plasma oscillations in hall thrusters - princeton...

PHYSICS OF PLASMAS VOLUME 8, NUMBER 4 APRIL 2001

Plasma oscillations in Hall thrustersE. Y. ChoueiriElectric Propulsion and Plasma Dynamics Laboratory, Princeton University, Princeton, New Jersey 08540

~Received 27 September 2000; accepted 14 November 2000!

The nature of oscillations in the 1 kHz–60 MHz frequency range that have been observed duringoperation of Hall thrusters is quantitatively discussed. Contours of various plasma parametersmeasured inside the accelerating channel of a typical Hall thruster are used to evaluate the variousstability criteria and dispersion relations of oscillations that are suspected to occur. A band by bandup-to-date overview of the oscillations is carried out with a description of their observed behaviorand a discussion of their nature and dependencies through comparison of the calculated contours toreported observations. The discussion encompasses the excitation of low frequency azimuthal driftwaves that can form a rotating spoke, axially propagating ‘‘transit-time’’ oscillations, azimuthaldrift waves, ionization instability-type waves, and wave emission peculiar to weakly ionizedinhomogeneous plasmas in crossed electric and magnetic fields. ©2001 American Institute ofPhysics. @DOI: 10.1063/1.1354644#

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I. INTRODUCTION

A. Background and motivation

Hall thrusters, also called closed drift thrusters~CDT!,are coaxial plasma accelerators for spacecraft propulsionhave been optimized in Russia during the past 30 yearshigh thrust efficiency~near and above 50%). They are paticularly suited for station keeping and orbit transfer msions where they offer substantial propellant mass savover chemical thrusters. Since 1972, more than 110 Hthrusters have been flown on Russian spacecraft, and mthan 52 thrusters remain in operation.1 An increasing numberof commercial and scientific space missions in Russia,US, Europe, and Japan are planned with spacecraft uHall thrusters.

Some of the original work on the Hall thruster was coducted in the US in the early and mid-1960’s2–5 but interestin that accelerator was greatly diminished in favor of ithrusters. Most of the work since then has been conducteRussia until the early 1990’s when a resurgent interesthese devices in the US, Europe, and Japan brought aboincreasingly strong revival in related research and devement.

The name ‘‘closed drift’’ refers to the azimuthal drift oelectrons that is common to all variants of such thrusteThe two main modern variants are the stationary plasthruster~SPT! and the thruster with anode layer~TAL !. @Thestationary plasma thruster~SPT! is sometimes referred to amagnetic layer thruster, or thruster with extended acceltion zone, to differentiate it from the anode layer thrus~TAL !, which is also operated in a stationary mode.# Theformer differs from the latter by its extended channel anduse of insulator chamber walls. Although the two have dferentiating features in their operation and performance, trely on the same basic principles for ionizing and acceleing the propellant. In the US, these principles were first crectly described and verified experimentally throughseminal work of Janes and Lowder~1965!.5 These principles,

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and other features of modern Hall thrusters, are alsocussed in Refs. 6–9. A schematic of a Hall thruster ofSPT type~i.e., extended channel! is shown in Fig. 1. Briefly,the underlying processes are as follows. The electrons fthe cathode enter the chamber, and are subject to anmuthal drift as a result of the crossed~mainly radial! mag-netic and axial electric fields. The electrons in the closdrift undergo ionizing collisions with the neutrals~most typi-cally xenon! injected through the anode. While the magnefield is strong enough to lock the electrons in an azimutdrift within the chamber, it is not strong enough to affect ttrajectory of the ions which are essentially accelerated byaxial electric field. An axial electron flux equal to that of thions reaches the anode due to a cross-field mobility that oexceeds classical values, and the same flux of electronavailable from the cathode to neutralize the exhausted ioQuasineutrality is thus maintained throughout the chamand exhaust beam, and consequently no space-charge lition is imposed on the acceleration, allowing relatively hithrust densities compared to conventional electrostatic ppulsion devices. Nominal operating conditions of a commflight module~e.g., the Russian SPT-100! operating with xe-non are 2–5 mg/s mass flow rate, 200–300 V applied vage, yielding a plasma exhaust velocity of 16 000 m/s, anthrust of 40–80 mN, at efficiencies of about 50%. Measuperformance is reported in Refs. 10–12.

Because of the stringent requirements of commerspacecraft, which often require trouble-free thruster opetion for more than 8000 hours, various aspects of Hthruster performance can benefit from further improvemeThese include efforts to lower beam divergence, characteelectromagnetic interference, lower erosion rate, lengtlifetime, further raise the thrust efficiency and thrustpower ratio and scale these devices down or up in powAlso, more compact power processing units~PPU! with spe-cific masses around 5 kg/kW are sought to enhance the cpetitiveness of these thrusters. Aside from low mass, a mrequirement of the PPU is its ability to handle or suppre

1 © 2001 American Institute of Physics

t to AIP copyright, see http://ojps.aip.org/pop/popcr.jsp

and

1412 Phys. Plasmas, Vol. 8, No. 4, April 2001 E. Y. Choueiri

FIG. 1. Schematic of a Hall thruster with an extended insulator channel~SPT! showing the external cathode, the internal anode, the radial magnetic fieldtypical particle trajectories.

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oscillations in the total circuit. The interaction between tPPU and its load can be quite complex due to the ubiquitpresence of oscillations.

As we shall see in the following sections, the Hathruster plasma has a very rich and complex wave and ncharacteristics over a wide frequency spectrum.

The character of these oscillations range from narroband and very coherent waves with well-defined propagaangles, to broadband turbulence. Aside from their rolePPU design, oscillations in Hall thrusters can play a rolethe divergence of the ion beam and can even lead toextinction of the discharge. More fundamentally, somethese oscillations play a direct role in setting the performalevel and efficiency of these devices. Many of the osciltions are inherent to the ionization, particle diffusion, aacceleration processes of the device and can be consider‘‘natural’’ modes excited by the plasma to ‘‘self-regulatethe charged particle production and diffusion processeorder to adjust to a particular imposed operating mode.

The earliest studies specifically addressing these osctions are represented in Refs. 5, 9, 13, and 14. More recewith the surge of interest in the US, Europe, and Japanumber of Hall thruster oscillation studies15–26 have beencarried out, many of which are ongoing. In an earlier confence paper27 we presented an overview of the observedcillation modes along with a description of their nature adependencies. In the present paper we present an upd


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overview that accounts also for recent measurementsstudies of Hall thruster oscillations.

