Deterministic Chaos in Alcator C-Mod Edge
Turbulence
Victoria R. Winters
Submitted to the Department of Nuclear Science and Engineering
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Nuclear Science and Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
_______
MASSACHUSET Ts INStTU'rE
OF TECHNIOLOGY
June 2014
JUL 2 S 2014
LIBRA RIES
@2014 Victoria R. Winters. All Rights Reserved.
The author hereby grants to MIT permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document in whole or in
part.
Signature redacted
..........
........
...........
.....
Department of Nuclear Science and Engineering
May 9, 2014
redacted
.
Author....
Signature
Certified by ......
...............................................................
Anne White
Assistant Profess r of Ifclear Science and Engineering
*
W~
Thesis Supervisor
7.h
ignadU1
Accepted by ......
I
reac
1ure
e-
............ !..........................
Richard Lester
Professor of Nuclear Science and Engineering, Department Head
Thesis Reader
1
Acknowledgements
I would like to thank Professor Anne White for all her help with understanding basic
plasma physics, learning IDL, and overall guidance through my research, as well as being
a wonderful mentor throughout my undergraduate years at MIT.
2
Deterministic Chaos in Alcator C-Mod Edge
Turbulence
by
Victoria R. Winters
Submitted to the Department of Nuclear Science and Engineering
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Nuclear Science and Engineering
Abstract
Understanding the underlying dynamics of turbulence in magnetic confinement
fusion experiments is extremely important. Turbulence greatly reduces the confinement time of these devices and therefore greater knowledge of turbulent dynamics
can help with its mitigation. Experiments from the Alcator C-Mod tokamak [18]
provide support for a theory that edge turbulence in tokamak fusion plasmas is
the result of deterministic chaos, rather than stochastic processes [15]. Using readily available reflectometer data from Alcator C-Mod (C-Mod), analysis of C-Mod
edge turbulence in Ohmic plasmas and Ion Cyclotron Range of Frequencies (ICRF)
heated L-Mode plasmas shows that density fluctuations just inside or at the Last
Closed Flux Surface (LCFS) exhibit exponential power spectra. Theoretically, the
characteristic slope of the data on a semi-log plot gives the full width of the underlying Lorentzian pulses, which give rise to the exponential power spectra due
to the dynamics of deterministic chaos. Using a separate fitting routine, individual Lorentzian pulses in the reflectometer time series data are identified, and the
widths of the Lorentzian pulses match the inverse characteristic frequency of the
exponential power spectra. Analysis of the waiting times between pulses and the
pulse amplitudes indicate these are randomly distributed yet the pulse widths have
a narrow distribution. These characteristics are consistent with a chaotic process.
There is also a preliminary comparison of GPI data and a discussion of limitations
of the analysis presented here and plans for future work. Overall, the experimental
results in this study are consistent with edge turbulence that is at least partially
generated by chaotic dynamics.
Thesis Supervisor: Anne White
Title: Assistant Professor of Nuclear Science and Engineering
3
Contents
9
1 Introduction
9
9
10
11
12
12
3 Methods
3.1 Power Spectra ..........................
3.2 Time Series Data ........................
3.3 Relationships between Plasma Parameters and Characteristic Slope ....
13
13
14
15
4 Results and Discussion
4.1 Power Spectra and Connection to Pulses ...................
4.2 Trends among Plasma Parameters, Pulse Widths, and the Characteristic
Slope .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Comparison of Experimental Data to Past Experiments and Theory . . .
4.4 Overview of other Plasma Modes . . . . . . . . . . . . . . . . . . . . . .
15
16
5 Conclusion
27
.
.
.
.
2 Background and Previous Work
2.1 The Alcator C-Mod Tokamak .................
2.2 History of Searching for Universality in Turbulence.....
2.3 A Stochastic Process versus a Deterministic Chaos Process
2.4 The 0-Mode Reflectometer ...................
2.5 Plasma Regimes in Magnetic Fusion Systems . . . . . . . .
5
21
23
25
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A typical magnetic field shape for a tokamak. Image from [8} . . . . . . . .
homodyne reflectometry signals at low amplitude and high amplitude density fluctuations, in both cases homodyne most consistent with Langmuir
probe data. Figure from [6] . . . . . . . . . . . . . . . . . . . . . . . . . .
Power spectra of different simulated pulses. It is clear that only the power
spectrum from the simulated Lorentzian pulse gives a straight line on the
semi-log plot. Figure from [20] . . . . . . . . . . . . . . .. .. . .. .. ..
Comparison between a similar Gaussian versus Lorentzian curve. The key
distinguishing feature is the longer tails of the Lorentzian. . . . . . . . . .
The power spectrum for shot number 1120224028 (Ohmic) plotted on a
linear, semi-log and log-log graph along with its exponential fit, Ae-0"'.
Additional exponential power spectrum from shot 1120224019 (L-Mode)
plotted on a linear, semi-log and log-log graph along with its exponential
fit, Ae-0'. The peak shown at approximately 200kHz is noise. ......
The distribution of pulse widths found for experimental shot 1120224019,
with the highest amplitude being at around .0048. The characteristic slope
found from a separate fit to the power spectrum matches at .0048. . . . . .
An example from experimental shot 1110215001 that shows a Lorentzian
pulse identified in the time series and its exponential power spectrum. With
a pulse width of .00424 and a characteristic slope of .00419, the width of
the pulse almost matches the slope of the power spectrum. . . . . . . . . .
Two pulses found in the time series for experimental shot 1120224019, each
fitted to both a Lorentzian (red solid line) and a Gaussian (blue dashed
line). Both fits appear to be good, however it is only the Lorentzian fit
that captures behavior near the tails of the pulses. . . ............
A plot of the distribution of the absolute value of the pulse amplitudes.
The distribution appears random. . . . . . . . . . . . . . . . . . . . . . .
The time series data as a whole has a Gaussian probability distribution
function, with equally likely positive and negative fluctuations. . . . . . . .
The characteristic slope of the power spectra plotted against the line averaged density for the corresponding shot. A clear inverse relationship is
seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The characteristic slope plotted against: upper-left Ne, upper-right VNe,
lower-left Te and lower right VTe. It appears as though the local density
and local density gradient have a weak inverse relationship to the characteristic slope of the power spectrum. . . . . . . . . . . . . . . . . . . . . .
The distribution of waiting times for C-Mod experimental shot number
1120224019 plotted semi-log. The independent best fit is shown with a blue
dashed line, while the best fit line from the model proposed by Hornung
is shown as a solid green line. Although the amplitudes of the two fits are
different, it is clear that the characteristic slopes are nearly identical.
