Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
187–191
J. Plasma Fusion

Excitation of Convective Cell Modeby Universal Drift Wave Turbulence
SAITO Tsuyoshi.1, ITOH Sanae -I.2, YAGI Masatoshi2, HAHM Taik Soo3 and CHENG Chio Z.3
1

Interdisciplinary Graduate School of Engineering Sciences Kyushu Univesity, Kasuga, 816-8580, Japan
2
Research Institute for Applied Mechanics, Kyushu University Kasuga, 816-8580, Japan3
Princeton Plasma Physics Laboratory, Princeton University Princeton NJ 08543-0451, USA(Received: 9 December 2003 / Accepted: 6 July 2004)

Abstract
The interaction between the universal drift wave and the convective cell mode in the shearless slab geometry isinvestigated using a kinetic-fluid model based on nonlinear gyro-kinetic equations. It is found that the convective cellmode is nonlinearly excited in a way, which is similar to the zonal flow generation of ITG turbulence. On the other
hand, if we introduce the large ion-ion collision effect, the zonal flow is damped and instead the streamer and finite k ymodes are excited in the convective cell mode. As the result, the flux is larger than that without collision. The forwardcascade is observed in k z space and the forward/inverse cascade is observed in k y space in the time evolution ofenergy spectrum.
Keywords:
drift wave, convective cell, inverse cascade, zonal flow, kinetic fluid model, parametric decay instability

1. Introductionin the short wavelength region. These difference make it
difficult to compare simulation results of universal drift wave
turbulence with the theory directly, in which the linear
growth rate is a given parameter and is not described selfconsistently by the model itself [2]. Nevertheless, there is still
an academic interest to investigate the development of
universal drift wave turbulence and the excitation of
convective cell mode in the three dimensional configuration
space, especially in terms of cascade and inverse cascade due
to not only ion nonlinearity but also electron nonlinearity,
which is not included in the adiabatic electron response. This
paper is organized as follows. In the next section, the ChengJohnson kinetic-fluid model is explained and simulation
results are discussed in section 3. The conclusions are
summarized in section 4.

To understand anomalous transport in high temperatureplasma is a crucial issue in nuclear fusion research. ITG (iontemperature gradient driven drift wave) is a possible
candidate for anomalous ion heat transport. The gyro-kinetic
simulations of ITG turbulence demonstrate that zonal flows
play a crucial role in regulating drift wave turbulence and thelevel of the anomalous transport [1]. The theoreticalexplanation for the generation of zonal flow is proposed
based on parametric instability in shearless slab geometry[2,3]. Simulations in such a simplified drift wave systemwere performed long time ago[4], although parameter
dependence was not examined systematically. In addition,
direct comparisons of these theory to the gyro-kinetic
simulation results were possible in some idealized test caseonly [1]. In this paper, we perform the simulation study ofuniversal drift wave turbulence and investigate the excitationof convective cell mode in detail using a kinetic-fluid modelin the shearless slab geometry. This is an extension of themodels proposed by Smolyakov [5] and Hinton [6]. Ourmodel describes the universal drift wave turbulence andnonlinear excitation of convective cell mode [3,4], which
include zonal flows and streamers generated by drift waves.
Simulation results show that no saturated state is attained in
the universal drift wave turbulence. This is attributed to thefact instability source exists in the short wavelength region ofk y space. For ITG turbulence, the instability source exists inthe long wavelength region and there is enough energy sink

2. Models
To investigate the interaction between universal drift
wave and convective cell in shearless slab geometry (x, y, z)( x is the radial direction, y is the poloidal direction and z is
the direction of ambient magnetic field), a kinetic-fluid
model is derived integrating the nonlinear gyro-kinetic
equation in the velocity space and using linear closure to
reproduce the linear dispersion relation [7,8]. The ion
continuity equation is written by©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: tsuyoshi@riam.kyushu-u.ac.jp
1187Saito T. et al., Excitation of Convective Cell Mode by Universal Drift Wave Turbulence

2 n - i~ C g Z z + x (1 + C g Z ) 2 z
0k i
0k i
i k
i
*
2t k
2t k
(1)
+ x ! ( b $ kll# kl) z k l | k ll (n k ll + xz k ll ) = 0,
2 k = k l+ k ll

EDW =
ECC =

! [1 + k m + x (1 - C )] z ,
2

2

2

0k

k x , k y , kz ! 0k! [ k m + x (1 - C ) ] z .
2

2

2

0k

k x , k y , kz = 0kand the electron continuity equation is by

2 n - i~ g Z z - (1 + g Z + k 2 m 2 ) 2 z
e k
e
e
* e
2t k
2t k
+ x ! ( b $ kll # k l) z k l (n k ll - kll 2m 2 z k ll ) = 0,
2 k = k l+ k ll

