J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)245–248
000–000

New Possibilities of Mathematical Modeling ofTurbulent Transport Processes in Plasma
SKVORTSOVA Nina, BATANOV German, PETROV Alexander, PSHENICHNIKOV Anton, SARKSIAN Karen,KHARCHEV Nikolai, BENING Vladimir1, KOROLEV Victor1, MARAVINA Tatiana1, NASYROV Farit2,
SANCHEZ Joaquin3, OHKUBO Kunizo4, SHIMOZUMA Takashi4, IDEI Hiroshi4,
YOSHIMURA Yasuo4 and KUBO Shin4
General Physics Institute, Moscow, Russia
Moscow State University, Moscow, Russia
2Ufa State Aviation Technical University, Ufa, Russia
3
EUROATOM-CIEMAT, Madrid, Spain
4
National Institute for Fusion Science, Toki 509-5292 Japan
1(Received: 9 December 2003 / Accepted: 29 April 2004)

Abstract
Measurements in the LHD, L2-M, and TJ-II stellarators show that plasma density fluctuations have non-Gaussian
distributions with heavy tails and sharper vertices. Non-Gaussian probability densities of stochastic plasma processes
indicate non-Brownian character of the motion (diffusion) of particles. The role of rare events related to stochastic
plasma processes with larger spatial and temporal scale becomes important. It is shown that the first-order differencesof fluctuation samples are stochastic and their probability distribution is a mixture of Gaussians with different scales.
Subordinated Lèvy process can be used to describe the turbulent transport process.
Keywords:
plasma turbulence, fluctuation, non-Gaussian process, Lèvy processes, stellarator, scattering technique

1. Introduction

autocorrelation functions with oscillating tails and waveletspectra show distinct harmonics. At the same time, theprobability density function (PDF) of the time sample ofmagnitudes differs from a Gaussian distribution [7,8].
According to the traditional viewpoint on what goes onin plasma turbulence, the distributions of processes registeredin experiments in stellarators should be normal. However, thestatistical analysis of experimental data demonstrates that inpractice this is far from being so. The obtained empirical
distributions are usually far from being normal possessing
densities with tails heavier and vertices sharper than thoseinherent in the normal law [7]. This means that the registeredprocesses are far from being successfully modeled by theBrownian motion. Moreover, it was statistically made evident
that these processes possess a kind of memory exhibiting thepresence of long-range dependence of their values [8].
Recently, many alternative models were proposed forthese processes. First of all, it should be mentioned that,
following the ideology of Hurst and Mandelbrot [9], in many
papers the authors investigated the possibility of applicationof the so-called fractional Brownian motion as an alternative
to pure Brownian motion. We should even say that thefractional Brownian motion is a very popular model. Unlike

Recently, low-frequency (LF) plasma turbulence inclosed magnetic confinement systems attracted considerable
interest of plasma physicists. The 2003 EPS conference
demonstrated that experimental research of this kind is now
being carried out in all of the existing tokamaks andstellarators: T-10, LHD, TJ-II, DII-D, JET, etc. (see [1], reportnos. P-2.56, P-3.121, P-4.5, O-2.1A, and P-120). A reason
for the increase of interest in LF turbulence is that many
experimental facts directly indicate its influence on themacroscopic plasma parameters in closed magnetic
confinement systems.
In many toroidal devices LF turbulence in core and edge
plasmas is strong structural [2,3]. The term “strong structuralturbulence” means that on the background of strong
turbulence due to the onset of plasma instabilities, there exists
an ensemble of the interacting stochastic plasma structures.
Various types of stochastic structures (such as solitons,vortices and MHD structures) were observed experimentallyin the edge plasma [4-6]. It was found that the strong
structural turbulence existing in the edge plasma possesses
some general properties irrespective of the turbulence source
and the type of stochastic plasma structures. The turbulenceof this kind is characterized by slowly decreasing©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: nina@fpl.gpi.ru
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Skvortsova N. et al., New Possibilities of Mathematical Modeling of Turbulent Transport Processes in Plasma

radiation of the heating gyrotron [12]. The correlation andstatistical characteristics of time samples of amplitudes ofdensity fluctuations and their first-order differences areanalyzed. The first-order difference of amplitudes of plasma
density fluctuations in a time sample is defined as ∆Xj = Xj (tj)– Xj–1(tj–1) [7]. The first-order differences of the random value∆Xj depend on the linear and nonlinear growth and decay
processes involved in the formation of plasma density
fluctuations under conditions of a particular experiment.
Figure 1 shows the autocorrelation functions (ACFs) ofamplitudes of plasma density fluctuations and their first-order