6
10
12
14
15
17
18
19
19
20
21
22
22
23
. . . 24
15
16
The power spectra for (a)an ELM-free H-mode, (b)an EDA H-Mode, and
(c) an I-mode. The ELM-free H-mode appears to be largely exponential, while the EDA H-mode is only exponential in the range from 300 to
700kHz. The I-mode appears to be exponential with a Gaussian riding on
top, consistent with [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A Comparison of power spectra of I-Mode and L-Mode. (a) Shows the
power spectrum at all times throughout the experimental run. The power
spectra dip at lower frequencies is only seen during the times during I-mode.
Parts (c) and (d) show I-mode power spectra during this time window.
They have a dip around 0-200kHz that is consistent with QCM and WCM.
From (b) and (e), it is seen that L-mode sections of this same run do not
have this feature. Figure provided by Arturo Dominguez [6]. . . . . . . . . 27
7
List of Tables
1
2
3
4
. .. . .*. .. 25
5
... ...
10
Typical C-Mod Parameters. Data from [18]........... .
Examples of the variety of shots analyzed. All shots with RF power are
L-Mode, and all without are Ohmic. All power spectra were calculated
using a 100ms time window and all pulses found in the time series data
were taken from that same 100ms time window. . . . . . . . . . . . . . . . 16
A comparison ofR2 values between the Lorentzian and Gaussian fits. It is
clear that both pulses provide great fits, however overall the Lorentzian fits
are slightly better. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Comparison of various other theories to the deterministic chaos model and
how both this study (Winters experiment) and the TJK Stellarator exper.
iment fit to the different theories. .
The reflectometer channels, where the numbers correspond to the number
locations in the MDS tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8
1
Introduction
As technology grows and is distributed throughout the world, the demand on the energy
grid continues to climb. Currently the primary fuel supplying power to the energy grid are
fossil fuels [1]. However, concerns over the large amounts of carbon dioxide produced by
these sources of energy and their effect on climate change have caused scientists to look for
newer and cleaner alternatives. While there are many such sources, such as geothermal,
wind, solar and water, nuclear sources are extremely attractive because of their enormous
energy density. On average, the fission of one uranium produces 210MeV of energy [91,
one fusion of two hydrogen atoms produces around 17.6MeV [9], while one combustion
reaction of methane produces only 8.4eV[5}.
Fission energy production has already been realized and implemented throughout the
world; however nuclear fusion energy production remains outside of society's grasp. Fusion
occurs naturally in stars and supernovae through these systems' huge gravitational field
that serves to both confine the plasma and keep it at a high enough density to overcome the
electrostatic Coulomb repulsion between ions needed to produce this reaction.On earth,
such a gravitational field is impossible. Therefore, one commonly used substitute is to
confine the plasma via magnetic fields. However, these magnetic fields cannot sustain an
extremely dense plasma, and so a less-dense but hotter plasma is created. One of the most
effective devices for containing this plasma is a tokamak, which confines the plasma in a
toroidal shape via a helically shaped magnetic field generated by both external magnets
and an internal plasma current.
Because no system is capable of perfectly confining a plasma and the plasma itself is
not perfect; the tokamak has its own instabilities and is turbulent. Turbulence causes
enhanced transport of particles and energy across magnetic field lines, which lowers the
time a plasma can be confined. Therefore, understanding the underlying mechanisms of
turbulent processes is imperative for its mitigation. The focus of this thesis is to look
for experimental evidence of a possible deterministically chaotic generation of turbulence
at or just inside the last closed flux surface (the edge plasma) of the Alcator C-Mod
tokamak. The experimental evidence required to support a deterministic chaos model is
an experimental power spectrum in edge density fluctuations caused by Lorentzian-shaped
pulses in the raw temporal fluctuation data. Should this theory be correct, it could support
a larger, more general underlying universal aspect of plasma edge turbulence.
2
2.1
Background and Previous Work
The Alcator C-Mod Tokamak
A tokamak is a toroidally-shaped device that uses a combination of external magnetic
fields and internal plasma current to create an overall helically-shaped magnetic field
used to confine high temperature plasmas(see Figure 1). There are two plasma regimes
inside a tokamak that are relevant to this thesis: Ohmic plasmas and L-Mode plasmas.
An Ohmic plasma is heated by the internal current inside the plasma alone, whereas an L9
Mode plasma is generated by using both ohmic heating and ion cyclotron radio frequency
(ICRF) heating.
Coils
ANWI
W0
Blanket
Plasma
_M
field line
Figure 1: A typical magnetic field shape for a tokamak. Image from [8]
Alcator C-Mod, the tokamak at MIT, was used to produce both the Ohmic and LMode needed for this study. Alcator C-Mod is a compact tokamak that contains plasmas
with high magnetic fields. Its parameters are given in Table 1
Table 1: Typical C-Mod Parameters. Data from [18]
Value
Parameter
8.1T
Maximum Magnetic Field
0.68m
Major Radius
0.22m
Minor Radius
Maximum Plasma Current 2.01MA
8MW
ICRF Source Power
2.2
History of Searching for Universality in Turbulence
The origin of the search for universality of turbulence across different systems began with
Kolmogorov's 1941 pioneering paper, which predicted that the energy of a turbulent system with a very high Reynolds number is proportional to the wavenumber to the -5/3
power when plotted on a log-log scale [13]. Since then, the power spectra of turbulent systems, including fusion systems, have been traditionally plotted in the log-log format and
fit to power laws. However, it is commonly seen that the power spectra from turbulence
in plasmas cannot be fit with one power law. Often, they are fit to several different power
laws [21].
It was recently proposed by Dr. Maggs and Dr. Morales at UCLA that turbulent power
spectra from tokamak edge plasmas are exponential and are caused by the presence of
10
Lorentzian pulses in edge density fluctuations, and that these Lorentzians are a signature
of deterministic chaotic generation of edge turbulence [15]. Experiments at the Large
Plasma Device Upgrade (LAPD-U) at UCLA have shown that exponential power spectra
in plasmas can be generated by a series of Lorentzian pulses in the time series data [15] [20]
and can arise from nonlinear interactions of electromagnetic drift waves [24]. In addition,
experiments at the low magnetic field TJK stellarator in Germany have also identified
the presence of Lorentzian pulses and exponential spectra in the edge[12]. If tokamak
edge turbulence is a deterministic and chaotic rather than stochastic and random, it
could significantly alter the current theoretical, simulation and experimental approaches
currently used to study and predict edge turbulence in tokamaks.
