εc. For ε > εc the turbu-lent flow transitions to an inhomogeneous S3T turbulentfixed point state consisting of zonal jets and their associ-ated perturbation field. The formation of finite amplitude

equilibrium jets under homogeneous forcing in this baro-clinically stable regime occurs through a bifurcation similarto that discussed previously in the example of barotropicjet formation. When ξ > ξc the only homogeneous statesare the (unstable) laminar equilibria with ε = 0 and Cek = 0and the flow if perturbed transitions to an inhomogeneousstate, which equilibrates to an inhomogeneous statisticalstate with zonal jet flows3.

We wish to examine the equilibria that are predicted byS3T both in the baroclinically stable and unstable regimesand compare them with the turbulent states in correspond-ing QL and NL simulations. We consider two example cases,one with ξ = 0, a stable case with no temperature differenceacross the channel, and another with ξ = 2 in which casethe temperature gradient is being relaxed to a baroclinicallyunstable shear.

When ξ = 0 the S3T dynamics of the two layer model isfound to be essentially barotropic, similar to the examples insection 1.6, with the difference being that the deformationradius is finite. This case models the atmospheres of theouter planets which have small temperature gradients andare maintained turbulent by injection of energy by smallscale convective excitation. The same stochastic excitationis applied to the S3T and QL and NL simulations by forc-ing the 14 gravest zonal wavenumbers, k, with the forcingstructure Fk chosen so that the (i, j) element of the exci-tation is proportional to exp[−(yi − yj)2/(2s2)], as in (30),with s = 1/

√2. The total energy input by this excitation

is 2.5 Wm−2 distributed equally among the excited zonalcomponents. The domain is a doubly periodic channel withLx = Ly = 40. Dissipation parameters for the NL, QL andS3T simulations are r = rT = 0, rp = 1/5 and rm = 1/2000.An example of emergence of zonal flows in this system canbe seen in Fig. 7 in NL, QL and S3T and a comparisonof the corresponding zonal mean flows is shown in Fig. 8.S3T predicts that for these parameters the statistics of theturbulent flow are attracted to an equilibrium with a jet andassociated consistent eddy structure. The equilibrium meanflow is barotropic and barotropically stable with maximumgrowth rates plotted as a function of k in Fig. 8. Boththe NL and QL simulations reflect the predictions of S3Tand the equilibrated flow has the characteristic complexstructure of the 23 N jet on Jupiter with the sharp pro-grade jets and the rounded retrograde jets (Sanchez-Lavegaet al. 2008; Farrell and Ioannou 2008). The structure of theretrograde jets is not quite consistent with potential vortic-ity homogenization (Dritschel and McIntyre 2008), as thepotential vorticity gradient of the equilibrium flow in Fig. 8is slightly negative in limited locations of the retrograde jets

3In this regime the S3T self-sustains in the absence of forcing.In the absence of forcing the S3T reduces to the corresponding QLsimulation. NL self-sustains when ξ > ξc, but we note that self-sustaining turbulent states have been found for subcritical parametersclose to criticality (Lee and Held 1991).

15

NL upper layer U (y , t ) , ξ = 0

t (day)

y

1000 2000 3000 4000 5000 60000

10

20

30

S3T upper layer U (y , t ) , ξ = 0

t (day)

y

1000 2000 3000 4000 5000 60000

10

20

30

−10

0

10

−10

0

10

QL upper layer U (y , t ) , ξ = 0

t (day)

y

1000 2000 3000 4000 5000 60000

10

20

30

−10

0

10

Figure 7: Hovmoller diagrams of jet emergence in NL, QLand S3T simulations under Jovian conditions with ξ = 0.Shown for the NL, QL and S3T simulations are the flow,U1(y, t), in the upper layer (the flow in the lower layer is notshown as the flow is almost barotropic). After a series ofmergers S3T is attracted to a statistical steady state with an = 2 jet. The whole process of jet mergers and S3T equili-bration is accurately reflected in the QL and NL simulations.This figure shows that S3T predicts the structure, growth andequilibration of strongly forced jets in both the QL and NLsimulations. The dissipation parameters in all simulations arer = rT = 0, rp = 1/5 and rm = 1/2000. The channel hasLx = Ly = 40 and global zonal wavenumber k = 1, . . . , 14are equally excited in energy. The total energy input by thestochastic excitation is 2.5 W m−2.

which is indicative of dynamical processes beyond mixing(cf. also Fig. 3 in Farrell and Ioannou (2008)).

Consider now the baroclinically unstable case, ξ = 2,which represents midlatitude Earth like conditions. Weagain choose a doubly periodic channel with Lx = 40 andLy = 20 and impose dissipation parameters r = 1/10,rT = 1/20 and rp = rm = 1/5. For these dissipation pa-rameters the shear HT is baroclinically unstable (cf. a plotof the maximum growth rate of this flow as a function ofzonal wavenumber is shown in Fig. 10c) and a self-sustainedturbulent state develops in NL, QL and S3T. In the absenceof excitation the NL simulation equilibrates to a 3 jet struc-ture as shown in the top panel of Fig. 9 with associatedjet structure shown in Fig. 10a. To obtain correspondencewith the NL we parameterize the eddy interactions in theQL and S3T as in DelSole and Farrell (1996), using a state-independent stochastic forcing, with the same structure as