B. Organization

We start in Sec. II with a quantitative characterizationthe plasma in a typical Hall thruster. We use contoursvarious plasma parameters measured inside the accelerchannel of the SPT as the starting point of our investigatand calculate the magnitude and spatial distribution of vous associated natural and collision frequencies, characttic lengths and velocities. This leads to a thorough characization of the plasma in the channel under typical operatwhich paves the way for a quantitative band by band ovview of the observed oscillations presented in Sec. III alowith theoretical models for these oscillations and the assated stability criteria. The theoretical models are evaluausing the plasma characterization of the preceding sectio

II. HALL THRUSTER PLASMA CHARACTERIZATION

Before we proceed with the discussion of the variotypes of processes in the Hall thruster, and in order to renthat discussion quantitative, we conduct a thorough and stially resolved characterization of the plasma inside the chnel. For all our subsequent calculations, we use as ininput, the contour plots measured inside the channel by Baev and Kim.28 Similar measurements were also reported


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1413Phys. Plasmas, Vol. 8, No. 4, April 2001 Plasma oscillations in Hall thrusters

Bishaevet al. in Ref. 29. These measurements were cducted in an SPT with a 10 cm diameter at a discharge vageUd5200 V, a mass flow ratem53 g/s of xenon, and adischarge current ofI d53 – 3.2 A. The authors reported29

that the dimensions of the probes used for these measments were chosen to satisfy the requirements of the ptheories adopted for data reduction and that the measureaccuracy was checked using optical diagnostics. Othertails concerning the experimental techniques can be founthese references.

A. Input contours

First, the magnetic field, the radial variation of whicmay be neglected, was mapped axially with a Hall probeis shown in Fig. 2.

We digitized the contour data using a grid of 92 bypoints with a spacing of 0.5 mm between points~correspond-ing to the resolution of the measurements!. The resultingmatrices were then used along with an image processinggorithm that was developed specifically for this task, to gerate 8-bit color raster images showing the magnitudespatial distribution of the measured parameters. An imenhancement algorithm was then used to smooth thetours for better visualization. Consequently, in all the coraster plots in this paper, the details of any structure whdimension is smaller than 0.5 mm cannot be taken as phcal. The purple background surrounding each of the imacorrespond to regions where no experimental data wavailable. This is especially the case for many paramenear the walls. Also, the accuracy and reliability of the cotours near the inlet or anode~specifically for axial locationsx<15 mm! are not good.

Panels~a!–~e! of Fig. 8 below show the resulting contours for electron temperature, charged species density, fling potential, charge density production rate and xenon ntral gas density consecutively. The inlet and anode are onleft-hand side of the graphs, and the two parallel bottomtop gray lines represent the inner and outer insulator warespectively.

The region of the electron density map that has the hiest density@Fig. 8~b!# is shown to be uniform at 631017

FIG. 2. Axial profile of radial component of the magnetic field,Br normal-ized by its maximum valueBr* ~with Br* 5180 G! as measured in Ref. 28


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m23. In reality, only the outer contour of the red regionthat map was measured at that value. The lack of data inthat contour would artificially result in an infinite characteistic length for the density gradient in that region~i.e., uni-form density!. This is to be kept in mind later when lookinat maps calculated using the density gradients.

B. Calculated plasma characteristics

The above-mentioned measurements are used to calate various parameters of interest. This creates a thoroand spatially resolved characterization of the plasma.calculate all the natural frequencies~ion and electron cyclo-tron frequenciesvci , vce ; ion and electron plasma frequenciesvpi , vpe ; the lower hybrid frequency,v lh! many col-lision frequencies~electron-neutral collision frequencyne ;ion-neutral collision frequencyn i , Coulomb collision fre-quencies; frequency of ionization by electron impact, et!.All the characteristic length scales corresponding to thfrequencies are also calculated. We also calculate all thesociated particle velocities: the electron thermal velocv te5(Te /me)

1/2; the Alfven velocity; the ion velocity andthe electron azimuthal and axial drift velocities.

In calculating the collision frequencies involving eletrons at each grid point, we use experimentally measucross sections for xenon with the full energy dependencecalculate the convolution integrals using a Maxwellian vlocity distribution for the electrons. For instance, for the toelectron-neutral collision frequency,ne , we use

ne5naE FM~v !Qe~v !v rel dv, ~1!

where v rel is the relative velocity between the two speciand Qe is the total collision cross section obtained, for xnon, from the experimental measurements in Ref. 30shown in Fig. 3.

When calculating the electron drift velocities we use tfollowing standard expressions:

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ne]x2me'Ex , ~2!

FIG. 3. Total collision cross section for electrons in xenon from Ref. 30.a0

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For Eqs.~2! and~3! we treat each grid cell as imbedded inrectangular coordinate system shown in Fig. 4, withx-axis pointing downstream along the thruster axis~i.e.,along the applied electric fieldEx! and thez-axis along the

FIG. 4. Adopted rectangular coordinate system. Thex axis is along thethruster axis~i.e., along the applied electric fieldEx) and thez axis is alongthe thruster radius~i.e., along radial magnetic fieldBr). They axis thereforecorresponds to the azimuthal dimension.


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thruster radius~i.e., along radial magnetic fieldBr). udey

therefore corresponds to the azimuthal electron drift velocwhich would be in the negativey direction. The applicationof this coordinate system to describe the entire channelries the assumption of small channel curvature:h/R!1,whereh is the channel ‘‘height’’ andR its mean radius.

All the above-mentioned parameters were calculatedeach point of the grid. In order to illustrate the orderingthe magnitudes of these parameters, the parameter fromspatial matrix was spatially averaged between the crosstions at axial positionsx152.5 cm andx254 cm and plottedin Figs. 5–7. These three plots show the magnitude ordeof the relevant frequencies, characteristic lengths, and velties, respectively.~The neutral gas temperature was assumto be at the anode temperature of 1000 K.! In the character-istic length plot of Fig. 6, we also show the calculated chacteristic lengths of the axial gradients of electron denstemperature, and magnetic field, which will be importantour oscillation analysis. In calculating the characterislength of the electron density gradient we neglectedweight of the central region@red spot in Fig. 8~b!# which, asmentioned earlier, would have yielded very large lengtThe characteristic gradient length,L

“a , of a parametera isgiven by

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wherex is the axial dimension.For further illustration, the spatial maps of the followin

parameters: electron-neutral total collision frequencyne ,ionization frequencyn ioniz , the axial electric fieldEx ~ob-tained by calculating the gradient of the applied potentia!,the ion velocity ui , the electron azimuthal drift velocityudey, and the electron Hall parameterVe are shown in Figs.8~f!–9~e! consecutively.

The total electron-neutral collision frequencyne , shownin Fig. 8~f!, was calculated using Eq.~1! and its spatial dis-

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FIG. 5. Magnitude ordering of rel-evant frequencies.~Spatially averagedbetween the cross sections at axial psitionsx152.5 cm andx254 cm.!


