David Pace

Thesis - Appendix B: Pulse Detection Techniques

Sat, 24 Sep 2011 12:43:52 +0000

Pulse Detection Techniques

Overview

The study of exponential spectra and accompanying Lorentzian pulses in time series data requires that the pulses be isolated for determining their characteristic widths. A variety of identification techniques have been applied to the large data set of this experiment, and notes on two of these methods are provided here so that they may be further developed in future studies. The brute force method of sliding a Lorentzian shape through an entire time series is described in Section 4.3. The two other methods described here are the wavelet phase method in which constant phase over all frequencies identifies a pulse event, and the amplitude threshold method in which events of statistically significant amplitude are identified as pulses.

Wavelet Phase Method

Some aspects of wavelet transform techniques, including a comparison of their power spectra results to those of the more familiar Fourier methods, are discussed in Appendix A. The phase of a time series as calculated with wavelet techniques can be used to identify pulses because any singularity exhibits a constant phase [Farge, 1992] over all frequencies. Lorentzian pulses in this experiment are not necessarily singularities, and those with larger time widths are less so than narrower pulses. In spite of this difficulty, the wavelet phase method still appears to identify pulses fairly well.

Using the CWT as given in Eq. (A.2), which is a complex valued function for the Morlet wavelet basis, the phase, φ, is determined from,

where I and R signify the imaginary and real parts of the CWT respectively. The value of the phase at any frequency ranges from -π to +π.

Test Lorentzian Pulses

Figure B.1 presents the phase contours of two test signal Lorentzian pulses. One of these is narrower than the other, τ = 0.5 μs compared to τ = 4 μs. Both contours clearly identify a pulse by the symmetry of the phase values. The narrower pulse of the left panel more accurately reproduces the behavior of a singularity and the phase contour is remarkably sharper than that of the wider pulse in the right panel.

Figure B.1: Phase contours of test signal Lorentzian pulses. The left panel shows the well defined phase pattern of a narrow (τ = 0.5 μs, overplotted) Lorentzian pulse. The right panel shows that wider pulses (τ = 4 μs, overplotted) still exhibit symmetry about the event, but that constant phase as a function of frequency is not as evident as for the narrower pulse. [Full Size]

Phase as a function of frequency for test signal Lorentzian pulses is shown in Fig. B.2. It is observed that narrower pulses demonstrate a more constant phase across the range of frequencies returned.

Figure B.2: Phase versus frequency for test signal Lorentzian pulses of varying width. The narrowest pulse, τ = 0.5 μs, results in the most constant value of phase. [Full Size]

Measurements of Lorentzian pulses show that they can be either positive or negative polarity. Figure B.3 compares the phases of test signal Lorentzian pulses for each polarity. Both phase contours exhibit symmetry about the center of the pulse, but the negative polarity pulse features a much narrower region of zero phase. This result suggests that negative polarity pulses can be accurately identified by the wavelet phase technique, and even if this proves prohibitively difficult it is a simple task to invert the raw signals and apply the standards developed to isolate positive polarity events.

Figure B.3: The wavelet phase is shown for positive (left panel) and negative (right panel) Lorentzian pulses. In both panels the test signal Lorentzian features τ = 1 μs and is overplotted in solid black. [Full Size]

Application to Experimental Results

Figure B.4 presents a measured pulse and its corresponding wavelet phase representation. The measurement is made in the outer region at r = 0.55 cm and features a large amplitude positive polarity pulse. The phase contour demonstrates excellent symmetry across the time of t = 7.17 ms where the pulse event is centered. This particular pulse exhibits a phase relationship that indicates it is similar to a singularity.

Figure B.4: A pulse event from the experiment (left panel) and its wavelet phase representation (right panel). The pulse is clearly identified in the phase representation at t = 7.17 ms. [Full Size]

Limitations of the Wavelet Phase Detection Technique

An immediately apparent limitation of this technique is that it identifies only singularities, not Lorentzian pulses. In a best case scenario, the wavelet phase could be examined to determine the most likely temporal locations of Lorentzian pulses. These events could be extracted and then passed through a secondary analysis to determine whether they have any particular shape. Such a procedure essentially employs the same curve fitting technique as described in Section 4.3. That method simultaneously identifies events and classifies them as Lorentzian or non-Lorentzian, removing the need for the initial wavelet phase calculation. It is possible that the wavelet phase is useful in selecting out events that are embedded in background coherent fluctuations. In this case, the phase may be better suited for identifying a non-coherent mode related pulse which can then be further studied to determine whether it is a Lorentzian. This overcomes a limitation to the curve fitting technique in that fitting errors may incorrectly identify parts of a coherent oscillation as a unique Lorentzian pulse.

