J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Turbulence Diagnostic Simulator for Analyzing Structural Formation in
Magnetically Confined Plasmas
N. KASUYA, S. NISHIMURA, M. YAGI 1), K. ITOH, S.-I. ITOH 1) and N. OHYABUNational 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: 19 November 2009 / Accepted: 8 January 2010)

Turbulent structures affect the level of anomalous transport, and their formation mechanism is one of the crucial
issues in the fusion plasmas. The structural formation mechanism in cylindrical plasmas is discussed in detail to
identify the bifurcation phenomena of the turbulent structure. A three-field reduced MHD model was extended todescribe the resistive drift wave turbulence, and a zonal flow and a poloidally localized turbulent structure, which
has the typical temporal scale of the streamer, are formed selectively by nonlinear couplings of unstable modes. We
carried out analyses to simulate the measurement with electrostatic probes, and find out the characteristic featuresof the turbulent states. This is the first step for integrated comparison between experiments and numerical simulations
by using a long series of three-dimensional turbulence data to understand the global transport phenomena.
Keywords: turbulence simulation, diagnostic, structural formation, zonal flow, streamer, linear device, spectrum
analysis

carried out numerical simulations in cylindrical plasmas, asa minimal model for analyzing the structural formation
mechanism in magnetically confined plasmas by mode
coupling [6][17]-[21]. Turbulent structures, such as zonal
flows and streamers have been obtained in the nonlinear
saturated states, and their formation mechanisms have beenstudied [6]. In this paper, we analyze the time series of thephysical variable from three-dimensional numerical data,and carry out correlation and spectrum analyses as carriedout with electrostatic probes in experiments. The
spatio-temporal profiles are obtained, which identify thebifurcation phenomena of the turbulent structure.
The paper is organized as follows. In Sec. 2, the set ofmodel equations and parameters for the analyses aredescribed. A zonal flow and a streamer are formed in the
nonlinear simulations, and the formation mechanism is
explained in Sec. 3. The correlation and spectrum analyses
are carried out on the numerical turbulence fields, andcharacteristic features of the formed structures are
discussed in Sec. 4. Then, we summarize our results in Sec.
5.

1. Introduction

The theoretical researches on plasma turbulence
reveal that meso-scale structures, such as a zonal flow anda streamer, play an important role on turbulent transport [1].
Therefore, quantitative analyses of their formation andself-regulated mechanism must be carried out forunderstanding the global transport phenomena.
Development of the experimental diagnostics can give highresolution measurements of fluctuations in space and time[2][3], and integration of several measurements will givedetailed profiles of the turbulence in magnetized plasmas.
In numerical simulations, improvement ofcomputation can produce a long series of turbulent data
[4][5], and turbulence analyses can be carried out on the
numerical turbulence fields [6][7]. We have beendeveloping a turbulence diagnostic simulator, which
numerically simulates plasma turbulence and data analyses
on the simulation data, as same in the experiments. The
simulator studies the fundamental mechanisms of turbulent
structural formation, and aids the development of the data
analysis technique to deepen our physical understandings.
Plasma experiments in a simple linear configuration
have been revisited for quantitative understandings of thestructural formation mechanism by turbulence [8]-[12].
These laboratory plasmas are suitable for carrying out
detailed
measurementsoffluctuations,andtwo-dimensional measurements reveal the feature of
turbulent structures [13][14] and their formation
mechanisms by nonlinear mode coupling [15][16].
For the first step to study fusion plasmas, we have

2. Drift Wave Simulation

2.1. Model
We 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 [18]). The
three-field (density, potential and parallel velocity ofelectrons) reduced fluid model is adopted here. The plasma

author’s e-mail:kasuya@nifs.ac.jp523©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

N. Kasuya et al., Turbulence Diagnostic Simulator for Analyzing Structural Formation in Magnetically Confined Plasmas

carried out, where the time independent source profile isgiven byhas 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 [22], so theeffect of neutral particles is taken into consideration. The
continuity equation, the vorticity equation and Ohm's law
can be used to obtain the fluctuating density, potential andparallel velocity of electrons [23]:dN2   //V  V // N   N  N  S ,dtd   2d   2 N    in     in   
dt
dt 


S r  2
  r 2 
4S0  N   r      ,

1
exp
2
LN   LN    LN  

(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 [18]. The density profile peaked at r
= 0 destabilizes the resistive drift wave.

