Measurement of ion motion in a shear Alfvén wave -...

Measurement of ion motion in a shear Alfvén waveNathan Palmer, Walter Gekelman, and Stephen VincenaDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095

sReceived 3 March 2005; accepted 22 April 2005; published online 20 June 2005d

In this study, the technique of laser-induced fluorescencesLIFd has been used to meausureTi and theE3B0 and polarization drifts of shear Alfvén waves in the Large Plasma Device at UCLAfW.Gekelman, H. Pfister, Z. Lucky, J. Bamber, D. Leneman, and J. Maggs, Rev. Sci. Instrum.62, 2875s1991dg. The waves were launched by an antenna located at the end of the device and were observedto propagate along the axis of a 9 m long, 40 cm diameter cylindrical argon plasma in the kineticregime fbe<9.5sme/midg, with fwave/ fci<0.8. Care was taken to record the measurements fromvarious diagnostics at the same spatial positions on four cross-sectional planes along the length ofthe plasma. Two-dimensional LIF measurements of the ion drifts perpendicular toB0 wereundertaken. Ion drifts were observed to be as large as 14% of the ion thermal speed. The ionpolarization andE3B0 drifts were distinguished by their phase relation toBwave. The measureddrifts are compared to kinetic theory.E' sthe transverse component ofEwaved was computed fromthe drift velocities, andEi was estimated fromE'. © 2005 American Institute of Physics.fDOI: 10.1063/1.1930796g

I. INTRODUCTION

In space physics, the high resolution measurements ofspacecraft such as Freja1 and Fast Auroral Snapshot ExplorersFASTd sRef. 2d have generated great interest in the study ofAlfvén waves in the auroral ionosphere. The Freja missionobserved strong low frequency electromagnetic spikes withDE/DB<vA/c in conjunction with deepsdn/n<0.7d spa-tially narrow swidth,d=c/vped density depressions.3 Theauthors propose that highly nonlinear Alfvén waves with anelectric field parallel to the ambient magnetic field were re-sponsible for the density depletion.

The FAST satellite has observed strong wave-particle in-teractions between energetic electrons and intense “electro-magnetic ion cyclotron waves”si.e., shear Alfvén wavesd.4

The interactions include the coherent modulation of field-aligned electron fluxes as well as the acceleration of second-ary electrons to form counterstreaming field-aligned fluxeswith energies up to 300 eV. In another mission, the Polarspacecraft measured intense electric and magnetic field struc-tures associated with Alfvén waves at the outer boundary ofthe plasma sheet. The events were observed when the space-craft was field aligned with intense auroral structures belowsdetected by an onboard UV imagerd. The authors concludethat the Poynting flux associated with these waves is of oneto two orders of magnitude larger than previously estimatedand may be sufficient to power the acceleration of auroralelectronssas well as the energization of upflowing ions andJoule heating of the ionosphered.5 A related study6 correlatesthe observations of the FAST and the Polar satellites to iden-tify the principal magnetospheric drivers of auroral accelera-tion mechanisms. The authors identify large-amplitudeAlfvén wavesspropagating from the magnetotail into the au-roral regiond as one of three main drivers of auroral accel-eration, the other two being field-aligned currents and earth-ward flows of high-energy ion beams. The GEODESICsounding rocket made simultaneous observations of two-

dimensional ion distribution functions and plasma waves inthe high-latitude topside ionosphere.7 The rocket encoun-tered large amplitude, short perpendicular wavelength Alfvénwaves, and the correspondingE3B0 drifts were measured.

Until recently Alfvén waves have been difficult to ob-serve in laboratory devices because of their low frequenciesand long wavelengths. Early experiments were limited bynoisy sources, high collisionality, and physical size. A reviewof the progress of laboratory experiments on Alfvén wavesand their relationship to space observations was given byGekelman.8

Since the introduction of the Large Plasma DevicesLAPDd at UCLA, detailed experimental studies of Alfvénwaves have been carried out in tandem with the developmentof a detailed theoretical description. Morales, Loritsch, andMaggs9 analytically described the structure of shear Alfvénwaves with small transverse scaleson the order of d=c/vped for the inertial regime. Their theory was directlycompared to experimental observations of shear Alfvén wavepropagation in the LAPDsRef. 10d with excellent agreement.Morales and Maggs extended their theory to describe shearAlfvén waves in the kinetic regime,11 and other LAPDAlfvén wave experiments have been conducted12–17 with anemphasis on their relevance to space plasmas and auroralprocesses.

