David Pace

Thesis - Chapter 2: Experimental Setup and Overview of the Temperature Filament

Sat, 03 Oct 2009 20:47:41 +0000

Experimental Setup and Overview of the Temperature Filament

Large Plasma Device

The Large Plasma Device (LAPD-U), part of the Basic Plasma Science Facility (BaPSF) at UCLA, provides an ideal parameter regime in which to perform this experimental investigation. The present device, as shown in Fig. 2.1, is a larger version of the one described in Gekelman et al. [1991].

Figure 2.1: Dimensional schematic of the LAPD-U, illustrating the long axial extent that is vital to the parallel heat transport studies performed. [Full Size]

The LAPD-U produces plasma by discharging a cathode-anode pair. A barium-oxide coated, nickel cathode is heated to emission temperature near 850 °C. A wire mesh molybdenum anode is positioned 50 cm away from the cathode. A large capacitor bank connects the pair and discharges a few thousand Amperes of current for some fixed bias voltage (typically in the 65 V range). The anode mesh is 50% transparent, allowing half of the electrons emitted by the cathode to pass through and travel down the length of the device. This produces plasmas that fill the 20 meter-long vacuum chamber. During the main discharge (a flat-top current pulse between the cathode-anode pair lasting up to 12 ms), the resulting plasma column extends up to 60 centimeters in diameter.

The LAPD-U provides a wide range of plasma parameters for study. The solenoidal magnetic field may be set to any value between 500 and 2500 Gauss. Magnetic coil elements are separately connected, allowing for various field configurations (mirror, beach, etc.). Inert gases are typically used to generate plasmas since these do not chemically interact with the oxide coating of the cathode. Table 2.1 lists typical plasma parameters for the LAPD-U.

Table 2.1: LAPD-U Parameters

Temperature Filament Setup

Afterglow Plasma

Figure 2.2: Time evolution of the discharge current through the anode-cathode pair (black, Idis) and the line-averaged column electron density (red, ne) during an LAPD-U plasma-pulse. [Full Size]

The principal results of this thesis are obtained in the afterglow phase of the LAPD-U plasma. The afterglow is a common term used to describe the plasma that remains in the vacuum chamber after a cathode discharge has been turned off. Figure 2.2 is a plot of the LAPD-U discharge current (black trace) and the line-averaged electron density (red trace) as measured by an interferometer. A double-sided blue arrow extends across the post-discharge regime in which the plasma is in the afterglow phase. When the discharge shuts off there is no longer a heating source to maintain plasma temperature. Thermal conduction to the ends of the device causes the electron temperature to decay rapidly, on the order of microseconds. Diffusion is a much slower processes, however, so the plasma density decays on the order of milliseconds. As a result, the LAPD-U afterglow plasma features a very low temperature (Te, Ti) < 1 eV) while maintaining a moderate density (ne ≈ 1 × 1012 cm-3) for the first 30 ms after the primary discharge ends. Most of the temperature filament experiments begin at a time 0.5 ms after the discharge current ends and last for 10 to 15 ms.

Basic Overview

A small heat source is applied at one end of the LAPD-U during the early afterglow phase in order to create a temperature filament that is approximately five times hotter than the surrounding background. Plasma properties are measured throughout the resulting three dimensional structure over time scales that allow the relevant transport and turbulence behaviors to arise.

A schematic of this system (not to scale) is given in Fig. 2.3. A cylindrical coordinate system is shown and is referenced throughout this treatment. The axial coordinate is labeled z, the radial coordinate r, and the azimuthal coordinate is θ. The electron beam is located at (r,z) = (0,0) cm. The heat source is modeled as a one meter long, 3 mm across rigid cylinder of hot plasma that is placed 16 m away from the main cathode/anode. The heating done in this region is classically transported to setup the filament structure that may reach 12 m in length and 1 cm in diameter. Further details of this system are presented in the following sections.

Figure 2.3: Not to Scale. Schematic of the temperature filament experiment highlighting the extended structure parallel to the background magnetic field. [Full Size]

Heating Source: Lanthanum Hexaboride Electron Beam

Figure 2.4: Photograph of the LaB6 electron beam. The beam is a single crystal of LaB6 permanently fixed in a mounting structure. The crystal is held in place by Mo-Re posts. A common ballpoint pen is provided for scale. [Full Size]

Plasma heating is accomplished by the injection of energetic electrons from a crystal of lanthanum hexaboride (LaB6). The work function of LaB6 is low enough that direct heating of the crystal, accomplished by running current through it, provides adequate energy for thermionic emission. An operating temperature of 1800 °K is used. Such crystal temperatures are achieved by running a current through the crystal itself. Figure 2.4 shows the electron beam device as purchased from Applied Physics Technologies (a-p-tech.com): a single crystal of LaB6 is held in place above a ceramic washer by molybdenum-rhenium (Mo-Re) posts. On the underside of the washer, the posts provide the access point for driving the heater current.

