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PHYSICS OF PLASMAS VOLUME 9, NUMBER 8 AUGUST 2002

Experimental study of lower-hybrid drift turbulence in a reconnectingcurrent sheet

T. A. Carter,a) M. Yamada, H. Ji, R. M. Kulsrud, and F. Trintchoukb)

Princeton University, Plasma Physics Laboratory, P.O. Box 451, Princeton, New Jersey 08543

~Received 18 March 2002; accepted 22 May 2002!

The role of turbulence in the process of magnetic reconnection has been the subject of a great dealof study and debate in the theoretical literature. At issue in this debate is whether turbulence isessential for fast magnetic reconnection to occur in collisionless current sheets. Some theories claimit is necessary in order to provide anomalous resistivity, while others present a laminar fastreconnection mechanism based on the Hall term in the generalized Ohm’s law. In this work, athorough study of electrostatic potential fluctuations in the current sheet of the magneticreconnection experiment~MRX! @Yamadaet al., Phys. Plasmas4, 1936~1997!# was performed inorder to ascertain the importance of turbulence in a laboratory reconnection experiment. Usingamplified floating Langmuir probes, broadband fluctuations in the lower hybrid frequency range( f LH;5 – 15 MHz) were measured which arise with the formation of the current sheet in MRX. Thefrequency spectrum, spatial amplitude profile, and spatial correlation characteristics of the measuredturbulence were examined carefully, finding consistency with theories of the lower-hybrid driftinstability ~LHDI !. The LHDI and its role in magnetic reconnection has been studied theoreticallyfor decades, but this work represents the first detection and detailed study of the LHDI in alaboratory current sheet. The observation of the LHDI in MRX has provided the unique opportunityto uncover the role of this instability in collisionless reconnection. It was found that:~1! the LHDIfluctuations are confined to the low-beta edge of current sheets in MRX;~2! the LHDI amplitudedoes not correlate well in time or space with the reconnection electric field, which is directly relatedto the rate of reconnection; and~3! significant LHDI amplitude persists in high-collisionality currentsheets where the reconnection rate is classical. These findings suggest that the measured LHDIfluctuations do not play an essential role in determining the reconnection rate in MRX. ©2002American Institute of Physics.@DOI: 10.1063/1.1494433#

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I. INTRODUCTION

Magnetic reconnection1 is a fundamental process iplasma which is thought to play important roles in both labratory and natural plasmas through affecting magnetic toogy and through heating and particle acceleration. It is gerally accepted that the process of reconnection dependthe formation of sharp gradients in the magnetic field, calcurrent sheets,2 which exist on scales where dissipation bcomes important. The earliest quantitative model of currsheet reconnection was presented by Sweet and Parker3 us-ing the resistive magnetohydrodynamic~MHD! approxima-tion. This model provides a mechanism by which magnetopology can change much faster than would be allowedsimple resistive diffusion, but fails to explain the rapid timscales observed in natural and laboratory plasmas. The scoming of the Sweet–Parker model is the crucial dependeof the reconnection rate on the plasma resistivity. In cosionless plasmas where the Spitzer resistivity is small, vthin current sheets are needed in order for resistive diss

a!Present address: Department of Physics and Astronomy, University offornia, Los Angeles, CA 90095-1547. Electronic [email protected]

b!Present address: Cymer, Inc., San Diego, CA.

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tion to become important. The thinness of the current shimpedes the flow of mass through the current sheet, slowthe reconnection process significantly.

One of the earliest proposals for speeding up SweParker reconnection was the inclusion of a turbulent anolous resistivity in the resistive MHD model.4 Strong currentsand density gradients found in current sheets can drivecroinstabilities which could provide sufficient anomalodissipation to increase the Sweet–Parker rate to match obvations. In addition to increasing the Sweet–Parker rasimulations have found that using a current-density depdent turbulent resistivity may allow reconnection at neaAlfvenic rates via the development of slow-mode shocks,5 asfirst proposed by Petschek.6 While models using anomalouresistivity succeed in achieving fast reconnection, a cruquestion to answer is: What microinstabilities operate inconnecting current sheets, and can they provide sufficresistivity to justify these models? A number of instabilitihave been proposed to produce turbulence and anomaresistivity in current sheets, including Buneman,7 electroncyclotron drift,8 ion acoustic,9 and lower-hybrid drift.10 Dueto the plasma parameters expected or observed in cusheets~Ti /Te*1, b*1, j /ne;v th,i!, many of these insta-bilities have been effectively ruled out as candidates forsistivity generation. Of the remaining candidates, the low

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3273Phys. Plasmas, Vol. 9, No. 8, August 2002 Experimental study of lower-hybrid drift turbulence . . .

hybrid drift instability ~LHDI ! is considered by some to bthe ‘‘best bet’’ for providing anomalous resistivity in reconnecting current sheets.11 However, it is well known that theLHDI is linearly stabilized by large plasma beta,12 suggest-ing that the instability might have difficulty being excitenear the center of high-beta current sheets where resistgeneration is desired. Simulations have shown that the LHcan grow in the edge of the current sheet, but disagree oneffectiveness of the instability in penetrating high betagions and providing resistivity.13,14

Recently, an alternative theoretical picture of fast recnection has emerged from simulations employing two-fleffects, particularly effects embodied in the Hall terms in tgeneralized Ohm’s law.15 In these simulations, the relaxatioof reconnected field lines is seen to be governed by the wtler wave, rather than by the Alfve´n wave as in MHD recon-nection. Due to the dispersive nature of the whistler wathis results in a low-aspect-ratio current sheet geomwhich does not impede mass flow and hence allowsreconnection.16 The reconnection rate in these simulationsfound to be independent of the type of dissipation availain the current sheet, making the model very attractivereconnection in collisionless plasmas. Simulations supping this theoretical picture have been primarily twdimensional, artificially suppressing instabilities and perhsuppressing anomalous resistivity which could potentiadominate over other effects in setting the reconnection rHowever, initial three-dimensional Hall MHD simulationindicate that while turbulence does develop, the laminar twdimensional~2-D! picture of reconnection remains intact anthe turbulence only serves to slow the reconnectprocess.14