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FIG. 6. Magnitude ordering of rel-evant lengths.~Spatially averaged be-tween the cross sections at axial postions x152.5 cm andx254 cm.!

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tribution, to a large extent, reflects the neutral xenon disbution in the channel shown in Fig. 8~e!. This frequency, aswe shall see later in Sec. III D 2, sets an upper limit for otype of oscillations. Its average betweenx152.5 cm andx2

54 cm is 1.53 MHz. The xenon ionization frequencyn ioniz

by electron impact, shown in Fig. 9~a!, is also importantsince it represents the characteristic oscillation of the iontion instability discussed later in Sec. III B. The axial electfield shown in Fig. 9~b! shows regions with values as high20 kV/m, which explains the electron heating seen insame regions in the measured electron temperature ploFig. 8~a!. The ion velocity map of Fig. 9~c! shows that theions can reach velocities as high as 14 km/s inside the cnel. It is interesting to note from that plot that the highevelocities are attained near the downstream end of the in


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insulator wall. This fact combined with the flow directiomeasurements in that region shown in Ref. 28, hint torole played by ion sputtering in insulator erosion. The eletron drift velocity in the azimuthal directionudey, calculatedwith Eq. ~3! is shown in Fig. 9~d!. Its average betweenx1

52.5 cm andx254 cm is 690 km/s. Again, the regions osubstantial acceleration in that map corresponds to thoshigh electric field in Fig. 9~b!. Finally, the electron Hall pa-rameterVe is shown in Fig. 9~e!. It is much larger than unityover the entire grid and averages about 286 betweenx1

52.5 cm andx254 cm. The corresponding average for thion Hall parameterV i ~not shown here! is 0.15. This is char-acteristic of Hall thrusters whose design obeys the inequties V i,1!V i . This high value of the Hall parameter implies, through Eq.~3!, that the cross-field drift velocity

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FIG. 7. Magnitude ordering of rel-evant velocities.~Spatially averagedbetween the cross sections at axial psitionsx152.5 cm andx254 cm.!



FIG. 8. ~Color! Contours of plasma parameters based on the measurements of Ref. 28. Averages betweenx152.5 cm andx254 cm are in parentheses.~a!Electron temperature~12.8 eV!. ~b! Charged species density (3.331017 m23). ~c! Floating potential~73 V!. ~d! Charge production rate (3.33104 A/m3). ~e!Neutral xenon density (231019 m23). ~f! Total electron-neutral collision frequency~1.53 MHz!.

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towards the anode is, on the average, about 300 tismaller than the azimuthal drift velocity. This estimatebased on a classical treatment of cross-field mobility adiffusion. The real value of the axial electron current canestimated by subtracting the measured ion current fromdischarge current and is often much higher than the classvalue given by Eq.~2!. Quite often, one must invoke neawall collisional effects9,22,20or plasma oscillations5,23,20,31toexplain the actual cross-field mobility and diffusion of thelectrons when the magnetic field is high.

With the above characterization, we have a detailed


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experimentally based picture of the plasma in the Hthruster with which we can proceed to a quantitative studythe oscillations.

III. OVERVIEW OF OBSERVED OSCILLATIONS

Oscillations over a wide spectrum of frequencies haalways been observed in the terminal characteristics of Htype thrusters. During the 1970’s several experimental st



FIG. 9. ~Color! Results of calculations. Averages betweenx152.5 cm andx254 cm are in parentheses.~a! Ionization frequency~35.3 kHz!. ~b! Axial electricfield ~6.6 kV/m!. ~c! Ion velocity ~9.3 km/s!. ~d! Electron azimuthal drift velocity~690 km/s!. ~e! Electron Hall parameter~286!. ~f! Frequency of unstablewaves according to Eq.~22! for the first mode (k51/r ) with largely azimuthal propagation (ky510kx). Purple5stable.

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ies dedicated to the observation of oscillations and the msurements of their characteristics were conductedRussia.9,13,14,32

The amplitude and frequencies of observed oscillatiwere found to be strongly dependent on operating contions:

~1! mass flow rate and propellant type;~2! applied voltage;~3! initial and time-evolving geometry;~4! degree of contamination of the discharge chamber;


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~5! cathode characteristics~mass flow rate and location!;~6! PPU characteristics and configuration;~7! and most strongly the magnetic field profile and mag

tude.

Furthermore, when spatially resolved probing of the oscitions was conducted32 inside the channel it was revealed ththe oscillation spectra also depend on the axial locationside the channel.

This multiparameter dependence renders the oscilla


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TABLE I. Character of measured oscillation spectra as a function of the maximum value of the magneti

Br* normalized by its optimal value, (Br* )opt.170 G, form54 mg/s,Ud5200 V and an SPT with a diameteof 9 cm operating with xenon. The numbers represent the relative importance of a given frequency banoverall spectrum. The scale is 1 to 10 where 1 is the weak amplitude~below the sensitivity of the diagnostics!and 10 is the dominant amplitude of the same order as the applied voltage. 0 means that the oscillationband were absent and NA denotes unavailable data. The digits are relative within a given band of freqand not across the bands. Based on experimental data and oscillograms from Refs. 9, 13, 14 and espec32.

Regime I II IIIa IIIb IV V VIBr* /(Br* )opt ,0.38 0.38–0.47 0.47–0.6 0.6–0.76 0.76–1 1–1.35 1.35<

1–20 kHz 1 1 8 8 3 10 420–60 kHz~Azimuthal 0 6 0 4 2 0 0waves!20–100 kHz 1 5 4 6 7 6 470–500 kHz 1 4 4 7 7 6 82–5 MHz~Azimuthal 1 3 3 4 5 1 1waves!0.5–10 MHz 1 3 3 4 5 5 610–400 MHz 1 3 2 3 4 5 5.GHz NA NA NA NA NA NA NA

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picture too complex for straightforward description. However, the strong dependence on the magnetic field offersopportunity to describe the oscillation spectra as a funcof the maximum value of the magnetic fieldBr* ~usuallyreached a few millimeters upstream of the exit! with all otheroperating parameters~geometry, mass flow rate, dischargvoltage! held constant.~This leaves the discharge currentbe a unique function of the magnetic field.! This was at-tempted by Tilinin32 who conducted extensive measuremeof the spectra in the range 10 kHz–400 MHz at four axlocations in the channel, and under a wide range of operaconditions. Tilinin identified six major regimes of operatiodepending on the range ofBr* , with each regime characterized by a general spectrum of oscillations and a range fordischarge current.

A review of these measurements and others9,13,14is sum-marized in Table I. In the following subsections we discuthese observations band by band.

It is relevant to mention here that a survey of the Hthruster oscillations can be made by varying the dischavoltage instead of varying the magnetic field. This wasroute followed by the recent experimental characterizatcarried out by Chestaet al.16 and Gasconet al.18 Such sur-veys offer a different view of the dependencies of somethe modes described below.

