Plasma Physics and Controlled Fusion
Plasma Phys. Control. Fusion 57 (2015) 045004 (16pp)
doi:10.1088/0741-3335/57/4/045004
Chaotic density fluctuations in L-mode
plasmas of the DIII-D tokamak
J E Maggs, T L Rhodes and G J Morales
Physics and Astronomy Department, University of California, Los Angeles, CA 90095, USA
E-mail: maggs@physics.ucla.edu
Received 2 December 2014, revised 15 January 2015
Accepted for publication 3 February 2015
Published 5 March 2015
Abstract
Analysis of the time series obtained with the Doppler backscattering system (Hillsheim et al
2009 Rev. Sci. Instrum. 80 0835070) in the DIII-D tokamak (Luxon 2005 Fusion Sci. Technol.
48 828) shows that intermediate wave number plasma density fluctuations in low confinement
(L-mode) tokamak plasmas are chaotic. The supporting evidence is based on the shape of the
power spectrum; the location of the signal in the complexity-entropy plane (C-H plane) (Rosso
et al 2007 Phys. Rev. Lett. 99 154102); and the population of the corresponding Bandt–Pompe
(Bandt and Pompe 2002 Phys. Rev. Lett. 88 174102) probability distributions.
Keywords: chaos, plasmas, spectral density, pulse structure
(Some figures may appear in colour only in the online journal)
1. Introduction
fluctuations. One important and very useful feature of the
DBS is that it provides the simultaneous, remote sampling of
the fluctuations at several radial locations without the need for
temporal averaging. Another aspect of the DBS instrument is
that it allows, in principle, the simultaneous determination of
the flow of the density fluctuations (where the flow is due to
a combination of the E × B velocity and the inherent velocity
of the fluctuations). The role of flows is a delicate issue in the
interpretation of measurements with material probes inserted
in the plasma (e.g. Langmuir-probes), which have been a
major tool for extracting information [17–19] about the nature
of edge fluctuations.
This paper reports the analysis of time series of DBS
measurements for L-mode plasmas. Three analysis techniques are applied: conventional power spectra with complex time signals; complexity-entropy (C-H) plane [20];
and, Bandt–Pompe [21] probability distribution functions.
The Bandt–Pompe probability and the C-H plane analysis
are powerful tools developed by applied mathematicians to
distinguish between stochastic and chaotic processes, but
have not received much attention by the plasma/fusion community. Recently this methodology was applied to a basic
electron heat transport experiment [22] to identify the chaotic
dynamics associated with pressure gradient instabilities, and
has also been used in the study of interacting flux ropes in a
laboratory plasma [23]. The results of these three signal analysis techniques consistently demonstrate that the fluctuations
It is widely recognized that non-equilibrium fluctuations at the
plasma edge can have important consequences for the transport properties of magnetic confinement devices. Unraveling
the underlying dynamics responsible for these fluctuations is
a challenging research topic that has received considerable
attention, as documented by the extensive literature quoted
in various surveys [1–5]. A basic issue in the development
of a proper description of the transport associated with the
edge fluctuations is whether the processes are stochastic [6]
or chaotic [7, 8]. Generally speaking, stochastic physical systems have many degrees of freedom and are often described
by random variables and probability distributions. In contrast,
chaotic systems are deterministic, and are often represented
by a few coupled modes. The differences between stochastic
and chaotic processes are addressed in more detail in section 5. The present study addresses the question of the statistical nature of edge fluctuations by performing an analysis
of the time signals obtained with the Doppler backscattering
system (DBS) [9] in the DIII-D tokamak [10]. The Doppler
backscattering technique is sensitive to plasma turbulence
levels and flow, and has been utilized to measure radial electric fields, geodesic acoustic modes, zonal flows, and intermediate scale, k ~ 1–6 cm−1, density turbulence [11–16]. This
diagnostic instrument has properties that are well suited to
the exploration of the dynamical origin of plasma density
0741-3335/15/045004+16$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
sampled in the plasma edge region by the DBS diagnostic are
chaotic.
The manuscript is organized as follows. Section 2 presents
the tokamak plasma parameters corresponding to the signals
analyzed. Section 3 explains the basic features of the DBS
diagnostic system. Section 4 provides a brief overview of
the Bandt–Pompe probability and the C-H plane. Section 5
presents detailed results of the signal analysis for L-mode
plasmas. Section 6 discusses the findings and conclusions are
given in section 7.
1.0
(a)
0.5
0.0
3
Ip(MA)
shot 150141
(b)
Te(0) (keV)
5
4
3
2
1
0
(c)
Ti(0) (keV)
5
4
3
2
1
0
4
(d)
2
1
0
2. Plasma conditions
The experiments presented here were performed on the
DIII-D [24], a medium-sized tokamak with major radius R
= 1.7 m, minor radius a = 0.6 m, and vertical elongation ~2.
Data from four L-mode, deuterium plasmas are included in
this study. All time signals in the data sample set consist of
20 000 points. Three of the plasmas use an upper single-null
diverted discharge with BT = 2.1 T and comprise a ‘current
scan’: shot 150 141 with Ip = 0.76 MA; shot 150 142 with
Ip = 1.0 MA; and, shot 150136 with Ip = 1.4 MA. The corresponding safety factor values at the scaled radial position,
ρ = 0.95 (i.e. q95) at time 2500 ms, are: 6.47, 4.9, and 3.58,
respectively. For the three L-mode plasmas in the ‘current scan’
data set, plasma confinement improves with increasing plasma
current. For each plasma, three 4 ms time intervals, beginning
at 1250, 2500 and 3525 ms, are included in the study. The digitization rate is 5 MHz. The current scan cases are referred to
by citing the plasma current and time interval associated with
the data. The fourth L-mode plasma, labeled ‘L-mode 1’, is
taken from shot 155 674 for which the magnetic field configuration is a lower single-null with BT = 2.0 T. The plasma current is 1.3 MA, injected neutral beam power Pinj = 1 MW, and
the line-averaged density is ne = 2.6 × 1019 m −3. A sample of
20 000 points is acquired with 10 MHz digitization rate in a
2 ms time interval beginning at t = 924 ms. For brevity, the
detailed temporal and radial behavior of plasma parameters is
given for only one case. However, all of the examples chosen
for inclusion in this study are considered representative of
L-mode plasmas in the DIII-D tokamak.
The case whose parameters are presented in detail is from
the current scan with Ip = 0.76 MA. Figure 1 shows the time
histories of plasma current Ip, chord-averaged density n chord,
power injected by the neutral beams Pinj, and central electron
and ion temperatures, Te(0) and Ti(0). The L-mode, line-averaged density is ne = 1.8 × 1019 m −3. Approximately 1.9 MW
of neutral beam heating power was applied continuously
throughout the shot starting at 300 ms. The application of
neutral beams results in increased plasma temperature as well
as increased toroidal rotation due to the momentum input.
As seen in figure 1, the plasma current, Ip, and plasma density, nchord, were constant after 533 and 940 ms, respectively.
Three time intervals of 4 ms duration are considered in the
current scan cases, and the starting points are indicated by
vertical dotted lines in figure 1: 1250 ms (before MHD oscillations known as sawteeth begin), 2500 ms (during sawtooth
3
(e)
ne(0) (1019 m-3)
Pinj
2
1
0
0
1000
Time (ms)
2000
3000
Figure 1. Temporal evolution of (a) the plasma current, (b) central
electron and (c) central ion temperatures, (d) chord-averaged
density and (e) neutral beam power for the 0.76 MA case. Three
time intervals of 4 ms duration are considered in the signal analysis,
the starting points are indicated by the vertical dotted lines.
oscillations), and 3525 ms (during the higher injected neutral
beam power level, figure 1(e), just before a low confinement to
high confinement transition or L- to H-mode transition) with
the primary focus on the 1250 ms time period. The initiation
of the sawteeth can be seen as the periodic oscillations in the
electron temperature beginning around 1450 ms (figure 1(b)).
