Shear Alfve´n waves in a magnetic beach and the roles of...

PHYSICS OF PLASMAS VOLUME 8, NUMBER 9 SEPTEMBER 2001

Shear Alfve n waves in a magnetic beach and the roles of electronand ion damping

S. Vincena,a) W. Gekelman, and J. MaggsDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095

~Received 29 January 2001; accepted 11 June 2001!

Experiments are performed in the Large Plasma Device~LaPD! @Gekelmanet al., Rev. Sci. Instrum.62, 2875~1991!# at the University of California, Los Angeles to study the propagation of the shearAlfven wave in a parallel gradient of the background magnetic field. The waves are excited bymodulating a field-aligned electron current drawn to a disk antenna with a radius on the order of theelectron skin-depth,d5c/vpe . The resulting shear waves have a nonzero parallel electric field andpropagate both parallel and perpendicular to the background magnetic field. In this experiment, thewave is launched in a region where its frequency,v equals one-half the local ion-cyclotronfrequency,vci and the local Alfveń speed,vA , is approximately equal to the electron thermal speed,

ve . The wave propagates along a slowly decreasing background field to wherev5vci and vA

' ve/2. The wave thus propagates from a region where Landau damping is significant to whereion-cyclotron damping dominates. Detailed two dimensional measurements of the wave magneticfield morphology are presented. The measured wavelength decreases in accord with WKB solutionsof a modified wave equation. Wave damping is also observed and dissipation by both ions andelectrons is required in the WKB model to fit the data. Suppression of the damping via electrons inthe model results in a predicted wave magnetic field amplitude twenty times larger at theion-cyclotron resonance point than observed. ©2001 American Institute of Physics.@DOI: 10.1063/1.1389092#

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I. INTRODUCTION

The shear Alfveń wave is an electromagnetic wave inmagnetized plasma which propagates at frequencies bthe ion-cyclotron frequency. This type of wave may be foueverywhere from the Sun, the magnetosphere of the E~and other planets! to fusion research plasmas. In the myrienvironments where the shear Alfveń wave is found, manycontain spatial nonuniformities in a variety of plasma paraeters. The study of waves in the presence of nonuniformihas added greatly to our understanding of basic plasma pics and, as plasmas in the laboratory and in space are prwith ever-increasing spatial and temporal resolution, nmysteries arise to stimulate further research. This is a stof the propagation from a localized source of an axisymmric (m50 mode! shear Alfven wave in a laboratory plasmin the presence of a parallel gradient in the background mnetic field. Alfven waves produced by small-scale currefluctuations appear to be a ubiquitous and important pnomenon for understanding the complex dynamics ofEarth’s ionosphere. Although the studies of Alfveń wavesfrom small sources are still relatively recent, it is the scarcof laboratory work aimed at understanding how these wabehave in the presence of spatial nonuniformities which mtivates the present experiments.

Some of the earliest work on Alfveń waves in nonuni-form magnetic fields was performed by researchers1,2 study-ing the excitation of radial eigenmodes of a cylindric

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plasma which then propagated axially into a region of dcreasing magnetic field to the point where the wave fquency matched the local ion-cyclotron frequency—thecalled ‘‘magnetic beach.’’ Much later, Amagishiet al.3

observed the mode conversion of an axisymmetric glocompressional Alfveń mode to the shear wave along an icreasing parallel magnetic field. The launching mechanfor the compressional wave was a Helmholtz-like coil withdiameter equal to the plasma diameter. The results weresistent with the fluid-theory predictions of Woods4 for oscil-lations in a bounded, cylindrical plasma.

The development of mirror machines for fusion reseaprovided a convenient arena to study the propagationAlfven waves in parallel magnetic field gradients. In particlar, research in the heating of plasmas to attain fusion teperatures prompted the study of Alfveń waves near the ioncyclotron resonance—termed ICRF heating. Naturally, sithese experiments were intended to heat the bulk plasmafocus was on global eigenmodes and not on controlled wexcitation by localized sources. For example, ICRF heatwas studied in the THM-2 mirror machine5 using aStix-type1 coil exciter, and in the Phaedrus-B tandem mirro6

using a rotating field antenna7,8 which selectively excited aglobal,m521 eigenmode of the shear Alfveń wave and wasfound to efficiently transfer energy to the ions. Additionalthe loss of wave energy to the electrons was pointed outtheoretical study of Alfveń wave heating in open confinement systems by Zvonkov and Timofeev;9 there, the authorsconsidered the mode conversion of compressional to swaves at the Alfveń resonance layer.

4 © 2001 American Institute of Physics

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3885Phys. Plasmas, Vol. 8, No. 9, September 2001 Shear Alfven waves in a magnetic beach . . .

Few studies to date have been done on the propagaof Alfven waves from small, localized sources. Spontanefluctuations were observed by Zwebenet al.10 in the Macro-tor tokamak. An investigation of localized shear Alfveńwaves including the effects of parallel wave electric fiewere made by Borget al.11 in the TORTUS tokamak12 usingsmall, magnetic dipole antennas. The wave magnetic fifrom the localized source were observed to spread acrosbackground field lines, in contrast with the ideal magnehydrodynamic picture. The electron–neutral collision ratethis experiment was high enough that the radial propagawas dominated by resistive spreading of the wave fields

In a pair of theoretical papers beginning in 1994, Mrales et al.13,14 studied the radiated azimuthal shear Alfvńwave magnetic field from small perpendicular scale sour

in two parameter regimes:vA@ ve andvA! ve . Here,vA isthe Alfven speed (vA5B/A4pnimi , B is the backgroundfield strength,ni andmi are the ion density and mass, respe

tively! and ve[A2Te /me is the average electron thermspeed, withTe the electron temperature andme the electron

mass. The first case (vA@ ve) is relevant to the conditionsnear the edge and limiter regions of tokamak plasmas anthe Earth’s ionosphere. In this limit, the shear wave is termthe ‘‘inertial Alfven wave.’’ The propagation of the inertiawave from a source having transverse scale on the ordethe electron skin-depth (c/vpe) was found to be governed ba collisionless divergence determined by propagation cothat emanate from the edges of the exciter. In the second

