PHYSICS OF PLASMAS
VOLUME 6, NUMBER 12
DECEMBER 1999
Characterization of the frequency ranges of the plasma edge
fluctuation spectra
B. A. Carreras
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-8070
R. Balbin, B. van Milligen, M. A. Pedrosa, I. Garcia-Cortes, E. Sanchez, and C. Hidalgo
Asociación Euratom-Ciemat, 28040 Madrid, Spain
J. Bleuel, M. Endler, and H. Thomsen
Max-Planck-Institut für Plasmaphysik, Euratom Association, 85740 Garching, Germany
A. Chankin, S. Davies, K. Erents, and G. F. Matthews
JET Joint Undertaking, Abingdon, Oxon, OX14 3EA, United Kingdom
共Received 14 May 1999; accepted 16 August 1999兲
Frequency spectra of fluctuations for the ion saturation current, floating potential, and turbulent
transport measured in the plasma edge of plasma confinement experiments 共tokamaks and
stellarators兲 have been analyzed to identify the frequency ranges characterized by a power
dependence. Three main regions can be identified. For the intermediate frequency region, the decay
of the spectra is close to 1/f , as is expected in self-organized criticality systems. This region is
particularly important for the role that it plays in plasma transport and the self-similarity of the
fluctuations and fluxes. The effect of plasma rotation on the decay indices has also been studied.
© 1999 American Institute of Physics. 关S1070-664X共99兲00212-8兴
I. INTRODUCTION
we extend the analysis to fluctuation measurements in TJ-IU,
W7-AS, and JET fusion devices and to the identification of
the different frequency ranges. We also consider the effect of
the plasma poloidal velocity on the change of the spectral
decay indices.
We dedicate particular attention to the 1/f frequency region. Apart from the reason of a possible connection to the
self-organized criticality 共SOC兲 behavior, there are other reasons to focus on this frequency range. One is that the turbulent particle flux is self-similar over these time scales, as
becomes evident from Rescaled–Range analysis3 of the time
records or from analyzing the probability distribution function 共PDF兲 of the time averaged flux,11 which shows an algebraic tail for large flux events. A second reason is that the
cross-power spectral function12 between the density fluctuations and poloidal component of the electric field fluctuations
peaks in this frequency range. For all these reasons, the 1/f
range of the spectrum is one of the most interesting ones for
understanding the plasma transport.
The rest of this paper is organized as follows. In Sec. II,
we describe the way to identify the different spectral regions
using fluctuation data close to a point with V ␪ ⫽0. The results for this analysis on three different plasma confinement
devices are reported. In Sec. III, the effect of the plasma
rotation on the decay indices is considered, and the analysis
of the available data is extended to V ␪ ⫽0 radial regions.
Finally, in Sec. IV, the conclusions of this paper are presented.
Comparative studies of plasma edge fluctuations carried
out in different magnetically confined devices 共tokamak and
stellarator兲 support the view of the universality of edge
plasma turbulence and the associated turbulent transport.
Studies of the fluctuation level and velocity shear layer structure gave a strong indication of this similarity.1,2 Recently,
the study of the self-similarity properties of the fluctuations
has led to very close values of the self-similarity parameter
for different devices.3 It has also been found that the power
spectra of the saturation current, electrostatic potential fluctuations, and the turbulence-induced flux measured in the
TJ-I4 and TJ-IU5 stellarators, Wendelstein 7 Advanced Stellarator 共W7-AS兲,6 and the Joint European Torus 共JET兲7 fusion devices show the same functional dependence over the
whole frequency range.8 These results may be an indication
of edge plasma turbulence evolving into a critical state independent of the size and plasma characteristics of the device.
We will show in this paper that the experimentally measured plasma edge fluctuation spectrum is consistent with the
existence of three distinct frequency ranges, each with characteristic power dependence. For a long time, the existence
of a high-frequency asymptotic power fall-off of the fluctuation spectra with characteristic decay indices close to ⫺2 or
higher has been recognized.9 Also at the lowest frequency
end of the spectrum, the fluctuation spectrum is a weak function of frequency. On these initial studies of the fluctuation
spectra, little attention was dedicated to a possible intermediate frequency region. This range is characterized by a decay index close to ⫺1. This dependence was noticed in the
DIII-D tokamak fluctuation measurements when they were
carried out in the plasma rest frame,10 that is, at the point
where the poloidal flow velocity is zero, V ␪ ⫽0. In this paper,
1070-664X/99/6(12)/4615/7/$15.00
II. FREQUENCY RANGES IN THE SPECTRUM OF
PLASMA EDGE FLUCTUATIONS
Plasma edge fluctuation measurements discussed in this
paper have been carried out by means of Langmuir probes in
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© 1999 American Institute of Physics
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Phys. Plasmas, Vol. 6, No. 12, December 1999
Carreras et al.
