J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Analyses of Nonlinear Coupling for Turbulent Structural Formation in
Magnetized Cylindrical Plasmas
Naohiro KASUYA, Masatoshi YAGI 1), Kimitaka ITOH and Sanae-I ITOH 1)
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan
1)

Research Institute for Applied Mechanics, Kyushu University, 6-1 Kasuga-kouen, Kasuga, Fukuoka 816-8580, Japan(Received: 1 September 2008 / Accepted: 26 December 2008)

Turbulent plasmas form a variety of meso-scale structures such as a zonal flow and a streamer, which regulates
micro-scale fluctuations. The mechanism of nonlinear structural formation has been studied with a three-field(density, potential and parallel velocity of electrons) reduced fluid model, which describes the resistive drift wave
turbulence in magnetized cylindrical plasmas. The turbulent structure, streamer, is selectively formed by changing
the ion-neutral collision frequency, which is the damping parameter of the zonal flow, in the target plasma. The
turbulent structures are formed by nonlinear wave coupling of eigenmodes. The possible coupling can be controlled
by restricting some modes in numerical simulations, and it is found that the same structural formation mechanism isdominant within several kinds of coupling paths.
Keywords: turbulence, structural formation, resistive drift wave, zonal flow, streamer

specifies the dominant three-wave coupling for the
structural formation. To understand the self-organized
selection mechanism, the number of modes included innumerical simulations is changed in this paper. The
characteristic nonlinear energy transfer is compared toclarify the selection rule of the formed structure
depending on mode coupling paths. 12,13,14,15,16
The paper is organized as follows. In Sec. 2, the set ofmodel equations for the analyses are described.
Simulations using this model are explained, and showformation of turbulent structures, a zonal flow andstreamer in Sec. 3. The magnitudes of the nonlinear
coupling are calculated to study the formation mechanismof the streamer, and calculation with additional modes isalso carried out for comparison in Sec. 4. Then, wesummarize our results in Sec. 5.

1. Introduction

Turbulent plasmas form a variety of meso-scale
structures such as a zonal flow and a streamer [1,2]. A
streamer, which is a localized vortex in the azimuthal
direction, is generated by nonlinear wave coupling, and issustained for a much longer duration than the oscillation
period of micro instability. The turbulent structure affects
the level of the anomalous transport in fusion plasmas byregulating micro-scale fluctuations, therefore, theformation and self-regulated mechanism of the turbulent
structures should be taken into consideration tounderstand the transport processes.
Plasma experiments in a simple linear configuration
have been carried out recently for quantitative
understandings of the structural formation mechanism byturbulence [3-7]. A density gradient drives the drift wave
turbulence in these plasmas. Two-dimensional
measurements reveal the feature of turbulent structures[8-9] and their formation mechanisms by nonlinear mode
coupling [10-11]. 3,4,5,6,7,8,9,10,11
Numerical simulations of drift wave turbulence inlinear devices have been carried out to understand thefundamental mechanism of structural formation bycomparison with experiments [12-16]. We havedeveloped a three-dimensional numerical simulation code,
which simulates the resistive drift wave turbulence inmagnetized cylindrical plasmas [12]. Selective formationof the turbulent structure can be identified by changing a
damping parameter of the zonal flow [14]. There exist
several paths for energy transfer by nonlinear couplingsin plasmas. Each magnitude of the nonlinear coupling

2. ModelWe have been developing a three-dimensional
numerical simulation code of the resistive drift wave
turbulence in a linear device, called ‘Numerical Linear
Device’ (NLD, details are described in [13]). The
three-field (density, potential and parallel velocity ofelectrons) reduced fluid model is adopted. The plasma has
a simple cylindrical shape, and the magnetic field has only
the component in the axial direction with the uniform
intensity. According to experiments, high density (ne >1×1019[m-3]) and low temperature (Te < 5 [eV]) plasmas inan argon discharge are analyzed. The density of neutral
particles is high even in the plasma core region [17], so theeffect of neutral particles is taken into consideration. The

author’s e-mail: kasuya@nifs.ac.jp

77©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

N. Kasuya et al., Analyses of Nonlinear Coupling for Turbulent Structural Formation in Magnetized Cylindrical Plasmas= 0 destabilizes the resistive drift wave.

continuity equation, the vorticity equation and Ohm's law
can be used to obtain the fluctuating density, potential andparallel velocity of electrons [18]:dN
2= − ∇ //V − V∇ // N + µ N∇ ⊥ N + S ,dtd ∇ ⊥ 2φd ∇ ⊥φ ⎞⎛2= ∇N ⋅ ⎜ −ν in ∇ ⊥φ −⎟ −ν in ∇ ⊥ φ
dt
dt ⎠
⎝