E DW represents the energy of drift wave (k z ! 0 ) and E CC
represents the energy of convective cell mode (k z = 0 ) [3,4].
In addition, the analytical dispersion relation of universal
drift wave is obtained in this limit v i % ~/k z % v e , keeping
collionless dissipations as(2)where k is the wave number vector, n k is the fluctuating ion
density and z k is the electrostatic potential. ~ *= xk y t 2i /( 2L n ) is the normalized diamagnetic drift
frequency by the ion cyclotronfrequency X i , where t i is theion gyro-radius, x = T e /T i is the ratio of electron and ion
temperature, L n = - (d/dx ln n) - 1 is the density gradient scale
length. m = x Xi /( 2 ~ pi ) is the normalized Debye length
by the ion Lamor radius, where ~ pi is the ion plasma
frequency. C 0k = I 0 (b k ) e - b represents the finite Lamor
effect, where I 0 is the zeroth order modified Bessel function
with b k = k 92 t 2i /2 and k 92 = k x2 + k y2 . Z i, e are plasma
dispersion3 functionwith the argument g i, e = ~/( k z v i, e ) and|k ll = 2 # d xxe - x J 0 (b 1k/2 x) J 0 (b 1k/l 2 x) J 0 (b 1k/ll2 x)/ C 0k ll , where0v i, e = 2T i, e /m i, e are ion and electron thermal velocity, T i, e are
ion and electron temperature, m i, e are ion and electron massand J 0 is the zeroth order Bessel function.The factor 1/2 ineqs.(1) and (2) appears due to the definition of thermalvelocity. The Poisson equation is used to eliminate thefluctuating electron density in the electron continuity
equation. The energy conservation reration is obtained fromeqs.(1) and (2) as~* C0k
1 + x (1 - C0k )~=+i(1 + x) ~* r
(1 - C0k ) g e e- g - C0k g i e- g
[1 + x (1 - C02k )]2

[

2
e

2
i

]

(4)

Using this model, the excitation of convective cell isinvestigated in the next section.

k

3. Simulation results
In present numerical simulations, the following
parameters are used: 16 # 32 # 8 Fourier modes are used in
the wave number space (k x , k y , k z ) . The smallest wave
numbers are chosen as k x0 = k y0 = 0.15 , k z0 = 0.02 . The realmass ratio is used as m i /m e = 1836 , then X 2i /~ 2pi= 10/( m i /m e ) . The density gradient scale length is chosen as
t i /L n = 0.2 (the wave number is normalized by ion gyroradius). And the effect of ion-ion collision is also
investigated. The ion-ion collision is taken into account byreplacing 2 " 2 + o ii in the ion continuity equation, where2t2to ii is the normalized ion-ion collision frequency by the ion
cyclotron frequency. Such a simplified Krook-like collision
term does not conserve particle, momentum and energy,
therefore, the saturation amplitude of turbulence is not
obtained accurately. However, it might be still applicable todamp the zonal flow ( k x ! 0, k y = 0, k z = 0 ). In this paper, welimit our discussion on the excitation of convective cell modein the development of turbulence not on the statisticalproperty of turbulence in the saturation phase.
Figure 1 and 2 show dependence of the real frequency
and the growth rate of universal drift wave on thewavenumber k y . The thick line corresponds to the case with2R

! S Re (s ) 22t z
k

2

2
k

+ 2 Re (s1 ) z k

2

T
V
(3)
- 2 Im (s 2 ) Re (z k ) Im (zo k ) - Im (z k ) Re (zo k ) W = 0
X

where s1 = i~ * e (g e Z e - C 0k g i Z i ) , s2 = 1 + g e Z e + k 2 m 2+ x (1 + C 0k g i Z i ) . In the fluid ion, 1 % g i and near adiabatic
electron response, g e % 1, i.e., v i % ~/k z % v e and neglecting
the collisionless dissipations, it is reduced to
d/dt (E DW + E CC ) = 0 with

0.06

7 10-3kx=0.15

0.04

6 10-3ωr

kx=0.15

5 10-30.02
ω0kx=2.40-0.02

4 10-3i

3 10-3kx=2.40

-3

2 10
-0.04
-0.06

1 10-3
-5

0

0 100

5

ky

0
ky

Fig. 1 Dependence of real frequency on k y .

Fig. 2 Dependence of growth rate on k y .

