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Phase and amplitude evolution in the network of triadic interactions of the

Hasegawa-Wakatani system

Ö. D. Gürcan, J. Anderson, S. Moradi, A. Biancalani, P. Morel

CNRS, LPP, Ecole Polytechnique

Hasegawa-Wakatani system, commonly used as a toy model of dissipative drift waves in fusion

devices is revisited with considerations of phase and amplitude dynamics of its triadic interactions.

It is observed that a single resonant triad can saturate via three way phase locking where the phase

diﬀerences between dominant modes converge to constant values as individual phases increase in

time. This allows the system to have approximately constant amplitude solutions. Non-resonant

triads show similar behavior only when one of its legs is a zonal wave number. However when an

additional triad, which is a reﬂection of the original one with respect to the yaxis is included, the

behavior of the resulting triad pair is shown to be more complex. In particular, it is found that

triads involving small radial wave numbers (large scale zonal ﬂows) end up transferring their energy

to the subdominant mode which keeps growing exponentially, while those involving larger radial

wave numbers (small scale zonal ﬂows) tend to ﬁnd steady chaotic or limit cycle states (or decay

to zero). In order to study the dynamics in a connected network of triads, a network formulation

is considered including a pump mode, and a number of zonal and non-zonal subdominant modes as

a dynamical system. It was observed that the zonal modes become clearly dominant only when a

large number of triads are connected. When the zonal ﬂow becomes dominant as a ’collective mean

ﬁeld’, individual interactions between modes become less important, which is consistent with the

inhomogeneous wave-kinetic picture. Finally, the results of direct numerical simulation is discussed

for the same parameters and various forms of the order parameter are computed. It is observed

that nonlinear phase dynamics results in a ﬂattening of the large scale phase velocity as a function

of scale in direct numerical simulations.

I. INTRODUCTION

Two dimensional Hasegawa-Wakatani equations[1]

with proper zonal response consist of an equation of

plasma vorticity

∂

∂t ∇2Φ + ˆ

z× ∇Φ· ∇∇2Φ = Ce

Φ−en+DΦ∇2Φ(1)

and an equation of continuity

∂

∂t n+ˆ

z× ∇Φ· ∇n+κ∂yΦ = Ce

Φ−en+Dn(n), (2)

with the E×Bvelocity deﬁned as vE=ˆ

z× ∇Φin

normalized form and e

Φ=Φ− hΦiwhere hΦidenotes

averaging in y(i.e. poloidal) direction. Here nis the

ﬂuctuating particle density normalized to a background

density n0,Φis the electrostatic potential normalized to

T/e,κis the diamagnetic velocity normalized to speed of

sound, Cis the so called adiabaticity parameter, which

is a measure of the electron mobility and DΦand Dnare

dissipation functions for vorticity and particle density re-

spectively. For ﬂuctuations we have DΦ∇2e

Φ=ν∇4e

Φ

from kinematic viscosity, whereas for the zonal ﬂows

DΦ∇2Φ=−νZF ∇2Φfrom large scale friction. Unless

the system represents a renormalized formulation, Dn

should actually be zero, however here we include it for

completeness and numerical convenience and take it to

have the same form as the vorticity dissipation with diﬀu-

sion Dn(en) = D∇2enand particle loss D(n) = −DZF n.

The Hasegawa Wakani model, was initially devised as

a simple, nonlinear model of dissipative drift wave tur-

bulence in tokamak plasmas. It has the same nonlinear

structure as the passive scalar turbulence [2] -with vortic-

ity evolving according to 2D Navier-Stokes equations- or

more complex problems such as rotating convection [3, 4].

From a plasma physics perspective it can be considered

as the minimum non-trivial model for plasma turbulence,

since it has i) linear instability (e.g. Hasegawa-Mima

model does not [5]), ii) ﬁnite frequency (so that resonant

interactions are possible [6]), and iii) a proper treatment

of zonal ﬂows[7]. The model is well known to gener-

ate high levels of large scale zonal ﬂows, especially for

C&1[8–10]. It has been studied in detail for many prob-

lems in fusion plasmas including dissipative drift waves in

tokamak edge [11, 12], subcritical turbulence[13], trapped

ion modes [14], intermittency [15, 16], closures [17–19],

feedback control [20], information geometry [21] and ma-

chine learning [22]. Variations of the Hasegawa-Wakatani

model are regularly used for describing turbulence in ba-

sic plasma devices[23–25].

Formation of large scale structures, in particular Zonal

ﬂows in drift wave turbulence is one of the key issues in

the study of turbulence in fusion plasmas, which can be

formulated in terms of modulational instability of either

a gas of drift wave turbulence using the wave kinetic for-

mulation [26] or a small number of drift modes[27], result-

ing in various forms of complex amplitude equations such

as the celebrated nonlinear Schrödinger equation (NLS)

[28]. It is also common to talk about zonal ﬂows as re-

sulting from a process of inverse cascade[29, 30]and their

back reaction on turbulence[31, 32] results in predator-

prey dynamics, possibly leading up to the low to high

arXiv:2202.09593v1 [nlin.CD] 19 Feb 2022

2

conﬁnement transition in tokamaks[33, 34]. While the

role of the complex phases in nonlinear evolution of the

amplitudes, especially in the context of structure forma-

tion, for example as in the case of soliton formation in

NLS, was always well known, its particular importance

for zonal ﬂow formation in toroidal geometry has been

underlined recently[35].

Here we revisit the Hasegawa-Wakatani system, with

proper zonal response, as a minimum system that allows

a description of zonal ﬂow formation in drift wave tur-

bulence, and study interactions between various number

of modes from three wave interactions to the full spec-

trum of modes described by direct numerical simulations,

focusing in particular on phase dynamics and the possi-

bility of phase locking and synchronization. It turns out

that while resonant three wave interactions involving un-

stable and damped modes favor phase locking (i.e. a state

where the diﬀerences between individual phases remain

roughly constant as they increase together), interactions

involving zonal ﬂows (i.e. four wave interactions includ-

ing the triad reﬂected with respect to the yaxis), seems

to have a complicated set of possible outcomes depending

on if the zonal ﬂow wave number is larger or smaller than

the pump wave-number. It therefore becomes critical to

study a “network” of connected triads in order to see the

collective eﬀects of a number of triads on the evolution

of zonal ﬂows and of the relative phases between modes.

Two diﬀerent network conﬁgurations are considered: that

with a single kyand many diﬀerent q’s, and that with a

single qbut many diﬀerent ky’s. Note that the algorithm

that we use computes all possible interactions between

the modes in a given collection of triads and then com-

putes the interaction coeﬃcients and evolves the system

nonlinearly according to those.

