Drift waves and chaos in a LAPTAG plasma physics experiment Walter Gekelman
Drift waves and chaos in a LAPTAG plasma physics experiment
Walter Gekelman and Patrick Pribyl
Department of Physics and Astronomy, University of California (UCLA), 1000 Veteran Avenue Rm.15-70
Los Angeles, California 90095
Henry Birge-Lee
North Hollywood High School, 5231 Colfax Avenue, North Hollywood, California 91601
Joe Wise
Wildwood School, 11811 Olympic Boulevard, Los Angeles, California 90066
Cami Katz
Harvard Westlake, 3700 Coldwater Canyon Avenue, Studio City, California 91604
Ben Wolman
Palisades Charter High School, 15777 Bowdoin Street, Pacific Palisades, California 90272
Bob Baker
University High School, 11800 Texas Avenue, Los Angeles, California 90025
Ken Marmie
Roosevelt Middle School, 222 E. Acacia Avenue, Glendale, California 91250
Vedang Patankar
North Hollywood High School, 5231 Colfax Avenue, North Hollywood, California 91601
Gabriel Bridges
John Marshall High School, 3939 Tracy Street, Los Angeles, California 90027
Samuel Buckley-Bonanno
Lincoln Middle School, 1501 California Avenue, Santa Monica, California 90403
Susan Buckley and Andrew Ge
Westlake High School, 100 Lakeview Canyon Road, Westlake Village, California 91362
Sam Thomas
John Marshall High School, 3939 Tracy Street, Los Angeles, California 90027
(Received 14 April 2015; accepted 10 November 2015)
In a project involving an alliance between universities and high schools, a magnetized plasma column with a steep pressure gradient was established in an experimental device. A two-dimensional probe measured fluctuations in the plasma column in a plane transverse to the background magnetic field.
Correlation techniques determined that the fluctuations were that of electrostatic drift waves. The time series data were used to generate the Bandt-Pompe entropy and Jensen-Shannon complexity for the data. These quantities, when plotted against one another, revealed that a combination of drift waves and other background fluctuations were a deterministically chaotic system. Our analysis can be used to tell the difference between deterministic chaos and random noise, making it a potentially useful technique in nonlinear dynamics.
V
2016 American Association of Physics Teachers .
[ http://dx.doi.org/10.1119/1.4936460
]
I. INTRODUCTION
About twenty-three years ago, a few authors on this paper attended an NSF-sponsored meeting encouraging the formation of alliances between universities and high schools. As a result of this meeting, LAPTAG (Los Angeles Physics
Teachers Alliance Group) was formed. The motivation was to use the resources that a university could provide to enhance the science education of high school students and provide some continuing education in physics for teachers.
The LAPTAG philosophy was to have small (10–15 member) groups of students closely interact with high school and university faculty. Today, LAPTAG still exists and is open to anyone that is interested, but only highly motivated students continue to come on a weekly basis for durations of one to three years. In these visits, students see how “real” science is done, as the experiments are not “cook book” projects. Students experience detector probes breaking, and witness times when various experimental conditions prohibit any progress at all. Students get to see experts puzzled from time to time, and the process by which problems are resolved.
We soon realized that the alliance would not last unless it was centered on a project in which everyone could participate. The first project was in seismology and a grant from the University of California was used to purchase ten seismometers that were distributed to high schools. Lectures on earthquakes and wave propagation were delivered, and the instruments were buried in various schoolyards. Data on the endless small (and sometimes large) temblors that routinely occur in southern California were compared.
Since the first author of this paper is a plasma physicist, the next project was the study of ion sound waves in a plasma. To accomplish this project, a small device was
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constructed at UCLA in 1999. The LAPTAG students and teachers played a large role in the fabrication of the device, with the help of their faculty advisors, and the research resulted in a publication.
In 2007 a magnetized plasma device was constructed in the LAPTAG lab to replace the original machine, and is there for the sole use of high school students. The addition of a magnetic field opened up the possibility for the study of a large variety of waves including drift waves, which were observed on the plasma edge. The data acquired for this publication were taken on another device constructed (by the first and second authors) for the
UCLA undergraduate plasma physics laboratory course.
That device is longer than the LAPTAG machine and has a larger magnetic field, which made it easier to study the waves. The drift wave experiment was performed during the summer and during the months when the machine was not used for teaching undergraduates. In all our projects, concepts that the students learn in high school are extended, and additional information is supplied to allow us to tackle far more sophisticated problems. The experiments run in parallel with a lecture series on the physics of the waves, the principles behind the plasma diagnostics, and the mathematical concepts required to understand them.
