J. Plasma Fusion Res. SERIES, Vol.4 (2001) t6t-166

Self Organization oI Zonal Flows:
Some Physics Behind the Color Viewgraphs
CHAMPEAUX Stephanie, DIAMOND Patrick H.* and MALKOV Mikhail
University of Califurnia, San Diego, ln Jolla, CA 92093-03j9, USA(Received: 5 December 2000 / Accepted:27 August 2001)

Abstract
Two model calculations which elucidate the basic physics of zonal flow generation are presented.
The first uses statistical ray optics to clarify the physics of the modulational instability process. The
second uses the envelope formalism to obtain and solve a simple system ofequations for the zonal flow
and drift wave envelope. This simple system is closely analogous to the Zakharov equations forLangmuir turbulence.

Keywords:
zonal flow, modulational instability, drift wave turbulence

1. Introduction

attractively packaged as striking color viewgraphs, forexample. However, there is still a need for physicalinsight, as gleaned from simplified, model problems.
Moreover, such model problems often prove useful inbenchmarking complex codes. In this paper, we discuss
two model calculations pertinent to zonal flow physics.
In Section 2, a physical picture ofzonal flow generation,using coherent and statistical ray theory, is presented.
This picture confirms and elucidates results obtained
previously using modulational analyses. In Section 3,

Zonal flows are shear layers or strongly anisotropic
vortices with fte = 0, ftrr 0, kLp, finite and (nearly) zero=frequency, Q = 0. Since fte = 0,i, = 0, so that zonal

flows are intrinsically incapable ofdriving transport, andthus represent a reservoir of benign fluctuation energy.
Zonal flows may be thought of as a particular limit (ft.
) ks= 0) of the more general convective cell structure.
In contrast to the familiar mean E x 86 shear flows, withdirection determined by profile structure andcharacterized by a single scale, zonal flows are ofenvelope perturbation theory is used to derive a"minimal" system of equations for the drift wave +coherent flow problem. This system is analogous to theindeterminate direction and exhibit a spectrum of scales
producing a spatially complex flow profile. Zonal flows

are nonlinearly generated by drift waves via

Za|,harov equations familiar from Langmuir turbulence,and may be used to examine both zonal and streamer
flow growth.

modulations of the radial flux of vorticity (i.e. charge
separation current [1,2]) and are damped by ion-ion
collisions [3], nonlinear feed-back on the underlying
drift waves [4] or (possibly) by Kelvin-Helmholtz typeinstabilities which disrupt them [5].
Several aspects of zonal flows mentioned above
have been extensively (indeed exhaustively) studied,especially via numerical simulations. Such simulations
are powerful tools, and yield results which can be*Corresponding author's

2. Physical Picture of Zonal Flow
Generation
It is illuminating to present a short, 'back-of-anenvelope' type demonstration of zonal flow instability.
Consider a packet of drift waves propagating in an

e-mail: pdiamond@ physics.ucsd.edu
161@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Champeaux S. et al., Self Organization of Zonal Flows: Some Physics Behind the Color Viewgraphs

ensemble of quasi-stationary, random zonal flow shear

calculated. To this end, we use the Wave Kinetic

layers. Take the zonal flows as slowly varying withrespect to the drift waves (i.e. Q .. ,), etc. quasistationary. The spatially complex shearing flow will
result in an increase in (kT, the mean-square radial
wave vector (i.e. consider a random walk of k,, asdescribed by eikonal theory). In turn the drift wave
frequency a"*l(l + ft1p3) must then decrease. Since O

Equation (W.K.E.)

4dt * t v.+ V )'VN - ox
3- 1ot + krv') +ok= yrN - L.{D1, Ntl No,
and the methodology of quasilinear theory to obtain:< ok, the drift wave action density N(&) = e(L)lar,isku(i', N | =o*conserved, so that drift wave energy must also decrease.
As the total energy of the system of waves and flows is

also conserved (i.e. €*un" * tFlow = const.), it thus
follows that the zonal flow energy must, in turn,increase. Hence, the initial perturbation is re-enforced,
suggestive of instability. Note that the simplicity andD*= k'eflow generation is a robust and ubiquitous phenomenon.

