Emergence of non-zonal coherent structures

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by  Nikolaos A. Bakas

Emergence of non-zonal coherent structures
Nikolaos A. Bakas
Department of Physics, University of Ioannina, Ioannina, Greece
Petros J. Ioannou∗
Department of Physics, National and Kapodistrian University of Athens, Athens, Greece
5
1
0 Planetary turbulence is observed to self-organize into large-scale structures such
2
as zonal jets and coherent vortices. One of the simplest models that retains the

n relevant dynamics of turbulent self-organization is a barotropic flow in a beta-plane
a channel with turbulence sustained by random stirring. Non-linear integrations
J
of this model show that as the energy input rate of the forcing is increased, the
1
homogeneity of the flow is first broken by the emergence of non-zonal, coherent,
2
westward propagating structures and at larger energy input rates by the emergence

] of zonal jets. The emergence of both non-zonal coherent structures and zonal jets
h
is studied using a statistical theory, Stochastic Structural Stability Theory (S3T).
p
S3T directly models a second-order approximation to the statistical mean turbulent
-
o state and allows the identification of statistical turbulent equilibria and study of
a their stability. Using S3T, the bifurcation properties of the homogeneous state in
.
s barotropic beta-plane turbulence are determined. Analytic expressions for the zonal
c
andnon-zonallarge-scalecoherentflowsthatemergeasaresultofstructuralinstability
i
s are obtained and the equilibration of the incipient instabilities is studied through
y
numerical integrations of the S3T dynamical system. The dynamics underlying
h
p the formation of zonal jets are also investigated. It is shown that zonal jets form
[ from the upgradient momentum fluxes that result from the shearing of the eddies

by the emerging infinitesimal large-scale flow. Finally, numerical simulations of the
1
v nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of
0 the structures predicted by S3T, even in highly non-linear parameter regimes such as
8 the regime of zonostrophic turbulence.
2
5
0
.
1
0 Atmospheric and oceanic turbulence is commonly ob- et al. 2014) and non-zonal coherent structures (Bakas and
5 served to be organized into spatially and temporally co- Ioannou 2014). This argues that the emergence of coherent
1 herent structures such as zonal jets and coherent vortices. structures in a homogeneous background of turbulence is a
:
v A simple model that retains the relevant dynamics, is a bifurcation phenomenon.
i barotropic flow on a β-plane with turbulence sustained by An advantageous method to study such a phenomenon,
X
random stirring. Numerical simulations of the stochasti- is to adopt the perspective of statistical state dynamics of
r
a callyforcedbarotropicvorticityequationonthesurfaceofa the flow, rather than look into the dynamics of sample real-
rotatingsphereoronaβ-plane,haveshownthecoexistence izations of direct numerical simulations. This amounts to
ofrobustzonaljetsandoflarge-scalewestwardpropagating studythedynamicsandstabilityofthestatisticalequilibria
coherent structures that are referred to as satellite modes arising in the turbulent flow, which are fixed points of the
(Danilov and Gurarie 2004) or zonons (Sukariansky et al. equationsgoverningtheevolutionoftheflowstatistics. This
2008; Galperin et al. 2010). Emergence of these coherent approach is followed in the Stochastic Structural Stability
structuresinbarotropicturbulencehasalsoanotherfeature. Theory (S3T) (Farrell and Ioannou 2003) or Second Order
As the energy input of the stochastic forcing is increased or Cumulant Expansion theory (CE2) (Marston et al. 2008).
dissipation is decreased, there is a sudden onset of coher- This theory is based on two building blocks. The first is
ent zonal flows (Srinivasan and Young 2012; Constantinou to do a Reynolds decomposition of the dynamical variables
1
into the sum of a mean value that represents the coherent this assumption. The second goal is to study in detail the
flow and fluctuations that represent the turbulent eddies eddy-mean flow dynamics underlying the S3T instability
andthenformthecumulantscontainingtheinformationon focusing on the example of jet formation. And the third
the mean values (first cumulant) and on the eddy statistics goal is to compare the characteristics of the structures that
(higher order cumulants). The second building block is emergeinS3Tagainstnon-linearsimulations,eveninhighly
to truncate the equations governing the evolution of the non-linear regimes that at first glance present a challenging
cumulants at second order by either parameterizing the test for the restricted dynamics of S3T.
terms involving the third cumulant (Farrell and Ioannou
1993a,b,c;DelSoleandFarrell1996;DelSole2004)orsetting 1. Formulation of Stochastic Structural Stability
the third cumulant to zero (Marston et al. 2008; Tobias Theory under a generalized average
et al. 2011; Srinivasan and Young 2012). Restriction of
Consider a nondivergent barotropic flow on a β-plane
the dynamics to the first two cumulants is equivalent to
with cartesian coordinates x = (x,y). The velocity field,
neglectingtheeddy-eddyinteractionsinthefullynon-linear
u=(u,v), is given by (u,v)=(−∂ ψ,∂ ψ), where ψ is the
dynamics and retaining only the interaction between the y x
streamfunction. Relative vorticity ζ(x,y,t)=∆ψ, evolves
eddies with the instantaneous mean flow. While such a
according to the non-linear (NL) equation:
second order closure might seem crude at first sight, there
√
is strong evidence to support it (Bouchet et al. 2013). (∂ +u·∇)ζ+βv =−rζ−ν∆2ζ+ εfe, (1)
t
Previous studies employing S3T have already addressed
the bifurcation from a homogeneous turbulent regime to where ∆=∂2 +∂2 is the horizontal Laplacian, β is the
xx yy
a jet forming regime in barotropic β-plane turbulence and gradient of planetary vorticity, r is the coefficient of linear
identifiedtheemergingjetstructuresbothnumerically(Far- dissipationthattypicallyparameterizesEkmandraginplan-
rellandIoannou2007)andanalytically(BakasandIoannou etaryatmospheresandν isthecoefficientofhyper-diffusion
2011;SrinivasanandYoung2012)aslinearlyunstablemodes that dissipates enstrophy flowing into unresolved scales.