The controversy over the role of turbulence in reconntion is obvious in the differences between these theoretmodels. Motivated by the goal of resolving this controverwe present in this paper experimental studies of fluctuatiin the current sheet of the Magnetic Reconnection Expment~MRX!.17 The goal of this work was to:~1! identify andcharacterize any microinstabilities present in the MRX crent sheet, and~2! determine the role of these instabilitieduring reconnection. Measurements of fluctuations in MRwere done primarily using floating Langmuir probes, but itial studies with magnetic pick-up probes were also pformed. The result of these studies was the observatiobroadband electrostatic and magnetic fluctuations in therent sheet which are identified as lower-hybrid drwaves.18,19 This identification is made following careful examination of the frequency spectrum, spatial amplitude pfile, and spatial correlation characteristics of the turbulenThe fluctuations are observed to develop with the formatof the current sheet in MRX and have a frequency spectcentered near the local value of the lower hybrid frequenThe radial profile of the fluctuation amplitude is found topeaked on the inner edge of the MRX current sheet, content with the computed profile of the LHDI linear growtrate. Two-point correlation measurements reveal a decorrtion length in the fluctuations comparable with the predicwavelength of the LHDI, consistent with the expected strolinear growth rate in MRX current sheets (g;v r). The iden-


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tification of the LHDI has provided an opportunity to expementally investigate the role of this instability in reconnetion. The radial amplitude profile, time behavior, ancollisionality dependence of the fluctuations suggest thatLHDI is not contributing significantly to determining the reconnection rate in MRX.

This paper is organized as follows: Section II offersreview of theories of the lower-hybrid drift instability, including the development of a model to be used to compwith measurements in MRX. In Sec. III the experimenapparatus and diagnostics used in this work are descriSection IV presents a detailed analysis of the fluctuatmeasurements and a discussion of their implications forconnection in MRX. A summary of the work is offered iSec. V.

II. REVIEW OF THE LOWER-HYBRID DRIFTINSTABILITY

The lower-hybrid drift instability has been studied theretically for decades, motivated by its possible role in manetic reconnection,10 theta-pinches and other fusiodevices,20 and space plasmas~e.g., the magnetosphere21!. Inthe following, a review of the theory and prior experimenstudies of this instability is presented. Section II A reviewthe theory of the LHDI, including the derivation and discusion of a local, linear, electrostatic model of the LHDI. Thmodel will be utilized to explain features of the experimendata in Sec. IV. Section II C briefly discusses prior obsertional studies of the LHDI.

A. Linear LHDI theory

In order to develop a model to be used to compare wdata taken on the MRX experiment, we present a derivaof a local, linear, electrostatic theory of the lower-hybrid drinstability. The following closely follows the procedureused by Krall and Rosenbluth,22 Davidsonet al.,12 and Hubaand Wu23 in deriving the dispersion relation for the lowehybrid drift instability. We consider a local model of thcurrent sheet, assuming that the wavelength of the modinterest is much smaller than the gradient scale length inplasma,l!(d ln n/dx)21, (d ln B/dx)21. We therefore use aslab model in the derivation, with density and magnetic fiegradients in thex direction~corresponding tor in MRX! andcurrent in they direction ~corresponding tou in MRX!. Asthe MRX current sheet thickness is comparable tor i ,24 wetreat the ions as unmagnetized with a cross-field flow velity V. The derivation will take place in the frame where thelectric field is zero, and the ion flow velocity,V, thereforerepresents both the ion diamagnetic drift speed and anE3B electron current. The equilibrium ion distribution function is chosen to be a shifted Maxwellian:

f i05

n

p3/2v th,i3 expS 2

vx21~vy2V!21vz

2

v th,i2 D ,

wherev th,i5A2Ti /M andn are evaluated locally. The electrons are magnetized, and we write the equilibrium distribtion function as a function of the constants of the electrmotion:v2, pz , andpy5mvy2eAy(x)/c. If we assume that


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3274 Phys. Plasmas, Vol. 9, No. 8, August 2002 Carter et al.

the gradient in the magnetic field is weak, we can appromate py'mvy2eB0x/c. The electron distribution functionis chosen to be

f e05

n~X!

p3/2v th,e3 expS 2

v'2 1vz

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v th,e2 D ,

whereX is related to the canonical momentum in they di-rection, X5x2vy /Ve52(eB/c)py . We choose a locamodel and expand aboutx50 to find

f e0'S 12en

vy

VeDFM,e ,

whereen5d ln n/dx and FM,e is a Maxwellian electron dis-tribution.

Using the electrostatic approximation, the perturbelectron density can be shown to be~see the Appendix!

dne522qn0

mv th,e2 f2

2qn0

mv th,e2 f~v2kyvD,e!

2

kiv th,e

3E xdxexp~2x2!J02~k'rex!ZS v2kyV“Bx2

kiv th,eD .

The perturbed ion density is straightforward to calculin the limit of unmagnetized, drifting ions, yielding25

dni52qn0

Mv th,if@11z iZ~z i !#, ~1!

FIG. 1. Real frequency and growth rates for the LHDI using paramerelevant to the MRX experiment.


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where z i5(v2kyV)/kv th,i . We can then use Poissonequation to relate the density perturbations to the potenperturbation:

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2ld,i2 Z8~z i !1

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kiv th,e


kiv th,eD D J .

The dispersion relation can then be obtained from roots

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2k2ld,i2 Z8~z i !1

1

k2ld,e2 ~11c!,

~2!

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kiv th,e


kiv th,eD .

The value of the electron“B drift velocity can be shownto depend on the value of the total plasma beta. If we assequilibrium between the magnetic and plasma pressure,also that the temperature is spatially uniform, we find

“S B2

8p D52“p,

2B2

8p

] ln B

]x52n~Te1Ti !

] ln n

]x,

eb52 12 ben ,

[V“B52

b

2vD,e ,

whereb is the total plasma beta,b58pn(Te1Ti)/B2. Thus

the plasma beta enters into Eq.~2! through the“B drift termin the plasma dispersion function.