A. Bulk discharge oscillations in the 1–20 kHz band

1. Observed behavior

Oscillations in this frequency band are often referredin the Russian literature as ‘‘loop,’’32 ‘‘circuit’’ or‘‘contour’’ 26 oscillations. WhenBr* is increased to about haits optimal value~Regime IIIa in Table I! these oscillationsbecome prominent and their amplitude can reach 10% oftotal voltage amplitude. ForBr* /(Br* )opt between 0.8 and 1these oscillations are relatively damped. This regime oferation is called ‘‘optimal’’32 ~Regime IV in Table I!. Theoscillations become violent as soon asBr* is increased a little

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above its optimal value~Regime V in Table I!. Their ampli-tude can reach as high as 100% of that of the dc voltaoften causing the discharge to be extinguished. Since atBr*5(Br* )opt the ratio of the ion current at the exit to the dicharge current reaches a maximum and any further increin Br* causes a severe onset of oscillations in this frequeband, the Hall thruster is typically operated just below thonset in the so-called ‘‘optimal regime’’~Regime IV inTable I!.

2. Physical description

By operating in the‘‘optimal’’ regime~Regime IV inTable I! which, for a fixed thruster, mass flow ratem anddischarge voltageVd can be reached by changingBr* , theviolent oscillations in this band can generally be avoidedis known that these oscillations can be quite sensitive toentire circuitry including the PPU~hence their name ‘‘cir-cuit’’ or ‘‘loop’’ oscillations ! and that the use of the propematching filter in the circuit can be instrumental in furthreducing their amplitude. A recent numerical model devoped by Boeuf and Garrigues,20 however, has shown that it inot necessary to invoke interaction between the thrusterthe external circuit to explain these oscillations. Rather,model, which consists of a one-dimensional~1D! transienthybrid treatment of electron and ion transport~electron fluidand collisionless kinetic equation for the ions! points to aninstability caused by a periodic depletion and replenishmof the neutral near the exit. Since the magnetic field in tregion is large, the associated low electron conductivity leto an increase in the electric field required to maintain crent continuity. The resulting enhanced ionization deplethe neutral density causing the downstream front of the ntral flow to move upstream into a region where the ionizatrate is lower. This decrease in upstream ionization ratecause a decrease in the flux of electrons to the exit whcauses the ionization there to abate and effectively bri


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back downstream the neutral gas front. The completed cwill then restart causing an oscillation whose frequency fain the 15–22 kHz range. The authors termed this fluctuaa ‘‘breathing’’ mode. The model’s predictions of these osclations seem to be in good qualitative agreement withrecent measurements of Darnonet al.21 and conform to theobserved behavior described in the preceding subsecwith the exception that the model does not show a dampof these oscillations in the optimal regime described abo

The detailed picture derived from the model of Boeand Garrigues substantiates the appellation ‘‘instabilitythe position of the zone of ionization’’ sometimes used inRussian literature to refer to these oscillations. This also croborates the results obtained independently from themerical model of Fifeet al.22 who used a two-dimensiona~2D! description in which the electrons are treated as a flua particle-in-cell~PIC! code was used for the heavy specand the effects of wall conductivity were included. Thesimulations when compared to a simple analytical modethe fluctuation of the ionization zone, revealed a ‘‘predatprey’’ type fluctuation in the right frequency range thatessentially equivalent to the picture described above.analytical model is based on writing the following specconservation equations:22

]ni

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L, ~10!

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vn

L, ~11!

wherek is the ionization rate coefficient that depends onelectron temperature,L is the extent of the zone,ni , v i , nn ,vn are the densities and velocities of the ions and neutrrespectively. Linearizing the above two equations for smperturbations,ni8 and nn8 in the densities and combininyields

]2ni8

]t21kni ,0nn,0ni850, ~12!

where the 0 subscripts denote unperturbed quantities.equation represents an undamped harmonic oscillator wfrequency

v5~k2ni ,0nn,0!1/25S ni ,0

ni ,0nn,0D 1/2

~13!

which can be readily estimated for our particular case stuUsing the average values shown in the captions of panels~b!,~d!, and~e! of Fig. 8, we have for the charge production rani ,053.33104 A/m35231023m23 s21, and the densitiesni ,053.331017m23, nn,05231019m23, yielding an oscilla-tion frequency of about 12 kHz.

Furthermore, since to a zeroth order in the perturbatwe haveknn,05v i /L and kni ,05vn /L, the frequency canalso be expressed as

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which shows that it scales inversely with the length of tionization region. This was indeed observed experimentby Schmidtet al.33 in a Hall discharge. A recent experimental characterization of plasma fluctuations of a Hall thrusby Chestaet al.16 noted the existence of these oscillationsan axial mode in the ionization zone at high operating voage conditions.

A consistent experimental and theoretical descriptionthis mode has emerged despite the fact that rigorous exnations for the stability criteria and the abatement of tmode for a certain range of magnetic field strength, as noin the preceding subsection, have not yet been fully mad

In sum, there is now little doubt as to the primary origof these oscillations.

B. Rotating spoke and related oscillations in the5–25 kHz band


It is important to differentiate within that band betweetwo overlapping modes: low frequency~5–25 kHz! azi-muthal oscillations, which act as a rotating spoke andrelated to the ionization process, and higher freque~20–60 kHz! azimuthal modes which are caused by dritype instabilities associated with the gradients of densitymagnetic field.

The former are anchored in the anode region and munder some conditions~e.g., low discharge voltage! extendthroughout the discharge while the latter, described lateSec. III C, typically appear in the region of the discharwhere the magnetic field is high.

The earliest experimental study5 using azimuthally posi-tioned probes, already demonstrated that density fluctuatform a rotating spoke~in the same direction as that of thelectron drift! with a phase velocity about 0.23Ex /Br . Thespoke, which has one end near the anode and fundamfrequencies between 5 and 25 kHz, is also tilted azimuthby about 15 to 25 degrees. The frequency of these fieextend to a higher part of the spectrum than the fundamefrequency of the main spoke component but drop signcantly in amplitude. Observations of these modes wereported in Refs. 5 and 13 and most recently in Ref. 16.

The behavior of these ionization-related azimuthal oslations were not studied thoroughly as a function of the mnetic field. For a fixed magnetic field~profile and magni-tude!, the appearance and amplitude of these oscillatidepend on the location of the operating point alongcurrent–voltage curve of the thruster.13 They are dominant alow discharge voltage, tend to diminish at higher voltagand become very weak in the current saturation part ofcurrent–voltage characteristic, except in the vicinity of tanode.