Gekelmanet al.18 made measurements of the propaga-tion of large-amplitudesBwave/Bo,10−3d Alfvén waveslaunched by an inductive loop antenna in an argon plasma.The ion temperatures and drifts of the waves were measuredusing laser-induced fluorescencesLIFd at a limited number ofspatial locations. LIF has previously been used by others tomeasure the properties of electrostatic waves in plasmas.19,20

In this work we present the first detailed LIF measurementsof ion motion in a shear Alfvén wave.

PHYSICS OF PLASMAS12, 072102s2005d

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http://dx.doi.org/10.1063/1.1930796

II. SHEAR ALFVÉN WAVES

The shear Alfvén wave propagates along the ambientmagnetic field andsto first orderd causes magnetic field fluc-tuations orthogonal to the direction of the background mag-netic field. When the frequency of the wave is a substantialfraction of the ion cyclotron frequency and the parallel phasevelocity of the wave falls within the electron velocity distri-bution so that Landau damping is important, a kinetic de-scription of the wave dispersion is necessary.16 This experi-ment was performed in the kinetic regime for shear waveswhere vA!vTe and be.me/mi, with vA=B0/Î4pnMi, be

=8pnTe/B02, andvTe=Î2Te/me. In this case the kinetic dis-

persion relation for shear Alfvén wavessneglecting colli-sionsd reduces to

v2

ki2 = vA

2s1 − v2 + k'2 rs

2d, s1d

where rs=cs/vci is the ion sound gyroradius, withcs

=ÎTe/mi and v=v /vci. Note that in the limitsk'→0 andv→0, Eq. s1d reduces to the pure magnetohydrodynamicsrelation. In previous experiments21,22 wave propagation inboth the kinetic and inertial regimes has been observed. Therelation between the parallel and perpendicular componentsof the wave electric field in the kinetic regime isfsee Ref. 12,Eq. s9dg

Ei =ikik'rs

2

s1 − v2dE'. s2d

Collisions affect shear wave propagation in two impor-tant ways. First, they cause additional dissipationsdampingdas the wave propagates axially.12 Second, large perpendicularwave numbers are more heavily damped, which tends tosmooth out and broaden the transverse structure of thewaves.9 Because the LAPD plasma is highly ionized,electron-ion Coulomb collisionssnei<2.53106 s−1d are thedominant collision mechanismssee Table Id. The experi-ments reported in this work were done in the kinetic regimesvA,vTed and the plasma was made as collisionless as pos-sible. Since the LIF diagnostic required the use of an argonplasma to make the parallel wavelength as short as possiblewithout experiencing cyclotron damping, the wave frequencywas set to 80% offci. The experimental parameters are sum-marized in Table I.

III. EXPERIMENTAL SETUP

The experiments for this work were carried out in theoriginal LAPD at UCLA.23 The plasma column was 9 m longand 40 cm in diameter, withne,1012 cm−3 andTe of severaleV. Since this work was completed, the LAPD has been up-graded to be even larger and more flexible.24 The plasma iscreated by a pulsed dc electrical discharge between an oxidecoated cathode and nearby anode.

The experimental setup used to launch Alfvén waves andrecord diagnostic data is summarized in Fig. 1. As shown inthe right side of the figure, Alfvén waves were launched intothe plasma with an inductive loop antenna. The antenna is

TABLE I. Plasma parameters, derived quantities, and discharge characteristics.

Description Symbol Value

Gas fill pressuresArd PAr 5310−5 torr

Discharge current Idis 2800 A

Discharge voltage Vdis 32 V

Discharge duration Dtdis 5.5 ms

Magnetic field Bo 1200 G

Plasma density n <1.331012 cm−3

Electron temperature Te 3.6 eV

Ion temperature Ti <0.8 eV

Ion-cyclotron frequency fci 46.1 kHz

Wave frequency fwave 36.0 kHz

Wave-to-cyclotron-frequency ratio v 0.78

Ion thermal speed vTi 2.03105 cm/s

Ion sound speed cs 2.93105 cm/s

Electron thermal speed vTe 1.13108 cm/s

Alfvén speed vA 3.63107 cm/s

Scaled plasma beta besme/mid 9.5

Debye length lD 1.6310−3 cm

Collisionless skin depth d=c/vpe 0.47 cm

Ion sound gyroradius rs=cs/vci 1.0 cm

Perpendicular wavelength l' 10 cm

Parallel wavelength li 890 cm

Perpendicular wave number k' 0.63 cm−1

Parallel wave number ki 7.1310−3 cm−1

Electron-ion Coulomb collision frequency nei 2.53106 s−1

Collision-to-angular-frequency ratio G=nei/v 11

072102-2 Palmer, Gekelman, and Vincena Phys. Plasmas 12, 072102 ~2005!