The experimental apparatus consists of a stainless steel chamber surrounded by electromagnets. The students must learn a bit about vacuum technology and how to prevent any air from getting into the machine during the course of the experiment. The magnets are run in a steady state using three dc power supplies. The plasma is produced with fast electrons that partially ionize a gas bled into the machine. The students learn how to operate the machine as part of the course of their studies.
Drift waves occur spontaneously in a magnetized plasma with a pressure gradient. They were the first wave extensively studied in the context of plasma transport, that is, the transport of heat and plasma density across a strong background magnetic field. This is a key issue in exploration of confinement of plasmas for magnetic fusion devices. Drift wave studies began in the 1960s
and continue to this day.
A radial pressure gradient occurs when the center of a plasma column is denser and hotter than its edge. The pressure gradient is the source of free energy, which ultimately drives the wave. In this experiment, drift waves spontaneously arose, propagated, and decayed in a magnetized plasma column.
Although the waves have a well-defined frequency, their phase and amplitude varied from shot to shot and therefore cannot be studied using simple averaging techniques. In order to verify the dispersion and confirm that the observed fluctuations were indeed drift waves, correlation techniques were employed. The thermodynamic entropy is not directly measured, instead the permutation entropy H(t) of the time series data of density fluctuations is determined. This was used to study the chaotic property of the waves. The usefulness of the concepts we describe go far beyond the study of plasma waves. The entropy as well as the complexity presented here can be used to explore any time series data (stock market prices, spread of disease, temperature variations, heartbeat
) and can be used to distinguish between ran-
dom noise and deterministic chaos in any situation.
II. EXPERIMENTAL SETUP
The drift wave experiment was performed in a magnetized plasma column. The plasma sources were two cathode/anode pairs, one on each end of the cylindrical chamber. The chamber was surrounded by solenoidal magnets, which provided a constant axial magnetic field. Figure of the device.
The Argon plasma was pulsed using two capacitor banks and transistor switches, one for each source. The pulsing arrangement is shown in Fig.
, which is a block diagram
I of the apparatus. The plasma is pulsed ( V dischage
Discharge
¼ 3 A, Argon pressure P ¼ 6 10
4
¼ 94 V,
Torr) with the left- and right-hand sources synchronized. The cathodes were made of Lanthanum Hexaboride (LaB
6
) heated to
1750
C, their operating temperature for emission.
,
plasma was on for 8.5 ms with a repetition rate of 4 Hz. A pulse generator drove the transistor switches and provided a trigger for the digital oscilloscope to acquire the signals.
When the computer controlled probe reached the next position on a grid in the xy -plane (determined by the user), the software allowed the scope to accept the next trigger. The probe signals were then digitized and transferred to a computer. A second, fixed probe allowed for the measured signals to be correlated. The probes went through double sliding vacuum seals, which were pumped through a manifold using a mechanical pump. This arrangement maintains vacuum integrity as the probe was moved.
III. DRIFT WAVES
shows a photograph
The
Drift waves can be derived from the ion continuity equation and the equation for the force on a small element of plasma fluid. The equations are nonlinear and assumptions have to be made to derive a wave equation. The assumptions are that the waves have no intrinsic fluctuating magnetic field and since the waves are low in frequency we can express the electric field as
~ ¼ r / . The electron density is determined by the local electric potential. The equations are linearized, and Fourier analysis converts them to algebraic equations.
lead to a simple dispersion relation for a “slab model” in cylindrical geometry x k h r P
¼ enB
0
; (1)
Fig. 1. (Color online) Drift wave experimental device. The plasma is made with two cathodes 2.54 cm in diameter. The solenoidal coils surrounding the vacuum vessel produce a uniform magnetic field of 160 G. The system base pressure is 1.0
10
6
Torr using a turbo pump. The feed-through attached to a 2D probe drive is shown on the left. The Argon plasma column is also visible.