3.2. Nonlinear simulation

Linear analyses in the cylindrical geometry give linear
growthrates and eigenfrequencies [13]. Only n = 1 modes
can be unstable with these parameters. The dispersion
relation of the linear eigenmodes with n = 1 shows weak
dispersion in small m (m < 4), and ∂ω / ∂kθ ~ 0 with m = 4– 6. The initial condition is given to be f = 0 for (m, n) = (0,0) and f = 1 × 10-8 sin (πr / a) for all the other modes,
where f implies {N, φ, V}. Simulations are performed with
256 grids in the radial direction. Fourier modes (m, n) = (0,0) and m = ±1 - ±16, n = ±1 - ±16 are taken (m × n = 16 ×
16). Modes with (m, n) = (3 – 6, 1) have the largest
amplitudes in nonlinear phases, so this number of modes
must be taken in calculations at least. The time evolution ofeach mode is calculated with Eqs. (1) – (3). Figure 1 showsan energy spectrum in a nonlinear saturation state, where2
Eφ ( m ) = ∫ dr 3 ( ∇ ⊥φm ) 2 , and the dependency of the linear
growthrate of n = 1 mode on m. The growthrates arecalculated with the instantaneous density profile. Theenergy input comes from unstable modes with m = 3 – 6,
where the instantaneous growthrate is positive, and istransferred to the higher m region by nonlinear mode
coupling to be dissipated, and to the lower m region to becollisionally damped. The saturation state is sustained withthese energy balances.

(1)− ∇ //V − V ∇ // N + µ W ∇ ⊥ 4φ

,
(2)
dV
M
(∇//φ − ∇ // N ) −ν eV + µV∇⊥ 2V ,
=
(3)
dt
me
where N = ln (n / n0), V = v// / cs, φ = e ϕ / Te, n is thedensity, n0 is the density at r = 0, v// is the electron velocity
parallel to the magnetic field, cs is the ion sound velocity, ϕis the electrostatic potential, Te is the electron temperature,d / dt = ∂ / ∂t + [φ, ] is the convective derivative, S is aparticle source term, M / me is mass ratio of ion andelectron, νin is ion-neutral collision frequency, νe = νei + νenis the sum of ion-electron and electron-neutral collision
frequency, and µN, µV, µW are artificial viscosities. The ion
cyclotron frequency Ωci and Larmor radius measured by
the electron temperature ρs are used for the normalizationsof the time and distance, respectively. The equations are
solved in the cylindrical coordinate with spectral expansion
in the azimuthal and axial directions assuming periodic
boundary condition, where m and n are the azimuthal andaxial mode number, respectively. The boundary conditionin the radial direction are set to f = 0 at r = 0, a when m ≠ 0,and ∂f / ∂r = 0 at r = 0, f = 0 at r = a when m = 0, where f
implies {N, φ, V}, and r = a gives an outer boundary of theplasma column.

1.E+000.02

Eφ

3. Turbulent Structure
3.1. Simulation parameters

A nonlinear simulation has been performed to
examine the saturation mechanism of the resistive drift
wave turbulence. The following parameters are used: B =0.1 [T], Te = 2 [eV], a = 10 [cm], length of the device λ =1.7 [m], µN = 1 × 10-2, µV = µW = 1 × 10-4. Using theseparameters, νe is estimated to be νe = 310 [13]. The
electron collisions (νei and νen) destabilize and theion-neutral collisions (νin) stabilize the resistive drift wave
[13]. Therefore, the drift wave can be excited with large νeand small νin. There is ambiguity of the value of collision
frequency νin, which depends on the neutral density.
Therefore, νin is used as a parameter for controlling the
instability in our simulations. The calculation with a fixed
particle source has been carried out, where the time
independent source profile is given by2⎡ ⎛ r ⎞2 ⎤4S0 µ N ⎡ ⎛ r ⎞ ⎤⎢1 − ⎜ ⎟ ⎥ exp ⎢− ⎜⎜ ⎟⎟ ⎥ ,S (r ) =2
LN ⎢⎣ ⎜⎝ LN ⎟⎠ ⎥⎦⎢⎣ ⎝ LN ⎠ ⎥⎦growthrate

1.E-01
1.E-020.01

1.E-03
1.E-04

0
0

4

8

m 12

16

Fig. 1: Potential energy spectrum and instantaneous linear
growthrate dependence on m. These are the case with νin = 0.1 at t
= 7500. Unstable modes with m = 3 - 6 drive the turbulence.