tothehomogeneousturbulentstateequilibrium. Itwasalso The exogenous forcing term fe, parameterizes processes
shownthattheresultingdynamicalsystemfortheevolution such as small scale convection or baroclinic instability, that
of the first two cumulants linearized around the homoge- are missing from the barotropic dynamics and is necessary
neous equilibrium possesses the mathematical structure of to sustain turbulence. We assume that fe is a temporally
the dynamical system of pattern formation (Parker and delta correlated and spatially homogeneous random stir-
Krommes 2013). Comparison of the results of the stability ring injecting energy at a rate ε and having a two-point,
analysis with direct numerical simulations have shown that two-time correlation function of the form:
the structure of zonal flows that emerge in the non-linear
simulationscanbepredictedbyS3T(SrinivasanandYoung (cid:104)fe(x ,y ,t )fe(x ,y ,t )(cid:105)=δ(t −t )Ξ(x ,x ,y ,y ),
1 1 1 2 2 2 2 1 1 2 1 2
2012; Constantinou et al. 2014). However, these research (2)
efforts, have assumed that the ensemble average employed where the brackets denote an ensemble average over the
inS3Tisequivalenttoazonalaverage,asimplificationthat different realizations of the forcing.
treats the non-zonal structures as incoherent and cannot S3T describes the statistical dynamics of the first two
address their emergence and effect on the jet dynamics. In same time cumulants of (1). The equations governing the
addition, the eddy-mean flow dynamics underlying the S3T evolution of the first two cumulants are obtained as follows.
instability even in the jet formation case, that involve only We decompose the vorticity field into the averaged field,
the interactions of small scale waves with the large-scale Z =T[ζ], defined as a time average over an intermediate
coherent structures are not clear. time scale and deviations from the mean or eddies, ζ(cid:48) =
So the goals in this article are the following. The first ζ−Z. The intermediate time scale is larger than the time
goal is to adopt a more general interpretation of the ensem- scale of the turbulent motions but smaller than the time
ble average, in order to address the emergence of coherent scale of the large scale motions. With this decomposition
non-zonal structures. We adopt the more general inter- we rewrite (1) as:
pretation that the ensemble average is a Reynolds average
over the fast turbulent motions (Bernstein 2009; Bernstein (∂t+U·∇)Z+βV =−∇·T[u(cid:48)ζ(cid:48)]−rZ−ν∆2Z, (3)
and Farrell 2010). With this definition of the ensemble
where U = [U,V] = [−∂ Ψ,∂ Ψ] and u(cid:48) = [u(cid:48),v(cid:48)] =
mean, we obtain the statistical dynamics of the interaction y x
[−∂ ψ(cid:48),∂ ψ(cid:48)] are the mean and the eddy velocity fields
of the coarse-grained ensemble average field, which can y x
respectively. The mean vorticity is therefore forced by the
be zonal or non-zonal coherent structures represented by
divergence of the mean vorticity fluxes. The eddy vorticity
their vorticity, with the fine-grained incoherent field repre-
sented by the vorticity second cumulant and we revisit the
structural stability of the homogeneous equilibrium under
2
ζ(cid:48) evolves according to: where
(∂t+U·∇)ζ(cid:48)+(β+∂yZ)v(cid:48)+u(cid:48)∂xZ = A =−U ·∇ −(β+∂ Z)∂ ∆−1+∂ Z∂ ∆−1−r−ν∆2,
i i i yi xi i xi yi i i
=−rζ(cid:48)−ν∆2ζ(cid:48)+fe+T[u(cid:48)·∇ζ(cid:48)]−u(cid:48)·∇ζ(cid:48), (4) (11)
(cid:124) (cid:123)(cid:122) (cid:125) governs the dynamics of linear perturbations about the in-
fnl
stantaneousmeanflowU. Therighthandsideof(10)isthe
where fnl is the term involving the non-linear interactions correlation of the external forcing with vorticity, which for
among the turbulent eddies. A closed system for the sta- deltacorrelatedstochasticforcingisindependentofthestate
tistical state dynamics is obtained by first neglecting the oftheflowandisequalatalltimestotheprescribedforcing
√
eddy-eddy term fnl to obtain the quasi-linear system, covariance: ε(cid:104)feζ(cid:48) +feζ(cid:48)(cid:105) = ε(cid:104)fefe(cid:105) = εΞ. Therefore
1 2 2 1 1 2
The second cumulant evolves then according to:
(∂ +U·∇)Z+βV =−∇·T[u(cid:48)ζ(cid:48)]−rZ−ν∆2Z, (5)
t
(∂ +U·∇)ζ(cid:48)+(β+∂ Z)v(cid:48)+u(cid:48)∂ Z = ∂ C =(A +A )C+εΞ, (12)
t y x t 1 2
√
=−rζ(cid:48)−ν∆2ζ(cid:48)+ εfe, (6)
and forms with Eq. (9) the closed autonomous system of
In order to obtain the statistical dynamics of the quasi- S3T theory that determines the statistical dynamics of the
linear system (5)-(6) we adopt the general interpretation flow approximated at second order.
that the ensemble average over the forcing realizations TheS3Tsystemhasboundedsolutions(cf.AppendixA)
is equal to the time average over the intermediate time and the fixed points ZE and CE, if they exist, define sta-
scale (Bernstein 2009; Bernstein and Farrell 2010). Under tistical equilibria of the coherent structures with vorticity,
this assumption, the slowly varying mean flow Z is also ZE, in the presence of an eddy field with second order
the first cumulant of the vorticity Z = (cid:104)ζ(cid:105), where the cumulant or covariance, CE. The structural stability of
brackets denote the ensemble average. The time mean of these statistical equilibria addresses the parameters in the
the vorticity flux is equal to the ensemble mean of the physical system which can lead to abrupt reorganization of
flux T[u(cid:48)ζ(cid:48)] = (cid:104)u(cid:48)ζ(cid:48)(cid:105). The fluxes can be related to the the turbulent flow. Specifically, when an equilibrium of the
second cumulant C(x ,x ,t)≡(cid:104)ζ(cid:48)(x )ζ(cid:48)(x )(cid:105), which is the S3T equations becomes unstable as a physical parameter
1 2 1 2
correlation function of the eddy vorticity between the two changes, the turbulent flow bifurcates to a different attrac-
points x = (x ,y ), i = 1,2. We hereafter indicate the tor. In this work, we show that coherent structures emerge
i i i
dynamic variables that are functions of points x =(x ,y ) as unstable modes of the S3T system and equilibrate at
i i i
with the subscript i, that is ζ(cid:48) ≡ ζ(cid:48)(x ). By making the finite amplitude. The predictions of S3T regarding the
i i
identification that the fluxes at point x are equal to the emergence and characteristics of the coherent structures
value of the two variable function (cid:104)u(cid:48)ζ(cid:48)(cid:105) evaluated at the are then compared to the non-linear simulations of the
1 2
same point x=x =x , we write the fluxes as: stochastically forced barotropic flow.