Using Eq.~2!, we can explore the linear characteristiof the LHDI using parameters relevant to the MRX expement. The relevant dimensionless parameters in MRXenr i /2;1 ~density gradient scale length is roughly 2r i!,V/v th,i;2.5 (j /ne2nevD,e;2.5v th,i), andTi /Te;1. Figure1 shows the real frequency and growth rate as a functionnormalized wave number,k're , for these parameters and foa few values of the normalized parallel wave numbeki /k'AM /me, for the caseb50. The frequencies obtainefrom the roots of Eq.~2! are Doppler shifted byk'V in thisplot in order to show the frequency in the ion rest frame.the ion rest frame, positive real frequency is found fork' inthe electron diamagnetic direction, indicating that the ustable waves propagate in that direction. The growth ratethe LHDI is found to be quite strong, and peaked nek're;1 andv;vLH . Significant growth is found at a widerange ofk're , translating to a range of real frequenciesto two to three times the lower hybrid frequency. In the irest frame, the phase velocity of the waves at peak gro~v;vLH , k're;1! is

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Therefore the strongest growth is found where ion Landdamping of the waves is strongest. This actually drivgrowth of the LHDI, as it is a negative energy drift wavethe ion rest frame.26 For ki50, the growth rate of the waveis due to2(] f i

0/]v)v/k and no damping is provided by thelectrons~for b50!. As ki is acquired by the wave, accesselectron Landau damping along the field line is provideresulting in a lowering and eventual suppression ofgrowth rate, at very small values ofki /k' .

Figure 1 is for the case of zero plasma beta. Significbeta values are found in the MRX current sheet, froroughly 10% to 100% at the edge to infinite local beta atfield null in guide-field-free current sheets. Figure 2 shothe effect of increasing plasma beta on the peak growthfor the LHDI under the same conditions in Fig. 1 and fki50. As beta is initially increased there is little changethe peak growth rate, but afterb'1, the peak growth ratedrops dramatically.

The local, electrostatic, linear model of the LHDI prsented in this section shows that we should expect the LHto be fairly strongly growing in conditions similar to thosfound in MRX, with g;vLH at k're;1. It is interesting tonote that the marginal state of the LHDI is predicted to ocat significantly shallower density gradients than thoseserved in MRX. An estimate of the critical density gradie~assuming the cross-field current to be entirely diamagne!is r ien/2;2V ivLH;1/20.12 The predicted LHDI growthrate drops dramatically as beta is raised, and the instabililikely to be suppressed in the center of the current shwhere beta is locally infinite. Electromagnetic correctionsthe LHDI were first explored by Davidson,12 who foundthese corrections to be destabilizing in regimes similarthose found in MRX (V/v th,i*1). These destabilizing effects were found to lead to the restoration of growth at lonwavelengths, but only increased the value of the peak grorate slightly compared to the electrostatic case. Thus,overall effect of beta is a stabilizing one, however the eltrostatic model presented in Sec. II slightly overestimatesdegree of stabilization. For this reason, as well as for s

FIG. 2. Peak growth rate for the LHDI as a function of beta.


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plicity, the above-presented electrostatic model, which ctures the dominant finite-beta effect of resonant“B stabili-zation, is used instead of a fully electromagnetic theorycalculate peak growth rates for comparison with data psented in Sec. IV. Forki50, the electromagnetic LHDI isflute-like, only generating perturbations in the backgroumagnetic field component~Bz in MRX!.

B. Nonlinear effects and simulations

1. Saturation mechanisms.

The anomalous transport properties of the LHDI habeen of great interest in the theoretical literature, especias applied to theta pinches27 and magnetic reconnection.28

The starting point for estimates of transport coefficientsthe determination of the saturation level of the LHDI. Thearliest estimate of this level was done by Davidson,29 wherequasilinear theory was employed to determine the efficieof saturation by plateau formation and current relaxatiThe former is unlikely to be important in MRX plasmas,collisions are likely to maintain Maxwellian particle distrbution functions ~this has been observespectroscopically30!. It has been pointed out26 that currentrelaxation does not provide a realistic bounds on the srated amplitude, as the energy in the field is tied to the crent, and this thermodynamic estimate should be basedthe total magnetic energy in the system. Ion trapping wobserved as the saturation mechanism for the LHDI in simlations by Winske.31 This mechanism is effective when thLHDI spectrum is nearly monochromatic, as was observethese simulations at moderate drift velocityV/v th,i*3.Huba32 considered the effect of electron resonance broading on the saturation of the LHDI. In this study, the stabiliing electron“B resonance was shown to be nonlineabroadened, allowing a larger population of electrons to inact with and damp the LHDI waves. A saturation estimatethis process~however forV/v th,i&1! was made by Gary:33

S EnTi

D'2

5

me

M

Ve2

vp,e2 S Ti

TeD 1/4 V2

v th,i2 . ~3!

However, this saturation mechanism, which is similarelectron trapping, might be hampered in MRX by high eletron collisionality. Finally, a numerical calculation of the efect of nonlinear Landau damping~or mode–mode coupling!on the saturation of the LHDI was performed by Drake.26 Inthis case, nonlinear transfer of energy from growing lowavelength modes (kre;1) to damped short wavelengtmodes provided a saturation mechanism. This calculayielded an estimate for the saturation level of the LHDI:

ef

Ti'2.4S 2me

M D 1/2 V

v th,i. ~4!

This calculation ignored any nonlinear coupling into dampmodes with finiteki , and therefore is likely to be an overetimate of the saturation amplitude.

2. Quasilinear resistivity

Davidson27 presented a calculation of the anomaloussistivity and heating rates of the LHDI which will be re


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viewed here. The quasilinear equation for the evolutionthe zero-order distribution function of speciesj due to thepresence of waves in the plasma is

S ]

]t1v"

]

]x1

qj

mjS vÃB

c D " ]

]vD f j

5S ] f j

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52qj

mjK dE"

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The anomalous momentum exchange rate between specjand the fluctuations can be calculated by taking the firstlocity moment of (] f j /]t)anom for velocity in the currentdirection:

S ]

]tnjmjVy, j D

anom

5qj^dEydnj&. ~6!

Equation~6! can then be used to compute an effective cosion rate due to the waves:

neff5qj

njmjVy, j^dEydnj&. ~7!

Equation~7! provides an instability-model-independent wto experimentally determine the effective collision rate dto a measured spectrum of electric field and density fluctions. However, simultaneous measurement of the amplitand phase of both density and electric field fluctuations iplasma is quite a difficult task, and was not attempted asof this work. A simpler, yet model-dependent, expressionthe effective collision rate can be obtained through usinglinear theory for the LHDI to compute the density perturbtion as a function of the electric field perturbation,dnj

52x j ikydEy,ky/4pqj . Using the expression for the ion den

sity perturbation in Eq.~1!, the effective collision rate estimate reduces to27

nLHDI5ImF k'

4vp,i2

k'2 v th,i

2 z iZ~z i !Gk',max

Ti

meV

EnTi

, ~8!

wherek',max indicates that the expression should be evaated at the frequency and wave number at peak growth,E5(dE)2/8p. Experimental evaluation of Eq.~8! can beperformed with knowledge of only the amplitude of the eletric field fluctuations in the plasma.