0 5 10 15 20 25 30 35 40−20

−15

−10

−5

0

5

10

15

20

25

y

U

0 0.5 1 1.5 2 2.5−0.2

−0.1

0

k

kci

Figure 8: Comparison of the S3T equilibrium jet in the upperlayer for ξ = 0 (solid blue) with instantaneous realizations ofthe jet in corresponding NL (solid black) and QL (solid red)simulations. The jet is barotropic and the jet in the lowerlayer is not shown. The growth rate of the least dampedeigenmode of the S3T equilibrium jet is shown as function ofzonal wavenumber in the lower panel. The circles in this plotindicate the growth rate for each of the 14 harmonics retainedin the S3T simulation. The parameters are as in Fig. 7.

that used for ξ = 0. A total injection rate of 0.5 Wm−2 inthe 14 gravest zonal components and diffusion with ν = 0.02bring the S3T and the QL simulations in good agreementwith the NL. The agreement in the emergence of the jetsand in the maintained flows are shown in Fig. 9 and Fig. 10.S3T predicts that for these parameters the statistics of theturbulent flow are attracted to an equilibrium with a jet andassociated eddy structure. The mean flow is by necessitystable with growth rates shown in Fig. 10b. To avoid insta-bility the jets become increasingly east/west asymmetricas criticality is increased with the eastward-portion equili-brating by zonal confinement (Ioannou and Lindzen 1986;James 1987; Roe and Lindzen 1996), and the westwardjets equilibrating more barotropically approximately closeto the Rayleigh-Kuo stability boundary while maintainingsubstantial baroclinicity.

A characteristic of the equilibrated jets in all cases is thatwhile stable, they are highly non-normal and support strongtransient growth. This non-normal growth is succinctlysummarized in Fig. 11 (a,b) by comparison between theFrobenius norm of the resolvent of the jet perturbationdynamics ∥∥R(ω)

∥∥2F

=∑k

trace(Rk(ω)Rk(ω)†

), (47)

whereRk(ω) = − (iωI + Ak(Ue))

−1, (48)

16

t (day)

NL upper layer U (y , t ) , ξ = 2y

100 200 300 400 500 6000

5

10

15

y

S3T upper layer U (y , t ) , ξ = 2

t (day)0 100 200 300 400 500 600

0

5

10

15

QL upper layer U (y , t ) , ξ = 2

y

t (day)100 200 300 400 500 600

0

5

10

15

−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

−3

−2

−1

0

1

2

3

Figure 9: Hovmoller diagrams of jet emergence in NL, QLand S3T simulations under Earth-like conditions with super-critical ξ = 2. Shown for the NL, QL and S3T simulations arethe flow, U1(y, t), in the upper layer. The flow has substantialbaroclinicity and mean flow in both layers is shown in Fig. 10.This figure shows that S3T predicts the structure, growthand equilibration of strongly forced jets in both the QL andNL simulations of self-sustained baroclinic turbulence. Thedissipation parameters are r = 1/10, rT = 1/20, rp = 1/5and rp = 1/5 and the channel size is Lx = 40 and Ly = 20.In the QL and S3T simulations global zonal wavenumberk = 1, . . . , 14 are stochastically excited equally in energywith a total energy injection of 0.5 W m−2. In QL and S3Tdiffusive damping is included with ν = 0.02.

and its equivalent normal counterpart, with resolvent thediagonal matrix, Sk, of the eigenvalues of Ak:

R⊥k (ω) = − (iωI + Sk)−1

. (49)

The square Frobenius norm of the resolvent shown as afunction of frequency, ω, in Fig. 11 is the ensemble meaneddy streamfunction variance, 〈|ψ|2 + |θ|2〉, that wouldbe maintained by white noise forcing of this equilibriumjet. Non-normality increases the maintained streamfunctionvariance 〈|ψ|2 + |θ|2〉, over that of the equivalent normalsystem 〈|ψ⊥|2 + |θ⊥|2〉 and the extent of this increase is ameasure of the non-normality (Farrell and Ioannou 1994;Ioannou 1995). The non-normality of the equilibria, whichis associated with both baroclinic and barotropic growthprocesses, increases as ξ increases as shown in Fig. 11.

The maintained eddy streamfunction variance, 〈|ψ|2〉, inthe equilibrium jet and for comparison the eddy streamfunc-

0 2 4 6 8 10 12 14 16 18−1

0

1

2

3

4

y

U1,U

2

0 0.5 1 1.5 2 2.5−0.4

−0.2

0

0.2

0.4

k

kci

(a)

(b)

Figure 10: (a) Upper and lower layer flow in S3T (blue), QL(red) and NL (black) for supercritical equilibria with ξ = 2.(b) Growth rate as a function of zonal wavenumber k (bluecurve) for the S3T equilibrated flow shown in (a) and in blackthe corresponding growth rate of the meridionally uniformbaroclinic flow with ξ = 2 with rT = 1/15 and r = 1/5. Withthese dissipation parameters the flow becomes unstable forξ > ξc = 0.875. In the absence of dissipation the shortwavecut-off occurs at k = λ2 = 2. The circles indicate the growthrate of the k associated with the 14 harmonics used in theS3T calculations. The parameters are as in Fig. 9.