Finally we consider the results from direct numerical

simulations (DNS) using a pseudo-spectral 2D Hasegawa-

Wakatani solver. The DNS and the network models

correspond exactly, in the sense that if we consider an

Nx×Nygrid and consider all the possible triads in such

a grid and solve this problem using our network solver, we

obtain exactly the same problem (including the boundary

conditions that are periodic) as the DNS. The results of

the DNS show qualitatively similar behavior to the two

network models that we considered. However looking at

the evolution of phases, we observe a nonlinear ﬂatten-

ing of the phase velocity for large scales computed as a

function of x, suggesting nonlinear structure formation

in the classical sense of nonlinearity balancing dispersion

resulting in a constant velocity propagation at least for

large r scale structures. These vortex-like structures that

move at a constant velocity are also clearly visible in the

time evolution of density and vorticity ﬁelds.

The rest of the paper is organized as follows. In the

remainder of the introduction, the Hasegawa-Wakatani

system is reformulated in terms of its linear eigenmodes

and the amplitude and phase equations for these eigen-

modes, writing out explicitly the nonlinear terms that

appear in this formulation. In Section II, diﬀerent types

of interactions among dissipative drift waves are consid-

ered using these linear eigenmodes, starting with the ba-

sic three wave interaction. After showing that there is

no qualitative diﬀerence between a near resonant and

an exactly resonant (within numerical accuracy) triad,

the details of the phase dynamics of such a single triad

are discussed. In Section III, the interaction with zonal

ﬂows are considered. It is noted that when we consider

a triad and its reﬂection with respect to its pump wave-

number together as a pair, the behavior of the system is

qualitatively diﬀerent from the single triad case. After

a discussion of order parameters for this system, a net-

work formulation is considered and the results from such

a network model is presented. Finally the reslts from

direct numerical simulations of the Hasegawa-Wakatani

system is discussed and compared with those earlier re-

sults based on reduced number of triads. Section IV is

conclusion.

A. Linear Eigenmodes

We can write the Hasegawa-Wakatani system in

Fourier space for non-zonal modes (i.e. ky6= 0) as fol-

lows:

∂tΦk+ (Ak−Bk) Φk=C

k2nk+NΦk(3)

∂tnk+ (Ak+Bk)nk= (C−iκky) Φk+Nnk (4)

where

Ak=1

2Dk2+C+C

k2+νk2 (5)

and

Bk=1

2Dk2+C−C

k2+νk2. (6)

using the simpler notation Φk→Φkand deﬁning:

Nnk =1

2X

4

ˆ

z×p·qΦ∗

pn∗

q−Φ∗

qn∗

p(7)

and

NΦk=1

2X

4

ˆ

z×p·qq2−p2Φ∗

pΦ∗

q

k2. (8)

Diagonalizing the linear terms we can write:

∂tξ±

k+iω±

kξ±

k=N±

ξk (9)

3

with the complex eigen-frequencies ω±

k=ω±

rk +iγ±

kthat

can be written as:

ω±

k= Ω±

k−iAk

with

Ω±

k=± σkrHk−Gk

2+irHk+Gk

2!(10)

where σk=sign (κky),

Hk=qG2

k+C2κ2k2

y/k4, (11)

and

Gk≡B2

k+C2

k2. (12)

This allows us to write the two linear eigenmodes as:

ξsk

k=nk+k2

C[Bk−iΩsk

k] Φk. (13)

where sk=±. The nonlinear terms in (9) become:

Nsk

ξk =Nnk +k2

C(Bk−iΩsk

k)NΦk, (14)

and the inverse transforms can be written as:

Φk=i

2

C

k2X

sk

ξsk

k

Ωsk

k

(15)

nk=−i

2X

sk

1

Ωsk

k

(Bk+iΩsk

k)ξsk

k. (16)

Considering the inviscid limit, {D, ν } → 0and ky→

O(), where we keep terms only up to O()we obtain:k

ξ+

k=nk+k2−iκky

2A2

kΦk

ξ−

k=nk−1 + iκky

2A2

kΦk,

which means that one could loosely refer to these two

modes as the potential vorticity mode (i.e. ξ+

k=nk+

k2Φk) and the non-adiabatic electron density mode (i.e.

ξ−

k=nk−Φk), somewhat similar to the real space decom-

position used in Ref. 36. Since the equations are already

diagonal for ky= 0 modes, we can use ξ+

k=k2Φkand

ξ−

k=nkfor those (or Φkand nkexplicitly as we will do

below).

Notice that the two eigenmodes in (13) are not orthog-

onal. They have the same frequencies (in opposite direc-

tions) but diﬀerent growth rates with γ+

k> γ−

k(with

γ−

k<0, while γ+

kcan be positive or negative depend-

ing on the wave-number). The full nonlinear initial value

problem can be solved using linear eigenmodes by ﬁrst

computing ξsk

k(0) from (13), and then advancing those

to ξsk

k(t)using (9), where the linear matrix is now diag-

onal (but the nonlinear coupling terms are rather com-

plicated), and ﬁnally going back to compute Φk(t)and

nk(t)using (15-16). Obviously, this approach does not

involve any kind of approximation.

B. Amplitude and Phase Equations

Substituting ξ±

k=χ±

keiφ±

kinto (9), we get:

∂tχ±

keiφ±

k+iω±

kχ±

keiφ±

k=N±

ξk eiφNξ±

k, (17)

taking the real part we obtain the amplitude equations:

∂t−γ±

kχ±

k=N±

ξk cos φNξ±

k−φ±

k(18)

and taking the imaginary part and dividing by χ±

kwe get

the phase equations:

∂tφ±

k=−ω±

kr +N±

ξk

χ±

k

sin φNξ±

k−φ±

k. (19)

The form of the amplitude equation (18) means that the

ﬁxed point for the amplitude evolution is determined

by the phase diﬀerence between Nsk

ξk and ξsk

kfor each

sk. However such a ﬁxed point keeps evolving since the

phases themselves increase linearly with the linear fre-

quency while being deformed by the nonlinear terms.

Note that if the nonlinear phase is dominated by a slowly

evolving mean phase (could be the case if the nonlin-

ear interactions are dominated by the theractions with

a zonal ﬂow), the individual phases will be attracted to

this nonlinear mean phase, since if the individual phase

is behind the nonlinear phase, the sin φNξ±

k−φ±

kwill

be positive, causing the individual phase to accelerate,

whereas if the individual phase is ahead of the nonlinear

phase it will be slowed down. However since we have

linear frequencies it is impossible for individual phases

to become phase locked directly with the slow nonlinear

phase. Instead the nonlinear term plays a role akin to

that of the ponderomotive force in parametric instabil-

ity.