(n)

(4a)

ak,

V n'l i'r,l' n(t, a

(4b)

tr)

@c)

In the broadened resonant response R(\,d, A is

A slightly larger envelope is required for a'physical argument' which is also quantitatively

neglected compared to qVr,,, since zonal flows are
slowly varying with respect to the drift waves. It follows
that fluctuatin g zonal flow band width is much narrower
than drift wave frequency. The term LaoN2lNo
represents drift wave nonlinear damping via selfinteraction of drift waves (i.e. cascade by local

predictive. Consider a drift wave packet propagating in asheared flow field. Take a4 ,lr'rl ano l& l>lv'"tvul,
so that wave action density is conserved (i.e. N(ft) = ttfo,

a constant). Thus, wave energy density evolves
according to:interaction).
Here q is the radial wave number of the zonal flow*"1'=N'+and equilibrium balance in the absence of flow has been="(+.v,*.# +)used to relate La1,to 1. The wave energy then evolves

according to:

(1)

d€(L)

Lu

_ zp? pK k,
dt (r * *', p1)' d k,_[ztr,kup?\,,,^.
lvrt(k).

\ t+kipl Ian ( t, a) = r,, ((nr,,,)'.

clarity of this argument supports the assertion that zonal

=l

(3)

-

(s)

As total energyrof the stationary wave-flow system isHere, we have assumed stationary, isotropic turbulenceand have used the eikonal equation dk,ldt = -a&ev)l?-x. Eqn. (l) just states that the drift wave packet loses
or gains energy via work on the mean flow via waveinduced Reynolds stress. (lndeed, k,ke€(&) - |i,iu)t1.

i.- """:;iJilk1" :::;1,."[l;;_, _r>-& !1 ^x,(t __, c:
To=-2q,(r**ip1),

Note as well that the factor k,kse(b)V's is rathert. -)q'lo, *oK, (n)obviously suggestive of the role of triad interactions incontrolling fluctuation - flow energy exchange. Forn(\-zonal flows, the shear is random and broad-band, so that
Hence,(N) + fr, and Nsv'p -)v'B -->t'p, N-;(ui").

(

Eqn. (1) may be re-written as:

d- r(L)=-vr.,kr(i'rN),
dt-us'[he z'na,'r)=(t +t, pz) Q) .

,

(6a)

(6b)

Here (4) is the mean potential enstrophy density of thedrift wave turbulence.
The result given above in Eqn. (6a), obtained bytransparent physical reasoning, is identical to that

Q)

where Vr,, is the drift-wave radial group velocity. To
complete the argument, the correlator (V'BN) must bederived previously by formal modulational stabilityr62Champeaux S. et al., Self Organization of Zonal Flows: Some Physics Behind the Color Viewgraphs

arguments [l]. Note that a(n]lak, < 0 (a condition
which is virtually always satisfied in drift wave
turbulence) is required for zonal flow growth. In
addition, the argument above reveals that drift wave ray

based on a two scale analysis where the density andelectrostatic potential fields are split into small scale ( i )chaos provides the key element of irreversibility, whichassociated with drift-wave propagation and large scaleis crucial to the wave-flow energy transfer dynamics.
Here, ray chaos requires overlap of the 9/q = Vr,,
resonances in D6 a condition easily satisfied for finite
lifetime drift wave eddys and (nearly) zero frequency
zonal flows (i.e. Aa4 >> O). Under these conditions a(flows [1,12]. In order to describe nonlocal interactions

in drift-wave turbulence, the approach adopted here isl) fluctuations n = fi + n and Q =Q *@-. The small scale