tion variance, of the equivalent normal jet system, 〈|ψ⊥|2〉,as a function of the criticality measure of the equilibratedflow ξ is shown Fig. 12. The eddy variance maintainedby the equilibrium jet increases as ξ4 while the equivalentnormal system eddy variance, 〈|ψ⊥|2〉, does not increaseappreciably. This increase in variance with criticality is dueto the increase in the non-normality of the equilibrated jet.The heat flux, which is proportional to 〈θ∂xψ〉, exhibits aξ7 power law behavior implying an equivalent higher orderthermal diffusion. While such power law behavior is recog-nized to be generic to strongly turbulent equilibria (Heldand Larichev 1996; Barry et al. 2002; Zurita-Gotor 2007), itlacked comprehensive explanation in the absence of the SSD-based theory for the structure of the statistical equilibriumturbulent state and its concomitant non-normality that isprovided by S3T. Coexistence of very high non-normalitywith modal stability has been heretofore regarded as highlyunlikely to occur naturally and such systems have beengenerally thought to result from engineering contrivance.In fact the goal of much of classical control theory is to sup-press modal instability by designing stabilizing feedbacks.That this process of modal stability suppression sometimesresulted in modally stable systems that were at the sametime highly vulnerable to disruption due to large transientgrowth of perturbation led to the more recent developmentof robust control theory which seeks to control transientgrowth associated with non-normality as well as modalstability (Doyle et al. 2009). A widely accepted argument

17

−4 −2 0 2 4 610

2

103

104

105

ω(day−1)

||R(ω

)||2 F

−50 −25. 0 25 50

101

102

103

104

ω(day−1)

||R(ω

)||2 F

(b)(a)ξ = 2ξ = 0

Figure 11: Panel (a): the Frobenius norm of the resolventassociated with the eddy dynamics about the equilibriummean flow of Fig. 8 that obtains at ξ = 0 as a function offrequency and the Frobenius norm of the resolvent of thecorresponding equivalent normal eddy dynamics (dashed).Similarly in (b) for the equilibrium flow shown in Fig. 10 forξ = 2.

for the necessity of engineering intervention to produce asystem that is at the same time modally stable and highlynon-normal follows from the observation that in the limitof increasing system non-normality, arbitrarily small per-turbations to the dynamics result in modal destabilization.If the dynamics is expressed using a dynamical matrix, aswe have done here, this vulnerability of the non-normaldynamics to modal destabilization finds expression in thepseudo-spectrum of the dynamical matrix (Trefethen andEmbree 2005). It is remarkable that the naturally occurringfeedback between the zonal mean flow and the perturbationsin turbulent systems stabilizes these systems.

In summary, we have provided an explanation for theobservation that adjustment to stable but highly amplify-ing states exhibiting power law behavior for flux/gradientrelations is characteristic of strongly supercritical baroclinicturbulence. One consequence of stability coexisting withhigh non-normality in the Earth’s midlatitude atmosphereis the association of cyclone formation with chance occur-rence in the turbulence of optimal or near optimal initialconditions (Farrell 1982, 1989; Farrell and Ioannou 1993a;DelSole 2004a). We have explained why a state of high non-normality together with marginal stability is inherent: it isbecause the SSD of the turbulence maintains flow stabilityby adjusting the system to be in the vicinity of a specificstability boundary, which is identified with the fixed pointof the S3T equilibrium, while retaining the high degreeof non-normality of the system. Such a state of extremenon-normality coexisting with exponential stability is anemergent consequence of the underlying SSD of baroclinicturbulence and can only result from the feedback controlmechanism operating in baroclinic turbulence which hasbeen identified by SSD analysis using S3T.

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

104

105

ξ

〈|ψ|2〉

,〈θ

∂xψ〉

,〈|ψ

⊥|2〉

ξ 4

ξ 7

Figure 12: For the equilibrated jets shown are: the eddybarotropic streamfunction variance 〈|ψ|2〉 (solid line), theeddy heat flux 〈θ∂xψ〉 (dashed line), and the eddy barotropicstreamfunction variance maintained by the equivalent normalsystem 〈|ψ⊥|2〉 (dash-dot line) as a function of the criticalityparameter ξ. The eddy barotropic streamfunction varianceincreases as ξ4, the heat flux increases as ξ7, while the equiv-alent normal variance is nearly constant. The dissipationparameters are r = 1/10, rT = 1/20, rp = 1/5, rm = 1/5,ν = 0 and the channel size is Lx = 40 and Ly = 20. The 14gravest zonal components are excited.

8. Application of S3T to study the SSD of wall-bounded shear flow turbulence

Consider plane Couette flow between walls with veloci-ties ±Uw. The streamwise direction is x, the wall-normaldirection is y, and the spanwise direction is z. Lengths arenon-dimensionalized by the channel half-width, δ, and ve-locities by Uw, so that the Reynolds number is R = Uwδ/ν,with ν the coefficient of kinematic viscosity. We take forour example a doubly periodic channel of non-dimensionallength Lx in the streamwise direction and Lz in the span-wise (note we have adopted the wall-bounded turbulenceconvention for the vertical coordinate which differs fromthe meteorological convention).

In the absence of turbulence the equilibrium solution isthe laminar Couette flow with velocity components (y, 0, 0)which has been shown to be linearly stable at all Reynoldsnumbers (Romanov 1973) and globally stable for R < 20.7(Joseph 1966). However, experiments show that plane Cou-ette flow can be induced by a perturbation to transition toa turbulent state for Reynolds numbers exceeding R ≈ 360(Tillmark and Alfredsson 1992). During transition to tur-bulence and in the turbulent state a prominent large scalestructure is observed in the flow. This structure, referredto as the roll/streak, comprises a modulation of the stream-wise mean flow in the spanwise direction by regions of highand low velocity, referred to as streaks, together with a setof nearly cylindrical vortices in the wall-normal/spanwiseplane, referred to as rolls. The roll circulation is such that

18

the maximum negative wall-normal velocity is coincidentwith the maximum positive streamwise velocity of the streakand the maximum wall-normal velocity with the minimumstreak velocity. This roll circulation serves to amplify thestreak by advecting the mean shear; a mechanism referredto as lift-up. The streak is analogous to the jet that developsin barotropic and baroclinic flows and roll-streak structurearises naturally from S3T instability of the spanwise ho-mogeneous shear turbulence. To show this we formulatethe S3T dynamics for this flow and demonstrate that theS3T homogeneous turbulent equilibria become unstablewith the eigenfunctions of maximum growth rate being theroll-streak structures.