Shortly after the increase in Pinj at 3500 ms there is a transition
to a higher confinement regime, known as H-mode, at about t
~ 3558 ms. The increase in Te(0), Ti(0), and ne at that time is an
indication of this transition (figures 1(b)–(d)).
Radial profiles of several parameters of interest: electron
and ion temperatures, electron density, and magnetic safety
factor q, are shown for time t ~ 1240 ms in figure 2. The
parameters are displayed as functions of the normalized radial
coordinate, ρ, defined as the normalized, square root of the
toroidal magnetic flux. While the 1250 ms time interval is the
main focus of the paper, data from the other time intervals are
presented as needed to help illustrate various points. While the
radial profiles shown in figure 2 are somewhat different for the
later time intervals, the profiles shown can be considered as
representative of the ‘current-scan’ L-mode dataset. Figure 2
indicates that there is a gradual, but substantial, increase in
Te and Ti towards the center of the tokamak ( ρ = 0). Both
experimental data and spline fits with error bars are shown.
The density also increases slowly inside of the last closed flux
surface ( ρ = 1). Outside of ρ = 1, the density drops rapidly as
the vacuum vessel is approached.
3. DBS diagnostic
This study concentrates on the plasma dynamics as revealed
by DBS measurements of density fluctuations. Doppler
2
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
particular polarization and probe frequency (e.g. for X-mode
polarization it depends on both local magnetic field strength
and plasma density, while O-mode depends only on local density). It is important to note that the Bragg relation is a vector
relation, so that scattering occurs only if k n∼ is both present and
aligned along klocal.
A feature of the backscattering process is that the backscattered power is related to the local fluctuation level at that
particular k n∼. A flow velocity of the density fluctuations in
the direction of k n∼ creates a Doppler shift in the signal scattered from the density fluctuations. If required, the local
flow velocity V of the fluctuations can be obtained from
the measured frequency shift, fDoppler, using the relation,
fDoppler = 2 klocal • V /2π . The local wave number, klocal, and
radial location of the DBS signal are determined using 3D
ray-tracing calculations, utilizing the GENRAY code [25] that
uses density profiles and equilibrium magnetic field information as inputs. The Doppler shift frequency results from
both the background E × B velocity and intrinsic propagation
velocities of the fluctuations. In cases for which the E × B
velocity is dominant, the Doppler frequency measurement
provides a good estimate of the local radial electric field.
The Doppler backscattering data analyzed in this manuscript was obtained from a system consisting of eight fixedfrequency channels (ranging from 55 to 75 GHz) that generally
cover a large radial range in cutoff layer locations. Details of
the DBS system can be found in reference [26]. The radial
locations of the cutoff layers for each of the frequency channels used in the 0.76 MA case are indicated in figure 2(c), with
each position marked by the channel number. The system was
operated in X-mode polarization and utilized the right-hand
cutoff, so that the cutoff locations depend upon both density
and magnetic field. As discussed earlier, the wave number
probed by the DBS system depends upon the probe frequency
(or, equivalently, the vacuum wave number), the angle of incidence with respect to the plasma cutoff surfaces, and the local
plasma parameters. For the results shown here, the probed
wave numbers range from k θ = 5–7 cm −1 or k θρs = 0.75 to 2,
where ρs is the ion gyroradius (for deuterium) evaluated using
the local electron temperature and magnetic field, and k θ is the
poloidal wave number at the cutoff layer. These wave numbers
are in the range typically associated with the trapped electron
mode instability, although a value of k θρs = 0.75 could be considered in the higher wave number region of the ion temperature gradient instability ([27, 28], and references therein).
Figure 3 shows time-frequency spectrograms of the DBS
data. The three time intervals examined in this study are indicated by vertical dashed lines. The spectrograms are generated by computing short-time-interval power spectra for each
channel of the DBS system. Complex time signals are constructed from the in-phase and quadrature components of each
channel and then multiplied by a Hanning window 3.28 ms in
length (16384 data points at 5 MHz digitization rate). The time
for which the power spectrum is representative is taken as the
center time of the Hanning window. The power spectra are
computed with a temporal resolution of 3.28 ms by moving
the center of the Hanning window by this amount. After
150141, 1240 ms
2.
Te (keV)
1.
0
2.
Ti (keV)
1.
0
8 7
6 5
3.
2.
ne (10 m )
19
1.
4 3
-3
0
2
1
safety factor q
12
8
4
0
0.
0.2
0.4
0.6
0.8
1.0
normalized radius ρ
Figure 2. Spatial profiles of (a) electron temperature (from
Thomson scattering and ECE), (b) ion temperature (from charge
exchange recombination spectroscopy), (c) plasma density (from
Thomson scattering and profile reflectometry) and (d) magnetic
safety factor (from the magnetic equilibrium fitting code, EFIT) as
functions of normalized radius at a time 10 ms before the early time
interval of the 0.76 MA case. In addition, (c) shows the cutoff layer
locations of the various DBS channels labeled by channel number.
backscattering is a microwave scattering technique that is
sensitive to the magnitude and flow (generally poloidal) of
coherent and turbulent density fluctuations. It is widely utilized to determine radial electric field profiles and intermediate wave number scale (k ~ 1–6 cm−1) density turbulence
levels, n∼. DBS has also been used in other experiments to
study the physics of geodesic acoustic modes and zonal flows
[11–14]. In Doppler backscattering, a probe beam, of a given
frequency and polarization, is launched into a plasma containing a cutoff for that frequency and polarization. The probe
beam is injected at an angle with respect to the cutoff layer
and the beam is refracted and reflects out of the plasma upon
encountering the cutoff. Along the path of the beam, power is
scattered by density fluctuations, and it is the radiation scattered back along the beam path that is detected and analyzed.
Due to a combination of wave number matching and probe
electric field swelling near the cutoff, the strongest backscatter
often (but not always) occurs in the vicinity of the cutoff layer.
In the scattering process, the density fluctuation wave number
satisfies the familiar Bragg-scattering relation, k n∼ = 2 klocal,
where klocal is the probe-beam, wave number vector within
the plasma. The probe wave number, klocal, varies within the
plasma due to the variation of the index of refraction for that
3
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
Figure 3. Contours of spectral power (logarithmic scale) versus time are shown for each channel of the DBS system. The peak of the
power, as indicated by the brightest color, tracks the Doppler shift. The three time intervals examined are indicated by vertical dashed lines.
Note that each spectral contour graph is auto-scaled for display, so they cannot be directly compared. A generic scale is shown to the right
for reference.
smoothing over a 10 kHz window in frequency, the results of
the computation are displayed as contour plots of the log of
the spectral power as a function of frequency and time, over
the frequency range from −2.5 to 2.5 MHz. In figure 3, bright
yellow indicates high power and blue denotes low power. It is
of interest to examine this figure closely. The plasma current
and density are replicated for convenience of reference at the
top of panels (a) and (b), respectively. Channels 1 through 4
(probe frequencies 55, 57.5, 60, 62.5 GHz respectively) run
from the top to the bottom of panel (a) and 5 through 8 (probe
frequencies 67.5, 70, 72.5, 75 GHz respectively) from the top
to bottom of panel (b). Note that channel 1 is the radially outermost channel, while channel 8 is the innermost one (refer to
figure 2(c) for radial locations of channels). Early in the discharge, before 300 ms, all channels show relatively narrow frequency band signals, with little or no Doppler shift. At 300 ms
the neutral beam heating is initiated and there is a subsequent
increase in temperature (figure 1), as well as, an increase in
amplitude and mean frequency and frequency bandwidth of
the power spectra (figure 3). As time progresses, the Doppler
shifts, as indicated by the frequency locations of the ‘yellow’
band, increases for channels 3–8, culminating with a roughly
steady-state condition starting around 2000 ms. Starting at
~1000 ms for channel 8 and ~1300 ms for channel 7 there is
a decrease in magnitude of the fluctuation-scattered signal as
evidenced by a reduction in brightness of the spectrograms
for these channels. Channels 1 through 6 remain relatively
unchanged throughout the time period 300–3450 ms. The
reduction in scattered power in channels 7 and 8 is interpreted
as being due to a reduction in the level of density fluctuations,
n∼, at the radial locations of the cutoff layers for these channels. Even with a reduction in signal levels, the backscattered
power remains large enough to identify clear Doppler shifts
in the power spectra for channels 7 and 8. In contrast to this
particular case, other cases at higher plasma current indicate
a reduction in fluctuation levels on the inner channels to such
an extent that no clear Doppler shift can be discerned from the
DBS data. The significance of these observations is clarified
by the data analysis presented in the following sections.