made of electrically insulated copper tubing bent into a longloop, 46 cm alongB0 and 6.5 cm across. When current isdriven through the loop, the antenna currents alongB0 andthe azimuthal antenna currents inductively couple into theplasma to drive shear Alfvén waves. The rf input signal wasproduced by an arbitrary function generator, and the antennacurrent was driven by a rf amplifier through an impedance-matched, frequency-resonantLC network. The wave drivingsignal for these experiments was a tone burst of ten sinu-soidal cycles at 36 kHz with a maximum amplitude of 600 Apeak-to-peak through the antenna.

A three-axis induction coil probe was used to measurethe magnetic fields of the propagating shear Alfvén waves.25

The probe utilized shielded back-to-back coils to minimizeelectrostatic pickup noise. The magnetic field signals wererecorded over a period of 1024ms at a sample rate of 1 MHzwith a 30 shot average at each position. The data window forthe magnetic field signals begins 100ms before the rf toneburst. A Langmuir probe was used to measure the densityand temperature of the plasma. The magnetic field probe,Langmuir probe, and ion saturation current measurementswere all taken at multiple positions on four cross-sectionalplanes along the plasma column, as shown in Fig. 2. Thedistances of the fourx-y planes from the antennaslabeled asDz in the figured were 66 cm, 223 cm, 349 cm, and 475 cm,respectively.sTheir positions were chosen to provide mea-surements at regular intervals along thez axis.d All dataplanes were centered on the samex-y position. Magneticfield measurements were taken on 20320 cm2 planes with 1cm spacing between points.

To make reliable spatial correlations between wave andplasma properties that were measured with different probes apulsed electron beam was used to reproducibly align thevarious diagnostics to the same magnetic field linesthe same

x-y coordinatesd on each measurement plane. The electronbeam was a tightly wound tungsten coil mounted 443 cmfrom the antenna. It emitted a field-aligned beam of electrons<5 mm in diameter. The pulsed beam produced a readilyobservable signal on all planes. For each data plane eachdiagnostic probe was moved with stepper motors and thedata plane aligned with the beam. The magnetic field dataplanes were aligned with an accuracy of ±3 mm and the LIFand Langmuir data planes with an accuracy of ±2 mm.

IV. USING LIF TO MEASURE VELOCITYDISTRIBUTIONS

A. Basic principle

LIF is a technique which enables highly localized andtime-resolved meaurements of the ion velocity distribution ina plasma.26–28 A tunable narrow-band laser is directedthrough a selected volume in the plasma and scanned over afrequency range that spans the spectral linewidth of a suit-

FIG. 1. Experimental setup for Alfvénwave studies in the LAPD.

FIG. 2. Positions of the fourx-y planes where the data were recorded. Thelarge cylinder represents a portion of the plasma column, andDz in eachcase is the distance from the antenna. Thez axis has been scaled downrelative tox andy for better visualization.

072102-3 Measurement of ion motion in a shear Alfvén wave Phys. Plasmas 12, 072102 ~2005!


able ion transition. Ions will be excited by the laser andfluoresce if their velocity satisfies the Doppler relation

k l ·vi = 2psnl − nod, s3d

wherek l and nl are the laser wave number vector and fre-quency,vi is the ion velocity, andno is the frequency of theion transition in the rest frame. The fluorescence at eachfrequency of the laser is proportional to the number of ionsin the detection volume that have a specific velocity compo-nent alongk l. Scanning the laser over the entire spectral lineand recording the fluorescent intensity at each frequency stepthus maps out the distribution functionf isr ,v ,td at positionr . Under ordinary conditionsf i will be Maxwellian-like themeasured fluorescent light profiles shown in Fig. 3. The iontemperature can be computed from the width of the distribu-tion, while the bulk ion drift velocity is derived from thecenter offset of the distribution relative tono. Two- andthree-dimensional velocity distributions can be obtained byredirecting the laser to pass through the detection volumealong orthogonal axes.