3. Review of simulations of the LHDI

Although predictions of strong anomalous resistivity dto the LHDI have been made, the usefulness of this resisity in reconnection is questionable if the LHDI is suppressat the center of high-beta current sheets, where it wouldneeded to provide dissipation. Several simulations have bperformed to study the LHDI and investigate the likelihoof the it penetrating to the center of a current sheet@see, e.g.,Refs. 13, 14, 34–37#. Two recent simulation efforts addresing the LHDI during magnetic reconnection disagree onimportance of the instability in determining the reconnectrate. The first effort involves three-dimensional particsimulations of reconnection performed by Horiuchi aSato.13 In these simulations the LHDI grows up early on t


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edges of the current sheet, consistent with finite-beta stazation. In the case that no external driving electric field wpresent, the LHDI did not penetrate further into the curresheet, but instead resulted in a modification of the currsheet profile which in turn drove a low-frequency magneinstability. In this case, reconnection electric field was ninduced at the null by the LHDI but was instead providedthe low-frequency instability, which was seen to genersignificant anomalous resistivity. The low-frequency instabity was identified as the drift-kink instability~DKI !, whichwas so named by Zhu and Winglee38 after observations insimulations of the magnetotail, but which was perhaps fistudied analytically by Yamanaka39 ~and later by Winske40!.When a driving electric field was applied in the simulatioby Horiuchi and Sato,13 the LHDI was found to penetrate tthe magnetic null and provide anomalous dissipation priothe triggering of the DKI. In either case, the LHDI was seas quite essential to determining the reconnection ratethese simulations, either through penetration to the null lor through nonlinearly driving the DKI. It should be notethat the importance of the DKI in current sheets is currenthe topic of much theoretical debate. The DKI has beenmarily observed in low mass ratio particle simulations, aDaughton41 has shown that while the growth rate of thinstability can be large when the mass ratio is artificiasmall, the DKI should have negligible growth rate at realismass ratios in Harris equilibria. Daughton does however sgest that other equilibrium profiles, especially those with snificant background density, may increase the growth ratethe DKI.41,42

A second recent simulation effort has shown that effeassociated with the Hall term in the generalized Ohm’s lcan result in fast reconnection in laminar current sheet15

The simulations supporting this fast reconnection mechanhave been almost exclusively done in two dimensions~thex–z plane in the model presented in Sec. II A!, artificiallysuppressing instabilities like the LHDI. However, recethree-dimensional simulations by Rogerset al.14 using a HallMHD model have shown that while LHDI does developthe edge of the current sheet, it does not dramatically athe physical picture of fast reconnection found in the 2simulations. In fact, development of the LHDI was observto slow the reconnection rate relative to the rate foundlaminar 2-D simulations. The differing conclusions reachby these two simulation efforts demonstrate the theoretcontroversy over the role of turbulence, specifically duethe LHDI, in reconnection.

C. Prior experimental studies of the LHDI

There have been very few experimental observationsthe LHDI, and none in previous laboratory reconnection eperiments. The earliest report of an experimental observaof the LHDI was made by Gurnettet al.,43 who studied sat-ellite measurements of fluctuations in the Earth’s magnetail. Analysis by Hubaet al.28 suggested that the frequencspectrum and amplitude of the waves was consistent withoperation of the LHDI in the magnetotail. Shinoharaet al.11


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also presented an analysis of recent satellite measuremethe magnetotail, suggesting that the observed fluctuatwere due to the LHDI. An estimate of the anomalous restivity due to these fluctuations was made, but it was fouthat the value of this resistivity was not enough to increthe growth rate of the tearing mode to the level necessarexplain the triggering of an associated substorm. HoweShinoharaet al. suggested that the computed anomaloussistivity might still be enough to be essential to magnereconnection in the tail. In these satellite measurements,tailed observation and analysis of the LHDI is quite difficuas the profile and location of the tail current sheet is not wmeasured simultaneously to the fluctuation measurementboth cases, however, the data suggested that electrofluctuations might be strongest away from the center ofcurrent sheet. There have been experimental studies oLHDI in other plasma configurations which are not direcrelevant to the problem of magnetic reconnection. A C2

laser scattering measurement of fluctuations in a theta-pplasma was made by Fahrbachet al.44 These measuremenprovided some limited information on the wavelength afrequency spectrum of fluctuations in the plasma, andcharacteristics were shown to be consistent with linearnonlinear theories of the LHDI.26 Measurements in magnetoplasmadynamic thrusters, which involve strong cross-ficurrent and density gradients, have also revealed evidefor the LHDI.45,46

III. EXPERIMENTAL APPARATUS

The measurements reported in this paper were takethe Magnetic Reconnection Experiment~MRX!17 at Princ-eton Plasma Physics Laboratory. MRX was constructedthe purpose of studying magnetic reconnection in a wcontrolled laboratory plasma where MHD is satisfied in tbulk of the plasma~Lundquist number (S)@1, r i!L!. Aschematic drawing of the MRX apparatus, showing locatof the current sheet and fluctuation measurement geomis shown in Fig. 3.

Current sheets in MRX are formed between two coil scalled ‘‘flux cores,’’ shown in Fig. 3. The current sheetindicated by crosses in Fig. 3, surrounded by a representafield line. The primary sheet current flows in the toroidal~u!direction, the reconnecting field is in theZ direction, and thedensity and magnetic field gradients are in theR direction.The work reported in this paper was done in current she

FIG. 3. Schematic of the MRX device, including representative currsheet location and fluctuation measurement geometry.


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where no macroscopicBu ~sometimes called ‘‘guide’’ field!is present during reconnection~‘‘null-helicity’’ reconnection,as opposed to ‘‘co-helicity,’’ where a guide field is presen!.