At approximately 3558 ms there is a transition to a high
confinement regime, or H-mode. This transition is evidenced
by the increase in chord-averaged density, as seen in the top
panel of figure 3(b) (also by the increase in Te, Ti, and ne(0) at
that time as seen in figures 1(b)–(d) and the rapid change in
the spectral shape and magnitudes of channels 1 through 4.
Channels 5 through 8 change later in time as they are initially
far from the edge, but the cutoff layers for these channels are
moved outward into the vicinity of ρ = 1 by the increased density. The H-mode and the H-mode transition, while of significant interest, are not addressed in this manuscript.
4. Bandt–Pompe probability and C-H plane
In addition to their power spectra, time signals are distinguished by the types of structure they contain. A technique
for quantifying structure in time signals is provided by the
Bandt–Pompe (BP) probability [21]. The BP probability
is the probability distribution of amplitude orderings that
occur in a time signal, T (t ), measured at N , evenly spaced,
discrete points. Computation of the BP probability requires
use of an ‘embedding space’. An embedding space of dimension, d , consists of d values of the time signal in the order
in which they appear in the time record. These d values are
called ‘d-tuples’. In the signal T (t ) the ‘d-tuple’ at the point,
tn, is: [T (tn ), T (tn + 1), … , T (tn + d − 1)]. The set of ‘d-tuples’
associated with the signal are found by letting n range from:
1 ≤ n ≤ N − d + 1.
Thus, for embedding dimension, d , there are N − d + 1
‘d-tuples’ associated with a signal of length N . Each one of
these d-tuples contains d , consecutive (in time) values of the
signal T (t ). The BP probability depends upon the ordering of
4
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
the amplitudes in the set of d-tuples. A group of elements of
length d can be ordered in d ! combinations, so that the BP
probability space has dimension d !. This set of d ! permutations in the amplitude orderings in the d-tuples associated with
a time signal, constitute the basis set of the BP probability
space. The value along each direction in the BP probability
space can range from 0 to 1. The values of all probabilities
must sum up to unity.
The BP probabilities are presented, herein, plotted as probability versus ‘bin number’, where ‘bin number’ represents
one of the d ! permutations of the amplitude ordering. The BP
distributions are ordered by probability value, with the highest
probability first, and then the other probabilities in descending
order. Therefore, in the BP plots, the bin numbers do not refer
to a particular amplitude permutation, but rather the rank of
probability in descending order. Thus, in comparing two probability distributions, it should be kept in mind that the state
with the highest probability in each distribution is not necessarily the same amplitude ordering in the d-tuple.
The BP probability is capable of detecting structure in
time signals because structure preferentially populates a set
of amplitude permutation states. It turns out that chaotic and
stochastic dynamics produce distinctly different BP distributions. Generally speaking, chaotic distributions have a range
of permutation states with high occupation numbers, along
with a rather broad range of unoccupied permutation states.
In contrast, stochastic processes have most permutation states
occupied and the distributions tend to be more uniform. These
distinctions can be quantified by use of the ‘C-H plane’.
The C-H plane was introduced by Rosso, et al [20] as a
method for distinguishing chaotic and stochastic time signals. The C-H plane comprises two statistical measures: the
normalized Shannon entropy, H , and the Jensen-Shannon
Complexity, C. Both of these measures are computed from
the BP probability. Denoting the set of d ! probabilities for
embedding space of dimension d as, P = {pi }, the normalized
Shannon entropy, H , is defined as,
that the chosen embedding dimension produce statistically
reliable results. For example, in the maximum entropy case,
the average occupation number for the probability distribution, Pe, is N/d!, and should be much larger than unity to give
robust results. This can be accomplished by choosing a small
d. On the other hand, the sensitivity of the analysis to different
structures depends upon the number of amplitude orderings represented in the embedding space. Larger embedding
spaces are capable of representing many more structures and
this consideration argues for a large d. Freedom in this regard
is highly restricted, however, because the number of amplitude orderings increases factorially with embedding dimension. The data analyzed in this study consists of 20 000 points
and the embedding dimension is chosen to be d = 6, because
N/d! = 28, and 720 different amplitude orderings are sampled.
In addition, the temporal duration of structures sampled at the
data sampling rate of 200 ns and d = 6, (1.2 μsec) is of primary interest to this study. We mention, for completeness, that
larger scale temporal structures can be investigated, for a fixed
embedding dimension, by using the method of sub-sampling
[22], but this technique is not needed for the data analyzed
here.
BP probabilities that have the same entropy do not necessarily have the same complexity, and, therefore, C and H can
be treated as independent variables to create the C-H plane.
For a given value of the entropy, H , the complexity, C, has a
maximum and minimum value, and the locus of these extrema
trace out two curves for 0 ≤ H ≤ 1. In plots showing the C-H
plane (refer to figure 6 as an example), these two curves are
labeled ‘maximum complexity’ and ‘minimum complexity’.
All points in the C-H plane are located between these two
curves. The probability distributions that correspond to
‘maximum complexity’ have a limited number of uniformly
occupied states with the remainder of the states unoccupied.
Distributions that correspond to ‘minimum complexity’ have
one occupied state at high probability and the rest are uniformly occupied at lower probability.
Rosso, et al [20], have shown that the C-H plane is useful for
distinguishing chaotic processes from stochastic processes.
By comparing the C-H plane locations of signals generated
by known chaotic and stochastic processes, Rosso, et al demonstrated that they occupy different regions of the C-H plane.
While there is no ‘hard’, well-defined boundary between the
two, the stochastic process of fractional Brownian motion
(fBm) [29] serves as a useful demarcation between the chaotic and stochastic regions. As a general guideline, processes
that have C-H plane locations on or below (i.e. having lower
complexity at the same entropy) the locus of fBm in the C-H
plane are considered stochastic, while those that are above it
(i.e. having higher complexity at the same entropy) are considered chaotic. In cases where some ambiguity is present,
details of the Bandt–Pompe probability distributions are used
to resolve the issue. In the C-H plane, as a rule of thumb, chaotic signals are characterized by moderate values of entropy,
H , and high values of complexity, C. Stochastic processes are
characterized by high values of entropy and low values of
complexity.
d!
1
1
H≡
S,
∑[−pi log2(pi )] =
(1)
log2(d !)
log2(d !) i = 1
where S is the (un-normalized) Shannon entropy. One probability distribution of particular interest, denoted by Pe, is the
distribution with all equal probabilities: pi = 1/d ! ∀ i. The
distribution Pe has normalized entropy H = 1, and thus represents the highest possible entropy state.
The Jensen-Shannon complexity, C, is then defined as,
⎡S P + Pe − S(P ) − S(Pe) ⎤
⎣
2
2
2 ⎦
(2)
C
H (P ).
= −2 d ! + 1
log2(d ! + 1) − 2log2(2d !) + log2(d !)
d!
( )
The notation (P + Pe )/2 means the probability distribution
obtained by adding the BP probability, pi, to the Pe probability,
1/d !, and then dividing by 2, for all i: 1 ≤ i ≤ d !.