(vA! ve) the shear wave is called the ‘‘kinetic Alfveńwave,’’ and is more relevant to the physics of the Eartmagnetotail and to the interior regions of tokamak plasmThe radiation pattern of the kinetic wave from small sourcwas also found to spread radially, but in a more complicafashion than the inertial wave. In the magnetosphere,

transition region between these two regimes~wherevA5 ve

and Landau damping must be considered! is currently notwell understood by the space plasma community, but mayimportant in understanding the behavior of Alfveń reso-nances on auroral field lines.15

Concurrent with the publication of the theoretical invetigations by Morales et al., a series of experimentawork16–19 served to verify and expand on the understandof the Alfven wave in both of these regimes. In 1994, Gekman et al.16 used a skin-depth-scale disk antenna~of thesame type used in this experiment! to excite the inertial Al-fven wave and verified the predicted radiation pattern13 in-cluding the spreading of wave magnetic field energy perpdicular to the background field. In 1997, Gekelmanet al.17

made similar observations with the kinetic Alfveń wave.Leneman18 studied both the inertial and kinetic waves~usinga sandwiched composite of two disk antennas! for a widerange of plasma conditions including the effects of cosions.

Low-frequency electromagnetic wave emissions halong been associated with auroral phenomena. In 1972,polar-orbiting satellite, Injun 5 detected monochromatic E~extremely low frequency! radio bursts at frequencies belothe proton gyrofrequency; these bursts were associ


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with low-energy auroral electron precipitation.20 Statisticalmeasurements of the data from the S3-3 satellite in 19821

and the ISIS-1 and ISIS-2 satellites in 198722 showed thatthese emissions were consistent with electromagneticcyclotron waves—which is simply the shear Alfveń wave atfrequencies near the ion-cyclotron frequency. The obsetion of these low-frequency waves and their correlation welectron precipitation was subsequently made by the Viksatellite23 during nightside auroral field line crossings.

In observations from the FREJA satellite reportedLouarn et al. in 1994,24 strong low-frequency electromagnetic spikes were encountered withDE/DB on the order ofvA /c. The authors attributed the spike-like nature of the snals to the satellite passing through solitary structures wscale sizes on the order of the electron skin-depth. Thphenomena were dubbed SKAWS~solitary kinetic Alfvenwaves!. The ‘‘kinetic’’ portion of this nomenclature is unfortunate, since it was intended to merely distinguish it fromideal MHD limit, whereas the term ‘‘kinetic Alfveń wave’’ isproperly applied to the shear Alfveń wave in the limitvA

! ve . In the low altitude auroral ionosphere, the oppositegenerally true:vA@ ve , so that SKAWS are actually manifestations of the inertial Alfveń wave. Later investigations byVolwerk et al. in 199625 provided a more detailed examination of SKAWS. It was determined that the SKAWS wetubular current structures of skin-depth scale withDBi /DB'10%, and they were situated at the edge of large-scshear regions in the current.

More recently, shear waves have been observed byFast Auroral SnapshoT~FAST! satellite26 in regions of field-aligned electron fluxes and are believed to be responsibleboth modulating these electron fluxes27 and accelerating ionsperpendicular to the background magnetic field to energieseveral keV.28 A comprehensive review of laboratory experments on Alfven waves and their relationship to space obsvations may be found in the recent publicationGekelman.29

The remainder of the manuscript is organized as followin Sec. II a review is presented of the theoretical aspectthe shear Alfveń wave relevant to these experiments. In SIII we describe the experimental device, antenna and dacquisition. The experimental findings are presented in SIV followed by concluding remarks in Sec. V.

II. REVIEW OF SHEAR ALFVE N WAVES FROM SMALLSOURCES

The shear Alfveń wave radiation from small disk antennas~as used in the present experiments! has been previouslystudied both theoretically and experimentally. The sourcethe Alfven wave in either case is a harmonically modulatelectron current filament with transverse size on the ordethe electron collisionless skin-depth or the ion sound gyrodius. Moraleset al. have derived an integral expression fthe spatial dependence of the radiated magnetic field ininfinite, uniform plasma,13,14

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3886 Phys. Plasmas, Vol. 8, No. 9, September 2001 Vincena, Gekelman, and Maggs

where J1 is the first-order Bessel function, andA is theHankel transform of the field-aligned current density drato the antenna atz50 andu is the azimuthal unit vector. Incontrast to the standard MHD description, Alfveń waves ra-diated from small cross-field sources have a parallel elecfield and propagate perpendicular as well as parallel tobackground magnetic field.

Experiments by Gekelmanet al.16,17 have verified thepredicted spatial form of the radiation patterns. The waare observed to satisfy the cold plasma shear Alfveń wavedispersion relation. The waves are cylindrically symmetas expected from the symmetry of the exciter. The wamagnetic field is confirmed to be primarily in the azimuthdirection and the field has zero magnitude on axis withexciter. The field amplitude increases with radial distanfrom the exciter, reaches a peak value and then decays.pattern is analogous to the magnetostatic picture of the fifrom a current carrying wire of finite radius. The radial peof the field magnitude moves outward with an increasaxial distance from the antenna since the wave propagperpendicular as well as parallel to the magnetic field. Tangle of propagation with respect to the background fieldsmall ~a few degrees! since the perpendicular group velociof the wave is much smaller than the parallel group veloc

These previous experiments have focused on the sAlfven waves radiated by small sources in various limiticases:v!vci , vA@ ve and vA! ve . In the present experiment, we study this same short perpendicular wavelenradiation but in regimes where wave particle damping isportant:v'vci ~ion-cyclotron damping! and vA' ve ~elec-tron Landau damping!. To study the shear wave under theconditions, we must use a dispersion relation which incorrates these loss mechanisms. In an infinite, uniform plasthe general dispersion relation for azimuthally symmeshear Alfven wave propagation (m50 cylindrical modes, orequivalently,ky50 in Cartesian coordinates! may be writtenas

n'2 ni

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2 2e i!/~n22exx!. ~2!