Amplitude Threshold Method

The amplitude threshold method isolates pulse events (again, not necessarily Lorentzian pulses) that exhibit statistically significant large amplitude. This is sometimes called “conditional averaging” in cases where the large amplitude events serve as triggers that set a reference time for another phenomenon.

Figure B.5 presents two raw measurements from the same discharge but different spatial positions. The Isat trace is measured at (r,z) = (0.5, 192) cm and the Vf trace is measured at (r,z) = (0.35, 544) cm. With an axial separation of Δ z = 352 cm, the correlation between the turbulent pulse region on both traces is interesting. The amplitude threshold method of pulse detection is used to study such correlations by allowing for the generation of ensemble profiles from a moving probe (in this case, the Vf measuring probe) based on the occurrence of similar pulses measured with a fixed Isat probe.

Figure B.5: Same shot time series highlighting the correlation between pulses separated in position. The Isat trace (solid black, top) is placed in the outer region and detects little other than pulses. The Vf trace (solid blue, bottom) is placed in the gradient region and measures both coherent modes and turbulent pulses. [Full Size]

The identification of pulse events begins by calculating the standard deviation (σ), or root-mean-square (RMS), of the fluctuations in a signal. Figure B.6 is a plot of both the fluctuating component of the Isat signal (originally seen in Fig. B.5) and the RMS level as computed by a sliding window. The sliding window computes the RMS level for a limited time range of the entire signal and then translates the center of the window to calculate subsequent values. In this example, the RMS trace is useful for determining the likely beginning of the turbulent regime of the system. The increase in fluctuation level just after t = 7 ms suggests some type of transition has occurred, though it does not necessarily indicate anything about the presence of Lorentzian pulses.

Figure B.6: The fluctuating component of the Isat signal from Fig. B.5 (black, bottom) and the RMS value (red, top) calculated with a sliding window. [Full Size]

An RMS value calculated over the entire time series of a single discharge is used to search for pulses. From a computational perspective, once the threshold has been assigned, any points above that value are collected as possible pulse peaks. For any sets of points that occur in sequence (i.e., any continuous collection of values that are all above the threshold), only points that represent local maxima are returned. This allows for the detection of overlapping pulses so long as they are minimally distinct.

Figure B.7 shows spikes in the data of Figs. B.5 and B.6 as found by the threshold technique. In this example the threshold level is set at 2.0, meaning that only features extended greater than two standard deviations above the fluctuation level for the entire signal are kept.

Figure B.7: A narrow time series of the previously shown Isat measurements with pulse peaks identified according to the amplitude threshold method. [Full Size]

Figure B.8 highlights a weakness of the threshold method. If the level is set too low, then common oscillatory features are likely to be branded as pulses. In Fig. B.8 this is seen near the time t = 8.085 ms. At that time position there is a small oscillation (similar to those seen in the vicinity of t = 8.12 ms) that results in a peak. While the true fluctuation amplitude of this peak is small, by riding on top of the larger oscillation it satisfies the detection criteria and is labeled a pulse peak. Other similar looking features within this time range may appear as peaks, but actually feature flat regions without a singular maxima. The vector representation of the plot trace may result in the appearance of sharp peaks when none are present. In order to minimize erroneous spike detection the threshold should be set as high as possible.

Figure B.8: Zoomed in view of Isat trace highlighting a false positive from the amplitude threshold detection method. The middle pulse peak does not correspond to an actual pulse event.

While the shape of the pulses is not determined by this method, it may be safely assumed that a confirmation of their shape can be coupled with this conditional analysis to begin a study of the transport caused by pulses. At the very least, this type of analysis helps to determine the amount of transport that leads to the generation of outer region pulses.

Thesis - Appendix A: Wavelet Analysis to Calculate Power Spectra

Sat, 17 Sep 2011 20:34:15 +0000

Thesis - Table of Contents

Wavelet Analysis to Calculate Power Spectra

Overview

Wavelet analysis techniques, while not as commonly understood as Fourier analysis, are nonetheless frequently applied to problems in which time and frequency information are desired simultaneously. Analysis suites such as IDL (popular in plasma physics circles) provide complete libraries for easily incorporating these techniques into a research program. One of the best introductory pieces on wavelet analysis [Torrence and Compo, 1998], serves as the foundation for the IDL implementation (see notes in source file wv_cwt.pro, a routine used for the analysis presented here).