Choosing the embedding dimension, d, depends upon the
length of the time signals analyzed, N, and the size (in time),
dΔt, of the structures one wishes to investigate. It is desired
5
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
5. Signal analysis
The question addressed in this section is whether the fluctuations observed in the L-mode plasmas of the DIII-D tokamak
are stochastic or chaotic in nature. Techniques have been
developed by researchers in diverse fields that permit this
question to be addressed by analysis of the temporal behavior
of fluctuations. First, the shape of the power spectrum of a
signal is a strong indicator of the dynamical nature of the
signal. Studies of nonlinear dynamics models [30–32] have
established that time signals whose power spectrum exhibits
an exponential form (i.e. proportional to exp(−aω), with ω the
angular frequency) most likely arise from chaotic dynamics.
In contrast, the statistical tools used to investigate stochastic
time signals presuppose that their power spectra have a powerlaw dependence (i.e. are proportional to ω−b) [33]. However,
the shape of the power spectrum of a particular time signal
is not dispositive as to the nature of the dynamics producing
it, and additional tests are needed. Two additional tests of the
dynamical nature of a time signal are provided by the analysis
techniques discussed in section 4: the Bandt–Pompe probability and the C-H plane.
The combination of: (1) the shape of the power spectra;
(2) the details of the Bandt–Pompe probability distributions;
and, (3) locations in the C-H plane are used to test the statistical nature of DBS time signals. As discussed in section 3,
DBS time signals arise from density fluctuations. Thus, the
statistical nature of the time signals measured by the DBS
system are taken as an indicator of the statistical nature of the
density fluctuations in the DIII-D plasma. These three techniques, applied to the DBS time signals, demonstrate that the
density fluctuations measured in the edge region of the DIII-D
tokamak, under the operational conditions described in section 2, are chaotic.
Figure 4. Power spectrum, displayed in log-linear format, from
channel 3 of the ‘L-mode 1’ case is compared to a Doppler-shifted
‘exponential tent’ (red curve) with frequency shift fD = 65 kHz and
characteristic time-scale τ = 765 ns. The inset over the frequency
range from −850 to 950 kHz illustrates the degree to which the
spectrum is exponential. The epsilon value for this fit (equation (4))
is 4.1%.
fluctuation spectrum in the plasma rest frame. To accomplish
this goal, plasma locations with low flow, or small Doppler shift,
are investigated first in section 5.2. Once the basic form of the
rest frame fluctuation spectrum is established, cases with strong
flow, which typically have more complicated power spectra, are
investigated. Section 5.3 presents a strong flow case in which
reduced levels of the DBS Doppler shifted density fluctuation
component of the backscattered signal, relative to a ‘noise’ component, results in chaotic signals with increased entropy for the
central DBS channels. In contrast, section 5.4 presents a strong
flow case in which the most central Doppler shifted signals are
so reduced, relative to the noise signal, that the resultant backscattered signal is stochastic. The plasma conditions used in this
study are described in section 2.
5.1. Power spectra
As noted previously, the DBS instrument produces both, an
‘in-phase’, and a ‘quadrature’ signal for each of its eight separate frequency channels. This feature allows for the creation
of a complex time signal for each channel. The Fourier transform of a complex time signal exists over negative and positive frequencies. The power spectra presented here are created
from such complex time signals. The data used contain 20 000
points (4 milliseconds at 5 MHz digitization rate or 2 ms at
10 MHz), and average power spectra are obtained by dividing
the original signal into 10 sub-signals of 2000 points each.
The displayed power spectrum is obtained by averaging over
the spectra of the ten sub-signals. The spectra are obtained
using rectangular time-windows. Data acquired with a 5 MHz
digitization rate, has a Nyquist frequency of 2.5 MHz, while
data acquired at 10 MHz has a Nyquist frequency of 5 MHz.
With complex time signals, the DBS system is well suited for
measuring plasma flows by tracking the frequency shift (either
positive or negative) in the power spectrum that arises from such
flows. The details of the power spectra change with plasma conditions, and a proper interpretation of individual power spectra can
present a challenge. It is important to first determine the density
5.2. Small doppler shifts
Of particular interest in this study are cases for which the
observed Doppler shifts are small. Presumably these small
Doppler shifts are due to weak plasma flow. The cases in
which the Doppler shifts are small give the clearest picture
of the fluctuation spectrum in the rest frame of the plasma.
An example of a power spectrum from the ‘L -mode 1’ case
at a radial location with weak flow is shown in figure 4 in
6
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
fitting exponential tents. The value of ‘const.’ in equation (3)
is too small to affect the value of ε, and it is used only for aesthetics in the graphic presentation.
The spatial location sampled by each channel of the DBS
instrument is dependent upon the density and magnetic field
profiles, which are determined, a posteriori, as discussed in
section 3. From the density profile obtained for the ‘L-mode
1’ case, it is found that the cutoff layer of channel 3 is very
near the last closed flux surface (LCFS). Thus, this particular
example represents fluctuations at the plasma edge. The power
spectrum for channel 3 of the ‘L-mode 1’ case has a Doppler
shift of 65 kHz and a characteristic time of 765 ns. The bottom
panel of figure 4 shows a blowup of the channel 3 power spectrum over the frequency range from −850 to 950 kHz in order
to demonstrate the exponential behavior of the power spectrum over 30 dB range in power for this case.
Further examples of power spectra with exponential
dependence are shown in figure 5. These examples are from
the ‘current scan’ data set described in section 2. All examples
are from channel 1 of the DBS instrument, and are chosen
for exhibiting relatively small Doppler shifts. Doppler shifts
range from 10 to 170 kHz, and the characteristic times range
from 560 to 815 ns. Smaller characteristic times correspond
to shallower slopes (covering a wider frequency range per
decade of power) in a log-linear plot. There is some departure from the exponential fit in the vicinity of the Doppler frequency, as the peak of the data is ‘rounded’ while the peak
of the fit is sharp. The rounding effect, evident in the data,
occurs from fluctuations in the mean flow with an additional
contribution arising from the finite spread of wave numbers or wave number resolution of the DBS diagnostic. The
‘rounding effect’ is less apparent at smaller Doppler shifts.
These examples show that, in cases that exhibit small Doppler
shifts, DBS power spectra are modeled very well by the symmetric, Doppler-shifted, ‘exponential tent’ form. Evidently,
the fluctuation power spectrum in the plasma rest frame (i.e.
the frame moving with the flow) is symmetrically exponential
about the Doppler frequency.
Figure 6 shows the location in the C-H plane of the time
signals associated with the power spectra shown in figures 4
and 5. The locations of both the ‘in-phase’ and ‘quadrature’
signal components are shown for each case. The C-H plane
locations of these two components are often so close that the
two, separate symbols denoting each component appear to
be only one. For reference, the locations of known chaotic
and stochastic processes are also indicated. The location of a
time signal from the chaotic Lorenz model [34] is shown as an
‘open’ circle. Also shown are the locations of signals generated by the ‘logistic map’ [35], a recurrence map obeying the
relation: xn + 1 = r xn(1 − xn ), for various values of r between
3.65 and 4, at which the map exhibits chaotic behavior. It is
seen from figure 6 that the DBS signals shown in figures 4
and 5, all of which exhibit exponential power spectra, are also
located in the same region of the C-H plane as these examples
of chaotic processes.
Stochastic processes are associated with signals generated
by fractional Gaussian noise (fGn) and fractional Brownian
motion (fBm) [29]. fGn is a stationary statistical process with
log-linear format. This example from channel 3 of the DBS
system exhibits a relatively small Doppler shift. The power
spectrum is very clearly ‘exponential’ in nature, at both positive and negative frequencies. Specifically, significant portions
of the spectrum are linear in frequency when plotted using a
log-linear format. That is, over a wide frequency range, the
power spectrum is proportional to exp(−2ωτ ), and thus the
log of the power spectrum is proportional to ω. The parameter
τ is a time scale characterizing the slope of the linear portion
of the spectrum.