Consider the vector velocity field ~U to be decomposedinto a streamwise mean, with components, (U, V,W ), andperturbation from this mean with components (u, v, w).The pressure gradient is similarly decomposed into itsstreamwise mean, ∇P , and perturbation from this mean,∇p. All streamwise averaged quantities are denoted withcapitals and the streamwise averaging operation is denotedby an overline. In these variables a unit density fluid obeysthe non-divergent Navier-Stokes equations:

~ut + ~U · ∇~u + ~u · ∇~U +∇p−∆~u/R =

= −(~u · ∇~u − ~u · ∇~u) + ~e , (50a)

~Ut + ~U · ∇~U +∇P −∆~U /R = −~u · ∇~u , (50b)

∇ · ~U = 0 , ∇ · ~u = 0 , (50c)

with boundary conditions ~u = 0 and ~U = (±1, 0, 0) aty = ±1 and periodicity in x and z.

In the perturbation equation (50a) allowance is made forspecifying an explicit external perturbation forcing, ~e. Astochastic parameterization is now introduced to account forboth this external forcing and the perturbation-perturbationinteractions, ~u · ∇~u − ~u · ∇~u. With this parameterization,the perturbation equation becomes:

~ut + ~U · ∇~u + ~u · ∇~U +∇p−∆~u/R = ~E . (51)

Perturbation equation (51), coupled with the mean flowequation, (50b), form the quasi-linear (QL) form of theNavier-Stokes equations. This approximate set of equationsis also referred to as the restricted nonlinear (RNL) system(Thomas et al. 2014).

It is convenient to eliminate the pressure and expressequation (51) in terms of cross-stream velocity v and cross-stream vorticity, η = ∂zu− ∂xw. The equations then takethe form:

∆vt + U∆vx + Uzzvx + 2Uzvxz − Uyyvx − 2Uzwxy−− 2Uyzwx −∆∆v/R = ∆Ev , (52a)

ηt + Uηx − Uzvy + Uyzv + Uyvz + Uzzw −∆η/R = Eη .(52b)

z

y

dU = cos( / y / 2) sin( 2 pi z / Lz)

ï1.5 ï1 ï0.5 0 0.5 1 1.5ï1

ï0.8

ï0.6

ï0.4

ï0.2

0

0.2

0.4

0.6

0.8

1

z

y

dU = cos( / y / 2) sin( 4 pi z / Lz)

ï1.5 ï1 ï0.5 0 0.5 1 1.5ï1

ï0.8

ï0.6

ï0.4

ï0.2

0

0.2

0.4

0.6

0.8

1

z

y

dU = cos( 3 / y / 2) sin( 4 pi z / Lz)

ï1.5 ï1 ï0.5 0 0.5 1 1.5ï1

ï0.8

ï0.6

ï0.4

ï0.2

0

0.2

0.4

0.6

0.8

1

z

y

dU = cos( 3 / y / 2) sin( 6 pi z / Lz)

ï1.5 ï1 ï0.5 0 0.5 1 1.5ï1

ï0.8

ï0.6

ï0.4

ï0.2

0

0.2

0.4

0.6

0.8

1

Figure 13: The rate of change of streamwise roll accelerationinduced by a streak perturbation to a Couette flow that ismaintained turbulent by stochastic forcing. Distortion ofthe turbulence by the streak perturbation induces Reynoldsstresses that force roll circulations supporting the streak viathe lift-up mechanism. Shown are contours of the imposedstreak perturbations, δU = cos(πy/2) sin(2πz/Lz), with δU >0 in z > 0, and vectors of the resulting rate of change of rollacceleration, (V , W ). The Reynolds number is R = 400,Lx = 1.75π and Lz = 1.2π. Adapted from “Dynamics ofstreamwise rolls and streaks in turbulent wall-bounded shearflow”, by B. F. Farrell and P. J. Ioannou, 2012, Journalof Fluid Mechanics, vol. 708, pp. 149-196. c© CambridgeUniversity Press. Reprinted with permission.

where Ev and Eη are the stochastic excitation in thesevariables (cf. Schmid and Henningson (2001)). In theperturbation equations (52), advection of perturbations bythe small V and W components of the streamwise meanvelocity has been neglected4. Using nondivergence the meanflow equation (50b) can be written as:

Ut = UyΨz − UzΨy − ∂yuv − ∂zuw + ∆1U/R , (53a)

∆1Ψt = (∂yy − ∂zz)(ΨyΨz − vw)−− ∂yz(Ψ2

y −Ψ2z + w2 − v2) + ∆1∆1Ψ/R .

(53b)

In (53b), ∆1 ≡ ∂2y+∂2zz and V andW have been expressed interms of the streamfunction, Ψ, as V = −Ψz and W = Ψy.

We next Fourier expand the perturbation fields in x: v =<[∑

k vk(y, z, t)eikx], η = <

[∑k ηk(y, z, t)eikx

], and write

the equations for the evolution of the Fourier componentsof (52) in the matrix form

dφkdt

= Ak(U)φk +√εFkdBtk , (54)

where the state of the system φk = [vk, ηk]T comprises thevalues of the vk and ηk on the N = NyNz grid points of

4The results presented are not affected by neglecting the advectionof the perturbation field by V and W velocities in the perturbationequations, cf. Thomas et al. (2014).