It is not the aim of this thesis to reproduce basic introductory material that is readily available, so the details of wavelet analysis are left to the many textbooks and other materials that exist. In simplified terms, a wavelet analysis is the application of a bandpass filter with logarithmic spacing in the frequency domain. The discussion in this Appendix concentrates on comparison to and validation against the more well established Fourier techniques that can be applied to the same data.

A wavelet as a function of time, Ψ(t), is defined as (Eq. 6.1.4 of Debnath [2002]),

where a is a scaling parameter that sets the frequency represented by the wavelet and b determines the time center of the wavelet. The Continuous Wavelet Transform (CWT), W, of a function of time, f(t), is (Eq. 6.2.4 of [Debnath, 2002]),

where the scale and time center is still determined by the a and b parameters of the wavelet. The wavelet power spectrum, PW, is therefore given by PW = |W(f)|2.

A full spectrogram is generated through wavelet analysis by setting the scale (a) to a constant value and solving across all time values (b). Repeating this process for all scales that translate to a relevant frequency completes the analysis.

Parameters of Wavelet Analysis

The description of wavelet analysis as a logarithmically spaced comb filter (i.e., picking out the power in a specific, non-continuous array of frequencies) easily incorporates the concept of basis functions. The basis function is the specific filter used in the analysis. Wavelet basis functions come in a wide variety and it is left to the user to determine the best option for any given analysis. This is a difficulty in wavelet analysis that has no analog in Fourier analysis because that transformation is unique.

Wavelet basis functions can be real or complex valued, discrete or continuous, and orthogonal or non-orthogonal. The turbulence study presented here depends on the availability of phase analysis, which immediately requires that only complex valued basis functions be used. The ability to review specific ranges of frequencies is also necessary, suggesting the use of continuous bases. These two desired properties are the most important features to consider. The primary difference between the orthogonal property choices of a wavelet basis is that a non-orthogonal basis will “double-count” the power contribution of some frequencies, thereby returning inaccurate absolute amplitudes. The basis ultimately chosen for this work, the Morlet wavelet, is non-orthogonal. The potentially inaccurate amplitudes of the power spectra are overcome by only applying this technique to study the temporal behavior of modes and not their spatial structure or other features that rely on amplitude measures.

The Morlet wavelet, ΨM, is written as (Eq. 6.2.12 of [Debnath, 2002]),

where ωo is a variable describing the frequency at which the Fourier transform of the Morlet wavelet demonstrates peak amplitude.

To review the behavior of wavelets used in various analyses it is helpful to utilize test signals. Figure A.1 is a plot of pure sine waves of 5 kHz (top) and 50 kHz (bottom). Both signals are centered on zero for the spectral analysis and only offset to provide for clear review in the figure.

Figure A.1: Pure sine waves of 5 kHz (top) and 50 kHz (bottom) used to demonstrate the resolution of wavelet power spectra.

The 5 kHz test signal is used to highlight the frequency resolution of the Morlet wavelet. Figure A.2 compares the power spectra of the 5 kHz test signal for three wavelet basis functions: Morlet, Gaussian, and Paul. The Gaussian and Paul wavelet return similar results that exhibit poor frequency resolution compared to the Morlet. This early benchmarking of wavelet performance is necessary because of the numerous options in setting the parameters of the analysis. Deciding on the Morlet basis is relatively simple, but it remains to compare the Morlet wavelet results to those computed with standard Fourier methods.

Figure A.2: Wavelet power spectra to identify the test signal's 5 kHz mode using different wavelet basis functions. The Morlet wavelet results in the best frequency resolution.

Fixed Frequency Test Signals

The most significant difference between a wavelet spectrum and that of an FFT is that the wavelet technique results in a bandwidth that is frequency dependent. The bandwidth, or frequency resolution, of an FFT is determined by the total length of the input, l. The frequency resolution is constant and given by Δf = 1/l where l is the time length of the data window over which the FFT is performed. For a time series of N data points, output power spectrum provides a value for every frequency from f = 0 to f = (N/2)(1/l).

Wavelet analysis returns spectra in which the value of Δf/f is constant [Farge, 1992]. For an FFT, Δf/f decreases with increasing frequency because Δf is constant. In a wavelet result, the resolution of higher frequency terms is worse than that of lower frequencies. An illustration of this is shown by analyzing the test signals of Fig. A.1.