Janes and Lowder,5 who were the first to measure thmode in a Hall accelerator, focused on providing a quanttive description of the role these oscillations play in anomlous electron diffusion, and onlyspeculatedon the origin andmechanisms behind the rotating spoke itself and its ass


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casitheteth

ino

ntte

leai

re,o

she

th

zion

th

e-ro-ldf the

r:

in

halen-,ovde-

id,l-

t

fe

city

ic

ndby


ated azimuthal electric fields. They showed that azimutelectric fields associated~and in phase! with the spoke existand can explain the measured cross-field electron transpothe anode. This anomalous transport or diffusion can beplained on the basis of a cross-field drift due to the crossof Br and Eu , where the latter is the azimuthal oscillatinelectric field associated with the potential oscillations.deed, measurements show that the electron currentsduced by this drift generally agree with the experimentainferred axial electron current.

Their implication of an ionization-related mechanislargely stems from a corollation they found between the amuthal speed of the spoke and the so-called critical iontion velocityuci5A2e i /M wheree i andM are the ionizationpotential and atomic mass of the propellant. This correlatwas found to hold for operation with xenon, krypton, aargon for whichuci is 4.2, 5.7, and 8.7 km/s, respectively

This rotating spoke and its associated oscillations walso observed and measured by Esipchucket al.13 and mostrecently by Chestaet al.16 where an axially tilted azimuthallypropagating disturbance in the 5–10 kHz range was obseto have a phase velocity nearuci for xenon.

The mechanism behind the formation of the spokebe qualitatively attributed to a coupling between the dennonuniformities and the ionization process with the tilt of tspoke determined by how far an ionization wave propagaalong the anode while an ion is accelerated throughdischarge.5 The spoke can thus be thought of as resultfrom incomplete ionization of the gas and rapid lossnewly born ions.13 With increasing power this competitioabates, the ionization region spreads azimuthally aroundanode and the amplitude of the low frequency spoke-relaoscillations decreases considerably.

Aside from such qualitative explanations, the detaiphysics of this mode in the Hall thruster plasma remlargely unexplored.

C. Gradient-induced oscillations in the 20–60 kHzband


These azimuthal oscillations have relatively higher fquency than those described in the preceding subsectionmore broadband and their amplitude is a strong functionthe magnetic field profile. Specifically, Morozovet al.9

showed that the amplitude of these oscillations increaseas much as a factor of 5 to 8 if the axial gradient of tmagnetic field is reversed from positive (Br increasing withdistance from the anode! to negative, while keeping thepower constant. Their phase velocities are typically inrange 0.2–0.8Ex /Br .


Unlike the modes described in Sec. III B, these amuthal oscillations are not directly related to the ionizatiprocess. Indeed one of the earliest theories9 that successfullymodeled their stability criteria, frequencies, and growrates, was based on a simple nonreacting~i.e., no ionizationor recombination! two-fluid ideal MHD formulation (Ve


al

tox-g

-ro-

i--

n

e

ed

ny

se

gf

hed

dn

-aref

by

e

-

5`) without an energy equation. By linearizing the corrsponding equations for small amplitude planar waves, Mozov et al.9 find a simple dispersion relation which can yieunstable modes depending on the sign and magnitude oion drift velocity ui , the electron azimuthal drift velocityude

and the following relative inhomogeneity scale paramete

Br

ne

]

]x S ne

BrD . ~15!

For typical Hall thruster parameters, withudey,0 and ui

.0, they find that an instability in the frequency bandquestion can exist for cases where]Br /]x,0.

In order to carry a quantitative study of these azimutoscillations for our case, we choose to apply the more geral dispersion relation derived by Esipchuck and Tilinin14

who also show that the dispersion relation of Morozet al.,9 derives from a special case of their more generalscription.

Again, the starting equations are those of a two-fluMHD, collisionless, unbounded plasma which, after the folowing assumptions:~1! unmagnetized ions~i.e., r ci@Lwhere r ci is the ion cyclotron radius andL is the devicelength!, ~2! relatively weak inhomogeneity

L,L“Br

,L“ne

, ~16!

and for the frequency range,vci!v!vce ,vpe ~wherev isthe frequency of the oscillations in question!, can be linear-ized using oscillatory disturbances of the form

a5a01a exp~kxx1kyy2 ivt !, ~17!

wherea is any of the variables of the problem,a0 its steady-state value,a is the amplitude of its oscillatory componenand a!a0 . This will yield the following dispersionrelation:9

12vpi

2

~v2kxui !2

1vpi

2

vcivce1

vpi2

k2vA2

2vpi

2 ky~udey2uB!

k2ui2~v2kyudey!

50,

~18!

wherevA is the Alfven velocity vA5B2/m0Mini , Mi is theion mass,k is the wave numberk52p/l (l is the wave-length of the oscillations!, kx andky are the components othe wave vectork along the applied electric field and thazimuthal direction, respectively~see Fig. 4!. An importantparameter in the above equation is the magnetic drift velouB ,

uB[ui

2

vciL“B, ~19!

where, again,L“B is the characteristic length of magnet

field gradient as defined in Eq.~9!. In order to study thestability of low frequency electrostatic waves, Esipchuck aTilinin14 considered the case of the frequencies bounded

~v2kxui !2!

k2vA2vcevci

k2vA21vcevci

, ~20!


s

ith


FIG. 10. ~Color! ~a! Phase velocity from Eq.~22!, normalized byEx /Br (k51/r , ky510kx!. ~b! Minimum frequency of density-gradient-driven oscillation~with l,L). ~c! Minimum wavelengths of density-gradient-driven oscillations~with l,L). ~d! Criterion for the ionization instability from Eq.~30!. ~e!Frequency of unstable waves from Eqs.~22! and ~33! for ‘‘transient-time’’ oscillations (kx510ky). ~f! Frequency of unstable high frequency waves wazimuthal propagation.

r

and furthermore added one more assumption~3!

v lh.~vcivce!1/2!vpi , ~21!

wherev lh.(vcivce)1/2 is the lower hybrid frequency. Unde

all these assumptions the dispersion relation Eq.~19! has thefollowing roots:


v5kxui2k2ui

2

2ky~udey2uB!6

kui2

2~udey2uB!

3F S kx

kyD 2

24kx~udey2uB!

kyui14

udey~udey2uB!

ui2

11G 1/2

.

