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Phase and amplitude evolution in the network of triadic interactions of the Hasegawa-Wakatani system

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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 $y$ axis 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.
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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
Φen+DΦ2Φ(1)
and an equation of continuity
∂t n+ˆ
z× ∇Φ· ∇n+κ∂yΦ = Ce
Φen+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 DΦ2e
Φ=ν4e
Φ
from kinematic viscosity, whereas for the zonal flows
DΦ2Φ=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) = D2enand 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+ (AkBk) Φk=C
k2nk+NΦk(3)
tnk+ (Ak+Bk)nk= (Ciκky) Φk+Nnk (4)
where
Ak=1
2Dk2+C+C
k2+νk2 (5)
and
Bk=1
2Dk2+CC
k2+νk2. (6)
using the simpler notation ΦkΦkand defining:
Nnk =1
2X
4
ˆ
z×p·qΦ
pn
qΦ
qn
p(7)
and
NΦk=1
2X
4
ˆ
z×p·qq2p2Φ
pΦ
q
k2. (8)
Diagonalizing the linear terms we can write:
tξ±
k+±
kξ±
k=N±
ξk (9)
3
with the complex eigen-frequencies ω±
k=ω±
rk +±
kthat
can be written as:
ω±
k= Ω±
kiAk
with
±
k=± σkrHkGk
2+irHk+Gk
2!(10)
where σk=sign (κky),
Hk=qG2
k+C2κ2k2
y/k4, (11)
and
GkB2
k+C2
k2. (12)
This allows us to write the two linear eigenmodes as:
ξsk
k=nk+k2
C[Bkisk
k] Φk. (13)
where sk=±. The nonlinear terms in (9) become:
Nsk
ξk =Nnk +k2
C(Bkisk
k)NΦk, (14)
and the inverse transforms can be written as:
Φk=i
2
C
k2X
sk
ξsk
k
sk
k
(15)
nk=i
2X
sk
1
sk
k
(Bk+isk
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+k2iκky
2A2
kΦk
ξ
k=nk1 + 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=nkΦk), 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±
ke±
kinto (9), we get:
tχ±
ke±
k+±
kχ±
ke±
k=N±
ξk eNξ±
k, (17)
taking the real part we obtain the amplitude equations:
tγ±
kχ±
k=N±
ξk cos φNξ±
kφ±
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±
kφ±
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±
kφ±
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= 103is 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 ω0(with q=kp). 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 q2Bqisq
q
p2Bpisp
pq2p2(Bkisk
k)
(21)
where
mskspsq
kpq Cˆ
z×p·q
4Ωsp
psq
qq2p2
and ±
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
pφsq
qφsk
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+±
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+±
pξ±
p=N±
ξpqk +δN±
ξpqk (24)
tξ±
q+±
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+
kω
pω
q= 0 or ω
kω+
pω+
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= 103, 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. ω2×1015) 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 ω0.01. 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. ω2×1015 vs. ω0.01 ) does
not make much difference in terms of time evolution (see
figure ), while picking something like p= (0.5,1.5),
which gives ω0.07 (with ωk0.1for 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. ωqωpωk) 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±
k0
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
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±
k2γ±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
γ4.2×1033.1×1031.8×1041.81.84.8
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
kγsk
kχsk
k
=X
σpq
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
σpq
mskσpσq
kpq sin δskσpσq
kpq ψskσpσq
kpq χσp
pχσq
q
χsk
k
+X
σqk
mspσqσk
pqk sin δspσqσk
pqk ψspσqσk
pqk χσq
qχσk
k
χsp
p
+X
σkp
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·qq2p2C2
4Ωsp
psq
qk2p2q2(30)
Mnspsq
kpq =ˆ
z×p·qC
4Ωsp
psq
qp2q2Bqisq
qq2
Bpisp
pp2(31)
Mskφsq
kpq =iˆ
z×p·q
2Ωsq
qq2Bqisq
qq2
(Bkisk
k)q2p2(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+sk
kξsk
k=X
sp
Mskspφ
kpq ξsp
pΦ
q+Mskspn
kpq ξsp
pn
q
(36)
tξsp
p+sp
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=kqand 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
p1ξsk
kξsp2
p2(38)
tnq+DZnq=X
sk,sp
Mnsksp
qkp ξsk
kξsp
p1ξsk
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+sk
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+sp
p1ξsp
p1=X
skMspskφ
ξp1kq Φ
q+Mspskn
ξp1kq n
qξsk
k
(41)
tξsp
p2+sp
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 qkywe 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 sk
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 q1, 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 ±
`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`+`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=re=1
NX
`
e`(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=re=P`χ`e`
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`
(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 = 103.
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`e`=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`+ 2.
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 = 103, 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 = 103.
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 N1
xN1
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= 104, 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±
kye±
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.
[1] A. Hasegawa and M. Wakatani, Pys. Rev. Lett 50, 682
(1983).
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  • M Wakatani
A. Hasegawa and M. Wakatani, Pys. Rev. Lett 50, 682 (1983).
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  • J Herring
M. Lesieur and J. Herring, Journal of Fluid Mechanics 161, 77 (1985).
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  • K E Heikes
F. H. Busse and K. E. Heikes, Science 208, 173 (1980).
  • L K Currie
  • S M Tobias
L. K. Currie and S. M. Tobias, Physics of Fluids 28, 017101 (2016).