Proceedings of ITC/ISHW/2007

Frequency Spectra and Statistical Characteristics of Plasma Density
Fluctuations Measured by Doppler Reflectometry in ECR HeatedPlasma in the Presence of Induction Current in the L-2M Stellarator
Nina N. SKVORTSOVA1), Vyacheslav V. SAENKO1), Viktor Yu. KOROLEV2),
German M. BATANOV 1), Dmitrii V. MALAKHOV1), Anton A. PSHENICHNIKOV1),
Aleksandr E. PETROV1), Karen A. SARKSYAN1), Nikolai K. KHARCHEV1)
1)

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

Moscow State University, Moscow, 119899 RussiaData on the parameters of plasma density fluctuations in the edge plasma of the L-2M stellarator are presented.
The density fluctuations were measured by Doppler reflectometry. Studies were made in the basic magnetic
configuration of L-2M under ECRH conditions and in modified magnetic configurations where the rotationaltransform at r/a ≤ 0.6 (a being the radius of the last closed flux surface) was decreased to negative values by
exciting induction current in the plasma. It is shown that induction current results in a shear of the poloidal plasma
velocity at the edge of the plasma column. The probability density functions of plasma fluctuations are found to bedifferent from normal (Gaussian) distributions and have heavy tails in both regimes. This type of turbulence isfairly described by shift-scale mixtures of 3 - 4 stochastic plasma processes, each with a normal probability densityfunction. The induction current affects shift-scale mixtures, and components with high mean velocities (convectivecomponents) appear in some processes. The experiments showed that the characteristics of turbulent fluctuations inthe edge plasma are sensitive to changes in the magnetic structure of the core plasma.
Keywords:

plasma turbulence, non-Gaussian process, stellarator, Doppler reflectometry.

1. Inroduction
Experimental studies of low-frequency fluctuations ofplasma density, plasma potential and particle flux in theL-2M stellarator have revealed the state of structural
turbulence over all the volume of the plasma column [1].
Thestateofstructural
turbulencecombines
determinate-chaotic nature with the existence of ensemblesof stochastic structures (vortices, solitons, wave packages)
is superimposed on strong plasma turbulence [2].
The present paper reports on experimental studies ofspectra and probability characteristics of structural
turbulence in the edge plasma of the L-2M stellarator.
Experiments were carried out under conditions that the"standard" magnetic configuration was modified byexciting an additional induction current. Plasma density
fluctuations were studied with the use of a Doppler
reflectometer [3].

2. Experimental conditions
The L-2M stellarator has two helical windings (l = 2),
a major radius R = 100 cm, and a mean plasma radius<a> = 11.5 cm [4]. The plasma was created and heated by
one or two 75-GHz gyrotrons (at the 2nd harmonic of theelectron gyrofrequency). The magnetic field at the center
author’s e-mail: nina@fpl.gpi.ru

of the plasma was Bt = 1.3–1.4 T. The gyrotron power was
P0 = 150–300 kW, and the microwave pulse duration wasup to 15 ms. Measurements were performed in a hydrogenplasma with mean density <n> = (0.8–2.0)⋅1013 cm–3 andcentral temperature Te = 0.6–1.0 keV. In the edge plasma atradius r/a = 0.9, the density was at a level of n (r) = (1–2)
1012 cm–3 and the electron temperature was Te (r) = 30–40
eV. The duration of the steady-state phase of the discharge
was 10 ms.
In the magnetic configuration of the L-2M stellarator,
an induction current excited in the opposite direction to thetoroidal magnetic field (hereafter referred to as a negativecurrent) was used to change the value and sign of rotationaltransform and to increase the shear of magnetic lines. In [5],the radial profiles of rotational transform i/2π werecalculated for various values of the induction current.
According calculations, the induction current of 10-15 kA
may result in two large magnetic islands located in thecentral region, whereas at the plasma edge at r/a = 0.8, themagnetic structure changes only slightly.
The probing frequencies used in our experimentcorrespond to the peripheral regions where the electron
density drops from ~1.7⋅1013 cm-3 to ~1.1⋅1013 cm-3. The
angles of incidence of the probing beam of thereflectometer are 4°, 8° and 12° with respect to the normal

Proceedings of ITC/ISHW/2007

3. Experimental results
As an illustration, Fig. 1 shows a time sample of thereflectometer signal in an ECRH discharge with anadditional induction current.
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1000455055606570t, ms
600
4002007HPSHUDWXUH

0455055606570t, ms

0-4-8-12-16&XUUHQW455055606570t, msFig.1. Time behavior of the refectometer signal,
average plasma density, gyrotron power, central
electron temperature, and plasma energy in adischarge with current I= –15 kA.
Also shown are time dependences of the average

Spectral density, arb.un.

plasma density, gyrotron power, central electron
temperature (ECE signal), and plasma energy. Note that theelectron density is maintained during more than 10 ms
after the gyrotron power is switched off. A similar
behavior of the electron density and fluctuations isobserved in discharges without induction current.
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Frequency, kHz