turbulence is considered within the adiabatic regime d =vlo"kfllvao > I with a the parallel electron diffusion in acharacteristic wave period at6. In this regime, electrons
tend to Boltzmann distribution it = e6lT".In conrrast,
the large scale fields associated here to the limit a -+ 0
correspond to ft;1 - 0 modes. The Boltzmann responsethen does not hold on large scale, (i.e. therc n +"$n").
Therefore a reductive perturbation theory [13] is applied
to the coupled Hasegawa-Mima [14] equation for drift

positive Lyapuanov exponent is present and neighboring

drift waves rays diverge exponentially in time. Ray
chaos in turn assures that zonal flow shearing and wave

refraction are random, thus validating the use of thestochastic methodology employed here. In the casewhere rays are not chaotic, envelope perturbation
formalism, methods from the theory of trapping, orparametric instability theory must be used to calculate
zonal flow generation.

waves including a modified plasma response to themodulations of the density and electrostatic potential,
and the 2D Euler equation (equivalent to the meanpolarization current) and continuity equation for themean flow. The dynamics of ft1; - 0 flow and drift wave
envelope structures with arbitrary anisotropy in thepoloidal plane is described, after moving to drift-wave
group velocity reference frame by

Recent gyrokinetic simulations have demonstrated

that modulational instability growth of zonal flow
perturbations in fully developed drift-ITc turbulence
can occur. In particular, recent simulation results showthat the evolution of a (seed) zonal flow perturbation

t\
zldk:
\t
+plQ,i oinitialized in a bath of quasi-stationary ITG turbulence
which had already saturated by other processes. The
seed perturbation clearly grows exponentially and its
growth induces a further decrease in the ion thermal

io,N+ +lI$do+t_2dn+2J?. a," lru

ok,oky

dk:

I

*\ zr

\- "v I"J

diffusivity, as measured by the simulation. Nevertheless,this evidence for the viability of the modulational
instability of zonal flows in turbulence, as well as theirrole in regulating transport, is quite compelling.

**v,\.zN=0,- l+4!,"(
P?k'\-(10)J

3. Envelope Equation Theory of Zonal Flow

Irdr - vsand Streamer Generationv)(a* *a*)+' le/^ dxx- \+2p? Qt k, ky-

In this section we explore the generation of large
scale anisotropic flow, zonal flows and streamers (zonal
flow counterpart), by examining the modulational

ldw

)

\

stability of mono-chromatic drift-wave packet envelope.
As recent simulations [6] suggest that streamers are+lk:-k; D,,lllrl'=9,(1 1)(ed"-,,, .v )" +r.U,

(r2)

J'

quasi-flute nonlinear structures, we will focus on ftr - 0
large scale flow dynamics. The processes of zonal flow

and streamer formation may be viewed as different
manifestations of a nonlinear'self-shearing' instability
process, which is somewhat akin to self-focusing or
caviton formation. This investigation also complementsI

t6 =0,

where y* = plQ/L,. The drift-wave envelope N(r, X, y)is slowly varying, on the variation scales of thesecondary mean field 1 . ;. Long scales ? = €2t, (X, y) =(u, ey) have been introduced with a small parameter

previous works on convective cells excitation by Alfven

waves [7,8], as well as works on density andtemperature gradient drift wave soliton within adiabatic
regime [9,10] and previous theoretical works on zonal

measuring the relative magnitude of drift-wave andmean field perturbations. Secondary mean flows stirredby ponderomotive type force react back on the driftt63

Champeaux S. et aI., Self Organization of Zonal Flows: Some Physics Behind the Color Viewgraphs

wave envelope dynamics through nonlinear coupling via

4a?p!,10?,(r* p?k')'E x B sheared flow and polarization drift. Equations(10)-(12) constitute a unified 2D system which
describes mean flow generation via anisotropict^\o?