TABLE I. Machine parameters and plasma edge flow shear for the analyzed
discharges.
Device
JET
TJ-IU
W7-AS
W7-AS
a 共m兲
R 共m兲
B (T)
dV ␪ /dr 共s⫺1兲
1.2
0.10
0.172
0.178
2.9
0.60
2.0
2.0
2.6
0.67
1.2
2.5
1⫻105
4⫻104
1⫻10 5
1⫻105
a tokamak 共JET兲 and two stellarators 共TJ-IU and W7-AS兲.
Table I shows the main characteristics of the devices and the
plasmas under investigation. Plasma edge fluctuations and
their induced turbulent flux have been determined from the
measurement of the ion saturation current, I s , and of the
floating potential, V f , fluctuations. The experimental techniques used in performing those measurements have been
described elsewhere.13,14 The ion saturation current fluctuations are related to density and electron temperature fluctuations: I s ⬇nT 1/2
e ; whereas the floating potential fluctuations
depend on both plasma potential and electron temperature
fluctuations, V f ⬇V p ⫺3T e . The time evolution of the radial
turbulent particle flux has been computed as ⌫(t)⫽ñẼ ␪ /B,
where E ␪ is the poloidal electric field and B is the magnetic
field. The poloidal electric field has been deduced from floating potential signals measured by poloidally separated
probes. The density fluctuations have been approximated by
the saturation current fluctuations and the plasma potential
has been approximated by the floating potential. We have
used these measurements to determine the power spectrum
of the ion saturation current fluctuations, electrostatic potential fluctuations, and fluctuating induced flux.
In experimental studies of plasma fluctuations, the existence of a high-frequency asymptotic power fall-off of the
fluctuation spectra with characteristic decay indices close to
⫺2 or higher has already been recognized.9 Also at the lowest frequency end of the spectrum, the fluctuation spectrum
is found to be weakly dependent on frequency. In the initial
studies of the fluctuation spectra, little attention was dedicated to a possible intermediate frequency region. Recently,
the existence of a 1/f frequency region has been reported10,15
in the DIII-D tokamak fluctuation measurements when they
are carried out in the plasma rest frame, that is, at the point
where the poloidal flow velocity is zero, V ␪ ⫽0. Similar dependence was also reported in Ref. 16 for W7-AS plasma
edge fluctuation data. However, direct examination of the
fluctuation spectra does not always show, in an obvious way,
a distinct region with this power dependence. Sometimes the
intermediate region seems to have a continuous change in
exponent. We must realize that a variation in plasma conditions during the measurements can in time change the breaking frequencies and round up the points in the spectrum. The
need for high statistics to make a good determination of the
power dependence may work against keeping the plasma
conditions uniform. Measurements in well-controlled discharges would be very desirable.
In examining multiple plasma discharges, the importance of looking near V ␪ ⫽0 becomes evident. Otherwise, the
power associated with the intermediate region can be consid-
FIG. 1. Spectrum of the ion saturation current fluctuation from a measurement in W7-AS near V ␪ ⫽0. A 20 ms data record has been used. The
spectrum S( f ) is multiplied by f ␣ to have a flat intermediate region and to
identify the three spectral regions.
erably higher than ⫺1, and at first sight, it is not clear
whether the change in the spectral index is caused by a
change in turbulence dynamics or just by a Doppler shift
effect. Therefore, it is important to know the plasma velocity
at the position at which the fluctuations are measured. In the
discharges considered here, the poloidal velocity is not directly measured. We take the phase velocity of the fluctuations as an estimate of V ␪ . This is not always an accurate
estimate and may present problems in the interpretation of
the spectra, as will be discussed later. However, it is a good
first approximation in carrying out this type of analysis. In
the next section, we will address the issue of the nonzero
velocity situations.