To explicitly display the exponential nature of the observed
power spectrum, it is compared, or ‘fit’, to an analytic form
comprising a Doppler-shifted exponential plus a white ‘noise
floor’ (represented by the addition of a constant value). The
‘fit’ is obtained from the relation,
A exp[−2( ω − ωD )τ ] + const.
(3)
Note that this form produces a ‘fit’ symmetric about the frequency, ωD. The angular Doppler shift frequency, ωD is related
to the Doppler frequency, fD, as, ωD = 2π fD. The characteristic time-scale associated with the exponential fit has a direct
connection to a Lorentzian pulse (in time) with the form:
[τ 2 /(τ 2 + t 2 )]. The power spectrum of this Lorentzian pulse
is exp(−2ωτ ). The ‘fits’ obtained from equation (3) have the
appearance of a ‘tent’, and in the following they are referred
to as ‘exponential tents’.
As a measure of how well the ‘exponential tent’ fits the
observed power spectrum, the root-mean square of the difference is computed and compared to the maximum amplitude,
to form the measure, ε,
(PSdata − PSfit )2 1/2
ε=
,
(4)
max(PSdata )
where, PSdata is the power spectrum of the data and PSfit is
the power spectrum of the fit (the ‘exponential tent’). The
brackets, , indicate the mean, or average, over all the points
in the frequency domain. To determine the parameters of the
‘exponential tent’ (equation (3)) that best fit the data, the characteristic time, τ, is first found by fitting a line to the linear
region of log(PSdata ), with a standard fitting routine (IDL,
poly_fit). Depending upon the Doppler shift associated with
the power spectra, a limited region of positive or negative frequencies, over which log(PSdata ) is linear, is used in the fitting
routine. This procedure determines a nominal value for the
characteristic time. The value of the characteristic time can
typically vary by ±5% of the nominal value and still produce
a linear fit that (within visual resolution) falls within the range
of fluctuations present in the power spectra over the region in
which it exhibits exponential behavior. Once the characteristic
time is chosen, the amplitude, A, and the Doppler shift, ωD, are
then adjusted to minimize ε (equation (4)). The amplitude of
ε for the fits shown in figures 4 and 5 varies between 4–7 percent. The 4 percent level of ε is the best that can be achieved,
because the power spectra of the data have fluctuations (due to
phase interference effects), whereas, the ‘exponential tent’ is
a smooth function. Thus, there will unavoidably be a non-zero
difference between the observed power spectra and the best
7
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
Figure 5. Power spectra from the current scan case, chosen because they exhibit relatively small Doppler shifts, are compared to Dopplershifted exponential fits (epsilon values, equation (4), are given for each case). Characteristic times τ range from 560–815 ns and Doppler
shifts fD from 10–170 kHz.
The stochastic process of fractional Brownian motion is nonstationary. The scale along the fBm curve shown in the C-H
plane indicates the value of the Hurst exponent. The point
with Hexp = 0 is at the extreme right (highest entropy) and the
point with Hexp = 1 is at the extreme left (lowest entropy). The
values are not uniformly distributed along the curve. Note that
the Hurst exponent does not characterize the power spectra
without additionally specifying the process, because fGn
(Hexp = 1) is the same point as fBm (Hexp = 0).
Figure 7 shows the BP distribution of the 1.4 MA channel
1 ‘in-phase’ time signal from the 1250–1254 ms time
interval. The power spectrum for this case is shown in panel
(f) of figure 5 and its location in the C-H plane is displayed
power-law, power spectra proportional to f β, with −1 ≤ β ≤ 1.
The fGn time signals used to generate the points labeled ‘fGn’
in the C-H plane are generated numerically starting from
power spectra with slopes given by the relation, β = 1 − 2Hexp,
where Hexp is the Hurst exponent. The Hurst exponent ranges
in value from 0 to 1. Beta equal to 1 corresponds to Hexp = 0,
and β = −1 corresponds to Hexp = 1. The fGn signal with
Hexp = 0.5 is located at C = 0, H = 1, and represents ‘white
noise’. The curve labeled fBm represents the stochastic process of fractional Brownian motion. The curve is the locus of
points obtained by time integration of the fGn signals. The
power spectra of fBm time signals are power laws with slopes
given by, β = −1 − 2Hexp, and thus beta ranges from −1 to −3.
8
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
0.6
Current scan
C
d=6
blue - 0.76 MA
green - 1.0 MA
red - 1.4 MA
star - 1250-54 ms
triangle - 2500-04 ms
square - 3525-29 ms
0.5
0.4
logistic
map
Maximum
complexity
L mode 1
0.3
.8
1
Lorenz
model
fBm
.6
0.2
.4
Minimum
complexity
0.1
.2
fGn
0.0
0.0
0.2
H
0.4
0.6
0.8
0
1.0
Figure 6. The C-H plane (of embedding dimension d = 6) showing the location of the signals producing the power spectra displayed in
figures 4 and 5. All DBS signals are in the ‘chaotic’ region of the C-H plane. The ‘L-mode 1’ label refers to the case shown in figure 4
while all other cases are shown in figure 5. Location of chaotic signals from the Lorenz model (open circle) and the logistic map (solid
maroon diamonds) are included for reference. The yellow-orange curve corresponds to fractional Brownian motion (fBm) and the green
dots to fractional Gaussian noise (fGn). The scale on the fBm curve is the value of the Hurst exponent.
0
Bandt-Pompe
distribution
log probability
−1
Maximum
complexity
d=6
Logistic map
−2
Minimum
complexity
−3
fBm (Hexp = .95)
−4
−5
1.4 MA
1250-54 ms
channel 1
0
200
bin number
400
600
Figure 7. The Bandt–Pompe distribution for the channel 1 ‘in-phase’ signal from the 1.4 MA, 1250–1254 ms case is compared to four
other distributions that have the same entropy. The probability distributions shown correspond to maximum complexity (red), minimum
complexity (blue), the logistic map (violet) and fBm with Hurst exponent Hexp = 0.95 (orange).
as one of the red ‘stars’ in figure 6. The BP distribution for
this case is compared to BP distributions for known processes
in figure 7. All of the distributions have the same value of
normalized entropy, H , only the values of the complexity
distinguish them in the C-H plane. In fact, this particular
example from the data is chosen because it has an entropy
value large enough to compare it to an example of fBm. The
distribution with ‘maximum complexity’ has a limited range
of populated states with uniform, non-zero probability with
the remainder of the states unoccupied (zero probability).
In contrast, the ‘minimum complexity’ distribution has one
state with a high probability, and all remaining states populated uniformly at lower probability. The distribution for the
logistic map resembles the ‘maximum complexity’ distribution in the number of unoccupied states, but the probabilities
are not uniformly distributed. The fBm distribution has a few
states with high probability, but most states are occupied at
considerably lower probability. The BP distribution for the
1.4 MA channel 1 case displays chaotic features, in that,
it has a limited range of occupied states at relatively high
9
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
−5
fDS = 75 kHz
0.76 MA
1250-1254 ms
fits
channel 2
log Power
−10
fDS = 175
channel 4
fDS
= 750
channel 6
−15
fDS = 1400
channel 8
−20
−2
−1
0
1
2
Frequency (MHz)
Figure 8. Power spectra that exhibit a progressively larger Doppler shift as the probing beam frequencies increase from channel 2 to
channel 8 (0.76 MA, 1250–1254 ms case). The high frequency side of the exponential tent can be clearly identified for each channel shown.
probability and a broad range of unoccupied states (similar
to the logistic map).
an exponential with characteristic times within 20% of the
channel 1 characteristic time. The characteristic time for the
exponential fit increases from 850 ns for channel 1 to 960 ns
for channel 6 and then drops to 750 ns for channel 8. Below
the peak, the spectra depart from the exponential tent ‘fit’ as
shown by the dashed (blue) lines. The larger the Doppler shift,
the greater the difference. The power in the frequency range
below the Doppler-shifted peak may result from the accumulation of scattering events along the beam ray path at locations
not in the immediate vicinity of the reflection region of the
beam. This extraneous power is usually (but not always) much
weaker than that returned from the reflection region.