The free energy sources available to drive instabilitiesMRX current sheets can be at least partially revealed throthe measurement of profiles of magnetic field~and hencecurrent! and plasma temperature and density. The knowleof these profiles will also allow theoretical prediction of instability characteristics in MRX for comparison with fluctution measurements. The bulk of the diagnostics in the Mdevice is for the measurement of magnetic fields, with cloto 180 magnetic pickup coils in the vacuum vessel. Thcoils are hand wound, using 80 turns of 38 gauge magwire on 3-mm-diam, 3-mm-long cylindrical plastic formThe coils are distributed among three probes; two of thprobes~90-channel and 60-channel 2-D probes! are for thepurpose of measuring all three vector components of the fiin a coarse grid spacing~4 cm near the current sheet, and 68 cm at radii well inside the current sheet location! in onetoroidal plane of the experiment. Using these magnetic msurements and assuming axisymmetry, the poloidal fluxelectric field can be calculated:c5*0

R2prBz(r )dr, Eu

521/(2pR)(]c/]t). A high-resolution ~0.5 cm spacing!one-dimensional~1-D! magnetic probe is used to measureBz

along ther direction. The magnetic field profile in MRX iswell described by the Harris sheet theoretical equilibriuprofile.24,47 The measuredBz field in MRX is fit to this the-oretical profile (B}tanh((r2r0)/d)) and the current density isderived analytically from the fit. The thickness of the curresheet in MRX is found to be comparable to both the ion sdepth and the ion gyroradius (d;r i;c/vp,i).

24 A tripleLangmuir probe48 is employed to measure density (ne), elec-tron temperature (Te), and floating potential (Vf) profiles inthe current sheet.

Collisionality in MRX current sheets is characterizedthe parameterlmfp /d, whered is the width of the currentsheet andlmfp is the electron mean free path against Colomb collisions. Two observations which motivate the stuof turbulence and anomalous resistivity have been madthe collisionality was lowered~lmfp /d is increased! in MRX.The first observation is that the measured toroidal reconntion electric field,Eu , is no longer balanced by classiccollisional drag at the center of the current sheet,Eu /hSpj u

@1, where hSp is the classical Spitzer perpendiculresistivity.49 If the measured ratioE/ j is defined as an effective resistivity, h* 5E/ j , reconnection data from MRX isfound to agree with a generalized Sweet–Parker thebased on this effective resistivity~and also including com-pressibility and downstream pressure!.50 The second observation is that of direct, nonclassical ion heating during reconection in MRX current sheets.51 One possible explanationfor these two observations is the presence of turbulenclow-collisionality MRX current sheets, which creates a tubulent anomalous resistivityh* .hSp ~so thatEu /h* j u51!and directly heats the ions.

The initial search for high-frequency fluctuationsMRX using probes revealed broadband noise generatedimpedance mismatches in both power transmission lifrom the MRX capacitor banks and in transmission lines

t


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the diagnostics themselves. This noise precluded the detion of signals from the plasma and had to be addresseorder to study fluctuations in the MRX current sheet. In ordto improve immunity to noise generated by the power ccuitry in MRX and to facilitate active impedance matchingthe diagnostics, small broadband buffer amplifiers were binto probes used for the fluctuation studies reported here.use of a miniature SOT-23 surface mount package forbuffer ~Burr–Brown OPA682!, along with 0805 package suface mount capacitors and resistors, allowed the placemof all components on a double-sided printed circuit board4.5350 mm approximate dimensions. The boards are plainside 1/4 in. ~0.635 cm! to 3/8 in. ~0.9525 cm! stainlessprobe shafts, which allows the leads connecting the prtips to the amplifier to be only several millimeters~5–10mm! long. The amplifier allows an easy transition from thigh-impedance probe tip into a 50V transmission line,eliminating impedance matching issues. While the ovevoltage gain is unity, the amplifier does boost the signal crent to assist in noise immunity. A high-bandwidth ferricore 1:1 pulse transformer is used to provide isolation frthe plasma in electrostatic~Langmuir probe! diagnostics, butis not present in amplified magnetic pickup coil diagnostiThe magnetic field value of the core saturation is well abothe fields used in these experiments (;200 G). Perturbationof the background field due to the presence of a high-m fer-ritic material is negligible due to the size and toroidal geoetry of the transformer. Signals are propagated downprobe shafts using low-loss semirigid coaxial line~UT-85LL!. Signal transport from the probe to the digitizing ocilloscope~approximately 12 m away! is accomplished usinglow-loss RG8 coaxial cable. The bandwidth of the syst~amplifier input to RG8 output! is measured to be 100 kH& f &125 MHz when the transformer is used for isolatiand f &300 MHz when no transformer is used.

Fluctuations in the plasma floating potential were msured using differential floating Langmuir probes. Floatipotential measurements were chosen over ion saturationrent measurements due to the difficulty of measuring hfrequency current signals accurately in the presence of ccapacitance. Differential measurement was performed inder to remove low frequency floating potential signals, whcan be on the order of 100 V~fluctuating signals are on thorder of 1 V!. Single floating probe measurements usivoltage division were not practical due to the limited dnamic range of the data acquisition system~8 bit! and noisegenerated by the high-power pulsed electronics. Differenfloating Langmuir probes are constructed using two spatiseparated cylindrical tungsten wires sheathed in alum(Al2O3). The diameter of the tungsten tips varied frommil ~0.762 mm! to 5 mil ~0.127 mm!, and in all probe tips a1 mm length of the wire is exposed to the plasma. Studiespatial correlations were performed using three-tip probwhere two of the tips are used to make spatially separadifferential floating potential measurements using the thtip as a common reference. Correlation probes were cstructed with 1, 3.5, and 10 mm separation. Magnetic fifluctuations are measured using magnetic pickup coil baprobes. The coils are placed inside small glass tubes co


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with graphite to provide electrostatic shielding. The pickprobes are coupled directly to a buffer amplifier at the endthe probe shaft, with no transformer. The bandwidth ofprobe is set by theL/R time of the coil (L'10 mH) basedon the input impedance of the buffer amplifier~500 V!,which is around 20 ns~making the bandwidthf &50 MHz!.The probes were inserted radially into MRX plasmas,shown in Fig. 3.