B. LIF in argon

Input laser light at 611.492 nm excites argon ions in themetastable 3d 2G9/2 state to the 4p 2F7/2 state. Within a fewnanoseconds29 they spontaneously decay to the 4s 2D5/2 stateemitting fluorescent light at 460.957 nmswavelengths mea-sured in aird. The spectral profile can thus be interpretedsolely as a Doppler broadened Gaussian as long as otherpossible sources of spectral broadening are accounted for. Inthis experiment, the natural linewidth,29 pressurebroadening,30,31 Stark broadening,30,31 Stark splitting,32,33 la-ser bandwidth, and instrumental bandwidth of the detectionsystem were determined to be negligible. To minimize theeffect of Zeeman splitting the input laser was polarized par-allel to B0 to exclude theDmj = ±1 transitions. When thecomposite profile includes only theDmj =0 peaks atB

=1200 G, the measured full width at half maximum of theprofile differs from the purely Doppler-broadened width byless than 0.3%.

Power broadening is an artificial broadening of the spec-tral line shape that occurs when the laser intensity becomeshigh enough to saturate the atomic transition, i.e., when theoptical pumping rate and stimulated emission dominate overspontaneous emission.34–36Goeckner, Goree, and Sheridan36

measured the spectral line broadening as a function of laserintensity for the same Ar+ transitions used in these experi-ments. Their results indicate that the laser intensity used inthese experiments was close to power broadening and theapparent ion temperature may be inflated by a factor of 2. Noobvious evidence of saturation was observed during the ac-quisition of LIF data and the ion temperatures measured inthese experiments wasTi <0.8 eV. As no evidence of powerbroadening was observed when the laser power was de-creased, we will assume the ion temperature is the measuredvalue. In these experiments the maximum laser intensity wasneeded to optimize the ratio of LIF to spontaneous light. Themeasurement of ion temperature in these experiments wassecondary to the measurement of ion drifts which are unaf-fected by slight power broadening.

Figure 4 shows a schematic of the laser setup. The ionexcitation light was produced by a Coherent 899-21 tuneabledye laser and Rhodamine 6G dye, with a typical output of500 mW. The 899-21 was optically pumped with 6–8 Wfrom an argon ion laser. The other pictured componentsserved to calibrate, modulate, or monitor the laser output,and will be described subsequently.

The first attempts at detecting laser-induced fluorescentlight in the LAPD plasma followed the conventional ap-proach implemented by other labs—i.e., completely nonin-vasive external optics.27,37 This approach works well forplasmas in which the intensity of the laser-induced fluores-cence is comparable to the spontaneously emitted ion light,or the LIF signal can be extracted through extensive signalaveraging. The plasma used in this experiment satisfied nei-ther criterion. Because the plasma is pulsed approximatelyonce per second, signal averaging over more than,100shots is time consuming and impractical. The LAPD plasmasproduced by fast electronsd also has a bright background ofspontaneously emitted light at the same wavelength as thefluorescent light. This problem is exacerbated by the large

FIG. 3. sColord. LIF profiles with least-squares best-fit Gaussian curves attwo different time steps on measurement plane 2. The two curves havesimilar widthsssame ion temperatured but are horizontally offset, indicatingdifferent measured drift velocities. The reduced chi-squares are 1.8 att=192ms and 1.3 att=206ms. The two curves were taken one half waveperiod apart.

FIG. 4. Schematic of the laser setup for LIF in argon.



diameter of the plasmas<40 cmd. To image a fluorescentspot near the center of the plasma, the detection optics mustpeer through<20 cm of bright background light. Even whenthe best available spatial-filtering optical techniques wereemployed, the laser-induced fluorescent light was only asmall fractions<7%d of the total light collected. The solu-tion to this problem was to move the detection optics insidethe plasma, as close as possible to the fluorescent spot. Thisled to the development of a robust optical probe which couldmeasure two components of the ion velocity vector in theplane perpendicular to the axis of the background magneticfield. By mounting a single input beam at an angle of 45°with respect to the detection axisssee Fig. 5d measurementscould be made along two orthogonal axes simply by rotatingthe entire probe shaft 180°.