Figure 2.5: Circuit diagram of the electron beam heat source. Beam current is driven between the LaB6 crystal and the LAPD-U anode through the afterglow plasma. [Full Size]

The LaB6 crystal is mounted on the end of a standard LAPD-U probe shaft and placed in the plasma away from the main cathode/anode (drawn in Fig. 2.3). The main LAPD-U anode also serves as the anode for beam circuit, as shown in the diagram of Fig. 2.5. A heater current is continually run through the crystal to maintain emission temperature at all times during an experiment. The anode connection includes a pulse circuit that applies the beam voltage, Vbeam, at a prescribed time within each LAPD-U discharge. The applied voltage set by Vbeam determines the energy of the emitted electrons, which allows the input heating power to be varied. Electron beam current is measured by monitoring the voltage across a 1 Ω resistor connected within the current path. Unless otherwise specified, the beam energy is 20 V for all experiments performed in helium. The ionization energy of helium is approximately 24.6 eV, which allows these experiments to be conducted with minimal concern for density production from the injected electrons. Some experiments are conducted at energies above 24.6 V, but these are not presented as part of the heat transport study.

Figure 2.6: The LaB6 crystal is seen glowing white hot (center) during a discharge pulse in the LAPD-U. The purple background seen inside the small windows is the typical LAPD-U helium plasma. [Full Size]

Figure 2.6 shows the heated crystal embedded in a plasma discharge. The distance between the yellow magnet coils is approximately 32 cm.

Diagnostics

The length of the LAPD-U plasma along the applied background magnetic field is approximately 16.6 m. Diagnostic ports are placed every 32 cm along this entire length, allowing for excellent axial resolution in this transport experiment.

Langmuir probes are used because they provide excellent spatial resolution (determined by the size of the probe) and time resolution (limited by the acquisition electronics). Furthermore, probes made out of nearly any metal can withstand the 5 eV plasma temperature throughout an entire experimental run while accessing the entire plasma volume of the device. The probes utilized in this study are made of either tantalum or nickel. Spatial access is provided by probe drives that move the probes in a full two-dimensional plane at any one axial position. Further details of Langmuir probe usage will be presented immediately preceding the display of results so that the method used to obtain the measurement may be explained in context.

In addition to the fundamental quantities of temperature, density, and flow velocity, the Langmuir probes are used to measure ion saturation current, Isat. Ion saturation current is related to electron temperature and density by Isat ∝ ne \√Te, meaning that fluctuations in these quantities are mixed by an Isat diagnostic. It is shown later that the coherent modes are accurately represented by an Isat measurement in situations where pure density or temperature measurements are not possible.

Measuring Electron Temperature

Swept Probe Technique

Electron temperature measurements using Langmuir probes most commonly employ the sweep method. Varying the applied voltage to a probe, V, while recording the current collected, I, results in a current versus voltage characteristic, the I-V curve, that can be fit to determine the electron temperature, electron density and plasma potential. This method is widely known and has been explained in the plasma physics literature (see Ch. 2 of Hutchinson [2002]).

Figure 2.7: Traces of the I-V characteristic are presented from the four distinct regions of plasma heating. The main discharge (black) represents the standard LAPD-U plasma as generated by the cathode-anode system. The early afterglow (red) occurs after the main discharge has ended, but before the beam heating begins. The heating stage (green) represents the primary experiment during which the temperature filament is maintained. Finally, the late afterglow (blue) represents the stage occurring after the beam heating has ended. [Full Size]

One problem in applying the swept probe method in fluctuation studies is that the parameters obtained represent averages over the time of the complete voltage sweep. A useful measurement of any wave requires the sweep to be performed at well over twice the frequency of the wave so that the Nyquist frequency of the resulting data also remains above that of the wave. A reproducible and established system for producing such a sweep (with frequency > 50 kHz) is difficult to build due to capacitive effects. An alternate method for measuring electron temperature using the I-V curve is developed.

From Isat measurements it is apparent that the plasma discharge and resulting temperature filament are highly reproducible. This technique involves maintaining a fixed probe voltage for a series of 20 discharges and then incrementing the voltage for the next 20 discharge series. After repeating this procedure for 100 voltages, an ensemble of 2000 discharges is used to construct an I-V characteristic corresponding to every time point of the acquisition. These characteristics allow for the calculation of plasma parameters with time resolution equivalent to that of the common Isat~data sets.