19

z

δU

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.5

0

0.5

1

(δV, δW)

z

y

−1

−0.5

0

0.5

1

Figure 14: The most unstable streamwise roll and streakeigenfunction of the S3T system linearized about the span-wise uniform equilibrium at supercriticality ε/εc = 1.4. Thegrowth rate of this mode is λr = 0.014. Shown are velocityvectors (δV, δW ) (left) and streamwise velocity δU (right).The ratio of the maxima of (δU, δV, δW ) is (1, 0.06, 0.03).Other parameters are as in Fig. 13. Adapted from “Dynamicsof streamwise rolls and streaks in turbulent wall-boundedshear flow”, by B. F. Farrell and P. J. Ioannou, 2012, Journalof Fluid Mechanics, vol. 708, pp. 149-196. c© CambridgeUniversity Press. Reprinted with permission.

the (y, z) plane and

Ak(U) =

(LOS LC1

LC2 LSQ

), (55)

with

LOS = ∆−1[−ikU∆ + ik(Uyy − Uzz)− 2ikUz∂z −

−2ik(Uz∂3yyz + Uyz∂

2yz)∆

−12 + ∆∆/R

], (56a)

LC1= 2k2∆−1 (Uz∂y + Uyz) ∆−12 , (56b)

LC2 = Uz∂y − Uy∂z − Uyz + Uzz∂2yz∆

−12 , (56c)

LSQ = −ikU∆ + ikUzz∆−12 + ∆/R , (56d)

being the conventionally designated Orr-Somerfeld, cou-pling, and Squire operators respectively. In equations (56),∆−1 and ∆−12 are the inverses of the matrix Laplacians,∆ and ∆2 = ∂2x + ∂2z , which are rendered invertible by en-forcing the boundary conditions. The boundary conditionssatisfied by the Fourier amplitudes of the perturbation fieldsare: periodicity in x and z and vk = ∂y vk = ηk = 0 aty = ±1.

The ensemble average perturbation covariance, Ck =〈φkφ†k〉, evolves according to the time-dependent Lyapunovequation:

dCkdt

= Ak(U)Ck + Ck A†k(U) + εQk , (57)

in which: Qk = FkF†k. If then, as previously, we make

the ergodic assumption that streamwise averages are equalto ensemble averages all the quadratic fluxes that enterinto the streamwise averaged flow equations (53) becomelinear functions of the Ck and the mean flow evolutionequations (53) can be written concisely in the form:

dΓ

dt= G(Γ) +

∑k

< (LRSCk) , (58)

z

y

−1.5 −1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure 15: The finite amplitude S3T equilibrium stream-wise roll and streak resulting from the equilibration ofthe eigenmode shown in Fig. 14 at supercriticality ε/εc =1.4. Shown are the streamwise averaged streamwise flow,U(y, z), (contours) and the streamwise averaged velocities,(V,W ) (vectors). The maxima of the fields (U, V,W ) are(0.26, 0.02, 0.009). Adapted from “Dynamics of streamwiserolls and streaks in turbulent wall-bounded shear flow”, by B.F. Farrell and P. J. Ioannou, 2012, Journal of Fluid Mechan-ics, vol. 708, pp. 149-196. c© Cambridge University Press.Reprinted with permission.

where Γ ≡ [U,Ψ]T determines the three components of thestreamwise averaged flow, the term

∑k < (LRSCk) produces

by multiplying Ck with matrix LRS the forcing of the meanequations by the perturbation field and G(Γ) is the nonlin-ear term representing the self-advection of the streamwiseaveraged flow. Equations (57) and (58) comprise the S3Tsystem for the Couette problem. The forcing covariances,Qk, are chosen to be spanwise homogeneous. Under thisassumption spanwise homogeneous S3T equilibrium statesexist. For further details on the formulation see Farrell andIoannou (2012).

The Couette flow is a laminar equilibrium of the S3Tsystem with excitation ε = 0. For any ε and any spanwisehomogeneous Qk there are always spanwise independentS3T equilibria having spanwise independent streamwiseaveraged flow Ue(y) and Ψe = 0 and spanwise homogeneousperturbation covariances Cek. These are equilibria because,consistent with the spanwise independence of both theequilibrium mean flow and the imposed excitation, Cek is alsospanwise independent and this results in ensemble mean uw,v2 and w2 that are independent of z, and vw that identicallyvanishes by symmetry. Consequently, (53b) admits Ψe = 0as a solution as the ensemble mean perturbation forcingvanishes. However, the ensemble mean Reynolds stressdivergence ∂yuv in (53a) does not vanish and Ue(y) satisfies∆1U

e/R = ∂yuv indicating that the presence of externalexcitation induces a modification to the laminar Couetteprofile.

20

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

0.05

0.1

0.15

0.2

0.25

0.3

ǫ/ǫ c

r.m

.s.velocity

ǫu/ǫ c

r .m.s . streak ve loc ity

10 × r .m.s . rol l ve loc ity

Figure 16: Typical S3T bifurcation diagram for the Couetteproblem. Shown are the RMS streak velocity (solid) and 10×the RMS streamwise roll velocity (dashed) as a function of theperturbation forcing amplitude, ε. For ε/εc < 1, the spanwisehomogeneous state is S3T stable. At εc the spanwise uniformequilibrium bifurcates to an equilibrium with a streamwise rolland streak. Stable streamwise roll and streak equilibria extendup to εu/εc = 2.1 beyond which the streamwise roll and streaktransitions to a time-dependent state which can self-sustainand the amplitudes of the roll and streak become independentof ε. Shown for reference are the r.m.s. velocities of the streakand roll in the self-sustaining state. The Reynolds number isR = 400, Lx = 1.75π and Lz = 1.2π.