The output frequency of the dye laser was tracked andmeasured two ways—with a wave metersa Fizeau interfer-ometer, model 7711 from New Focusd and by comparisonwith the absorption spectrum from an iodine cell. A beamsplitter was used to sample<100 mW from the dye laseroutput beam which was then steered through the iodine celland into the wave meter, as shown in Fig. 4. The wave meterwas useful for coarse frequency tuning because it providesnearly instantaneous readings of the wavelength with a pre-cision of 0.001 nmsdf <0.8 GHzd or a measured velocityresolution which is<20% of vTi for 1 eV ions.

Higher precision frequency calibration was achieved us-ing the iodine cell. Using tabulated wavelengths of iodineabsorption peaks, the laser frequency was calibrated with anaccuracy of<0.1 GHz. This corresponds to a resolution invelocity space of 0.063105 cm/s or<3% of vTi for 1 eVions.

Because of the large number of shots required for signalaveraging and the long durations,1 hd of each frequencyscan, LIF data could only be acquired at a fraction of the

spatial positions where other diagnostics were recorded. Oneach of the four planes where data were takenssee Fig. 2d,LIF data were taken with both the 45° downward and 45°upward probe orientations at 9 positions on a 333 cm2 gridat the center of the planesexcept for plane 4, where onlythree positions were recordedd. Each LIF data run consistedof one full frequency scan at a single position with the probein either orientation. The data recorded included the LIF sig-nal, the laser power, the Alfvén antenna current, and thewavelength reading from the wave meter.

V. WAVE MAGNETIC FIELD DATA

Since the signals from the magnetic field coils are pro-portional to the time derivative of the magnetic field an in-tegration technique was used to obtain the magnetic field. Itsabsolute value was obtained by using the probe to measurethe field close to a long straight wire through which a knownoscillating current close to the experimental frequencyflowed. A rotation corrected the probe data for both themounted angle of the probe coils and the angle of the probeshaft at each position.

Figure 6 shows a sample ofBwave data from measure-ment plane 2. Successive frames in the figure represent timeincrements of 4.0ms. The full wave period is 28mssfwave

=36 kHzd. The basic pattern of the magnetic fields is that oftwo counterstreaming axial currents. TheBwave vectors addconstructively in the center of the pattern and destructivelytowards the edges. This wave field structure is very similar tothat of previous studies in which shear Alfvén waves werelaunched by axial currents from two small disk antennasdriven 180° out of phase.12,13 The wave fields have a broadtransverse spatial pattern that is largely independent of smalls,1 cmd inhomogeneities in plasma density. An interestingfeature is that although the entire pattern oscillates predomi-

FIG. 5. sColord. Optical and mechani-cal design of the 2D LIF probe head.



nately in a linear fashion along a nearly 45° diagonal, it alsorotates as the wave fields pass through their minima and theaxial currents twist around one another. For each of the fourplanes and for the temporal interval in which the wave waspresent a two-dimensional Fourier transform was used to de-termine the perpendicular wave number, which wask'

<0.628±0.30 cm−1. The large error reflects the large per-pendicular wavelength with respect to the grid over whichdata were acquired. This error made it impossible to measuresmall changes in wave number from plane to plane.

To illustrate the propagation of the wave alongB0, Fig. 7shows a snapshot in time of theBwave field on the four cross-sectional measurement planes. The parallel phase velocitywas measured by tracking the axial propagation of a fixed

point in phase along a field line. The average parallel phasevelocity for these experiments wasvf,i=33107 cm/s withan uncertainty of<10%. Table II lists the amplitude of thewave magnetic field on each of the four measurement planes.

Using the measured components ofBwave the axial cur-rent density can be computed as follows:

jz =c

4ps= 3 Bwaved · z.

For these low frequency waves, the displacement current isnegligible. Figure 8 shows a sample of the computed axialcurrent density along withBwave at a time when the fields aremaximum on measurement plane 2.

FIG. 6. A time series showingBwave at4 ms intervals on a single measure-ment planesplane 2d. The full waveperiod is 28ms. The plot axes indicatex andy position in centimeters.



VI. LIF DATA

For each position the digital LIF signals were acquiredas a time series at each laser frequency step. A time serieswith the laser off was also acquired as a background refer-ence. To reduce extraneous noise the data were smoothedover three time stepssout of 1024d and digitally filteredsalow-pass filter with cutoff atfcid. The laser-off signal wassubtracted from the laser-on signals and normalized to themeasured instantaneous laser power. Finally, a nonlinearleast-squares fitting algorithm was used to fit a Gaussiancurve to the profile of LIF versus ion velocity at each timestep. Examples of such fits are shown in Fig. 3. The param-eters of the fitted curves then yield the total ion drift velocity

vdrift and ion temperatureTi at each time step. The averageion temperature measured on each plane is shown in TableIII. There was no observable ion heating due to the Alfvénwaves.