~22!


dp

,ea

amionrer tigonai

u

-

a

e,

s

mre

elof-R

sin.nhe

-

0.th

hindzileld

agrtto

ta-in

i-n

inp-al

eir

fallhatthebedy in

inl

ibleve

ene-thebyralrals

eit

kly

hese

la-ncil-

ga-a-

lic-o-ajortricical

e-


Before we apply the above equation to our particular stuwe should discuss briefly Esipchuck’s and Tilinin’s assumtions as listed above. Assumption~1! is quite valid for Hallthrusters. Assumption~2!, namely that of weak homogeneityis not amply satisfied over the channel and, as it can be sfrom Fig. 6, on the average, the gradient lengths aresmaller than the device length. They are however of the sorder of magnitude. Assumption 3, namely that theplasma frequency is much larger than the lower hybrid fquency, is also, on the average, not strongly the case fotypical SPT case considered here as can be seen from FBearing this in mind, we proceed with a stability evaluatiusing Eq.~22! which, as we shall see, does predict the mfeatures of the experimental observations.

For our case study, we calculate the roots in Eq.~22!which are of the form

v5v r1 ig, ~23!

wherev r is the frequency, andg, the imaginary part, is thegrowth rate. We select the grid points where the above eqtion yields roots with positive imaginary solutions~i.e., g.0) which, from Eq.~17! correspond to growing oscillations, i.e., instability.

The frequencies of unstable waves were calculatedcording to Eq.~22! for the first mode, i.e.,k52p/l51/rwhere r is the radius of a given grid point. Furthermorsince we are looking for azimuthal modes we chooselargely azimuthal propagation withky510kx . The resultingmap is shown in Fig. 9~f! where the color purple denotestability. It can be readily seen that the plasma is stablethese disturbances for most of the channel except for a sregion which clearly falls in the part of the plasma whe]Br /]x,0, as can be noted by looking at the magnetic fiprofile in Fig. 2. The existence of an instability in regionsnegative axial gradient ofBr , is in agreement with the experimental observations mentioned above as reported in9. Indeed, the choice of aBr profile with ]Br /]x.0, such asthe one applied to the device that produced the data uhere, was motivated by stability considerations resultfrom studies with variousBr profiles such as that of Ref. 9The range of excited frequencies as calculated and showthat map, is 25 to 55 kHz which is in agreement with tmeasured frequencies of these azimuthal waves.

A map of the phase velocity,vf , of the unstable wavesvf5v/k5 f l is shown in Fig. 10~a! where the phase velocity is normalized by the azimuthal electron driftudey @shownin Fig. 9~d!# which is very close toEx /Br ~becauseVe

@1). The calculated phase velocities are between 0.1 toE/B, which are in the lower end of the range reported inabove mentioned experimental studies.

Conclusions regarding the nature of oscillations in tband: Oscillations with azimuthal propagation in this baare low frequency electrostatic waves with strongly amuthal propagation (kx!ky). They are rendered unstab~i.e., excited! by the presence of gradients of magnetic fieand density. These gradients can have the signs and mtudes to present a free energy source for electrostatic pebations to grow. The free energy source can be relateddrift velocity associated with the gradient@such as in Eq.


y,-

enlle

-he. 5.

n

a-

c-

a

toall

d

ef.

edg

in

22e

s

-

ni-ur-

a

~19!# and the instability can therefore be called a drift insbility. Furthermore, since the general dispersion relationEq. ~18!, with uB andudey both set to zero, is often assocated with fast magnetosonic waves, Esipchuck and Tilini14

proposed the name ‘‘drift magnetosonic waves.’’From, a practical point of view, these waves do not

principle represent a formidable problem to Hall thruster oeration since the thruster is nominally operated in the optimregime with a magnetic field profile that does not favor thprominence throughout most of the discharge.

D. Oscillations in the frequency band 20–100 kHz


Aside from the azimuthal waves whose frequenciesin within this range, there are other types of oscillations tcan contribute to this band of the spectrum. Even inabsence of the low frequency azimuthal waves descriabove, the measured spectra show considerable energthat range~cf. Table I!. This is the case even for operationthe ‘‘optimal’’ regime. Since the well-defined azimuthawaves, described above, fall within that band, other posscontribution from nonazimuthal oscillations did not receias much characterization in the literature.


From Fig. 5, it can be seen that this band falls betwethe ion collision frequency and the electron collision frquency. Also, from the same figure, we can see thatcharacteristic frequency for the incoming xenon neutralselectron impact falls also in that range. It is therefore natuto suspect that mechanisms related to collisions with neut~which dominate in weakly ionized plasmas! and/or ioniza-tion are playing roles in this band. We shall look, albbriefly, at each of these two possibilities.

a. Instabilities related to an inhomogeneous and weaionized plasma.The ionization fraction was calculated fromthe data shown in panels~a! and ~e! of Fig. 8 and averagedover all grid points betweenx152.5 cm andx254 cm is0.028. For xenon, the cross sections are such that, at telectron temperatures@averageTe is about 13 eV from Fig.8~a!#, collisions with neutrals dominate. The dispersion retion in Eq. ~19! was derived from a collisionless descriptioand although it is capable of describing the azimuthal oslations in a slightly lower frequency band~and as we shallsee in Sec. III F, in a higher band!, it does not lead to un-stable modes in this band other than for azimuthal propation. Adding the effects of collisions to the governing equtions that yielded Eq.~19!, would greatly complicate thederivation of the dispersion relation. For the sake of simpity, and to illustrate the possible instabilities of an inhomgeneous weakly ionized plasma, we shall make one massumption, namely the neglect of the applied dc elecfield. Under this assumption there exist several classtreatments of the problem.34–36

In Ref. 36 the relevant case of oscillations in the frquency range

n i,v,ne ~24!


n

l-

bicd

iesth

i-as

an-enrgerhefp

hee

aetensllithi

is

ter,ing

r-

ntoeofbee-ingostosi-tivethe

ityforehon-hent-orth

ed

eresip-is-flyingd

entev-hason

er-es-chf

i-


is treated. It is shown that oscillations with frequencies agrowth rates given by

v r.g.kS Te

MiD 1/2

~25!

@where one recognizes the ion acoustic velocityuia

5(Te /Mi)1/2] can become unstable if the following inequa

ity regarding the strength of the density gradient holds

L“ne

<Te

vcemeS Mi

TeD 1/2

, ~26!

which is equivalent to stating that the inhomogeneity muststrong enough so thatL

“neis smaller than a characterist

cyclotron radiusr ca based on the ion acoustic velocity anthe ion cyclotron frequency (r ca[uia /vci). This, along withEqs.~24! and ~25!, gives oscillations bounded by

n i,v r<v“ne

,ne , ~27!

wherev“ne

is the electron drift frequency

v“ne

[Tek

mevceL“ne

. ~28!