2.3
A Stochastic Process versus a Deterministic Chaos Process
A stochastic process is by definition random, meaning that there is no way of exactly
predicting how that process will behave. It can be described by probabilities and nothing
more. An example of a stochastic process is radioactive decay, where a certain element
has a probability of decaying within a certain amount of time. However, it is impossible
to correctly predict exactly when each decay will happen.
A deterministic chaotic process, however, is different. This process in theory is governed by a set of differential equations, thereby making it completely deterministic. However, it is extremely sensitive to initial conditions, thereby making the process in reality
completely unpredictable [11]. A good example of this kind of system is a double pendulum, where there is a governing set of differential equations but, with completely relaxed
assumptions, it is impossible practically to predict exactly its trajectory at every point in
time [28].
Lorentzians and the Power Spectrum
According to the recently proposed theory by Maggs and Morales [15], Lorentzian-shaped
fluctuations (Lorentzians) are a signal of deterministic chaos in the edge turbulence of
tokamak plasmas. The Lorentzians must also have a narrow distribution of widths. If
this is the case, then the Lorentzians should have a significant effect on the shape of the
power spectrum of the edge turbulence. A power spectrum of the turbulence is created by
performing a fourier transform on the temporal data into frequency space. A Lorentzian
is described by the following equation temporally [29]:
1
.51'
L(t) = 1(1) to)2 + (.5r)2
7(t where P is the full width at half maximum of the Lorentzian, and to is the center of the
pulse. The fourier transform of a Lorentzian is of the form [20]:
F(w) oc exp(-w/wda,)
(2)
Where w is the frequency, wh, is the characteristic frequency (the slope on a semilog portion of the graph), and the inverse of wch., is the full-width of the Lorentzian.
11
Therefore, if Lorentzian pulses in the timeseries indeed have a large effect on the power
spectrum, one expects the power spectrum to be exponential in shape with an inverse
characteristic frequency equal to the full width of the pulses found in the time series data.
2.4
The 0-Mode Reflectometer
The O-Mode polarized reflectometer diagnostic on Alcator C-Mod was used to study the
plasma edge density fluctuations. The reflectometer is capable of measuring fluctuations
by sending a polarized probing electromagnetic wave of a given frequency into a plasma.
The polarization and frequencies are chosen such that the wave reaches a propagation
cutoff in the plasma and is then reflected back towards the detection antenna [6]. Because
the index of refraction of the plasma is a function of magnetic field and density, one can
determine density fluctuations knowing the value of the magnetic field and the relation
between the transmitted and received waves.
For the purposes of this thesis, only the homodyne signals were used and analyzed,
because the homodyne signals most closely match Langmuir probe measurements [6], see
Figure 2.
a)
S*'
b)
80.0000 ms
100.000 me
ISI EIodyne
1
10
1000
freueny
100
cIWay
(~z)frt
1000
(MIL)
Figure 2: homodyne reflectometry signals at low amplitude and high amplitude density
fluctuations, in both cases homodyne most consistent with Langmuir probe data. Figure
from [6]
2.5
Plasma Regimes in Magnetic Fusion Systems
While there are a variety of different types of plasma regimes in magnetic fusion systems,
the two most important in this study are Ohmic modes and L-modes. Ohmic mode
plasmas are heated purely by the current induced inside the plasma in a tokamak. Lmode plasmas are heated both through Ohmic means, but also by radio-frequency (RF)
heating.
There are 3 other plasma modes which were looked at in this study, but due to time
constraints were not capable of being studied thoroughly. These modes are EDA H-Mode,
the ELM-Free H-mode and the I-mode. The ELM-Free H-mode is a highly confined mode
12
that contains no Edge Localized Modes (ELMs) [10]. These ELMs are an instability that
causes sudden bursts of particles and energy out from the edge of a plasma [17]. The EDA
H-Modes are similar to ELM-Free H-Modes, and typically don't contain ELMs either,
but they have slightly lower energy confinement than ELM-Free H-modes but reduced
impurity confinement [10]. The I-mode is an improved confinement regime (compared to
Ohmic and L-Mode plasmas) that does not have the build-up of impurities that H-Mode
plasmas have [17]. I-Modes and H-modes, unlike Ohmic and L-modes, tend to have various
coherent modes that affect the shape of the power spectrum.
3
Methods
Alcator C-Mod has a large library of Reflectometer data from many different types of
plasmas from previous years. Data samples that were either Ohmic or L-mode plasmas
were selected from the library and analyzed to determine whether or not there is support
for the deterministic chaotic theory proposed at UCLA.
3.1
Power Spectra
In order to calculate the power spectra, the raw homodyne fluctuation data from the
reflectometer was imported into IDL, a vectorized programming language used for data
analysis, and was then fourier transformed and graphed versus frequency on a linear plot,
a semi-log plot and a log-log plot. Each power spectrum was then fit using the CURVEFIT
routine in IDL to an exponential curve. The range of frequencies over which this curve fit
well to the data was then evaluated and compared to the range of frequencies over which
a power law fit. In Figure 3 are examples of different types of simulated pulses and their
power spectra. Only the power spectrum from the simulated Lorentzian pulse yields a
straight line on the semi-log plot.
The slope of the power spectra was then recorded. This slope was recorded as 1/fha,
the inverse of the characteristic frequency. This slope was then compared to the full
width at half maximum of the pulses found in the time series data. According to the
deterministic chaotic model, this characteristic slope would be equal or near the half
maximum of the pulses.
13
W/Ai
40
0.79
0.53
0.26
10
Lorentzian
Gaussian
.
10
sech 2
10
10 0
100
200
f (kHz)
300
Figure 3: Power spectra of different simulated pulses. It is clear that only the power
spectrum from the simulated Lorentzian pulse gives a straight line on the semi-log plot.
Figure from [20]
3.2
Time Series Data
In addition to analysis of the power spectra of specific experimental shots (with each run
of Alcator C-Mod called a "shot"), the raw density fluctuation data from the reflectometer
was analyzed. Because pulses are often buried in the time series data, candidate pulses
were found by manually going through the data. Once found, a separate CURVEFIT
procedure was run to fit the candidate pulse to a Lorentzian curve. The full width at half
maximums were recorded, as well as the amplitudes. The full widths were then compared
to the characteristic slopes of the power spectra.
Comparison to Gaussian pulses
Because Lorentzian and Gaussian curves look very similar (see Figure 4), it is important
to fit the pulses found in the raw time series data to both Lorentzian and Gaussian curves
to see whether or not Lorentzians indeed provide a better fit. Therefore, after a Lorentzian
fit was conducted, the pulses were fit indepedently to a Gaussian curve using the function
GAUSSFIT in IDL. The R2 value for each curve was then compared to see which had the
better fit.The R2 value is determined by Equation 3 [4]:
R=
1 -(y
- fi) 2
E(y, - y)2
(3)
where y is ith experimental data point, fi is the value produced by the fitted function,
and y is the average value of the observed data.
