J. Plasma Fusion Res. SERIES, Vol.4 (2001) 5tg_523

Nonlinear stationary structures in a Drfit wave Turbulence
Modulated by Zonal FlowsSINGH Raghvendra* and KAW predhiman
Institute for Plasma Research, Bhat, Gandhinagar 3g2 42g, India(Received: 5 December 2000 / Accepted: 24 August 2001)

Abstract
We investigate nonlinear stationary structures in a system of coupled equations describing
drift wave
turbulence and associated self-consistent zonal flows. The short scale drift wave turbulence
is describedby a kinetic wave equation for the action density of drift waves, whereas the longer
scale zonal flows aredescribed by a dynamic equation for the toroidally and poloidally symmetric componentof the potential.
A variety ofradially propagating coherent structures such as nonlinear periodic wave trains, solitons
andshocks in the modulation envelope are shown to be possible.

Keywords:
drift wave, zonal flow, coherent structure, stationary structure, turbulence

1. lntroduction

amplitudes well below the typical mixing length

Recently, there has been a resurgence of interest in

estimates 1e$*f1 - (p"lL,)(p" is the ion Larmor radius

the study of zonal flows generated by drift wave

at electron temperature and is a measure of typicalscale-length of turbulence and Z, is the scale-length ofdensity gradient driving the turbulence and subscript M
refers to mixing length saturation level), and to avalue
determined by the neoclassical damping rate v of zonal

turbulence in toroidal magnetically confined plasmas [l-41. ZonaI flows are radially structured poloidal flows
with toroidal and poloidal symmetry (m = n = 0), which
are generated by a modulational instability mechanism

operating on the turbulence. Physically, energy isflows [5], viz. (e$rlT) - (p,/L,)(v/a.)r/2 (where vla_ <I, ar. being the typical drift wave frequency andpumped into longer scales because of an inverse cascadein the turbulence and m = n = 0 perturbations are

subscript S on fr refers to new saturation level). The
associated anomalous transport coefficients also show

preferred because they are unaffected by magnetic shear

in the confinement device. The sheared zonal flows are

the above improvement. It is easy to understand theimprovement on physical grounds. As the damping

typically randomly oriented and induce a strong
refractive effect on the propagating drift waves. In the
weakly turbulent limit, this random shearing refraction
leads to a quasilinear diffusive spreading of the drift
wave turbulence spectrum in the radial wavenumber

parameter y drops, the zonal flows become stronger andlimit the turbulent fluctuations to lower amplitudes. An
obvious question arises as to what happens as thedamping parameter y -+ 0. As the zonal flows getstronger and stronger, the assumption of a weaklyturbulent state and quaslinear diffusion of the drift wave

space, finally producing a nonlinear stationary state withrandom phased, near Gaussian, turbulence. This limit
has been extensively studied in recent years [4] and has

turbulence spectrum become questionable. Instead, weled to the conclusion that the saturated drift wave
fluctuation level is quenched by zonal flow effects to*enter a regime of strong turbulence where coherent@2001 by The Japan Sociery of plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: rsingh@ plasma.ernet.in519

Stationary Structures in a Drfit Wave Turbulence Modulated by Zonal Flows

Singh R. e/ al., Nonl

zonal flow structures are fo'trapping' of significant nu

dx'l

stresses due totories: in their turn, collective'trapped' and'untraPPed'sustain the zonal flow.

_u), dk, =Q (3)4!'=r4(,*d2\ro",,dx\dx(Uand produce aof drift wave trajecwave distributions
coupled nonlinear

Eq. (3) may be readily integrated to give the constantsof motion

classical Bernstein structures are similar in nature
been extensively
Greene - Kruskal waves which
studied for Vlasov - Poisson t of equations in one-w =v[6] and have also been
dimensional electrostatic Plas
phenomena [7].
extended to wave-wave i
stationary
nonlinearinvestiga
weletterIn thisdescrbing
equationsstructures in a system ofdrift wave turbulence andiated self-consistent

zonal flows. A variety of radial

propagating coherent

structures such as nonlinear
solitons and shocks in theriodic wave trains,ulation envelope areshown to be possible. We esuch structures to beand suggests an exact solution to eq. (1)Nk(k,, k y x) = Nt(W(k,, x), k r)'[8] etc. Many of thestudies here shouldfeatures of drift wave turbu
in the atmosphere,also be important for RossbY
these twodescribingsequati
fundamental
since theimilarvery[2,9].
varieties of turbulence areintermittency, bursty I