Figure 10~b! shows the minimum bound on the frequencof oscillations whose wavelengths are contained insideSPT. Only grid points satisfying the criterion in Eq.~26! areshown. The rest~purple color! correspond to stable condtions. It is clear that oscillations in the band in question cbe produced. In Fig. 10~c!, the corresponding wavelengthare shown.

b. Instabilities related to ionization.The ionization in-stability in a weakly ionized gas can be describedfollows.37 Due to ionization, the electron density may icrease in a certain region. Due to the presence of currthrough that region, the evolution and absorption of eneare altered. If the energy increase is higher than the entransferred from the electrons through collisions with tneutrals, the electron density may grow further because odependence on the temperature, leading to an unstablecess. A similar type of instability was invoked to explain tlow frequency bulk discharge oscillations described in SIII A.

Smirnov37 treats the case of the ionization instability ofweakly ionized plasma in a crossed electric and magnfields. By setting up a balance equation for the electronergy per unit volume, taking into account the energy traferred from electrons to the neutrals through elastic cosions, and the relationship between perturbations oftemperature and density, the following rate equationfound:

d

dt S 3

2neTeD5nemeude

2 neF ~vce2 1ne

2!1/2

ne212

Te

e iG , ~29!

wheree i is the first ionization potential of the neutrals. Thyields the following instability criterion:

S 2Te

e iD 1/2

<Ve . ~30!


d

e

e

n

s

tsygy

itsro-

c.

icn---es

Although the case seems quite similar to the Hall thrusthere are two major differences. First there was a simplifyassumption thatne is largely independent ofTe . Second, inthat derivation, there was an allowance for ay component ofthe electric field of the orderVeEx which would force theelectron current to flow almost axially. Keeping these diffeences in mind, we evaluate the criterion in Eq.~30! in theform Ve /(2Te /ee)

1/2>1, over the entire grid. The map iFig. 10~d! shows the regions of instability and the extentwhich the criterion is satisfied. Most likely, the quantitativrelevance of this map to our problem is not good in viewthe two differences listed above, but if the picture is totrusted qualitatively, it would be interesting to note the rgion where the criterion is most satisfied. This region, benear the exit of the inner insulator is the region that is msubjected to geometrical changes, due to erosion or deption of sputtered material. There may therefore be a tentalink between the appearance of these oscillations andtime-dependent geometry.

The time scales associated with this kind of instabilare of the same order as the ionization time scales. Therewe can refer to Fig. 9~a! for the relevant frequencies whicfall within the frequency band being considered here. Cclusions regarding the nature of oscillations in this band: Toscillations in this band may be related to either a gradiedriven instability that is peculiar to weakly ionized plasmaan ionization-type instability. It is also possible that bomechanisms contribute to the spectrum in this band.

E. Oscillations in the frequency band 70–500 kHz


Oscillations in this frequency band are often call‘‘transient-time’’ oscillations in the Russian literature13 be-cause they have frequencies that roughly correspond toui /L,the ion in-chamber residence time scale. These waves wfirst measured and characterized experimentally by Echucket al.13 and their experimentally observed charactertics are well documented in that paper. Here we list briesome of their major features. They are quite active duroperation at the ‘‘optimal’’ regime. With all parameters fixeandBr increasing, these oscillations first become prominas Regime IIIb is reached with their amplitude reaching seral volts ~cf. Table I!. They increase in importance witincreasingBr* and their amplitude can become as high30% of the discharge voltage. Their amplitude distributiover the channel strongly depends on the profile ofBr . Theiroverall spectrum seems to be quite independent ofm at fixedUd . They are largely turbulent but still preserve some detministic space-time correlations. They are believed to besential for turbulence-driven or anomalous diffusion whibecomes necessary at higherBr due to the inadequacy oclassical mobility and diffusion.


The name ‘‘transient-time’’ oscillation is quite approprate since a frequencyv r of


n

th

e-bect

th

nit

hivbanticyia-

e,tio-eie

es

c

n-teanew

he

-azi-thalde to

a-

s-at-

ar-

ua-as-

delyalh

d

hatyti--ma.and

tric

rip-

be

y-

i-ient

ses,n ate


v r.kxui

b

b11, ~31!

where b[uB /uudeyu, can be obtained from the dispersiorelation in Eq.~19! for the following condition:14

ky2!kx

2 , ~32!

which implies an almost axially propagating wave, andinequality

uB. 12 @~udey

2 1ui2!1/22uudeyu#. ~33!

In Fig. 10~e! we show the calculated frequencies in rgions of instability. The excited waves, have frequenciestween 60 and 800 kHz for our case study. It can be direnoted that the region with negativeBr gradient is stablewhich is in agreement with observations. Furthermore,highest frequencies are confined just to the left ofBr* . Thisis also the case for the growth rate which is given by14

g.kxui

Ab

b11, ~34!

and therefore scale similarly to the frequencies. This confiment is in good agreement with what was observed wspatially resolved probing.13,32 Finally, the ‘‘quenching’’ ofthese oscillations with diminishingBr reported in the experi-mental literature is also apparent from Fig. 10~e! where thefrequencies~and hence the wavelengths! diminish with de-creasingBr to the left of the maximum value.

Conclusions regarding the nature of oscillations in tband: These oscillations are quasiaxial electrostatic wawith a relatively broad and mixed band. They tend torelatively turbulent and are presumed to play an importrole in regulating the plasma transport. Their characterisand dependencies are well predicted by the linear theorgradient driven magnetosonic waves with almost axpropagation (ky!kx). Physically they are due to the coupling of the ‘‘beam mode’’kxui to the oscillations driven bythe inhomogeneity. Due to their strong scaling withkui ,they are often called ‘‘transient-time’’ oscillations.

F. Oscillations in the frequency band 0.5–5 MHzand higher


Like the ‘‘transient-time’’ oscillations discussed abovoscillations at higher frequencies become more prominenBr* is increased, as illustrated in Table I. Even for operatat the optimal regime, oscillations and ‘‘noise’’ with frequencies near and higher than those of the ‘‘transient-timoscillations are more important than the lower frequencsuch as the ‘‘circuit’’ and low frequency azimuthal wavwhich become very weak.

It is interesting to note that one mode of high frequenoscillations seems to have been discovered theoreticallyEsipchuck and Tilinin14 before it was observed experimetally. The same authors looked for the theoretically predicmode using probes and high frequency instrumentationfound it to exist. This mode has the following observed bhavior. The propagation is mostly azimuthal like the lo


e

-ly

e

e-h

sesetsofl

asn

’’s

yby

dd

-

frequency azimuthal drift wave described in Sec. III A. Tfundamental frequency is~for optimal operation with xenonand nominal conditions! in the 2–5 MHz range. In that frequency range, it is possible to see this mode by usingmuthally spaced probes and most often the first azimumode is observed~i.e., k51/R) and sometimes the seconazimuthal mode is detectable. The phase velocity is closudey and is in the negativey direction. A reversal of thedirection of the field would reverse the direction of propagtion. While the amplitude is raised with increasingBr* , thefundamental frequency drops.

