Proceedings of ITC/ISHW2007

Analysis of Temporal Evolution of Dynamic and Diffusive Components in
Drift Particle Fluxes in the Edge Plasma of the L-2M Stellarator Operated inDifferent Confinement Regimes
Nina N. SKVORTSOVA 1), German M. BATANOV1), Nikolay K. KHARCHEV1),
Yuryi V. KHOL’NOV1), Dmitryi V. MALAKHOV1), Alexander E. PETROV1),
Vyacheslav V. SAENKO1), Karen A. SARKSYAN1), Dmitryi G. VASILKOV1),
Victor Yu. KOROLEV2), Yuryi V. ZHUKOV2)
1)

A.M. Prokhorov General Physics Institute, Moscow, 119991 Russia
2)

Moscow State University, Moscow, 119899 Russia

The article presents an analysis of dynamic and diffusive components in turbulent fluxes measured in the edge
plasma of the L-2M stellarator. It is shown that a shift-scale mixture of several (three-four processes, depending on
macroscopic characteristics of the plasma) normal processes adequately describes the measured particle fluxes. The
Estimation Maximization algorithm is applied to probability densities of flux increments in order to determine thenumber of mixture components and the weight of each component in the mixture. The method of sliding separationof mixtures is applied to separate between the dynamic and diffusive components of volatility and to trace theirtemporal evolution. Some characteristic L-2M regimes are examined. It is shown that turbulent particle transport in
the edge plasma occurs by both the ballistic and diffusive mechanisms, and their contributions are of the same orderof magnitude.
Keywords:

turbulence, turbulent flux, stellarator, non-Gaussian statistics, volatility, diffusion.

1. Introduction
Strong structural low-frequency turbulence inmagnetoactive plasma is frequently observed in the courseof investigations of low-frequency fluctuations in toroidal
magnetic confinement systems [1]. A characteristic featureof strong structural low-frequency turbulence is thepresence of ensembles of stochastic plasma structures andprobability density functions (PDF) for all plasma
variables differ from the normal distributions. In particular,PDFs of turbulent particle fluxes are leptokurtic and haveheavy-tailed distributions.
The approach based on inhomogeneous random walks
allows the PDF of increments of amplitudes of turbulent
particle fluxes to be represented in the form of shift-scale
mixtures of normal distributions. In this case, it is possible
to separate dynamic (convective) and diffusive components
in turbulent particle fluxes. This approach was applied toanalyzing turbulent fluxes in the FT-2 tokamak and L-2M
stellarator [2].
The article presents an analysis of dynamic anddiffusive components in turbulent fluxes measured in theedge plasma of the L-2M stellarator. For analysis we use awide database of experiments under diversified operating
conditions (in the basic and modified magnetic
configurations, the use of boronization of the chamber wall
author’s e-mail: nina@fpl.gpi.ru

surface to suppress impurity fluxes from the wall).

2. Estimation of turbulent processes in plasma interms of volatility
Turbulent processes in the edge plasma of the L-2M
stellarator were estimated in terms of volatility. Volatility,
as applied to turbulent fluxes, means "the degree ofunpredictable change over time of edge turbulent fluxes".
The probabilistic concept of volatility has long been in use
in financial statistics; however, to our knowledge, thischaracteristic has never been employed in papers devoted
to plasma turbulence. Volatility of a random process isdetermined by at least two types of factors. The first typecan be characterized as a dynamic (convective) factor. The
influence of the dynamic factor makes itself evident in thefact that the process varies because of the presence of acertain trend or a combination and, hence, interaction oftrends that indicates the presence of structures in a plasma.
The dynamic component corresponds to a drift (which canoccur by a variety of processes), for example, to ballistictransport. The second type of factors is termed stochastic
or diffusive. With the use of special mathematicalstatistical procedures, the diffusive component can berepresented as a sum of subcomponents, each being related
to a particular type of stochastic structures, their interaction,