2. Nonlinear Theory=tq - Q + u2l4q). This is of the standard form of asolutions for theWe look for nonlinear
in the absence
coupled drift-wave zonal flow
(1wavesfordriftsinks
andsources= Aarr = 0).
ofwith radial
ineframe
Going to amoving
frame, weinthat
itY
velocity U and assuming stati
equawave
may replace dldt by -Ualax i the kinetic
ity N* = lecprlTl2 ll +tion for dimensionless action
near equation for thek2d)2 of drift waves andobtain the set:
zonal flow potential V = eQlT"

kinetic energy term (K]) and a potential energy term(/(x)) combining to give a constant of motion. Thisequation indicates that if/has a maximum valuefi, wemust distinguish between 'trapped' drift waves with IV <fi (which sample only parts of x-space because in otherparts K? < 0 and 'untrapped' drift waves with W > f^(which sample all of x-spacebecause K'2 > 0everywhere). The expansion of k2 << 1 used above isonly a simplification to make the algebra less tediousand is no way necessary physically for the trapping ofdrift waves. The physical process responsible fortrapping of drift waves near the minimum of a velocity
shear layer may be readily studied by writing the(u,,-4#*0,*(, #)d=o (1)23!+*r e)v)'+ur'&=- Dx k

where v = dvldx is thecharacteristic ray equations for drift wave trajectories
viz. (in dimensionless variables), dxldt = -zk'kyt(l + l&)zand dk,ldt = ky[ (dvld.x) + (d3vldx3)] and dkrldt = 0'
Taking v near aminimum as v = vm + v'l(x - x^)212 we
get the equation @2k,ldf) + U4v'1,10 + P)21k, = 0. For[1+/<'J

itude of zonal flow

normalizations: -x tovelocity. We have introduced:pJL,)c,, t to p,lv*, Npp,, k to p;r, v to drift sPeed =L)2 and lL, v to cslL,.
to the mixing length estimatev* is the radial grouP veloci of drift waves given bY
the normalized expression - ,krl(l + k2)'z. Eq. (1)conserves N1 along the c(5)krW may be physically interpreted as the frequency ofthe drift wave as seen from a frame moving radiallywith velocity U, corrected for effects due to velocity anddensity perturbations - (v + d2vld*) terms - associatedwith the zonal flow; this frequency is an invariant (aconstant of motion) because in the moving frame, thezonal flow structures are independent of time. We thus
expect, that as the drift wave encounters regions withdifferent v(x) and v"(x) (prime denotes an -r derivative),
its t, will change in such a way as to keep the wave
number k, and the frequency (krW) constant. When ft,'k, <. l, we may rewrite the constant of motion W as
W K +/(.r) where K"= k'+ Ul2lg and/(;r) = v + v" +ved non-Gaussian
relevant for interpretation offeatures of the turbulence such as coherence,l'#-+#-#*ftu'k, =constant (4)v; > O, this is the equation for a nonlinear oscillator as
may be readily verified by multiplying the equation withdk*ldt and integrating once. The solution may bedescribed in terms of complete elliptic integral of thesecond kind

given by520

Singh R. et al., Nonlinear Stationary Structures in a Drfit Wave Turbulence Modulated by Zonal FlowsThe entire nonlinearity in eq. (8) comes from thedependence of the right side on/= v + v" + td - t - v2t
41fi which, in turn, is determined by the choice of thetrapped and untrapped action densities Nu,r(W, ky).

f ("or-' k,lk,, k "/(l + t', + *';')= (zt r/'^)"'t t1t + kzr + k 'o)(6)where we have used the boundary condition, k" = ko,Since any function of the constants of motion W, ft, is anexact solution of eq. (l), there is an arbitrariness in the
choice of Nr.r. This arbitrariness is well known in thestandard BGK problem also. Ultimately, the choice of a)k,/a = 0 at t = 0. When t3, t rt .. 1, the solution may be
written as[,-^',,,,suitable Ny,1 will be decided by conditions ofaccessibility as one follows up on an initial value
problem and/or questions of stability of the nonlinear
stationary states being discussed here. But these

1k, = k ocnlext of {zt',vi t(t + k', * k'.\'\"',1-J]'=k."o"(\E4r'i['tle)

where K denotes the complete elliptic integral of firstkind. Thus k, and x are both periodic functions of rquestions are beyond the scope of the present
calculation and have to be treated separately. Here, toillustrate the nature of typical nonlinear structures, weshall make choices of N7, Ny on intuitive grounds. We

which become simple trignometric functions in the limitk", kr 11 1. Physically, as the drift wave creeps up thevelocity shear layer from its minimum, it loses its k, (xmomentum) in the process of satisfying the WKB
conditions, the ray eq. (7). Finally ft,2 hits zero and thewave gets reflected getting trapped in the trough,oscillating back and forth between the two reflection
points with a typical bounce frequency f{247^> tinterms of unnormalized variables (c L,)(lg p")(2 d[d2v"l^ldx21lv-1r/21 which is related to curvature of the shear
layer near its minimum.
Following up on the discussion after eq. 2 we may
now argue that solution for stationary eqs. (1) and (2)The two distributions are chosen to be continuous atl=

may be obtained by solving the nonlinear self-

L and the ft, dependence is chosen in such a manner thatassumeNu(W, kr) =Nr(W, k r) =('.__ ) lr*'r,-p"'f'Nor(f^- W)tt2 + Nou