1 2
(cid:104)u(cid:48)ζ(cid:48)(cid:105)=(cid:104)u(cid:48)ζ(cid:48)(cid:105) . (7)
1 2 x1=x2 2. S3T instability and emergence of finite ampli-
Expressing the velocities in terms of the vorticity [u(cid:48),v(cid:48)]= tude large-scale structure
[−∂ ∆−1,∂ ∆−1]ζ(cid:48),where∆−1istheintegraloperatorthat
y x The homogeneous equilibrium with no mean flow
inverts vorticity into the streamfunction field, we obtain
the vorticity fluxes as a function of the second cumulant, Ξ
ZE =0, CE = , (13)
in the following manner: 2r
(cid:104)u(cid:48)ζ(cid:48)(cid:105)=(cid:2)(cid:104)u(cid:48)1ζ2(cid:48)(cid:105)x1=x2,(cid:104)v1(cid:48)ζ2(cid:48)(cid:105)x1=x2(cid:3) is a fixed point of the S3T system (9) and (12) in the
=(cid:104)−(cid:10)∂ ∆−1ζ(cid:48)ζ(cid:48)(cid:11) ,(cid:10)∂ ∆−1ζ(cid:48)ζ(cid:48)(cid:11) (cid:105) absence of hyperdiffusion (cf. Appendix B). The linear
y1 1 1 2 x1=x2 x1 1 1 2 x1=x2 stability of the homogeneous equilibrium can be addressed
=(cid:104)−(cid:0)∂ ∆−1C(cid:1) ,(cid:0)∂ ∆−1C(cid:1) (cid:105). (8) byperforminganeigenanalysisoftheS3Tsystemlinearized
y1 1 x1=x2 x1 1 x1=x2 aboutthisequilibrium. Theeigenfunctionsinthiscasehave
Consequently, the first cumulant evolves according to: the plane wave form
∂tZ+UZx+V(β+Zy)+rZ+ν∆2Z = δZ =Z einx+imyeσt , δC =C (x˜,y˜)einx+imyeσt,
nm nm
=∂ (cid:0)∂ ∆−1C(cid:1) −∂ (cid:0)∂ ∆−1C(cid:1) . (9) (14)
x y1 1 x1=x2 y x1 1 x1=x2 where x˜ = x −x , x = (x +x )/2, y˜ = y −y , y =
1 2 1 2 1 2
Multiplying (A5) for ∂tζ1(cid:48) by ζ2(cid:48) and (A5) for ∂tζ2(cid:48) by ζ1(cid:48), (y1+y2)/2, n and m are the x and y wavenumbers of the
adding the two equations and taking the ensemble average eigenfunction and σ =σ +iσ is the eigenvalue with σ =
r i r
yields the equation for the second cumulant C: Re(σ), σ = Im(σ) being the growth rate and frequency
i
√
∂ C−(A +A )C = ε(cid:104)feζ(cid:48) +feζ(cid:48)(cid:105), (10) of the mode respectively. The eigenvalue σ satisfies the
t 1 2 1 2 2 1
3
non-dimensional equation: minimum energy input rate required is ε˜ =67 and occurs
c
at β˜ =3.5, while the minimum zonostrophy parameter
ε˜ (cid:90) ∞ (cid:90) ∞ min
dk˜d˜l(1−N˜2/K˜2)Ξˆ(k˜,˜l)× required for the emergence of coherent flows is Rβ =0.82
2πr3L2f −∞ −∞ and occurs for β˜→ 0. For β˜≤ β˜min, the structures that
(m˜k˜−n˜˜l)(cid:104)n˜m˜(k˜2 −˜l2)+(m˜2−n˜2)k˜ ˜l (cid:105) first become marginally stable are zonal jets (with n=0).
+ + + + The critical input rate increases as ε˜ ∼β˜−2 for β˜→0 and
× (cid:16) (cid:17) = c
iβ˜ k˜K˜2−(k˜+n˜)K˜2 +(σ˜+2)K˜2K˜2 the homogeneous equilibrium is structurally stable for all
s s
excitation amplitudes when β˜=0. However, the structural
=(σ˜+1)N˜2−in˜β˜, (15) stability for β˜=0 is an artifact of the assumed isotropy of
the excitation and the assumption of a barotropic flow. In
where L is a characteristic length scale, σ˜ = σ/r and
f the presence of even the slightest anisotropy (Bakas and
(n˜,m˜)=L (n,m) are the non-dimensional eigenvalue and
f Ioannou 2011, 2013b), or in the case of a stratified flow
wavenumbersrespectively,ε˜=ε/(r3L2)isthenon-dimensional
f (Parker and Krommes 2015), zonal jets are S3T unstable
energy injection rate of the forcing, β˜=βLf/r is the non- and are expected to emerge even in the absence of β. For
dimensional planetary vorticity gradient, β˜> β˜ , the marginally stable structures are non-zonal
min
and ε˜ grows as ε˜ ∼ β˜1/2 for β˜ → ∞. Since the critical
1 (cid:90) ∞ (cid:90) ∞ c c
Ξˆ(k,l)= Ξ(x˜,y˜)e−ikx˜−ily˜dx˜dy˜, (16) forcing for the emergence of zonal jets (also shown in fig-
2π −∞ −∞ ure 1), increases as ε˜ ∼β˜2 for β˜→∞, for large values of
c
β˜non-zonalstructuresfirstemergeandonlyatsignificantly
is the Fourier transform of the forcing covariance, K˜2 =
higher ε˜zonal jets are expected to appear. Investigation of
k˜2+˜l2,K˜2 =(k˜+n˜)2+(˜l+m˜)2,N˜2 =n˜2+m˜2,k˜ =k˜+n˜/2
s + these results with other forcing distributions revealed that
and ˜l+ =˜l+m˜/2 (cf. Appendix B). For a forcing with the the results for β˜(cid:29) 1 are independent of the structure of
mirror symmetry Ξˆ(k,−l)=Ξˆ(k,l) in wavenumber space
the forcing (Bakas et al. 2015).
and for n˜ (cid:54)=0, the eigenvalues satisfy the relations:
The parameter regime of S3T instability is now related
to the results of previous studies and to geophysical flows.
