Thesis – Chapter 1: Introduction

Thesis – Table of Contents


1.1 Motivation

Anomalous transport is an area of great interest within the plasma physics research community. The field of magnetically confined thermonuclear fusion may benefit significantly from an improved understanding of this topic. It has already been shown that turbulent fluctuations increase the transport of mass and energy [Horton, 1999] in magnetically confined laboratory plasmas. Improved confinement can expedite the development of fusion reactors as controllable energy sources.

Space plasma research also encounters anomalous transport [Committee on Solar and Space Physics, 2004] across naturally occurring boundaries in temperature, density, and magnetic field. The modeling of space weather can be beneficially impacted by improvements in plasma transport understanding.

coronal loops

Figure 1.1: Coronal loops on the Sun as imaged by the Transition Region and Coronal Explorer (TRACE) satellite. Filamentary structures form along the magnetic field lines emanating from the Sun. Image taken from Stanford-Lockheed Institute for Space Research.

Filamentary pressure structures, meaning structures that are aligned along magnetic field lines with narrow radial extent compared to their length, are prevalent in both space and fusion plasmas. Figure 1.1 is a satellite photograph of the solar corona in which bright filaments are seen flowing along looping magnetic field lines. Energy transport along these filaments, and even through the solar wind en route to interaction with the Earth”s magnetic field, are ongoing areas of research. An example of filamentary structures from a fusion device is seen in Fig. 1.2, a photograph from the MAST fusion device. This image shows bright filaments in the outer edges of the device. They are the manifestation of edge-localized modes (ELMs) that transport hot plasma from the center of the device out to the walls. Controlling ELMs to minimize their transport or to avoid them altogether is presently a major effort within the fusion community [Evans et al., 2008].

Many features of filamentary structures remain unknown, including their capacity for transport, mechanisms leading to their generation, and which plasma waves they may produce. A difficulty in studying these issues within the space and fusion examples above is that the resulting systems are actually a mixture of many individual filaments. Interactions between the filaments and the existence of background instabilities complicate the interpretation of observations. This thesis utilizes an experiment in which a single filamentary structure is generated in the background of a quiescent plasma. The resulting system may be imagined as the isolation of one of the many structures seen in the previous two images. The fluctuation spectra and associated transport generated by this single filament proves rich with dynamic behavior. Studies related to plasma turbulence, spontaneously generated temperature waves, and non-linear interactions of drift-Alfvén waves are all performed within this configuration.

MAST filaments

Figure 1.2: Photograph of filamentary
structures in the MAST fusion device. Edge-localized modes appear as bright filaments as they conduct large amounts of energy and particles out of the confinement region. Image taken from UKAEA.

The experimental configuration used in this project was originally motivated by the desire to present experimental evidence for classical heat transport in magnetized plasmas. A summary of that successful effort is available as a Ph.D. thesis [Burke, 1999]. The theory of heat transport due to Coulomb collisions [Landshoff, 1949; Spitzer and Härm, 1953a; Rosenbluth and Kaufman, 1958; Braginskii, 1965] was developed nearly 50 years before it was quantitatively validated in a laboratory plasma [Burke et al., 1998; 2000b] using this configuration. The experiment consists of a narrow cylindrical region of warm plasma (Te ≈ 5 eV) embedded in a cold background plasma. The heated filament of plasma is manipulated to control the temperature gradient, thus driving classically described heat transport. Classical transport is always initially observed in this experiment, but if the heating is applied over a longer time interval or above a certain temperature threshold, the system transitions to a regime of enhanced, or anomalous, transport greater than that predicted by classical theory. Turbulent fluctuations are observed in this regime, and while some of their features have been investigated [Burke et al., 2000a], a mature understanding requires more detailed experimentation.

A summary of the transition from classical to anomalous transport in this experiment is provided by Fig. 1.3, a spectrogram of Isat power spectra (color contour) with the fluctuating component of a single Isat trace (I~sat, solid white) overplotted. The heated filament is generated at time t = 0 ms and maintained until t = 12 ms. Prior to t = 6 ms there is one well defined mode between 25 and 45 kHz. This is a drift-Alfvén eigenmode that has been detailed extensively both theoretically [Peñano et al., 2000] and experimentally [Burke et al., 2000a]. The presence of this coherent mode does not alter the transport levels, i.e., the observed transport remains classical during the presence of the drift-Alfvén wave. After t = 6 ms, a transition from coherent spectra to broadband spectra occurs. The transition is delineated by the disappearance of the coherent drift-Alfvén line into a broad region of power spread across many frequencies. Transport levels are enhanced, or anomalous, during times after this transition. All of this behavior occurs within a range corresponding to low frequency turbulence. The low frequency range is an area of active research within plasma physics, as discussed in the following section.

wavelet-based spectrogram

Figure 1.3: Time evolution of the power spectrum (color contour) with an overplot of the fluctuating component of an Isat signal (solid white) from the same spatial position. Coherent fluctuations of the drift-Alfvén eigenmode are visible for t ≤ 5.5 ms. After 5.5 ms there is a sharp transition to broadband spectra that correlates with the appearance of large relative amplitude pulses in the Isat signal.