Figure A.3 plots the resulting power spectra from performing transforms using either FFT or Continuous Wavelet Transform (CWT) techniques. Peaks in the FFT spectra accurately identify the center frequencies of the test signals. The peak in the 5 kHz result (green) is just as narrow as the peak corresponding to the 50 kHz result (blue). Peaks from the CWT analysis demonstrate a reduction in frequency resolution for the higher frequency wave. The ratio of the frequency spacing to the center frequency is constant for the CWT result. Figure A.3, can be used to calculate these values,

This demonstrates that wavelet techniques are comparable to Fourier techniques for calculating power spectra. In the context of this thesis, wavelet analysis is used to provide a broad picture of the time evolved spectra. Any determination of a particular mode frequency is done with Fourier analysis in order to more precisely determine the value.

Figure A.3: Power spectra of the test signals. The FFT analysis is clearly peaked at the frequencies of the test signals. The CWT results provide a less clear peak for the higher frequency signal. [Full Size]

Evolving Frequency Test Signals

Figure A.4 is a narrow time view of the test signal used to demonstrate the ability of the wavelet analysis technique to discern the temporal behavior of coherent modes. This signal is a regular 5 kHz sine wave onto which a 17 kHz sine is added, beginning at t ≈ 5.25 ms. A spectrogram of this test signal is expected to reveal the constant presence of the 5 kHz and the sudden onset of the 17 kHz oscillation.

Figure A.4: The amplitude, A, of the test signal featuring a secondary frequency. This narrow time region of the complete signal highlights the activation of a 17 kHz signal at t ≈ 5.25 ms. The background, constant, oscillation is a 5 kHz wave.

Figure A.5 is a wavelet spectrogram of this test signal. The constant 5 kHz wave is identified throughout the entire time. The exact turn-on of the 17 kHz mode is difficult to discern based on the broad time range returned. It is noteworthy that the time resolved nature of the sudden turn-on is given accurately by the higher frequency behavior. The immediate injection of the 17 kHz signal carries with some type of &delta-function influence that causes power to be calculated for all frequencies. In this way, any sudden event in a time series can be determined to reasonable accuracy through the wavelet method, regardless of any particular frequency that may be associated with the phenomenon.

Figure A.5: Spectrogram calculated using the CWT technique on the evolving frequency test signal. The appearance of the 17 kHz oscillation is evident, though the best time resolution for marking this abrupt change comes at higher frequencies. [Full Size]

Experimental Data

Figure A.6 presents spectrograms (power spectra over time) from the temperature filament experiment that demonstrate the differences between CWT and FFT analyses. These spectra are calculated from measurements of electron temperature in the outer region of the filament in an experiment for which the background magnetic field is Bo = 900 G.

Figure A.6: (a) Spectrogram calculated using a sliding FFT window to compute the spectrum for individual times. The central time of the FFT window is used to position the result. The transition to broadband turbulence appears as a sharp event at t = 9 ms. (b) Spectrogram calculated using the CWT method. The transition to turbulence appears as a process beginning near t = 7 ms. [Full Size]

In Fig. A.6a a coherent drift-Alfvén mode (f ≈ 30 kHz) and its harmonics is observed to Doppler shift during the first 9 ms of the temporal evolution. This same mode is clearly visible in the CWT representation of Fig. A.6b. Any differences in frequency resolution between these methods is insignificant. In fact, the CWT results in a seemingly better identification of mode frequency for the drift-Alfvén wave. The delay between time zero (beginning of beam heating) and the appearance of features within the spectrograms is due to the time it takes for these features to appear at the measurement position 256 cm axially separated from the beam.

The most striking difference between these two results concerns the temporal features of the transition to broadband spectra. The FFT result of Fig. A.6a shows a sharp transition from the coherent mode to a broadband spectra at t = 9 ms. The CWT result of A.6b indicates that this transition may not be as dramatic and that it begins earlier than t = 9 ms. The sharp transition shown in the FFT spectrogram results from the windowing method employed. The windowed FFT computes a handful of individual spectra over user-defined time regions of the input time series. The larger the time window used, the better the frequency resolution of the spectra according to the Δf relation given earlier. Improved frequency is achieved by reducing temporal resolution. The sharp transition in the FFT spectrogram appears because one window covers none of the broadband fluctuations while the next window is the first one to detect them. The window that first detects them returns a significantly different result from the preceding window. The CWT method does not window across the data and therefore provides a better temporal resolution in this case (the poor frequency resolution at high frequencies is the tradeoff for this method). It is generally the case that a combination of FFT and CWT techniques provides the most complete analysis of spectral features in an experiment.