To identify the frequency breaking points that separate
the different frequency regimes in the plasma edge fluctuations data, it is useful to first make a guess as to where the
1/f region in the spectrum is located. In this analysis, we
have used a logarithmic binning of the spectrum to get rid of
what Mandelbrot calls ‘‘Spanish moss.’’17 This rebinning of
the spectrum allows a more precise determination of the frequency breaking points. After the breaking points have been
tentatively identified, we do a tentative fit to the spectrum in
this region using a simple power function such as S 2 ( f )
⫽A 2 f ⫺ ␣ 2 . We then multiply the power spectrum, S( f ), by
f ␣ 2 and plot the resulting function. Using this method, the
two frequency-breaking points and the corresponding threefrequency regions are well identified. This is illustrated in
Fig. 1, where we have plotted S( f ) f ⫺ ␣ 2 as a function of the
frequency. In this figure, S( f ) is the power spectrum of the
ion saturation current fluctuation for a W7-AS discharge.
The figure shows a fairly flat region between 10 and 100
kHz, which indicates that the power law is accurate in this
frequency region. Figure 2 includes several spectra corresponding to different discharge parameters in W7-AS. For
these discharges, the sampling frequency is 2.0 MHz, and the
fluctuation record length used in the calculation of the spectrum is 40 000 points. In the other devices, the record length
for the available measurements is shorter. Because of this, it
Phys. Plasmas, Vol. 6, No. 12, December 1999
FIG. 2. Spectra of the ion saturation current fluctuation taken from 20 ms
data sections near the flow shear layer at W7-AS. The spectra S( f ) are
multiplied by f ␣ to have a flat intermediate region and are plotted with an
arbitrary offset.
is more difficult to get a determination of the frequency
breaking points. Still, using this method, it is possible to
make a good determination of them. This is the case of the
TJ-IU stellarator, where the sampling frequency is 1 MHz
and the record length in calculating the spectrum is 4096
points, an order of magnitude less than the case of W7-AS.
Figure 3 shows that the three regions are well separated. The
transition points between the three frequency regions are not
FIG. 3. Spectra of the ion saturation current fluctuation taken from 4 ms
data sections near the flow shear layer at TJ-IU. The spectra S( f ) are multiplied by f ␣ to have a flat intermediate region. They are plotted with arbitrary offset and correspond to phase velocity of ⫺200, ⫺100, and 3030 m/s,
respectively.
Characterization of the frequency ranges of the plasma . . .
4617
FIG. 4. Cross-power spectral function for 20 ms data sections near the flow
shear layer at W7-AS. The different curves correspond to the different
power spectra shown in Fig. 2.
always sharp 共Figs. 2 and 3兲. This could be a consequence of
the limited statistics, and/or it could be the result of slow
changes in plasma parameters during the measurements. In
general, the separation between the second and third region
is better defined than the separation between the first and the
second regions. Furthermore, the lowest frequency region
has stronger oscillations and they may be due to the limited
statistical information for this region. It is not always apparent whether the lowest frequency region is well described by
a single power law. The possible presence of magnetohydrodynamic 共MHD兲 activity, plasma column drifts, and motion
of the probes can often distort the low-frequency range of the
fluctuation spectra. All of these effects further complicate the
picture for this frequency range. Therefore, we only discuss
the properties of the fluctuation spectra in the second and
third regions.
The main reason for identifying the intermediate frequency region is the possible critical role that it plays in
plasma transport. For the same fluctuation data used in calculating the spectra in Fig. 2, we have calculated the crosspower spectral function12 between the density fluctuations
and poloidal component of the electric field fluctuations. The
results are plotted in Fig. 4. The figure shows that crosspower spectral function peaks in the intermediate frequency
range. This correlation between the 1/f fluctuation range and
transport dynamics is under investigation.
To summarize the results of the analysis of the ion saturation current fluctuation spectra near V ␪ ⫽0, in Fig. 5 we
have plotted the values of the spectral decay index in the
second and third frequency regions for several discharges of
W7-AS and TJ-IU. Because we do not have the data exactly
at V ␪ ⫽0, we have plotted the exponent vs the phase velocity
for the particular time sequence considered. We have taken
the values of the velocity as small as possible to be close to
V ␪ ⫽0. The values of ␣ 2 are scattered between ⫺1 and ⫺1.5.
For ␣ 3 , the values range between ⫺3 and ⫺5. In this figure,
no error bars are assigned to the plotted values of ␣ 2 and ␣ 3
because of numerous sources of error and the difficulty in
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Carreras et al.
Phys. Plasmas, Vol. 6, No. 12, December 1999
FIG. 5. Decay index in the second 共full symbols兲 and third frequency regions 共open symbols兲 for the ion saturation current fluctuation spectra near
V ␪ ⫽0. These values are for several discharges of W7-AS 共circles兲 and
TJ-IU 共squares兲.
estimating them with accuracy. We estimate that a rough
value for the error bars would be 30% of the value of ␣.