pure Brownian motion, fractional Brownian motion has anindisputable advantage, which consists in that this process cansuccessfully model the long-range dependence of the valuesof registered data (time series). However, we dare say thatalong with this advantage, fractional Brownian motion has ano less indisputable drawback, which consists in that all its
finite-dimensional distributions remain Gaussian,contradicting to what is observed in practice. Therefore,
alternative models have to be designed. Recently it wasproposed to model plasma turbulence by the so-called
subordinated Lèvy processes. Recall that a Lèvy process is astochastic process with stationary independent first-orderdifferences. A subordinated process is a stochastic process ofthe form S (t) = X (M (t)), t ≥ 0, where X (t) and M (t) arestochastic processes such that X (t) is measurable and thetrajectories of the subordinator M (t) start from the origin anddo not decrease. In this setting the process M (t) plays therole of random time. The application of subordinated Lèvy
processes to the interpretation of the structural plasma
turbulence was proposed by V. Korolev and was justified inthe corresponding limit theorems, see [10, 11]. Here we only
mention two attractive properties of subordinated Lèvy
processes, namely, their self-similarity and the representabilityof their finite-dimensional distributions as scale/location
mixtures of Gaussian laws.
In the problems under discussion the intensity ofprocesses substantially varies in time so that the notion ofsubordination seems to be very adequate. Thus, theamplitudes of plasma density fluctuations and their first-orderdifferences in the structural turbulence observed in the hightemperature plasma of L-2M, TJ-II, and LHD [3] vary
substantially in time. Heavy tails in this case can arise as aresult of a certain influence function (with memory) or as aresult of a great number of events with abnormally large
amplitudes in first-order differences [11]. Samples of firstorder differences of such processes are usually independentand homogeneous. The adequacy of the proposed models isconfirmed by the results of statistical analysis giving a good
fit of mixtures of normal distributions with real data. In ouranalysis we apply a well-developed technique of statisticalseparation of finite normal mixtures, in particular, theEstimation Maximization (EM) algorithm.
In this paper we consider the statistical characteristics oftime samples of amplitudes and their first-order differencesof the core plasma density fluctuations in stellarators LHD,
L-2M and TJ-II and relation of these characteristics with thenon- Brownian particles motion.

Fig. 1

Autocorrelation functions of amplitudes of density
fluctuations (a) and their first-order differences (b) in
LHD (the upper plot) and L-2M (the lower plot).

Fig. 2Probability densities of amplitudes of fluctuations (theupper plot) and their first-order differences (the lowerplot) in successive time intervals in LHD.

2. Experimental results
The parameters of the L-2M, LHD and TJ-II devices aredescribed in detail in [3]. Density fluctuations in hightemperature plasma at the middle radius in TJ-II were studiedby 2-mm scattering technique (k = 3 cm–1 and k = 6 cm–1).
Plasma density fluctuations in the heating region near the axisof the plasma column in L-2M (k = 20 cm–1 and k = 40 cm–1)
and LHD (k ~ 30 cm –1) were measured from scattered2246

Skvortsova N. et al., New Possibilities of Mathematical Modeling of Turbulent Transport Processes in Plasma

differences in L-2M (k = 40 cm–1) and LHD (k ~ 30 cm–1). It
is well known that, for the structural plasma turbulence, thetime sample is not homogeneous and independent (the ACF
demonstrates the presence of a long-lived component). The
independence of the first-order differences in the structural
plasma turbulence is evidenced by the absence of long-termcorrelations in their ACF.
The scale mixtures of Gaussian laws model experimental
probability density functions of the first-order differences. As
an illustration, we only present PDF figures of strong
structural turbulence in LHD, similar results were obtainedin other two devices.
In Fig. 2, one can see the PDFs of time samples offluctuation amplitudes measured in successive time intervals
during an LHD discharge. Note that a non-Gaussian PDF offluctuation amplitudes, which is typical in the structural
plasma turbulence, varies noticeably during the discharge (theupper plot in Fig. 2). Unlike the PDF of the amplitudes, thePDF of the first-order differences in the steady-state dischargevaries only slightly (the lower plot in Fig. 2). The
independence of the time samples of the first-order differences
and the stability of their PDF suggest that subordinated Lèvy
processes can model the structural plasma turbulence. Below,
it will be shown that mixtures of normal distributions give agood fit to the PDF of the first-order differences of thefluctuation amplitudes in the structural plasma turbulence.
The longest time samples of fluctuations up to 200 ms
were measured in TJ-II, where measurements were made at
the middle radius (far from the heating region and the plasmaboundary). The analysis of samples (histograms) of the firstorder differences of the plasma density fluctuations
demonstrated the dependence of the fitted distribution on theFig. 3