C. Nonlinear Terms

In order to compute N±

ξk in terms of ξ±

k, we need to go

back to Φkand nkusing (15-16), compute the nonlinear

terms (7-8) using those and combine them as in (14).

They can then be written in the form:

4

Figure 1. The resonance manifold ∆ω=ω+

kr+ω+

pr+ω+

qr= 0 of

the Hasegawa-Wakatani system for the case C= 1.0,κ= 0.2,

ν=D= 10−3is shown corresponding to the wave vector

k= (5,5) that is shown explicitly. Any pthat falls onto the

region inside the resonance manifold (shown here with a ﬁnite

width of ±0.04 with ∆ω > 0in red and ∆ω < 0in blue if in

color) gives ∆ω≈0(with q=−k−p). As discussed in the

text, because of the fact that the (+) and (−)modes have

the same frequency (but opposite direction of propagation in

ydirection) all possible combinations of (+) and (−)modes

resonate on the same manifold.

Nsk

ξk =1

2X

4X

sp,sq

Mskspsq

kpq ξsp∗

pξsp∗

q(20)

in terms of the linear eigenmodes, where the sum is over

sp, sq={(+,+) ,(+,−),(−,+) ,(−,−)}for sk= (+,−).

The nonlinear interaction coeﬃcients in (20) can be writ-

ten (i.e. between 3 non-zonal modes) as:

Mskspsq

ξkpq =mskspsq

kpq q2Bq−iΩsq∗

q

−p2Bp−iΩsp∗

p−q2−p2(Bk−iΩsk

k)

(21)

where

mskspsq

kpq ≡Cˆ

z×p·q

4Ωsp∗

pΩsq∗

qq2p2

and Ω±

kis given in (10).

Note that these coeﬃcients are complex, and have dif-

ferent phases in general. In other words the explicit forms

of (19) can be written as:

∂tφsk

k=−ωsk

kr +X

4X

sp,sq

Mskspsq

ξkpq ξsp

pξsq

q

|ξsk

k|

×sin θskspsq

Mkpq −φsp

p−φsq

q−φsk

k(22)

a) b)

d)c)

Figure 2. The resonance manifold, shown on top of the growth

rate where red corresponds to γ+

k>0and blue to γ+

k<0for

a) the most unstable model on the grid k= (0,1.125), b) a

nearby mode with a small kxcomponent k= (0.250,1.125),

c) a mode with kx=kythat is k= (1.125,1.125) and ﬁnally

d) a mode that has kxkywith k= (1.125,0.125).

where θskspsq

Mkpq is the phase of the nonlinear interaction

coeﬃcient Mskspsq

ξkpq .

II. INTERACTIONS AMONG DRIFT WAVES

A. Three wave interactions

Consider three separate modes k,pand qthat satisfy

the triadic interaction condition k+p+q= 0, possibly in

the presence of other modes. The nonlinear term for the

wave number kcan then be divided into the interaction

with the pair pand q, and the interaction with the rest

of the modes (if they exist). If the three wave interac-

tion that we consider is resonant, slightly oﬀ-resonance,

or completely non-resonant, its evolution is likely to be

diﬀerent, which can be considered as diﬀerent scenarios.

It may also be possible to model the eﬀects of rest of the

modes as background forcing, modiﬁcation of the linear

terms (à la eddy damping) or simply as stochastic noise.

Thus separating the nonlinear term into the interaction

with the pair pand q(i.e. N±

ξkpq ) and the interaction

with the rest of the modes (i.e. δN ±

ξkpq ), we can write:

∂tξ±

k+iω±

kξ±

k=N±

ξkpq +δN±

ξkpq (23)

where

N±

ξkpq =M±++

ξkpq ξ+∗

pξ+∗

q+M±+−

ξkpq ξ+∗

pξ−∗

q

+M±−+

ξkpq ξ−∗

pξ+∗

q+M±−−

ξkpq ξ−∗

pξ−∗

q

5

with M±±±

ξkpq being (complex) nonlinear interaction coef-

ﬁcients, and

δN ±

ξkpq =N±

ξk −N±

ξkpq .

The pand qmodes evolve similarly:

∂tξ±

p+iω±

pξ±

p=N±

ξpqk +δN±

ξpqk (24)

∂tξ±

q+iω±

qξ±

q=N±

ξqkp +δN ±

ξqkp (25)

Notice that, since there are two eigenmodes (23-25) rep-

resent 6 equations. One can therefore consider reso-

nances between 3growing modes, 2growing modes and a

damped mode, or a growing mode and 2damped modes

etc. However since the frequencies are the same with op-

posing signs, and due to the condition that the ﬂow ﬁeld

is real, we have both kyand −kycomponents, whenever

we have a resonance say of the form ω+

k+ω+

p+ω+

q= 0,

(with k+p+q= 0), we also have ω−

k+ω−

p+ω−

q= 0,

ω+

k−ω−

−p−ω−

−q= 0 or ω−

−k−ω+

p−ω+

q= 0 etc. In other

words, whenever we have a resonance for three +modes,

we also have all the other combinations. The form of the

resonance manifold can be seen in ﬁgures 1 and 2, for

C= 1,κ= 0.2, and ν=D= 10−3, which we will refer

to as the “C= 1 case”.

The three wave interaction system (23-25) can be im-

plemented numerically without much diﬃculty by drop-

ping the δNξterms above. One can also formulate the

same three wave interaction problem in the original vari-

ables Φk,Φp,Φq,nk,npand nqusing the form (3-4)

before the transformation, and then transform the result

using (13). Obviously those two approaches are numeri-

cally equivalent and naturally they give exactly the same

results. We used this to verify that the eigenmode com-

putation was correct. While in general it is unclear if

the eigenmode formulation provides any concrete advan-

tage apart from diagonalizing the linear system, the ad-

vantage becomes self-evident if the resulting ﬂuctuations

have ξ+

kξ−

kand we can drop the ξ−

kmode for example.

1. Is there a diﬀerence between exact and near resonances?

We ﬁrst pick a primary wave-number k= (0,1.125)

which is the linearly most unstable mode on a grid with

dkx=dky= 0.125 for the C= 1.0case and consider the

resonance manifold as shown in ﬁgure 2a in order to pick

a second wave-number p= (−0.5,−1.0) as the point on

the k-space grid that is closest to the resonance manifold.

The third wave-number qis computed from k+p+q= 0.