Figure 2.8: Electron temperature in the radial center of the filament and axial position z = 224 cm as measured using the swept probe technique. [Full Size]

Sample I-V traces are shown in Fig. 2.7. Traces from four different time periods are plotted because the voltage range over which the collected current is exponential (i.e., the range over which a fit is inversely proportional to the electron temperature) is found to vary with time. In order to automate the analysis of thousands of data signals it is necessary to determine the voltage range over which the temperature fits should be performed. This required manually examining selected traces from each probe position. The relevant voltage range is different for each region: main LAPD-U discharge (black), early afterglow without beam heating (red), beam heating (green), and late afterglow after the beam is shut off (blue). Electron temperature is calculated through linear fitting of the types of curves shown here, generally within the range -15 ≤ Vprobe ≤ 5 V.

Calculation of the temperature is not possible in the late afterglow because the exponential range falls between two of the recorded voltages. There is no possibility of an accurate linear fit since the voltage range is smaller than the voltage resolution of the sweep.

Figure 2.9: Head of a two-sided triple probe placed against a ruler (cm scale). Three disc tips are visible, with another set of three on the other side. [Full Size]

An electron temperature measurement acquired from the swept probe technique is shown in Fig. 2.8 for the radial center of the filament. There is no direct filtering of this trace, though fluctuations are reduced by the ensemble nature of the data collection. The steady-state temperature reached in the later half of the time series is also observed with the companion technique for measuring electron temperature, the triple probe technique.

Triple Probe Technique

Electron temperature measurements are also made with triple probes in this thesis. The triple probe technique [Chen and Sekiguchi, 1965] uses three probe tips to simultaneously observe different regions of the I-V characteristic. This provides a continuous measure of temperature, floating potential and ion saturation current. Figure 2.9 shows the three tips of one of the triple probes used. These are 1 mm diameter discs embedded in a low-outgassing epoxy that is plasma facing.

Figure 2.10 is a circuit diagram illustrating the connections between the tips. Tips 1 and 2 are connected in fashion typical of Isat measurements. A fixed bias is applied between them and the voltage across a resistor in series is acquired. The unique aspect of the triple probe is that tip 3 is used to measure floating potential, Vf. The potential between the tip measuring Vf and the electron collecting tip (2) is labeled as VTe and is related to the temperature according to, Te = VTe / ln(2) (Eq. 2.1), which is also derived from Chen and Sekiguchi, [1965].

Figure 2.10: Circuit diagram of the triple probe system. The probe tips are numbered and the measurement points are labeled according to whether they record electron temperature (VTe), floating potential (Vf), or the voltage across the resistor corresponding to ion saturation current (VR). [Full Size]

An example temperature trace from a triple probe measurement is presented in Fig. 2.11. This result is in qualitative agreement with the swept probe result of Fig. 2.8, even though the measurements are performed in different parameter regimes within different experimental run days. Both temperature traces identify the low frequency, coherent oscillations of the thermal wave (seen at t = 5 ms in Fig. 2.11) and a nearly steady-state temperature during the latter half of the heating cycle. Differences between these two results stem from the different background parameters (neutral fill pressure, axial position, heating power, etc.). The agreement between these very different methods suggests that both are valid in studying the filamentary system.

2.2.6 Measuring Parallel Flow

The plasma flow measurements employ the Mach probe method of comparing Isat collection on probe faces oriented in opposite directions. That is, the surface normals of the two probe collection areas point in opposite directions. For two such probe faces, the parallel flow Mach number, M||, is (see page 85 of Hutchinson [2002]),

where 1 and 2 denote the probe faces and the Mach number is defined as M = v/Cs where Cs is the ion sound speed,

in which γ is the adiabatic index, Z is the charge factor, and μ is the ion mass factor.

Figure 2.11: Electron temperature as measured using the triple probe technique at the filament center and axial position z = 384 cm. While this is a different parameter regime compared to the swept probe measurement of Fig. 2.8, the result is qualitatively similar with respect to the detection of thermal waves and a steady-state temperature at the end of the heating cycle. [Full Size]

It should be noted that parallel flow results presented here are likely to be underestimates of the actual value. The factor of 0.45 given in Eq. 2.2 may be further modified due to the small size of the Langmuir probe tips compared to the ion gyroradius [Shikama et al., 2005]. Errors associated with probe measurements due to calculation of the effective collection area and pitch-angle with respect to the background magnetic field prevent a significant improvement in the accuracy of the reported flow Mach numbers. Regardless, the qualitative behavior of these results serves as a useful motivation for future efforts.

This analysis can be applied to probe faces with any orientation to the magnetic field, though the parallel orientation is the simplest theoretically. Perpendicular flows are difficult to measure in the temperature filament experiment through the Mach probe method because a probe with full radial and azimuthal diagnostic capability is also large enough to cause a disruptive perturbation. Parallel velocity measurements are made with Langmuir probes featuring the smallest possible surface area. In most cases this is a one millimeter diameter disk or a rectangular tip with the largest dimension on the order of one millimeter.