sample size (the length of the corresponding time series). The
histograms for relatively short time samples (103 points,
which corresponds to the interval of 2 ms) are well describedby mixture of two Gaussian distributions

f ( x; p, µ1 , µ 2 , σ 1 , σ 2 ) =p( x − µ1 )2+exp −2σ 122π σ 11− p( x − µ2 )2,exp −2σ 22
2π σ 2p, 1 – p are probabilities (weights), µ1, µ2 are drift rates andσ1, σ2 > 1 are diffusion rates of these two distributions. Thecoincidence of the model with experimental data was tested
by Kolmogorov-Smirnov goodness-of-fit test and reached P= 0.9 for equal intervals throughout discharge duration (P isthe test characteristic which means the probability to attainworse coincidence in the next experiment as compared to theobserved one: the closer P to 1, the better the attainedcoincidence; the closer P to 0, the worse the coincidence; see,e.g., [13]). For the longer samples up to 3 × 104 points, amixture of three Gaussian distributions

f ( x; p, µ1 , µ 2 , µ3 , σ 1 , σ 2 ,σ 3 )
3

=∑
k =1

pk( x − µ k )2
exp −
2σ k2
2π σ k(p k > 0, p 1 + p 2 + p 3 = 1) provides good fit withexperimental data (P ≈ 0.8).
The length of stationary time samples of plasma density
fluctuations in the heating region of LHD did not exceed 5·103
points. Figure 3 shows the histogram of a time sample
consisting of 2,000 points. In the same figure one can see amodel normal distribution which fails to agree withModeling of the PDF of a time sample of first-order difference of fluctuations in LHD by mixing three Gaussian distributions,labeled by 1,2 and 3, respectively.

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Skvortsova N. et al., New Possibilities of Mathematical Modeling of Turbulent Transport Processes in Plasma

This work was supported in part by the RussianFoundation for Basic Research (project nos. 03-02-17269, 0402-16571, 02-01-01080a, 02-01-00949a) and the LIME
Program (Japan).

experimental data: the Kolmogorov—Smirnov test for fitting
one normal distribution with parameters µ = –0.000148, σ =0.029439 gives a significance value P = 0.0017. This timesample is modeled by a mixture of three Gaussian processes
with weights p1 = 0.482032, (µ1 = –0.001758, σ1 = 0.036352);
p 2 = 0.400771 ( µ 2 = –0.000051, σ 2 = 0.015801); p 3 =0.117197 (µ3 = 0.006141, σ3 = 0.032432). The Kolmogorov—
Smirnov test proves that this model mixture fits processadequately: P = 0.8176. The probability density function forthis mixture is also shown in the figure. The representationof the PDF of first-order differences of fluctuation amplitudes
by scale mixtures of normal processes corresponds to theLèvy process, which allows one to estimate the relative
contribution from different turbulent diffusion mechanisms totransport processes as well as diffusion coefficientsthemselves.

References[1] 30th EPS, St. Petersburg, 2003. http://eps2003.ioffe.ru/
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3. Conclusions
From this analysis we make three definite conclusions:(i) the normal model fails to fit the PDF of first-orderdifferences of fluctuations in the structural plasma turbulencewith a very high significance level; (ii) even two-component
normal mixtures demonstrate very good fit with very high Pvalue for rather short samples of fluctuations; (iii) for longer
samples, a slight complication of the model (introduction ofthe third component) practically solves the problem.
The main conclusion is that the registered process canbe successfully modeled by a combination of a finite numberof diffusive processes each of which corresponds to a certain
diffusive mechanism related to the structural turbulence in aplasma transport process. This combination is a subordinated
Lèvy process (more precisely, a subordinated Wiener process)with a subordinator having a discrete distribution. As this isso, the characteristics of the combined process (the shares ofeach diffusion component as well as their drift and diffusiverates) can be estimated automatically, say, by the EM
algorithm.

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