(1)

3. Turbulent Structural Formation

Using NLD, selective formation of the turbulent
structures, zonal flow and streamer, has been studied.
Linear analyses give linear growthrates andeigenfrequencies [18]. Only n = 1 modes can be unstablewith these parameters. The dispersion relation of the linear
eigenmodes with n = 1 shows weak dispersion in small m(m < 4), and / k~ 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 (r / a) for all the other modes,
where f implies {N, , 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).
In the nonlinear saturation states, two kinds ofturbulent structures have been obtained [6]; 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. Figure 1(a) shows the snapshot of the contour of the potential in thecase with in = 0.02, where the (0, 0) component isdominant. The amplitude of the (0, 0) mode oscillates, andthose of the other modes varies, accordingly. The limit
cycle oscillation between the (0, 0) mode and theperturbations contributes to the saturation [19]. Figure 1(b) shows the time evolution of the contour of thefluctuation, where the (0, 0) components are subtracted.
The perturbation structure in the zonal flow case is a
mixture of some modes. The dominant mode is replaced
from (4, 1) to (2, 1), and the vortex structures are tilted
and broken by the (0, 0) shear flow in this duration.
If the collision frequency is large, the zonal flow
remains stable, owing to strong collisional damping, andparametric coupling with modes, which have neighboringm and the frequency close to each other, forms a streamer.
Figure 1 (c) shows the time evolution of the contour offluctuation in the case with in = 0.1. Strong vortex islocalized in the poloidal direction, and rotates to theelectron diamagnetic direction. The localized structure issustained for much longer duration than the typical time  //V  V  // N   W   4

,
(2)
dV
M
2
//  // N   eV  V V ,

(3)
dt
me
where N = ln (n / n0), V = v// / cs,  = e  / 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, 
is the electrostatic potential, Te is the electron temperature,d / dt =  / t + [, ] is the convective derivative, S is aparticle source term, M / me is mass ratio of ion andelectron, in is ion-neutral collision frequency, e = ei + enis the sum of ion-electron and electron-neutral collision
frequency, and N, V, W are artificial viscosities. The ion
cyclotron frequency ci and Larmor radius measured by
the electron temperature s 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, , V}, and r = a gives an outer boundary of theplasma column.
2.2. Simulation parameters
A nonlinear simulation has been performed to
examine the saturation mechanism of the resistive drift
wave turbulence. The relation f (-m, -n) = conj(f (m, n)),where f implies {N, , V}, is used in the calculation tosatisfy the physical quantities to be purely real in the real
space. The following parameters are used: B = 0.1 [T], Te =2 [eV], a = 10 [cm], length of the device  = 1.7 [m], N =1  10-2, V = W = 1  10-4. Using these parameters, e is
estimated to be e = 310 [18]. The electron collisions (eiand en) destabilize and the ion-neutral collisions (in)stabilize the resistive drift wave. Therefore, the drift wave
can be excited with large e and small in. There isambiguity of the value of collision frequency in, which
depends on the neutral density. Therefore, in is used as aparameter for controlling the instability in our simulations.
The calculation with a fixed particle source has been524

N. Kasuya et al., Turbulence Diagnostic Simulator for Analyzing Structural Formation in Magnetically Confined Plasmas

scale of the drift wave oscillation (more than 3000 ci-1).
Modes with (m, n) = (4, 1) and (5, 1) are dominant inthis case, and their nonlinear coupling with the mediatormode (1, 2) is the origin of the sustainment of the streamer[6].The streamer in our simulation consists of rather small
number of dominant Fourier modes, (4, 1) and (5, 1),
which are coupled with each other by means of themediator mode (1, 2), and form a quasi-mode like structure
localized in the direction. The poloidally localized
turbulent structure, which has the typical spatial andtemporal scale of the streamer without the spatial scale in
the radial direction, are formed by self-bunching of acouple of unstable modes in nonlinear saturation states.
The nonlinear formation mechanism was quantitatively
analyzed by calculating the nonlinear energy transfer toeach mode [24].
The self-organized structures depend on ion-neutral
collision frequency, which is the damping parameter of thezonal flow in this model. If the collision frequency is small,a zonal flow is generated, which suppresses turbulent
transport. The fluctuation amplitudes are larger withsmaller in, but the turbulent fluxes have the same level,
not depending on in, because of the formation of the zonal
flow [6]. On the other hand, if the collision frequency islarger than a critical value, a zonal flow is damped, and a
streamer is formed. Figure 2 shows the dependencies offluctuation energies of the potential on in in nonlinear
states. The energy components of (0, 0) mode, which is themode energy

0.15

Zonal
flow

0.1

Streamer(4,1) + (5,1)

0.05

(0,0)