Identification of the processes underlying low frequency turbulence in magnetized plasmas is an ongoing challenge within plasma physics [Krommes, 2002]. By “low frequency” it is meant that the frequency of the fluctuating quantity, ω, is less than the ion cyclotron frequency, Ωi. This topic is relevant to the magnetically confined fusion research community because turbulent fluctuations can enhance the transport of mass and energy [Horton, 1999], thereby degrading tokamak performance. The topic is also of interest in space plasma efforts [Committee on Solar and Space Physics, 2004] in which enhanced transport across naturally occurring boundaries in temperature, density, and magnetic field can result in major effects observable by space and ground-based instruments.1.2 Low Frequency Turbulence

A significant effort has been devoted to the identification of universal behaviors in the spectra of turbulent fluctuations. A rich literature exists for both laboratory [Chen, 1965, Kamataki et al., 2007, Labit et al., 2007, Škoric and Rajkovic, 2008, Budaev et al., 2008, Pedrosa et al., 1999, Stroth et al., 2004, Carreras et al., 1999, Zweben et al., 2007] and space [Tchen, 1973, Kuo and Chou, 2001, Milano et al., 2004, Zimbardo, 2006, Bale et al., 2005] plasmas. The cited references are merely a representative sample of the available literature. Kolmogorov”s early contribution [Kolmogorov, 1941] has had a major influence in these activities [Frisch, 1995]. In particular, that pioneering work makes a general prediction of algebraic spectral dependencies that has resulted in most modern spectral results being presented in a log-log format. Piecewise fits are then applied in order to extract power-law values for comparison to the Kolmogorov prediction. A large dynamic range is compressed by the log-log display, however, and important features related to the turbulence may be obscured. An exponential frequency spectrum is one such important feature, and its presence and underlying mechanism are described in this thesis.

1.3 Summary of Thesis Results

In the following an abbreviated description is presented regarding the major results obtained in this thesis. These are:

  • Confirmation of previous results due to filamentary geometry.
  • Observation of a spontaneous thermal wave in the absence of an externally driven source.
  • Observation of exponential power spectra associated with anomalous transport that are generated by Lorentzian pulses in measured time series data.
1.3.1 Confirmation of Physics Results Due to Plasma Geometry

The previously cited work of Burke, et al. was performed in the LAPD device prior to 2000. The present studies are performed in the machine that replaced the original LAPD, which has been named the LAPD-U, signifying it as an “upgrade” over the original. With similar plasma production sources and plasma properties, the major difference between these two machines is their length along the applied background magnetic field. The LAPD featured a plasma of less than 9.4 m in length. The LAPD-U plasma length is approximately 15 m. Throughout this thesis the LAPD-U designation will be used to emphasize the completely different linear device used for this work compared to the foundational efforts conducted on the LAPD.

Precisely because the LAPD-U is a different machine, all of the results in this thesis confirm that fundamental plasma physics is responsible for the observed phenomena, rather than the geometry of a particular device. The LAPD-U provides boundary conditions that were not present in the previous device, yet the coherent modes observed are the same, along with the important features of transport that have been re-observed.

1.3.2 Thermal Wave

The existence of low frequency, coherent, fluctuations is documented in the earlier work within this experimental environment [Burke et al., 2000a]. Observations show this is a coherent mode that, while seemingly unrelated to the generation of low frequency turbulence, is capable of strongly modulating the drift-Alfvén modes that are excited by the filament. These fluctuations are identified here as representing a spontaneously excited thermal wave. A thermal wave is the diffusive propagation of a temperature oscillation driven by a similarly oscillating source. Although thermal waves in plasmas have been studied [Gentle, 1988, Jacchia et al., 1991], and even manipulated to deduce subtle issues of anomalous transport in tokamaks [Mantica et al., 2006, Casati et al., 2007], controlled experiments in basic plasma devices are made difficult by the geometry of a magnetized plasma. The complexity arises due to the large difference in the thermal conductivities along and across the magnetic field, κ|| >> κ, requiring plasmas with significant length along the magnetic field direction.