14
-Gaussian
Lorentzian
0.90.80.70.6'U
0.5L
0.40.30.20.1
0
-5
0
arb. units
5
Figure 4: Comparison between a similar Gaussian versus Lorentzian curve. The key
distinguishing feature is the longer tails of the Lorentzian.
3.3
Relationships between Plasma Parameters and Characteristic Slope
In order to determine possible relationships between the characteristic slope of the power
spectra and the plasma parameters, dwscope, a shot analysis tool used to see the values
of plasma parameters for a given shot, was used to record the magnetic field, the plasma
current, and the line averaged density for each experimental shot. In addition, the local
density, the local density gradient, the local electron temperature and the local electron
temperature gradient were also recorded from the Thomson Scattering diagnostic system
[3].
These parameters were then plotted for each experimental shot with the characteristic
slope in order to determine the relationship between the parameter and the slope of the
power spectrum, if any.
4
Results and Discussion
In total, approximately 45 (half Ohmic and half L-mode plasmas) shots were studied and
analyzed at Alcator C-Mod. A sample of some of these are seen in Table 2
15
Table 2: Examples of the variety of shots analyzed. All shots with RF power are L-Mode,
and all without are Ohmic. All power spectra were calculated using a 100ms time window
and all pulses found in the time series data were taken from that same 100ms time window.
Shot
1120224019
1110201004
1110309012
1110309013
1110215001
1110201009
1120224028
1110201026
1120224030
1120224015
T1(ms) I T2(ms)
1000
1100
880
980
1300
1400
1000
1100
800
900
500
600
900
1000
770
870
900
1000
850
950
Bt(T)
5.25
5.39
-5.76
-5.77
-5.85
5.39
4.6
5.39
4.6
5.98
Te local(eV)
185
120
180
800
190
176
126
138
87
130
RF(MW)*
1.88 2.18
1.6
3.64 3.967
3.25 3.56
.6
0
0
0
0
0
NL..04 (m-3)
1.1e20
1.5e20
1.78e20
1.57e20
1.24e20
1.34e20
1.58e20
1.03e20
1.3e20
1.01e20
Radius=()
.878 .0095
.890 .00098
.883 .0024
.877 .0068
.880 .0046
.89 .0038
.854 .0077
.882 .0032
.867 .0037
.876+.0024
.
Overall, in the plasmas studied at C-Mod, RF heating ranged from 0.6 < Pap < 4.0
MW, the magnetic field ranged from 4.6 < Bt < 5.86 T, and the average density ranged
from 1 x 102 < (ne) < 1.8 x 10" m-3
4.1
Power Spectra and Connection to Pulses
Of the 45 Ohmic and L-mode shots studied, all of them when plotted semi-log was a
straight line, consistent with exponential behavior. As examples, in Figures 5 and 6, the
power spectrum is plotted on all three types of graphs and its fit is overplotted. It is clear
to see that the exponential curve fits over a broad range of frequencies. In general, the
exponential provided a good fit for frequencies ranging from 200 < f < 800 kHz. One
possible explanation for the poor fit at high frequencies is that the signal falls below the
noise floor of the reflectometer. Therefore, the spectrum flattens to the noise floor at these
higher frequencies. A possible explanation for the poorer fit at frequencies below 10kHz
is that the reflectometer is picking up MHD fluctuations in addition to the broadband
turbulence.
16
1120224028, ti =900, t2=1000
0.008
,
0.006
0.004
0.002
a.
0.000
0
200
400
600
800
1000
frequency (kHz)
0.010
L0.001--
0
400
600
hreuency (kHz)
200
800
0.1000
E 0.0100
0.0010
0.0001
0.1
10.0
1.0
100.0
1000.0
Figure 5: The power spectrum for shot number 1120224028 (Ohmic) plotted on a linear,
semi-log and log-log graph along with its exponential fit, Ae.0 39w
In addition to discrepancies at very low and very high frequencies, occasionally a large
spike in the power spectrum is seen at approximately 200kHz. This spike can be seen in
Figure 6 and is well documented noise in the reflectometer [6].
17
1120224019, t1 1000, t2=1100
0.008
E
0.006L
0.004
CL
0.000-
0
200
400
600
frequency (kHz)
800
1000
0
200
400
600
frequency (kHz)
800
1000
0.0100
CU 0.0010
1C
-
CO0.01
U
I
e
W3
0.1
1.0
10.0
frequency (kHz)
100.0
1000.0
Figure 6: Additional exponential power spectrum from shot 1120224019 (L-Mode) plotted
on a linear, semi-log and log-log graph along with its exponential fit, Ae004"'. The peak
shown at approximately 200kHz is noise.
Using the same 45 experimental shots with these exponential spectra, Lorentzian fits
were calculated to candidate pulses in the raw fluctuation data. Because the exponential
spectrum has one characteristic slope, it is necessary to have a narrow distribution of
pulse widths with the same value as the characteristic slope in order to have them be the
dominant factor in the power spectrum. As shown in Figure 8, the pulses found in the
time series data have similar widths to the characteristic slope found through the separate
fitting to the power spectrum.
In the cases analyzed, there was a narrow distribution of pulse widths, with the distribution centered around the characteristic slope of the spectrum, as in Figure 7.
18
IO~5
OF .
0.0030
....
B
0.0006
0.0040
00045
0.0050
0.0055
Figure 7: The distribution of pulse widths found for experimental shot 1120224019, with
the highest amplitude being at around .0048. The characteristic slope found from a
separate fit to the power spectrum matches at .0048.
1110215001
0.2
0.0
C
'0
-0.2
Pulse width: .00424
-0.4 1- .801434 .801435 .801436 .801437
.1
.801438
.801439
time (s)
cdE
0.010
ts
CL
U,
a)
0
a-
0.001
1/fchar= .00419
0
200
400
600
800
frequency (kHz)
Figure 8: An example from experimental shot 1110215001 that shows a Lorentzian pulse
identified in the time series and its exponential power spectrum. With a pulse width of
.00424 and a characteristic slope of .00419, the width of the pulse almost matches the
slope of the power spectrum.