While a direct numerical simulation only has the wave-

numbers on grid points, a three wave equation solver is

not constrained to such a grid. We can instead com-

pute pto be exactly on the resonance manifold -at least

t

Figure 3. Comparison between exact or near resonances, with

real parts of each eigenmode shown for each wave number as

labeled on the left side of the ﬁgure. The solid line is the exact

(i.e. ∆ω≈2×10−15) resonance of k= (0,1.125) with p=

(−0.5,−1.0632325265492) whereas the dashed line is the near

resonance with p= (−0.5,−1.0) and ∆ω≈0.01. While some

details change, the overall behavior, and saturation levels are

actually very similar.

within some numerical precision- for example by choosing

p= (−0.5,−1.0632325265492). Solving the three wave

equations numerically, using these slightly diﬀerent sets

of wave-numbers, we ﬁnd that having exact resonance or

near resonance (i.e. ∆ω≈2×10−15 vs. ∆ω≈0.01 ) does

not make much diﬀerence in terms of time evolution (see

ﬁgure ), while picking something like p= (−0.5,−1.5),

which gives ∆ω≈0.07 (with ωk≈0.1for comparison)

gives a completely diﬀerent time evolution, where one of

the modes keeps growing linearly without being able to

couple to the other two. We veriﬁed this for a bunch of

diﬀerent sets of wave numbers, and while there are some

diﬀerences in detail, generally both exactly resonant or

near resonant triads lead to saturation but non-resonant

triads can not saturate, possibly due to lack of eﬃcient

interactions. The boundary between what can be con-

sidered a near resonant vs. non-resonant interaction can

actually be deﬁned using this criterion. In particular, it

seems that the triads with one of the frequencies much

smaller than the other two (i.e. ωqωp∼ωk) tend to

support larger overall ∆ω, and nonetheless reach satura-

tion. However it is not clear whether these observations

from a single triad continue to hold when many triads

are interacting with each other.

B. Phase Evolution

Considering the (unwrapped) phase evolution of each

of the modes of the near resonant triad with k=

(0,1.125) and p= (−0.5,−1.0), we observe that while

6

Figure 4. Time evolution of the amplitudes of the eigenmodes

for C= 1 case with k= (0,1.125) and p= (−0.5,−1.0). We

have a “saturated” state with oscillating amplitudes. It seems

that as kand p(the two unstable modes and the two larger

legs of the triads) exchange energy, qplays the role of the

mediator.

Figure 5. Time evolution of the phases ϕ±

kand their sums

ψspsqsk

kpq for C= 1 case with k= (0,1.125) and p=

(−0.5,−1.0). Saturation of the amplitudes as seen in ﬁgure 4

is accompanied by a nonlinear frequency shift as shown in the

top plot and the saturation of the ψspsqsk

kpq ’s as shown in the

bottom plot. Note that ψspsqsk

kpq =const. would correspond to

phase locking.

some amplitude evolution continues, the phases converge

towards straight lines, implying more or less constant fre-

quencies in the ﬁnal stage. These nonlinear frequencies

are substantially shifted with respect to the initial linear

frequencies due to the eﬀect of nonlinear terms. How-

ever it appears that the system remains in resonance as

the sum of the ﬁnal nonlinear frequencies remain very

close to zero. In fact, it appears that the “near resonant”

system evolves towards resonance as a result of these non-

linear corrections, since ∆ωdecreases from the beginning

towards the end.

Using (18) and (19) with the assumption that ∂tχ±

k≈0

Figure 6. Time derivatives of the phases ϕ±

kfor C= 1 case

with k= (0,1.125) and p= (−0.5,−1.0), corresponding to

nonlinear frequencies. Notice that while dϕ−

q/dt appears to

oscillate wildly, since its amplitude χ−

qis vanishingly small,

as can be seen in ﬁgure 4, these oscillations are not important

for the rest of the dynamics.

and ∂tφ±

k=−ω±

k,nl is a constant, we obtain the nonlinear

frequency shift, i.e. δω±

kr =ω±

k,nl −ω±

kr as:

δω±

kr =sign (ωkr )v

u

u

u

tN±

ξk

2

ξ±

k2−γ±2

k, (26)

which can be computed given the ﬁnal amplitudes and

the nonlinear interaction coeﬃcients (21). For example

for the case above the smoothed saturated amplitudes

are shown in the table:

In order to elucidate dynamics of the phases in a triad,

we deﬁne the sums of phases as a separate variable fol-

lowing Ref. 37:

ψskspsq

kpq ≡ϕsk

k+ϕsp

p+ϕsq

q. (27)

We observe that while the phases keep increasing in time,

for a steady state, the phase diﬀerences should remain

bounded. We can write the equations for the amplitudes

as

k, +p, +q, +k , −p, −q, −

|ξ|0.89 0.93 0.52 0.041 0.040 0.0017

ωr0.099 −0.088 −0.020 −0.099 0.088 0.020

γ4.2×10−33.1×10−3−1.8×10−4−1.8−1.8−4.8

ωnl 0.20 −0.19 −0.016 0.20 −0.19 −0.016

δω 0.11 −0.11 −0.075 −1.06 1.12 23.2

Table I. Saturated amplitudes, linear frequencies, linear

growth rates, the ﬁnal nonlinear frequencies and the δω’s that

are computed from (26), rounded to two signiﬁcant ﬁgures

for the C= 1 case with k= (0,1.125) and p= (−0.5,−1.0).

Note that the basic assumption of (26) works only for linearly

unstable modes, and for those δω is not far from ωnl −ωr.

7

∂tχsk

k−γsk

kχsk

k

=X

σp,σq

mskσpσq

kpq cos δskσpσq

kpq −ψskσpσq

kpq χσp

pχσp

q

(28)

which contain the phases only through their sums (i.e. ψ

variables). We can also write an equation for the ψskspsq

kpq

explicitly as:

∂tψskspsq

kpq +ωsk

k+ωsp

p+ωsq

p

=X

σp,σq

mskσpσq

kpq sin δskσpσq

kpq −ψskσpσq

kpq χσp

pχσq

q

χsk

k

+X

σq,σk

mspσqσk

pqk sin δspσqσk

pqk −ψspσqσk

pqk χσq

qχσk

k

χsp

p

+X

σk,σp

msqσkσp

qkp sin δsqσkσp

qkp −ψsqσkσp

qkp χσk

kχσp

p

χsq

q

.