Thesis - Chapter 7: Conclusions

Sat, 03 Sep 2011 14:12:23 +0000

Thesis - Table of Contents

Conclusions

Studies of filamentary pressure structures are an ongoing work within the LAPD-U laboratory. The work of Burke, et al. [1998, 2000a, b] (and [Burke, 1999]) established the ability of this experimental geometry to observe classical transport in magnetically confined plasmas. An increase in transport and the loss of classical confinement for larger pressure gradients and/or longer heating periods naturally provided an opportunity to study turbulence. A reproducible characteristic of power spectra from measurements made during the anomalous transport phase of the experiment led to the identification of Lorentzian pulses in time series data. Large amplitude examples of these pulses emphasized the role of low frequency oscillations and led to the identification of a spontaneous thermal wave.

Spontaneous Thermal Waves

The temperature filament acts as a resonance cavity for spontaneous thermal waves [Pace, et al., 2008b]. Thermal waves result from the diffusive propagation of thermal energy across boundaries that separate regions of largely differing thermal conductivity. In the filament, this wave manifests itself as coherent fluctuations in electron temperature near the filament center. A drive source has not been identified, though it is clear that the input heating does not oscillate in such a manner as to be solely responsible.

The wave number vectors of thermal waves depend on the thermal conductivity of the medium. Wavelength measurements, in the form of phase velocity or amplitude decay measurements, allow for calculation of these plasma parameters. From Eq. 3.5 it is seen that the electron collision frequency can also be calculated based on knowledge of the thermal wave's properties. Given that the measurement of temperature fluctuations due to the presence of a thermal wave can be considerably simpler than measurement or modeling of fundamental plasma parameters, it is natural to suggest that a purposely driven thermal wave may be useful as a diagnostic instrument. Such a wave can be driven by external heating (e.g., electron cyclotron resonance heating) or with an electron beam setup similar to the one used here.

Exponential Power Spectra Related to Lorentzian Pulses

The power spectra calculated from time series measurements of plasma properties are found to exhibit an exponential dependence in frequency that is the result of Lorentzian shaped pulses in the raw signals [Pace et al., 2008a]. The exponential constant of the spectral shape is found to agree with the time width of the generating pulses, thereby providing support for the relation between the pulses and spectra. In the temperature filament experiment it is observed that the pulses appear only after the system transitions from classical transport into a turbulent regime of enhanced transport. Observations of exponential power spectra from many different plasma experiments suggest that Lorentzian pulses are a universal feature of plasma turbulence driven by cross-field pressure gradients. Comparison with a density gradient experiment of different scale length shows similar observations and demonstrates that this phenomenon is a general consequence of systems featuring pressure gradients.

Coherent drift-Alfvén eigenmodes present in the temperature filament suggest a generation mechanism for the Lorentzian pulses. The pulses appear to result from convective bursts of a nonlinear interaction between two drift-Alfvén modes of different m-number. Work on this topic is detailed in Shi [2008].

Future Work

Just as this thesis extends the earlier work performed by Burke [1999], it is possible that future thesis projects remain to be completed within this versatile experimental geometry.

Thermal Waves

The initial study of the thermal wave presented here should be expanded. Diagnostic difficulties prevented the complete elucidation of the wave's properties by limiting measurements of the wave vectors. In order to minimize perturbations, an imaging diagnostic should be considered. There is a considerable amount of visible light emanating from the temperature filament and it may be possible to detect oscillations caused by the thermal wave. It is unclear how this may be implemented in a manner that provides axial resolution.

During this study of the thermal wave, multiple attempts were made to forcibly drive the wave by modulating the input beam heating. These attempts were unsuccessful due to the difficulty in modulating the beam current during the afterglow phase. This difficulty might be overcome by building an anode onto the existing beam structure. The LAPD-U anode is 16 m away from the crystal, creating a weak electrical connection even in the presence of the plasma. Finer control of beam emission will likely be achieved by bringing the anode closer to the crystal. Since the LAPD-U anode cannot be adjusted to this end, a change to the beam structure is warranted. Controlled experiments of driven thermal waves will be useful for further developing the diagnostic capabilities of this wave.