A sample of the >100 pulses found were also fit to Gaussian curves to determine
19
.
which curve provided a better fit to the data. Overall, although both fits gave good fits,
as can be seen from Figure 9, the Lorentzians provide a slightly better fit. Because of the
sampling rate of the reflectometer, it is hard to determine whether the very good fit to
the Gaussian is truly because the pulses could be equally Gaussian or Lorentzian or if it
is because there is not a good resolution on the tails of the pulses. Overall, the Lorentzian
provides a better fit specifically near the tails of the candidate pulses. Table 3 provides a
sample of some of the R2 values of each pulse for experimental shot number
1120224019
0
00
0.10
0a
0.05- 000
M
I:
00
0
0
0
0
0
0
wf
C
0
0%
36
00
00
0
-0.05
1.00135
1.00136
1.00137
0.15.
0.1003
00
0.05-
i
0.00
0
'coj
0
0
0
P
b0
pFj
C3:q7~
0P
0
rip.
-0.101
1.001385 1.001390 1.001395 1.001400 1.001405 1.001410 1.001415 1.001420
time(s)
Figure 9: Two pulses found in the time series for experimental shot 1120224019, each
fitted to both a Lorentzian (red solid line) and a Gaussian (blue dashed line). Both fits
appear to be good, however it is only the Lorentzian fit that captures behavior near the
tails of the pulses.
Table 3: A comparison ofR2 values between the Lorentzian and Gaussian fits. It is clear
that both pulses provide great fits, however overall the Lorentzian fits are slightly better.
Lorentzian R Gaussian R2 time(ms)
.998
.998
1001.0952
.992
.925
1001.3585
.995
.981
1001.4015
20
4.2
Trends among Plasma Parameters, Pulse Widths, and the
Characteristic Slope
After finding both exponential spectra and a narrow distribution of pulse widths at or near
the characteristic slope of the spectra, an investigation was made to determine if there
were any relationships between the characteristic slopes and the plasma parameters. In
addition, the amplitude and width distributions of the temporal pulses were analyzed.
A plot of the distribution of the absolute value pulse amplitudes was made. Figure 10
shows that the pulses have no bias toward a certain amplitude. In addition, the amplitudes
are such that they are buried in the time series data, where the fluctuations can be equal
or greater in amplitude. One possible reason for the lack of large and small amplitude
pulses is that they are obscured because of the small sampling rate of the reflectometer.
The pulses themselves are equally likely to be negative or positive, consistent with the
time series data as a whole which has a Gaussian probability distribution function (see
Figure 11).
8
6
4
0
0
.00
13
1
E
2
0
0.15
0.20
0.25
0.30
Amplftude (&u.)
0.35
0.40
Figure 10: A plot of the distribution of the absolute value of the pulse amplitudes. The
distribution appears random.
21
a
I.
-.
.
0.3
0.2
0.1
0.0
-0.1
~1 U~PVIFT
.-
1.02
1,00
1.06
1.04
1.08
1.10
time(s)
Figure 11: The time series data as a whole has a Gaussian probability distribution function, with equally likely positive and negative fluctuations.
The characteristic slope of each power spectra was plotted against the line average
density for the plasma, shown in Figure 12. There is a clear inverse relationship between
the characteristic slope and the line averaged density.
x10A4
I,
a
a
5.5
5 *
a
a
a
OE~evimutOga
a
a
a
aa
a
aa
a
a
a
a
a
a
a
a
a
3
2
a
2S
a
2
a
__-10
U
12
13
1
Line AynWg Oensky 91 partksae
1is
16
17
Wigl)
Figure 12: The characteristic slope of the power spectra plotted against the line averaged
density for the corresponding shot. A clear inverse relationship is seen.
After finding the inverse relationship between the characteristic slope and the line averaged density, the local density (Ne), the local density gradient (VNe), the local electron
temperature (Te) and the local electron temperature gradient (VTe) were also investigated. From Figure 13, there appears to be a weaker correlation between the characteristic slope and Ne and VNe. However, there is no correlation between the characteristic
slope and Te or VTe.
22
00
o
[00
r
ra
13
0.04
]
~~
:0
0.
*
0.0021
1.0
1.5
U.
LOCal No M9% *Au VMSacu
000I- - - - - - -
oM
M.004
Oa
0
G0
--oOMk
o
Oar'
-
on00
0.006
200
2.0
-
0.0
0
0.
0~o
0mom
50
100
Local Ta
150
200
250
300
0
(95% kmamc
__
__
_
50
100
10
Local Gad T (95% &iawda
200
)
0
_
Figure 13: The characteristic slope plotted against: upper-left Ne, upper-right VNe,
lower-left Te and lower right VTe. It appears as though the local density and local
density gradient have a weak inverse relationship to the characteristic slope of the power
spectrum.
4.3
Comparison of Experimental Data to Past Experiments and
Theory
According to the proposed deterministic chaos theory of [15], the characteristic slope
obtained from an exponential fit to the experimental power spectrum should correspond
to the average full width of a collection of Lorentzian pulses that would occur in the
fluctuation time series data. We do see such a correspondence in our data set. According
to the theory proposed by Maggs and Morales [16], a key feature of the deterministic chaos
model is a narrow distribution of pulse widths. Similar to the results found in the TJK
stellarator experiment, the distribution of pulse widths in the C-Mod tokamak is narrow,
with a sharp peak at the value of the characteristic pulse width found from the power
spectrum data fitting routine. Also consistent with theory, it appears that individual pulse
amplitudes in the C-Mod data are random; however more data are needed to confirm this
absolutely. Due to the sampling rate and the noise of the reflectometer, pulses with
extremely high and extremely low amplitudes are harder to find manually.
There are strong similarities between the TJK Stellarator data and the data from the
C-Mod tokamak. The distribution of waiting times between individual Lorentzian pulses
in C-Mod was found to exhibit an exponential shape, consistent with what was found for
the TJK Stellarator. Using an independent fitting routine to the exponential shape, the
characteristic slope of the exponential was found to be 12.71kHz.
In addition to the independent fit of the C-Mod data, an additional fit was done using
23
the model Hornung proposed in [12], of the form:
(4)
PDF(At) ~-fwte-flAt
Where f,, is approximately the inverse of the average waiting time between the pulses, and
is Equation 5 in [12]. The plot predicted by the Hornung model gave a characteristic slope
to be 12.918kHz, extremely close to the independent fit (see Figure 14 . The characteristic
slope is very close to the average of the individual waiting times.
Furthermore, the distribution of pulse waiting times, and the distribution of pulse
widths found in C-Mod are similar to those found in the TJK Stellarator, which is surprising for two devices of vastly different parameters. Both C-Mod and TJK found an
exponential dependence on the distribution of waiting times between pulses, with the
slope of the distribution roughly equal to the average waiting time between two individual pulses. However, the TJK Stellarator results showed a dependence of pulse widths on
magnetic field, but the C-Mod data show no such relationship. Interestingly, the typical
widths of Lorentzian pulses found in the edge plasma at C-Mod are of the same order
of magnitude as pulses found in the TJK Stellarator and in the LAPD device (a linear,
non-fusion plasma device) [20].