For the elements of the dielectric tensor, we use thepropriate terms provided by Stix:30

exx51

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where v i is the average ion thermal speed (v i252Ti /mi).

The functionZ is the plasma dispersion function31 which isretained to include the effects of ion-cyclotron dampiwhenv'vci . Numerical values forZ are obtained using analgorithm developed for computation of the complex erfunction.32 The first two bracketed terms inexy are due to theion response and the final term: 2v/vci is a contributionfrom the electrons. Bothexx andexy include simplificationsbased on the assumption that (k'v i /vci)

2/2!1.


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The usual parallel element~which is dominated by theelectron response! is given by

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whereze is the ratio of the parallel wave phase speed toaverage electron thermal speed (ze[v/kive) and Z8 is thederivative of the plasma dispersion function with respectits argument.

In order to estimate the effects of collisions~which areprimarily due to electron–ion Coulomb interactions undour experimental conditions!, e i is modified by including aKrook collision operator in the linearized Vlasov equatioresulting in

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where the derivative ofZ is now with respect to the newargument:he[ze(11 iG); G is the ratio of the collision fre-quency to the angular wave frequency:nei /v, andnei is theelectron–ion Coulomb collision frequency as given by Koand Horton:33

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3 ,

with the Coulomb logarithm lnL having the approximatevalue of 11 in these experiments.

III. EXPERIMENTAL ARRANGEMENT

A. Experimental device

These experiments are performed in the Large PlasDevice ~LaPD! at the University of California, Los Angele~UCLA!.34 The device is a stainless steel cylindrical vacuuchamber which is 10 m in length and 1 m in diameter. Thechamber is surrounded by 68 pancake electromagnets feseven separate power supplies—this allows for a varietyaxial field configurations. The plasma is produced by meof a pulsed electron discharge~between a heated nickel shecathode and a planar copper mesh anode! into a backgroundof neutral gas~helium in these experiments! at pressures ofapproximately 131024 Torr. The cathode is coated withthin ('531023cm! circular layer of barium-oxide whichreduces the work function of the electrons within the nick

The cathode and anode are located at one end ofdevice and are separated by a distance of 94 cm. The plawhich extends beyond this cathode–anode source regioncomes nearly fully ionized within several milliseconds aftwhich the main discharge plasma can be made to last u10’s of milliseconds. Outside the source region, the typihelium plasma density is approximately 231012 cm23 ~asmeasured with a 56 GHz swept homodyne interferomete35!,the electron temperature is 6–8 eV~measured with Lang-muir probes!, and the ion temperature is 160.5 eV ~mea-sured using a Fabry–Pe´rot interferometer!.


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B. Wave launching and detection

The wave launching antenna is the same design asin previous shear Alfveń wave experiments16,17,19and is sim-ply a 1 cm diameter circular copper mesh of 50% optictransparency. The antenna is inserted into the plasma annormal to the plane of the antenna is aligned with the baground magnetic field. A positive bias pulse is applied toantenna for several hundred microseconds during the pladischarge; the antenna bias is with respect to a floating cper end plate which terminates the plasma column. A typexperimental setup is shown schematically in Fig. 1. Tbias voltages used are between 15 and 25 Volts which rein currents of approximately one-tenth of the electron saration value and depletions of field-aligned density of aproximately 5%. The waves are launched by modulatingeffectively steady state current~through an isolation transformer! with a phase-locked tone burst from a rf amplifierfrequencies below the ion-cyclotron frequency. The veccomponents of the radiated wave magnetic fields aretected using a probe comprising three sets of opposiwound induction coils. The component signals are first dferentially amplified to remove common mode pickup athen digitized with a computer controlled data acquisitisystem.36 This system is also used to automatically moprobes in multiple dimensions, control the signal souwave form ~frequency, duration, etc.! and vary the experi-mental timing.

FIG. 1. Schematic for the wave launching antenna, magnetic field probedata acquisition system. All unlabeled cables are 50V coaxial cables.


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IV. EXPERIMENTAL RESULTS

A. Uniform plasma dispersion relation

The ability of the disk exciter to launch an azimuthalsymmetric (m50) shear Alfven wave in the LaPD has beeestablished in previous experiments16,17 for frequencies be-low v50.8vci . This was accomplished through measuments of the azimuthal magnetic field,Bu in an axially uni-form background magnetic field. What has not yet beestablished is that this same antenna also radiates a deaxial wave magnetic field (Bz) at frequencies below the ioncyclotron frequency which arises from the relative slippain the E3B drifts of the ions and electrons as part of thshear wave collective particle motion. Before conductingwave experiments in a nonuniform field, we find it prudentbegin by comparing measurement to theory of the parawave phase speed in a uniform plasma over the rang

scaled frequencies (v[v/vci) which will be present in thenonuniform experiment.