To distinguish the ion drifts due to the waves from ran-dom fluctuations, correlation techniques were employed.First the cross correlation between the filteredvd,wave signaland the antenna signal was used to synchronize those twosignals in time. Then, using a sliding window four waveperiods in width, a cross correlation coefficient was com-puted at each time step for the synchronizedvd,wave and an-tenna signals. The correlation coefficients from each timestep were then turned into a weighting function with valuesranging from zero to one, indicating the probability that thetwo signals were related. Finally, the filteredvd,wave signalwas multiplied by the weighting function to attenuate theuncorrelated noise. A series of plots of the resultingmeasured ion drifts due to the waves on plane 1 is shown inFig. 9.

The drifts measured on the other planes were similar tothose of plane 1 but smaller in amplitude and with morerandom scatter in the orientation of the vectors as the wavesdecayed.

To quantify the uncertainty in the drift measurements the

FIG. 7. sColord. Bwave on the four measurement planes att=271ms. The color bar scale is in units of Gauss. Note that the distance from plane 1 to plane 4supper rightd is <li /2.

TABLE II. Wave magnetic field vs axial position.Dz is the axial distancefrom the antenna.Bwave is averaged over a few central positions on eachplane. The average wave numberk' is 0.628 +/−0.30 cm−1.

Dz scmd Bwave sGd

66 0.597

223 0.433

349 0.358

475 0.304



value ofvd,wave in proportion toBwave was examined at onewave period during the most steady-state period of wavepropagation. The average and standard deviation ofvd,wave

for those time steps was computed at each position. These, inturn, were averaged over the four positions in the LIF mea-surement plane which best represented the center of the wavefield. The values thus computed for each measurement planeare summarized in Table IV.

Figure 10 shows the comparison of the experimentallymeasured value ofv' with the theoretical value for all fourmeasurement planes. The error bars onv' are based on un-certainties inBwave andk', and represent the largest experi-mental uncertainty. The predicted values are derived fromthe plasma dispersion relation in the kinetic limit andAmpère’s law.

The ion drift formulas for finite frequencysv,vcid are

vE3B0=

c

B02

E' 3 B0

s1 − v2d, s4d

vp =c

vciB0s1 − v2d]E'

]t. s5d

The two drifts have comparable magnitudesthe polariza-tion drift is smaller by a factor ofvd but can be distinguishedby their phase relationship toBwave. The two drifts have a90° phase difference—vE3B0

is maximum whenE' andBwave are maximum, whilevp is maximum whenE' andBwave are passing through zero. At the axial position wherevE3B0

is maximum, it is antiparallel toBwave at the midpointbetween the axial current channels. From this point at a dis-tance ofli /4 farther alongz, vp is maximum and carriespolarization current acrossB0 from one axial current channelto the other. Thus, the orientation of thevd,wave vectors alsohelps to distinguish the two drifts, in conjunction with theirtemporal relationship toBwave.

Figure 11 showsBwave andvd,wave measured at two dif-ferent moments in time on plane 1. In the upper frame att=265ms, Bwave has just passed through a minimum andvd,wave is directed from one current channel to the other—anexample of polarization drift. In the lower frame, approxi-mately one-quarter wave period later att=271ms, Bwave ismaximum andvd,wave is antiparallel toBwave. The drift in thatcase isE3B0.

From the dispersion relation and Ampère’s law the ratioof the shear wave perpendicular electric to magnetic field inthe kinetic limit is

E'

Bwave=

vA

c

1 − v2

s1 − v2 + k'2 rs

2d1/2. s6d

Comparison of the thoretical and experimental values ofE'

is shown in Table V.By substitutingE' in Eq. s4d the E3B0 drift may be

written as

FIG. 8. sColord. The axial wave cur-rent density j z computed fromBwave

sat a moment in time whenBwave wasmaximumd on measurement plane 2.

TABLE III. The average ion temperature measured with LIF on each plane.Dz is measured from the antennasat the far end of the machined; the warmertemperatures at largerDz values are closer to the source region.