(29)

While the form of (29) looks terribly complicated (e.g.

when we expand the sums we have 8equations, each of

whom having 12 terms on their right hand side) it is use-

ful for insight into phase locking. For example by setting

∂tψskspsq

kpq = 0 in (29), and ∂tχsk

k= 0 in (28), we can ob-

tain constant amplitude, phase locked solutions, if such

solutions exist. Unfortunately, even the computation of

this “ﬁxed point” requires numerical analysis. We can

also integrate (28-29) numerically, which gives exactly

the same result as the system in terms of ξ±

k.

III. INTERACTIONS WITH ZONAL FLOWS

When two non-zonal modes interact with a zonal one

the evolution equations and the nonlinear interaction co-

eﬃcients are diﬀerent from non-zonal three wave interac-

tions discussed in the previous section. Using the origi-

nal variables Φkand nkas in (3-4), zonal and non-zonal

modes interact with the same nonlinear interaction coef-

ﬁcients but diﬀerent linear propagators. However, when

we diagonalize the linear propagator, the nonlinear inter-

action coeﬃcients for zonal and non-zonal modes diﬀer-

entiate.

In particular we have

Mφspsq

kpq =−ˆ

z×p·qq2−p2C2

4Ωsp∗

pΩsq∗

qk2p2q2(30)

Mnspsq

kpq =ˆ

z×p·qC

4Ωsp∗

pΩsq∗

qp2q2Bq−iΩsq∗

qq2

−Bp−iΩsp∗

pp2(31)

Mskφsq

kpq =iˆ

z×p·q

2Ωsq∗

qq2Bq−iΩsq∗

qq2

−(Bk−iΩsk

k)q2−p2(32)

Msknsq

kpq =iˆ

z×p·qC

2Ωsq∗

qq2(33)

so that for three waves k,pand qwith qy= 0, we can

write:

∂tΦq+νZF Φq=X

sk,sp

Mφsksp

qkp ξsk∗

kξsp∗

p(34)

∂tnq+DZF nq=X

sk,sp

Mnsksp

qkp ξsk∗

kξsp∗

p(35)

∂tξsk

k+iωsk

kξsk

k=X

sp

Mskspφ

kpq ξsp∗

pΦ∗

q+Mskspn

kpq ξsp∗

pn∗

q

(36)

∂tξsp

p+iωsp

pξsp

p=X

sp

Mspφsk

pqk Φ∗

qξsk∗

k+Mspnsk

pqk n∗

qξsk∗

k.

(37)

We can write these in the form (23-25) by letting ξ+

q= Φq

and ξ−

q=nqand paying attention to the form of the

interaction coeﬃcient Mskspsq

ξkpq when one of the legs is

zonal.

In order to study the interactions between two modes

with a zonal ﬂow in the Hasegawa-Wakatani system nu-

merically, we pick a primary wave-number k= (0,1.125)

which is the linearly most unstable mode on a grid with

dkx=dky= 0.125 for the C= 1.0case. We choose

p= (−0.5,−1.125) so that q= (0.5,0) is a zonal wave

number. The 6ﬁeld variables are now ξ±

k,ξ±

p,Φqand

nqwhose evolutions are shown in ﬁgure 7 for the case

C= 1,νZ=DZ= 0 and γk&γp>0. In the ﬁnal state,

the system ﬁnds a ﬁxed point characterized by constant

nonlinear frequency shifts, constant amplitudes and con-

stant ψkpq’s. However this kind of steady state solution

seems to be exclusive to the single triad case.

A. Triad pairs

Because of the symmetry of the system, if we consider

two wave-numbers p1=−k−qand p2=−k+qwith

kin ˆ

yand qin ˆ

xdirections, we get two triads that are

reﬂections of one another with respect to the axis deﬁned

by k. Such a system involves four diﬀerent wave-numbers

connected with two diﬀerent triads. Including the pq

transformation we have four triads as shown in ﬁgure 8.

However as long as we use symmetric forms for the inter-

action coeﬃcients, we can drop the two triads we obtain

8

Figure 7. Time evolution of the three wave equations involv-

ing a zonal mode q, for the case C= 1,νZ=DZ= 0 and

γk&γp>0with ky= 1.125 and q= 0.5[i.e. k= (0, ky),

p= (−q, −ky)and q= (q, 0)]. The system reaches a steady

state by introducing nonlinear frequencies in order to arrive

at a state where the sums of phases ψkpq ’s are constant. Note

that it is p, which becomes the dominant mode in the ﬁ-

nal state and the existence of zonal ﬂows does not lead to a

complete suppression of turbulence. Instead the zonal ﬂow

acquires a constant nonlinear frequency.

from the pqtransformation and count only two tri-

ads. Since the two triads of such a pair are reﬂections of

one another, the nonlinear interaction coeﬃcients diﬀer

only in sign while the complex frequencies are the same,

and as there are two eigenmodes for each wave-number,

we have 8equations. The equations for zonal modes can

be written from (34-35) as:

∂tΦq+νZΦq=X

sk,sp

Mφsksp

qkp1ξsk∗

kξsp1∗

p1−ξsk

kξsp2

p2(38)

∂tnq+DZnq=X

sk,sp

Mnsksp

qkp ξsk∗

kξsp∗

p1−ξsk

kξsp

p2, (39)

which is possible since Msksp{n,φ}

ξkp2q=−Msksp{n,φ}

ξkp1qbe-

cause p2

2=p2

1and p2y=p1ywhile p2x=−p2x. The

equation for the primary mode, can be written as:

∂tξsk

k+iωsk

kξsk

k=X

spMskspφ

ξkpq Φ∗

qξsp∗

p1+ Φqξsp∗

p2

+Mskspn

ξkpq n∗

qξsp∗

p1+nqξsp∗

p2, (40)

Figure 8. All the four triads involved in the interaction be-

tween the most unstable mode with k=kyˆ

ywith ky= 1.125

and a given zonal mode with q= 1.0, obtained by reﬂection

with respect to kand the exchange of pand qof the primary

triad, which is shaded. The existence of the reﬂected triad is

indeed important as it changes the qualitative behavior with

respect to the single triad case.

and the remaining two equations are the same as (37)

but with diﬀerent signs and conjugations:

∂tξsp

p1+iωsp

p1ξsp

p1=X

skMspskφ

ξp1kq Φ∗

q+Mspskn

ξp1kq n∗

qξsk∗

k

(41)

∂tξsp

p2+iωsp

p2ξsp

p2=−X

skMspskφ

ξp1kq Φq+Mspskn

ξp1kq nqξsk∗

k

(42)

where ωsp

p2=ωsp

p1. Notice that this is also equivalent to

one of the radial Fourier modes of a quasi-linear (e.g.

zonostrophic) interaction, where for each ﬁeld one would

consider a single pybut the full spatial dependence in x.