A limitation of this measurement is that it provides scaled information about the flow velocity with respect to the ion sound speed instead of the absolute velocity. In some instances a temperature measurement is available for the same set of discharges and the calculation of an absolute velocity is possible. In most cases presented in this thesis, however, the Mach probe measurement is performed without an auxiliary temperature measurement and the study focuses on the presence of supersonic flows and their relation to the other processes of the system.

Figure 2.12 demonstrates the calculation of plasma flows using Isat measurements from a double sided, or Janus, probe. These measurements are made at (r,z) = (0,64) cm, which places the probe within the heating region of beam. Recall that the heating source of this experiment is modeled as a one meter long, three millimeter wide cylindrical plasma made hot by the electrons emitted from the LaB6 crystal. The orientation of the probe tips in the figure is labeled according to which end of the LAPD-U they face. One tip, the Isat Beam Facing tip, faces the LaB6 crystal. The other tip, the Isat Anode Facing tip, faces the main anode of the LAPD-U. Flow analysis is performed with the anode facing signal in the denominator of Eq. 2.2, meaning that positive flow values correspond to plasma flowing toward the main LAPD-U anode. This convention is chosen because we expect the flow to dominate in this direction since the pressure gradient source is located at the opposite end of the device.

Figure 2.12: (r,z) = (0,64) cm, Bo = 900 G. The Isat traces from both sides of a Janus probe (red and blue) are plotted in comparison to the parallel flow (black) calculated from them. [Full Size]

Coherent oscillations in the parallel flow are observed in Fig. 2.12. From the raw Isat traces it can be seen that the flow oscillations do not simply mirror the Isat measurements. In particular, the pulse-like event at t = 1.06 ms results from the large dip in Isat measured on the anode facing tip and the lack of a similar reduction at the beam facing tip. Later in time at t = 1.18 ms a coherent oscillation appears on the anode face and is delayed on the beam face. This behavior results in the calculation of another oscillating flow.

2.2.7 Transport Modeling

The previous work using this experimental geometry to verify the existence of classical heat transport compared measured results with those of a code that modeled classical transport (see Section IV. of Burke et al., [2000b]). A similar effort is incorporated into this thesis.

A thesis in plasma theory by Shi [2008] is a companion to this experimental treatment. The transport code is implemented by Shi is based on the equations of Braginskii [1965]. This set of equations includes the plasma continuity equation, the momentum equation, and the power balance equations. After applying the restraints of quasi-neutrality and cylindrical geometry along with the boundary conditions of the present experiment, these equations reduce to,

where Vz is the flow velocity parallel to the background magnetic field, Rin is a collision operator for ion-neutral collisions, Qb is the heat input from the electron beam, τe is the electron collision period, and σn is the ion-neutral collision cross-section. The model provides expected classical behavior for the density, electron temperature, and parallel flow evolution. This thesis is concerned with the experimental aspects of the temperature filament environment. The thesis written by Shi is concerned with the theoretical issues of the temperature filament, including heat transport and non-linear interactions between drift-Alfvé waves. Various figures within this thesis will compare measurements with Shi's model results.

2.3 Filament Behavior

2.3.1 Temporal Behavior

The control circuit of the electron beam allows both the heating start time and its duration to be adjusted. The temporal evolution of the temperature filament exhibits three stages, as illustrated by Fig. 2.13. The solid black curve in Fig. 2.13 corresponds to the electron temperature measured, using a small triple probe, at an axial distance 384 cm from the beam injector at the radial center of the filament. For this figure the strength of the confinement magnetic field is 900 G.

Figure 2.13: Electron temperature and beam current at (r,z) = (0, 384) cm. For the heating case (solid black), the electron beam is activated 0.5 μs after the LAPD-U discharge ends. The heated filament reaches temperatures comparable to that of the main plasma and is much hotter than the background afterglow plasma (dashed red). In this representation the beam current (dotted blue) is an ensemble trace while the heating is from a single shot. [Full Size]

The injected beam current is represented in Fig. 2.13 by the dotted blue curve, with t = 0 corresponding to the time when beam injection starts. The dashed red curve shows the decay of the electron temperature in the absence of beam injection, i.e., during the afterglow. It is seen that the heat source supplied by the beam causes a significant increase in temperature. For an interval of about 2 ms after the beam injection begins, a quiescent temperature evolution is observed. This behavior corresponds quantitatively to that predicted by the classical theory of heat transport based on Coulomb collisions [Braginskii, 1965; Spitzer and Härm, 1953b], as has been documented extensively in previous work [Burke et al., 2000b, 1998]. A second stage can be identified approximately 2 ms after beam injection when high-frequency oscillations become apparent in the electron temperature. These oscillations correspond to coherent drift-Alfvén waves driven unstable by the electron temperature gradient [Peñano et al., 2000] and have been described in detail in a previous publication [Burke et al., 2000a]. A third stage appears after 5 ms of beam injection. At this later time, low-frequency oscillations mixed with incoherent high-frequency oscillations are observed.