The wave dispersion measurements are performeddetermining the phase delay of thez-component of the wavemagnetic field between four axial locations, all of which aon the same field line as the antenna. To accomplish thidisk antenna of radius,r s50.5 cm ('c/vpe) is first placedat the radial center (r 50) of the 35 cm plasma column at aaxial location defined to bez50 as shown in Fig. 2. Thediameter of the plasma column is determined by takingfull-width at half the maximum value of a density profile.three-axis magnetic field probe~of transverse size less thar s) is also placed at the pointr 50, but displaced axially bya distance ofz1594 cm. The point,r 50, is determined bycentering the probe at the point of the minimum receivsignal from theBx and By probe coils while a 200 kHz~chosen arbitrarily! Alfven wave is launched from the antenna. The perpendicular components of the wave fieldmeasured to be zero atr 50 while Bz is at or near its maxi-mum value. By measuringBz at this point, it is assured thathe probe will be located on the field line connecting to tcenter of the antenna and therefore all axial measuremlocations will be field-aligned; additionally, atr 50 there isno need to account for measurement errors in the parphase speed due to radial propagation since the radial pshifts enter aseik'r . With the probe thus aligned, a 20-cyclphase-locked tone burst is launched from the antenna

frequency ofv50.5, and the resultantBz time series atz1 isdigitized and stored. The measurements are repeated unensemble average over 40 plasma discharges is recoThe wave frequency is then increased and the process

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FIG. 2. Overhead view of the LaPDshowing the position of the wavelaunching antenna~at z50) and thefour B-field probe measurement locations (z5z1 ,z2 ,z3 ,z4) during theuniform-background-field parallelphase velocity measurements.


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peated until 25 uniformly spaced values ofv ranging from0.5 to 1.1 are recorded. The probe is then moved to an alocation z25220 cm, ther 50 location is found with thesame method as described above and the frequency screpeated. Additional frequency scans using this procedare then performed atz35346 cm andz45472 cm. The axialspacing between successive measurement points is conDz5125 cm. The parallel phase speed of the wave is demined by measuring the delay time,Dt, between points ofconstant phase in the pairs of signals: (z1 ,z2), (z2 ,z3) and(z3 ,z4) for each launched frequency and computingvp

5Dz/Dt. In all cases, the measurement ofDt is made attimes just after the active broadcast of the antenna is tenated. This is done to eliminate the possibility of signal cotamination by direct pickup from the launching circuitry. Additionally, for each frequency,v, the phase velocitydetermination is made using only those pairs of axial potions which both lie beyond one parallel wavelength of tantenna. This is determined by solving forv in the cold-plasma shear Alfveń wave dispersion relation:

v/ki5vAA12v2, ~3!

which yields

v5@11~l i /2pl i !2#21/2. ~4!

Here the ion inertial length,l i5c/vpi528.7 cm. Settingl i5z1594.3 cm in this expression shows that the closmeasurement point lies within one wavelength of the antefor v,0.89. With l i5z35346 cm, this drops to 0.46, sthat the measurements of relative phase speeds betweecationsz3 andz4 are performed at an axial distance greathan one parallel wavelength from the antenna for all valof v. Figure 3 shows the measured parallel wave phaselocity as a function of frequency. The phase velocitiesnormalized to the Alfveń speed and the frequencies are nmalized to the ion-cyclotron frequency. The vertical errbars are due mainly to the uncertainty inDt while the hori-zontal error bars are from the uncertainty in the frequency

FIG. 3. Parallel phase velocities for the shear Alfveń wave in a uniformplasma. Shown are~a!–~c! measurements,~d! results from kinetic theory,and ~e! results from cold-plasma theory. The measurements are all madleast one parallel wavelength from the exciter and are coded to indwhich z-locations~see Fig. 2! were used to determine the phase velocitie


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a finite ~20-cycle! tone burst. The dashed curve in Fig.shows the phase velocities determined from Eq.~3! whichexhibits the cold-plasma resonance at the ion-cyclotronquency.

The solid curve shows the numerical solutions of tgeneral dispersion relation@Eq. ~2!# as computed from theexperimentally measured plasma parameters and a sperpendicular wave number:k'50.3 cm21; this single valuewas determined by selecting the dominant peak from Bedecompositions of magnetic field profiles,Bu(r ). This peakcontribution proved to be roughly invariant with respectthe wave frequency. The kinetic and cold-plasma curves bprovide acceptable fits to the experimental data untilproximately v50.95, above which the ion kinetic effectmust be included to match the observed minimum phspeed near the cyclotron frequency. Additionally, measuments of the~highly damped! shear Alfven wave above theion-cyclotron frequency are reproduced by the kinetic theo

B. Nonuniform background field

1. Setup

The magnets of the LaPD are configured to produc‘‘step’’ magnetic field profile—with one-half of the devicoperating at approximately 1800 G and the other half at 9G. A graph of the axial field profile atr 50 is shown in Fig.4; this figure also shows a drawing of the device with tsame axial coordinate scaling.

With this field configuration, axial scans of the plasmare made using a Langmuir probe on opposite sides ofmaximum field gradient as depicted in Fig. 4 and the resare shown in Fig. 5. The density on the low-field side of tgradient is 25% greater than on the high-field side withaverage value of 2.531012 cm23 and the temperature on thlow-field side is approximately 10% larger than the higfield side~6.3 eV vs 5.7 eV!. Although there are differencein both quantities on opposite sides of the field gradient, thappear to be localized near the region of maximum gradz'157 cm ~relative to the antenna, located atz50!. Axialmeasurements on the high-field side betweenz5110 cm andz5140 cm show both density and electron temperature tonearly constant, while on the low-field side of the gradiebetweenz5175 cm andz5205 cm, the values of densitand temperature are again nearly flat in this region, but wdifferent values. In order to have some model for the coplete axial variation of density and temperature, which wbe needed later, we assume that in the range 0 cm,z,140cm the values are constant and equal to those measuretween 110 cm,z,140 cm, and from 175 cm,z,300 cm,the values are constant and equal to the measurements175 cm,z,205 cm. To model the discontinuity betweethese regions, which cannot be measured due to physconstraints of the experimental device, we choose to jointwo regions with a linear ramp. The resulting axial profilesdensity and electron temperature are also shown in FigFinally, the low-field side plasma potential is measured togreater by 2.2 V in the bulk of the plasma than on the higfield side, indicating an ambipolar electric field of magnitu

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FIG. 4. Side view of the LaPD show-ing the magnitude of the axial magnetic field profile ~at r 50) for theparallel-gradient magnetic field experiments. Also indicated is the regioin which detailed measurements werobtained.