0
0

0.05

0.1

0.15

0.2

in /  ci

Fig. 2: Dependencies of the fluctuation energies of theelectrostatic potential on in. Energies of (0, 0) mode andthe sum of (4, 1) and (5, 1) modes, which are the zonal
flow and the streamer component, respectively, are shown.
zonal flow, and the sum of (4, 1) and (5, 1) modes, which
form the streamer, are shown. The zonal flow amplitude issmall and the streamer modes are excited when in > 0.052.
As in is decreased, the zonal flow begins to be excited.
There are two dominant energy exchange paths from m  0mode by mode coupling in this case. One is that to (0, 0)
mode to form the zonal flow, and the other is that to themediator mode ((1, 2) in this case) to form the streamer.
These two kinds of structural formation mechanisms are
involved, but only one of the structures can appear instationary states from their competitive nature. When thezonal flow is formed, the E  B shearing of the zonal flow
breaks the phase locking of the modes, so the streamer is
not formed, even though amplitudes of the modes are large.
The streamer is selectively formed to keep the particle
balance as long as the drift wave is unstable. Thisprediction has guided the experimental discovery of thestreamer on the experimental device LMD-U [25].

4. Comparison of Spatial and Temporal Structures

In laboratory plasmas, the electrostatic probe
measurement is useful to obtain fluctuation data with highspatial and temporal resolutions in experiments [25]. A
transition between two turbulent states has been observedin the linear device, and the spatio-temporal spectra were
measured with a multiple probe array [26]. Two kinds ofturbulent structures are obtained in our simulations. We
carry out analyses resemble to the single and multi probe
measurements on the time series of three-dimensional
fields, as described in the previous section, to clarify thedifference of the turbulent structures. This is preparation of
more sophisticated comparison between experiments andnumerical simulations of plasma turbulence.

Fig. 1: Snapshots of the contour of the electrostatic
potential, which is normalized by the maximum value atthat time. The typical structure and the time evolution of
the contour of the fluctuation in the zonal flow case with
in = 0.02 are shown in (a) and (b), respectively. The time
evolution of the contour of the fluctuation in the streamer
case with in = 0.1 is also shown in (c).

4.1. Temporal structure from a signal at a single point525

N. Kasuya et al., Turbulence Diagnostic Simulator for Analyzing Structural Formation in Magnetically Confined Plasmas

Firstly, we show the time evolution of the potential ata fixed point (Fig. 3). This corresponds to a signal from asingle probe. The fine scale oscillation and its envelope
correspond to the drift wave oscillation and the turbulent
structure, respectively. The cases with in = 0.045 and 0.1
are shown as typical examples of the zonal flow andstreamer, respectively. In the zonal flow case, themagnitude of the envelope abruptly changes in a few timesof the drift wave oscillation period. On the other hand, thetime evolution is rather periodic in the streamer case. The
period of the envelope variation is about 70 times of thedrift wave oscillation period, corresponding to the rotation
period of the poloidally localized structure. When thestreamer comes to the observation point, the amplitude
becomes larger. To describe the difference of the temporalvariations, the auto-correlation of the signal is calculated(Fig. 4). The correlation time is much larger in the case of
the streamer than that of the zonal flow.
In the rotating frame with the velocity of the streamer
propagation, the streamer stands the same poloidal position,and the life time of the streamer can be estimated. Figure 5
shows the auto-correlations of the potential fluctuation on
the streamer frame when in = 0.1 by calculating theauto-correlation in the position where the envelope of thefluctuation has the maximum amplitude. Parameter scan
showed that the life time of the streamer is longer in theless unstable case with larger in [24].

Fig. 3: Time evolutions of the potential at r/a = 0.5,  = 0
and z/ = 0.375 when in = (a) 0.045 (zonal flow case)and (b) 0.1 (streamer case).

4.2. Radial correlation from signals at two points
Next, we show the correlation between two radialpoints to clarify the difference of the radial structure
between the zonal flow and streamer case. This
corresponds to the analysis using two probe signals at
different radii. Figures 6 (a) and (b) show the temporal

Fig. 4: Auto-correlations of the potential fluctuations at r/a= 0.5,  = 0 and z/ = 0.375 when in = (a) 0.045 and (b)0.1.
1.5
Auto-correlation1

in = 0.1

0.5
0
-0.5
-1
-1.5
-2000

-1000

0
Time

1000

2000

Fig. 5: Auto-correlation of the potential fluctuations,calculated on the streamer frame at r/a = 0.5 and z/ =0.375 when in = 0.1.

Fig. 6: Time evolutions of the radial correlation of thepotential fluctuations between r / a = 0.5 and 0.75 at  = 0,z /  = 0.375, when in = (a) 0.045 and (b) 0.1.