2
188

-5

5

Saito T. et al., Excitation of Convective Cell Mode by Universal Drift Wave Turbulence

100

100

-2

-2

Potential Energy

10

10

-4

10

10-6

EDW

Eφ(kz)EDW(kZ=0.06)
EDW(kZ=0.02)

-8

10
10

ECC

10-6

t=1000
t=500
t=100

10-12
-0.16

200 400 600 800 1000 1200 1400 1600
t

Fig. 3 Time evolution of energy for drift wave andconvective cell mode.

-0.08

0
kz

0.08

0.16

Fig. 4 Time evolution of power spectrum for potential
energy E x (k z ) .
7 10-310-3t=1500

6 10-3

t=1200

5 10-3Eφ(kx,ky,kz=0)

10-5
10-7

t=1000

10-9

t=500

10-11

t=100

4 10-3k =-0.15

k =0.15x

x

3 10-3k =0x

2 10-3
1 10-3

10-13
10-15

t=120010-10EDW(kZ=0.04)

10-12
0

10-4

10-8

-10

Eφ(kx=0.15,ky)

t=1500

0 100
-5

0
ky

-1 10-35

k =0.30

k =-0.30

x

-1

-0.5

x

0

0.5

1

ky

Fig. 5 Time evolution of power spectrum for potential
energy E z (k x , k y ) with k x = 0.15 .

Fig. 6 Power spectrum of potential energy for convective
cell mode E z (k x , k y , k z = 0) at t = 1500.

k x = 0.15, k z = 0.02 and the dotted line to k x = 0.24 ,k z = 0.02 , respectively. It is shown that the universal drift
wave is unstable in the short wavelength region of k y . It isfound that for larger k z , the universal drift wave is stable.
Figure 3 shows the time evolution of energy for drift
wave E DW and the convective cell mode E CC . The thick line
corresponds to the energy for drift wave and the dotted line tothe energy forconvective cell mode.The energy for each
Fourier mode of drift wave with k z = 0.02, 0.04, 0.06 is also
shown. The dashed line indicates the mode with k z = 0.02 ,the long dashed dotted line, the mode with k z = 0.04 and theshort dashed dotted line, the mode with k z = 0.06 . The
maximum growth rate of the drift wave is given by thelongest wavelength mode with k z = ! 0.02 . It is seen that theconvective cell and the drift wave with short wavelength ink z are nonlinearly excited. The amplitude of these modes
reaches at almost the same level as k z = ! 0.02 at t $ 1300
(inverse of ion cyclotron frequency).
Figure 4 shows the time evolution of power spectrum forthe potentialenergy in k z space defined by E z (k z )2= !k , k z k . Time slices of t =100, 500, 1000, 1200, 1500
are shown. The convective cell mode glows and the normalcascade occurs in k z space, simultaneously.
Figure 5 shows the time evolution of power spectrum for2
the potential energy defined by E z (k x , k y ) = !k z k in thecase with k x = 0.15 . Time slice of t =100, 500, 1000,1200,
1500 are shown. The cascade and invers cascade occurs in k yspace and the spectrum tends to be flat at t = 1500 and thek y = 0 mode is nonlinearly excited. Figure 6 shows the power
spectrum of potential energy for convective cell mode2
defined by E z (k x , k y , k z = 0) = z k at t = 1500 . The caseswith k x = 0, ! 0.15, ! 0.30 are plotted.The long dasheddotted line with diamonds indicates the mode with k x = 0 , thethick line with black circles, the mode with k x = 0.15 ,thethick line with black squares, the mode with k x = 0.30 , thedashed line with circles, the mode with k x = - 0.15 and thedashed line with squares, the mode with k x = - 0.30 . It isclearly seen that the components of zonal flow withk x = ! 0.15, k y = 0 are dominant in the convective cell modeof potential energy.
Figure 7 shows the time evolution of energy for drift
wave E DW and convective cell mode E CC in the case witho ii = 0.01. It is found that the energy for convective cellmode is larger than that of the drift wave compared with thecollisionless case. Fig. 8 shows the power spectrum ofpotential energy for convective cell mode E z (k x , k y , k z = 0)
at t = 800 in the case with o ii = 0.01. It is seen that the zonal
flow is damped by the collision, however, finite k y modes
with k x = ! 0.30, k y = " 0.15 are excited instead. The
amplitude of potential energy for drift wave is also larger
than that obtained in the collisionless case.
Figure 9 shows the time evolution of energy forconvective cell mode E CC and particle flux defined by
Ck = 0 = !k = k l+ k ll n k l (- ik y ll z k ll ) in cases with o ii = 0 and

x

y

z

3
189

Saito T. et al., Excitation of Convective Cell Mode by Universal Drift Wave Turbulence
3.5 10-2
1

10

3.0 10-2-1

2.5 10-2
Eφ(kx,ky,kz=0)

Potential Energy

10

10-3
EDW

10-5
ECC
-7

x

x

1.5 10-2
k =0

1.0 10-2

x

-3

10

5.0 10EDW(kZ=0.02)-9

0.0 100

EDW(kZ=0.06)

10-5.0 10-3-110

k =0.15

k =-0.15x

EDW(kZ=0.04)

10

k =-0.30

k =0.30

2.0 10-2

100 200 300 400 500 600 700 800
t

-1

-0.5

x

0

0.5

1

ky

Fig. 7 Time evolution of energy for drift wave andconvective cell modewith o ii = 0.01.