100..........
0.00
0.02
0.04
0.06
0.08
0.10
0.12
waiting time (ins)
Figure 14: The distribution of waiting times for C-Mod experimental shot number
1120224019 plotted semi-log. The independent best fit is shown with a blue dashed line,
while the best fit line from the model proposed by Hornung is shown as a solid green line.
Although the amplitudes of the two fits are different, it is clear that the characteristic
slopes are nearly identical.
In addition to comparison with the TJK Stellarator Case, the results found in this
study were compared with other existing edge turbulence theories, namely the Deterministic Chaos theory [16], a dissipation range cascade theory proposed by Terry [26] and a
self organized criticality theory proposed by van Milligen et al [19]. The results of these
comparisons are shown in Table 4.
24
Table 4: Comparison of various other theories to the deterministic chaos model and how
both this study (Winters experiment) and the TJK Stellarator experiment fit to the
different theories.
Maggs[15]
Terry26]
van Milligen(19]
(deterministic
chaos)
(dissipation range
cascades)
(Self Organized
criticality)
Winters
experiment
(tokamak)
Hornung[12]
experiment
(stellarator)
Lorentzian pulses
n/a
Lorentz./Gauss.
yes
yes
Exponential
spectra over
broad range of
frequency
LP width
Exponential
spectra over
broad range of
frequency
LP width
in time series
Power spectra
are exponential
LP width
Power spectra are
exponential at high-f
power law at low-f
pulses in time series
Power spectra are
exponential
n/a
no relation
matches
matches 7
n/a
Pulse widths
randomly distr.
waiting times
narrow
distribution
waiting times
narrow
distribution
waiting times
random
exponentially
exponentially
n/a
distributed
pulse amplitudes
distributed
n/a
matches
LP widths narrowly
distributed
n/a
n/a
n/a
n/a
randomly
distributed
4.4
Overview of other Plasma Modes
In addition to the Ohmic and L-modes studied in depth here, a cursory look at both
I-Mode and H-Mode plasmas was also done. Some examples of the power spectra are
shown in Figure 15. Although the ELM-free H-mode appears to be largely exponential
here, other examples did not appear to be exponential and the shape of any H-mode
power spectrum varied widely. The shape of the I-mode power spectrum from this study
can be compared to the shape shown in Figure 15c. The dip in the power spectra from
0 < f < 200 is consistent with previous results from the reflectometer at Alcator C-Mod
[6]. In [6], Dominguez explains the Gaussian dip as the quasi-coherent mode (QCM) and
weakly coherent mode (WCM), whose features in a power spectrum are expected to be
seen in this range of frequencies, riding on top of a background exponential (see Figure
16).
25
10~
E
2
0
O10
0
200
400
600
frequency (kHz)
800
1 0
200
400
600
frequency (kHz)
800
1
200
400
600
frequency (kHz)
800
1000
E
14
a
0
0d
E
00
10
2
10
-
10
o10
0
Figure 15: The power spectra for (a)an ELM-free H-mode, (b)an EDA H-Mode, and (c)
an I-mode. The ELM-free H-mode appears to be largely exponential, while the EDA
H-mode is only exponential in the range from 300 to 700kHz. The I-mode appears to be
exponential with a Gaussian riding on top, consistent with [6].
26
10-3
400
3000
0
N
g200
100
0
0..5
33
1.0
time[s]
(b) L-mode
1.5
l6
(c)I-mode
-2
t=0.40s
5-2
-5
0
500
(d) I-mode
t =0.82 s
0
Data
Fit
Freq[kHz]
Data
Fit
Freq[kHz
-5
0
=-3
-
.3 0
Data
Fit
Freq[kHz
500 Im
0
500
(e) L-mode
t = 1.50 s
Data
Fit
Freq[kHz
500
Figure 16: A Comparison of power spectra of I-Mode and L-Mode. (a) Shows the power
spectrum at all times throughout the experimental run. The power spectra dip at lower
frequencies is only seen during the times during I-mode. Parts (c) and (d) show I-mode
power spectra during this time window. They have a dip around 0-200kHz that is consistent with QCM and WCM. From (b) and (e), it is seen that L-mode sections of this same
run do not have this feature. Figure provided by Arturo Dominguez [6].
Because of time constraints, it was not possible to analyze in-depth any other plasma
modes besides the Ohmic and L-modes described previously.
5
Conclusion
Using the edge density fluctuation data from the reflectometer diagnostic taken just inside
or at the Last Closed Flux Surface, an analysis of Ohmic and L-Mode plasmas in the
Alcator C-Mod tokamak was conducted to search for evidence of exponential spectra and
Lorentzian pulses. The power spectra of density fluctuations clearly exhibit an exponential
shape. Lorentzian-shaped pulses are identified in the time series data as well. In all
plasmas analyzed, the widths of the Lorenzian pulses found in the time series are found to
match the characteristic width extracted from the slope of the exponential power spectrum
data plotted semi-log. The Lorentzian pulse width is shown to vary most strongly with the
line averaged density, although a weak correlation with local densities and local density
gradients is also observed. The pulse width does not depend on magnetic field, local
electron temperature, and local temperature gradient. The widths of the Lorentzian pulses
are narrowly distributed, with a sharp peak at the value of the characteristic width, but
27
the pulse amplitudes are randomly distributed. The waiting times between Lorentzian
pulses follow an exponential distribution. Overall, the observations of Lorentzian pulses
and exponential spectra in reflectometer edge fluctuation data from the C-Mod tokamak
are consistent with measurements of edge plasma turbulence made at the TJK Stellarator
[121 and the LAPD plasmas [20], and are consistent with the deterministic chaos theory
proposed by Maggs and Morales [16].
If the deterministic chaos theory is correct, as evidence from this study seems to suggest, there could be profound implications for the way edge turbulence theory is conducted
for plasma systems. Understanding the underlying damage of at least part of the edge
turbulence in a fusion system increases knowledge of the system as a whole and could
therefore lead to better, more accurate modeling. A more in-depth knowledge of the
underlying dynamics of the turbulence combined with more accurate modeling can help
with increasing the confinement time in a magnetic confinement device.
