An experimental investigation is presented into the wave-driven electron transport in the near-field plume of a hollow cathode operating in a 300 V, 4.5 kW magnetically shielded Hall thruster. Correlational analysis of probe measurements in the cathode plume shows two types of electrostatic waves: ion acoustic turbulence propagating along the applied longitudinal magnetic field at frequencies from 500 to 1250 kHz and coherent, azimuthal anti-drift waves with a fundamental frequency of 95 kHz and mode numbers from $m=1\u22124$. A quasilinear analysis is applied to quantify the impact of each wave on the electron transport in the near-field plume. It is found that the ion acoustic modes give rise to an enhanced effective collision frequency in the direction parallel to the applied magnetic field that exceeds the classical collision frequency by two orders of magnitude. The anti-drift waves promote an anisotropic collision frequency that depends on the direction of the electron drift. While the enhanced collision frequency from these waves is comparable to the classical frequency for motion along the applied magnetic field, the effective collision frequency in the azimuthal direction exceeds the classical by three orders of magnitude. These results are discussed in the context of their impact on the steady-state plasma gradients in the near-field cathode plume. Closure models for incorporating the effective collision frequencies from both types of waves into fluid-based codes are derived and shown to agree with the measured wave-driven collision frequencies.

## I. INTRODUCTION

The efficient operation of the Hall effect thruster, a widely used type of plasma propulsion, strongly depends on the ability of its electron source, the hollow cathode, to couple electrically to the thruster's plasma discharge.^{1} These thermionically emitting devices enable the electrons to both ionize the thruster's propellant and neutralize the positively charged ion exhaust after it has been accelerated out of the thruster. If changes to the cathode operation or environment impede the ability of these electrons to connect to the thruster's main discharge, performance and stability can suffer. Indeed, it has been shown that varying key cathode parameters such as flow rate, temperature, orientation,^{2–10} and test environment^{11–19} can negatively impact the cathode coupling and by extension thruster operation. Although these effects have been well-documented, there are aspects of the underlying physical processes governing the cathode coupling that remain poorly understood. This lack of understanding continues to present an obstacle for the modeling, design, and optimization of Hall thrusters.

The major challenge in building a first-principles description of the cathode coupling stems from the existence of non-classical processes in the plumes of these devices. Most notably, the electron resistivity is orders of magnitude higher than can be explained by friction from ordinary collisions.^{20–28} It thus is not possible to model or predict the local plasma environment using a classical plasma approach. In an effort to address this shortcoming, there has been a concerted effort to identify potential mechanisms that may drive this enhanced resistivity and apply these to our understanding of hollow cathode operation. These studies have shown that the onset of propagating electrostatic waves in the form of current-driven ion acoustic turbulence (IAT) is the dominant contributor to an effective, “anomalous” collision frequency on the electrons.^{21–23} Subsequent efforts have built on these findings to incorporate non-classical, wave-driven effects self-consistently into numerical results. They ultimately have yielded improved predictions both for the cathode plume distribution and dynamical evolution.^{28–31}

A major caveat for these previous studies—both numerical and experimental—is that they have focused primarily on stand-alone configurations. These are experimental domains in which the cathode is tested in a dedicated facility and not operated as part of a thruster discharge. These stand-alone configurations have the advantages of easier accessibility and controllability to facilitate basic plasma studies, but there are substantial differences in their test environments and the environment of a hollow cathode operating with an actual Hall thruster. These differences may fundamentally change the electron dynamics in these systems. For example, it is not clear if IAT will have the same degree of influence on hollow cathode plumes operating in thrusters. A potential reason for this is that cathodes in stand-alone configurations have only one source of neutral gas (through the cathode bore), while cathodes in thrusters are subject to neutral influx from the surrounding main discharge.^{32} Since neutral gas has been shown to be a major contributor to the damping of IAT,^{20,25} the enhanced resistivity from turbulence may be curtailed or even non-existent when the cathode is operated in conjunction with a thruster. As another potential discrepancy, while most work on stand-alone devices to date has been for unmagnetized plumes, hollow cathodes in Hall thrusters are subject to strong magnetic fields. As a result, hollow cathode plumes in thrusters have been shown to exhibit gradient-driven, rotational (“anti-drift”) instabilities not present in unmagnetized devices.^{33,34} If these modes contribute to electron transport, they could fundamentally change the non-classical effects governing the electron dynamics.

In light of the importance of cathode coupling to thruster operation and these pressing questions about the nature of the electron dynamics, the goal of this paper is to investigate experimentally wave-driven, non-classical resistivity when these devices are implemented in a thruster environment. We focus on examining the role of the two classes of oscillations that we anticipate should exist in the plume: rotational anti-drift waves^{33} and IAT.^{23} To this end, this paper is organized in the following way. In Sec. II, we employ a quasilinear approximation to relate wave amplitude and dispersion of electrostatic waves in a cathode plume to an enhanced electron collision frequency. We then describe the dispersion relations of the anti-drift wave and ion acoustic turbulence and relate these to effective collision frequencies. In Sec. III, we describe the experimental setup and diagnostics employed to characterize the wave and plasma properties in the cathode coupling region. In Sec. IV, we present results for measurements of the wave properties, calculate an effective collision frequency from the two modes, and compare these results to the classical electron collision frequency. In Sec. V, we discuss our results in the context of previous studies on cathode coupling dynamics and motivate simple closure models for the anomalous collision frequency.

## II. WAVE DISPERSION AND TRANSPORT COEFFICIENTS

The goal of this investigation is to evaluate the role of plasma waves in driving non-classical transport in the cathode plume of a Hall thruster. In order to establish a metric for assessing these effects, we present in the following a theoretical framework for relating a key property governing the transport of the electrons, their resistive drag, to the properties of the plasma oscillations (frequency, amplitude, etc.). We then apply linear dispersion theory to derive expressions for wave-driven “collision frequencies” associated with this enhanced resistive drag.

Figure 1 shows a cylindrical coordinate system as applied to the hollow cathode and Hall thruster we employed in this investigation (discussed further in Sec. III). The Hall thruster is a crossed field (E × B), axisymmetric device in which the main plasma discharge is confined to an annular channel. The hollow cathode is mounted along the thruster centerline and concentric with the discharge channel. As indicated in Fig. 1, the axial direction, $z\u0302$, is along thruster centerline with the origin coincident with the cathode exit plane. For the domain of interest, the cathode plume is assumed to be subject to a strong and approximately uniform magnetic field oriented in the axial direction, $B\u21920=B0z\u0302.$ Electric fields and pressure gradients can be both axial and across the magnetic field in the radial direction, $r\u0302$. This crossed-field configuration can give rise to drifts in the $\theta \u0302$ direction.

We adopt the approach of Davidson and Krall^{35} to relate the wave properties to electron drag in the coordinate frame shown in Fig. 1. Following this work, if we assume the waves of interest are electrostatic (as is consistent with the anti-drift and ion acoustic modes), the Boltzmann equation for the electron distribution function can be expanded to second order and averaged over the fast time scale of wave propagation. This yields an effective additional force density on the electrons that arises from wave propagation,

where *q* denotes fundamental charge, $\delta ne$ is the perturbation to the electron density introduced by the propagating waves, $\delta E\u2192$ is the perturbation to the electric field associated with wave propagation, and the brackets $\u27e8..\u27e9$ denote an average over the characteristic time scale of the plasma waves. Equation (1) shows that if the density and electric field perturbations associated with a wave have components that are in phase, they can positively couple to act as an additional forcing term on the electrons. This forcing term in turn scales directly with the amplitude of the waves.