Dz scmd Ti seVd

66 0.77

223 0.89

349 1.04

475 1.10



FIG. 9. sColord. A time series showingvd,wave vectorsspinkd overlaid onBwave at 4 ms intervals on measurement plane 1. The largest drift measured on plane1 wasvd,wave=2.13104 cm/s, which is<11% of vTi for 0.77 eV ions.



vE3B0= SBwave

B0D vA

s1 − v2 + k'2 rs

2d1/2s− ud. s7d

The expression forvE3B0considers only the dominant

k' mode and strictly speaking is the Fourier component ofthe velocity for thisk'. The wave number spectra does havea single mode, broad enough to yield a 50% uncertainty ink'. The sensitive dependence onk' in the denominator ofEq. s7d results in a 50% uncertainty in the theoretical predic-tion when the experimental values are inserted. With this inmind the curves in Fig. 10 are in agreement. The measuredperpendicular ion drift velocity on plane 1 is compared to thetheoretical estimate of the polarization andE3B0 drift ve-locity in Fig. 12. The ion drift should oscillate in time be-tween theE3B0 and polarization drifts. In Fig. 12 the maxi-mumE3B0 agrees well with the prediction but not with thepolarization drift. The polarization drift for anm=0 mode isexpected to be largest at the moment that the wave magneticfield goes through zero. In the data the magnetic field neverhas a null between the current channels. Figure 11, top panelis a snapshot at a time where the wave field should be zero,yet this is clearly not the case. The polarization drift closest

to the midpoint of the current channels at that time points inthe correct direction, i.e., polarization current flows from oneto the other. The pattern should not rotate for anm=0 mode.This implies the existence of higher orderm numbers notincluded in the theory. The amplitude as a function of time ofthe vE3B0

is reasonable.To make a precise prediction of the measured quantities

in this experiment one must use a kinetic theory with theinclusion of collisions in conjunction with a model of theantenna including the near field. This is beyond the scope of

TABLE IV. The average amplitude and standard deviation ofvd,wave on thefour measurement planes. For comparison,vTi=1.923105 cm/s for 0.77 eVions. As the wave fields decrease away from the antenna, random noiseincreasingly obscures the measured drifts.

Dz scmdAverage

vd,wave scm/sdStandard devsvd scm/sd

Ratiosvd/vd,wave

66 1.333104 1.883103 14%

223 8.243103 1.843103 22%

349 4.913103 1.663103 34%

475 6.193103 3.433103 55%

FIG. 10. sColord. Axial dependence ofv' taken from averages from thecenter of each plane. The red curve forv',expt is the experimentally derivedvalue, and for comparison, in blue forv',theory, see formulafEq s7dg. Thedashed curves are the error ofv',expt. They are large because of the uncer-tainty in l'.

FIG. 11. sColord. Polarization drift on measurement plane 1 att=265mssupper framed and E3B0 drift on the same plane att=271ms slowerframed, approximately one-quarter wave period later. The pinkvd,wave driftvectors are overlaid onBwave.The wave is traveling into the page. The back-ground magnetic field points into the page.



this work. It is encouraging that Eq.s7d predicts velocitiesclose to those measured. There is an excellent agreement inthe observed directions of the polarization andE3B0 driftsof the wave. For the first time, the cross-field ion motion thatcloses the three-dimensional Alfvén wave current system hasbeen demonstrated with LIF. Comparison of the axial currentcarried by the electrons is consistent with the ion polarizationcurrent densitys<90 times smaller thanjzd multiplied by themuch larger cross-field area through which the polarizationcurrent flows.

Ei was estimated fromE' using the kinetic dispersionrelation fEq. s2dg. Inserting the experimental values for theparameters, we obtain

Ei =ikik'rs

2

s1 − v2dE' < S 1

110DE'.

This is derived for a collisionless plasma, however, inour caseG=nei/v=11. Table VI is a summary of the driftvelocities and the electric field on the four planes on whichdata were acquired. The parallel electric fields are over ahundred times smaller than the perpendicular ones but, nev-ertheless, this can have great consequences. The parallel fieldof the kinetic wave affects the background electrons and cancause either heating or electron acceleration depending uponthe details of the electron distribution function.