σ˜ =σ˜∗ , and σ˜ =σ˜ , (17)
(−n˜,m˜) (n˜,m˜) (n˜,−m˜) (n˜,m˜) Previous studies have identified a parameter regime which
is distinguished by robust, slowly varying zonal jets as
implying that the growth rates depend on |n˜| and |m˜|. As
well as propagating, non-dispersive, non-zonal coherent
a result, the plane wave δZ =cos(nx+my) and an array
structures(Galperinetal.2010). Thisregimethatistermed
of localized vortices δZ =cos(nx)cos(my), have the same
as zonostrophic, is in a region in parameter space in which
growth rate, despite their different structure. For zonally
the zonostrophy parameter is large (R ≥ 2.5) and the
symmetric perturbations with n˜ =0, only the second rela- β
scale k = 0.5(β3/ε)1/5 in which anisotropization of the
tionin(17)holdsand(15)reducestotheeigenvaluerelation β
turbulent spectrum occurs is sufficiently larger than the
derived by Srinivasan and Young (2012) for the emergence
forcing scale (k /K ≤ 1/4). This regime is shown in
of jets in a barotropic β-plane. β f
Weconsiderthecaseofaringforcingthatinjectsenergy figure 2 to be highly supercritical for all β˜. In addition,
at rate ε at the total wavenumber K : Bakas and Ioannou (2014) calculated indicative order of
f
magnitude values of β˜ and ε˜for the Earth’s atmosphere
(cid:112)
Ξˆ(k,l)=2K δ( k2+l2−K ), (18) andoceanaswellasfortheJovianatmosphere. Fromthese
f f
values we calculated the relevant zonostrophy parameter
and obtain the eigenvalues σ˜ by numerically solving (15). R and indicated the three geophysical flows in figure 2.
β
For small values of the energy input rate, σ˜ < 0 for all We can see that all three cases are supercritical: the Jovian
r
wavenumbers and the homogeneous equilibrium is stable. atmosphere is highly supercritical and is well within the
At a critical ε˜ the homogeneous flow becomes S3T unsta- zonostrophic regime, while the Earth’s atmosphere and
c
ble and exponentially growing coherent structures emerge. ocean are slightly supercritical (at least within the context
The critical value, ε˜ , is calculated by first determining the of the simplified barotropic model).
c
energyinputrateε˜(n˜,m˜)thatrenderswavenumbers(n˜,m˜) We now examine the growth rate and dispersion prop-
t
(cid:0) (cid:1)
neutral σ˜ =0 , and then by finding the minimum erties of the unstable modes for ε˜> ε˜ and consider first
r(n˜,m˜) c
energy input rate over all wavenumbers: ε˜ = min ε˜. the case β˜=1, with ε˜=2ε˜ . The growth rate of the maxi-
c (n˜,m˜) t c
Thecriticalenergyinputrateε˜c asafunctionofβ˜isshown mally growing eigenvalue, σ˜r, and its associated frequency
in figure 1. In addition, the corresponding critical zonos- of the mode, σ˜i, are plotted in figure 3(a) as a function of
trophy parameter R = 0.7(ε˜ β˜2)1/20 which was used in |n˜| and |m˜|. We observe that the region in wavenumber
β c
previous studies to characterize the emergence and struc- space defined roughly by 0<|n˜|<1/2, and 1/2<|m˜|<1
ture of zonal jets in planetary turbulence (Galperin et al. is unstable, with the maximum growth rate occurring for
2010), is shown as a function of β˜ in figure 2. The absolute zonal structures (n˜ = 0) with |m˜| (cid:39) 0.8. The frequency
4
(a)
1 0.4
0.8 0.4
0.3 0.3
0.2
0.6
105 σ˜rjets≥σ˜rnon−zon σ˜rjets≤σ˜rnon−zon ββ˜˜22 ˜m||0.4 0.1 00..12
0.2
104 β˜−2 00 0.1 0.2 0.3 0|n˜.4| 0.5 0.6 0.7 0.8 0
˜ǫc (b)
103 1 7
6
0.8
0.4 0.2 5
102 β˜1/2 0.6 0.6 4
˜m|
|0.4 3
2
10110−2 10−1 100 101 102 103 0.2 1
β˜ 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
|n˜|
Figure 1: The critical energy input rate ε˜c for structural
Figure 3: Dispersion relation of the unstable modes for
instability(thicksolidline)andthecriticalenergyinputrate
β˜=1(panela)andβ˜=10(panelb). Thecontoursshowthe
forstructuralinstabilityofzonaljets(solidline)asafunction
of β˜. The behavior of these critical values for β˜ (cid:28) 1 and growthrateσ˜r andtheshadingshowsthefrequencyσ˜i ofthe
β˜(cid:29)1isindicatedwiththedashedasymptotes. Inthelight unstablemodes. Forβ˜O(1),stationaryzonaljetsaremore
unstable and for β˜ (cid:29) 1, westward propagating non-zonal
grayregiononlynon-zonalcoherentstructuresemerge,while
structures are more unstable. For both panels, the energy
inthedarkgrayregionbothzonaljetsandnon-zonalcoherent
structures emerge. The thin dotted vertical line β˜=β˜min inputrateisε˜=2ε˜c.
separatestheunstableregion: forβ˜<β˜min zonalstructures
grow the most, whereas for β˜>β˜min non-zonal structures
growthemost. of the unstable modes is zero for zonal jet perturbations
(n˜ =0)andnon-negativeforallotherwavenumbers(n˜ =(cid:54) 0).
Using the symmetries (17), this implies that real unstable
mean flow perturbations δZ propagate in the retrograde
direction if n˜ (cid:54)= 0 and are stationary when n˜ = 0. As ε˜
increases the instability region expands and roughly covers
the sector 1/2 < |N˜| < 1, with zonal structures having
4.5 a larger growth rate compared to non-zonal structures, a
Jupiter result that holds for any ε˜when β˜<β˜ .
4 min
For β˜ > β˜ the non-zonal structures have always
min
3.5 zonostrophic regime largergrowthrate. Thisisillustratedinfigure3(b),showing
the growth rates and frequencies of the unstable modes
3
for β˜ = 10. For larger β˜ values there is a tendency for
Rβ2.5 the frequency of the unstable modes to conform to the
corresponding Rossby wave frequency
2
ocean
1.5 β˜n˜
atmosphere
σ˜ = , (19)
1 R n˜2+m˜2
0.5 atendencythatdoesnotoccurforsmallerβ˜. Acomparison
10−2 10−1 100 101 102 103
β˜ betweenthefrequencyoftheunstablemodesandtheRossby
wave frequency is shown in figure 4 in a plot of σ˜ /σ˜ . For
Figure 2: The critical zonostrophy parameter R = i R
β slightly supercritical ε˜, the ratio is close to one and the
0.7(ε˜cβ˜2)1/20 for structural instability (thick line) and the
correspondingcriticalparameterforstructuralinstabilityof unstable modes satisfy the Rossby wave dispersion relation.
zonaljets(thinline)asafunctionofβ˜. Theshadedregionde- At higher supercriticalities though, σ˜ departs from the
i
notesthezonostrophicregimeforwhichboththeinequalities Rossby wave frequency (by as much as 40% for the case of
R ≥2.5 and k /K ≤1/4 are satisfied. The stars denote
β β f ε˜=50ε˜ shown in figure 4(b)).
thepositionoftheEarth’satmosphereandoceanaswellas c
theJovianatmosphereintheR ,β˜parameterspace.