The locations, in the C-H plane, of the time signals associated with the power spectra shown in figure 8 are given in
figure 9. All channels (1 through 8) are displayed in figure 9
and they all fall in the chaotic region of the plane. It should be
emphasized that channels 1 through 4 are located towards the
edge of the plasma, as seen in figure 2(c), while 5 through 8
are located in the core of the plasma. The first four channels
are clustered together in the C-H plane, but the last four follow
the arc of the maximum complexity curve as the entropy progressively increases with increasing channel number (i.e.
at locations deeper into the plasma). The channels with the
largest Doppler shifts have the largest entropy. As the power
returned from the reflection layer becomes weaker the entropy
of the detected signal increases. The reduction in signal levels
is a result of larger antenna to plasma distance as well as the
often observed reduction of fluctuation levels of the core
plasma as compared to the edge. However, the chaotic nature
of the signal is still manifest for all channels.
The details of the BP distributions of channels 2 and 8 are
shown in figure 10. The increase in the number of occupied
states is the cause of the increased entropy for channel 8 relative to channel 2. Note that both signals have about the same
value of complexity, as seen in figure 9. The BP distribution
of the fBm time signal with the same entropy as the channel
5.3. Large doppler shifts
In regions of the plasma with strong flow, the DBS signals
exhibit large Doppler shifts. It should be noted that the lower
frequency channels (e.g. 1 through 4 in panel (a) of figure 3)
show relatively little change in magnitude as the shot develops.
The higher frequency channels (especially 7 and 8, in panel
(b) of figure 3) show a decrease in magnitude after approximately 1000 ms for channel 8 and approximately 1300 ms for
channel 7. This reduction in signal strength for the inner channels is interpreted as a reduction in the amplitude of density
fluctuations at the respective cutoff layers (refer to figure 2
for radial locations). The reduction in signal amplitude for the
innermost channels has a significant consequence for the statistical nature of the backscattered signals. Sufficient reduction in the amplitude of the backscattered signals results in
a change from a chaotic to a stochastic nature. The details of
this change are presented at the end of this section, but first,
the case illustrated by figure 3 is examined in detail, and it
is demonstrated that the signals from all channels exhibit a
chaotic nature.
Figure 8 shows an example of progressively larger Doppler
shifts detected as the frequency of the probing beam increases
from channel 2 through channel 8 for the 0.76 MA case. For
clarity, each power spectrum is offset from its immediate
neighbor by a factor of 0.001, and only the even numbered
channels are displayed. These power spectra are viewed as
Doppler-shifted exponential tents with increasing Doppler
shift and decreasing amplitude as the successively higher frequency probe beams (higher frequency channels are labeled
by higher numbers) penetrate deeper into the plasma. The
power spectra for channels 2 through 8 at frequencies on
the high side of the Doppler peak are fit reasonably well by
10
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
0.6
C
d=6
0.76 MA
1250-1254 ms
0.5
channels
1- 8
0.4
1
0.3
2
3
Maximum
complexity
6
5
4
7
fBm
8
0.2
Minimum
complexity
0.1
0.0
0.0
0.2
H
0.4
0.6
0.8
1.0
Figure 9. C-H plane for signals that progressively display large Doppler shifts. The power spectra of all channels from the 0.76 MA, 1250–
1254 ms case are shown. All signals are in the chaotic region of the C-H plane.
0
Bandt-Pompe
distribution
0.76 MA
1250-1254 ms
−1
log probability
d=6
−2
fBm (Hexp = .5)
−3
−4
channel 2
channel 8
0
200
bin number
400
600
Figure 10. The BP distribution for channel 2 and channel 8 of the 0.76 MA, 1250–54 ms case. The probability distributions for fBm with
Hurst exponent Hexp = 0.5 has the same entropy as the channel 8 signal, but lower complexity.
8 signal (Hexp = 0.5) is shown for comparison. The number of
occupied states for the stochastic, fBm process is larger (706
of 720 bins are occupied), and the probabilities are more uniformly distributed as compared to the probability distribution
for channel 8.
spectrum is offset from its immediate neighbor by a factor
of 0.001. For this example, it appears that the Doppler shifts
of the higher frequency DBS channels (channels 5 and 7)
are large enough to move the ‘exponential tent’ feature out
of the sampled frequency range, −2.5 to 2.5 MHz. However,
the absence of an identifiable exponential feature in the
higher frequency channels is attributed to a weak backscattered signal from the reflection point for these channels. The
fluctuation levels in the inner regions of the 1.4 MA plasma
sampled by the DBS are very small. The power spectra for
channels 3 and 5 have large Doppler shifts and a limited
frequency range over which they are exponential (linear in
log-linear display). The characteristic times associated with
the exponential features vary from 625 to 500 ns. The power
spectra of the channel 7 signal does not have an exponential
5.4. Stochastic signals
A contrasting case to that shown in figure 8 is shown in
figure 11. The power spectra of channels 1, 3, 5 and 7, for
the case with 1.4 MA current during the 1250–1254 ms time
interval, are shown in a format similar to that of figure 8. The
channel 1 power spectrum for this case is the same as shown
in panel (f) of figure 5, and the ‘exponential tent’ fit shown in
that figure is reproduced in figure 11. For clarity, each power
11
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
log Power
−4
1.4 MA
1250-1254 ms
−6
channel 1
−8
−10
channel 3
−12
channel 5
−14
channel 7
−16
−2
−1
1
0
2
Frequency (MHz)
Figure 11. The power spectra for the 1.4 MA current in the same time interval as in figure 8. No exponential feature is evident in the
channel 7 power spectrum. The channel 7 time signals are stochastic.
0.6
C
d=6
1.4 MA
0.5
1250-1254 ms
0.4
channels
1-8
3
2
1
4
Maximum
complexity
fBm
0.3
0.2
5
8
Minimum
complexity
0.1
7
6
0.0
0.0
0.2
H
0.4
0.6
0.8
1.0
fractional
Gaussian noise
(fGn)
fBm
0.12
0.10
0.08
0.06
7
6
0.04
0.02
0.96
0.97
0.98
0.99
1.00
0.00
Figure 12. Locations in the C-H plane of the signals from all channels for the 1.4 MA, 1250–1254 ms time interval. Channels 6 and 7 (red
triangles) lie in the stochastic region of the C-H plane characterized by fGn while channel 8 is located on the fBm curve.
feature and is approximately uniform over the frequency
range from −1 to 2 MHz.
The locations in the C-H plane of all the signals for the 1.4
MA, 1250–1254 ms time interval, case are shown in figure 12.
The entropy of the signals systematically increases as the
location of the reflection layer moves deeper into the plasma
(i.e. as the beam frequency increases). Channels 1 through 5
are in the chaotic region of the C-H plane, although channel 5
12
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
0
Bandt-Pompe
distribution
log probability
−1
d=6
fGn
Hexp = .95
−2
channel 7
−3
fBm
−4
−5
channel 1
0
200
channel 5
channel 3
bin number
400
600
Figure 13. The Bandt–Pompe distributions corresponding to the C-H plane locations shown in figure 12 progressively change character
with increasing channel number, starting out with clear chaotic characteristics at channel 1 and progressing to purely stochastic
characteristics at channel 7.
(β = 1 − 2Hexp ), the power spectrum of the fGn signal has a
slope of β = −0.9.