Figure 2.14: Oscillations in ion saturation current due to (a) coherent drift-Alfvén mode and (c) thermal wave. (b) Two-dimensional contour of the cross-covariance, R12, between two probes with axial separation Δz = 160 cm. [Full Size]

Figure 2.14 provides a useful summary of the individual characteristics of the low frequency thermal waves and the higher-frequency drift-Alfvén waves. The time range sampled overlaps between the second and third stages identified in Fig. 2.13, before the broadband develops. The center panel is a two-dimensional (across the magnetic field) spatial domain that covers the full radial extent of the filament. The color contour represents the cross-covariance between Isat signals from Langmuir probes separated by an axial distance of Δz = 160 cm. One probe is fixed at a position z = 384 cm away from the beam injector and at a radial position within the region of measurable drift-Alfvén wave activity. The other probe is placed at z = 544 cm and it scans a two-dimensional plane across the confinement magnetic field. Measured signals are digitally band-pass filtered to isolate oscillations due to the drift-Alfvén modes. The cross-covariance is calculated at zero time delay for each plasma discharge over a restricted time interval that captures the largest drift-Alfvén wave. The resulting signal is averaged over 20 independent discharge pulses at every spatial location sampled. The result is an amplitude-dependent measurement of the phase difference between the signals of the two probes, and therefore contains an imprint of the spatial structure of the coherent drift-Alfvén wave at one time in the plasma discharge. The light circular region at the center of the drift-Alfvén wave depicts the highly localized thermal oscillation. The trace in the top panel of Fig. 2.14 shows the time evolution of the ion saturation current at a location r = 5 mm, i.e., within the gradient region. It shows the coherent oscillations associated with the two-dimensional snapshot seen in the center panel. These oscillations have typical frequencies in the range of 25-40 kHz. The bottom panel shows the temporal evolution of the ion saturation current at the center of the filament. It is clear that the oscillations here have much lower frequency (by a factor of 5-8) than those in the gradient region. A cross-modulation that shows the faint presence of the drift-Alfvén mode can be seen in portions of the trace.

Figure 2.15: z = 384 cm (a) Radial temperature profile of the filament at t = 3.0 ms. (b) Two-dimensional contour of Isat illustrating the symmetric nature of the filament during early times. [Full Size]

Finally, at the latest times of the beam heating, 8 ≤ t ≤ 10 ms, the fluctuations become turbulent and there are no observable coherent modes in the spectra. Figure 2.13 illustrates that the amplitude of these fluctuations is large, exhibiting δTe/Te ≈ 50%. Transport is enhanced above classical levels, or “anomalous”, during this turbulent phase of the filament's evolution.

2.3.2 Spatial Behavior

Radial profiles at z = 384 cm are shown in Fig. 2.15. Panel (a) presents a profile at t = 3.0 ms during the classical transport stage. Within only 0.5 cm the filament has nearly reached a uniform temperature. Panel (b) is a two-dimensional contour of Isat that highlights the symmetric nature of the filament. This result is taken from t = 1.0 ms, also during the classical transport stage. Comparisons between the radial profiles during classical and anomalous transport regimes elucidate the difference between classical and anomalous thermal transport.

Figure 2.16: Diagram highlighting the different spatial regimes in the radial profile of temperature. The thermal wave is confined to the center of the filament while the drift-Alfvén eigenmode is an extended structure. The drift-Alfvén mode exists as a global feature, but its amplitude peaks in the gradient region. [Full Size]

The radial symmetry shown in Fig. 2.15 is observed at all axial positions. This regularity during the early time evolution of the filament allows for the summary drawing given in Fig. 2.16. In this figure, a mapping of fluctuation behavior is given in terms of a typical radial temperature profile. The LaB6 crystal is shown for perspective. The thermal wave is strictly confined to the center of the filament, though it does overlap with the global structure of the drift-Alfvén eigenmode. In Fig. 2.16 the gradient region is labeled as the location of the drift-Alfvén wave because that is where it exhibits its largest amplitude.

2.3.3 Spatiotemporal Evolution

Figure 2.17 illustrates the marked difference in the filament profile before and after the transition to anomalous transport. In panel (a) a cylindrically symmetric profile is observed at t = 1 ms. Well after the transition away from this classical regime, panel (b) presents the resulting profile at a time of t = 8 ms. Both of these contours are actually ensemble results over a few thousand discharges.

Figure 2.17: Isat contours at z = 544 cm (a) The t = 1.0 ms time is the standard used to experimentally determine the filament center. (b) At t = 8 ms the filament has transitioned to a turbulent state in which both classical transport and cylindrical symmetry are no longer present.