The parallel current densityjz can be derived from datasuch as those in Fig. 8. Since the derived parallel electricfield was spatially averaged, the axial current was averagedas well. The currentj isrd in each of the two channels of Fig.8 was fitted to a parabola with a base 3 cm in radius, and theaverage current density in each was evaluateds j i=5.6310−2 A/cm2d. The current parallel to the magnetic field ina shear Alfvén wave is carried by electrons and the cross-field current by the ion polarization drift. These currentsmust be continuous,j iA1= j'A2. Herej i is the current densityin either channel.A1 has a normal along the background fieldand is the region in Fig. 8 in either of the magneticO points.A2 can be approximated by a rectangle with one field-alignedside,dz, and another,L, in the transverse plane. In the case ofFig. 8, L is a line between the two current channels parallelto the magnetic field vectors. This is the line that the polar-ization current j' crosses. We estimateL<14 cm. Fromcontinuity, we estimate thatdz<1 m or about 1/8 of theparallel wavelength. The polarization current closes in therange of spatial locations whereBwave<0 and the ion drift isdirected from one current channel to another, as att=265ms in Fig. 11.

VII. SUMMARY AND CONCLUSIONS

This is a basic plasma physics experiment on the ionmotion, electric fields, and current systems of a shear Alfvénwave. While not reporting anything unanticipated, we mea-sure, for the first time to the authors’ knowledge, thevE3B0and ion polarization drifts of an Alfvén wave using laser-induced fluorescence. The drifts were measured at differenttimes during the wave cycle and the measurements agreedvery well with kinetic theory. The cross-field current of ashear Alfvén wave with finitek' is carried by the ions andlimits the final wave amplitude since the parallel electronmobility is very large in most cases. For example, in thisexperiment the cross-field ion drift was about 14% of the ionthermal speed. The wave magnetic fields were measured andthe parallel wave current derived from them was found to beconsistent with the cross-field current. The components ofthe perpendicular wave electric field were calculated fromthe drift velocity. The parallel wave electric field was evalu-ated using the dispersion relation and the value forE' andwas found to be very smalls<0.41 mV/cmd. Measurementof a predicted 0.4 V change in electron energy over the 10 mplasma column was not attempted.

Parallel wave electric fields are important for electronacceleration in the auroral ionosphere where parallel wave-

TABLE V. The average experimental perpendicular electric fieldE' andstandard deviation of the experimental values on the four measurementplanes. The error in the theoretical estimate is 50% due to the uncertainty ink'. The values of electric field are in mV/cm.

Dz scmd E'expt sexpt E'theory

66 75.9 10.3 90.3

223 47.1 9.0 67.7

349 28.0 8.3 56.8

475 19.6 22.1 52.4

FIG. 12. Solid line: measured magnitude ofvdrift as a function of time. Thedashed curve is the predicted amplitude of theE3B0 velocity fEq. s4dgusing the measured values of the dependent variables. The dotted line is thepredicted amplitude of the polarization driftfEq. s7dg at the center of mea-surement plane 1.

TABLE VI. Summary of the maximum values ofvE3B0, E', andEi on the

four measurement planes, and the estimated uncertainty inE'. For refer-ence,vTi=1.923105 cm/s for 0.77 eV ions.

Dz scmd vdrift vE3B0/vTi E' smV/cmd Ei smV/cmd

66 9.9% 68.7 0.40

223 6.1% 39.9 0.23

349 4.6% 24.6 0.14

475 7.1% 39.6 0.22



lengths are of order 100 km. Since both density and magneticfield vary along the wave trajectory effects do not cancelover a wave cycle. The wave behavior in nonuniform mediais not obvious. For example, a calculation of the propagationof a large transverse scale Alfvén wave with a densityfilament38 shows the formation of two current channelswhich appear like the ones shown in Fig. 8, but are createdby a beat between the polarization velocity of the ions andthe density perturbation instead of by an antenna. If the waveamplitude is large enough, ponderomotive effects appear.39

Nonlinear effects were not observed in this experiment butcould be expected if the wave fields were ten times larger.Although the Alfvén waves are a fundamental mode in plas-mas, their interaction with the plasma they travel through issubtle to say the least. The full implications of their behaviorin space, astrophysical and thermonuclear plasmas is just be-ginning to emerge.

ACKNOWLEDGMENTS

The authors would like to thank Professor Raul Stern forhis invaluable help at the outset of the experiment. The au-thors also acknowledge the many useful discussions withFrançois Anderegg, Masaaki Inutaki, George Morales, andJames Maggs.

This work was supported by the Office of Naval Re-search under Grant No. N00014-97-1-0167 and the NationalScience Foundation under Grant No. PHY-9309075.

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Measurement of ion motion in a shear Alfvén wave -...

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