Proceedings of ITC/ISHW2007
etc. In the basic model of nonhomogeneous random walk
described by a subordinate Cox process, volatility isnaturally represented as a sum of the dynamic and diffusive
components [3].
In previous studies of low-frequency turbulence, it
was shown that processes of turbulent particle transport aredescribed by subordinate Cox processes and is described,
to a high degree of accuracy, by shift-scale mixtures ofnormal distributions [4]. If X is an increment of a random
process under consideration, then its distribution is a
shift-scale mixture of normal distributions if
P ( X < x ) = ∫∫ Φ(x −VΕΦ(),Ux−v)dP(U < u ,V < v ) =u(1)x ∈ ℜ,

where U ≥ 0 and V are finite random variables. In thiscase, volatility is characterized by a vector containing
components of three types: a weight (positive numbers,
which add up 1), drift factors (averages), and diffusion
coefficients (standard deviations) of the corresponding
components. Each of these mixture components
corresponds to a particular mechanism and a particular
structure in plasma turbulence, which is characterized byparameters αj и σj, whereas the weight pj defines theproportion of structures of the jth type in the whole processof plasma turbulence.
In [5], it is shown that a finite shift-scale mixture of
normal distributions (1) is represented in the formx −ajk∑ p Φ( σjj =1) = ΕΦ (jx −V),U(2)where the pair of random variables U and V has adiscrete distribution

3. Analysis of dynamic and diffusive componentsof turbulent fluxes
The L-2M stellarator has two helical windings (l= 2),
a major radius R = 100, and a mean plasma radius<a> = 11.5 cm [6]. The plasma was created and heated by
one or two 75-GHz gyrotrons under conditions of electron
cyclotron resonance heating (ECRH) at the second
harmonic of the electron gyrofrequency. The magnetic
field at the plasma center was Bt = 1.3–1.4 T. The gyrotron
power was P0 = 150–300 kW, and the microwave pulse
duration was up to 15 ms. Measurements were carried out
in a hydrogen plasma with an average density of <n> =(0.8–2.0)⋅1013 cm–3 and central temperature of
Te = 0.6–1.0 keV. Turbulent flux was measured in the edge
plasma at a radius r/a = 0.9, the density was at a level ofn(r) = (1–2)⋅1012 cm–3 and the electron temperature wasTe(r) = 30–40 eV.
The local fluctuating particle flux as defined as%Γ = (δ ne ⋅ δ vr ) [4], where δ ne denotes the plasma
density fluctuations, and δ vr = δ EΘ B expressed
through the fluctuation of the poloidal electricfield δΕ Θ = (δϕ1 − δϕ2 ) r ∆Θ , where δϕ is thefloating-potential fluctuation in the plasma, and Θ is thepoloidal angular coordinate, r is the middle radius. Local
turbulent particle fluxes were measured in the edge plasmaof L-2M by probe systems consisting of three single
cylindrical probes. The flux was measured at different
distances of the last closed flux surface (LCFS).
Time samples of local fluxes in L-2M are nothomogeneous and independent. The PDFs of local fluxesin L-2M stellarator are leptokurtic and have heavier tails ascompared to a Gaussian distribution Statistical analysis ofcharacteristics of fluctuating particle flux is carried out
with time samples of flux increments, which arehomogeneous and independent.

kDV = ∑ p j (a j − a ) 2 ,(3)j =1kkj =1j =1

ΕU 2 = ∑ p jσ 2j , a = ∑ p j a j .

Probability

P ((U ,V ) = (σ j , a j ), j = 1,..., k, so that
0.20

Shot # 53168

Accordancewith model
97%

SumPDFs

0.15(4)

Expression (3) defines that volatility component
which is associated with the presence of local trends; it will
be referred below as a "dynamic component" of volatility.
Expression (4) defines a "diffusive component" ofvolatility. Total (multidimensional) volatility of the process
is the square root of the sum of two components, one ofwhich represents a spread of local trends, whereas the othercharacterizes diffusion.

20.1010.05

30.00-2-1012Increment, arb.un.

Fig.1. Modeling of PDF of increments of theturbulent flux measured in L-2M stellarator.

Proceedings of ITC/ISHW2007

Diffusion components, arb. un.

The finite mixtures of several (three or four, depending on
macroscopic characteristics of the plasma) normal
distributions adequately describe The PDFs of flux
increments [5]. As an illustration, Fig. 1 shows thehistogram of a time sample of the flux increments alongwith the result of processing. The Estimation Maximization(EM) algorithm was applied to probability densities of flux
increments in order to determine the number of mixture
components and the weight of each component in themixture. In this particular case, the PDF of flux increments
is a scale mixture of three normal distributions that
correspond to three stochastic plasma processes. Thisparticular experiment was conducted in the basic magnetic
configuration (rotational transform at the magnetic axis is0.175, mean plasma radius is 11.5 cm), with preliminary
boronization.
The method of sliding separation of mixtures wasused to separate between the dynamic and diffusive
components of volatility and to trace their temporalevolution. Time variations in the components of thesums (3) and (4) were calculated using a time window"sliding" over a time sample of flux increments. This
technique for analyzing the structural plasma
turbulence is described in more detail in the review [3].