Expanding on the interpretation of the wave propagation as a drag on the electrons, we can define effective wave-driven collision frequencies,

where $u\u2192e()$ and $\nu AN()$ denote the electron drift and enhanced collision frequency in each direction, *m _{e}* is the electron mass, and

*n*is the time-averaged local electron density. These expressions reflect the anisotropic effect propagating waves can have on the electron transport in the cathode plume. Physically, an enhanced collision frequency along magnetic field lines, $\nu AN(z)$, will act as an impedance on the electron motion giving rise to higher potential gradients to maintain a set discharge current. The collision frequency in the azimuthal direction, $\nu AN(\theta )$, however, promotes radial, cross field current. This stems from the de-magnetizing effect of the drag introduced by the waves. While collisions in the radial direction, $\nu AN(r)$, also can promote cross field current, this contribution scales inversely with the classical Hall parameter (ratio of cyclotron to classical collision frequency), which is on the order of 100–1000 for the plasma of interest. It is, therefore, assumed to be negligible compared to the cross field current induced by the azimuthal drag. For this investigation then, we confine our discussion to the effective collision frequencies acting in the azimuthal and axial directions.

_{e}To express the collision frequencies from Eq. (2) in terms of wave properties, we represent the electric field and density perturbations for the propagating waves as a Fourier composition in the cylindrical coordinates from Fig. 1:

where $c.c.$ denotes the complex conjugate, *k _{z}* denotes the axial component of wavenumber,

*m*is the mode number in the azimuthal direction,

*ω*is the frequency of each mode,

*t*denotes time, and $ne(kz,m)$ and $\varphi (kz,m)$ are the Fourier amplitudes of the density and the potential oscillations, respectively. Armed with Eq. (3), we thus can find expressions for the collision frequencies from Eq. (2):

where $Im[]$ denotes the imaginary component.

Equation (4) provides a prescription for evaluating the non-classical, wave-driven effects of the two modes of interest in the cathode plume—provided we can measure or estimate the perturbed quantities associated with the wave propagation. To this end, we consider the plasma properties and dispersion of both the acoustic wave and the anti-drift wave. In our treatment of these modes, we assume in the following—as is consistent with the near-field plume of our experimental setup (Sec. III)—that the magnetic field is sufficiently strong to magnetize the electrons but weak enough that ion dynamics are independent of the field.

### A. Ion acoustic waves

The ion acoustic wave is an electrostatic mode that has been observed extensively in the plumes of unmagnetized hollow cathodes. It is driven unstable in these devices in the direction of strongest electron drift. In magnetized plasmas, these waves can appear in the cross field direction provided there is a sufficiently high electron cross field drift (cf. Ref. 36). However, in the thruster cathode plasma, the highest drift is along magnetic fields where there is the lowest resistance path. Given the azimuthal symmetry of the system shown in Fig. 1, these modes thus preferentially will propagate in the $z\u0302$ direction with *m *=* *0. Similarly, consistent with the work from Ref. 23, we assume the wavelengths of the excited modes are long compared to the Debye length, $k\lambda de\u226a1$, and that the plasma is not in equilibrium such that the electron temperature is not equal to the ion temperature, $Te>Ti$. We also make the assumption that the sound speed is negligible compared to the electron drift $cs=Te/mi\u226aue(z)$. Together, these simplifications allow us to find expressions for the dispersion and the perturbed potential in terms of electron density,^{29}

where $ui(z)$ is the ion drift speed in the axial direction in the laboratory frame of reference and *v _{te}* is the electron thermal speed.

Equation (5) illustrates two key expected properties of the ion acoustic wave in the cathode plume. The first is that it has an acoustic dispersion in the ion frame of reference—the phase velocity is the ion sound speed. In the laboratory frame, the wave frequency is Doppler shifted by the ion drift in the axial direction. The second property is that there is a phase delay between the density and potential fluctuations introduced by inverse electron Landau damping [the imaginary component in the second line of Eq. (5)]. As noted in Ref. 37, other parameters can contribute to this phase delay including classical electron collisions, but for a plasma characterized by a strong electron drift, these other effects generally are negligible compared to the phase introduced by this inverse damping. With these results in mind, we can see from Eqs. (4) and (5) that the IAT will only contribute to the longitudinal anomalous collision frequency,

where we implicitly have assumed the wave vector and electron drift are parallel. The physical implication from this result is that the anomalous collision frequency scales both with the amplitude of the fluctuations and the energy distribution across wavenumber.

### B. Anti-drift wave

The anti-drift wave is a rotational oscillation that propagates primarily in the azimuthal direction with small but finite components in the axial direction, parallel to the applied magnetic field. It derives its energy from both gradient-driven drifts in the azimuthal direction as well as the electron longitudinal drifts in the near-field magnetized cathode plume. This energy transfer can result in an effective drag and, therefore, enhanced collisionality on this species in both directions. The key assumptions underlying our treatment of this mode include $\Omega i<\nu e<\Omega e$ where $\Omega i,e$ denotes the species gyrofrequencies; an approximately constant electron temperature and negligible ion temperature, *T _{e}* >

*T*; a wave frequency that satisfies $\nu i<\omega <\nu e$ where $\nu i,\nu e$ denote the classical ion and electron collision frequencies; and a high phase velocity in the longitudinal direction, $\omega /kz\u226bcs.$

_{i}The anti-drift waves that were observed in Refs. 33 and 34 in the plume of a Hall thruster hollow cathode exhibited a *m *=* *1, bulk propagation in the azimuthal direction in the plasma. In the former study, a non-local dispersion relation analysis in cylindrical coordinates was employed to compare the measured wave properties to experimental results. For this work where our goal is to examine the potential impact of these oscillations on the electron dynamics, we forgo a global analysis in favor of a simpler plane wave, Cartesian formulation. The transformation between cylindrical and Cartesian coordinates can be approximated with the relation $k\theta \u2192m/r$ where $k\theta $ is the Cartesian representation of the wavenumber in the azimuthal direction, *m* is the mode number, and *r* is the radial coordinate. Adopting this convention and following the approach of Frias *et al.*,^{38} the real dispersion and electron density for the anti-drift wave can be written as

where we have introduced the diamagnetic drift $v*=Te/(qB0)ne\u2032(r)/ne$, the electric field driven drift $vE\xd7B=Er/B0$, and a collisional term, $\nu pl=kz2Te/me/\nu e$. We note that in a departure from the derivation of Frias *et al.*,^{38} we have neglected magnetic field gradient-driven drifts and introduced (following Ref. 33) the contributions from electron motion parallel to the applied magnetic field. Solutions to the first expression in Eq. (7) yield the relationship between frequency and wavenumber of the mode. The second expression shows that the mechanism for introducing phase delay between the potential and density fluctuations stems from electron collisions through *ν _{pl}*. This promotes the exchange of energy of the wave and the electron drifts.

Since the anti-drift wave propagates longitudinally and azimuthally, we anticipate there may be contributions to the anomalous collision frequency in both of these directions. From Eqs. (4) and (7), we find

In order to arrive at these forms, we have made the assumption that

where $Ln=|ne/ne\u2032(r)|$ denotes the characteristic length scale of the density gradient. We have found consistently that this criterion is satisfied for the hollow cathode plasma we investigate in this study (Sec. IV). Physically, the results in Eq. (8) show that with more energy present in the oscillation (greater the amplitude), the greater the rate at which energy is extracted from the electron drift. Similarly, the linear dependence on the electron collision frequency illustrates that it is this classical drag that gives rise to a phase delay between potential and density oscillations necessary to promote the growth of the waves.

In summary, we have shown in this section from a formulation for a simplified geometry that the oscillations we anticipate in the hollow cathode plume can impact the electron transport [Eqs. (6) and (8)]. The onset of ion acoustic waves in the longitudinal direction will lead to a higher effective collision frequency along magnetic field lines. The growth of the anti-drift waves similarly will contribute to higher resistivity along field lines, although they also will promote crossed field transport through an enhanced collision frequency in the azimuthal direction. The degree to which these two modes influence transport will depend on their amplitudes and dispersive properties. The remainder of this work is devoted to measuring these properties experimentally.