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Fig. 2. Block diagram of the experimental arrangement. Two capacitor banks ( C ¼ 19.8 mF) provide electrons for the dc discharges. They are in series with individual transistor switches that, when closed, complete the cathode-anode circuits. The solenoidal magnets are shown as grey rectangles and controlled by three power supplies to produce a uniform magnetic field (0 < B
0z
< 400 G). The oscilloscope data acquisition and probe drive are controlled by a computer running a customized
LABVIEW program. The magnetic field is aligned with the z -axis, which is the chamber axis. The fixed probe is at z ¼ 0.20 m and the transverse movable probe at z ¼ 0.60 m.
where P is the plasma pressure and k h
¼ 2 p = k h is the azimuthal wavenumber. Here, we assume the wavelength is infinite ( k k
¼ 0) along the background magnetic field and the perpendicular phase velocity of the wave is defined as v phase
¼ x = k h
. In the LAPTAG drift wave experiment, the ion radius of gyration about the background magnetic field
( B
Oz
¼ 160 G) is R
CI
¼ 1.3 cm ( T
I
¼ 0.1 eV), which is smaller than that of the plasma column diameter. The electron gyroradius is 0.26 mm, and Eq.
does not account for the finite gyro-radius. To do this, “warm plasma theory” must be
and for our conditions the dispersion relation becomes x k h r P
¼ enB
0
0 : 618
1 þ k 2 h q 2 s
; (2) where q s is the ion sound gyroradius, which is based on the electron temperature.
IV. CORRELATION ANALYSIS
Drift waves in a plasma occur spontaneously and do not need to be launched. Any given observed drift wave may start at a different position and at a slightly different time, every time the plasma is turned on. Figure
shows an oscilloscope trace of one data shot; the inset in shows two shots of ion current to a probe over a smaller temporal interval. It is clear that the signals are both sinusoidal but their phases differ. Figure
is an average over 50 shots; the wave component averages out and in fact Fig.
is now a trace proportional to the plasma density as a function of time.
Because the waves are random, all information about them is lost in the averaging process. To extract information about the waves, a correlation analysis must be employed. The signals are filtered at frequency x to isolate an individual wave at that frequency using the cross spectral function
C ð ~ ; x ; s Þ ¼
X
I sat – fix ; j ; x
ð ~ ; t Þ I sat – mov ; j ; x
ð ~ ; t þ s Þ v t
" j ¼ 1 t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
#
I sat – fix ; j ; x
ð ~ ; t Þ I sat – fix ; j ; x
ð ~ ; t þ s Þ v t
" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X X
#
I sat – mov ; j ; x
ð ~ ; t ; s Þ I sat – mov ; j ; x
ð ~ ; t þ s Þ
: j ¼ 1 t j ¼ 1 t
(3) n p
Here, the ion saturation current
T e
, where n
I sat is the plasma density and is proportional to
T e is the electron temperature, I sat-fix probe and I sat-mov is the ion saturation current to the fixed for the movable probe, j is the shot number, ~ ¼ r ð x ; y ; z Þ is distance from the fixed probe to the movable one, t is the time-step, and s is the delay time. For a given ~ and delay time, the digitally filtered time series of the fixed probe and movable probe (shifted by s ) are
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Fig. 3. Ion saturation current (in mA) as a function of time. (a) The ion saturation current I sat at a fixed location in a plane transverse to the magnetic field and at z ¼ 0.60 m. The inset, which spans 400 l s, illustrates an arbitrary phase shift that occurs for each of the ensemble of 50 shots. The dotted and dashed lines in the inset are data from two successive shots. (b) An average of the ion saturation current over 50 data shots.
multiplied and then summed over all the shots in the ensemble ( j ¼ 1, N shots
). The denominator in Eq.
normalizes the value of C from 1 to 1. The functions in the denominator are the autocorrelation functions.
Both a fixed and a moveable Langmuir probe
were used to measure the ion saturation current. The raw, nonaveraged, non-filtered data from both probes were stored. In addition, at each location 50 shots, called an ensemble, were recorded. First, the data from both probes were filtered by multiplying the spectrum of the data in complex Fourier space by a Gaussian centered at 18 kHz. This is the approximate frequency of the wave shown in the spectrum of our data. Then the data from the fixed probe were correlated with the data from the moveable probe.
The data were correlated by offsetting the time series data from the moveable probe by a lag time s , and then summing up the multiplication of the fixed probe values and the corresponding moveable probe values for every time step in the data. When s corresponds to the time it takes for a disturbance to get from the fixed to the movable probe the correlation will be largest. The fixed and moveable probe data were padded with zeros to make this possible. Finally, this summation was normalized by dividing by the square root of the autocorrelations of each time.