In analogy with the test function probe used to eluci-date the mechanism underlying jet formation in barotropicflow (cf. section a) we wish to determine the effect on thespanwise homogeneous field of turbulence of an infinites-imal spanwise-dependent mean-flow streak5 perturbationδUs(y, z) added to the equilibrium flow, Ue(y). We areparticularly interested to determine if distortion of the tur-bulence by the perturbation streak results in a positivefeedback on the perturbation streak, δUs(y, z). We deter-mine this feedback by calculating the change δCk in Cekresulting from the streak perturbation δUs(y, z) as in thebarotropic example. The divergence of the ensemble aver-aged perturbation Reynolds stresses resulting from this δCkproduce a torque in the y − z plane inducing a circulation,according to equation (53b), with streamfunction:

∂tδΨ = ∆−11

[−(∂yy − ∂zz) δ vw − ∂yz (δ w2 − δ v2 )

].

(59)An example streak perturbations, δUs, together with vec-tors of the induced streamwise roll circulation from (59),is shown in Fig. 13. Remarkably, this streak perturbationinduces a distortion of the perturbation field resulting in astreamwise roll forcing configured to amplify the imposedstreak perturbation through the lift-up mechanism i.e. pos-itive wall-normal velocity is collocated with the minimumof the streak and maximum negative wall-normal velocity

5The streak component, Us, is in general defined as the departureof the streamwise averaged flow U from its spanwise average [U ], i.e.Us = U − [U ].

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y

[U ](y)

Figure 17: Comparison of the spanwise and time averagedstreamwise flow, [U ](y), for the self-sustaining state (solid)with the mean flow obtained from a 128 × 65 × 128 directnumerical simulation (DNS) of Couette turbulence at R =1000 in a doubly periodic channel in x and z of length 4πin each direction. The S3T self sustaining turbulent stateproduces on average a friction velocity based Reynolds numberof Rτ = 64.9 while the DNS simulation has Rτ = 66.2.This Reynolds number indicates the turbulent productionand dissipation and is defined as Rτ = uτ δ/ν with δ the

channel half-width and uτ =√ν d[U ]/dy|y=1 the friction

velocity. This figure demonstrates that the S3T self-sustainingstate produces a streamwise averaged flow profile consistentwith simulations of Couette flow turbulence. (Courtesy of V.Thomas)

is collocated with the maximum of the streak. As a resultthis streak perturbation, when imposed on the initially ho-mogeneous field of turbulence, induces Reynolds stressesdriving a roll circulation producing through lift-up growth ofthe imposed streak. This robust Reynolds stress mediateddestabilizing feedback process operating on the streamwisestreak and roll structure has important implications forboth the transition to and the maintenance of turbulence inshear flows. We will show below that even when the streakstructure is highly complex and time-dependent, as in a tur-bulent shear flow, the streamwise roll forcing produced bythe perturbation Reynolds stresses remains collocated so asto amplify the streak. Moreover, this property of imposedstreaks to induce, through modification of the perturbationfield, streamwise roll forcing configured to reinforce the im-posed streak provides the mechanism for a streamwise rolland streak plus turbulence cooperative instability in shearflow. This emergent instability is especially interesting be-cause wall-bounded flows have laminar and turbulent meanvelocity profiles with negative curvature and as a result donot support fast inflectional laminar flow instability as amechanism for robustly transferring energy from the meanflow to the perturbation field as is required in order to main-tain the turbulent state. While most streak perturbationsorganize turbulent Reynolds stresses that do not exactly

21

0 200 400 600 800 1000 1200 1400 1600

0

0.05

0.1

0.15

t

d (

∫ Ω

2 x d

y d

z /(

2 L

y L

z)) /

dt

−50 0 50

−20

0

20

z+

y+

z+

y+

−50 0 50

−20

0

20

Figure 18: Streamwise roll forcing by perturbation Reynoldsstresses in the self-sustaining state with ε = 0. Top left: Vec-tors of instantaneous cross-stream/spanwise velocity accelera-tion, (V , W ), at time t = 980. Top right: Streamwise roll andstreak structure at the same time. Lengths are measured inwall units, y+ = Rτy and z+ = Rτ z. Bottom: Time series ofstreamwise roll forcing as indicated by the rate of change ofthe average square streamwise vorticity. It is remarkable thatthe perturbations, in this highly time-dependent state, actto maintain the roll circulation produce, not only on average,but also at nearly every instant. Adapted from “Dynamicsof streamwise rolls and streaks in turbulent wall-boundedshear flow”, by B. F. Farrell and P. J. Ioannou, 2012, Journalof Fluid Mechanics, vol. 708, pp. 149-196. c© CambridgeUniversity Press. Reprinted with permission.

amplify the streak that produced them, as is clear in thecase of the streak perturbation in Fig. 13, if a streak wereto organize precisely the perturbation field required for itsamplification then exponential modal growth of this streakand its associated streamwise roll and perturbation fieldswould result.