Exponential Spectra

While there is a wide range of theoretical work to be performed with regard to the exponential spectra, there is still a contribution to be made by experiments. The next phase of experimental research will likely focus on spatial measurements of the Lorentzian pulses. Some measurements indicate that these pulses are azimuthally localized, but the lack of a radial or azimuthal array of probes makes it difficult to reach any certain conclusions in this regard. The spatial behavior of the pulses is vital in order to understand their generation. This will also help in determining the similarities and differences between the time series pulses from other types of plasmas. Filamentary structures in fusion devices are generally known to be coherent structures that propagate radially outward. The pulses in the temperature filament experiment do not appear to feature similar behavior and it will be instructive to determine this with certainty.

Filamentary Structures

Nonlinear interactions between drift-Alfvén waves might be studied by the inclusion of a second filament. Experimentally, this is accomplished by adding a second electron beam to the existing setup. The two beams, with an adjustable radial separation, each generate a filamentary structure that supports coherent drift-Alfvén eigenmodes. Adjustments to the amount of overlap of these modes should allow for control over the types and amplitudes of their interactions. The resulting system may begin to accurately model that of the limiter-edge experiment discussed in Ch. 5. The pressure gradient of that experiment creates a full range of drift-Alfvén modes simultaneously. Plasma edges in other devices may have the same quality. As the single temperature filament aids in describing the behavior of these waves and plasma transport in general, so too might the double-filament experiment begin to reproduce the behavior of plasmas in which turbulence is always fully developed.

The temperature filament experiment provides a basic plasma environment that can simulate the situation encountered in other devices. The physics uncovered in this geometry is applicable to space and laboratory plasmas. Many research topics remain to be explored within this experimental setup and a wealth of “future work” will certainly add to the field of plasma physics.

Thesis - Chapter 6: Plasma Flow Parallel to Background Magnetic Field

Sun, 28 Aug 2011 14:09:28 +0000

Thesis - Table of Contents

Plasma Flow Parallel to Background Magnetic Field

Plasma flow is studied in relation to another aspect of this thesis, namely the thermal wave. This section examines describes flows aligned with the applied background magnetic field, i.e., parallel flows.

Spatiotemporal Behavior and Modeling

A comparison of the time evolution of parallel flow between measurements and a theoretical model is presented in Fig. 6.1. The model is designed using Eqs. (2.4)-(2.7). This analysis is restricted to the early times that correspond to the classical transport regime of the filament. The position of this measurement is (r,z) = (0, 384) cm. The rise in the measured flow velocity immediately after beam heating begins is not captured in the model result. It should be noted that differences in the numerical values between the flow measurements and the code are expected to be larger than those observed for temperature comparisons. The model calculates flow values by solving for the absolute value of the flow and then normalizing it to the sound speed as calculated using its own temperature value. Therefore, any differences present in the absolute flow speed (which cannot be measured by the Janus probe method) are compounded by the differences in temperature.

Figure 6.1: Comparison of measured and theoretical parallel flows in the filament center for early time evolution. The model result captures the qualitative behavior of the flow, except for the initial (and possibly transient) increase in magnitude. The measured trace has been smoothed to reduce higher frequency fluctuations. [Full Size]

Figure 6.2 compares two-dimensional flow profiles from the experiment with the results of the fluid code that is also used to compare classical temperature profiles. The result is shown for the axial position z = 384 cm. The qualitative agreement is excellent as both results identify a wide radial profile of plasma flow. One difference is the sharp edge that appears just after t = 1 ms in the measurement. It is not yet understood why the model would result in a softer edge profile. The direction of the flow also agrees within these two representations. Early in the filament evolution and far into the outer region of the radial profile, the flow is directed toward the LaB6 beam source. Some time after the heating process has begun, this direction is reversed. Pressure gradients naturally present in the afterglow plasma compete with those generated by the beam heating. This interaction may explain the sharp boundary observed in the experiment but not entirely reproduced by the model.

Figure 6.2: Parallel flow velocity as measured (color contour) and as predicted by classical theory (black lines). [Full Size]

Quantitatively, the agreement between the model and the measurements is nearly perfect at the far edges and early times, and better than 20% at the filament center. The gradient region exhibits larger discrepancy between the values (e.g., at t ≈ 2 ms and r > 1.5 cm). The theoretical result places the peak flow just off the center of the filament. It is noteworthy that the largest absolute value of flow is centered on the filament in the model. The off-center feature occurs because the model temperature profiles decrease faster than the absolute velocity. The temperature decline in the radial direction reduces the ion sound speed much faster than the corresponding reduction in absolute flow speed. Similar off-center peaks in Mach number are not observed in the measurements.