## III. EXPERIMENTAL SETUP

The test article we employed for this investigation was the H9 (Fig. 1), a 9-kW class device designed by the Jet Propulsion Laboratory in collaboration with the University of Michigan and the Air Force Research Laboratory.^{39,40} Although not a flight unit, the H9 represents the state of the art in modern flight thrusters^{41} intended for deep space exploration. Key features include a centrally mounted hollow cathode, graphite covers on the magnetic pole pieces, and a magnetically shielded discharge. Magnetic shielding is a design technique in which the magnetic field geometry in the annular discharge is configured to reduce contact of energetic ions with the chamber walls and therefore prolong thruster life.^{42,43} The cathode for the H9 employs a LaB_{6} emitter and a graphite keeper that can sustain currents up to 60 A. Its plume is subject to a strong axial magnetic field, consistent with the coordinate system defined in Sec. II. Electrically, following Ref. 44, the cathode body is tied to the thruster main body. All electrical potentials in the following work are referenced with respect to this potential.

We performed our experimental investigation of the cathode plume by operating the H9 on xenon at 4.5 kW power, a discharge voltage of 300 V, and a flow fraction through the cathode of 7%. This is one of the nominal operating conditions of the H9 with measured performance metrics that include 294 mN thrust, 1906 s specific impulse, and 60.4% efficiency.^{40} All testing was performed in the Large Vacuum Test Facility (LVTF), a 6 m × 9 m cryogenically pumped chamber located at the University of Michigan. The base pressure of this system was $10\u22127\u2009Torr-xenon$, but during the 4.5 kW operation, the facility pressure was maintained at $5\xd710\u22126\u2009Torr-xenon$ as measured in the plane of the thruster.

We employed two sets of diagnostics for this study: a cylindrical Langmuir probe to characterize the background plasma properties and an array of ion saturation probes to measure the plasma transients. Both sets of probes were mounted on high-speed translation stages and inserted axially with a reciprocating action into the near-field of the hollow cathode. The region of interrogation (shown graphically in Fig. 1) consisted of a 20 mm × 20 mm rectangle in r-z located 25 mm downstream of the cathode exit. The measurement resolution was 2.5 mm in the axial direction and 5 mm in the radial.

The Langmuir probe consisted of a cylindrical tungsten tip with an exposed surface area of $Ap=1\u2009mm2$. During measurements, we biased this probe to fixed voltage and monitored current as a function of probe position as it was inserted into the cathode plume. By varying the applied voltage from -25 V (ion saturation with respect to thruster body) to 25 V (in excess of local plasma potential) in 1 V increments for each insertion, we constructed spatially resolved current-voltage Langmuir traces. From these measurements, we in turn inferred several key properties including the plasma density, electron temperature, and plasma potential.

The wave probe array consisted of three cylindrical tungsten tips each 3.8 mm in length and with a 0.38 mm radius. The probes were spaced apart such that two were aligned in the axial direction but with a 5.5 mm separation and two were aligned in the azimuthal direction with a 5.1 mm gap. During measurements, we biased each probe to −36 V to ion saturation and measured the fluctuations in this current at a rate of 10 MHz. The relative magnitude of these fluctuations as compared to the local mean value of ion saturation served as a proxy for relative fluctuations in ion density. In our analysis, we performed both Fourier transforms and cross-correlations between the probe tip signals to estimate the frequency, amplitude, and local wavenumbers in the axial and azimuthal directions.

## IV. RESULTS

We present in this section measurements of both the time-averaged plasma properties and the wave properties in the near-field of the hollow cathode. We then use these measurements combined with the expressions from Sec. II to evaluate the effective electron collision frequencies from the measured waves.

### A. Background plasma properties

Figure 2 shows the time-average electron temperature, plasma potential, and plasma density in the plume of the thruster hollow cathode at the 300 V and 4.5 kW condition. We determined the electron properties through an analysis of the current-voltage Langmuir probe traces generated per the procedure outlined in Sec. III. We estimated the plasma potential, $\varphi $, with the first-derivative technique applied to these traces, and we calculated the electron temperature, *T _{e}*, from the slope of the log-linear plot of the IV trace.

^{45}The plasma density,

*n*, was inferred from the average measured ion saturation current,

_{e}*i*, by applying the thin sheath approximation—valid for our typical high density plasma conditions: $ne=isat/(q0.61ApTe/mi)$. In order to determine the uncertainty for these calculated parameters, we performed a bootstrap analysis for each IV trace, re-sampling the dataset and repeating the estimates of the parameters. This yielded typical uncertainties of 20% in both the electron temperature and plasma potential and 50% in the plasma density.

_{sat}The trends in Fig. 2 are consistent with previous work reported for centrally-mounted cathodes in Hall thrusters.^{46,47} The monotonic decrease in plasma density with position in Fig. 2(a) results from the expansion of the plasma as it is emitted from the cathode. The near alignment of the contours of constant density in the axial direction for $r>5\u2009mm$ is an indication of the anisotropy introduced to the plasma expansion by the strong confinement from the local axial magnetic field (∼250 G). The approximately constant electron temperature ($Te=4.6\xb11\u2009eV$) in Fig. 2(b) suggests a high degree of thermal conductivity and subsequent isothermality in this area. The existence of gradients in the potential well shown in Fig. 2(c) is indicative of electric fields pointing axially toward the cathode that drive electron current downstream.

The gradients in potential and density exhibited in Fig. 2 both are conducive to the onset of the waves discussed in Sec. II. The distinctive potential well can facilitate a strong, axial electron drift, a pre-requisite for current-driven IAT. Similarly, the marked radial gradients in potential and density in the region near the cathode centerline can promote azimuthal electron drifts, one of the requirements for the onset of the anti-drift mode (Sec. II). With this in mind, we next turn to investigating if there is evidence that these waves actually exist in this region.

### B. Wave measurements

We show in Fig. 3 power spectra and dispersion relations measured in the axial and azimuthal directions at two radial positions, *r *=* *0 and $r=20\u2009mm$, located $z=25\u2009mm$ from the cathode exit. The power spectra [Fig. 3(a)] have a frequency resolution of 5 kHz and were generated from a Fourier analysis of the relative fluctuations in the ion saturation current, $i\u0303sat/isat$. They thus are a direct indication of the magnitude of the density fluctuations in this region. The dispersion relation [Figs. 3(b)–3(d)] plots were created with a Beall analysis^{48} applied to the measurements of ion saturation current from the probe tips in the array described in Sec. III. They are histograms that represent the probability that a given frequency at which the density oscillates is correlated with a wavenumber in the direction defined by the line connecting the set of probes. The reported results consist of the average of 1000 correlations applied to the probe signals and have a maximum measurable wavenumber $kmax=\pi /\Delta x$, where $\Delta x$ denotes the distance between probes in the array. The frequency resolution for both figures is 5 kHz, and we have converted the azimuthal component measurements to mode number, *m*, with the relation $k\theta =m/r$ (Sec. II). In all of the histogram plots, we have normalized the intensity to unity but adjusted the color scale to illustrate the salient wave features.

The results in Fig. 3 show three non-trivial trends in frequency and wavenumber. The first, as can be seen from the power spectrum plot in Fig. 3(a), is a low frequency peak at 10 kHz. This is the breathing mode, which originates in the thruster channel (cf. Ref. 33). This oscillation is pervasive in Hall thruster discharges, but as can be seen from this plot, it is not dominant in the cathode plume. The second oscillation is characterized by a series of discrete peaks with a fundamental frequency at 95 kHz and three harmonics. The third notable feature exhibited by the power spectrum is a broadband structure between 500 kHz and 1250 kHz that has an inverse decay in amplitude with frequency. Along centerline (r = 0), this broad spectrum is punctuated by a small peak at $\u223c1200\u2009kHz$ that does not persist at larger radii.

We can map these features of the power spectra to wave properties in the cathode plume with the dispersion plots. We first focus on the lower frequency components in Figs. 3(b) and 3(c) where we show measurements from 0 to 500 kHz. The peak at 10 kHz, which we have identified with the thruster breathing mode, has no detectable dispersion in the azimuthal direction and a small but finite component of wavenumber in the axial direction. This is consistent with the fact that the breathing mode is believed to be primarily longitudinal. We also can see from Fig. 3(b) that the discrete peaks starting at 95 kHz in the power spectrum are aligned with mode order. The fundamental corresponds to an *m *=* *1 mode, the harmonic is aligned with *m *=* *2, etc., until the mode amplitude is no longer detectable at $\u223c\u2009500kHz$. The non-local behavior suggests that these oscillations represent bulk rotations of the plasma—azimuthally propagating eigenmodes—that are consistent with the anti-drift wave previously reported in Ref. 33. These rotational modes also exhibit axial wavenumbers [Fig. 3(d)] where the magnitude of the component in the axial direction increases approximately linearly with the mode order. The small magnitude of these wavenumbers indicates that the wavelengths of these axial components ($>10\u2009cm$) are comparable to the characteristic axial length of the cathode plume. We note here as well that although we only have shown dispersion relations at two radial locations, we found both quantitatively and qualitatively that this rotational dispersion persisted throughout the plume. Most notably, by employing the transformation $k\theta =m/r$ (Sec. II), we found at different radii the same correlation between mode number and frequency as shown in Fig. 3(b).