β
5
and Young 2012). Therefore another mechanism should be
(a)
1 responsible for producing the upgradient fluxes in the case
0.8 0.99 of an isotropic forcing.
0.98
˜m|0.6 In order to investigate the eddy-mean flow dynamics
| 0.97
underlyingtheS3Tinstability,wecalculatethevorticityflux
0.4 0.96
divergence that is induced when the statistical equilibrium
0.2 0.95
0 0.1 0.2 0.3 0.4 0.5 (13) is perturbed by an infinitesimal coherent structure δZ.
|n˜|
(b) For an S3T unstable structure, the induced flux divergence
1.5
tends to enhance the coherent structure δZ producing the
1.4
1 positivefeedbackrequiredforinstability. Sothegoalofthis
1.2
˜m| sectionistoilluminatetheeddy-meanflowdynamicsleading
| 1
0.5
to this positive feedback and to understand qualitatively
0.8
0 0.6 why the homogeneous equilibrium is more stable for small
0 0.2 0.4 0.6 |n˜| 0.8 1 1.2 1.4 and large values of β˜.
For zonal mean flows (9), (12) are simplified to:
Figure 4: Ratio of the frequency of the unstable modes
tσ˜hieovsaemrtehweacvoernreusmpbonerdsinσ˜gRfraetqu(ae)ncε˜y=of2aε˜cRaonsdsb(yb)wεa˜v=ew50itε˜hc ∂tU =−∂y(cid:104)u(cid:48)v(cid:48)(cid:105)−rU =∂y(cid:0)∂y22x1∆−11C(cid:1)x1=x2−rU, (20)
whenβ˜=100. Valuesofonedenoteanexactmatchwiththe and
Rossbywavefrequency.
∂ C =(A +A )C+Ξ, (21)
t 1 2
where
3. Analysisoftheeddy-meanflowdynamicsunder-
lying jet formation A =−U ∂ −(β−∂2 U)∆−1∂ −r, (22)
i i xi yiyi i xi
In this section, we investigate the eddy-mean flow dy- respectively. Asaresultthezonalmeanflowisdrivenbythe
namics leading to jet formation. These dynamics should momentum flux divergence of the eddies. The perturbation
have the property of directly channeling energy from the in vorticity covariance δC that is induced by the mean
turbulent motions to the coherent flow without the pres- flow perturbation δU can be estimated immediately by
enceofaturbulentcascade. Previousstudieshaveidentified assuming that the system (20)-(21) is very close to the
such mechanisms for the maintenance of zonal jets. Huang stability boundary, so that the growth rate is small. In this
and Robinson (1998) showed that shear straining of the case the mean flow evolves slow enough that it remains in
turbulent field by the jet produced upgradient momentum equilibrium with the eddy covariance, that is dδC/dt(cid:39)0.
fluxes that maintained the jet against dissipation. A sim- BakasandIoannou(2013b)showedthattheensemblemean
ple case that clearly illustrates the physical picture for the momentumfluxinducedbyaninfinitesimalsinusoidalmean
mechanism of shear straining is to consider the evolution of flow perturbation δU = (cid:15)sin(my), where (cid:15) (cid:28) 1 (i.e the
eddies in a planar, inviscid constant shear flow. The eddies eigenfunction of (B4)), is equal in this case to the integral
are sheared by the mean flow into thinner elliptical shapes, over time and over all zonal wavenumbers of the responses
while their vorticity is conserved. For an elongated eddy to all point excitations in the y direction:
this implies that the eddy velocities decrease and the eddy
energy is transferred to the mean flow through upgradient δ(cid:104)u(cid:48)v(cid:48)(cid:105)= 1 (cid:90) ∞ (cid:90) ∞ (cid:90) ∞u(cid:48)v(cid:48)(t)dtdξdk, (23)
momentum fluxes. This mechanism can operate when the 2π
−∞ −∞ 0
time required for the eddies to shear over is much shorter
than the dissipation time scale. The reason is that in this where u(cid:48)v(cid:48)(t) is the momentum flux at time t produced by:
limit even the eddies with streamfunctions leaning against
G(k,y−ξ)=B(k)h(y−ξ)eikx+il0(y−ξ). (24)
the shear that initially widen significantly gaining momen-
tum, have the necessary time to shear over, elongate and The Green’s function G has the form of a wavepacket with
surrender their momentum to the mean flow. Given that an amplitude B(k) and a carrier wave with wavenumbers
for an emerging jet the characteristic shear time scale is (k,l )thatismodulatedintheydirectionbythewavepacket
0
necessarily infinitely longer than the dissipation time scale, envelope h(y). The characteristics of the amplitude, the
it needs to be shown that shear straining can produce up- wavenumber and the envelope depend on the forcing char-
gradient momentum fluxes in this case as well. In addition, acteristics, but in any case the calculation of the ensemble
previousstudieshaveshownthatshearingofisotropiceddies mean momentum fluxes is reduced to calculating the mo-
on an infinite domain does not produce any net momentum mentum fluxes over the life cycle of wavepackets that are
fluxes (Shepherd 1985; Farrell 1987; Holloway 2010) and initially at different latitudes and then adding their relative
should have no effect on the S3T instability (Srinivasan contributions.