The results of the system (38-42) are shown in ﬁgure 9

for the C= 1 case with ky= 1.125 [i.e. k= (0, ky),

p1= (−q, −ky),p2= (q, −ky)and q= (q , 0)] for

q= (1.0,1.5,2.0,4.0) from top to bottom respectively.

For q≤kywe have instability and pkeeps growing expo-

nentially whereas for q > kywe get some sort of steady

or limit cycle state. Performing a scan of kyand qfor

this two triad system (keeping in mind that for ky>2we

have no instability and therefore the pump mode decays)

we observe that we can deﬁne a four wave interaction con-

dition of the form ωsk

kr +ωsp1

p1r+ωsp2

p2r+ωsq

qr = 0, which turns

into Ωsk

k+2Ωsp

p= 0 since ωqr = 0 and ωp1r=ωp2r=ωpr .

There seems to be 3distinct regions in ﬁgure X: for q < 1,

the ξ+

pmodes grow exponentially as in the top plot of ﬁg-

ure 9, for the central region where q≈1, we have satu-

ration and then somewhat chaotic evolution, and ﬁnally

for q1, we observe limit cycle oscillations between ξ+

k

9

Figure 9. Evolution of a triad pair with the same parameters

as ﬁgure 7, no zonal ﬂow damping νZ=DZ= 0 and ky=

1.125 [i.e. k= (0, ky),p1= (−q, −ky),p2= (q, −ky)and

q= (q, 0)] for four diﬀerent values of q= (1.0,1.5,2.0,4.0)

from top to bottom for which the growth rates of the subdom-

inant modes are γp= (0.00099,−0.0016,−0.0042,−0.017) re-

spectively. Note that apart from the second plot, which dis-

plays some chaotic behavior, the curves for ξ+

p1and ξ+

p2overlap

almost exactly.

and ξ+

pmodes, mediated by zonal ﬂows.

One is tempted to argue that since the pwith px< py

wins the competition to attract more energy, the cascade

will proceed in this direction, and in the next step we can

consider the interaction of this ξ+

pas the pump mode for

the next triad etc. However, since each mode interacts

with many triads simultaneously, the fact that ξ+

pwins

the competition in the single triad (or one triad and its

reﬂection) conﬁguration does not really mean the energy

will indeed go this way.

B. Triad Networks

In order to study the fate of the cascade, we need to

consider multiple triads that are connected to one an-

other. However as we add more zonal and non-zonal

modes, it becomes quite complicated to keep track of all

the interaction coeﬃcients, conjugations etc. In order to

simplify this task, we can divide the problem into two

steps i) construction of a network of three body inter-

actions and ii) computation of the evolution of the ﬁeld

variables on this network. For example for the above

problem we need to consider a network of Nk= 4 wave

Figure 10. The structure of the network with a single ky

with ky= 1.125 shown as a ﬁlled (red if in color) node. A

reduced version with qvalues that only go up to 0.5is shown

for clarity. Notice that in this network while all of the 26

triads involve one of the zonal modes, only 8of them involve

the q= 0 mode.

number nodes, coupled to Nt= 2 triads, with Nf= 2

ﬁelds in each node, with an interaction coeﬃcient of the

size Nf×Nf×Nffor each connection. Since a network

in Fourier space is made up of three body interactions,

for each node, we can compute a list of interacting pairs

and the interaction coeﬃcients, so that we can write

∂tξi

`+Lij

`ξj

`=1

2NX

`0,`00=i`

Mijk

``0`00 ξj

`0c`0ξk

`00 c`00 (43)

where i`is the list of precomputed interaction pairs for

the node `. The indices i,jand kcorrespond to dif-

ferent ﬁelds (eigenmodes or Φkand nk), the matrix Lij

`

is the linear matrix in kspace (i.e. diagonal with the

elements iω±

`for the eigenmodes), the Mij k

``0`00 is the in-

teraction coeﬃcient for each interaction and Nis the

number of independent wave number nodes so that when

we reach the full grid, we have exactly the same interac-

tion coeﬃcients as the system formulated using discrete

fast Fourier transforms (i.e. divided by Nx×Ny). Fi-

nally if we write the triad interaction condition in the

form k`+σ`0k`0+σ`00 k`00 = 0 , where σare ±1, the

ξj

`0c`0are deﬁned as:

ξj

`0c`0

=(ξj

`0σ`0=−1

ξj∗

`0σ`0= +1

This is necessary unless we have the negative of each wave

number vector as a separate node in the network.

Notice that when computing the nonlinear interaction

coeﬃcients for the eigenmodes, we would use (21) if all

the nodes have nonzero ky. In contrast we would use (30)

10

and (31) if the receiving node (i.e. node `) is zonal or

(32) and (33) if one of the interacting pairs (i.e. `0or

`00) are zonal. Two or more zonal mode do not interact

because of the geometric factor ˆ

z×p·q, which appear

in front of all the interaction coeﬃcients.

Finally, if it makes sense to zero out some of the ﬁelds

at a given wave-number (e.g. in eigenmode formulation,

we may decide to throw away some damped modes),

one may switch to a formulation where each node corre-

sponds to a wave-number/ﬁeld variable combination via

{kx, ky, sk} → `. In this case, assuming that the linear

matrix Lij

`in (43) diagonal takes the form:

∂tξ`+iω`ξ`=1

NX

`0,`00=i`

M``0`00 ξc`0

`0ξc`00

`00 (44)

C. Order Parameters

The phases of wave-number nodes in Hasegawa-

Wakatani turbulence evolve according to (19) or written

explicitly as (22). This suggests that one can possibly de-

ﬁne some kind of order parameter for this system. The

usual deﬁnition of the Kuramoto order parameter can be

written for the network formulation of (44) as:

z=reiψ =1

NX

`

eiϕ`(45)

without explicitly distinguishing +or −modes. How-

ever this order parameter based on an unweighted sum

is probably relevant only if all the oscillators were identi-

cal with all-to-all, unweighted couplings of the Kuramoto

type. Instead we can use an amplitude ﬁltered Kuramoto

order parameter (i.e. the sum is computed only over the

oscillators with an amplitude larger than a threshold), or

deﬁne a weighted version of (45) as:

z=reiψ =P`χ`eiϕ`

P`χ`

(46)

whose absolute value would tends towards 1if the rele-

vant phases (i.e. those that have large amplitude) are the

same. However note that the weighted order parameter

tends towards 1also when one of the modes dominate

over the others, while ψas deﬁned in (46), can still be

used as a mean phase.