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35 mV/cm within the gradient. As a summary of the aboresults a selection of plasma parameters and quantitiesrived from them is presented in Table I.

2. Wave launching and measurement

A disk antenna is placed at the radial center of thelindrical plasma column at the axial location defined toz50 as depicted in Fig. 6. At this position, the launchifrequency is approximately one-half of the local cyclotrfrequency (v50.53!. A tone burst of 10 cycles is launchefrom a positively-biased disk antenna. A plot of the modlated antenna current is shown in Fig. 7. The lower axisthis figure (t) shows the timing of the wave experimentunits of wave periods from the beginning of the antencurrent modulation. The frequency,f 5355 kHz, is chosen sothat the pointv51 lies well within the measurement regioA tone burst~rather than a continuous wave! is employed toinvestigate the possibility of wave reflection from the fie

FIG. 5. Experimental electron~a! density and~b! temperature data pointsand extrapolated axial profiles covering the range from the antenna (z50)to the farthest wave field measurement point,z5300 cm. The models arecreated from measurements in two regions:~110 cm,z,140 cm! and(175,z,205 cm! where the density and temperature are approximaflat. Since it is not mechanically possible to make measurements betwz5140 cm andz5175 cm, a linear ramp is chosen to join the two regio


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gradient and the ends of the device. The first and lastcycles of the input signal are attenuated with a one-quaperiod sine envelope to reduce bothdB/dt noise and har-monic generation at the source.

Ensemble averages~over 20 plasma discharges! of thetime series of magnetic field data are acquired at six raaccess ports using a triaxial magnetic induction probe.each port, the probe samples anxz-cross-section of theplasma column. Figure 6 is an overhead view of the Lashowing each spatial location visited during the data runthe probe. The cross-sections (xz-planes! are shown to scalewithin the LaPD and also enlarged for easier viewing. Tnumber of spatial locations in each plane is not constant,varies between 372 and 561 points. The maximum parafield gradient occurs atz5157 cm, where there is no avaiable radial access port, due to a vacuum chamber seam

3. Wave magnetic field morphology

The magnetic field data acquired on the spatial gshown in Fig. 6 may be visualized by creating twdimensional images of the data planes, with the shadingthe images proportional to the strength of the magnetic fieFigure 8 shows how the data planes would look if the msured field were zero everywhere—all points in the ima

yen.

TABLE I. Measured parameters and derived quantities at the locationthe antenna (z50 cm! and the ion-cyclotron resonance point (z5260 cm!.

Value at Value at

Parameter Symbol z50 cm z5260 cm

Axial magnetic field~Gauss! B0 1744 926Electron density (31012 cm23! ne 2.2 2.8Electron temperature~eV! Te 5.7 6.3Alfven speed (3108 cm/s! vA 1.3 0.61Electron thermal speed (3108 cm/s! ve

1.4 1.5

Ratio of wave frequency to ion-cyclotron frequency

v 0.53 1.0

Ratio of collision frequency toangular wave frequency

G 1.60 1.75

Electron skin-depth~cm! d 0.36 0.32Ion gyroradius~cm! r i 0.12 0.22Ion sound-gyroradius~cm! rs 0.28 0.55


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FIG. 6. Overhead view of the LaPDfor the parallel gradient magnetic fieldexperiments. Each spatial location visited by the magnetic field probe duringthe experiment is marked with a dowithin the device and in an enlargeview above. All spatial locations lie inthe plane y50, in which By

5sgn(x)Bu . The arc-like shape of thedata planes is a consequence of tmotion of the probe.

outte

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n

kin

fo

th

, t

eeneca

is

.8

Toof

ter

utseor

ughap-pe-

theve

reInea-

oncelly

Bin

ed

theg-

r a

ne

have the same green color. This view also shows varielements of the experimental setup which, to reduce cluare not present in subsequent figures.

The instantaneous spatial patterns of they- andz-components of the wave are displayed as color imageFig. 9 for the timet55. This time is at the middle of theantenna modulation and well into an observed CW-patterthe fields. The radiated magnetic field is primarilyBu , sothat By changes sign aboutx50. The phase fronts of thewave are identifiable as continuous color regions~either redor blue!, and zero field as green. Ther-component of thewave field is not discernible from the combination of bacground noise and probe misalignments. The maximumstantaneous absolute value ofBy at this time is approxi-mately equal to 40 mG—notice that the colorbar valuesBy in this figure extend from220 mG to 120 mG. Thishalf-max scaling emphasizes phase front information rathan the precise location of wave-field extrema. Figureconveys a sense of the general wave behavior, namelyfollowing.

~1! During the active broadcast of the antenna, the wavobserved on both sides of the maximum field gradi(z5156 cm! but does not exist significantly beyond thlocation where the wave frequency matches the loion-cyclotron frequency (v51 at z5261 cm!.

~2! On the incident side of the field gradient, the wavealmost entirelyBu , with Bz only significantly noticeableon the low-field side of the gradient in the region 0&v&1.0.