We next turn to examining the nature of the higher frequency oscillations in the power spectrum from Fig. 3(a). To this end, we show in Fig. 3(d) the dispersion relation measured in the axial direction on cathode centerline. Since the higher frequency, broadband oscillations (500 kHz–1250 kHz) are an order of magnitude lower in amplitude than the rotating modes [Fig. 3(a)], we have adjusted the color scale on this plot to highlight the dispersive nature of the high frequency content. As a result, the harmonic structure for the 0–500 kHz has been obscured. We label this saturated region on the plot as anti-drift waves and draw a dotted line to remind the reader of the weak linear trend exhibited in Fig. 3(c). We also have adjusted the wavenumber scale and data depicted in Fig. 3(d) in order to correct for probe aliasing. This was done following the procedure outlined in Ref. 23. As was pointed out in this previous work, if the wavelengths of the propagating waves are smaller than the distance between the probes, they will appear in dispersion analyses as unphysical wavenumbers that have been shifted to negative values by $\u22122\pi /\Delta x$. This is evidenced by the negative wavenumbers from −500 to 0 $rad/m$ in Fig. 3(d). This aliasing can be corrected by shifting the wavenumber dependence of the measured dispersion by one full phase ($2\pi /\Delta x$) and concatenating with the original dataset.

From this result, we immediately can see that the dispersive nature of the higher frequency content is different than the lower frequency rotational modes. In particular, there is an evident linear relationship between frequency and wavenumber (emphasized with a drawn line and labeled as “IAT”) that has a shallower slope than the rotational modes. The small amplitude high-frequency peak exhibited in Fig. 3(a) at 1200 kHz also follows the same linear trend as the broadband content. The range of frequencies where this linear dispersion exists as well as the slope are consistent in magnitude with the previous studies of stand-alone cathodes where they were identified as ion acoustic turbulence.^{23,25} This suggests correlationally that the power spectral content in this frequency range is the IAT. Moreover, as we would expect for these unmagnetized waves (Sec. II), we have found that the high frequency content is dispersionless in the azimuthal direction, suggesting these modes are purely in the r-z plane.

Although not shown here, we found that this linear dispersion at high frequency persisted in the axial direction for all measurements performed along the cathode centerline—the slope and frequency range remained unchanged. We did not, however, observe this dispersion in the Beall analysis for locations at radii greater than $r>5\u2009mm$. While this could be an indication that these modes do not exist at these larger radii, as can be seen from Fig. 3(a), the power spectrum shape from 500 to 1250 kHz remained qualitatively the same throughout the plume. The suggests the IAT may still persist, but its dispersion is obscured by the large disparity in the magnitude of the oscillations of the rotating modes and the acoustic content at larger radii. With this in mind, for the remainder of this investigation, we proceed under the assumption that even though we did not explicitly resolve the dispersion at these off-axis locations, the energy in the spectral range of 500 kHz–1250 kHz is associated with longitudinal acoustic modes.

We show in Fig. 4 the spatial dependence of the amplitudes of the *m *=* *1, *m *=* *2, and broadband acoustic modes in the plume. For the rotational modes, this plot represents the peak from the power spectrum [e.g., Fig. 3(a)] corresponding to the mode frequency at each spatial location. For the acoustic-like modes, the plotted quantity is the summation over the power spectrum from 500 kHz to 1250 kHz. The results shown in Fig. 4 illustrate several notable features about the observed oscillations. Both rotational modes exhibit a maximum in amplitude that occurs off-axis from the centerline. The *m *=* *1 mode, which has the highest amplitude, peaks at approximately $r=5\u2009mm$ while the *m *=* *2 mode peak occurs between *r *=* *5 and $10\u2009mm$. This non-monotonic dependence on radius is consistent with the results presented in Ref. 33 where high-speed imaging was applied to a comparable plasma plume. The fact that the location of the peak moves radially outward with mode number is also a feature that has been noted for drift-waves in low-temperature plasmas.^{49} The amplitude of the broadband fluctuations for the acoustic oscillations exhibits a peak off centerline as well, although the gradients are more gradual. Indeed, with the exception of points located at the downstream extremes of the domain, the amplitude varies less than $50%$ over the domain. This result is reminiscent of the findings from Ref. 25 that showed that the acoustic turbulence in the plume of a stand-alone cathode will saturate to a thermal limit after growing over a short spatial distance from the cathode exit.

### C. Comparison of wave measurements to linear theory

We quantify the observations from Sec. IV B explicitly here by evaluating Eqs. (5) and (7) with the background plasma parameters shown in Fig. 2 and comparing the predicted dispersion to measurement. For the ion acoustic waves, we estimate the phase velocity by fitting a line in Fig. 3(d) to the linear region of the dispersion relation and multiplying the slope by $2\pi $ to account for the conversion to angular frequency. This yields a phase velocity of $\omega /k=6\xb11\u2009km/s$. Following Eq. (5) and using the average value of electron temperature of $Te\u22484.6\u2009eV$, linear theory indicates that ion acoustic waves in this plasma environment should propagate with a phase velocity of $\omega /k\u22482\u2009km/s+ui(z)$. Comparing to the theoretical prediction, this result would suggest that the local velocity of ions on centerline is $ui(z)\u22484\xb11\u2009km/s$. This estimate for ion speed is in keeping with the typical values that have been measured in the plumes of hollow cathodes employed for electric propulsion^{25,50} and thus provides strong correlational evidence that the measured waves in this frequency range are ion acoustic in nature. With that said, the discrete peak at $\u223c1200\u2009kHz$ we observe in the dispersion on centerline is a notable departure from previous works on stand-alone cathodes^{23} where the acoustic waves exhibited a smooth power law decay with frequency. The mode associated with this peak still appears to be acoustic in nature, following the general dispersion shown in Fig. 3(d), yet it is not evident from the simplified linear theory (Sec. II) why energy should be concentrated locally at this frequency. While it is possible that higher order effects such as reflections off the density gradient in the axial direction could promote a resonant frequency like the one exhibited here,^{51} exploring these higher order effects ultimately is beyond the scope of this investigation. Indeed, in the following treatment, the fine details of the structure of the power spectrum become immaterial as we integrate over all the frequencies to assess the average impact [Eq. (6)] of the waves on the electron transport.

We next compare the measured wave properties to the linear theory predictions for the rotational, anti-drift mode. To this end, we first outline a range of values for the background plasma properties in Eq. (7) based on the measurements in the near-field region of the cathode in the H9 Hall thruster where we observed the rotational waves to have the highest amplitude ($r<10\u2009mm$). We then evaluate the linear dispersion relation numerically over this range of values. As shown in Table I, we use the range of $Te=4\u22126\u2009eV$ for the electron temperature and a local axial value of the magnetic field of $B0\u2248250\u2009G$. We assume there is negligible ion swirl in the azimuthal direction as they are unmagnetized and approximate the ion velocity as strictly axial with a range informed by the ion acoustic measurements, $ui(z)=3\u20135km/s$. For the electron drift in the axial direction, we assume that since this is a highly magnetized environment, the electron current primarily will be confined to motion along magnetic field lines. We thus can calculate an average axial drift by relating it to the total current, $ue(z)\u2248I/(q\u222bnerdr)$, where the integral is applied at fixed axial location and $I=15\u2009A$ denotes the electron current. Using our measured density profiles from Fig. 2(a), this yields a range of possible local speeds of $ue(z)=1000\u20133000\u2009km/s$.