IV. MEASUREMENTS OF FLUCTUATIONS IN THEMRX CURRENT SHEET

Measurements of fluctuations in the current sheetMRX are reported in the following. While fluctuations havbeen studied in current sheets previously,52 the measure-ments reported here are the first to be done in a current sformed in a plasma where, on the global scale, ions are mnetized (r i!L) and the MHD approximation is satisfied (S@1). In addition, fast reconnection, enhanced resistivity, anonclassical ion heating have been observed in MRX,50,51

providing an opportunity to determine if turbulence playsessential role in these phenomena. Measurements of hfrequency fluctuations were performed in the current sheeMRX, with the following goals:~1! identify any instabilitiespresent in the current sheet and determine their characttics and~2! determine the influence of these instabilitiesthe process of reconnection in MRX. These measuremresulted in the first observation of the lower-hybrid drift istability in a laboratory current sheet. This instability hbeen studied theoretically for decades in the context of crent sheets and magnetic reconnection, yet no detailedperimental investigation of the instability has been possiuntil this work.

A. Observation of the lower-hybrid drift instability

Measurements using amplified floating double Langmprobes placed on the edge of current sheets in null-helidischarges in MRX have revealed the presence of broadbfluctuations near the lower hybrid frequency. In this sectievidence supporting the identification of these fluctuationslower-hybrid drift waves is presented. The evidence is pvided by detailed studies of the frequency spectrum, raamplitude profile, and spatial correlations and propagatcharacteristics of the fluctuations. These observations wilshown to be consistent with theoretical predictions forlower-hybrid drift instability, using the theory developedSec. II for comparison.

Figure 4 shows an example of a differential floating ptential signal (df f) taken atr 50.34 m ~refer to Fig. 3 formeasurement geometry!, along with a time trace of the totatoroidal plasma current during a hydrogen dischargeMRX. The plasma current rises during formation of the crent sheet in MRX and then typically flattens in time durinthe quasi-steady period of magnetic reconnection. The fltuations are seen to arise with the formation of the currsheet and persist for 10–20ms. The amplitude of the measured fluctuations is typically several hundred millivolts, bcan be as high as 1–2 V. A normalized fluctuation amplitucan be constructed by comparing the amplitude to the m


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sured electron temperature,edf f /Te . This normalized am-plitude is typically found to be several percent~Te

;5 – 10 eV, edf f /Te&10%!. A windowed fast Fouriertransform~FFT! of the shown example signal is inset in Fi4, with a vertical line marking the position of the time aveaged lower hybrid frequency,f LH;16 MHz. The FFT is per-formed using a Hanning window, 8ms wide about t5252 ms, and the plot is made using a linear vertical axThe lower hybrid frequency is determined from measuments of the magnetic field near the fluctuation probe usvLH5AVeV i .

1. Frequency spectrum

The LHDI is expected to produce fluctuations whose fquency spectrum is located near the lower hybrid frequeThe detailed dependence of the frequency spectrum ofmeasured floating potential fluctuations on the lower hybfrequency was explored through varying the peak field incurrent sheet and the mass of the working gas (f LH

}B/AM ). The peak magnetic field value was varied throuraising or lowering the voltage on the capacitor bank usegenerate the poloidal field. Using this technique, the pfield was scanned from roughly 100 G~using 10 kV/8 kV onthe toroidal field/poloidal field bank! to 300 G ~14/12 kV!.Both hydrogen and helium were used as working gaseslowing for a factor of 2 change in the lower hybrid frequendue to ion mass.

Figure 5~a! shows a set of example average floating ptential fluctuation power spectra~linear vertical axis, loga-rithmic horizontal axis! at different local field values in hydrogen. Each plot is generated through averagingspectrum of ten discharges whose local magnetic field vafalls within a 50 G window of the magnetic field value anotating the graph. There is an upward shift evident inpower spectrum with increasing field strength, consistwith the shift in the local lower hybrid frequency. Figure 5~b!shows the frequency of peak fluctuation amplitude verlocal magnetic field for discharges in hydrogen. While theis some scatter, the peak frequency is seen to increaseincreasing magnetic field in a manner consistent withlower hybrid frequency. Figure 6 shows the average pospectrum of the fluctuations for hydrogen and helium d

FIG. 4. Traces of plasma current and measured floating potential salong with a FFT of the signal. Current sheet formation and reconnecoccur roughly fromt5240 to 280ms.


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charges with similar local magnetic field. A clear downshin the location of the fluctuation spectrum is observedhelium discharges, again consistent with the change inlocal lower hybrid frequency.

The theory of the LHDI predicts that the peak of thgrowth rate should occur at a wave number associated wreal frequency of roughly the lower hybrid frequency. Tobserved frequency spectrum is consistent with the lintheory in this regard, as the peak is near the lower hybfrequency. The LHDI theory also predicts a fairly wide ranof wave numbers where appreciable growth is found,shown in Fig. 1, which is consistent with the observationa wide frequency range in the fluctuation spectra.

2. Spatial amplitude profiles and time behavior

The LHDI is expected to be driven by density gradienand cross-field currents, which would suggest that it mibe localized near these energy sources in MRX currsheets. In order to determine if the observed fluctuationsconsistent with these expectations, a study of the radialplitude profile was performed. A comparison with the linetheory developed in Sec. II is presented, and provides fursupport for the conclusion that the measured fluctuations

aln

FIG. 5. ~a! Average floating potential power spectrum in hydrogen at diffent average field strengths.~b! Frequency at peak fluctuation amplitudehydrogen vs magnetic field.

FIG. 6. Average fluctuation spectra in hydrogen and helium for simmagnetic field values.


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due to the LHDI. A discussion of the observed time behavof the fluctuations, also based on the linear theory, is pvided.

Radial profiles of the amplitude of the floating potentfluctuations were constructed through shot-to-shot positing of the probe and averaging over many shots at eachsition. Figure 7 shows average radial profiles of the romean-square fluctuating floating potential amplitusuperimposed on the computed average current densityfile at four times during a set of more than 200 lowcollisionality (lmfp /d;5210) MRX discharges~12/10 kV,

FIG. 7. Radial profiles of rms fluctuation amplitude atz50 and currentdensity in the MRX current sheet at four times.


r-

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4 mT fill pressure, hydrogen!. The current densities showare computed by first fitting the average measured magnfield profile to a Harris sheet profile, then deriving the curent density from the fit. The magnetic measurements shoare made at a small toroidal separation (10° – 15°) fromfluctuation diagnostic. As the current sheet is formed,width of the sheet thins to be comparable to the ion sdepth while the radial position of the current sheet movoutward~due to the hoop force! in order to establish equilib-rium with an applied steady-state magnetic field in thezdirection. The plotted fluctuation amplitude is determinthrough first high-pass filtering individual fluctuation mesurements~digitally!, then averaging the square amplitudeeach radial position. The error bars represent shot-to-svariations in the measurement. The fluctuations are obseto grow up on the inner edge of the current sheet, thstrengthen and track the current sheet as it moves towarequilibrium position. Later in time, the amplitude decafairly rapidly even though the current sheet persists andconnection continues.