Figure 6.3 is a plot of the measured and theoretical electron temperatures and flow profiles corresponding to time t = 1.3 ms in Fig. 6.2. This display clarifies the evaluation of that preceding contour. Both radial profiles are broader than typical temperature profiles for the filament. The measured profile exhibits a steady decay of the flow and appears to never reach a lowest level. This contrasts the result from the model in which the profile is still broad but does flatten out near r = 2 cm.

Figure 6.3: Radial profiles taken from time t = 1.3 ms for z = 384 cm. (a) Electron Temperature (b) Flow [Full Size]

An interesting overview of flow behavior across multiple axial positions is shown in Fig. 6.4, which presents contours of the parallel flow as functions of radial position and early time. Notice that panel (b) was acquired with a radial range that differs from the other two contours. Panel (a) displays the flow from within the heat source region, Q, at z = 64 cm. Oscillations arise near t = 1 ms that are comparable to those shown in the time trace of Fig. 2.12. These same oscillations are also observed at the other axial positions of z = 224 cm in (b) and z = 448 cm in (c). The downstream regions demonstrate the coherent fluctuations prior to their appearance in the Q region. Panels (b) and (c) exhibit striations (which are representative of oscillations in this contour display) before t = 1 ms and they do not appear until after this time in (a). This observation suggests that the cause of the oscillations is within the heated filament and not driven by the beam heating directly, i.e., there are no such oscillations in the applied heating power.

Figure 6.4: Parallel flow contours at different axial positions for the same experimental parameters, Bo = 900 G, Vbeam = 20 V. These are not from the same discharge, however, because a number of discharges is necessary to obtain the contour measurement across the radial dimension (note also that (b) is acquired across a narrower radial span than the other panels). (a) z = 64 cm (b) z = 224 cm (c) z = 448. [Full Size]

Figure 6.4 also highlights the time evolution of the broad flow profile. This is determined by the interface of the opposing flows in the r > 0.1 cm regions. The negative flow values of (a) indicate that the plasma is flowing toward the LaB6 electron beam. Plasma flowing away from the beam is restricted to a region narrower than the beam diameter. At z = 224 cm the flow direction changes suddenly near r = 0.1 cm and t = 1 ms. The profile of flow away from the LaB6 crystal extends out to r > 0.2 cm. Finally, at z = 448 cm the flow directed away from the heat source reaches out to r > 1.0 cm, far wider than at the other axial positions. Similar profiles are predicted by the code because the parallel gradients are largest away from the LaB6 crystal. Closer to the LaB6 source, the parallel pressure is flat and the flow drive is reduced. Furthermore, the background electron temperature is lower farther from the beam, leading to a much lower ion sound speed and greater Mach number even in instances in which the absolute flow speed is constant.

An example of the time evolution of the Mach number at different axial positions is given in Fig. 6.5. The time evolution of the flow at the filament center confirms the larger Mach numbers corresponding to larger z values. Large amplitude fluctuations appear first at the z = 384 cm position (t ≈ 4.5 ms). Similar fluctuations next appear on the z = 576 cm trace just after 4.5 ms and then finally appear on the middle axial position near 5.5 ms.

Figure 6.5: Parallel flow at the filament center for three different axial positions. The axial positions are z = 576 cm (black, top trace at t = 8 ms), z = 480 cm (red, middle trace at t = 8 ms), and z = 384 cm (blue, bottom trace at t = 8 ms). [Full Size]

Relation to Thermal Wave

Measurements of parallel flow in the center of the filament indicate that the thermal wave modulates the flow. Figure 6.6 is a plot of the parallel flow at the filament center in the Q region at z = 64 cm. The time separation between the largest peaks in the flow correspond to the period of the thermal wave measured in the same parameter regime (but not in the same discharges).

The higher frequency oscillations in Fig. 6.6 are approximately 35 kHz, placing them within the frequency range of the coherent drift-Alfv$eacute;n mode. The earliest range of the plot shows growing amplitude for these oscillations with a culmination in the large peak that sets the beginning time for comparison to the thermal wave.

Figure 6.6: Time trace of parallel flow at the filament center in the Q region. The time period between large amplitude peaks is highlighted for comparison to the similar period of the thermal wave. The faster oscillations correspond to the drift-Alfvén wave. [Full Size]

Figure 6.6 represents typical fluctuation behavior of the parallel flow. These oscillations exhibit an amplitude of,

As seen in the figure, the flow exhibits periodic behavior on both a slower timescale (due to thermal wave modification), and a faster timescale (the higher frequency component unlabeled in the figure). The higher frequency component corresponds to the drift-Alfvén frequency, f ≈ 40 kHz. Both the thermal wave and the drift-Alfvén wave modulate the parallel flows.