Plasma property . | Range of values . |
---|---|

T _{e} | 4–6 eV |

B_{0} | 250 G |

$ui(z)$ | 3–5 km/s |

$ue(z)$ | 1000–3000 km/s |

ν _{e} | $(2\u221215)\xd7106\u2009s\u22121$ |

$vE\xd7B$ | 10–40 km/s |

$v*$ | 10–80 km/s |

r | 5 mm |

Plasma property . | Range of values . |
---|---|

T _{e} | 4–6 eV |

B_{0} | 250 G |

$ui(z)$ | 3–5 km/s |

$ue(z)$ | 1000–3000 km/s |

ν _{e} | $(2\u221215)\xd7106\u2009s\u22121$ |

$vE\xd7B$ | 10–40 km/s |

$v*$ | 10–80 km/s |

r | 5 mm |

For the classical electron collision frequency, we consider both Coulomb and electron-neutral interactions (cf. Ref. 52),

where Λ denotes the Coulomb logarithm, *n _{n}* is the local neutral density, and

*T*is given units of energy. We have all the required plasma measurements to estimate the range of these collision frequencies in the near-field cathode plume with the exception of the neutral density. To approximate this parameter, we follow the prescription of Goebel and Katz

_{e}^{52}and later Spektor

*et al.*

^{24}in assuming that the neutral plume from the cathode expands at a fixed angle,

*α*, from the orifice. The neutral gas at axial location

*z*thus can be approximated as

where $m\u0307$ denotes the mass flow rate through the cathode, $vt(n)$ is the neutral gas thermal temperature, and *r _{d}* denotes the diameter of the cathode keeper orifice. Assuming a typical neutral temperature of 1000 K exiting the cathode (consistent with the assumption that the neutrals are at equilibrium with the cathode's emitter temperature) and allowing for possible expansion angles ranging from $\alpha =30\xb0\u201370\xb0$, we find at the axial location where the measurements were performed a range of neutral densities of $nn=4\u201330\xd71018\u2009m\u22123$. With these values and the plasma properties in Fig. 2, we evaluate the expressions in Eq. (10) to find a range for total electron collision frequencies of $\nu e=\nu ei+\nu en=(2\u221215)\xd7106\u2009s\u22121$.

For the azimuthal drift velocities, we use the background properties in the near-field region (Fig. 2) at $r<10\u2009mm$ to establish the range $vE\xd7B=10\u221240\u2009km/s$ and $v*=10\u221280\u2009km/s$. Finally, we note that in evaluating Eq. (7), we need to make an assumption about the value of the radius, *r*. This free parameter was introduced by our decision to employ a Cartesian formulation for the dispersion instead of a full cylindrical analysis (recall $k\theta =m/r$). We follow the precedent of Ref. 49 in evaluating the dispersion at the radial location co-located with the maximum amplitude of the dominant, *m *=* *1 rotational mode, $r=5\u2009mm$.

Armed with these experimentally informed estimates for the background plasma parameters, we explicitly can evaluate the predicted frequency and wavenumbers of the excited waves from Eq. (7). To illustrate the approach, we first show in Fig. 5 an example of the numerical predictions from Eq. (7) for a set of representative drifts and plasma parameters. Here we have plotted both the frequency of oscillation and the predicted growth rate as a function of axial wavenumber, *k _{z}*, for four different mode numbers. As this plot shows, the expected frequency increases linearly with mode number. Moreover, as an encouraging preliminary result, we see that the magnitude of the frequency at each mode number is comparable with the measured values shown in Fig. 3. The growth rate, on the other hand, exhibits a non-monotonic dependence, showing an axial wavenumber that corresponds to maximum growth in each case. Physically, the existence of a maximum growth rate stems from the fact (Sec. II) that it is electron collisions in the longitudinal direction that provide the phase delay necessary for the anti-drift waves to extract energy from the electron flow. As $kz\u2192\u221e$, the characteristic length scale of the oscillation in the longitudinal direction becomes negligible compared to the mean free path such that collisions have a vanishingly small effect on the wave dynamics. This drives the growth to zero. Conversely, at asymptotically long wavelengths ($kz\u21920$), there is no wave motion along the longitudinal direction to facilitate the necessary phase delay between density and potential for growth. Between these extremes in wavenumber, there is an optimal phasing between density and potential oscillations that promotes maximal energy exchange with the background drift. With this in mind, in order to map the linear predictions to experiment, we assume that the mode frequency and axial wavenumber that will actually appear in the plasma will correspond to this maximum growth rate. For example, in Fig. 5 the fastest growing

*m*=

*1 mode would have an axial wavenumber of $kz\u224810\u2009m\u22121$ with a corresponding real component of frequency of $70\u2009kHz$. In order to account for the variance of experimental measurements, we sampled 600 times randomly from the range of values shown in Table I and generated dispersion relations similar to Fig. 5 for each case. This yielded a dataset of predicted axial wavelength and frequency for each mode number. We report the average values from this dataset as well as confidence intervals that represent the variance.*

Figure 6 shows the theoretical predictions against experimental measurements of the wave properties. The measured values are drawn from Figs. 3(b) and 3(c) where we have assigned error bars based on the full-width half maximum in the dispersion plots. Comparing these results, we see both qualitative and quantitative agreement within uncertainty of both the frequency and wavenumber. The frequencies exhibit particularly good quantitative agreement within error bars, increasing as a function of mode number. Similarly, the axial component for both the predicted and measured values increases linearly with mode number, although, quantitatively, the predicted values are lower. This discrepancy may stem from a number of non-ideal factor in the actual plume not included in the theoretical formulation. These include the existence of axial gradients, non-uniform temperatures and drifts, the spatial dependence of the plasma properties, and the fact that our analysis is inherently Cartesian, while the actual oscillation is non-local. In spite of these discrepancies, the agreement with prediction is still marked and provides additional quantitative support that these modes are described by the simple dispersion presented in Eq. (7) for drift-driven waves in the plasma. This is in keeping with the more detailed non-local analysis applied in Ref. 33 (though only a *m *=* *1 mode was considered in the previous work).

In summary, we have shown in the preceding evidence that both ion acoustic waves and anti-drift waves exist in the near-field plume of a hollow cathode operating in a Hall thruster. This is the first direct confirmation of these ion acoustic modes for a cathode actually operating in this environment. And, it is the first quantitative measurement of greater than *m *=* *1 modes for the drift oscillation. With this in mind, we next consider the central question as to the degree to which these oscillations may impact the electron dynamics in the near-field plume.

### D. Contributions to non-classical transport

We leverage in this section the formulation from Sec. II as presented in Eqs. (6) and (8) to quantify the impact of the IAT and anti-drift waves on electron transport. Before proceeding, however, we note that in order to estimate the amplitude of the fluctuations in plasma density in both of these expressions, we make the assumption that we can linearly relate this quantity to the ion saturation current: $(ne(kz,m)/ne)2\u2248(i\u0303sat(kz,m)/i\xafsat)2$. This is a valid approximation provided two key conditions are satisfied. The first, which is consistent with the derivations in Sec. II, is that quasineutrality applies on the time scale of both oscillations of interest such that fluctuations in the ion density mirror fluctuations in the electron density, $(ne(kz,m)/ne)=(ni(kz,m)/ni)$. The second assumption is that the relative fluctuations in ion saturation current are proportional to relative fluctuations in the ion density: $(i\u0303sat(kz,m)/i\xafsat)=(ni(kz,m)/ni)$. The validity of this equivalence is contingent on the probe sheath being able to adjust faster than the characteristic time scale of the plasma oscillation and the electron temperature remaining constant during the oscillations. Given that the formation of the Langmuir probe sheath is assumed to occur on the ion plasma frequency time scale, we assume the former criterion is met. For the latter criterion, we did not have the capability to resolve the temperature explicitly at these high frequencies. However, we note that since the dispersion relations for both the acoustic and anti-drift modes are contingent on the assumption of constant electron temperature and we have found close agreement between measurement and the linear dispersion, we proceed under the assumption that the temperature is also constant as these waves propagate. With that said, we recognize that this relationship between density and ion saturation measurements is an approximation and the results presented should be understood with this caveat.