Figure 14 demonstrates that the power spectra of the fGn
signal and the channel 7 signal are not the same. In figure 14,
the power spectrum labeled ‘channel 7’ is the average of the
power spectra of the ‘in-phase’ and ‘quadrature’ signals,
treated as separate, real, time signals. By construction, the
power spectrum of the fGn signal is a power law with slope
(−0.9) over the entire frequency range. The channel 7 powerspectrum exhibits this same slope only over the frequency
range from, approximately, 400 kHz to 2.5 MHz. Below
400 kHz, the channel 7 power spectrum has near zero slope,
and in that sense, resembles ‘white’ noise. The 6-tuples used
in computing the BP distributions for all channels spans a
time interval of 1.2 ms (six times the data acquisition time of
200 ns). The BP distributions presented in this analysis are not
particularly representative of structures with time scales larger
than a few times the 6-tuple span. Hence, the difference in
the power spectra at frequencies below 400 kHz (2.5 ms) is
not particularly important or meaningful in relation to the CH
plane analysis using a d = 6 embedding space.
is quite close to the fBm curve. The power spectra of these five
channels all have an exponential feature over some frequency
range. In contrast, the three channels whose power spectra do
not exhibit an exponential feature, channels 6 through 8, are
located in the stochastic region of the C-H plane. Channels
6 and 7, as shown in the inset of figure 12, are in the region
associated with the fGn stochastic process, while channel 8 is
located on the fBm curve. This difference in location is likely
due to the presence of a small instrumental noise signal in
channel 8. The lack of an exponential feature in these channels
is attributed to very weak backscatter from the cutoff layer.
Figure 13 shows the details of the Bandt–Pompe (BP)
distributions for the signals from channels 1, 3, 5 and 7. As
the channel number increases from 1 to 7, the characteristics of the BP distributions that distinguish them as chaotic,
namely, a region of relatively high, uniform probability and
many unoccupied states (bins), progressively changes to the
characteristics of the stochastic distributions, i.e. almost all
states occupied at a fairly uniform probability. Channel 5 has a
chaotic component even though its C-H plane location is very
close to the stochastic fBm curve. The location of channel 5
in the CH plane is consistent with adding a noise signal to a
chaotic signal as demonstrated by Rosso, et al [36]. Rosso,
et al find that the addition of a noise component to a chaotic
signal moves the C-H plane location towards the lower right
hand corner along a path roughly parallel to the maximum
complexity arc. As compared to the Bandt–Pompe distribution of fBm with the same normalized entropy (blue curve in
figure 13), the Bandt–Pompe distribution for channel 5 has
higher probabilities over a wide range of low bin numbers, and
lower probabilities at high bin numbers. On the other hand,
the Bandt–Pompe distribution for channel 7 (shown in red in
figure 13) clearly represents a stochastic process since it is
almost identical to the fGn distribution (shown in green) with
Hurst exponent, Hexp = 0.95. This value of the Hurst exponent
means that, due to the manner in which they are constructed
6. Discussion
In all of the cases investigated, the density fluctuations
measured by the DBS system in the outer region of the
DIII-D plasma (radial locations approximately ρ ≥ 0.8, see
figure 2(c)) have been identified as chaotic. The situation
at locations further into the plasma varies with plasma flow
conditions. Fluctuation spectra in the plasma rest frame (little
or no Doppler shift) are exponential and can be well modeled with the ‘exponential tent’ form given in equation (3).
Plasma flows, as measured by the DBS instrument, result
in a change in the frequency location of the ‘exponential
tent’ feature. However, strong flows are associated with the
13
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
−4
fGn ( f -0.9 )
log Power
−5
−6
channel 7
−7
−8
10
4
10
5
10
6
Frequency (MHz)
Figure 14. The power spectra of fGn signals (green curve) in the same location of the C-H plane as the channel 7 signals (red curve) have
different power spectra at low frequencies, but agree over the range 400 kHz to 2.5 MHz.
appearance of other features in the time signals that result in
higher entropy for channels with cutoff layers in the strong
flow region. These other features are generally noticeable in
the power spectra at frequencies below the Doppler shift frequency. Nonetheless, as long as the exponential feature associated with the rest frame dynamics can be identified in the
power spectrum, the time signals are found to have a significant chaotic component and are located in the chaotic region
of the C-H plane. An exponential feature in the power spectra
together with a location in the chaotic region of the C-H plane,
indicates that such signals arise from chaotic density fluctuations, even when the Doppler shifts are large. The example of
the 0.76 MA early time period presented in figure 8 illustrates
this behavior. Although not explicitly shown, the ‘L-mode 1’
case behaves very similarly. For that case, all power spectra
have clearly identifiable Doppler-shifted, exponential tent features and the locations in the C-H plane of signals from each
channel are very similar to those illustrated for the 0.76 MA
case in figure 9. Thus, when the DBS signal returned from the
cutoff layer is detectable, the density fluctuations at that radial
location are found to be chaotic even when located in the core
region of the plasma.
Only one example of stochastic behavior, as determined
by location in the C-H plane, has been displayed in detail
in figure 12, but other stochastic cases were investigated.
Stochastic behavior in the DBS signals, when it occurs, is found
in the higher frequency channels returned from the interior
regions of the plasma, and is not observed at the plasma edge.
These stochastic signals are due to the reduction, or absence,
of a strong signal from the reflection layer of the beam. The
lack of a strong signal from the reflection layer presumably
occurs because the amplitudes of density fluctuations in the
k-number range that backscatter the beam are too small, or not
correctly wave number-matched, to return a detectable signal.
The DBS data are consistent with having two contributions, an
intermediate wave number, Doppler-shifted signal originating
near the cutoff location, and a second, low amplitude and low
Doppler shift signal that extends over a broad range of frequencies. It is the first signal, the intermediate wave number,
Doppler-shifted component returned from the cutoff region,
that is normally considered to constitute the DBS signal,
while the second signal acts as a ‘noise’ background associated with the beam path through the plasma. Recognizing this
difference in the DBS signals, it is concluded that time signals
presenting a stochastic nature do not provide evidence that the
dynamics of the interior region are stochastic. When the signal
returned from the cutoff layer is too weak, no definitive conclusions can be drawn as to the statistical nature of the fluctuations at the cutoff location. Rather, stochastic signals arise
when the signal from the reflection region is very small as
compared to the beam path noise contribution to the DBS signals. It is important to note, however, that, when a stochastic
signal is returned by the diagnostic, the techniques employed
here can readily detect its presence.
Experimental studies [37] of electron heat transport and of
edge density fluctuations in a large linear device have identified the origin of the observed exponential power spectrum
to be individual Lorentzian pulses in the time series measured
with Langmuir probes. Similar results have been obtained in
the edge plasma of the TJ-K stellarator [38]. The origin of
Lorentzian pulses has been demonstrated [39] for the basic
models of chaotic dynamics. The pulses are a natural consequence of the chaotic dynamics in the vicinity of the separatrix
of elliptic regions in potential flow fields or, analogously, the
limit cycles of attractors in nonlinear dynamical models. Since
the DBS signals of L-mode plasmas exhibit clear exponential
features it is of interest to compare the statistical properties
of these features to those found in other experimental devices.
To this end, the probability distribution function (PDF)
of characteristic times found in all the DIII-D, L-mode cases
14
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
0.30
L-mode
plasmas
Probability
0.25
(a)
0.20
0.15
Mean
value
715 ns
0.10
0.05
0.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Characteristic time (µsec)
Signal (arb. units)
1.5
L- mode 1
1.0
(b)
channel 3q
Lorentzian
pulse
0.5
τ = 775 ns
0.0
−0.5
−1.0
−1.5
1445
1450
1455
1460
Time (µsec)
Figure 15. (a) The distribution of characteristic times, τ, extracted from both in-phase and quadrature power spectra for DIII-D L-mode
plasmas. (b) An example of a single Lorentzian pulse in the quadrature time signal from the channel 3 ‘L-mode 1’ case.
treated as real time signals. That is, all power spectra are
plotted over a positive frequency range. Frequency ranges,
over which the power spectra exhibit exponential behavior,
are identified and the logarithm of the power is fit to a straight
line (using IDL poly_fit) to extract characteristic times. If the
extracted characteristic time does not exceed twice the data
acquisition rate, it is not included in the database for the PDF.