In the nonlinear saturation states, two kinds ofturbulent structures have been obtained [14]; a zonal flow
and a streamer. If the collision frequency is small,compared with the growth rate of unstable modes insaturated states, modulational coupling of unstable modes
generates the (0, 0) mode, which is the zonal flow. If thecollision frequency is large, the zonal flow remains stable,
owing to strong collisional damping, and parametriccoupling with modes, which have neighboring m and thefrequency close to each other, forms a streamer. Snapshotsof the contours of the potential are shown in Fig. 2. Typicalstructure with the streamer is represented in (c), compared(4)

with S0 = 5.0, LN = 5 [cm]. This source profile gives adensity profile proportional to exp[-(r/LN)2] in a linear
phase, which is flattened for r / a = 0.2 - 0.8 in a nonlinear
phase, as described in [13]. The density profile peaked at r

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N. Kasuya et al., Analyses of Nonlinear Coupling for Turbulent Structural Formation in Magnetized Cylindrical Plasmas(±5, ±1) are dominant. A vortex structure localized in the θ
direction, shown in Fig. 2 (c), are formed by these modes,
and is sustained for a much longer duration than the drift
wave oscillation period (more than 3000 Ωci). This mode
matching comes from nonlinear mode coupling. Although
the phase velocities of the linear eigenmodes of (4, 1) and(5, 1) are different from each other, nonlinear frequenciesof (4, 1) and (5, 1) modes are downward shifted andbecome close [14].
To clarify the mechanism, the rate of the energy
transfer of each mode dE ( m, n ) dt is calculated.
Equations (1) – (3) can be divided into two parts: the partproportional to the own mode energy, and the othernonlinear coupling part. Linear (LT) and nonlinear (NT)contributions in dEφ ( m, n ) dt are calculated. The
magnitude of NT is comparable to LT, and LT drives themode variation, which is suppressed by nonlinear coupling
term NT [14]. To clarify which mode couplings are
dominant in nonlinear energy transfer, NT is decomposed
into components, corresponding to each three-wave
coupling (Fig. 3 (b)). Couplings with neighbouring modes,
such as (4, 1) ← (3, -1) + (1, 2) or (5, -1) + (-1, 2), aredominant. Although the amplitude of (3, -1) + (1, 2)
coupling is largest, the sign of the term NT is mainly
dictated by (5, -1) + (-1, 2) coupling, which is mostimportant for streamer formation in this case. If themediate mode (1, 2) is artificially removed after thesaturation, the streamer is not sustained and only a single
mode becomes dominant [14]. These results suggests thewith that with the zonal flow in (a), which shows thecontour of the (0, 0) mode. The perturbation structure in
the zonal flow case is a mixture of some modes, as shownin Fig. 2 (b).
Selective formation of the turbulent structure can be
identified by changing νin, which represents the strength ofthe damping force of the zonal flow [14]. The zonal flow
amplitude is small and the streamer modes are excitedwhen νin > 0.05. As νin is decreased, the zonal flow begins
to be excited. The inflection point is given to be νιnc ~0.052 with the parameters in Subsec. 3.1, which is the clear
indication to be nonlinear phenomena.

Fig. 2: Snapshots of the contours of the electrostatic potential,
which is normalized by the maximum value at that time. (a) The(0, 0) component and (b) other component in the case of zonal
flow formation with νin = 0.02, and (c) that of streamer formation
with νin = 0.1 are shown.

4. Energy Exchange Paths
4.1. Competition of mode coupling

In the cases discussed in Sec. 3, there are twodominant energy exchange paths from m ≠ 0 mode bymode coupling. One is that to (0, 0) mode to form the zonal
flow, and the other is that to the mediator mode ((1, 2) inthis case) to form the streamer. These two kinds ofstructural formation mechanisms are involved, but only
one of the structures can appear in stationary states fromtheir competitive nature. When the zonal flow is formed,
the E × B shearing of the zonal flow breaks the phaselocking of the modes, so the streamer is not formed, eventhough amplitudes of the modes are large.
Fig. 3: Time evolution of (a) Fourier modes of the fluctuation

4.2. Energy exchange for streamer formationenergy, and (b) rates of the nonlinear energy transfer decomposed

The detailed analysis of the energy exchange forstreamer formation is carried out. Figure 3 (a) shows thetime evolution of the fluctuation energy of the potential in
the case with νin = 0.1. Modes with (m, n) = (±4, ±1) andinto each three-wave coupling in the nonlinear steady state, whenνin = 0.1. This is the case when n = 0 modes are not included in
the calculation.