There is no correlation of the plotted values with the phase
velocity, and we should not expect one. At these low values
of the phase velocity, the discrepancy between the phase
velocity and the fluid velocity should dominate. We ignore
how large this discrepancy is because we do not have measurements of the poloidal velocity for these discharges.
There are a small number of data points in Fig. 5 because of
limited data available close to V ␪ ⫽0. In spite of the limited
number of cases, they suggest that the two indices are distinct in the two frequency regions. In the first one, it is about
⫺1.3⫾0.15 and the second one ⫺4.1⫾0.5. The scattering of
values may be due to the difference between fluid velocity
and phase velocity. This point will be discussed further in the
next section.
The floating potential fluctuation spectrum has the same
distinctive three frequency ranges as the ion saturation current fluctuations. For the data close to the zero velocity position, the spectral decay index in the intermediate frequency
region has more scattering than for the case of the ion saturation current. In general, ␣ 2 for the potential fluctuations is
somewhat lower than its value for the ion saturation current
in W7-AS data, but it can be higher for the TJ-IU and JET
data. In Fig. 6, we illustrate the difference using data from
edge plasmas in JET. Because the reciprocating probe stays a
very short time within the JET plasmas, the statistics are not
in good as in W7-AS. The spectra have been calculated over
a time interval of 10 ms, using 5000 points. Figure 6 shows
that both the ion saturation current and the potential fluctuations have a frequency range in which the spectra fall close
to 1/f . In the same figure, we have plotted the spectral flux
function for the same data set, showing that it peaks in the
intermediate frequency region, as was the case in Fig. 5. This
situation seems to be a general result for all the data considered here.
FIG. 6. Spectra of the ion saturation current and potential fluctuation taken
from 2 ms data sections near the flow shear layer at JET. The spectra S( f )
are multiplied by f ␣ to have a flat intermediate region and are plotted with
an arbitrary offset. Also in the plot is the spectral flux function for the same
discharge.
III. EFFECT OF A POLOIDAL FLOW VELOCITY ON
THE FLUCTUATION SPECTRUM
The presence of a plasma flow can significantly modify
the structure of the fluctuation spectrum. The theoretical predictions of the spectral power dependence are usually made
in the plasma rest frame. In doing a comparison with theory
or in just trying to determine the relevance of the 1/f regime
in plasma fluctuations, it is necessary to calculate how much
the spectrum changes when measured in the laboratory rest
frame. This is not a simple experimental task.
The simplest possible scenario that we can consider is
the presence of a poloidal flow. In this case, the simplest
assumption is that the flow causes a Doppler shift in frequencies,
共1兲
f̂ ⫽ f ⫹k̄ ␪ 共 f 兲 V ␪ .
Here, f̂ is the frequency in the laboratory frame, f is the
corresponding frequency in the plasma rest frame, and k̄ ␪ ( f )
is the mean value of the poloidal wave number at this given
frequency. If we make the assumption that each frequency
breaking point has a Doppler shift and that the spectrum
remains a power law in each frequency region, we can make
a simple estimation of the change in the spectral decay index.
This estimation gives the index in the moving frame as
␣ i共 V ␪ 兲 ⫽ ␣ i共 0 兲
ln
冉
ln共 f i / f i⫺1 兲
f i ⫹k̄ ␪ 共 f i 兲 V ␪
f i⫺1 ⫹k̄ ␪ 共 f i⫺1 兲 V ␪
冊
.
共2兲
Here, the subindex of f indicates each of the frequency values at the ends of the frequency regime considered. Equation
共2兲 can be tested using the results from the numerical calculations.
Phys. Plasmas, Vol. 6, No. 12, December 1999
In Ref. 18, we presented the results of numerical calculations of resistive pressure-gradient-driven turbulence with
a noisy density source. These results indicated that the fluctuation spectrum had a 1/f region. We can use the numerically calculated density fluctuations in the plasma rest frame
and determine the fluctuation measurement in a moving
frame by introducing a rigid rotation of the plasma, ␪ ⫽ ␪ 0
⫹ ␻ ␪ t. For different values of the rotation ␻ ␪ , we can calculate the power spectrum of the density fluctuations and
calculate the decay index in the intermediate frequency region, ␣ 2 (V ␪ ). Since we know all of the parameters, we can
also calculate ␣ 2 (V ␪ ) using Eq. 共2兲. There is good agreement
between the two calculations. These calculations show that
because of the Doppler shift, the measured decay index of
the intermediate frequency region may vary in the range between ⫺1 and ⫺3.