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Frequency, kHz

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Spectral density, arb.un.

to the last closed magnetic surface According to theBragg condition, the scattered signal corresponds to plasma
fluctuations with poloidal wavenumber k ≈ 2 cm-1.
The stray signal from heating gyrotron radiation wassuppressed with the help of a waveguide resonance filter,
which provided 30-dB attenuation at a frequency of
75.3 GHz. The polarization of the probing radiation in theplasma corresponded to an ordinary wave. We used
quadrature detection. Signals from two microwave diodes
were amplified in wideband amplifiers with gain of 10 andwere recorded by a 10-digit ADC operating at a sampling
rate of 5 MHz. The time resolution of the diagnosticsystem was 200 ns.
A Doppler shift, which corresponds to the poloidalrotation velocity, was determined from the shift of theFourier spectrum of a complex signal. In addition tocomplex Fourier spectra, we estimated and studied
autocorrelation functions, probability density functions offluctuation amplitudes and their increments for timesamples recorded in one of two reflectometer channels.

0.5
0.5
0.0-1500-1000 -500

0500 1000 1500

Frequency, kHz

0.0-1500-1000 -500

0500 1000 1500

Frequency, kHz
Fig.2. Spectra of the complex signals for two probing
frequencies 30.9 GHz and 37.6 GHz for an ECRH
discharge: (a, b) during and (b, d) after the gyrotron
pulse. Black segments on the frequency axis show
Doppler shifts.
Figure 2 shows spectra of the complex signals for two
probing frequencies 30.9 GHz and 37.6 GHz for an ECRH
discharge in the absence of induction current. Blacksegments on the frequency axis indicate Doppler shifts.
The scattered spectra are continuous with feebly marked
bands of width 50 kHz. It can be seen that the spectra arecontinuous with feebly marked bands of width 50 kHz.
The peak of spectral intensity is shifted from the zero(probing) frequency by nearly 250 kHz into the red region.
The spectrum is asymmetric: the spectral intensity falls
sharply in the red wing, whereas it decreases smoothly inthe blue wing of the spectrum. The shift of the peakintensity over frequency is usually interpreted as a Doppler
shift.
Figure 2 shows spectra of the complex signals fortwo probing frequencies 30.9 GHz and 37.6 GHz for an
ECRH discharge in the absence of induction current.
Black segments on the frequency axis indicate Doppler
shifts. The probing frequencies, it will be recalled,correspond to different plasma regions: the higher thefrequency, the higher the plasma density. It can be seenfrom these spectra that the Doppler shift is the same forboth probing frequencies during the heating pulse and in
the plasma decay phase. The poloidal velocity
corresponding to this shift is about 106 cm/s. In theabsence of an induction current, the shape of thespectrum does not change during the ECR heating pulse
and for 2-3 ms after its end. The Fourier spectra forProceedings of ITC/ISHW/2007

2.0,

1.5

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500 1000 1500

Frequency, kHz0
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density functions (PDFs) of fluctuation amplitudes andincrements differ from Gaussian distributions by heavytails. Previously, it was proposed to apply doubly
stochastic Poison process to describing the PDFs of theincrements of fluctuation amplitudes in structural plasma
turbulence [6]. Taking this approach, we used the model ofshift-scale mixtures of Gaussian distributions. A finite
shift-scale mixture of Gaussian (normal) distributions isrepresented in the form

500 1000 1500Frequency, kHz

1.5n x − aj , x∈8
P {X < x} = ∑ p j Φ  σ j j =1*+]Spectral density, arb.un.

Spectral density, arb.un.

probing frequencies 30.9 GHz and 37.6 GHz have thesame values of the width and Doppler shift, whichindicates that the poloidal plasma velocity is uniform ashear is absent in the edge plasma in the ECRH regime.

0.50.0-1500 -1000 -5000.50500 1000 1500

Frequency, kHz

0.0-1500-1000 -500

0500 1000 1500

Frequency, kHz
Fig.3. Spectra of the complex signals for two probing
frequencies 30.9 GHz and 37.6 GHz for an ECRH
discharge with additional induction current: (a, b)
during and (b, d) after the gyrotron pulse. Blacksegments on the frequency axis show Doppler shifts.
Figure 3 shows spectra of the complex signals for twoprobing frequencies 30.9 GHz and 37.6 GHz for an ECRH
discharge with additional induction current. Blacksegments on the frequency axis indicate Doppler shifts.
The spectra in this regime change markedly, depending on
the probing frequency. FWHM of the spectrum measured
with a probing frequency of 30.9 GHz is smaller. At thesame time, an intense spectral band (its intensity is nearly
equal to the intensity of the continuous spectrum) appears
in the frequency range 150-200 kHz. The red edge of thespectrum is shifts toward a lower frequency. In thescattered spectrum for a probing frequency of 37.6 GHz,
the shift of the red edge increases from -500 kHz to-750 kHz. The scattered spectrum for a probing frequencyof 34.8 GHz contains two intense bands in the range of-200 kHz and -400 kHz. Their intensity varies in time, but
their positions are fixed.
Hence, the generation of a magnetic island structurein the inner region of the plasma column involves a change
in behavior of the spectra measured in the narrow region
near the last closed magnetic surface.
The Doppler
shifts of the spectra corresponding to different radii prove
to be different, which is evidence for a radial shear of thepoloidal plasma velocity.
As noted above, the reflectometer data were also used
to study probability characteristics of plasma density
fluctuations. Characteristically, the autocorrelation
functions of fluctuation amplitudes demonstrated
long-lived tails. Histograms showed that the probability