2.4 Relationships Between Physics Results and Filament Behavior

2.4.1 Thermal Waves Appear in the Filament Center

The thermal waves discussed in Chapter 3 are only observed in the radial center of the filament. This is a result required by the nature of thermal waves because their drive source must be a heating source. While presently unknown, the heating source for the thermal waves must reside within the filament because the only energy source for all heating is the beam itself (the thermal wave may be excited by a heat source located axially away from the beam and does not have to be driven by the beam directly). Thermal waves are observed to initially appear at different times in the filament's evolution.

2.4.2 Exponential Spectra Occur in the Anomalous Transport Regime

The Lorentzian pulses and the resulting exponential spectra of Chapter 4 are observed at all spatial locations, but only late in time after the transition from classical to anomalous transport has occurred. This turbulent behavior is observed in many plasma devices, in all cases it appears driven by the pressure gradients existing in plasma boundaries (e.g., tokamak scrape-off layers and edge regions of linear devices).

Thesis - Chapter 1: Introduction

Sun, 08 Mar 2009 20:27:10 +0000

Thesis - Table of Contents

Introduction

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.

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. [Full Size]

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].

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. [Full Size]

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.

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.

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. [Full Size]

1.2 Low Frequency Turbulence

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.

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

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. [Full Size]

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.

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.

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.

Review: Exponential Frequency Spectrum and Lorentzian Pulses in Magnetized Plasmas

Thu, 08 Jan 2009 19:04:36 +0000

Sections

Details and Download

Previous Work

Modeling Lorentzian Pulses

What is the Relevance?

Discuss this Item

This is a review of a paper recently published by my group. Here, the paper is paraphrased to reach a wider audience. The plasma physics community can read the original publication, so my goal is to provide an example of current plasma physics research to the general public. The following review should be at a high school level and I appreciate any comments you may have regarding how to clarify it further. If you are interested in physics research, then I hope this helps to feed your curiosity.

Details and Download

This is a review of the paper titled, “Exponential Frequency Spectrum and Lorentzian Pulses in Magnetized Plasmas”. The published version of the paper can be downloaded from the following links.

Basic Information:

Citation: D. C. Pace, M. Shi, J. E. Maggs, G. J. Morales, and T. A. Carter, “Exponential Frequency Spectrum and Lorentzian Pulses in Magnetized Plasmas,” Phys. Plasmas 15, 122304 (2008)
Download: From DavidPace.com (free, no registration required)
Statements Required by the Publisher: Copyright (2008) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
The following article appeared in Phys. Plasmas 15, 122304 (2008) and may be found at http://link.aip.org/link/?PHP/15/122304.

Previous Work

Figure 14a from the paper, example of a pulse in the time series data.

An introduction to this work is provided on this page and the downloadable publication provided there. That first paper will hereafter be referred to as the PRL. The paper reviewed here (the POP) is longer and contains new material concerning the modeling of exponential power spectra as generated by the presence of Lorentzian shaped pulses in the time series data.

The basic idea behind all of this work is that we observe pulses, or spikes, in the plasma. These pulses are not always present, but whenever they are, the power spectrum of the data features an exponential dependence in frequency. The figure to the right provides an example of a pulse that was measured in one of the experiments. There are some background fluctuations, but the labeled pulse is clearly a unique feature of this signal. This is figure 14a from the paper.

Figure 6 from the paper, example of an exponential spectrum (such a spectrum appears as a straight line when plotted semi-log as in this figure).

In the next figure (figure 6 in the paper) an example of an exponential power spectrum is shown. The amplitude of the power spectrum decays exponentially with respect to the frequency. When such a spectrum is plotted in a semi-log format (logarithmic y-axis and linear x-axis), the exponential behavior appears as a straight region. In this particular example there are peaks in the spectrum that correspond to coherent waves. The exponential part of the spectrum is identified through the linear fit (a fit to the logarithmic format of the result) shown as the dashed red line.

The first paper (PDF) focused on demonstrating the connection between the pulses and the exponential spectra. The paper discussed in this review provides additional evidence for this connection by way of new modeling work. Some natural questions arise from the statements made thus far and are addresses by the more recent paper, including:

Can other pulse shapes generate exponential spectra (how do you know these pulses are Lorentzian)?
Is it possible to observe an exponential spectra with just about any combination of pulse-like features?
The data appear to show some non-symmetric pulses, can these be accounted for in the resulting spectra?

Modeling Lorentzian Pulses

The three questions above can all be addressed through analytic models of the pulses and spectra. Beginning with the question of whether non-Lorentzian shapes can generate exponential, the answer is yes. The argument in favor of the observed pulses being Lorentzian is that other pulse shapes, while potentially generating exponential spectra, do not produce the agreement between the pulse width and the exponential decay constant. As the PRL showed, there is a relationship between the temporal width of a Lorentzian pulse and the exponential decay constant of the power spectrum it produces. Presently, only the Lorentzian shape satisfies this relationship.