V. EXPERIMENTAL RESULTS
Two probes were used as discussed above. A fixed
Langmuir probe was positioned at the edge of the plasma column. A second movable probe penetrated the plasma chamber through a sliding seal and was positioned in the plasma column. The probes were separated by 30 cm along the axis of the machine. The movable probe traversed the x and y directions (transverse to B
Oz
) to typically 31 31 positions on a square grid ð D x ¼ D y ¼ 10 cm Þ , taking the ensemble of shots at each position, and acquiring data for 10 ms. A computer running a
LABVIEW
program written for the experiment managed data collection and controlled the motorized probe drive and a digital oscilloscope. The 2.5-
Gs/s (analog bandwidth 440 MHz) digital oscilloscope acquired the data at each spatial location. The probes could be either used to collect ion saturation current, or as swept
to determine the electron temperature.
We used the combination of these two signals to compute the pressure and density profiles.
The Langmuir probe tip (0.32-cm diameter, stainless steel disk) was attached to a tungsten wire, encased in a 1-mm diameter alumina ceramic tube, and placed inside a steel shaft that went through a double O-ring seal. The sliding seal allowed motion of the probes in the transverse plane within the vacuum system. A vacuum compatible ball valve
(visible in Fig.
1 ) allowed the probe to move continuously in a
plane without the use of vacuum bellows. The probe moved in a series of arcs but the radius of curvature was large enough so that correction from polar to Cartesian coordinates was not necessary. When the probe voltage was swept with respect to the chamber wall, the resulting current to the probe was measured. The drift waves occurred on the periphery of the plasma column. The data showed that the pressure and density were greatest at the center with a maximum gradient scale length of 1.5 cm.
To determine if there are discrete modes in a signal, we performed a Fourier analysis. The Fourier transform shows there is a dominant frequency at 18 kHz, which agreed with the approximate frequency of the time series in Fig.
.
The amplitude of this mode varies according to spatial position, with the maximum amplitude within the ring corresponding to the steepest gradient of the pressure profile in
Fig.
. This location was obtained by taking the FFT of the
48,050 shots collected during the data run by the fixed probe.
The frequency spectrum for the data in Fig.
is shown in
Fig.
5 . Aside from the peak at 18 kHz, the spectrum had a
low-frequency component as well as a long tail out to
30 kHz. The correlation analysis as described later revealed that these components are not associated with waves.
In order to check the drift wave theory, it was necessary to measure the plasma pressure. This is done using the electron temperature T e and density n derived from the swept
Langmuir probe; a cross section is shown in the upper inset of Fig.
. This data is used in the theoretical equation for drift waves described in Eq.
. The pressure was computed from P ¼ nkT , where k is Boltzmann’s constant. This data was available at every point of the cross-sectional measurement, and pressure contours are shown in Fig.
.
We performed the correlation analysis described by Eq.
on the time series data from each spatial location. The result for these experimental conditions is shown in Fig.
mode is an m ¼ 2 spiral structure, which has two positive lobes (red online) and two negative lobes (blue online). This spiral rotates in the clockwise direction (see video, enhancement online), the same as the electron diamagnetic drift velocity. The plasma column radius was a ¼ 1.5 cm, corresponding to the maximum gradient of the pressure. By using this radius, we determined that our drift waves had a wavelength of 4.7 cm. Note that the spiral extends into the lowdensity plasma several centimeters beyond the maximum in
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Fig. 4. Contour map of the pressure, in milliJoules per cubic meter. The inset shows the plasma pressure across the x -axis, at y ¼ 1.
the pressure gradient. The drift waves spin in a clockwise direction. It now remains to compare the data with theory to validate the presence of drift waves. If the data are not filtered, the correlation function has no obvious wavelike structure. All measured quantities on the right-hand side of Eq.
, including k h
, are displayed in Table
muthal velocity is f k h
¼ 886 m = s and that predicted by Eq.
is 894 ms. Predicted and observed azimuthal phase velocities agree within 6%. We are pleased with this agreement, considering the density derived from these probes is accurate
(in general) to within a factor of 2 and the electron temperature within 30%.
VI. ENTROPY
Entropy is a measure of disorder associated with a physical system. The system could be a gas or plasma with specific boundary conditions. There could be walls, for example, that isolate it from the rest of the world. For a plasma, the definition of a given microstate would consist of a list of all the momenta and positions of the ions and
Fig. 5. Frequency spectrum of fluctuations in the drift wave experiment. The probe was fixed at a location in the density gradient. The peak at 18 kHz is the frequency at which the drift waves appear. The spectra are in arbitrary units and shown on a linear scale. The spectra were averaged over all
48 10
6 shots in the experiment.