Fig. 8 Power spectrum of potential energy for convective
cell mode E z (k x , k y , k z = 0) with o ii = 0.01 at t = 800 .
10-1

0

10

10-3

y

x

-2

Eφ(kx,ky,kz=0)10Potential Energy(k x =0.15,k =0.30)
streamer(k =0,ky =0.30)

10-4
10-6

EDW

EDW(kZ=0.06)
EDW(kZ=0.02)

10-8
10-10

ECC

10-7

(k x =0.15,k =0.15)
y

10-9
10-11

EDW(kZ=0.04)

streamer(kx =0,k =0.15)
yzonal flow(k x =0.15,k =0)
y

10-13
400 450 500 550 600 650 700 750 800
t

10-12
0

10-5

200 400 600 800 1000 1200 1400 1600
t

Fig. 9 Time evolution of energy of convective cell mode andflux.

Fig. 10 Time evolution of power spectrum of potential energy
for convective cell mode.

o ii = 0.01. The thick lines represent the flux and the dashedlines, the energy of convective cell mode.In the case witho ii = 0.01, the growth of flux is rapid compared with thatobtained in the case without collision. It is found that fluxand energy of convective cell mode correlates with eachother for E CC & 10 - 2 . Figure 10 shows the time evolution of
power spectrum of energy for convective cell mode in thecase with o ii = 0.01 . It is found that the component of
streamer k x = 0.0, k y = 0.3 has the large contributioninconvective cell mode. The contribution of the zonal flow
k x = 0.15, k y = 0.0 is rather small. The electron nonlinearity
excites various Fourier modes in convective cell mode, which
affects the amplitude of drift wave turbulence as a backreaction, so that the simple picture such that the collisional
damping of zonal flow determines the saturation level of drift
wave turbulence might not be drawn for non-adiabatic
electron response.

saturation is attained in these simulations. This is attributed
from the fact that instability source exists in the shortwavelength region of k y space. For ITG turbulence,
instability source exists in the longwavelength region andthere is enough energy sink in the shortwavelength region.
Therefore, we limit our discussion on the excitation of
convective cell mode by the drift wave with k y + 1 in thedevelopment of turbulence not on the statistical property of
turbulence in the saturation phase. To attain the real
saturation of universal drift wave turbulence, we should
include the electron Lamor radius effect to stabilize universal
drift wave in the short wave length region k y & 1, which is
missing in our model. In addition,we should extend thesystem size at least 10 times larger than that in this study tocover enough dissipation range (up to the Debye scale). Sucha simulation with high resolution for k y spectral space isgigantic and beyond the scope of this study. It should be
investigated as a future work.The ion-ion collision effect onconvective cell mode is also investigated. The ion-ion
collision damps the zonal flow and instead excites streamer
and finite k y modes, which dominates the amplitude of
convective cell mode. As the result, the flux is larger thanthat obtained in the case without collision.The electron
nonlinearity excites various Fourier modes in convective cellmode, which affects the amplitude of drift wave turbulence asa back reaction, so that the simple pictures such that the

4. Conclusion
Nonlinear simulations of universal drift wave turbulence
are performed by Kinetic-fluid model in the shearless slab
geometry. The forward cascade is observed in k z space andthe forward/inverse cascade is observed in k y space in thetime evolution of energy spectrum.It is shown that theconvective cell mode is nonlinearly excited similar to thezonal flow generation of ITG turbulence, however, no4
190

Saito T. et al., Excitation of Convective Cell Mode by Universal Drift Wave Turbulencecollisional damping of zonal flow determines the saturationlevel of drift wave turbulence might not be drawn for the
non-adiabatic electron case. It should be clearified the effectof the electron nonlineairy on the excitation of convective
cell mode (including zonal flow, streamer, geodesic acoustic
mode etc.) in toroidal geometry. It is also left as a futurework.

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Symposium on 'Transport and Strctural Formation'
combined with US/Japan JIFT workshop ‘Strctural
Formation and Drift/MHD Turbulence’, Rep. of RIAM,
Kyushu Univ., No.S-1,157 (2003).

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