Figure 16 from the paper, showing the different spectral shapes resulting from different pulses.

The figure to the right is figure 16 from the POP. It plots the power spectra resulting from three different pulse shapes: Lorentzian, Gaussian, and hyperbolic-secant squared. While not shown in the paper, these three shapes are all very similar near their peaks. The differences between them are realized in the wings, away from their centers. Looking at the raw data from the experiments it would be reasonable to attempt to fit the pulses to any of these shapes. To compare their resulting power spectra, we analytically calculated the spectrum resulting from each pulse after giving them all the same temporal width. The figure shows that the Gaussian spectrum is not exponential across any frequency range. The hyperbolic-secant squared spectrum is exponential at the higher end of the frequency range, but not at the lower frequencies. The Lorentzian spectrum is exponential across all frequencies.

This initially seems to indicate that the hyperbolic-secant squared is a viable candidate for the pulse shape. It turns out that this is not the case, however, because the relationship between the pulse time-width and the exponential decay of the spectrum is not consistent. The spectra plotted in the figure all come from pulses with the same temporal width. The exponential parts of the Lorentzian and hyperbolic-secant squared spectra exhibit different slopes (the exponential part of the hyperbolic-secant squared spectrum is plotted as the dashed black line). This means that the spectrum produced by the hyperbolic-secant squared pulse would not allow for the calculation and prediction of the pulse width. In the experiments, however, we have shown that calculating the exponential behavior of the spectra allows for the determination of the widths of the pulses that generated it. Therefore, while this is not an exhaustive study of every single mathematical pulse shape in existence, it does show that there are no obvious examples of pulses other than Lorentzians that generate the same agreement between pulse and spectral characteristics.

Figure 17 from the paper, showing the different spectral shapes resulting from pulse sets with different distributions of time widths.

The next question addressed by the POP is whether exponential spectra are generated from any collection of pulse-like features. A model of Lorentzian pulses is constructed to test the sensitivity of the exponential spectral behavior with respect to the temporal widths of the pulses. In the figure to the left, two power spectra are plotted from different model sets. The Broad spectrum (dotted red line) is produced from a set of Lorentzian pulses in which the temporal widths vary between 2 and 10 microseconds. The Narrow spectrum (solid black line) is the result from a set of Lorentzian pulses in which the temporal widths vary only between 2.5 and 4.5 microseconds. For the same total number of pulses in each set, the Narrow set features a much more consistent range of pulses (i.e., the pulses are much more similar to each other within this set).

The figure shows that the power spectrum from the Narrow set exhibits exponential behavior over a greater frequency range than that of the Broad set. The Broad set loses its exponential behavior at lower frequencies. The relevance to the experiments is that since we measure exponential spectra over wide frequency ranges (see figure 6 as shown above), that implies the pulses in the experiment feature a narrow distribution of temporal widths. A consequence of this result is that if the pulses feature such consistent widths, then there is likely a generation mechanism that can explain the production of the pulses themselves. There are measurements of hundreds of pulses for a given experimental setup, and thousands of pulses overall, so the observation that exponential spectra are only observed when these pulses are reproducible and consistent in width is a powerful conclusion. In fact, it appears as though a completely random collection of pulses does not spontaneously result in an exponential power spectrum. The pulses need to feature the same shape and similar time widths.

Figure 14b from the paper, example of a Lorentzian fit to the pulse from figure 14a.

The final question covered in the POP concerns the observation of non-symmetric pulses. We claim that Lorentzian pulses generate the exponential power spectra, yet there are clearly examples of pulses that are not perfect Lorentzian shapes. One such example is shown in figure 14b from the paper, which is reproduced to the right. The measured pulse (solid red) is analytically fit to a Lorentzian shape (dashed black). It is clear that the fit is not perfect because the measured pulse is asymmetric. The leading edge (left side) of the measured pulse is steeper than a Lorentzian. While the trailing edge (right side) of the measured pulse is a very good fit, this still leads to the question of whether asymmetric, or skewed, Lorentzian pulse shapes can generate an exponential spectrum.

Figure 18a from the paper, fitting a skewed Lorentzian pulse to a measured pulse.

To model the behavior of non-symmetric Lorentzian pulses we modified the standard pulse shape with a multiplicative term shown as equation 9 in the paper. The result of this modification is that the pulse features a steepened edge. The figure to the left takes one of these modified Lorentzian fits as applied to a pulse from one of the experiments. Notice that even though this measured pulse is negative polarity (compared to the positive polarity of the pulse at the beginning of this review), the leading edge is once again steeper than the trailing edge. The measured pulse (solid black) is fit exceptionally well by the skewed Lorentzian (dashed red). Furthermore, figure 18b of the paper plots the resulting power spectra from models of either a pure or a skewed Lorentzian and shows that the resulting exponential shape is not changed by the skewing. The conclusion is that non-symmetric pulses in the measured data do not prevent the exponential spectrum from being observed.