modulation of quasi-monochromatic drift wave likestreamers (d" -+ 0) and zonal flows (dy -+ 0). Indeed
Eqns. (10) and (12) are the direct analogues for the drift
wave problem of the well- known Zakharov equations
for Langmuir Turbulence [15]. In what follows, weconcentrate on the sub-celeric limit (ed.( l) << vs'V
( : )), a regime analogue to the "subsonic" limit in

rci- o1*:,

Hl",l' (rs)Note that the most unstable scale Q-i then gives atypical scale for the poloidal extent of the streamer.
Indeed, the modulation undergoes a break up of theplane wave into a periodic pulse train, with Q-',r isapproximately the width of a pulse envelope soliton of
amplitude 11go | 1tO1 In the case of zonal flows (da -r0) generation, collisional damping of the poloidal flow
must be taken into account [17]. For damping ofpoloidal flows with the form of a drag, i = ev, where v
is the ion-ion collision frequency, the zonal flow

Langmuir turbulence, where the acoustic perturbations
are almost static and follow adiabatically the Langmuir
wave modulation.
We first discuss streamer generation (dx -+ 0) in
the sub-celeric limit. The streamer flow is then slaved toequation then becomes

the drift-wave stress, and is given by ayGQlT") =Qaiplk,kyl(dcoldkr) llrl' ano n = v*/ datldkre$1r".
Thus the drift-wave envelope dynamics is governed by a
cubic nonlinear Schr<idinger (NLS) equation(,d, -.33 a" * ,\arr "6\dkr^ I"'r"-2p? a, k,kyaxx I l" lt =0.
i d, lr+ L ?'9 a* ,, ''zal P! t" *'zd*l "v+ -M

r,__,i ,\"rjir=,| ot','
9'- { r +p3t'}
'"')
\

(16)

Concentrating on the sub-celeric limit, advection andcollisional damping compete to balance the nonlinear
Reynolds stress drive. Two regimes must then beconsidered, according to the length scale at which the

(r3)

l

zonal flow is excited.
In the weakly-collisional regime, corresponding toa modulation wavenumber Q, greater than vl(da/?ky),
the zonal flows are slaved to the drift-wave stresses by a

The NLS equation admits exact solutions of the form N= Noexp(iazl lfo l'zr) with a2 the nonlinear coefficientin Eqn. (13), which corresponds to a plane drift-wave
with frequency slightly shifted by the nonlinearity. It is

collisionless response, i.e. ax\eQIT") = -(2d)ip2,k,kyl(da/\kr) lArl'. Not" that, in contrast to the streamer
case, the mean density is weaker (Dr = 0) and its effecton drift wave dyramics is subdominant compare to themean potential @. As rather like the case of streamer
generation, the drift-wave envelope then obeys a onedimensional cubic NLS equation

straight-forward to show that this drift-wave solution isunstable for wavenumbers satisfyinC p?k'z, < 3(l +plk). As usual, the instability boundary is determinedby the competition between diffraction and self-focusing(here'focusing' corresponds to self-reinforcing
shearing). The streamer modulational instability growthp., such thatrate
| f.n(A) I is maximal for Q, =T-

I Dtq a"" N
id,l,r+" 2ak:

Q^=
2a?p?t ,t ,(t*p?k')1z}* (o? fi- o? 4- t) t + p? 4- p? r,l rzli

*zaip!k!,0,L'
ll- lrN=0.
da

l"'l'

(r7)

ek"

|

As previously, it can be straightforwardly shown that aplane drift-wave of amplitude Ne is modulationallyunstable for wavenumbers satisfying t + f,k'zr- P2"k2,) 0
tl8l. The most unstable scale is given by Q-"] , where(14)
The maximum growth rate is thent64

Champeaux S. et al., Self Organization of Zonal Flows: Some Physics Behind theQ^=tN.,(+!tg-)tzonal flows and streamers, is required to draw more far_reaching conclusions.