Aside from this azimuthal mode, the high frequency ocillations in the Hall thruster seem to have received lesstention than their lower frequency counterparts.


The theoretically predicted high frequency mode wasrived at also through the dispersion relation of Eq.~19! aftermore simplifying assumptions. The fact that the same eqtion, which as we discussed above, had some borderlinesumptions, could describe observed modes at three widifferent frequency ranges: lower frequency azimuthwaves, quasiaxial ‘‘transient-time’’ oscillations and this higfrequency azimuthal mode~all contained within the widerangevci!uvu!vce) is a testimony to its usefulness anoverall validity.

Instead of looking at these higher frequencies with tdispersion relation — an effort already accomplished analcally by Esipchuck and Tilinin14—we opt to use another formalism that emphasizes the collisional nature of the plasThis is one facet, the collisionless treatment of Refs. 1438 are missing.

The case of a weakly ionized plasma in a crossed elecand magnetic field has been treated by Simon.39 The formu-lation developed in that paper extends the simpler desction of an inhomogeneous weakly ionized plasma~that weused in Sec. III D to study the lower frequency waves! toinclude the effect of an applied electric field. The field canshown to be destabilizing for higher frequencies.36 We fol-low Simon’s assumption39 that the plasma is finite only inthex direction. This would not greatly affect our wave analsis since we will be looking only at ‘‘flute modes,’’ i.e.,kz

50 andk56ky which for our geometry correspond to azmuthal modes. By assuming that the most relevant gradis that of the charged particle density, we quote39 the mo-mentum equations for the ion and electron fluxesG,

Gx65S 2D'

6]n

]x6m'

6nExD7V6S 2D'6

]n

]z6m'

6nEzD ,

Gy652D i

6]n

]y6m i

6nEy , ~35!

Gz65S 2D'

6]n

]z6m'

6nEzD6V6S 2D'6

]n

]x6m'

6nED ,

where ions and electrons are denoted by pluses and minuV is again the Hall parameter and the formalism is cast idifferent axes convention. We go from Simon’s coordinasystem to ours in Fig. 4 by applying the transformationsx


a-

si

r-cewsorenuchn

ae

di

ase

zi-beighnd

erere

he

ther, afre-e

Hz,rvedszi-es-offernd

vertom-T asheatu-ve-henatu-

x-ld,forter-ob-ter,lowg

nkly


→x, y→z, andz→2y. The neglect of the temperature grdient in the above equations@cf. Eq. ~2!# greatly simplifiesthe problem as an energy equation is not needed. By uthe flux conservation equation in its unsteady form

]n

]t1“•G650, ~36!

one obtains a closed system of governing equations inn andV, whereV is the oscillating electrostatic field. After lineaizing with small perturbations the equations can be reduto two sets of coupled differential equations in the unknon andV. Simon uses the method of Kadmotsev–Nedospawhich requires trial solutions. The interested reader isferred to Ref. 39 for a discussion of the solution method athe boundary conditions. Having shown the governing eqtions, we now quote the resulting dispersion relation whiafter straightforward manipulation and the neglect of noflute modes~i.e., setkz50), gives the following frequencyand growth rate expressions:

v r5

Cky

dn0

dx@V im' i2Vem'e#2n0~Yi2Ye!kyQ

n02~Yi1Ye!

21ky2S dn0

dx D 2

@V im' i2Vem'e#

,

~37!

g5

Cn0~Yi1Ye!21ky

2Qdn0

dx@V im' i2Vem'e#

n02~Yi1Ye!

21ky2S dn0

dx D 2

@V im' i2Vem'e#

, ~38!

with

C52n0~YeXi1YiXe!, ~39!

Xs[L2D's , ~40!

Ys[L2m's , ~41!

L2[ky21S p

l D 2

, ~42!

Q[2E0xn0@V im' iYe2Vem'eYi # ~43!

1dn0

dx FVem'eXi2V im' iXe ~44!

11

2m' im'e~V i1Ve!

dE0

dxG , ~45!

where we have usedl, the finite spatial extent of the plasmin the x direction. A quantityf that is under the bar must bevaluated using the following integral:

f 52

l E0

l

f ~x!S px

l Ddx, ~46!

which results from the application of the boundary contions as discussed in Ref. 39.


ng

dnv-da-,-

-

We can now apply the above equations to our test cand use the numerator of Eq.~38! to find unstable roots (g.0).

The resulting frequencies of the unstable oscillations~fora case ofl 520 mm, and the first purely azimuthal mode! areshown in Fig. 10~f! only for regions of instability. We notethat according to the above formulation and calculations, amuthal oscillations in the range of few tens of MHz canexcited. It seems to be quite a different mode from the hfrequency azimuthal mode discovered by Esipchuck aTilinin14 since not only the frequencies are highfor our case but the mode is even excited in regions wh]Br /]x.0.

G. Recently observed modes

The recent experimental survey of oscillations in t2–100 kHz range, carried out by Chestaet al.,16 has revealedpreviously undocumented modes in addition to some ofbetter known oscillations discussed above. In particulaprominent axially propagating mode was observed at aquency significantly lower than that of the transient-timmode. Another mode in the frequency range above 20 kand which appears at high operating voltages, was obseto be an azimuthalm51 wave that is distinct in its propertiefrom previously observed azimuthal modes. Also, an amuthal (m512) mode that seems to be induced by the prence of the inserted probes was documented and mayinsight into the active control of Hall thruster oscillations aassociated transport.

IV. CONCLUDING REMARKS ON HALL THRUSTEROSCILLATIONS

We have considered oscillations in the Hall thruster ofive frequency bands ranging from very low frequency uptens of MHz. We used contours of various plasma paraeters measured inside the accelerating channel of an SPthe starting point of our investigation and calculated tmagnitude and spatial distribution of various associated nral and collision frequencies, characteristic lengths andlocities. This extensive plasma characterization was tused to study the various bands of oscillations that are nrally excited in such thrusters.

The complex nature of the oscillations, which are ecited by drifts and gradients of density and magnetic fiewas revealed by evaluating various dispersion relationsour numerical case study. Much of the calculated characistics had more than qualitative agreement with reportedservations. We graphically illustrated the nature, characand spatial dependencies, of various modes includingfrequency azimuthal drift waves which form a rotatinspoke, axially propagating ‘‘transit-time’’ oscillations, lowand high frequency azimuthal drift waves, ionizatioinstability-type waves, and wave emission peculiar to weaionized plasma in crossed electric and magnetic fields.


rac

-

n,

h.

-

H,

-

sC,

ical

Sci.


ACKNOWLEDGMENTS

This work was initially funded by Space Systems Loand subsequently supported by the Air Force Office of Sentific Research.

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