There are limitations and uncertainties in the present data and analysis that warrant
discussion. The main limitation is that only a relatively small data set has been analyzed,
and only L-Mode and Ohmic plasmas were studied in detail. A preliminary analysis of
I-mode data has been done. Data from I-Mode plasmas show that the power spectra are
best fit by a sum of an exponential (broadband turbulent background) and a Gaussian
(the Weakly Coherent Mode (WCM) feature). Given issues with automated pulse finding
methods [20], we chose to use a brute-force approach to find and fit pulses in the raw time
series. As a result, we analyzed all data sets by hand, and only analyzed 45 discharges,
which results in a data base consisting of a few hundred pulses. In future work, adding
an automated fitting routine could uncover more pulses, because it would allow a larger
set of C-Mod plasmas to be examined quickly. Building a larger data base of pulses will
improve the ability to identify the distributions of pulse characteristics (e.g. widths, amplitudes, waiting times). Another limitation is that only reflectometer data was analyzed
in detail. Some effects of the limited time resolution of the reflectometer on pulse fitting
was identified which needs to be better understood. It was found that pulses fit with a
Lorentzian shape are also reasonably well fit by a Gaussian shape, which could be caused
by the low sampling rate of the reflectometer data, and using Langmuir probe data could
help better discriminate between pulse shapes.
In terms of comparing reflectometer results with other fluctuation diagnostics, it was
found that in all plasmas where GPI is available, the GPI measured density fluctuation
power spectra are exponential. But we have not yet searched for Lorentzian pulses in
the GPI time series data. Additionally, there is a limited number of shots where GPI
and reflectometer are available simultaneously, making it difficult to directly compare the
characteristic frequencies of the exponential spectra. From the limited data set, there
are apparent differences between the GPI spectra characteristic frequency and the reflectometer spectra characteristic frequency. More work is needed to understand how this
difference may be related to differences between the diagnostics (measurement location,
time resolution, spatial resolution, wavenumber response, frequency response, etc.). An
important next step will be to make detailed comparisons among several edge fluctuation
diagnostics in dedicated experiments: reflectometer, Langmuir probes, new Mirror Probe
[14}, and GPI. Consistency among these diagnostics would provide better evidence that
28
the Lorentzians and power spectra studied here are indeed produced by the plasma itself,
and not just specific to the reflectometer. In the future, a search for exponential spectra
and Lorentzian pulses should also be conducted using I-Mode and H-Mode data from
C-Mod, and in Ohmic and L-Mode plasmas across a wider range of plasma parameters,
in order to determine whether or not these Lorentzian phenomena and exponential power
spectra are seen universally across all plasmas, or just in the special cases described in
this study.
29
References
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J. A. Casey et al. "Construction of a two-dimensional Thomson scattering system
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A. Coster. Goodness-of-Fit Statistics. Date Accessed: 5 May 2014. URL: http: //
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A. Courtney. Energy Prom Fossil Fuels. Date Accessed: 14 Mar 2014. 2005. URL:
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A. Dominguez. "Study of Density Fluctuations and Particle Transport at the Edge
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[7]
E. J. Doyle. "Chapter 2: Plasma Confinement and Transport". In: Nucl. Fusion
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[8]
M. Farrell. Tokamak: Future of Nuclear Power. Date Accessed: 2 Apr 2014. Dec.
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R. E. Faw and J. K. Shultis. Fundamentals of Nuclear Science and Engineering.
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M. Greenwald, J. Rice, and R. Boivin et al. H-Mode Regimes and Observations of
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http://www.psfc.mit. edu/-g/papers/iaea98-paper .pdf.
[11]
F. Heylighen. Deterministic Chaos. Date Accessed: 2 Apr 2014. 2002. URL: http:
//pespmc1.vub. ac .be/CHAOS.html.
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G. Hornung et al. "Observation of exponential spectra and Lorentzian pulses in the
TJ-K stellarator". In: Phys. Plasmas 18.082303 (2011).
[13]
A. N. Kolmogorov. "The local structure of turbulence in incompressible viscous fluid
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[14]
B. LaBombard and L. Lyons. "Mirror Langmuir Probe: a technique for real-time
measurement of magnetized plasma conditions using a single Langmuir electrode".
In: Rev. Sci. Instrum. 78.7 (2007), p. 073501.
[15]
J. E. Maggs and G. J. Morales. "Generality of Deterministic Chaos, Exponential
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[16]
J. E. Maggs and G. J. Morales. "Origin of Lorentzian Pulses in Deterministic
Chaos". In: Phys. Rev. E 86.015401 (2012).
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E. Marmar. "I-mode Powers Up on Alcator C-Mod Tokamak". In: Y12.00003. American Physical Society. 2011. URL: http: //www. aps . org/units /dpp /meetings/
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[19]
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J. J. Podesta, D. A. Roberts, and M. L. Goldstein. "Spectral exponents of kinetic and
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T. L. Rhodes et al. "Signal amplitude effects on reflectometer studies of density
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P. W. Terry. "Suppression of Turbulence and Transport by Sheared Flow". In: Rev.
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[26]
P. W. Terry et al. "Dissipation range turbulent cascades in plasmas". In: Phys.
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G. R. Tynan et al. "Turbulent-driven low-frequency sheared ExB flows as the trigger
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[27]
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31
Appendix
Both IDL and MATLAB were used to analyze the data and create the images used
in this thesis. All files and data used to make the images can be found in vwinters/UROP.Winters/UROP-Winters.
Figures Created in IDL
All figures that required the use of data stored on the C-Mod servers were created
in IDL through the methods that follow. For all procedures that require the use of
refLstructure.pro, there are several unnecessary output graphs that are made. However,
the graphs that each code were made to output are described in detail in each section. In
addition, they are saved on the user's home directory in a png format.
Power Spectrum Graphs
All figures illustrating the power spectra were created with the procedure pngtest.pro.
In order to run this procedure, two other procedures are needed for the code to function properly: refLstructure.pro and simple-png.pro. In addition, one code, known as
gfunct.pro will be needed in order to provide curve fitting. Running the procedure requires two initial inputs that indicate which frequency channel of the reflectometer to use.
The following numbers correspond to the different frequency channels used:
Table 5: The reflectometer channels, where the numbers correspond to the number locations in the MDS tree.
Reflectometer Frequency(GHz) Number Number2
75
'05'
'06'
88.5
112
'09'
'11'
'10'
'12'
Choosing which frequency of the reflectometer to use depends on the density of the
plasma on a particular shot. Lower density plasmas will have lower cutoff frequencies,
therefore a lower frequency of the reflectometer must be used in order to see the edge
plasma (see section 2.4). To run the code, first start idl in the command prompt. Once
idl is initialized, type the following command:
pngtest, a, b
Where a and b refer to Numberi and Number2 in Table 5, respectively.