FIG. 7. Modulated current signal to the antenna shown as a functioabsolute time~in micro-seconds! and as the time in wave periods since thbeginning of the pulse:t.


sr,

in

of

--

r

er9he

ist

l

~3! The parallel wavelength decreases with increasingv,reaching a minimum but nonzero value atv51.

One question which arises is that of wave reflection.study this,xz-cross-sections of the spatial wave patternsBy are displayed in Fig. 10 for times before, during, and afthe termination of the antenna current modulation:t5(7,8,9,10,11); the reader may wish to refer to the inpsignal graph of Fig. 7. As in the previous figure, the phafronts of the wave are identifiable as regions of either redblue. These phase fronts propagate from left to right althothe movement is not apparent in this figure since the snshots are taken at whole-number increments of the waveriod. At t57, the wave can be seen on both sides ofmaximum field gradient. The amplitude of the incident wadiminishes~as seen in the leftmost two planes! from t57 tot510. At t510 and later, the phase fronts of the wave aonly discernible on the low-field side of the gradient.quantitative terms, the ratio of the maximum amplitude msured on the high-field side of the gradient att57 to that att510 is 0.0460.02. Thus there is no appreciable reflectiof wave energy from the field gradient. Furthermore, sinthe region of a high magnetic field is approximately axiasymmetric about the antenna atz50 ~Fig. 4!, reflections ofthe wave from the regionz,0 may also be ruled out.

4. WKB model and wave damping

To study damping of the wave, we first develop a WKsolution for the expected wave behavior along a ray paththe axially varying plasma. This solution is then comparwith experimental data. An analysis of shear Alfveń wavepropagation in an axially nonuniform plasma leads tofollowing differential equation for the azimuthal wave manetic field:

S ]2

]z21ki

221

exx

]exx

]z

]

]zD Bu50, ~5!

whereki is the local parallel wavenumber as calculated founiform plasma. A WKB-type solution of Eq.~5! is soughtwhereBu has the general form

Bu5A~z!expF i E0

z

ki~z8!dz8G . ~6!

Inserting Eq.~6! into Eq. ~5! yields an approximate solutionfor A(z):

of



FIG. 8. ~Color! Orientation figure for the images of thexz-planes.

ta

izo

r-

turehe

be

is-sAd-eng

ri-to a

neper

A~z!5A0Aexx

ki. ~7!

To simplify the analysis and to compare with the dathe general solution is written in the following form:

Bu5A0Aexx

kie2k~z!cos@f~z!1f0#, ~8!

where

f~z!5E0

z

Re@ki~z8!#dz8

and

k~z!5E0

z

Im@ki~z8!#dz8.

The initial amplitude and phase (A0 and f0 , respectively!are free parameters of the model and will be used to optimthe fit to the data. The parallel wavenumber as a function


,

ef

z is numerically computed from local solutions of the dispesion relation@Eq. ~2!# using the axial magnetic field profileof Fig. 4 and the continuous density and electron temperaprofiles shown in Fig. 5. The only fixed quantities are twave frequency~355 kHz! and ion temperature~1 eV!. Thewave damping by electrons in the theoretical model maysuppressed by settingG50 in e i and by using the small-argument approximation for the derivative of the plasma dpersion function:Z8'22. With these limits, the electronrespond adiabatically to the presence of the wave fields.ditionally, collisional damping may be removed from thmodel while retaining electron Landau damping by makithe single approximation thatG50.

In order to compare these WKB results with the expement, we must interpolate the two-dimensional dataset onone-dimensional ray path. Figure 11 shows they-componentof the wave magnetic field in the region:x,0, at an arbitrarytime: t55.4. As can be seen in Fig. 6, this lower half-plabenefits from overlapping spatial data points, while the up

FIG. 9. ~Color! Cross-sections of the instantaneous wave magnetic fieldBy ~top! andBz ~bottom! at t55 in the planey50. The wave is launched from the

source atz50 and propagates from left to right. The maximum rms-amplitude ofBz is observed at approximatelyv50.85.


se:s


FIG. 10. ~Color! Cross-sections~in the planey50) of the instantaneous wave magnetic fieldBy at five different times near the end of the launched pult5~7,8,9,10,11!. The topmost image is att57 and time increases downward. The input modulation is completely terminated byt510. The wave propagatefrom left to right, and there is no noticeable evidence for a reflection of the wave from the parallel field gradient.

ineisceth

ticn

half plane would suffer from gaps in the data at the machport spacing. In using only thex,0 data we assume that thmagnetic field pattern is azimuthally symmetric, whichlargely supported by Fig. 10. Also shown in Fig. 11 is a blacurve representing the ray path along which the data arbe interpolated. This path is calculated by integratinggroup velocities parallel,

vg,i5vA

11k'2 rs

2 ~12v21k'2 rs

2!3/2,


e

ktoe

and perpendicular,

vg,'5v

k'

k'2 rs

2

11k'2 rs

2,

to the local magnetic field line. Here,rs is the ion soundgyroradius:rs[(Te /mi)

1/2/vci . Strictly speaking, the aboveexpressions for the group velocities are valid in the kinelimit ( vA, ve). Although the wave is launched in a regio

FIG. 11. ~Color! Measurements of the instantaneous spatial pattern of they-component of the wave magnetic field,By at time, t55.4. The black curveindicates the ray path along which the two-dimensional data were interpolated in order to compare with the WKB results.


-s

idge

d

ot

h

ed

ngnd

ek

ioth

tuthm

au

and

veichin-the

he

li-in

he

lli-m-rgyden-in-

ofsid-ed

nly

udehand.the

t ofws

-ese

c

r ofed byon of


wherevA' ve , in general the group velocity is nearly fieldaligned (vg,i@vg,') until v'1 by which point the wave haentered the kinetic regime.