Figure 8 shows a contour plot in ther – t plane of therms floating potential amplitude, along with the trajectorythe center of the current sheet and sheet thickness~R0 andR06d, determined from the fit of the average magnetic fieto a Harris profile!. Figure 8 shows in more detail how thfluctuation amplitude follows the trajectory of the curreprofile. As reconnection proceeds, the equilibrium is alteby the depletion of flux inside the current sheet. This lowtheB2 pressure pushing out on the current sheet, resultinan inward shift of the equilibrium position, as shown in thFig. 8 after t'258 ms. Reconnection continues as the curent sheet moves inward, until roughlyt5280 ms.

The radial profile measurements raise a key questWhy is the radial amplitude profile asymmetric? We waddress this question using the linear electrostatic mode

ent

FIG. 8. Contours of rms fluctuation amplitude in thr – t plane. Superimposed is the trajectory of the curresheet center (R0) and the current sheet thickness (R0

6d).


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the LHDI derived in Sec. II. Linear calculations of the locgrowth rate profile of the LHDI were performed basedmeasured profiles of density, electron temperature, and mnetic field. Electron temperature and density profiles wacquired in a similar fashion to the fluctuation profilthrough shot-to-shot positioning of a triple Langmuir proand averaging over several shots~at least ten! per position.The triple Langmuir probe measured density profile at5264 ms, along with a Harris sheet fit to the measurederage magnetic field profile, is shown in Fig. 9~a!. Both themagnetic field and the density are observed to be radiasymmetric with respect to the center of the current shThe magnetic field asymmetry is due to the cylindrical gometry of the field coils~flux cores! in MRX, which generatestronger fields inside the current sheet location than outsThe density asymmetry arises so that radial force balancebe achieved with this magnetic field profile.24 The densitygradient is a source of free energy for the LHDI, and a strger gradient on the inner edge implies the growth rate shobe larger there. In addition, the density asymmetry creatradially asymmetric cross-field electron-ion flow speed dference,Vd5 j /ne. This cross-field drift is also an importandrive for LHDI, and for a symmetric current density, th

FIG. 9. ~a! Radial profiles of fitted average magnetic field and densincluding a smooth fit to the density profile~two-temperature Lorentzian!.~b! Electron beta calculated from measured electron density, electronperature, and magnetic field.~c! Fluctuation amplitude and current densiprofiles att5264 ms. ~d! Computed peak growth rate profile for the LHDfor the measured profiles and forTi /Te51,2,3.


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larger density on the outer edge produces a smaller fldifference between the ions and electrons. The combinaof the density and magnetic field asymmetries producestrong asymmetry in the profile of the electron beta (be

58pnTe /B2), as shown in Fig. 9~b!. The beta on the inneedge of the current sheet is on the order of 10%, comparethe outer edge which has near unity beta. The large betathe outer edge should be has a significant stabilizing inence on the LHDI.

In order to compute a profile of maximum LHDI growtrate, a smooth fit to the density profile measurement@dottedline in Fig. 9~a!, arbitrarily using a Lorentzian with differen‘‘temperatures’’ on either side of the current sheet# along thefitted magnetic field and current profiles att5264 ms wereused to compute parameters in Eq.~2! ~assumingTi /Te

51,2,3!. The cross-field ion velocity (V) in this equationwas determined by equating the plasma current densityj5ne(V1vD,e), wherevD,e is the electron diamagnetic velocity. Dispersion relations and growth rates for the LHDwere then found through numerically finding roots of Eq.~2!,using model parameters determined from measured plaparameters at each radial location. Figure 9~d! shows theprofile of the maximum growth rate~maximized over wavenumber! which resulted from these calculations. The prdicted growth rate profile is quite asymmetric, in fact groing modes are only found on the inner edge of the currsheet. Growth is suppressed on the outer edge by the lbeta, low ion drift speed, and small normalized density gdient. The growth rate profile compares well with the mesured fluctuation amplitude profile att5264 ms, which isrepeated in Fig. 9~c! for clarity. There is no reason to expequantitative agreement between the saturated amplitudthe fluctuations and the linear growth rate in this case. Hoever, the linear growth rate profile should indicate wheredrive for the instability is strongest and hence should suggthe saturated amplitude might be largest.

The amplitude of the fluctuations in this set of dischargis observed to decrease rapidly shortly aftert5265 ms. Theradial profiles of measured plasma parameters change fsmoothly by comparison, and therefore do not seem to pvide an answer for the rapid time scale of the decrease.unknown parameter in these experiments in hydrogen ision temperature. It is expected that the ions should be heand the ion temperature should rise monotonically durreconnection, based on measurements in helium plasm30

This ion heating could increase theTi /Te ratio and also in-crease the total plasma beta. Davidsonet al.12 have shownthat at normalized drift speedsVd /v th,i*1 the critical beta atwhich the LHDI is suppressed can drop with increasiTi /Te . Figure 9~d! shows some support for this in MRXparameter regimes as the calculated linear growth rate dwith increasingTi /Te . We expect that the ion temperatushould be less than the electron temperature before recontion begins, again based on previous measurements inlium. An estimate of the ion temperature at late times canmade through considering a MHD force balance acrosscurrent sheet, resulting inTi /Te*2 for t5274 ms. The lin-ear growth rate should drop somewhat due to an increasthe temperature ratio toTi /Te;2 ~see Fig. 9!, but this may

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not fully explain the observed greater than fourfold dropthe fluctuation amplitude.