where Φ ( y ) is the normal distribution function; pj is theweight; n-number of component of mixtures, aj DQG 1j are
the mathematical expectation and dispersion of the normal
distribution, respectively.
Each component of the mixture is determined by acertain stochastic plasma process, such as the presence ofan ensemble of stochastic plasma structures, the interaction
between structures, etc. As applied to structural plasma
turbulence, the concept of volatility as a multidimensional
vector determining the behavior of fluctuations wasconsidered in [7]. When processing the increments offluctuations measured by the Doppler reflectometer, we
calculated the volatility components, in which values of aj
DQG 1j give information about the average rates anddispersion of rates of turbulence in each particular process
and their total effect. Modeling the PDFs of fluctuation
increments with the use of the Estimation Maximization
algorithm [8] allowed us to determine the number andweight of such processes and to trace behavior of thecomponents.
We studied how the mixture components and the PDF
fluctuations alter in response to a change introduced in themagnetic configuration by the induction current.
It was found that the turbulent process in the edge
plasma is fairly described by a mixture of 3 - 4 stochastic
plasma processes for both the ECRH regime and theregime with an additional induction current. The velocity
value deduced from the Doppler shift is a sum of twocomponents: the poloidal rotation velocity and the phasevelocity of the measured fluctuation. In our case, the lattercomponent is a sum of phase velocities of particular
processes:
nu = uE× B + ∑ u k phase ,k =1

where k is the number of processes determined followingkthe Estimation Maximization algorithm, and u phase are
their phase velocities. For plasma density fluctuations

Proceedings of ITC/ISHW/2007
measured in the regime with induction current, weobserved the mixture components characterized by larger
average velocities in comparison with similar components
in the regime without induction current.
Figure 4 shows the results of numerical simulationof the PDF of increments of density fluctuations in theregime with additional induction current. Turbulence isdescribed by a mixture of four processes of which one
has a large average velocity characteristic of regimes withcurrent. It is found that the characteristics of shift-scale
mixtures of Gaussians change substantially in thepresence of induction current.

Number of counts (*100)

10, N$8

6and the sliding separation of mixtures of Gaussiandistributions. It is shown that, in structural plasma
turbulence developing in the edge plasma of the L-2M
stellarator, there exist 3-4 independent Gaussian processes,
each having its particular propagation velocity. The
algorithm of sliding separation of mixtures works andmakes it possible to distinguish between the pure ECRH
regime and the ECRH regime with additional induction
current. A comparison between the pure ECRH regime andthe ECRH regime with additional induction current showsthat the behavior of subcomponents of volatility for theseregimes is different and is an inherent characteristic of theregime.
As a result, it has been established that the frequency
and probability characteristics of turbulent fluctuations in
the edge plasma are sensitive to changes in the magnetic
structure of the core plasma.

PV

4
20-0.15 -0.10 -0.05 0.00 0.05 0.10
Increment, arb.un.
Fig.4. Modeling of the PDF of density fluctuation
increments in the regime with additional induction
current. An arrow indicates the process characterized
by a large mathematical expectation.
The so-called "sliding separation of mixtures" [7] wasused to trace the temporal evolution of the components ofaverage velocity. In the ECRH regime, the averagevelocities of processes do not change during 3-5 ms after
the heating pulse is switched off. In the regime with current,
the average velocities in the mixture of processes drop by
an order of magnitude with 0.1-0.2 ms after the ECRH
pulse is switched off.

This work was supported by the Dynasty Foundation,
the CRDF Program for Basic Research, and the RF
Presidential Program "Support of Leading ScientificSchools" (project 5382.2006.2) and the RussianFoundation for Basic Research (project nos. 07-02-00455,07-01-00517).

References
[1]
[2]

[3]
[4]
[5]
[6]
[7]

4. Conclusions
The spectra of turbulent fluctuations in ECR regime
have nearly the same width and Doppler shift, whichindicates that the poloidal plasma velocity is uniform over
the region of observation. In the presence of the additional
induction current, the characteristics of Fourier spectra ofcomplex signals in the narrow near-separatrix regionchanged markedly and the Doppler shifts for three probing
frequencies became different, which is evidence for a shearof the poloidal plasma velocity.
The processing the results of measurements by theDoppler reflectometer demonstrated efficiency of statisticalmethods of shift-scale mixtures of Gaussian distributions[8]

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