To be precise, characteristic times smaller than 400 ns are not
included in the sample database. Some channels yielded no
results because there were no linear regions with characteristic time above 400 ns. These are the inner channels (typically
7–8) found to be in the stochastic region of the C-H plane. The
PDF displayed in figure 15(a) has a total of 144 samples.
The characteristic time distribution is very similar in shape
to pulse width distributions obtained in the TJ-K stellarator
(figure 5 in [38]), although the average characteristic time
for the DIII-D data is 4–10 times smaller than the average
studied, is presented in figure 15(a). Figure 15(b) shows an
explicit example of a single Lorentzian pulse in the DBS time
signals. Isolated pulses in the DBS signals are uncommon for
two reasons. First, the average time interval between pulses is
less than a factor of two larger than the average pulse width,
which means most pulses are overlapping. Second, pulses are
masked by Doppler shifts, which appear as coherent oscillations, at the Doppler shift frequency, in the DBS signals. The
example shown in figure 15(b) has a characteristic time of
775 ns, quite close to the mean value for the entire pulse width
distribution, 715 ns.
Due to the difficulty of extracting individual pulses from
the DBS signals, the characteristic times that comprise the
PDF displayed in figure 15(a), represent linear regions of
the power spectra of all L-mode cases plotted in a log-linear
format. The characteristic times are obtained from both the
‘in-phase’ and ‘quadrature’ power spectra for all channels
15
J E Maggs et al
Plasma Phys. Control. Fusion 57 (2015) 045004
pulse widths found in TJ-K. It is of interest to compare the
characteristic times to the ion gyrofrequency, fci, through the
product, τ fci. Langmuir probe data from the edge of TJ-K,
for the ‘high field’ case studied (0.244 T), yielded the range
(3.7 < τ fci < 5). Experiments on the linear LAPD device
have found τ fci = 4 for an ‘edge-limiter’ experiment and
τ fci = 6 for a temperature experiment [37]; both measurements were also obtained with Langmuir probes. For the
DIII-D cases investigated here, it is found that the value of
τ fci = 8.6, where the average of the gyrofrequency is over
radial location. The largest value of the product τ fci is only
about two times larger than the smallest value for all these
experiments. This range is relatively small as compared to the
wide variation of parameters, heating methods, and magnetic
configurations for all these experiments.
Sciences, using the DIII-D National Fusion Facility, a DOE
Office of Science user facility, under Awards DE-FC0204ER54698 and DE-FG02-08ER54984. DIII-D data shown
in this paper can be obtained in digital format by following
the links at https://fusion.gat.com/global/D3D_DMP. The
work of JEM and GJM is performed under the auspices of the
BaPSF at UCLA, which is jointly supported by a DOE-NSF
cooperative agreement.
References
[1] D’Ippolito D A 2011 Phys. Plasmas 18 060501
[2] Garcia O E 2009 Plasma Fusion Res. 4 019
[3] Budaev V P et al 2008 Nucl. Fusion 48 024014
[4] Zweben S J 2007 Plasma Phys. Control. Fusion 49 S1
[5] Sanchez E et al 2000 Phys. Plasmas 7 1408
[6] van Kampen N G 2007 Stochastic Processes in Physics and
Chemistry 3rd edn (Amsterdam: North-Holland)
[7] Schuster H G and Just W 2005 Deterministic Chaos
(Weinheim: Wiley-VCH)
[8] Brown R and Chua L O 1999 Int. J. Bifurcat. Chaos 9 785
[9] Hillsheim J C et al 2009 Rev. Sci. Instrum. 80 083507
[10] Luxon J L et al 2005 Fusion Sci. Technol. 48 828
[11] Hirsch M M, et al 2001 Plasma Phys. Control. Fusion
43 1641
[12] Conway G D et al 2008 Plasma Phys. Control. Fusion
50 124026
[13] Hennequin P et al 2004 Rev. Sci. Instrum. 75 3881
[14] Schmitz L et al 2008 Rev. Sci. Instrum. 79 10F113
[15] Hillesheim J C et al 2012 Phys. Plasmas 19 022301
[16] Estrada T et al 2012 Plasma Phys. Control. Fusion 54 124024
[17] Boedo J A et al 2001 Phys. Plasmas 8 4826
[18] Carreras B A et al 2001 Phys. Plasmas 8 3702
[19] Pedrosa M A et al 1999 Phys. Rev. Lett. 82 3621
[20] Rosso O A et al 2007 Phys. Rev. Lett. 99 154102
[21] Bandt C and Pompe B 2002 Phys. Rev. Lett. 88 174102
[22] Maggs J E and Morales G J 2013 Plasma Phys. Control.
Fusion 55 085015
[23] Gekelman W et al 2014 Plasma Phys. Control. Fusion
56 064002
[24] Luxon J L 2002 Nucl. Fusion 42 614
[25] Smirnov A P et al 2009 Proc. 15th Workshop on ECE and
ECRH (Singapore: World Scientific) p 301 (www.compxco.
com/genray.html)
[26] Peebles W A et al 2010 Rev. Sci. Instrum. 81 10D902
[27] Horton W 1999 Rev. Mod. Phys. 71 735
[28] Doyle E J et al 2007 Nucl. Fusion 47 S18–127
[29] Mandelbrot B B and Van Ness J W SIAM Rev. 10 422
[30] Ohtomo N et al 1995 J. Phys. Soc. Japan 64 1104
[31] Frisch U and Morf R 1981 Phys. Rev. A 23 2673
[32] Sigeti D E 1995 Physica D 82 136
[33] Davis A et al. 1994 J. Geophys. Res. 99 8055
[34] Lorenz E N 1963 J. Atmos. Sci. 20 130
[35] Strogatz S H 2001 Nonlinear Dynamics and Chaos
(Cambridge, MA: Westview Press)
[36] Rosso O A et al 2012 Physics A 391 42
[37] Pace D C et al 2008 Phys. Plasmas 15 122304
[38] Hornung G et al 2011 Phys. Plasmas 18 082303
[39] Maggs J E and Morales G J 2012 Phys. Rev. E 86 015401
7. Conclusions
The L-mode, edge plasma density fluctuation power spectra
presented here are found to be well fit by a Doppler-shifted
exponential (the ‘exponential tent’, equation (3)). In all cases,
the time signals associated with these power spectra have
locations in the C-H plane, and BP distributions, that indicate a strong chaotic nature. Taken as a whole, this evidence
indicates that the L-mode density fluctuation dynamics in the
edge of the DIII-D tokamak are chaotic. This finding sets constraints and provides guidance for theoretical and numerical
studies that aim to explain the turbulent nature of L-mode
plasmas.
Furthermore, when a clear DBS signal is returned from the
cutoff layer, it is found that the density fluctuations scattering
the beam are chaotic in nature even when located in the core
region. However, stochastic signals have been found for cases
in which the component of the DBS signal generated by scattering at the reflection layer, is very weak, or undetectable. For
the cases studied, all of the stochastic signals are associated
with the higher frequency channels that probe the core region.
Stochastic signals are presumed to result from noise generated
along the beam-path through the plasma. These signals do not
give information about density fluctuations at the reflection
layer and therefore are not used to draw conclusions about the
statistical nature of the fluctuations.
It is suggestive that it would be quite valuable to apply
the analysis techniques used in this study to other diagnostic
methods that sample fluctuations in confinement devices used
in fusion research.
Acknowledgments
This material is based upon work supported by the US Department of Energy, Office of Science, Office of Fusion Energy
16