With this in mind, we use our power spectral measurements for the ion saturation fluctuations as well as our measurements of the background and wave properties presented in Secs. IV A and IV B to estimate the anomalous collision frequencies for the propagating waves. For the ion acoustic contribution [Eq. (6)], we evaluate the summation over frequency by making the substitution (informed by dispersion relation measurements) that $kz\u2248\omega /(6000)m\u22121$ and summing over the measured power spectrum from 500 to 1250 kHz. For the contributions to the anomalous collision frequency from the anti-drift waves [Eq. (8)], we use the measurements of the background plasma parameters shown in Fig. 2 to estimate the drifts, and we evaluate at each location in the plume the summations over mode number. The amplitude of density fluctuations is inferred from the power spectra, and the measured axial wavenumber associated with each mode is determined from the Beall plots [e.g., Fig. 3(c)].

We show in Fig. 7 the results for the anomalous collision frequency from the ion acoustic waves in the axial direction, $\nu AN(z)(IAT)$, and for the anti-drift waves in the axial, $\nu AN(z)(AD)$, and azimuthal, $\nu AN(\theta )(AD)$, directions. The calculations are shown as a function of axial location at a position r = 5 mm from the cathode centerline. This is a region where we unambiguously detected the IAT and where the strong plasma gradients suggest the presence of non-classical effects. For comparison on this plot, we also show the estimated electron-neutral collision frequency, *ν _{en}*, electron-ion Coulomb collision frequency,

*ν*, and the electron-cyclotron frequency, Ω

_{ei}_{ce}. Axially, along the direction of applied magnetic field, the collision frequency from IAT is almost three orders of magnitude higher than the electron-ion collision frequency and up to an order of magnitude larger than the electron-neutral collisions. On the other hand, the anomalous collision frequency in the axial direction from the anti-drift mode, while higher than the electron-coulomb frequency for the region closer to the cathode, is less than the classical electron-neutral collision frequency. Taken together, these results suggest that the electron resistivity in the longitudinal direction in the near-field cathode plume is non-classical and is dominated by the action of the ion acoustic turbulence. In the azimuthal direction, the IAT collision frequency does not have a contribution (as this mode was assumed and later shown to propagate primarily in the longitudinal direction). However, we do find that the rotational mode collision frequency, $\nu AN\theta (AD)$, is three orders of magnitude higher than the Coulomb collision frequency and two orders of magnitude higher than the electron-neutral. This indicates that the electron dynamics in this direction (which has direct bearing on cross field transport) are non-classical and different in magnitude and scaling when compared to the longitudinal IAT-dominated motion. In both cases, longitudinal and azimuthal, we note that the electron cyclotron frequency still exceeds the collision frequency by an order of magnitude, suggesting that despite the enhanced collisionality, the plasma remains magnetized.

In summary, the above findings confirm experimentally for the first time that the transport in the strongly magnetized plasma of a hollow cathode operating in a Hall thruster is non-classical, driven by the presence of waves. However, the magnitude of this non-classical transport is anisotropic, exhibiting a different magnitude and spatial dependence in the azimuthal and longitudinal directions. Although not shown here, we also note that these trends in non-classical transport, as captured by these effective collision frequencies, persisted everywhere in the measurement domain. In Sec. V, we discuss these results in the context of previous work, their physical significance, and on-going attempts to model these non-classical effects self-consistently.

## V. DISCUSSION

### A. Implications for previous studies

Our experimental finding that wave-driven drag acts on the electrons in the cathode plume of a Hall thruster extends the conclusions from previous studies in two critical ways. First, although it has been shown that IAT exists in the plumes of stand-alone cathodes,^{23,25} it was an open question as to if the IAT would persist if the cathodes were operated in a thruster environment. This is because of the differences in magnetic field and background neutral density. Our study is the first demonstration to our knowledge that not only does IAT exist in a cathode operating in conjunction with a Hall thruster but that it can dominate the electron resistivity in the longitudinal direction. Second, we have shown that in addition to IAT, a second wave, the anti-drift mode, enhances the electron drag anisotropically and non-classically in both the azimuthal and axial directions. This suggests that in a treatment of a magnetized hollow cathode plume, the disparate effects of both the IAT and anti-drift waves must be considered.

In addition to this discussion of the cross field transport, we also can relate our work to the first experimental study of the anti-drift mode.^{33} It was noted in this previous investigation that even though the detected anti-drift wave in the cathode plume was propagating azimuthally, there was an oscillation in the total discharge current to the thruster at this same frequency. This was unexpected because the discharge current is an azimuthally integrated measurement. Any cross field driven current would be exhibited as a constant offset in the measured current instead of a varying value at the frequency of propagation. It was postulated in this previous work that this effect could be explained if the rotational mode somehow lost its azimuthal character and converted to a longitudinal oscillation as it propagated from the cathode into the thruster discharge chamber. Our results here, however, offer an alternative explanation as we have shown that the rotational wave has a finite axial component. This can give rise to a time-varying longitudinal current on the time scale of the oscillation. Given the magnitude of the oscillation (approaching 100% of the background in Fig. 4), it thus may not be surprising that this mode oscillation does appear in total current measurements.

### B. Implications for modeling non-classical transport

The prediction of the two-dimensional plasma state that results from the anomalous collision frequency effects would require a self-consistent model of this region. While this is beyond the scope of this investigation, we discuss here qualitatively how the measured plasma state is consistent with the presence of non-classical effects and how we can use our results to guide future modeling efforts. To this end, we first note that qualitatively we anticipate that the longitudinal enhanced resistivity will promote larger electric fields in the axial direction. This is what drives the steep gradients shown in Fig. 2(c) from $z=25\u221235\u2009mm$. The azimuthal collision frequency, on the other hand facilitates enhanced cross field transport radially, relaxing the gradients in this direction. This in turn can drive the plasma to equipotential—as exhibited downstream of $z=35\u2009mm$ in Fig. 2(c). The interplay between the longitudinal and cross field transport ultimately is what gives rise to the observed two-dimensional plasma state.

While we do not attempt to model this interplay directly here, we note that our results can be used to help guide on-going numerical efforts. Indeed, most modeling approaches for hollow cathodes to date have been fluid-based and formulated in the r-z plane. They thus cannot model the growth of kinetically driven IAT, the propagation of azimuthal modes, or the direct influence of these waves on the electron dynamics. These codes instead rely on the use of *ad hoc* transport coefficients, such as an anomalous collision frequency, to represent the kinetic and three-dimensional effects.^{21,26,27,30} While the use of these *ad hoc* terms may be effective in yielding agreement with experimental results, the challenge in adopting this approach stems from identifying closure models, i.e., expressions for the enhanced wave-driven collision frequency that accurately represent the non-classical processes. It also must be possible to solve for these coefficients self-consistently in a fluid framework. Specifically, the new, *ad hoc* terms must depend on fluid properties, e.g., $\nu AN(Te,ne,\u2026)$.

The closure problem as it relates to the IAT and anti-drift waves is illustrated by the forms of Eqs. (6) and (8). In particular, while these expressions do depend on properties such as temperature and drift that can be solved for in a fluid model, they also scale explicitly with the wave amplitude, $ne(kz,m),$ and depend on key wave properties such as wavenumber and frequency. Since these properties are not solved for in standard r-z fluid models, the closure problem thus becomes one of identifying a functional relationship between the amplitude of the waves and the plasma properties, e.g., $ne(kz,m)(Te,ne,..)$. This problem has been examined in detail for the IAT in cathode plumes in stand-alone configurations. Solutions have included adopting simplified models for the wave energy^{25} as well as implementing expressions for the wave amplitude based on the assumption that IAT growth has been saturated by nonlinear effects.^{21,26} This latter approach also has been applied to modeling the hollow cathode plume in a Hall thruster where the following expression was adopted:

where we have introduced the superscript “c” to denote closure, *ω _{pi}* is the ion plasma frequency, and

*α*is a constant of order unity. It should be noted that this expression was applied without experimental confirmation that IAT actually existed in the thruster/cathode environment. However, its use did yield predictions in qualitative agreement with the measured spatial profiles of the plasma properties in the plume of a centrally mounted cathode in a Hall thruster.