Fig. 6. The cross-spectral correlation function for data filtered at 18 kHz.
The maximum pressure gradient is at a ¼ 1.5 cm. The plasma density extends to þ / 4.5 cm. The azimuthal wavelength was derived from
2 p a ¼ 2 k h
, as there are two wavelengths in the system. Therefore, the wavelength of the drift waves is 4.7 cm. (enhanced online) [URL: http:// dx.doi.org/10.1119/1.4936460.1
].
electrons at a given moment in time, and there are an infinite number of possible microstates. Let P i be the probability that the plasma is in the i th microstate of the plasma. It could be that P i corresponds to all the particles in the plasma bunched into a small volume and moving at exactly the same speed in the x -direction. This is not very likely to occur. The second law of thermodynamics states that after the plasma system has time to interact, the most likely P i will be a state with the most disorder or highest possible entropy. The definition of entropy in terms of P is
X
S ¼ P i ln P i
: (4) i
Here, the sum is over all possible microstates. Each microstate has a certain probability of occurrence and can be just as well thought of as an element of a probability distribution.
Boltzman proved that the summation in Eq.
is identical to
the thermodynamic definition of entropy
ð
S ¼ dQ
;
T
(5) where Q is heat and T is temperature. In this experiment, we did not measure the probability distribution function, which would require tens of thousands of microscopic probes capable of measuring electron and ion velocity distribution functions at every angle. Instead, we had time series data of the
Table I. Measured values of various physical quantities.
Quantity k
?
r P
¼ k h h n i
B
0z e
1 þ k
2 h q
2 s v
?
Value
2.5
10
17 m
4.7
10
20 J/m
4
2 m
3
(at maximum pressure gradient)
1.6
0.016 T
10
19
C
24.2 ( q s
¼ 6 cm at maximum pressure gradient)
894 m/s
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Fig. 7. Time series of the data segment of the drift wave shown in Fig.
measured by the fixed probe with points connected by a line. There is a large amplitude variation as a function of time. The points in the center show the digitized data. The amplitude ordering of points P
1
–P
5 is discussed in the text. There are
6501 data points (indices) in every time series.
ion current at every location in the plasma. Fortunately, a technique exists for measuring the entropy of time series data—the Bandt-Pompe permutation entropy
S ð P Þ ¼ j p j
ð p Þ ln ½ p j
ð p Þ ; (6) with N ¼ n !. Here, p j
( p ) is a permutation of type p , which is a specific ordering of the n numbers, and the sum runs over all n ! permutations of order n . The integer n is called the embedding dimension. Note that this is of the same form as the standard thermodynamic definition of entropy, where p would be the number of states. Bandt and Pompe go on to show that the permutation entropy S is similar to the
for a chaotic system, such as the logistic map or tent map. The permutation entropy has been eval-
The definition of the time series entropy has the same form as Eq.
ity here is associated with the ordering of amplitude values.
Consider the data stream in Fig.
. The entropy associated with this data reflects the succession of amplitudes in the data stream. For example, suppose the data were reordered using the amplitudes to make it into a sine wave around the mean of the data. This would be a more ordered signal and would therefore have less entropy. If we consider the five larger point symbols ( n ¼ 5) in the center of Fig.
amplitudes are ordered by (1, 2, 5, 4, 3), where the first point has the largest amplitude and the third point the smallest.
Consider the amplitudes of any successive five points in the data stream. Since there are 5! (120) ways to order the amplitudes, we’ll break the entire time series up into bins of five points. This ordering happens 45 times in the series (the absolute amplitude of the points does not matter, only their amplitude ordering). The probability that this happens is
P ¼ 45/(6501–5). Using Eq.
, the entropy associated with this is S i
¼ 0.34.
We then have to find the frequency of occurrence of all the remaining 119 combinations throughout the data set and sum them up. This gives us the entropy of the time series.
For the time series in Fig.
7 , the entropy, normalized to
unity, is 0.497. For comparison, if we construct a sine wave of 16 cycles with the same number of points the entropy is
0.166. The entropy calculated for 6501 random numbers is
0.998. In our data, we have 50 shots at each location and 961 data points in the plane. The entropy calculation is repeated over 48,000 times.
The number n is called the embedding dimension. There have been studies of the optimum embedding dimension.