The discrepancy in thermal conductivities results in an extended structure that acts as the cavity for a thermal wave resonator [Shen and Mandelis, 1995]. The results presented here represent thermal wave oscillations that appear without the setting of a driver. Other experimental work involving this phenomenon, including those referenced, drive the wave with a controllable heat source. The drive source is as yet unidentified here, though it is demonstrated that the electron beam heating is not the direct cause, i.e., there are no coherent low frequency oscillations in the beam source. A possible candidate for the drive source is the heat-flux instability that is found in the solar wind [Forslund, 1970] and in laser-plasma interactions [Tikhonchuk et al., 1995]. This work has been summarized in Pace et al., [2008b].

1.3.3 Exponential Spectra

exponential power spectrum

Figure 1.4: Semi-log plot of an Isatpower spectrum. Coherent modes, due to the presence of drift-Alfvén waves, can be seen in the range 20 ≤ f ≤ 120 kHz and coexist with the exponential in the range 20 ≤ f ≤ 200 kHz.

Exponential spectra from a variety of experiments are found throughout the published literature. This is made possible by the semi-log display some researchers have chosen to use for the results. Figure 1a of Xia and Shats [2003] exhibits exponential behavior over four orders of magnitude from floating potential measurements. This experiment was performed in a helical device that reported proof of an inverse cascade. Figure 1 of Fiksel et al [1995] features an exponential dependence in an experiment observing magnetic fluctuation-induced heat transport. Figure 6b in Kauschke et al. [1990] shows an exponential spectrum with embedded coherent modes for a nonlinear dynamics experiment in a low pressure arc discharge plasma. Figure 7 of Maggs and Morales [2003] presents an exponential spectrum from magnetic fluctuations at the free edge of the LAPD-U. The exponential spectra in these examples are readily identified because of the semi-log plot display. The appearance of such spectra in a wide variety of experiments suggests that it may also be present in other results where it is simply compressed by a log-log display. Figure 1.4 provides an example of an exponential power spectrum from the experiment. In a semi-log display, an exponential dependence appears as a straight line. This behavior is used to calculate the scaling frequency (decay constant) of the spectra for comparison with the time width of the Lorentzian pulses. The coherent peaks in Fig. 1.4 (located at approximately f = 30, 60, 90, and 120 kHz) coexist with the exponential behavior that extends from 20 ≤ f ≤ 200 kHz.

The power spectra, P, of measured fluctuations display an exponential dependence in frequency, P(f) ∝ exp(-2f / fs), where fs is a scaling frequency. This exponential feature is only observed after the temperature filament transitions into the enhanced, or anomalous, transport regime. Concomitant with the exponential spectrum is the observation of pulses or spikes in the time series data. These pulses, which can be either upward or downward going in amplitude depending on the measurement location, are Lorentzian in temporal shape. A Lorentzian pulse has an exponential power spectrum, leading to the conclusion that the appearance of these pulses causes the exponential spectrum. A brief summary of this work may be found in Pace et al. [2008a].

1.4 Thesis Outline

This thesis is composed of five chapters. Chapter 2 presents the laboratory device in which this study is performed, along with a review of the various diagnostics employed to measure plasma properties. Chapter 3 details the results surrounding the identification of a spontaneously generated thermal wave in the filament. This is the culmination of an effort to identify coherent oscillations featuring a lower frequency than the other previously known modes of the system. The thermal wave is likely to be supported in many filamentary plasma systems including the solar corona. Modification of the temperature profile by the thermal wave leads to large amplitude pulses in time series signals. These pulses are discussed in Chapter 4, which also presents evidence for a universal characteristic of power spectra in turbulent plasmas. Such spectra exhibit exponential dependencies in frequency and are found to result from the Lorentzian shape of the measured pulses. Similar spectra, and in many cases similar pulses, are observed in the existing plasma literature and in ongoing research at linear machines and tokamaks. A density gradient experiment performed in the same device as this thesis work also exhibits these pulses and exponential spectra. Chapter 5 compares the density gradient experiment to the temperature filament experiment as part of the argument for the universal nature of the exponential spectra and Lorentzian pulses. Conclusions and a unifying summary of these topics are presented in Chapter 7. Finally, the appendices present results on plasma flows in relation to the primary topics, techniques of wavelet analysis that have been applied in power spectra calculations, and a summary of techniques employed to detect the Lorentzian pulses that generate exponential power spectra.

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