(18)

A second question concerns the scale and spatial
structure of the streamer and zonal flow patterns. Notethat the validity domain of the modulational analysis sits

The associated maximal growth rate is estimated to bey^= dJ? p? (r * p3 rr')'

ffi t * fwithin spatial and temporal scales larger or the order ofrespectively (6n/n;)-t and (6nlns)-2. In the case of the(1e)streamers, Eqn. (13) suggests that the cells, which may
be viewed as poloidally localized but radially extended

The zonal flow instability domain appears to be quite
extended in the (ft,, ft,) plane. It overlaps the instability

solitons, have poloidal extent LLe - (p2,/L^)(6n/ns)-l.F"(p,k), wherc (6n/ns) is the r.m.s. density fluctuationlevel of the drift wave, and F"(p,k) contains information
concerning dispersion, etc. Similarly, the maximal
growth rate for streamers scales as Eqn. (14) y (6n/no)2d2ikyLnc,(p,k), so, for mixing length levels,

domain for streamer generation defining a rather
localized domain. It follows then than drift-waves whose
wavenumbers satisfy the inequality plk? _ t < p1k? <3(l + dk?) can generate bothkjnds of anisorropic mean
flows via modulational instability.

streamer growth is quite virulent. For weakly collisional
zonal flows (poloidally extended solitons) the radial
exrent is LL, - (p2,/L,)(6nlns)-t Fry(p,k) (see Eqn.
(18)). Results indicate that M,- p,F,y(p,k)for 6nlns _The other regime to investigate for zonal flow
excitation corresponds to a collision dominated ordissipative limit [2], when e, the perturbation
wavenumber is smaller than the rutio v/(daldft,). The
zonal flow response to the drift wave stresses is thendissipative ,$lr" = (2plk,kre,/v) llrlr. m turn, thedrift-wave envelope obeys a NLS type equation withp rLn, so that a range of zonal flow scales can be

expected, depending upon the dispersion properties ofmode k. Note that both streamers and zonal flows will
be strongly localized near caustics where dza/lk1 -+ 0,

derivative nonlinearity

io,N+* *9 a*z dk,

Color Viewgraphs

a limit referred as the "semi-classical limit', andnumerically investigated in tl9l. Finite amplitude
oscillations interpreted as a "sea of solitons" develop on
intermediate scales shorter than the modulation scales,"

but still longer than the carrier wavelength.
A third question concerns the ultimate nonlinear
evolution and the structures. For the case of streamers,
which are described by a one dimensional cubic NLS-zP!ttn'o? nE" In l'=0.

Q0)
The case of dissipative dynamics turns out to besignificantly different from that of weakly collisional
zonal flow generation. The dynamics tied to this new
kind of nonlinearity will be addressed in detail in aequation, a streamer wave-train modulation can be
expected to break up [16] into train ofradially extended

future publication.
One question of considerable interest concerns

solitonic type structures. Such streamer-induced
solitons, then, are natural candidates for a dynamical
realization of the 'avalanche' concept. Indeed very
recent analyses [20] of numerical simulations indicate

the relative importance of, and competition between,

zonal flow and streamer structure formation.

Comparison of typical scales and maximal growth rates

that during transport'events' (i.e. avalanches), the radialassociated with streamer (e,, T) and weak collisionalcorrelation length of tbe intensity (i.e. spectrum

waves characterized by pr,l& <. l, secondary zonal flows

envelope) of the participating drift modes significantly
exceeds the correlation length of the drift wave eddy

zonal flow (Qrt, T,) generation show that for drift_

grow faster than streamer, while in the opposite limit
p2"k2

potential. This finding is thus consistent with the theory> l, streamers are more easily excited by drifr

presented here, in which streamer formation is tied towave modulation. The typical poloidal and radial extent

envelope modulation. Similarly, weakly collisional
zonal flow modulations can also be expected to break up
into a train of soliton-like poloidal shear flow cells. In
the case of strongly collisional zonal flow modulations,the nature of the nonlinear structure is not understoodof streamers and zonal flows will depend on the strengthof the dispersion in group velocity as compared tononlinear effects. A comparison of the robustness of thenonlinear soliton structures formed by streamers andzonal flows with respect to radial and poloidal longwavelength disturbances due respectively to growing

and must be await discussion in a future publication. Of

course, noise excitation must be added in order to165

Viewgraphs
Champeaux S. et al., Self Organizatiorof ZonalFlows: Some Physics Behind the Color

directly address many aspects of the fully turbulent
state.