Clearly, a very small value ( n ¼ 1, 2) would not work while a large value, such as n ¼ 8, would not be practical (as
8!
¼ 40320) to calculate in a reasonable amount of computer time. For data sets such as this n ¼ 5 or 6 are adequate. Both were tried with little difference. When the Bandt-Pompe entropy is divided by N , as in ln N
; (7) where N is the total number of possible permutations, the number of data points in the time series falls between 0 and
1. Figure
shows the entropy associated with the drift wave
Fig. 8. The Bandt-Pompe entropy H , associated with the entire spectrum
including the drift wave in the transverse plane corresponding to Fig. ( 6
).
The position ( x , y ) ¼ (0.5 cm, 0.5 cm) corresponds to the center of the plasma column. The normalized entropy falls between the values of 0.55
and 0.75. The entropy is largest in the center of the plasma column where the drift waves reside.
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(and all the additional noise in the transverse plane) shown in Fig.
.
VII. COMPLEXITY
The Jensen-Shannon Complexity C S
J
C
S
J
¼ 2
S
P e
þ
2
P 1
2
1
2
S P e
N þ 1 ln ð N þ 1 Þ 2ln 2 N Þ ln ð Þ
N
; (8) where S ( P ) and N are defined above and P e is the maximum entropy state for which every member of the probability distribution has the same value. Thus, S ( P e
) is the entropy of the time series in which every member has an amplitude of
1/ N . The Jensen-Shannon complexity defined in Eq.
is a comparison between the entropy of the time series data generated in the experiment and two other calculated time series.
The first has data with maximum entropy, and the second is constructed by adding 1/ N to the experimental data at every time step. Rosso et al .
showed that every chaotic map, as well as fractional Brownian motion,
appear at specific locations in the complexity plane. In this work, the Jensen-
Shannon Complexity and the Bandt-Pompe entropy are each an axis on a C-H diagram, on which the ordinate is complexity (C) and the abscissa entropy (H). Such C-H diagrams can discern between different degrees of periodicity associated with one of many routes to chaos, and can distinguish stochastic noise from chaos. As such, these diagrams are a prom-
ising tool for use in plasma physics.
Mathematicians have shown that data in the CH -plane are bounded by two curves,
one for each minimum and maximum complexity.
The exact shape of the minimum and maximum complexity curves is uniquely determined by the embedding number n .
For a given n all processes (stochastic, random, sinusoidal, etc.) must lie between these two curves. For example, a series of values from a random number generator will be a point on the lower right (large entropy, low complexity) in the CH plane, while a sine wave (low entropy and complexity) lies on the lower left. The great utility of this methodology is that chaotic processes that come from deterministic systems (for example, differential equations with chaotic solutions) can be immediately distinguished from utterly random noise—they occupy a position with moderate entropy and large complexity in the CH -plane.
The complexity and entropy in Fig.
is shown for every shot number and every position in the xy -plane for the drift waves. The data (dots) in the CH -plane indicate that the superposition of the drift waves and other fluctuations are chaotic, not simply stochastic or random processes. Also shown in the diagram are several iterative maps that lead to chaotic processes.
One such map is the Logistic map, given by the iteration
X n þ 1
¼ AX n
ð 1 X n
Þ : (9)
In the Logistic map shown in Fig.
, a value of A ¼ 3.05 was used (this value falls into a chaotic region). All iterative maps fall on the CH -plane. Processes that are highly chaotic, such as the logistic map (diamond) and the H enon map (triangle, plus sign), can be found in the region where the complexity is large and the entropy is moderate. The data from the experiment are chaotic, but not as much as the iterative maps mentioned. The continuous line in Fig.
is for frac-
in which the sequence representing the motion of a particle displays strong interdependence on distant samples, hence the system has “memory.” White noise (or 1/ f noise) would show up as a point where H is very large and C
S
J small, the lower right side of the C-H diagram. If the data were a perfect sine wave it would have low entropy and low complexity and appear on the lower left side of the plane.
The data filtered with a Gaussian filter centered at 18 kHz appear as a tiny cluster of points in Fig.
shots and positions are plotted. From its position in the CH
Fig. 9.