^{26}

Now that we have explicit measurements of the effective collision frequency driven by the IAT in the hollow cathode plume (Fig. 7) and measurements of the local plasma properties, we can compare them to the closure model, Eq. (12). Figure 8 shows this comparison as a function of distance from cathode exit plane at a location r = 5 mm from the thruster centerline. Here we have adjusted the coefficient, $\alpha =0.5$, to yield the best fit with the data. The resulting trend shows both quantitative and qualitative agreement in this region of the cathode suggesting that this closure, if not exact, is a reasonable approximation to be used in this class of device. This helps explain in part the relative success found in Ref. 26 in using this closure to model the cathode plume. It further suggests that this model may be applied for approximating the IAT in future fluid simulations.

With that said, while closure models have been proposed and applied in hollow cathode plume simulations to represent the influence of IAT, there has yet to be an attempt at closure based on the action of the anti-drift waves in the azimuthal direction. To motivate an expression, we re-consider Eq. (8) in light of our experimental measurements. First, as we note from Fig. 3, there is an evident linear relationship between frequency and mode number, *m*, and axial wavenumber, *k _{z}*. This suggests that the ratio of these quantities should be constant $(m/r)/kz=\delta >1$, where the inequality is greater than unity because the waves propagate primarily in the azimuthal direction. With this new variable, we can simplify Eq. (8) to

The summation over the wave amplitudes in Eq. (13) is not known *a priori*. However, we do note from Fig. 4 that the relative amplitude of fluctuations of these modes with the background is approaching the steady-state value, i.e., $ne(kz,m)/ne\u2192O[1]$. Physically, this large amplitude suggests that the wave growth may be approaching a nonlinear limit. Operating under this assumption, we consider two physically plausible mechanisms for the saturation of these drift-driven waves (cf. Ref. 35). The first is that the upper bound on the growth is limited by the amount of thermal energy available in the plasma such that $ne(kz,m)/ne\u223cq\varphi e(kz,m)/Te\u2248O[1]$. The second limits stems from the assumption that the upper bound on growth is limited by the amount of energy available in the drift that drives the mode unstable: $ne(kz,m)/ne\u223cq\varphi e(kz,m)/Te\u2248meue(\theta )2/Te$. Substituting these expressions in Eq. (13), we find

where we have introduced the labels of “thermal” and “drift” to denote the two proposed saturation mechanisms. Using the plasma measurements from Sec. IV as well as the measured collision frequency from Fig. 7, we show in Fig. 8 a comparison of Eq. (14) with the measured result. We have used values of *δ* = 2 and *δ* = 60 for the thermal and drift closures, respectively, in order to yield the best match with data. From this plot, we see that although both closures can be adjusted to agree in magnitude with the measured collision frequency, the $\nu AN(\theta )c(AD)|drift$ closure yields better agreement over the measurement domain. This latter result in particular provides at least correlational evidence that simplified closure may be appropriate and accurate in fluid based models. As a final comment, although we treated *δ* as a free parameter to match the data, we can use the measured ratio of wavenumbers (Fig. 6) to calculate a theoretical value. The result, $\delta \u224810$, is relatively close—within an order of magnitude—to the *δ* = 60 that yielded the best fit to the data in Fig. 8. The lack of exact quantitative agreement may be attributed to the several simplifying assumptions we employed in our analysis such as the adoption of a simplified canonical geometry and the use of simplified scaling law for the saturation energy.

### C. Implications for susceptibility of hollow cathodes to facility effects

The form of Eq. (14) has an implication in the context of one of the major outstanding challenges related to Hall thruster development: the problem of facility effects. Indeed, it is well known that Hall thruster operation will change as properties of the test environment, such as the ambient background pressure, vary. One of the most well-documented of these effects is the cathode coupling voltage, i.e., the potential increase from the cathode to the adjoining thruster plume will decrease with increased facility pressure. In an attempt to explain and predict the response of this parameter to facility pressure, Spektor *et al.*^{24} developed a quasi-1D model where he postulated that increasing neutral density would increase the electron-neutral collision frequency and, therefore, facilitate more cross field transport. This relaxes the potential and leads to lower coupling voltages. He showed that while his model did correctly capture trends in the coupling voltage with pressure, it was necessary to employ Hall parameters that were an order of magnitude lower than the actual values in the modeled thrusters. Recognizing that this was an unphysical assumption, Spektor suggested that there may be non-classical effects present contributing to an enhanced collision frequency. We have shown here that not only does such a non-classical mechanism exist but that it scales linearly with the electron-neutral collision frequency. This leads to an effective collision frequency that has the same dependencies on the background plasma as the electron-neutral collision frequency but with an order of magnitude higher amplitude (and therefore an order of magnitude lower Hall parameter). Coupled with Spektor's 1D model, our result suggests that the cathode's response to facility effects may indeed be explained by the enhanced non-classical transport that results from the presence of addition neutrals.

### D. Implications for ion energy in cathode plume

As the focus of this study has been on plasma waves in the near cathode plume and their impact on the electron dynamics, we have not performed measurements on the ion energies or attempted to correlate the wave properties with the ion dynamics. With that said, given that aspects of the ion energy distribution in stand-alone cathode experiments have been shown to be non-classical and wave-driven,^{20,22,50} we include here a qualitative discussion of the implications of our findings for the ion dynamics for a cathode in an actual Hall thruster.

In stand-alone experiments, the IAT has been shown to lead to enhanced heating of ions while the presence of low frequency, large amplitude waves has been correlated with transient potential structures that accelerate high energy ions back to the cathode surface. In both cases, these processes have been linked to enhanced erosion of surfaces in the near vicinity of the cathode. The presence of both IAT and large amplitude waves as measured in our cathode suggests that similar ion energization may be occurring. In correlational support of this hypothesis, two studies to date have indicated that non-classical ion acceleration does occur near cathodes operating as part of a thruster discharge.^{53,54} These investigations focused on the pole (the magnetic surface surrounding the cathode) in a magnetically shielded thruster with a centrally mounted cathode. It was found in both of these works that the ion energy distributions in the vicinity of the cathode were characterized by high energy tails (larger than the local plasma potential). It was hypothesized in Ref. 53 that these tails may be attributed to the presence of large amplitude oscillations—just as we have reported in this work. Our findings thus offer a possible explanation for non-classical ion properties that have been noted in Hall thruster cathode plumes.

Leaving aside non-classical ion effects, we also comment here on the strong assumption we have made that the cathode plasma is dominated by singly charged ions. This is justified by the observation that the electron temperature is relatively low in this region. With that said, previous investigations (cf. Ref. 55) have shown that there may be multiple charge states—though in small fraction—in stand-alone cathode plumes. If present, these ions will gain even more energy from potential gradients, further enhancing erosion of surfaces they impact. In sufficient quantity, the presence of multiply charged species may also alter the dispersion of the predicted waves. Exploring this effect ultimately is beyond the scope of this paper, but we remark that understanding the influence of higher charge states as well as how instabilities impact these species are critical questions for future investigations of these devices.

### E. Impact of magnetic field topology and cathode location

The measurements we have presented here were performed on a Hall thruster that is magnetically shielded with a cathode mounted on the axis of symmetry of the thruster. However, many conventional Hall thrusters currently in service rely on external cathodes (mounted outside the annular discharge) with a magnetic field geometry that is not shielded. This invites the question as to whether the same oscillations that have been reported here exist and impact the cathode dynamics in the same way for these other systems. We discuss here qualitatively the expected influence of both changing magnetic field configuration and cathode position.

As to the influence of the magnetic field topology, we note that the differences between shielded and conventional thrusters occurs primarily in the annular discharge region. Exterior to this location and at the magnetic poles, the field structure remains relatively unaltered. This similarity was remarked upon in Ref. 33. We, therefore, do not anticipate qualitatively that the differences that arise in a shielded configuration versus unshielded will fundamentally impact the conclusions we have drawn about the near-field environment of the cathode.