6
As the wavepacket propagates in the latitudinal direc- and slightly changes the amplitude of the fluxes as well as
tion, its meridional wavenumber and frequency are going slightly speeds up or slows down the wavepacket. The sum
to change due to shearing by the mean flow and due to of these two effects will produce the induced momentum
the change of the mean vorticity gradient β −U . The fluxes.
yy
resulting time variable momentum flux u(cid:48)v(cid:48)(t) can be calcu-
lated using ray tracing. According to standard ray tracing a. Thelimitofsmallscalewavepacketswithashortpropagation
arguments, the wave action is conserved along a ray (in the range
absence of dissipation) leading to the momentum flux:
In order to clearly illustrate the behavior of the eddy
fluxes, we consider the limit of β˜=βL /r (cid:28)1, where L
u(cid:48)v(cid:48)(t)=−|B|2A (t)e−2rt|h(y−η(t))|2, (25) f f
M is the scale of the wavepackets and in addition we assume
where A (t)=kl /(k2+l2)2 is the momentum flux of the that the scale of the mean flow, 1/m, is much larger than
M t t
the scale of the wavepackets mL (cid:28) 1. In this limit, the
carrier wave that determines the amplitude of the fluxes of f
wavepackets are dissipated before propagating far from the
the wavepacket and l , η(t) are the time dependent merid-
t
initial position and the effect of the change in the mean
ional wavenumber and position of the wavepacket respec-
vorticity gradient is higher order. As a result, Bakas and
tively(Andrewsetal.1987). Becauseofthesmallamplitude
Ioannou(2013b)showthatl andη decreasemonotonically
of the mean flow perturbation δU, the wavenumber and 1 1
with time with rates independent of δU and proportional
position of the packet vary slowly on a time scale O((cid:15)t) yy
to the shear δU (ξ) at the initial position ξ:
compared to the dissipation time scale 1/r and the domi- y
nant contribution to the time integral in (23) comes from (cid:18) (cid:19)
dA
small times. We can therefore seek asymptotic solutions of l =−δU (ξ)kt , η =−βδU (ξ) M kt2. (28)
1 y 1 y dl
the form t l0
Thatis,theamplitudeofthefluxA andthegroupvelocity
l =l +(cid:15)l +··· , η(t)=ξ+c t+(cid:15)η (t)+··· , (26) M
t 0 1 0 1 of the packets change only due to the shearing of the phase
where c = 2βkl /(k2 +l2)2 is the group velocity in the lines of the carrier wave according to the local shear.
0 0 0
absence of a mean flow and calculate the integral of u(cid:48)v(cid:48)(t) Consider in this limit the first term, u(cid:48)v(cid:48)S, arising from
the small amplitude change. Since the wavepacket is dis-
over time from the leading order terms. Substituting (26)
sipated before it propagates away, we can ignore to first
in (25) we obtain:
order propagation:
u(cid:48)v(cid:48)(t)=−|B|2A (0)e−2rt|h(y−ξ−c t)|2−
M 0 (cid:18) (cid:19)
dA
(cid:124) (cid:123)(cid:122) (cid:125) u(cid:48)v(cid:48) =−(cid:15)|B|2 M l (t)e−2rt|h(y−ξ−c t)|2
−(cid:124) (cid:15)|B|2(cid:18)ddAlMt (cid:19)ul(cid:48)0v(cid:48)lR1(t(cid:123))(cid:122)e−2rt|h(y−ξ−c0t)|(cid:125)2 S (cid:39)|B|2δUy(ξ)kdtlt(cid:18)ddAll0Mt 1(cid:19)l0e−2rt|h(y−ξ)|2,0 (29)
u(cid:48)v(cid:48)S so that the packet grows/decays in situ. Since the wave
d
−(cid:15)|B|2A (0)η (t)e−2rt |h(y−ξ−c t)|2. packet is rapidly dissipated, the integrated momentum flux
M 1 dy 0
over its life time will be given to a good approximation by
(cid:124) (cid:123)(cid:122) (cid:125)
u(cid:48)v(cid:48)β the instantaneous change in the flux1 that is proportional
(27) to (dA /dl ) . Figure 5 illustrates the amplitude of the
M t l0
momentum flux as a function of the angle θ =arctan(l /k)
t t
The first term, u(cid:48)v(cid:48) , arises from the momentum flux pro- of the phase lines of the carrier wave of the packet with the
R
duced by a wavepacket in the absence of a mean flow. y-axis. Itisshownthatthemomentumfluxofawavepacket
Because A (0) = kl /(k2 +l2)2 is odd with respect to with |θ |<π/6 (that is with phase lines close to the merid-
M 0 0 0
wavenumbers, this term does not contribute to the en- ional direction) excited in regions II or III, will increase
semble averaged momentum flux when integrated over all within the dissipation time scale. Compared to an un-
wavenumbers and will be hereafter ignored. The second sheared wavepacket, this process leads to upgradient mo-
term, u(cid:48)v(cid:48) , arises from the small change in the amplitude mentum flux. The opposite occurs for waves excited in
S
of the flux A over a dissipation time scale. The third regions I and IV (with |θ |>π/6) that produce downgradi-
M 0
term, u(cid:48)v(cid:48) , arises from the small change in the position of ent flux, as their momentum flux decreases.
β
the packet η compared to a propagating packet in the ab- We now consider the second term, u(cid:48)v(cid:48) arising from
β
senceofameanflow. Tosummarize,theinfinitesimalmean theeffectofpropagationonthemomentumflux. Thegroup
flow refracts the wavepacket due to shearing by the mean
1occurringoverthedissipationtimescale1/r thatisincremental
flow and due to the change of the mean vorticity gradient
insheartimeunits
7
(a)
0.5 0.2
I II
0.4 0.1
0.3 uv 0
0.2
−0.1
0.1
−0.2
−3 −2 −1 0 1 2 3
M 0 y
A
x 10−5 (b)
−0.1 5
−0.2
0
−0.3 uvβ
−5
−0.4 III IV
−0.5 −10
−80 −60 −40 −20 0 20 40 60 80 −3 −2 −1 0 1 2 3
θt=arctan(lt/k) y
Figure 5: Amplitude of the momentum fluxes, AM(t), of Figure 6: (a) Comparison of the momentum fluxes of an
wavepackets as a function of the angle θt = arctan(lt/k) unshearedwavepacketexcitedinregionsII(thicksolidline)
between the phase lines of the central wave and the y-axis. and III (solid line) to the momentum fluxes of a sheared
Theverticallinesseparatetheregionswith|θt|<π/6(IIand wavepacketshownbythecorrespondingdashedlines,when
III)and|θt|>π/6(IandIV). only the change in propagation is taken into account. A
snapshotofthefluxesatt=0.2/r isshown. Theplanetary
vorticitygradientisβ=0.1,thewavepackethasinitialvortic-
(cid:113)
velocityisgivenbycg =2βAM inthiscaseandasaresulta ityh(y)=e−y2, k2+l02=1,|θ0|=π/10and|B|=1. (b)
wavepacketstartinginregionIII,willpropagatetowardsthe Thedifferenceinmomentumfluxesbetweenashearedandan
north(c.f. figure5). Becauseshearingslowsdownthewaves unshearedwavepacketcalculatedovertheirlifecycle.