Though the magnitude of the flow fluctuates, it does not change direction. As in Fig. 6.6, the value of the Mach number remains positive throughout the fluctuations. This indicates that the direction of the flow is always toward the LAPD-U anode (i.e., away from the beam heat source). Flow oscillations therefore represent a decreasing/increasing response, and not a change in the absolute direction of the mass transport. These fluctuations do not alter the mean flow.

Flow as a Function of Input Power

Figure 6.7 presents time traces of parallel flow for two axial positions and various input heating powers. The nearly steady-state Mach number for 20 V in (a) is approximately 0.3, while that value approaches 0.6 further downstream at z = 576 cm in panel (b).

Figure 6.7: Parallel flow for multiple input heating powers (expressed as beam voltage) at (a) z = 480 cm and (b) z = 576 cm. [Full Size]

Larger Mach numbers for lower values of the heat input also support the model's finding. Lower values of heat input result in a shorter length of the resulting filament. For any fixed axial position within the filament for a 20 V experiment, a lower input heating brings the end of the filament closer to the measurement position (until the point at which the filament no longer extends to that position). Both panels show a maximum Mach number above 0.6, possibly indicating the end of the filament. In Fig. 6.7b it is possible that the 15 V experiment barely reaches the z = 576 cm position within the heating time interval. The 20 V case reaches this same maximum Mach number earlier in time and then features a decrease to a steady-state behavior. In contrast, the 12.8 V situation never reaches this maximum and is still increasing when the input heating is shut off. Perhaps longer heating times would allow this trace to match the behavior of the others.

The theoretical model accompanying this experimental work indicates that the filament length does not change appreciably with increased beam voltage. Furthermore, it suggests, by way of Fig. 6.8, that the end of the filament is well past the measurement locations presented in Fig. 6.7. Figure 6.8 is a plot of the theoretical axial electron temperature profile as a function of beam voltage. This plot represents the maximum length of the resulting filament. For the measurement positions of z = 480 cm and z = 576 cm, as shown in Fig. 6.7, the temperature filament certainly reaches the probe. Peaks in the parallel Mach flow are observed for beam voltages between 16 and 20 V in Fig. 6.7a. The model result concerning filament length contradicts the interpretation of the measurements that measured peaks in the flow correspond to the axial end of the filament passing through the measurement location. The temperature filament is predicted to extend beyond the axial measurement locations. Though not shown, the model also predicts that the temperature filament reaches its final length in a very short time, on the order of microseconds. The measured peaks in parallel flow occur much later in time compared to the development of the temperature filament.

Figure 6.8: Axial electron temperature profiles from a theoretical model for different values of electron beam voltage.

The model also suggests that the parallel flow will reach supersonic levels (i.e., M|| > 1) near the axial end of the filament. In Fig. 6.9 the parallel Mach number is plotted for a range of electron beam voltages. Comparing this with the theoretical axial electron temperature profiles of Fig. 6.8 shows that supersonic flow occurs near the end of the filament at z ≈ 800 cm. These results suggest that the axial extent of the filament may be determined by appropriately designed experimental measurements of the parallel flow.

Figure 6.9: Theoretical parallel Mach number for different values of the electron beam voltage. [Full Size]

Summary

Measurements of plasma flow is thus far limited to determination of the Mach number in the direction parallel to the applied background magnetic field. Heating from the electron beam results in a flow toward the main LAPD-U cathode-anode pair (i.e., away from the source that generates the temperature filament). Profiles of the parallel flow are considerably wider than those of electron temperature. A fluid model, the same one that provides descriptions of the classical temperature profile behavior, reproduces the qualitative behavior of the measured flow, though the quantitative agreement is not as strong as for the temperature comparisons.

The thermal wave modulates the parallel flow. This is observed in the appearance of flow pulses, or bursts, that occur with a time period corresponding to that of the thermal wave. The amplitude of flow fluctuations reaches values of δM|| / M|| ≤ 50%.

The fluid model of Shi produces axial temperature profiles that place the filament length within expectations based on the flow measurements. Furthermore, the model calculates that supersonic flows are achieved at the axial end of the temperature filament. The flows have been measured well short of the z ≈ 800 cm theoretical temperature filament length, possibly explaining the lack of a supersonic observation.