[2] Champeaux S. and Diamond P.H., Phys' Lett' A
288,2r4 (2001).

[3] Rosenbluth M.N. and Hinton F.L., Phys. Rev' Lett'

4. Gonclusion
In this paper, we have sought to elucidate certain
elements of the basic physics of zonal flow dynamics.
We have presented two simplified model calculations
which support, clarify and elucidate previous results
obtained by more formal methods. In the first of these(Section 2), statistical ray optics was used two describe

80.724 (1998).

[4] Malkov M. et al., Phys. Plasmas 58, 3996 (2001)'[5] Rogers B. et al., Phys. Rev. Lett. 85, 5336 (2000)'[6] P. Beyer et aI.,Phys. Rev. Lett. 85,4892 (2000).

l7l R.Z. Sagdeev et al., JETP Lett. 27,34O (1978).
t8l H. Sanuki and J. Weiland, J. Plasma Phys.23,29(1980).

t9l J.D. Meiss and W. Horton Jr., Phys. Rev. Letts' 48'

zonal flow generation. In the second (Section 3)'1362 (r98D.

envelope analysis techniques were used to derive a 2D,
anistropic system akin to that of the Zak'harov equations
from Langmuir turbulence. These general equations
were simplified to lD Nonlinear Schrodiner equations
for the case of zonal flows (ka = 0) and streamers (k, -+t10l V.P. Lakhin, A.B. Mikhailovskii and O'G'
Onishchenko, PPCF 30, 457 (1988).
tl ll A. Balk, V.E. Zakharov and S.V. Nazarenko, Sov'
Phys. JETP 71,249 (1990).

[12] L. Chen, Z. Lin and R'B. White, Phys. Plasmas 7,0), and used to compute the nonlinear growth rate andstability boundary of each type of structure in various
regimes. Ongoing work is focused on the problem of
zonal flow saturation. In particular, we are attempting tocompare the mechanisms of saturation by KelvinHelmholtz type instability and by feedback modulation

3129 (2000).

t13l A.C. Newell, Soliton in mathematics and physics'
Philadelphia, PA: Society of Industrial and Applied
Mathematics, P.32 (1985).
t14l A. Hasegawa and K' Mima, Phys. Rev' Lett' 39,
205 (19'11).
tl5l V.E. Zathatov, Sov. Phys. JETP 35, 9OB (1972)'
t16l M. Remoissenet, Waves Called Solitons (Springer-of the drift wave spectrum.

Acknowledgements
We acknowledge stimulating and helpful
discussions with X. Garbet, S. Benkadda, P. Beyer, L'
Chen. F.L. Hinton, T.S. Hahm, Z.Lin, W.M. Nevins,
M.N. Rosenbluth, V. Shapiro, P.L. Sulem and A'

Verlag. 1999) p.74.
Rosenbluth and F'L' Hinton, Phys' Rev' Lett'
M.N.
tlTl
80.724 (1998).
F.L. Hinton and M.N' Rosenbluth, PPCF 41' A653
(1999).

Smolyakov. This research was supported by U.S. D'O'E'
Grant No. FG03-88ER53275.

t18l A.I. Smolyakov, P.H' Diamond and V'I'
Shevchenko, Phys. Plasmas 7, 1349 (2000)'t19l J.C. Bronski and D.W. Maclaughlin' N'M'
Ercolani et al. eds, Plenum Press, New York,

References
et aI., in Plasma Physics andP.H.
Diamond[1](1994).

Controlled Nuclear Fusion Research, 18th IAEA
Fusion Energy Conference, Yokohama, Japan,
1998 (I.A.8.A., Vienna, 1998), pg. IAEA-CN-69/

t20l W.M. Nevins et al.,Bulletin of the A'P.S.' 44
Paper RM2 5,260 (1999).

TH3/1.

r66(7),