This result is encouraging; it suggests that the higher
values for ␣ 2 that are sometimes found in the experiment
may be only the consequence of the plasma rotation. However, it would be good to have some experimental tests of the
change in the decay indices. To do so, it is necessary to have
the same fluctuation measurement done in different reference
frames. This is not easy to do, but there is a possible solution. Near the plasma edge, there is a sharp shear flow layer.
Within a very short radial separation, about 1 cm, the poloidal flow velocity may change from 500 to ⫺500 m/s. In this
region, we expect that the plasma parameters change very
little in comparison with the change in the flow velocity. In
particular, the shear in the flow is almost constant, and the
ion saturation current changes by only 40%. In this figure,
we have plotted the phase velocity of the fluctuations as a
function of the radius for the discharge 35427 in W7-AS.
Each data point corresponds to a time series record of 40 000
points. We will assume that the probe measurements see the
same turbulence in the narrow radial band marked where the
shear flow is constant. For these time records, basically the
only thing that changes is the poloidal velocity, or equivalently, the reference frame in which the fluctuation measurements are done. For the four time records of 40 000 points,
we calculate the power spectra of the fluctuations. In the
spectra, we can identify the three frequency regions and the
corresponding breaking points using the method described
previously. When the frequency breaking points are plotted
as a function of the phase velocity, they scale approximately
linearly with velocity, as can be seen in Fig. 7共a兲. This scaling is derived from a Doppler shift, Eq. 共1兲. Using a twopoint probe, it is possible to make a determination of k̄ ␪ ( f )
in the moving frame. Therefore, using Eqs. 共1兲 and 共2兲 we
should be able to describe the frequency breaking points and
the exponents ␣ 2 and ␣ 3 as a function of the velocity. To fit
the nonzero velocity data, in general, it is necessary to introduce a shift in the velocity, ⌬V. This shift can be interpreted
as the difference between the real plasma poloidal velocity
and the measured phase velocity of the fluctuations. In Fig.
7, we show an example of such a fit. Because of the difficulty in calculating the frequency breaking points, the plots
show some degree of scattering. This fit gives a better estimate of ␣ 2 (0) than the one used in Sec. II. This new esti-
Characterization of the frequency ranges of the plasma . . .
4619
FIG. 7. Effect of a plasma rotation on 共a兲 the frequency breaking points and
共b兲 ␣ 2 . Comparison between experimental measurements and the results of
Eqs. 共1兲 and 共2兲.
mate brings the points of Fig. 5 closer to ⫺1. However, the
error bars involved in this process do not permit significant
improvement in the accuracy of the determination. Nevertheless, if future experimental measurements can be done with
higher level of control of plasma conditions and with higher
statistics, this method could be used to improve the accuracy
of the de termination of ␣ 2 (0).
Considering all of the data analyzed, we can use the
experimentally determined breaking points f 1 (V ␪ ) and
f 2 (V ␪ ) to calculate the terms ln(f 2(0)/f 1(0))/ln(f 2(V␪)/f 1(V␪))
and ␣ 2 (V ␪ )/ ␣ 2 (0). If Eq. 共2兲 is verified, these two quantities
should be equal. For different devices and plasma conditions,
we have plotted one versus the other in Fig. 8. The data
follow along a line that indicates the correlation between the
two. There is some degree of dispersion, but the errors are in
the determination of these points. A similar plot has been
done for ␣ 3 (V ␪ )/ ␣ 3 (0) in Fig. 9.
Equations 共1兲 and 共2兲 can be used to reconstruct the
spectrum in the plasma rest frame. To be able to do so, it is
necessary to have an accurate measurement of the poloidal
velocity. To use the phase velocity determined from the fluctuations is not reliable for the inverse transformation. In Fig.
10, we show the case of a reconstruction of the spectrum by
assuming a given poloidal velocity. The poloidal velocity has
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Carreras et al.
Phys. Plasmas, Vol. 6, No. 12, December 1999
FIG. 8. Normalized spectral decay index of second frequency region versus
Doppler shift prediction.
been estimated as the phase velocity plus the velocity shift,
⌬V, that has been found for this same discharge near the
V ␪ ⫽0 position.