in region III (η ∼−(dA /dl )), the wavepacket will flux
1 M t
its momentum from southern latitudes compared to when
proportional to the third derivative of δU yielding a hyper-
it moved in the absence of the shear flow. This is shown in
diffusive momentum flux divergence that tends to reenforce
figure6(a)illustratingthedistributionofmomentumfluxof
the mean flow and is therefore destabilizing. These destabi-
anunshearedanda shearedperturbationwhoseamplitudes lizing fluxes are proportional to β˜2 and as a result, the en-
are constant. Figure 6(b) plots this difference, u(cid:48)v(cid:48)β, and ergyinputraterequiredtoformzonaljetsincreasesas1/β˜2
shows that the flux is downgradient in this case. The same
inthislimit. Itisworthnotingthatthefirsttermintegrates
happens for waves excited in region II, while the waves
to zero only for the special case of the isotropic forcing, as
excited in regions I and IV produce upgradient flux.
even the slightest anisotropy yields a non-zero contribution
The net momentum fluxes produced by an ensemble of
from u(cid:48)v(cid:48) . For example consider the forcing covariance
S
wavepackets, will therefore depend on the spectral charac- Ξ(x1,x2,y1,y2)=cos(k(x1−x2))e−(y1−y2)2/δ2 that mim-
teristics of the forcing that determine the regions (I-IV), in
ics the forcing of the barotropic flow by the most unstable
which the forcing has significant power. Bakas and Ioannou
baroclinic wave, which has zero meridional wavenumber. In
(2013b) show that for the isotropic forcing (18):
this case the forcing that is centered at l =0 in wavenum-
0
ber space, injects significant power in a band of waves in
δ(cid:104)u(cid:48)v(cid:48)(cid:105)=
regions II and III and therefore u(cid:48)v(cid:48) yields upgradient
1 (cid:90) ∞ (cid:90) ∞ 1 (cid:90) ∞ (cid:90) ∞ S
= u(cid:48)v(cid:48) dξdk+ u(cid:48)v(cid:48) dξdk fluxes.
2π S 2π β
−∞ −∞ −∞ −∞
3ε˜β˜2r d3δU b. The effects of the change in the mean vorticity gradient and
(cid:39)0− . (30) the finite propagation range
32πK4 dy3
f
In order to take into account the effect of the change in
The first integral is zero, because the gain in momentum the vorticity gradient, we retain higher order terms with
occurring for |θ0|<π/6 (waves excited in regions II, III) is respect to mLf (cid:28) 1 in l1 and η1. In this case it can be
fully compensated by the loss in momentum for |θ0|>π/6 shown that l1 decreases with time at a rate proportional
(waves excited in regions I, IV) since for the isotropic forc- to U (ξ)+U (ξ) (Bakas and Ioannou 2013b). Since the
y yyy
ing all possible wave orientations are equally excited. The local shear and the local change in the vorticity gradient
net momentum fluxes are therefore produced by the u(cid:48)v(cid:48)β have different signs, the wavepacket is ’sheared less’ and as
term and are upgradient, because the loss in momentum a result we expect reduced momentum fluxes compared to
occurringfor|θ0|<π/6,isovercompensatedbythegainin the limit discussed in section 4.2. Indeed, for the isotropic
momentum for |θ |>π/6. The momentum fluxes are also
0
8
forcing:
(a) x 10−4 x 10−4 (b)
6 5 3
(cid:32) (cid:33)
3ε˜β˜2r d3δU 1 d5δU
δ(cid:104)u(cid:48)v(cid:48)(cid:105)(cid:39)− − . (31) 4 2
32πK4 dy3 4K2 dy5 y 0 E
f f
2 1
−5
That is, the change in the mean vorticity gradient has a
0 0
stabilizing effect. 0 x 5 0 50t0 1000
We finally relax the assumption that β˜ (cid:28) 1. In this (c) x 10−3 (d)
6 1040
case, l and η are affected by an integral shear and mean
1 1 5
1030
vorticitygradientovertheregionofpropagation. Forlarger 4
β˜, the wavepacket will encounter regions of both positive y 0 t1020
2
and negative shear and as a result, the momentum fluxes −5 1010
thatarequalitativelyproportionaltotheintegralshearover 0 1000
the propagation region will be reduced. In the limit β˜(cid:29)1, 0 x 5 0 2 x 4 6
the region of propagation is the whole sinusoidal flow with
consecutive regions of positive and negative shear and the Figure 7: EquilibrationoftheS3Tinstabilities. (a)Stream-
function of the initial perturbation. (b) Energy evolution
integralshearalongwiththefluxeswillasymptoticallytend
of the initial perturbation shown in panel (a) as obtained
to zero. As a result, the energy input rate required for
from the integration of the S3T equations (9) and (12)
structural instability of zonal jets increases with β˜ in this (dashedline)andfromtheintegrationoftheensemblequasi-
limit. linear(EQL)system(4)-(3)withNens=10(solidline)and
Nens=100(dash-dottedline)ensemblemembersthatisdis-
cussedinsection6. (c)SnapshotofthestreamfunctionΨeq
4. Equilibration of the S3T instabilities of the traveling wave structure and (d) Hovm¨oller diagram
ofΨeq(x,y=π/4,t)forthefiniteequilibratedtravelingwave.
We now investigate the equilibration of the instabilities
Thethickdashedlineshowsthephasespeedobtainedfrom
bystudyingtheS3Tsystem(9),(12)discretizedinadoubly thestabilityequation(15). Theenergyinputrateisε˜=4ε˜c
periodic channel of size 2π × 2π. We approximate the andβ˜=100.
monochromatic forcing (18), by considering the narrow
band forcing
form to the harmonic Z =cos(x)cos(5y) that propagates
√
(cid:26) westward. This is illustrated in the Hovm¨oller diagram of
K 1, for | k2+l2−K |≤∆K
Ξˆ(k,l)= f √ f f , (32) ψ(x,y =π/4,t) shown in 7(d). The sloping dashed line in
∆K 0, for | k2+l2−K |>∆K
f f f
thediagramcorrespondstothephasespeedofthetraveling
where k, l assume integer values, that injects energy at rate wave, which is found to be approximately the phase speed
εinanarrowringinwavenumberspacewithradiusK and of the unstable (n,m)=(1,5) eigenmode.