The cathode placement, on the other hand, can lead to more substantial differences in the local plasma environment. For externally mounted cathodes, axisymmetry is no longer valid, and the local magnetic field is no longer purely axial with respect to the cathode body. With that said, despite these differences, there is some correlational evidence to suggest that non-classical effects may still persist. For example, we anticipate that at least the same criteria for the growth of both instabilities will be met at external locations. There are still cross field gradients in the plasma properties in externally mounted cathodes^{7} that could drive drift-waves, and there is still a low impedance path along the magnetic field lines for electrons to acquire the high drift speeds that provide the energy source for acoustic modes. Similarly, the cathode coupling voltage in external cathodes has been shown to be even more susceptible to facility effects (Sec. V C) than cathodes mounted on thruster centerline.^{3,43,56} It has been suggested this is because the neutral density in the near-field of externally mounted cathodes is lower than for internal cathodes. Variations in background neutral density that result from changes in facility pressure thus produce higher fractional changes in the neutral environment for external cathodes. This high sensitivity to facility pressure is consistent with our finding that non-classical electron resistivity is directly linked to the neutral density environment and provides at least a qualitative indication that the electron dynamics in external cathodes are dominated by similar processes as the ones we have found in this work.

As an additional note, we have remarked how the potential gradients in both the cross field and longitudinal directions in the cathode plume can be attributed to non-classical effects. These potential structures in turn [Fig. 2(c)] lead to an electric field directed inward toward the cathode. Intriguingly, a previous study on a Hall thruster with an externally mounted cathode showed that there is a transverse drift in the ion population in the direction of the cathode^{57} that introduced a slight asymmetry to the discharge. It was suggested in this work that this drift may be the result of the presence of a weak electric field directed toward the cathode. Our findings support this conclusion, and as extension, we would anticipate that with decreasing facility pressure (and therefore a higher resistive path for electrons), this type of transverse drift may become even more pronounced.

### F. Relationship of findings to stand-alone tests

The question of whether stand-alone cathodes can faithfully recreate the behavior of hollow cathodes operating in Hall discharges has been the subject of extensive recent investigations (cf. Ref. 58) where different boundary conditions and flow environments have been explored. Our results suggest that in addition to the global operating conditions (current and flow), there are at least two critical elements that must be duplicated to attempt to map stand-alone experiments to cathodes operating in a thruster environment: the magnetic field and the neutral density. The former is critical as we have shown that the anti-drift wave (which only exists in the presence of a magnetic field) is a dominant oscillation in the plume driving the electron dynamics and the subsequent distribution of plasma background properties. The latter is also important as locally it is responsible for either damping (for the IAT) or driving unstable (for the anti-drift wave) the propagating waves. While recent efforts have focused on developing higher fidelity re-creations of the magnetic field environment, these same studies have also shown that the neutral density in stand-alone configurations can be 5–50 times higher than the neutral density experienced in Hall thruster environments.^{58} This stems from the fact that the stand-alone configurations employ a downstream anode for completing the electrical circuit. This structure has the unintended effect of blocking the flowing gas, artificially raising the neutral density in the near field. In light of our findings outlined above, we anticipate this higher neutral density environment may artificially damp the IAT through collisions while enhancing the effective collision frequency from the growth of the anti-drift waves. Both effects could reduce the electrical impedance along and across magnetic field lines, relaxing the potential gradients in the stand-alone configuration when compared to a real thruster environment. The major implication is that in the absence of a faithfully reproduced neutral and magnetic field environment, it may not be possible to re-create in a stand-alone configuration the behavior exhibited by hollow cathodes operating in conjunction with a thruster.

### G. Extensions beyond linear theory

As a final discussion point, we remark here on the validity of and extensions to our linearized analysis of the cathode waves and non-classical transport. To this end, one of the key tenets of the theory we have outlined for both the IAT and anti-drift waves is that the electron temperature remains constant on the time scale of the oscillations. Although there are techniques available for probing this quantity (or proxies for this quantity) in real time,^{59,60} we did not have these capabilities in our setup. If this plasma property does in fact vary, it would alter the dispersion and growth relations of both waves. However, the close correspondence we have found between our linear theory and experiment suggests that at least in the treatment of the waves, the approximation of constant temperature is appropriate.

Similarly, we note that although we have treated the IAT and anti-drift waves as separate oscillations in our analysis, it is possible that the two are coupled. Indeed, it is implicit in the formulation of the anti-drift wave in Sec. II that the electron collisions along magnetic field lines are dominated by classical effects. However, we have shown that the collisions in the longitudinal direction are non-classical. On the time scale of the slower frequency rotating oscillations, it thus is possible that the electron drag from the IAT along magnetic field could promote additional phase delay for the electron motion, thereby allowing further growth of this oscillation. Functionally, this might suggest that the transport coefficients for the IAT and drift waves are directly linked. For example, $\nu e\u2192\nu AN(z)(IAT)+\nu en$ in Eqs. (7), (8), and (14). With that said, this type of link between lower frequency, coherent oscillations and incoherent turbulence remains an active area of investigation.^{61}

## VI. CONCLUSION

In this work, we have investigated experimentally the role of electrostatic plasma oscillations in driving electron transport in the near-field plume of a hollow cathode operating with a Hall thruster. By employing a combination of ion saturation and Langmuir probes, we have measured the properties of the propagating waves and compared them to the predictions from linear dispersion relations. We have shown that both ion acoustic turbulence (IAT) (measured from 500 to 1250 kHz) and anti-drift waves (measured from 50 to 400 kHz) propagate simultaneously in the cathode plume. The IAT is primarily longitudinal and driven unstable by the strong electron drift along the magnetic field. The anti-drift waves propagate in the direction of diamagnetic drift, are rotational in nature, and derive their energy from azimuthal electron currents. In both cases, we have derived expressions for how the growth of these instabilities can impact the electron dynamics. Specifically, we have represented both effects with a transport coefficient, an anomalous collision frequency. By employing measurements of the wave dispersion and background plasma properties, we have demonstrated that both instabilities can give rise to effective collision frequencies that exceed the classical collision frequencies by 1–3 orders of magnitude. Significantly, we have found that this enhanced electron collision frequency is anisotropic. The contribution from the IAT collisions is dominant in the longitudinal direction while the collision frequency resulting from the rotational waves is dominant in the azimuthal direction. Whereas the IAT thus effects promote steeper gradients as they enhance the resistivity along magnetic lines, the rotational waves can facilitate transport across magnetic field lines, giving rise to more relaxed potential gradients. Both of these effects must be considered in arriving at a self-consistent description of the near-field plasma. We have expanded on this conclusion by proposing and validating simplified algebraic closure models that could be incorporated into self-consistent fluid-based models for the near-field cathode region. We also have discussed how our findings may help explain key aspects of cathode operation including the role of facility pressure in changing coupling voltage, the presence of anomalously high ion energies in cathode plumes, and the differences that exist between cathodes that are externally versus internally mounted in thrusters. We outlined as well how the transport in the thruster environment is fundamentally different and more nuanced than the non-classical transport that has been documented for stand-alone cathodes. And we noted that these differences pose a substantial challenge for re-creating thruster-like operation in a stand-alone configuration.

Taken together, the above results provide a comprehensive and, to our knowledge, first detailed experimental description of the wave-driven transport in the near-field of a hollow cathode operating in a Hall thruster. While we anticipate that these findings may pose a new challenge for the self-consistent modeling and prediction of the plasma dynamics in this region, the understanding of the non-classical effects and proposed closure models we have presented in this study ultimately may help guide the development and validation of these critical, higher fidelity simulations.

## ACKNOWLEDGMENTS

The authors also would like to acknowledge the technical support of Mr. Eric Viges in the operation and setup of the test facility. This work was in part supported through an Air Force Office of Scientific Research (AFOSR) Grant No. FA9550-17-1-0035. S. E. Cusson's effort was supported by the NASA fellowship NNX15AQ43H. Z. Brown’s contribution was supported by the National Science Foundation Graduate Research Fellowship Program Grant No. DGE 1256260. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.