IV. CONCLUSIONS
Analysis of the plasma edge fluctuation spectra in several confinement devices shows the existence of three distinct fluctuation ranges. In each of these ranges, and in the
absence of MHD activity, the spectrum is well described by
an algebraic function. In the lowest frequency range, the
spectral decay index is close to zero, and the spectrum is
basically flat. In the highest frequency range, the spectral
decay index is of the order of ⫺2 or higher, as has been
shown in the previous analysis.
The most interesting region is the intermediate frequency range. In this range, the spectral decay index is consistent with ⫺1 when the Doppler shift effects induced by
plasma velocity have been taken into account. This result
agrees with a similar result from the DIII-D tokamak.10,15
FIG. 10. Correction of frequency spectrum for Doppler shift using f̂ ⫽ f
⫺k̄ ␪ ( f )V ␪ ; a poloidal velocity of V ␪ ⫽800 m/s has been assumed.
The inclusion of the velocity effects in our analysis has allowed us to extend it into regions of the plasma edge where
the poloidal velocity is high.
Very high statistics in plasmas that maintain uniform
conditions are needed for a more accurate determination of
these spectral decay indices. These controlled experiments
are very important in resolving some of the critical issues
linked with the existence of the 1/f region of the spectrum
and should be pursued in the future.
ACKNOWLEDGMENTS
This research is sponsored in part by Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under Contract No. DE-AC05-96-OR22464. We also want to
acknowledge the Joint Commission for Scientific and Technological Cooperation between the United States and Spain
for its support during the completion of this research.
1
FIG. 9. Normalized spectral decay index of third frequency region versus
Doppler shift prediction.
A. J. Wootton, B. A. Carreras, H. Matsumoto, et al., Phys. Fluids B 2,
2879 共1990兲.
2
A. J. Wootton, H. Y. W. Tsui, and S. Prager, Plasma Phys. Controlled
Fusion 44, 2023 共1992兲.
3
B. A. Carreras, B. v. Milligen, M. A. Pedrosa, et al., Phys. Rev. Lett. 80,
4438 共1998兲.
4
I. Garcia-Cortes, M. A. Pedrosa, C. Hidalgo, et al., Phys. Fluids B 4, 4007
共1992兲.
5
E. Ascasibar, C. Alejaldre, J. Alonso, et al., in Plasma Physics and Controlled Nuclear Fusion Research 1994 共International Atomic Energy
Agency, Vienna, 1995兲, Vol. 1, pp. 749–756.
6
H. Renner, the WVII-AS Team, the NBI Group, et al. Plasma Phys. Controlled Fusion 31, 1579 共1989兲.
7
P.-H. Rebut and JET Team, in Plasma Physics and Controlled Nuclear
Fusion Research 1990 共International Atomic Energy Agency, Vienna,
1991兲, Vol. 1, pp. 27–48.
8
M. A. Pedrosa, C. Hidalgo, B. A. Carreras, et al., Phys. Rev. Lett. 82,
3621 共1999兲.
9
S. J. Levinson, J. M. Beall, E. J. Powers, et al., Nucl. Fusion 24, 527
共1984兲.
10
T. L. Rhodes, R. A. Moyer, R. Groebner, et al., Phys. Lett. A 253, 181
共1999兲.
Phys. Plasmas, Vol. 6, No. 12, December 1999
B. A. Carreras, B. v. Milligen, C. Hidalgo, et al., Phys. Rev. Lett. 共in
press兲.
12
E. J. Powers, Nucl. Fusion 14, 749 共1974兲.
13
M. A. Pedrosa, M. A. Ochando, J. A. Jiménez, et al., Plasma Phys. Controlled Fusion 38, 365 共1996兲.
14
J. Bleuel, G. Theimer, M. Endler, et al., in Controlled Fusion and Plasma
Physics 共European Physical Society, Petit-Lancy, 1996兲, Vol. 20C, pp.
727–730.
11
Characterization of the frequency ranges of the plasma . . .
15
4621
R. Moyer, R. Lehmer, J. A. Boedo, et al., J. Nucl. Mater. 266, 1145
共1999兲.
16
B. A. Carreras, B. Ph. van Milligen, M. A. Pedrosa, et al., Phys. Plasmas
5, 3632 共1998兲.
17
B. B. Mandelbrot, Multifractals and 1/f Noise 共Springer, New York,
1998兲.
18
B. A. Carreras, D. Newman, V. E. Lynch, et al., Phys. Plasmas 3, 2903
共1996兲.