We determine now the S3T stability of the spanwisehomogeneous equilibrium as a function of excitation am-plitude, ε, at a fixed Reynolds number, R, and show thatexponentially unstable streamwise roll and streak modesarise in a spanwise independent field of forced turbulenceif the perturbation forcing amplitude exceeds a threshold.The spanwise independent equilibrium is stable for ε < εc.At εc it becomes structurally unstable, while remaininghydrodynamically stable. The most unstable eigenfunction,which is shown in Fig. 14, consists of a roll circulation witha perfectly collocated streak. When this eigenfunction isintroduced into the S3T system with small amplitude, itgrows at first exponentially at the rate predicted by itseigenvalue and then asymptotically equilibrates at finiteamplitude. This equilibrium solution, shown in Fig. 15,is a steady, finite amplitude streamwise roll and streak.The bifurcation diagram of the S3T equilibria is shown inFig. 16 as a function of bifurcation parameter ε. The finiteamplitude streamwise roll and streak equilibria are S3T

stable for εc ≤ ε ≤ εu.At εu there is a second bifurcation in which the equilib-

rium becomes S3T unstable, while remaining hydrodynami-cally stable, and the SSD fails to equilibrate, instead tran-sitioning directly to a time-dependent state. Remarkably,the time-dependent S3T state that emerges for ε > εu self-sustains even when ε is set to 0. This S3T self-sustainingtime-dependent state produces realistic turbulence withmean turbulent profile [U ] shown in Fig. 17. Moreover,comparison with direct numerical simulations (DNS) ver-ifies that this S3T turbulence is similar to Navier-Stokesturbulence despite the greatly simplified S3T dynamics un-derlying it (Farrell et al. 2012; Constantinou et al. 2014b;Thomas et al. 2014).

Remarkably, the S3T self-sustaining state naturally sim-plifies further by evolving to a minimal turbulent system inwhich the dynamics is supported by the interaction of theroll-streak structures with a perturbation field comprising asmall number of streamwise harmonics (as few as 1). Thisminimal self-sustaining turbulent system, which proceedsnaturally from the S3T dynamics, reveals an underlyingself-sustaining process (SSP) which can be understood withclarity. The basic ingredient of this SSP is the robust ten-dency for streaks to organize the perturbation field so asto produce Reynolds stresses supporting the streak, via thelift-up mechanism as illustrated in Fig. 13. Although thestreak is strongly fluctuating in the self-sustaining state,the tendency of the streak to organize the perturbationfield is retained as illustrated in Fig. 18 in which a snapshotof the streamwise roll and streak is shown together withthe associated roll acceleration, (V , W ), arising from theperturbation Reynolds stresses (cf. equation (59)). Thetime derivative of the integral square streamwise vorticity,d/dt(

∫dy dz Ω2

x) with Ωx = Wy−Vz, provides a measure ofthe torque produced by the Reynolds stress divergences thatsupport the roll circulation. A times series of this diagnosticis also shown in Fig. 18. It is remarkable that the perturba-tions, in this highly time-dependent state, produce torquesthat maintain the streamwise roll not only on average butat nearly every instant. As a result, in this self-sustainingstate, the streamwise roll is systematically maintained bythe robust organization of perturbation Reynolds stress bythe time-dependent streak that was identified by SSD anal-ysis using the S3T system, while the streak is maintained bythe streamwise roll through the lift-up mechanism. Throughthe resulting time-dependence of the roll-streak structurethe constraint on instability imposed by the absence ofinflectional instability in the mean flow is bypassed and theperturbation field is maintained by parametric growth, thuscompleting the SSP cycle6.

6It has been shown that flows that at each time instant satisfy thenecessary for stability Rayleigh condition can still become exponen-tially unstable if the flow is time dependent (cf. Farrell and Ioannou(1999)).

22

We conclude that the dynamics of turbulence in wall-bounded shear flow can be understood at a fundamentallevel by using SSD and specifically by exploiting the directrelation between wall turbulence and the highly simplifiedS3T turbulence. Among the results obtained is that themechanism of turbulence in wall bounded shear flow is thesame roll/streak/perturbation SSP that has been shown tomaintain S3T turbulence.

9. Discussion

Although turbulence is commonly thought of as beingdefinitional of disorder, the turbulent state in shear flowpossesses an underlying order that is revealed by adoptingstatistical state variables to characterize the dynamics ofthe turbulent state. This fundamental order at the centerof turbulence dynamics has remained incompletely appre-ciated for lack of a conceptual basis as well as of methodsfor analyzing the order underlying statistical states of tur-bulence. In this review we have described an approach tounderstanding emergence of order in turbulence throughadopting the perspective of SSD. From this perspectiveorder is understood to arise in turbulence due to systematiccooperative interaction between large scale structures andthe field of small scale turbulence in which these structuresare embedded. Motivating examples of order emergencearise from considering a field of turbulence of some kind(GFD, MHD, Navier-Stokes) into which a trial perturbationof small amplitude but large scale is introduced. Whenintroduced, such a structure both alters the turbulence andis altered by it. Most such structures are not systematicallymaintained by this interaction. But suppose that we wereto continue trying different large scale perturbations untilwe hit on a perturbation structure that affected the turbu-lence in just such a manner as to produce Reynolds stressesconfigured to amplify this structure without changing itsform. Such a structure would naturally grow spontaneouslyout of the turbulence and would provide an explanationfor the observed emergence of large scale structure in tur-bulent flow. This concept of coherent structure emergencethrough cooperative multi-scale interaction in turbulencetakes analytical form through eigenanalysis of the SSD ofthe turbulence linearized about an equilibrium turbulentstatistical state. Bifurcations occur in association withthese instabilities as parameters of the problem, such as theamplitude of the turbulence excitation rate, the dampingrate of the flow, or the beta parameter, are varied. Ex-tensions of these instabilities into the nonlinear regime ofthe SSD reveal fixed point equilibria that predict the asso-ciated finite amplitude coherent structures e.g. the zonaljets in the case of barotropic and baroclinic turbulence inplanetary atmospheres. Moreover, these finite amplitudeequilibria of the nonlinear SSD have an interpretation thattranscends prediction of jet structure. The statistical state