The starting point for the ray path is chosen to coincwith a peak in the rms amplitude of the incident wave manetic field at (r ,z)5(22.5,75) cm. The group velocities arcalculated using a perpendicular wavenumber,k'50.7cm21. Since the azimuthal magnetic field can be expandea series of the Bessel functions,J1(knr ), this corresponds tothe third radial eigenmode in a 14.5 cm radius plasma cumn and was chosen to provide the best match betweendata and the WKB model. The sensitivity of the model to tchoice ofk' will be discussed in the next section.

The comparison of the WKB model with the interpolatdata is given in Fig. 12 which shows the following:

~a! Experimental data~b! Theory with kinetic ions (Ti51 eV!, adiabatic elec-

trons~c! Theory with kinetic ions and electron Landau dampi~d! Theory with kinetic ions, electron Landau damping a

electron–ion Coulomb collisions.

The initial amplitude and initial phase for all thretheory plots are adjusted to match the experimental peathe incident wave atz'110 cm.

The phase as a function ofz is fairly well matched by allthree theoretical profiles, with the most significant deviatoccurring at the point just after the wave emerges frommaximum field gradient region (z'175 cm!. Calculationsusing the average values for density and electron temperathroughout the measurement volume and only varyingbackground field strength produce plots with a phase juroughly twice as large~compared with Fig. 12! across themaximum field gradient region. Thus, the wave field mesurements are consistent with the Langmuir probe meas

FIG. 12. A comparison ofBy axial profiles between~a! measured datapoints, and the WKB model of Eq.~8!, with ki calculated considering~b!ion-cyclotron damping only;~c! ion-cyclotron damping plus electron Landau damping;~d! the same as~c! plus electron–ion Coulomb collisions. Thdata are taken at timet55.4 and interpolated along a ray path as discusin the text and shown in Fig. 11.


e-

as

l-hee

of

ne

reep

-re-

ments which show a rapid change in electron densitytemperature across the gradient.

Unlike the wave phase, fits of the axial decay in waamplitude show significant variations depending on whdamping effects are included, as shown in Fig. 13. Thestantaneous phase information has been removed fromdata by computing one-period rms values of ty-component of the wave magnetic field signal~multipliedby A2 to give the actual amplitude! centered about the timet55.4. The data are compared with the WKB model amptude envelope:A(z)e2k(z) for the three damping cases asFig. 12 with the values of the fitting parameter,A0 un-changed.

As was true with Fig. 12, the best theoretical fit to tamplitude data is given in Fig. 13~c! which includes the ki-netic effects of ions~ion-cyclotron damping forv'1) andelectrons~Landau damping and electron–ion Coulomb cosions.! All three theoretical amplitude curves show the copetition between the decrease in amplitude due to enetransfer from the wave to the particles and the amplituincrease resulting from the slowing group velocity and coservation of energy. The lack of a resonant amplitudecrease in the data nearv51 emphasizes the importanceincluding the electron damping mechanisms. When conering ion-cyclotron damping only, the theoretically expectamplitude of the wave atv51 is 86 mG, yet the amplitudeis 42 mG when electron Landau damping is added and o4.3 mG~coincident with the measured value! when Coulombcollisions are also included. Thus, the actual wave amplitis a factor of 20 times smaller at the resonance point twhat would be predicted if the electrons had been ignore

Of course, there is not a true resonance point sinceions have a finite temperature. The location of the onseion-cyclotron damping can be seen in Fig. 14 which sho

d

FIG. 13. Axial amplitude decay of they-component of the wave magnetifield. Shown are the experimental rms amplitude data~a! compared with theWKB results which include~b! ion-cyclotron damping only;~c! ion-cyclotron damping plus electron Landau damping;~d! the same as~c! pluselectron–ion Coulomb collisions. The wave amplitude is a free parametethe model and for a comparison, the theoretical curves have been scala factor which produces the best least-squares fit to the data in the regithe incident wave (z,150 cm!.


ef

Foac

nsauhl

thr ttharec

n-

, retticthre

Footpaum

thh

esms

e-

c-and

ail-be

for

n-

po-


the theoretical decay envelopes from the WKB modexp@2k(z)#. Curve~a! of Fig. 14 shows the spatial decay othe wave amplitude due to ion-cyclotron damping alone.helium ions at 1 eV in this experiment, the resonant intertion is still limited to a fairly narrow range:v.0.94. How-ever, curves~b! and ~c! show a steady loss to the electrowell before this point. In addition, both electron Landdamping and electron–ion Coulomb collisions show rougequal contributions to the axial amplitude decay.

5. Sensitivity of the model to k �

The model results of the previous section requireselection of a single perpendicular wavenumber in ordecompute both the ray path and the axial variation ofparallel wavenumber. But, what radial wavenumbersavailable for making this selection and how do they affthe fit of the WKB model to the data?

Figure 15 shows fits of the model~using all three damp-ing mechanisms! to the interpolated data for three perpedicular wavenumbers:k'5(0.48,0.70,0.92) cm21. Thesecorrespond to the 2nd, 3rd, and 4th radial eigenmodesspectively, of the azimuthal component of the wave magnfield. As before, the amplitudes and phases of the theoreprofiles are adjusted to match the experimental peak ofincident wave atz'110 cm. In each case, the data weinterpolated along the appropriate group velocity path.the casek'50.48 cm21, the interpolated data points do nextend axially as far as the other cases since the rayextends beyond the radial extent of the measurement volsoon afterv51. Since the valuek'50.7 cm21 was used inthe previous section, this figure shows the comparison offit to the two nearest allowable radial wavenumbers. Tchosenk'50.7 cm21 provides the best fit in a least-squarsense to the data. Higher wavenumbers exhibit greater daing and longer parallel wavelengths near the cyclotron renance. The opposite is true~less damping and shorter wavlengths! for wavenumbers smaller than the chosen one.