Fig.3. Time behavior of the diffusive component ofvolatility turbulent fluxes measured at different
radial distances from LCFS in three shots.
Figure 3 and 4 show how the diffusive (4) anddynamic (3) components vary with time in the radialturbulent flux measured at different distances from LCFS.
Measurements were performed from a shot to shot underidentical experimental conditions. Characteristically, thedynamic component (which is associated with convection)is more volatile than the diffusive component.

0,90,850,80,750,70,650,60,550,50,450,40,350,30,250,20,150,10,05

Fig.4. Time behavior of dynamic component of
turbulent fluxes measured at different radial
distances from LCFS in three shots.
2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000Number of points

Fig.2. Time behavior of three subcomponents of thediffusive component of flux for shot No. 55623.
Figure 2 shows the time behavior of three diffusive
subcomponents of volatility (i.e., summands in (3)) for theexperiment conducted in the modified configuration ofmagnetic field (rotational transform 0.082, mean plasma
radius 12.5 cm, the LCFS contacts with the chamber wall).
Note that the number of subcomponents remains
unchanged, but their intensity varies with time. Hence,each of the components reflects a temporally continuous
process, and the contribution from this process to the sum
varies with time.

Figure 5 shows the time behavior of the diffusive anddynamic components of volatility during the steady-statephase of the discharge. The flux radial turbulent wasmeasured at a distance of 6 mm from LCFS. The
experiment was conducted in the modified configuration ofmagnetic field (rotational transform 0.082, mean plasma
radius 12.5 cm, the LCFS contacts with the wall).
From the curves in Fig. 5 it will be noticed the dynamic
component compares with the diffusive component This isa typical example of discharges excited in the modifiedmagnetic configuration, where the LCFS contacts with thechamber wall, and is also observed when the quality of thechamber wall surface is insufficient, when the effect ofchamber wall boronization ceases.

Proceedings of ITC/ISHW2007
Dynamic component
Diffusive component

gradient increases a shear of the electric field, which
suppresses convection.
20.8

Diffusive, dynamiccomponents, arb.un.

Diffusive, dynamiccomponents, arb.un.

1.0

0.6

0.4

0.2

0.0
5

1015

20Time, ms

Fig.5. Time behavior of the dynamic and diffusive
components for shot No. 55623.
Figure 6 shows the results of measurements of thelocal turbulent flux (a) before and (b) after boronization.
The positive and negative values in the figure correspondto an outward and an inward flux, respectively. Afterboronization, the radial turbulent particle flux near theLCFS changes drastically and is preferentially directed
outward.

Fig.6. Comparison of the local turbulent flux (a)before and (b) after boronization, shot No.53163.
The time behavior of the dynamic and diffusive
components for this case is shown in Fig. 7. Note that thecomponent involved three turbulent processes in this casetoo. It can be seen that the diffusive component isdominant while the dynamic component is suppressed.
Recall that the dynamic component is associated withconvective transport, whereas the diffusive component
characterizes the stochastic component of volatility.
The effect may be explained as follows. Afterboronization, the flux of carbon and oxygen from thechamber wall decreases; accordingly, radiative losses at theedge also decrease. As a result, the electron temperature atthe plasma periphery increased to ~ 100 eV and a jump inTe forms near the LCFS in a narrow r/a = 0.05) layer, witha very large temperature gradient [7]. The temperature

Diffusive component
Dynamic component

101011121314Time, msFig.7. Time behavior of the dynamic and diffusive
components for shot No. 53163.

4. Conclusions
A new approach, based on the analysis of timesamples of the increments of turbulent particle fluxes, isused to estimate the dynamic and diffusive components inthe edge turbulent flux of the L-2M stellarator. It is shownthat a shift-scale mixture of several (three-four processes,
depending on macroscopic characteristics of the plasma)
normal processes adequately describes the PDFs of flux
increments. The number of mixture components and theweight of each component are determined using the EM
algorithm The method of sliding separation of mixtures isapplied to separate between the dynamic and diffusive
components of volatility and to trace their temporal
evolution. With the above methods of analysis, somecharacteristic L-2M regimes have been examined. It isshown that turbulent particle transport in the edge plasma
occurs by both the ballistic and diffusive mechanisms, andtheir contributions are roughly estimated.
This work was supported by the Dynasty Foundation,and the Russian Foundation for Basic Research (projectnos. 07-02-00455).

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