Once the code is initially run it will ask for several inputs. It will prompt the user for
information regarding the desired experimental shot number, the time range over which
the power spectrum should be calculated and the maximum power spectrum frequency
to plot. Once these inputs are given an image that contains 3 graphs will be outputted:
The power spectrum on a linear-linear plot, the same spectrum on a semi-log plot and
32
on a log-log plot. Each of these plots is fit to an exponential curve which is plotted over
the black experimental points with a dashed red line. In addition to the graphs, in the
terminal window there will be two outputs: the fitted parameters and the status of the fit
test. The fitted parameters will be ordered as: amplitude, slope. The amplitude is not a
physically important parameter, but the slope is used as comparison for the width of the
pulses found in the raw time series data.
Reading the Raw Time Series Data
Before fitting Lorentzian pulses in the raw time series data, this thesis required investigating the raw time series data by eye in order to find candidate pulses. This is done
by using plot.timeseries.pro. plot-timeseries.pro also needs refl-structure.pro in order to
function, since it is through refLstructure.pro that the reflectometer data is accessed.The
command to run the procedure is as follows:
plQt-timeseries, a, b, x1, x2
Where a and b are the two numbers used to specify the reflectometer channel and x1 and
x2 is the time (in milliseconds) for which the user wishes to plot the time series data.
Because plot..timeseries.pro uses the same base code for calculating the power spectra, it
will prompt the user for the same inputs as is prompted in the procedure png...test.pro.
Therefore, the desired shot number is needed and also a beginning time, an end time, and
the desired maximum frequency. For this code, only the desired shot number will actually
be used and the beginning time, end time and maximum frequency can be any value the
user wishes to put in that make sense. For example, a beginning time of 100, an ending
time of 200 and a maximum frequency of 800 is acceptable. The only necessary thing is
that the beginning time must be smaller than the ending time and within the range of
times where the reflectometer has taken data. Once all inputs have been entered, a graph
will output showing the raw time series data plotted from the beginning and ending times
the user specified in the initial command. The user can then identify candidate pulses
from the graph.
Fitting Pulses to a Lorentzian
Once candidate pulses are identified, the user can then fit the candidate pulse. All pulse
fits to the raw time series data were created using the procedure: png-testtimeseries.pro.
png-test.timeseries.pro requires two other procedures in order to work properly: refl-structure.pro
and simple.png.pro. reflstructure.pro is used once again to access the reflectometer data
and simple..png.pro is used to make the png. In addition, in order for the curvefit function used in png-test-timesereis to run correctly gfunct-timeseries.pro. This provides the
function that curvefit will fit to. The following command should used in order to run the
procedure:
png..test..timeseries, a, b, x1, x2, Al, A2, A3, A4
33
Where a and b are the two numbers used to indicate reflectometer channel (see power
spectrum fitting), x1 and x2 are the beginning and ending time in seconds for the pulse
to be fitted, and Al, A2, A3 and A4 are guesses for the parameters to be fitted for the
Lorentzian curve. Al is the amplitude of the pulse, A2 is the half-width of the pulse,
A3 is the location of the center of the pulse, and A4 raises the pulse either up or down
depending on sign. Again once the initial command is run, it will ask for the same inputs
as png-test.pro. However just like viewing the time series data, only the experimental
shot number is relevant.
Once all inputs have been entered, a graph of the time series data as well as three
outputs on the terminal will be shown. It will print the reduced chi square of the fit,
the fitted parameters as Al, A2, A3, A4 and finally whether or not the fit converged.
Since A4 is simply an offset parameter it's not particularly important, however the other
three fitting parameters are important for viewing the distribution of pulse amplitudes,
the width of the pulses fitted in the time series to compare with the characteristic slope of
the power spectrum, and the center to determine the time in between consecutive pulses.
The outputted graph will show the pulse with a red dashed fitted line over it.
Figures Created in MATLAB
All figures that did not require direct use of data stored at C-Mod were created in MATLAB. All graphs that were created in MATLAB used the data in the excel spreadsheet
entitled: data-for-graphs-for-paper.xlsx. Please see the Methods section for detailed information regarding how the excel spreadsheet was created. The following provides explanations of how the images were created with this data.
Dependence of the Characteristic Width on Line Averaged Density
In order to plot the dependence of the characteristic slope of the power spectra on the
line averaged density, run the code titled lineavgdens.m on MATLAB. All data points
were inputted by hand, from the excel spreadsheet called data-for-graphsfor-paper.xlsx.
Therefore, running the code will immediately output a graph of all data points with a fit
to the curve that was fitted using MATLAB's built-in "polyfit" function.
Best Fit line of the Consecutive Times between Pulses
Using the data from the excel spreadsheet, a number of other graphs were made in MATLAB, namely:
1. For experimental shot 1120224019, the distribution of Lorentzian pulse amplitudes.
2. For experimental shot 1120224019, the distribution of Lorentzian pulse widths
3. Characteristic power spectrum slope versus the local electron temperature (Te)
4. Characteristic power spectrum slope versus the local electron temperature density
(VTe)
34
5. Characteristic power spectrum slope versus the local electron density (Ne)
6. Characteristic power spectrum slope versus the local gradient of the electron density
(VNe)
7. The distribution of times between consecutive pulses
This last graph on the list also includes two best fit exponentials: the best fit line from
MATLAB (see "Dependence of the Characteristic Width on Line Averaged Density" and
read the MATLAB file there to see how polyfit works) and the best fit line from Hornung
et al (see section 4.3). For this thesis, all data was inputted by hand into MATLAB and
then run separately to make the graphs. Please see lineavgdens.m in order to plot fitted
lines over the experimental data set.
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Full list of experimental runs (shots) used
shot number
1110309005
1110309005
1120210026
1120210028
1120214002
1120224005
1120224006
1120224008
1120224009
1120224010
1120224010
1120224012
1120224012
1120224013
1120224014
1120224015
1120224016
1120224019
1120224020
1120224022
1120224027
1120224028
1120224028
1120224030
1110201004
1110201009
1110201010
1110201015
1110201019
1110201021
1110201024
1110201025
1110201026
1110309012
1110309013
1110114004
1110114005
1110114007
1110114011
1110114012
1120614002
1120614006
1120614007
1110215001
tl(ms)
900
1000
1000
1000
900
900
900
1000
1000
700
1200
700
1200
1200
1200
850
670
1000
900
650
1200
900
1200
900
880
500
500
550
500
560
600
600
770
1300
700
7
650
1400
900
700
1000
1000
1000
800
36
t2 (ms)
1000
1100
1100
1100
1000
1000
1000
1100
1100
800
1300
800
1300
1300
1300
950
770
1100
1000
750
1300
1000
1300
1000
980
600
600
650
600
660
700
700
870
1400
800
800
750
1500
1000
800
1100
1100
1100
900