f
width∆Kf. Weconsiderthesetofparametervaluesβ =10, Considernowtheenergyinputrateε˜=10ε˜c. Whilethe
r =0.01, ν =1.19·10−6, K =10 and ∆K =1, for which maximum growth rate still occurs for the (|n|,|m|)=(1,5)
f f
β˜ = 100. The integration is therefore in the parameter non-zonal structure, zonal jet perturbations are unstable
as well. If the S3T dynamics are restricted to account only
region of figure 1 in which the non-zonal structures are
for the interaction between zonal flows and turbulence by
more unstable than the zonal jets. The growth rates of the
employing a zonal mean rather than an ensemble mean,
coherent structures for integer values of the wavenumbers,
an infinitesimal jet perturbation will grow and equilibrate
nandmarecalculatedfromthediscreteversionofequation
at finite amplitude. To illustrate this we integrate the
(15) obtained by substituting the integrals with sums over
S3T dynamical system (20)-(21) restricted to zonal flow
integer values of the wavenumbers (Bakas and Ioannou
coherent structures. The energy of the small zonal jet
2013a).
perturbation δZ =0.1cos(4y) is shown in figure 8 to grow
We first consider the supercritical energy input rate
and saturate at a constant value and the streamfunction
ε˜=4ε˜ . Fortheseparametersonlynon-zonalmodesareun-
c
of the equilibrated jet is shown in the left inset in figure
stable, with the perturbation with (n,m)=(1,5) growing
8. However, in the context of the generalized S3T analysis
the most. At t=0, we introduce a small random perturba-
that takes into account the dynamics of the interaction
tion, whose streamfunction is shown in figure 7(a). After
between coherent non-zonal structures and jets, we find
a few e-folding times, a harmonic structure of the form
that these S3T jet equilibria can be saddles: stable to zonal
Z = cos(x)cos(5y) dominates the large-scale flow. The
jet perturbations but unstable to non-zonal perturbations.
energy of this large scale structure shown in figure 7(b),
To show this, we consider the evolution of the same jet
increases rapidly and eventually saturates. At this point
perturbation δZ = 0.1cos(4y) under the generalized S3T
the large-scale flow gets attracted to a traveling wave finite
dynamics(9),(12)andfindthattheflowfollowsthezonally
amplitude equilibrium structure (cf. figure 7(c)) close in
9
x 10−3 10−2
1 6 0.02
10−3 4 0
y
0.8 2
−0.02
E0.6 46 0.01 6 Ez10−4 00 x 5 6 0.02
0 0.01 4
y 0
2 4 y
0.4 00 5 −0.01 y2 −00.01 10−5 02 −0.02
0.2 x 0 0 x 5
0 5
x
0 10−6
0 500 1000 1500 2000 2500 3000 3500 0 200 400 600 800 1000
t t
Figure 8: Energy evolution of an initial jet perturbation Figure 9: Zonalenergyevolutionofarandomzonalpertur-
δZ = 0.1cos(4y) for the zonally restricted S3T dynamics bationimposedonthenon-zonaltravelingwaveequilibrium
(20)-(21)(dashedline)andthegeneralizedS3Tdynamics(9), shownintheleftinset. Thestreamfunctionoftheequilibrated
(12) (thin line). The insets show a snapshot of the mean structureisshownintherightinset. Theenergyinputrate
flowstreamfunctionatt=700(left)andthestreamfunction isε˜=30ε˜c andβ˜=100.
of the equilibrated structure at t = 3500 (right) under the
generalizedS3Tdynamics. Theparametersareε˜=10ε˜c and
β˜=100. compared to the mixed state in figure 9. Embedded in it
are non-zonal vortices with the same meridional scale and
with about 14% the energy of the zonal jet. These vortices
restricted S3T dynamics and equilibrates to the same finite
that are shown in figure 10(b) to have approximately the
amplitude zonal jet (cf. figure 8). At this point we insert a
compact support structure Ψ = cos(x)cos(4y) propagate
small random perturbation to the equilibrated flow. Soon
westwardasshownintheHovmo¨llerdiagraminfigure10(c).
after, non-zonal undulations grow and the flow transitions
to the stable Z =cos(x)cos(5y) traveling wave state that
5. Comparison to ensemble mean quasi-linear and
is also shown in figure 8. As a result, the finite equilibrium
non-linear simulations
zonal jet structure is S3T unstable to coherent non-zonal
perturbations and is not expected to appear in non-linear a. Comparison to an ensemble of quasi-linear simulations
simulations despite the fact that the zero flow equilibrium
Withinthecontextofthesecondordercumulantclosure,
is unstable to zonal jet perturbations. We will elaborate
the S3T formulation allows the identification of statistical
more on this issue in the next section.
turbulent equilibria in the infinite ensemble limit, in which
Finally,considerthecaseε˜=30ε˜ . Atthisenergyinput
c the fluctuations induced by the stochastic forcing are av-
rate,thefiniteamplitudenon-zonaltravelingwaveequilibria
eraged to zero. However, these S3T equilibria and their
become S3T unstable. To show this, we consider the non-
stability properties are manifest even in single realizations
zonal traveling wave equilibrium obtained by the evolution
of the turbulent system. For example, previous studies us-
ofthesmallnon-zonalperturbationδZ =0.01cos(x)cos(5y)
ing S3T obtained zonal jet equilibria in barotropic, shallow
to the homogeneous state that is shown at the left inset
water and baroclinic flows in close correspondence with
in figure 9 and impose a small random zonal perturbation.
observed jets in planetary flows (Farrell and Ioannou 2007,
2
The evolution of the zonal energy E =(1/2)U , where the
z 2008,2009b,a). Inaddition,previousstudiesofS3Tdynam-
overbar denotes a zonal average, is shown in figure 9. After
ics restricted to the interaction between zonal flows and
an initial transition period, the zonal perturbations grow
turbulence in a β-plane channel showed that when the en-
exponentiallyandtheflowtransitionstothejetequilibrium
ergy input rate is such that the zero mean flow equilibrium
state shown at the right inset in figure 9. Note however,
is unstable, zonal jets also appear in the non-linear simu-
that the jet equilibrium structure is not zonally symmetric.
lations with the structure (scale and amplitude) predicted
This is a new type of S3T equilibrium: it is a mix between
by S3T (Srinivasan and Young 2012; Constantinou et al.
a zonal jet and a non-zonal traveling wave with the same
2014).
meridional scale. These mixed equilibria appear to be the
A very useful intermediate model that retains the wave-
attractors for larger energy input rates as well. This is
mean flow dynamics of the S3T system while relaxing the
illustrated in figure 10 showing the structure of the mixed
infinite ensemble approximation is the quasi-linear system
equilibrium at ε=50ε . The equilibrium structure consists
c (5)-(6). Under the ergodic assumption, this can be inter-
of a large amplitude zonally symmetric jet with larger scale
10