FIG. 14. Spatial behavior of the exponential damping factor: exp@2k(z)#.Shown are the results of the WKB model which include~a! ion-cyclotrondamping only;~b! ion-cyclotron damping plus electron Landau damping;~c!the same as~b! plus electron-ion Coulomb collisions.


l:

r-

y

eoeet

e-icale

r

the

ate

p-o-

6. Parallel wave electric field

Since it is through the parallel electric field that the eletrons take energy from the wave, it is desirable to understthe axial variation inEi ; unfortunately, the parallel electricfields of the wave are too small to be measured with avable diagnostics. However, an indirect measurement canmade as follows: neglecting the displacement current

FIG. 15. A comparison of the WKB model fits for three different perpedicular wavenumbers:k'5(0.48,0.70,0.92) cm21. These correspond to the2nd, 3rd, and 4th radial eigenmodes, respectively, of the azimuthal comnent of the wave magnetic field.


as

n

s

rs

ens tfrricovmtrifoe

eiu

avetoith

ds.ticag-

ki-

th akedcastn--eed.asthe

theintoavetheer-

ndentocalthere-n-n-ven

value.

aresid-m-ronnasify

n-t byyso-fvellyesand

andon-

7-o.

el


these low-frequency waves, the azimuthal magnetic fieldfunction of k may be obtained14 using thez-component ofAmpere’s law:

Bu54p

ck'

j i ,

and ~with e i21'e i! the current density may be written iterms of the electric field by

j i5v

4p ie iEi .

Then, solving for the absolute value ofEi in terms ofBu ,and as a function ofz, we have

uEi~z!u5ck'

v

Bu~z!

ue i~z!u. ~9!

The predictions of Eq.~9! are plotted in Fig. 16 which showtwo cases for the magnetic field amplitude,Bu(z): ~a! themeasured rms values from Fig. 13~a!, and ~b! the WKBmodel amplitude envelope:A(z)e2k(z) from curve ~d! ofFig. 13.

Both the data and theory show three main features: fion the incident wave side (v,0.7) the parallel electric fieldis roughly constant which implies a steady loss of waveergy to the electrons through Landau damping; second, aparallel wavelength decreases near the ion-cyclotronquency (0.7,v,0.94) there is a region of increased electfield and, thus, increased electron dissipation; finally, abv50.94 the rapid decay of the azimuthal magnetic field aplitude results in a corresponding drop in the parallel elecfield. These results, of course, rely on the WKB modelthe axial variation ofe i and not direct measurements of thparallel electric field.

V. CONCLUSIONS

We have studied the propagation and damping of a shAlfven wave launched by a disk antenna of skin-depth rad

FIG. 16. Magnitudes of the parallel electric field computed from Eq.~9!.Shown are~a! results derived from the measured azimuthal magnetic fiamplitude, and~b! the WKB model using the amplitude from curve~d! ofFig. 13.


a

t,

-hee-

e-cr

ars

in a magnetic beach geometry. Disk antennas excite the wby modulating the field-aligned electron currents drawnthem and radiate azimuthally symmetric shear waves wfinite perpendicular wavelengths and parallel electric fielBefore performing experiments in a nonuniform magnefield, the wave dispersion was measured in a uniform mnetic field for a range of frequencies fromv50.5vci to v51.1vci . Phase velocity measurements agreed with anetic dispersion relation over the entire frequency range.

The same antenna was then placed in a plasma winonuniform background magnetic field and a phase-loctoneburst was applied to the antenna such that the broadfrequency was approximately one-half of the local iocyclotron frequency. In addition, the Alfveń speed at the antenna was approximately equal to the electron thermal spThe time evolution of the radiated wave magnetic fields wmeasured in a series of spatial planes which sampledwave magnetic field both parallel and perpendicular tobackground field. The wave was observed to propagatethe decreasing background field to a region where the wfrequency matched the local ion-cyclotron frequency andAlfven speed was approximately one-half the electron thmal speed. No reflected wave was observed.

The measured axial variations of the wave amplitude aphase were well reproduced by a WKB model of an incidplane wave which propagated and damped according to lsolutions of the same kinetic dispersion relation used inuniform field experiment. The satisfactory fit to the dataquired the inclusion of ion-cyclotron damping, electron Ladau damping, and electron-ion Coulomb collisions. By cosidering ion-cyclotron damping alone, the predicted wamagnetic field amplitude at the location of ion-cyclotroresonance was 20 times greater than both the measuredand the value predicted by also including electron losses

Although the shear Alfveń wave is primarily thought ofas being an ion wave, it is important to determine if thereany regions in which the electron dynamics must be conered. Failure to do so can result, for example, in an incoplete assessment of the efficiency of laboratory ion-cyclotheating experiments which rely on the placement of antenin cold, rarefied edge plasmas. In fact, the wave may modthe electron distribution function which could result in uwanted plasma perturbations. Furthermore, as pointed ouStreltsov and Lotko,15 wave–electron interactions may plaan important role in understanding the dynamics of renances in the Earth’s magnetosphere where standing Alńwaves along auroral magnetic field lines may continuamake the transition between the kinetic and inertial regimas they propagate through the magnetospheric plasmaare reflected between ionospheric endpoints.

ACKNOWLEDGMENTS

The authors wish to thank Professor George MoralesDr. David Leneman for their valuable discussions and ctributions.

This work was funded by ONR Grant No. N00014-91-0167, NSF Grant No. ATM-970-3831, and DOE Grant NDE-FG03-00ER54598.

d


l,

ds

ton

adR., DGndr

rts

lly

so

as

R

es.

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ck,

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es.

M.. C.es.

ggs,

s,


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