of the turbulence comprises both the mean flow and theperturbations which have mutually adjusted to produce theequilibrium statistical state so that these equilibria consti-tute a closure of the associated turbulence dynamics. Thesuccess of SSD in predicting the statistical mean state ofturbulence as S3T equilibria in the systems we have dis-cussed argues constructively that the nonlocal in spectralspace interaction between perturbations and large scalemean flows that is retained in S3T captures the physicalmechanism underlying the maintenance of the turbulentstate in these systems. In addition, this closure is deter-ministic so that the physical mechanisms producing theclosure are made available for study through analysis of theSSD underlying them. A straightforward example of suchan insight into closure in turbulence arises in the case ofbarotropic beta-plane turbulence in which the observed jetscale in strongly excited turbulence is linked mechanisti-cally through SSD with Rayleigh’s stability criterion; SSDanalysis makes the further and associated verifiable predic-tion of successive bifurcations to jet structure of smallerwavenumber with increase in turbulence intensity. This ex-ample shows the power of SSD to both predict and providephysical explanation for observed phenomena in turbulentflows.

Adopting the perspective of SSD also provides newconceptual insights into the dynamics of turbulence; anexample of this is the concept of the dynamical trajectoryof a statistical state. Consider a statistical state equilib-rium consisting of a fixed point. An example might be abarotropic jet together with its supporting turbulence. Thisfixed point corresponds to a probability density functionthat is stationary in state space. But just as sample statetrajectories, which are points in phase space, may convergeto a fixed point or follow a time dependent path through thestate space, so also statistical state trajectories may followmore or less intricate paths in phase space, taking theirprobability density function along with them. Interpreta-tion of the dynamics of these statistical state trajectoriesprovides a way to deepen understanding of the dynamicof turbulence. The successive bifurcations with increase inexcitation leading to lower and lower meridional wavenum-ber for the equilibrium jet in the example of beta-planeturbulence mentioned above can be viewed as a trajectoryof the deterministic SSD with the bifurcations arising frominstability of the evolving statistical state. These trajec-tories and instabilities have no analytical counterpart insample state dynamics. A familiar example of limit cyclebehavior of a statistical state trajectory is provided by theQBO of the Earth’s equatorial stratosphere which exhibitsnearly periodic 27 month cycles in phase space. The ana-lytical structure of the associated bifurcation only exists inthe SSD framework (Farrell and Ioannou 2003). Chaoticstatistical state trajectories are also found in jet dynamicsof plasma turbulence (Farrell and Ioannou 2009b). In the

23

case of wall-bounded shear flow turbulence the statisticalstate trajectory is also chaotic and the time dependenceof this statistical state, consisting of the streak structureand the associated perturbations it supports, which resultsfrom their cooperative interaction, is a fundamental com-ponent of the dynamical mechanism underlying the SSPmaintaining the turbulence. This is because the turbulenceis maintained by perturbations that result from parametricinteraction with the streak, the time dependence of whichis in turn maintained by interaction with the perturba-tions. This parametric process underlies energy transferfrom the inflectionally stable forced mean flow that is re-quired to maintain the turbulent perturbation variance inwall-bounded shear flows. This parametric growth processis intrinsically a property of the SSD and this cooperativeparametric process can only be understood through analysisof the dynamics of the statistical state of the turbulence.

Viewing turbulence from the perspective of SSD hasproven to be remarkably tractable and to provide a richnessof analysis and concept that has already allowed progress ina number of areas and holds promise of continuing insightinto the nature of turbulence.

APPENDIX A

The homogeneous equilibrium covariance

We prove that the equilibrium covariance (32) underhomogeneous stochastic excitation of a channel with pe-riodic boundary conditions produces zero mean vorticityfluxes, v′q′ = 0. This is required in order for Ue = 0 tobe an S3T equilibrium with perturbations with covariance(32). We provide a proof that uses technical argumentsthat are useful for exploring the properties of discrete S3Tdynamics and stability in periodic channels (cf. Bakas andIoannou (2011)). We use the property that in the matrixformulation of S3T in a periodic channel, in which thefields have been discretized on a grid, all matrices thatcorrespond to homogeneous continuous operators, as wellas all covariances of homogeneous fields, are circulant, i.e.each row is a cyclic shift of the row above it. They arecirculant because they commute with the matrix of spatialshifts on the grid lattice. Circulant matrices commute witheach other and their common eigenbasis is a unitary matrixconsisting of harmonics. Consequently the forcing covari-ance at wavenumber k is analyzed in Fourier componentsin y as Qkαβ =

∑l Qkle

il(yα−yβ), with l the y wavenum-

ber. Moreover, the Fourier coefficients Qkl must be realand non-negative coefficients in order to assure that Qkαβ ,for each k, is positive definite, Hermitian and a covarianceof a homogeneous field (this statement is the content of

Bochner’s theorem). Because for each k we have

n∑m=1

∆−1kαmQkmβ =∑l

Qkl

n∑m=1

∆−1kαmeil(ym−yβ)

= −∑l

Qkleil(yα−yβ)

k2 + l2

we obtain that the diagonal elements of∑nm=1 ∆−1kαmC

ekmβ

are equal and real and therefore the vorticity flux associatedwith the equilibrium covariance, which is proportional tothe imaginary part of the diagonal elements of this matrix,vanishes.

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