It would also make sense to look at the net eﬀect on

the nonlinear term on the phases instead. As discussed

in Section I B, since we can write:

∂tϕ`=−ω`+1

Nχ`

Im

X

`0,`00=i`

M``0`00 ξc`0

`0ξc`00

`00 e−iϕ`

(47)

Figure 11. Time evolution for a number of triad pairs (as de-

ﬁned in section III A) with diﬀerent values of qin the network

of interacting triads for C= 1 case with νZF =DZ F = 10−3.

A steady state turbulence level is observed, with elevated lev-

els of zonal ﬂows at large scales.

for the evolution of the phase, we can deﬁne:

Z`=R`eiψ`=1

Nχ`

X

`0,`00=i`

M``0`00 ξc`0

`0ξc`00

`00

(48)

with d`being the number of interactions for the node `

(i.e. length of i`), as some kind of local order parameter

for the node `, allowing us to write the phase equation

as:

∂tϕ`=−ω`+R`sin (ψ`−ϕ`), (49)

which attracts the system towards ϕ`=ψ`+ 2nπ.

D. Speciﬁc network conﬁgurations

1. Network with a single ky:

We consider a network of triad pairs as discussed in

section III A, with a single value of kyand qvalues that

go from 0.125 to 4.0in steps of 0.125. Notice that such

a network has many diﬀerent types of interactions as

shown in ﬁgure 10, but all of those involve one of the

zonal modes, which means that if we compute the inverse

Fourier transform in the xdirection, the network can be

seen to be equivalent to the single ky, full-x, quasi-linear

model [38, 39], since in both cases we have full spatial

evolution but only nonlinear coupling is with the zonal

ﬂow.

For the case C= 1, without zonal ﬂow damping (not

shown) we observe that the zonal ﬂows dominate and all

the other modes decay to zero. This may well be what

happens also in direct numerical simulations (DNS) even-

tually: what we observe in numerical simulations without

zonal ﬂow damping is a continual increase of zonal ﬂows

even for very long simulations.

In contrast, when we introduce zonal ﬂow damping by

letting νZF =DZ F = 10−3, we get dynamics and k-

11

Figure 12. The top plot shows the order parameter rde-

ﬁned in (45) or (46) as a function of time for a network with

single qand multiple ky. The two deﬁnitions are in reason-

able agreement apart from the peak around t= 2500 for

the weighted order parameter, which corresponds to the lin-

ear growth phase, where only a few modes around the most

unstable mode dominate. This can be seen at the bottom

plot where the amplitudes of a triad pair with q= 0.5and

ky= 1.125 are shown. Around t= 2500 the blue curve clearly

dominates.

Figure 13. The structure of the network with a single q= 0.5

zonal mode, shown as a ﬁlled (red if in color) node. A reduced

version with kyvalues that only go up to 0.5is shown for

clarity. Only 8of the full 26 triads involve the zonal ﬂow.

spectra which look more like fully developed Hasegawa-

Wakatani turbulence, as shown in ﬁgure 11, with high

levels of zonal ﬂows at large scales.

Figure 14. The top plot shows the order parameter rde-

ﬁned in (45) or (46) as a function of time for a network with

single qand multiple ky. The two deﬁnitions are in reason-

able agreement apart from the peak around t= 2500 for

the weighted order parameter, which corresponds to the lin-

ear growth phase, where only a few modes around the most

unstable mode dominate. This can be seen at the bottom

plot where the amplitudes of a triad pair with q= 0.5and

ky= 1.125 are shown. Around t= 2500 the blue curve clearly

dominates.

2. Network with a single q:

Here, we consider a network of triad pairs with a single

q, and a grid of values of kygoing from 0.125 to 4.0in

steps of 0.125. A reduced version of such a network is

shown in ﬁgure 13. Physically this network corresponds

to the opposite case where we consider a single qwith

the whole ydynamics if we compute the inverse Fourier

transform in y. Since it involves bunch of oscillators with

diﬀerent frequencies (as ωis mostly a function of ky) that

are coupled to each other and to a zonal mode that may

play the role of a dominant mean ﬁeld, it has the basic

ingredients that may lead to synchronization.

Nonetheless numerical observations suggest that there

is no obvious route to global synchronization in the three

body network of interacting triads consisting of a zonal

mode and drift waves of diﬀerent kyeither. The weighted

order parameter shows a brief increase during the nonlin-

ear saturation phase as the energy is transferred to the

zonal ﬂow, but otherwise remain close to zero, while the

Kuramoto order parameter simply remains close to zero

the whole time as can be seen in ﬁgure 14. Since we ob-

served no qualitative diﬀerence between the runs with or

without zonal ﬂow damping for this case, we only show

those with νZF =DZ F = 10−3.

12

t

Figure 15. The top plot shows the order parameter rdeﬁned

in (45) or (46) as a function of time for a DNS. The bottom

plot shows the amplitudes of a triad pair with q= 0.5and

ky= 1.125 in order to compare with the earlier plots. The

saturation levels for the amplitudes are diﬀerent because of

the normalization factor N−1

xN−1

yin front of the nonlinear

term implied in discrete Fourier transforms.

E. Direct numerical simulations

One can think of direct numerical simulation (DNS)

on a regular rectangular grid as a “network” in Fourier

space, in the sense that it consists of a collection of wave

number nodes connected to each other through triadic

interactions. In contrast to the networks that we con-

sidered that contain a single zonal mode, or a single

q= 0 mode, a regular rectangular grid has all the pos-

sible wave-numbers in a particular range, and it allows

using more eﬃcient methods for computing the convo-

lution sums. In practice, the high resolution direct nu-

merical simulations that we discuss here were performed

with a standard pseudo-spectral solver (i.e. with peri-

odic boundary conditions in both directions) using 2/3

rule for dealiasing and adaptive time stepping.

As with all the previous examples of single or multiple

triads, or networks with a particular selection of nodes

and triads, we use C= 1,κ= 0.2. Since we have a larger

range of wave-numbers, we choose ν=D= 10−4, with

a box size of Lx=Ly= 16πand a padded resolution of

1024 ×1024. The results show (see ﬁgures 15 and 16):

i. Initial linear growth followed by nonlinear satura-

tion.

ii. Formation and ﬁnally suppression of nonlinear of

convective cells that transfer vorticity radially.

iii. Consequent stratiﬁcation of vorticity leading to a

state dominated by zonal ﬂows (as in ﬁgure 16).

iv. Coherent nonlinear structures (e.g. vortices) that

are advected by the zonal ﬂows in regions of weak

zonal shear, get sheared apart if they fall into a

region of strong zonal shear.