C-H diagram associated with the drift waves. The entropy and complexity are calculated for all frequencies and appear as a cloud of dots around
H ¼ 0.6–0.7 (green online). There is a dot calculated for each of the 50-shot ensembles and for each of the 961 spatial positions in the plane. The embedding dimension is n ¼ 5. Shown on the diagram are minimum and maximum complexity curves, which bound all possible data. The black line starting at large values of H and low values of C
S
J is that for fractional Brownian motion, a stochastic process. The “point” labeled “filtered data” is for all the shots and positions calculated for the data filtered at 18 kHz. The point corresponding to a pure sine wave (with no phase or amplitude variations) is on the lower left. Finally, the dots located on the fractional Brownian motion curve (magenta online) are calculated for the same data set as the original cloud, but with the 18-kHz signal removed.
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plane, the filtered data are chaotic but not nearly as much as the unfiltered data, which have much larger amplitude variations and variations in phase. A pure sine wave at 18 kHz, with no phase or amplitude variations, is also shown in Fig.
and, as expected, is a point that lies on the lower left, close to the filtered data. The analysis indicates that the totality of drift waves and background fluctuations is chaotic and deterministic. In Fig.
, the entropy is largest in the center of the plasma; however, the complexity is smallest in the center and largest on the density gradient and beyond (this is where the chaos resides). Computer code for the calculation of C and H
is available as supplementary material.
To show that the data without the drift wave present was not chaotic the 18 kHz signal was removed from the Fourier transform for every case. This was accomplished by replacing the peak shown in Fig.
with a copy of the data adjacent to it for every time series. The data then correspond to a case of plasma noise and low frequency structures but no drift wave. When the complexity and entropy were calculated for this case, the data fall directly on the fractional Brownian motion curve, which represents stochastic noise. This implies that there is a nonlinear interaction between the drift wave and background noise that produces chaos. The powerful
CH -plane technique demonstrates that there is a nonlinear interaction that causes the chaos but does not shed light on what this mechanism may be.
VIII. SCIENTIFIC SUMMARY
Drift waves were spontaneously generated in a narrow magnetized plasma column. Because the amplitude and phase of the waves were random, correlation techniques had to be employed to measure their perpendicular wavelength and verify they are in accord with the dispersion relation.
Analysis of the entropy using the Bandt-Pompe formalism and the Jensen-Shannon complexity revealed that the data occupy a region of the CH -plane commensurate with deterministic chaos. Much as H-R diagrams are used to classify the age of stars based on their luminosity and temperature,
C-H diagrams are used to classify chaos. The C-H methodology does not identify the cause of the chaos, only its type.
The drift waves come and go, and each wave lasts for several cycles. The correlation technique allows for the determination of the wavenumber and spatial pattern, which in turn establishes them as drift waves. The low frequency part of the spectra in Fig.
corresponds to moving structures (density enhancements and depressions), and the high frequency part to stochastic noise. When the drift wave is removed from the experimental signal, the chaotic region in the CH plane vanishes, which is evidence for a nonlinear interaction between the wave and other modes.
IX. A MODEL FOR COLLABORATIONS BETWEEN
UNIVERSITIES AND SECONDARY SCHOOLS
This experiment involved a plasma device at UCLA and was a collaboration of university professors, high-school teachers, and a group of high-school students. Faculty able to make the commitment to form alliances involving students and their teachers can generate authentic research opportunities that benefits all parties involved.
Students enter LAPTAG with different backgrounds in math and physics and with diverse skill sets. We thus begin by carefully discussing and planning the research topic. The necessary diagnostics and machine adaptations are performed as a group effort. Throughout the process, lectures are given to help students deepen and refine their levels of understanding. Students identify the skills and knowledge they need to progress through the research, and peer-to-peer coaching helps to fill gaps in their proficiencies. Ultimately, students participate in the experiment and help analyze the data, and this may be followed by a conference presentation or by publication in a peer-reviewed journal. We have found that this process is very effective in allowing students to construct meaning for themselves throughout the learning process.
ACKNOWLEDGMENTS
The research presented here was conducted by the Los
Angeles Physics Teachers Alliance Group (LAPTAG), a group that has been active for the past 20 years. During that time, LAPTAG students have performed experiments ranging from seismology to plasma physics (LAPTAG has a plasma device at UCLA). The authors wish to acknowledge the technical contribution of Zalton Lucky and Marvin
Drandell for their valuable contributions. The authors thank
Steve Vincena, Peter Heuer, and Yuhou Wang for their assistance in the project. The work was done at the Basic
Plasma Science Facility at UCLA
supported by grants
DOE DE-FC02-07ER54918 and NSF-ATM-0531621.
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