What is the Relevance?

In a physics paper the relevance of the work is established in the beginning. In a review like this, however, it works better to explain the physics first and then put the results into context. This work experimentally observes pulse structures of a particular shape that only arise when the system has transitioned to a turbulent state (that is all shown in the PRL). The structures exhibit consistent time widths, implying that the generation mechanism is constant during their production. Turbulence is generally associated with the loss of coherent waves and the production of a broad spectrum of fluctuations. Here, however, it is shown that a very narrow-band behavior (the narrow distribution of pulse widths) is required to observe the broadband feature (the wide ranging exponential frequency behavior). This is a unique result that may be the beginning of new developments in plasma turbulence and transport. The bigger picture still needs to be determined. For example, what causes the pulses? If they are produced by a non-linear interaction of electromagnetic plasma waves, then it is likely that this same behavior occurs in other plasmas, both in space and in the laboratory. As such, we suggest that the pulses and exponential power spectra are a universal feature of plasma turbulence. If a Lorentzian shape is the result, then perhaps we can now work backwards to describe plasma turbulence with a set of solvable equations. What differential equations feature Lorentzian pulses as their solution? Perhaps these same equations can describe plasma transport in a new way. Plasma transport is important for understanding phenomena such as the solar wind and aurora in addition to the present challenge of developing nuclear fusion as an energy source.

My Thesis Project

Mon, 20 Mar 2006 08:18:57 +0000

I completed my Ph.D. thesis in December 2008 (the conferral date is 2009). Copies of it are available in both PDF and HTML format. The PDF is the source for the printed copies that were turned in to the UCLA Library as part of the requirements for completion. The web version omits the front matter but does present every chapter and the full appendices. A link to the downloadable PDF is provided below, along with the Table of Contents for the web version.

General Information

Title: Spontaneous Thermal Waves and Exponential Spectra Associated with a Filamentary Pressure Structure in a Magnetized Plasma
Department: Physics and Astronomy
Institution: University of California, Los Angeles
Download: PDF (5.5 MB, 162 pages)

This is a work in progress. I will upload sections as I finish converting them to HTML. The PDF download is the complete version, however, so the web version will not provide anything new.

Abstract

1 Introduction

2 Experimental Setup and Overview of the Temperature Filament

Abstract

An experimental study of plasma turbulence and transport is performed in the fundamental geometry of a narrow pressure filament in a magnetized plasma. An electron beam is used to heat a cold background plasma in a linear device, the Large Plasma Device (LAPD-U) [W. Gekelman et al. Rev. Sci. Instrum. 62, 2875 (1991)] operated by the Basic Plasma Science Facility at the University of California, Los Angeles. This results in the generation of a filamentary structure (~ 1000 cm in length and 1 cm in diameter) exhibiting a controllable radial temperature gradient embedded in a large plasma. The filament serves as a resonance cavity for a thermal (diffusive) wave manifested by large amplitude, coherent oscillations in electron temperature. Properties of this wave are used to determine the electron collision time of the plasma and suggest that a diagnostic method for studying plasma transport can be designed in a similar manner. For short times and low heating powers the filament conducts away thermal energy through particle collisions, consistent with classical theory. Experiments performed with longer heating times or greater injected power feature a transition from the classical transport regime to a regime of enhanced transport levels. During the anomalous transport regime, fluctuations exhibit an exponential power spectrum for frequencies below the ion cyclotron frequency. The exponential feature has been traced to the presence of solitary pulses having a Lorentzian temporal signature. These pulses arise from nonlinear interactions of drift-Alfvén waves driven by the pressure gradients. The temporal width of the pulses is measured to be a fraction of a period of the drift-Alfvén waves. A second experiment involves a macroscopic (3.5 cm gradient length) limiter-edge geometry in which a density gradient is established by inserting a metallic plate at the edge of the nominal plasma column of the LAPD-U. In both experiments the width of the pulses is narrowly distributed, resulting in exponential spectra with a single characteristic time scale. The temperature filament experiment permits a detailed study of the transition from coherent to turbulent behavior and the concomitant change from classical to anomalous transport. In the limiter experiment the turbulence sampled is always fully developed. The similarity of the pulse shapes and fluctuation spectra in the two experiments strongly suggests a universal feature of pressure-gradient driven turbulence in magnetized plasmas that results in non-diffusive cross-field transport. This may explain previous observations in helical confinement devices, research tokamaks and arc-plasmas.