526

N. Kasuya et al., Turbulence Diagnostic Simulator for Analyzing Structural Formation in Magnetically Confined Plasmas

evolutions of the radial correlation of potential fluctuations
between r / a = 0.5 and 0.75 at  = 0, z /  = 0.375 in thecase of the zonal flow and the streamer, respectively. In thestreamer case, the radial correlation becomes close to 1,
when the streamer comes to the observation point, t ~ 5700.
On the other hand, in the zonal flow case, the radial
correlation is smaller than that in the streamer case. Thereis a duration when the correlation becomes close to 1 evenin the zonal flow case. The streamer like structure appears
intermittently, but is sustained only for a short duration, so
temporally averaged correlation is smaller than that in thestreamer case. The dependency of the radial correlation on
in showed that the radial correlation becomes larger as inincreases, and close to 1 when the streamer is sustained
steadily [24].

Fig. 7: Time evolution of the  profile of the potential at
r/a = 0.5 and z/ = 0.375, when in = 0.1.

4.3. Spatio-temporal structure from multiple signals
Finally, we show the spatio-temporal structures fromthe poloidal profile. This corresponds to the analysis usinga poloidal probe array. The poloidal probe array is installed
to clarify the structural formation in the laboratory plasmas[25]. Figure 7 shows a time evolution of the  profile of theelectrostatic potential in the case with in = 0.1. The
localized vortex is formed (red region in Fig. 7), and its
poloidal position slowly rotates to the electron diamagnetic
direction. Mode matching by nonlinear coupling sustains
the localized structure for much longer duration than thedrift wave oscillation period.
Two-dimensional Fourier decomposition gives thespatio-temporal spectrum of the potential (Fig. 8). In thezonal flow case, low m modes with weak dispersion, which
are linearly unstable, are excited and show the broad band
spectrum. On the other hand, frequencies of the dominantmodes are close to each other, and their harmonics are
excited in the streamer case. The rather coherent spectrum
appears in this case. The spatio-temporal spectra show
clear difference between the zonal flow and streamer case.
The spatio-temporal spectrum shows the dispersion
relation of the excited modes. The instantaneous linear
dispersion relation is calculated with the density andpotential profile in the nonlinear saturated state, and iscompared with that of the excited modes in the streamer
case (Fig. 8 (c)). The modes with m = 3 - 5, which have thelargest amplitudes, are linearly unstable. A theoretical
study showed that unstable modes with dispersion relation
′(k) strongly induce modes with relation (k) = N′(k/N)(N=2,3,4…) by nonlinear coupling [27]. The sharp peaks
satisfy those relations, which are plotted by dashed curves
in Fig. 8 (c). In this way, diagnostics on numerical data isuseful to capture the fundamental characteristics ofturbulence.

(a)

(b)

(c)

Fig. 8: Spatio-temporal spectra at r/a = 0.5 and z/ = 0.375,
when in = (a) 0.02 and (b) 0.1. (c) Magnification of (b)with the dispersion relations of linear eigenmodes andquasi-modes induced by eigenmodes, which are indicatedby dashed lines 1 and 2-4, respectively.

5. Summary

We have carried out the nonlinear simulation of the527

N. Kasuya et al., Turbulence Diagnostic Simulator for Analyzing Structural Formation in Magnetically Confined Plasmas

resistive drift wave in cylindrical plasmas. Turbulence witha zonal flow and a streamer was obtained in the nonlinear
steady states. Analyses simulating the single and multiprobe measurements have been carried out to clarify thedifference of the spatio-temporal structures in thebifurcation. Intermittent behavior appears in the zonal flowcase. On the other hand, there is long sustainment of thebunched structure with a strong radial correlation in thestreamer case. A broad band and coherent spectrum appear
in the zonal flow and streamer case, respectively. The
frequencies of the nonlinearly excited modes agree withthe theory of induced quasi-modes. We showed theanalyses to clarify the difference of the turbulence
structures, which is observed in our numerical simulation,in this paper. Calculations to simulate each experimental
condition and detailed comparison with experiments will
be future tasks.
High-order correlation analyses make quantitative
estimation possible to find out which nonlinear couplings
are main contributors of the structural formation, and to
identify the direction of the energy transfer [28][29]. These
techniques will also be applied on the simulation data.
Applying the results obtained in the simulations incylindrical plasmas, we are developing the turbulence
diagnostic simulator in toroidal plasmas. The
spatio-temporal data of turbulent fields are generated by
global simulation, using a fluid model describing
drift-interchange instability. The analyses as same in theexperiments are made on the simulation data, andcomparison with experiments will be carried out.

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Acknowledgement
Authors acknowledge discussions with Prof. A.
Fukuyama, Prof. Y. Kawai, Prof. A. Fujisawa, Dr. S.
Shinohara, 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), for Young Scientists(20760581) and for Scientific Research (19360418,
21224014) of JSPS, by the collaboration program of NIFS
(NIFS09KNXN170, NIFS09KTAD009) and of RIAM of
Kyushu University.
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