1"+'f-w>f^

f<w<f(10)there is no symmetry in the +y directions. Thisasymmetry reflects the fact that drift waves typicallyconsistency condition:
1t(d2v ldxz)

Nou- w + (I (dv ldx) = (-l/2) (d ldx)

propagate in either the electron or ion diamagnetic drift

Itlrlr| | dkykyl
I dwJ(w,ky,f)Nu(w.ky)
'lJr^
lJ:

direction. It also ensures that the required integrals do

not vanish even when U -+ 0. For large k,, k, thedistributions drop-off as a power law which is typical oft\

many saturated turbulent states. The action density fortrapped waves is chosen in the so-called Bohm-Gross

L*['^tslr.v.)Nr(w,0,,]lrr awilw.k,.f".)1

form with a maximum at the bottom of the trough andthe phase space density dropping off to N,, at themaximum as a square-root function; this is extensively
used in the study of trapped particle effects in large
amplitude electrostatic waves [7]. With this choice, we

where Ny(IU, ftr) and Nr(W, ky), are the action densities

for untrapped and trapped parts of the stationary drift
wave turbulence, J(W, ftr, /) is the Jacobian forfindtransforming ft, variable to l,lz and is given by the

pv" -w+Ut/ =(ar- a,)(r/ +v")

express|+ r" +{tt-'"" ['{{w - r t'',,.1'- 0, [(u- - r)''' ]t.,^r
J=l(w-f)"'+U/2k,1/-(3a"t2- a,)fe^-r)'l (ll)where, ao = (3fNoull6)kr,L6, ay = (3LNorll6)1g,5 and. u''.,}']'l (e)a2= (nNo7/8)k,d. The coefficient ao arises from thecontribution of untrapped waves whereas the521

Stationary Structures in a Drfit Wave Turbulence Modulated by Zonal Flows

Singh R. et cl.,coefficients at a'i.d q2 arise
to trapped waves. We nowlimits. Let us first ignore v, trt,

the contributions due

solved to get the dispersion

ionU=(ar-a,)(l-The linearised I/ may be

shock-like structure with V going from 0 to v. after
ringing a few times periodically around the final value.
If the damping terms arelatge, there is no ringing andthe shock solution goes monotonically from 0 to v.. If
the dominant dissipative term is viscous (v = 0, 1t,eq. (11) in various
the nonlinear terms.

analyzed in x andq2) where 4 is the normalized wave number for theing to trapped andzonal flow velocity v. Thus,

untrapped drift waves cothe zonal flow

perturbations into radiallydispersive waves.

large), the monotonic shock solution may be analytically
written down in the form V = K4lF2{exp(Kz(x - Ut)ltt)l[ + exp1K2{x - ut ll2tt)12it.

ve-wave lnteractlons,veexhibited the usual
these dispersive waves wouldmodulational instability.
, we are looking forstationary solutions where the resonant waves' withsmall k"2 have been trapped in t modulation envelope,
thereby saturating the instabili . The structure of the3. Conclusions
In conclusion, we have looked for stationary

nonlinear stationary state isby the nonlinear

structures such as periodic nonlinear wave trains,trapped wave contribution I
Neglecting 4 p and higher-v)21',eq. (11) may bev)3t2)' to eq. (1 l).

solitons and shocks in the modulation envelopes may beformed. These solutions represent alternate saturated
states to modulation instabilities of drift wave
turbulence - nonlinear states which are dominated bycoherence and drift wave trapping in contrast to theusual saturated states dominated by random phases andquasilinear diffusive spreading of drift wave spectra.
Such nonlinear coherent structures may have alreadybeen observed in the collisionless (v = 0) strongly
turbulent drift waves imulation of Lin et al. [1] and may
also be responsible for observed phenomena such asIf we had retained, resonantv"-K2v+Bv
whereV=r^-r)0,K2=l(lazl@r - ao). Equation (12)^-solutions describing nonlinear coherent structures in thecoupled problem of drift wave turbulence and associatedself-consistent zonal flows and find that when drift wave
trappingis important, a variety of radially propagating

nonlinearity term [(v.

aso(12)(at-a")-llandB=
the quadrature ll2(dvl

the Sagdeev pseudodx)z + Y(V) = Constant w
potential takes the form V(V = -K2 Vl2 + 2l5Bst2.
This pseudopotential starts at V 0 with zero slope, has/82 and again takes onaminimum, -Y^,atV=rm=. An effective particle
the value O atV =V"=25116
pseudopotential withstarting at a V > 0 oscillates inintermittency, oscillations and bursty transport inexperiments on drift wave turbulence [8].

a frequency which is a func on of amplitude; thelinear periodic wave
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