Figure 16. Snapshots of vorticity and density at t=5000 from

DNS. The blue curve in both plots shows the zonal velocity

whose values are given on the right hand axes. An example

coherent vortex, that was moving upwards is encircled.

Since the wave-like dynamics seems to be primarily in

ydirection and reasonably localized in x, we can com-

pute the Fourier transform in yand plot phase of ξ±

ky=

χ±

kyeiφ±

kyat each x, compute ∂tφ±

ky(x, t)in order to com-

pute the phase speeds (see ﬁgure 17). We can also com-

pute an order parameter as a function of xand tfrom

this data.

While it is clear from 15 that there is no global syn-

chronization in direct numerical simulations, the plateau

form of the phase velocity as a function of kyat the radii

where it is positive for large scales, suggest that a pro-

cess of phase locking similar to soliton formation in non-

linear Schrödinger equation, where nonlinearity would

balance dispersion is at play for a range of kyvalues

around the linearly unstable mode. While ω/kybeing

the same across a range of xand kyvalues is obviously

very diﬀerent from ωbeing the same. However if we

note that the nonlinear dispersion relation takes the form

ω(x, ky) = vφ(x)ky, at the lowest order we can see that

the frequency in the frame moving with the zonal ﬂow ve-

locity becomes zero. This is roughly consistent with what

we see in time evolution, where coherent structures like

rotating vortices are advected by zonal ﬂows. In order

for such a detailed structure

IV. CONCLUSION

A detailed analysis of triadic interactions formulated

in terms natural frequencies reveals the complex nature

of the dynamics of the phases and amplitudes in the

Hasegawa Wakatani system. In particular, it is observed

that a single resonant (or near resonant) triad, includ-

ing a pump mode and two other modes, can saturate

by adjusting the sums of phases of its legs (ψskspsq

kpq =

φsk

k+φsp

p+φsq

q) to be asymptotically constant, resulting

in a set of nonlinearly shifted frequencies and constant

amplitudes. When the interactions with zonal ﬂows are

considered, a similar saturation is possible for a single

13

triad even without the condition of resonance. However

this solution breaks down when we add the triad, which

is the reﬂection of the original one with respect to the y

axis (or the wave-vector kvector). Instead we observe

three diﬀerent behavior for these triad pairs as a function

of the radial wave number.

i. For smaller radial wave numbers, we ﬁnd that the

subdominant mode becomes the dominant one and

grows exponentially. We call those unstable tri-

ads. They are associated with unstable subdomi-

nant modes.

ii. For medium radial wave numbers, after an initial

growth phase, the system saturates with a more or

less chaotic evolution, where the energy goes back

and forth between the modes. We call these satu-

rated triads. They are associated with weakly un-

stable, or weakly damped subdominant modes.

iii. For large radial wave numbers the system decays to

a steady state solution after a number of limit cycle

oscillations. In some cases, these limit cycle oscil-

lations can continue until the end of the simulation

time. We call these decaying triads (even though

they don’t decay to zero but to a constant). They

are associated with strongly damped subdominant

modes.

In order to study the dynamics when those triads are con-

nected to one another, we considered a network formu-

lation where the wave numbers (or wave number eigen-

mode combinations) are considered as nodes, and each

triad represents a three body interaction. It is shown

while the zonal ﬂow is almost never dominant in a single

triad, when the whole triad network with a large number

of triads is considered, the zonal modes become domi-

nant almost in each triad. Thus, the system can reach a

steady state where the zonal ﬂow dominates as the other

modes decay.

In terms of triadic interactions, as the zonal ﬂow be-

comes dominant, it plays the role of a collective mean

ﬁeld, in the sense that for each mode individual inter-

actions with non-zonal modes start to become less im-

portant compared to the interaction with the zonal ﬂow.

This happens only when the number of triads is large

enough so that the collective wins over the individual.

It is interesting to note that this picture is qualitatively

consistent with that of inhomogeneous wave-kinetic for-

mulation, where the zonal ﬂow is treated as a collective

mean ﬁeld, and the direct interaction between the modes

are either dropped or modeled with a diﬀusion operator.

This suggests that the wave-kinetic formulation may hold

beyond its range of validity.

Playing with the range of radial wave-numbers of the

network model, we observe that when the range includes

only unstable triads [i.e. (i) above], or unstable and sat-

1 3 5

14.74 29.48 40.54

Figure 17. Proﬁles of phase velocity as a function of ky, at

three diﬀerent values of x(i.e. 14.74,29.48 and 40.54) aver-

aged over t= [4500,5000] shown at the top plot. The three

plots that follow show the detailed time evolution (on the left

yaxes) of phase velocity as a function of xfor three diﬀerent

values of ky(i.e. 1,3and 5), together with the mean velocity

proﬁle shown for reference (on the right yaxes). The phase

velocity is computed using vφ=−∂tφ+

ky(x, t)/ky. The ky’s

for which the time evolution is given and the x’s for which the

phase velocities are shown are marked with horizontal lines

in the corresponding ﬁgures.

urated triads [i.e. (i) and (ii) above] the network sys-

tem remains unstable. It saturates only when we include

a suﬃcient range of decaying triads, with subdominant

modes with γ+

p<0. This means that ’local coupling to

damped modes’ (i.e. γ−

pmodes even though γ+

p>0) is

not a real mechanism for turbulent saturation. However

since the fact that γ+

p<0for those modes do not come

directly from dissipation but rather the detailed form of

the linear growth/damping whose form is determined by

various parameters including dissipation, it is correct to

argue that in contrast to the Kolmogorov picture where

there is an injection scale, a dissipation scale and the in-

ertial range in between, plasma turbulence can generate

and dissipate energy in much closer scales, even though

one may observe clear power law scalings.

One of the goals of the current paper was to study the

eﬀect of nonlinear synchronization of drift waves[40] on

the turbulent cascade using a framework similar to the

Kuramoto model[41], which has already been attempted

using simple models in fusion plasmas[42, 43]. We hoped

by considering a network of connected triads interacting

with zonal ﬂows we could setup a system that would tend

14

toward synchronization through slight nonlinear modiﬁ-

cations of the frequencies through their interactions with

the zonal ﬂow, playing the role of the control parameter.

However due to particular form of the systematic depen-

dency of the frequencies to the wave-numbers through the

dispersion relation, such a system does not seem to tend

towards synchronization. It should be checked whether

or not the discretization resulting from boundary condi-

tions, for example in cylindrical geometry change this pic-

ture drastically by impeding resonant interactions[44, 45]

especially among large scale modes.

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