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N. Kasuya et al., Analyses of Nonlinear Coupling for Turbulent Structural Formation in Magnetized Cylindrical Plasmas

case without (m, 0) modes. Modes with (m, n) = (±3, ±1)
and (±4, ±1) are dominant in this case, which form a
streamer. The structure sustains for much longer duration
than the drift wave oscillation period, but shorter than thatin the case without (m, 0) modes.
To confirm which mode couplings are dominant innonlinear energy transfer, the rate of energy exchange for(4, 1) mode is calculated, which is decomposed into
components, corresponding to each three-wave coupling(Fig. 4 (b)). Among the couplings with m ≠ ±4 modes,
couplings with neighbouring modes, such as (4, 1) ← (3,-1) + (1, 2) or (5, -1) + (-1, 2), are dominant for nonlinear
mode excitation, as is the same in the case without (m, 0)
modes. Couplings mediated by (m, 0) modes, such as (4, 1)← (-3, 1) + (7, 0) or (-5, 1) + (9, 0), give other paths forenergy transfer, which are main contributors of thedifference between the sum of the rate of energy exchange
and that from dominant coupling (4, 1) ← (3, -1) + (1, 2).
The couplings mediated by (m, 0) modes affect as dampingof the mode, whose rate of energy exchange has thenegative sign. It is found that the couplings mediated by (1,2) mode is still most important for streamer formation inthis case. The quasi-linear effect of the (0, 0) mode is also
important for mode saturation. The formation mechanismof the (0, 0) mode in the case with (m, 0) modes will be
studied in future works.

important role of the (1, 2) mode for the streamer
formation. The mediator mode number is (m, n) = (1, 2),
because most unstable modes with neighboring m and n =1 form the streamer. The axial mode number of themediator does not change with a different device length,
because n = 1 modes are still most unstable withparameters for experimental devices [13]. Note that mode
coupling in the kr space also affects on nonlinear saturation,
which suggests three-dimensional mode coupling isessential for preservation of the streamer [19].

4.3. Streamer formation with n = 0 modes

Calculations including modes with m ≠ 0 and n = 0
are also carried out. These axially homogeneous modes
contributes to the formation of the turbulent structure, so
inclusion of the modes is important for clarify the
structural formation mechanism in the experimental device.
In our model, stabilizing effect of modes with m = 0 and n≠ 0 is not included. This is because (0, n) modes show
bursty increase in the nonlinear phase, if these modes areadded in the calculation. Therefore, we only discuss therole of (m, 0) modes here. These modes are investigated interms of geodestic acoustic mode (GAM) oscillation in
tokamak plasmas [1].
Figure 4 (a) shows the time evolution of thefluctuation energy of the potential in the case with νin =0.15. Inclusion of (m, 0) makes the system more unstable,
and the zonal flow can be excited with larger νin than the

5. Summary

We have carried out the nonlinear simulation of theresistive drift wave in cylindrical plasmas. Turbulence witha zonal flow or a streamer was obtained in the nonlinear
steady states. Detailed analyses show that the energy
transfer between modes, which have neighboring m bymeans of the mediator mode (1, 2), is important for the
streamer formation. The streamer is sustained with balanceof some nonlinear mode couplings, and is selectively
formed with νin larger than the critical value, in spite of theother possible energy exchange paths. In this way, ourminimal model for analyzing the turbulent structuralformation mechanism by mode coupling represents theselection rule of the structure.

Acknowledgement

Authors acknowledge discussions with Prof. A.
Fukuyama, Prof. Y. Kawai, Dr. S. Shinohara, Dr. A.
Fujisawa, Dr. Y. Nagashima, Dr. T. Yamada, Dr. S.
Inagaki, Prof. P. H. Diamond, and Prof. G. R. Tynan. This
work is supported by the Grant-in-Aid for SpeciallyPromoted Research (16002005) and for Scientific
Research (20760581, 19360418) of MEXT, by Research
Fellowships of the Japan Society for the Promotion of
Science for Young Scientists, by the collaboration
program of NIFS (NIFS08KNXN138, NIFS06KDAD005,
NIFS07KOAP017) and of RIAM of Kyushu University.

Fig. 4: Time evolution of (a) Fourier modes of the fluctuationenergy, and (b) rates of the nonlinear energy transfer decomposed
into each three-wave coupling in the nonlinear steady state, whenνin = 0.15. This is the case when n = 0 modes are included in thecalculation.

80

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