We can offer some additional suggestions as tosource of the rapid decrease of the fluctuation amplitudeserved in these discharges. A nonlocal theory is likely tonecessary to fully describe the LHDI in MRX current sheedue to the presence of gradients in both ther and z direc-tions. In particular, it is important to note that the loctheory assumes that the strongest growing mode occuki;0, or at infinite parallel wavelength. The current sheetMRX is, of course, of finite length and largest parallel wavlength is likely set by this length. It is possible that plasmconditions away from the center of the current sheet~alongz! could have repercussions on the behavior of the instabnear the center of the current sheet, due to the tendencthe instability to grow at large parallel wavelength. For istance, it is known that the plasma pressure downstreathe MRX current sheet builds up during reconnection,49 andthis might lead to large downstream beta. This beta mistabilize the LHDI at largez, limiting the ki available formodes driven at the center. This effect, coupled with risbeta andTi /Te at the center, could possibly result in thobserved rapid drop of LHDI amplitude near the centerthe current sheet for the given parameters.

3. Spatial correlations and propagation characteristics

The linear theory provides predictions for wavelengand phase velocity of the LHDI, and further evidence for tpresence of this instability in MRX could be providethrough comparing measured spatial correlations in the fltuations with the theoretical predictions. In this section, sties of the decorrelation length in the measured fluctuatiare presented along with statistical dispersion relationsrived from the cross-spectrum of two spatially separatedferential probes.

Spatial correlations in the fluctuations were investigausing spatially separated double floating Langmuir probThree probes were constructed for this purpose, with proto-probe spacings of 1, 3.5, and 10 mm. The decorrelalength in the fluctuations was investigated through calcuing the coherency between separated differential probenals, which is defined as

g5uXa,bu

Audf f ,au2udf f ,bu2,

wheredf f ,a is the Fourier transform of signala, andXa,b

5df f ,adf f ,b* is the cross spectrum of signalsa andb. Fig-ure 10 shows the mean coherency, averaged over the Lfeature in the frequency spectrum and over 20 dischargesseparation, versus probe separation~normalized to the electron gyroradius!. Here the separation,Dx, is in the toroidaldirection, which is the current direction and the expecpropagation direction for the LHDI.

The signals are quite coherent at the smallest separa~1 mm!, but the coherency drops rapidly as the separabecomes larger. A decorrelation length for the turbulencebe estimated as the length at which the coherency drop


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1/e. From Fig. 10, a decorrelation lengthLc&10re is esti-mated. This length is comparable to the theoretically pdicted wavelength for the strongest growing portion of tLHDI spectrum (l;2pre). This estimate implies that significant new growth occurs over a single wavelength, anplication which is consistent with the predicted strong linegrowth rate for the LHDI (g;vLH).

Dispersion relations of the fluctuations were investigain the 1 mm separation case using the statistical methoBeall et al.53 Here the local wave number at each frequenis calculated from the phase in the cross-spectrum ofspatially separated signals. This computation was performusing two sets of data, one with the probe oriented inelectron diamagnetic direction and the second with the proriented in the ion diamagnetic direction. The distinction btween these two orientations is made by labeling one oftwo probes as primary~call it probe a, for instance!, andorienting the two probes such that probea is upstream withrespect to the second probe in a flow in either the ionelectron diamagnetic direction. This distinction is made pmarily as a test for any systematic asymmetries inprobes—if the two probes make measurements in an idecal fashion, rotating the probe should result in a positmeasurement ofklocal in the wave propagation direction ana negativeklocal measurement when oriented in the opposdirection. The statistical dispersion relations~v versus localwave number! resulting from orientations in the electron~la-beled 0°! and ion~labeled 180°! diamagnetic directions areshown in Fig. 11. The gray regions surrounding the blak', local curves represent the spectral width of thek', local cal-culation, which is quite large. The spectral width represethe spread in measuredk', local and the size of this spread idue to the observation of, on average, a large spread inphase shift in the cross spectrum at each frequency inturbulence. This fact precludes a statistically significanttermination of the wavelength and phase velocity of the fltuations. However, a preference for propagation in the etron diamagnetic direction is indicated by the measuremof primarily positive localk', local for orientation in the elec-tron diamagnetic direction, and negativek', local for the op-posite direction. This direction of propagation is consistewith the LHDI when observed in the ion rest frame. Spetroscopic ion flow velocity measurements have been p

FIG. 10. The mean coherency of spatially separated measurements of Lfluctuations in MRX.


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formed in the MRX which suggest that the ion rest framethe correct lab frame in the current sheet.30

The large spectral width of these local wave numbmeasurements could be attributed to several causes, ining deficiencies in the measurement technique and effecthe plasma. One possible deficiency in the measuremtechnique is the uncertainty in the instantaneous directiothe magnetic field during the measurement. The value oBz

is locally determined~within a few centimeters using the 1-Dmagnetic probe!, however the radial and toroidal fields ameasured on the other side of the torus~approximately 180°away in toroidal angle! and may not accurately represent tfields in the toroidal plane of the fluctuation measuremeThis may lead to projection effects which would make twavelength appear longer, but the effect should be protional to cosu and may not be large enough to explain tobserved width. In addition, if there is somekr to the wave,which we have assumed is zero in the theory presenteSec. II, we may be only measuring a projection ofk' in thetoroidal direction. This would lead to a smaller estimatethe wave number and a faster apparent phase velocity,sistent with the observations. Nonlinear effects may acontribute to the observed spectral width. The LHDI hafairly strong predicted linear growth rate in MRX, whichcomparable to the real frequency~a prediction supported bydecorrelation length estimates!. It is therefore not unreasonable to expect rapid nonlinear saturation of the instability anonlinear modifications to the wavelength spectrum ofturbulence. The linear characteristics of the instability, suas the phase velocity, may not be preserved in the nonearly saturated state, and this may be reflected in the msurement.

4. Comments on the saturated amplitude

In Sec. II a brief review of saturation mechanisms for tLHDI was offered, including plateau formation, current rlaxation, trapping, electron resonance broadening, and nlinear mode–mode coupling. Although collisional dissipatiis not enough to explain the rate of reconnection in thdischarges, sufficient collisions are available such thatteau formation and trapping might not be effective. We wtherefore compare the measured amplitude of the fluctuatto the theoretical predicted saturated amplitude due to

FIG. 11. Statistical dispersion relations for two probe orientatio~0°/180°5electron/ion diamagnetic direction!.


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models presented in Sec. II: electron resonance broade@Ref. 28, Eq.~3!# and nonlinear mode–mode [email protected], Eq. ~4!#. The peak amplitude~in both space and time!observed in the radial scan presented Sec. IV A 2 is roug^df f&max;0.4 V indicating a normalized potential fluctuation value of (Te'8 eV):

e^fp&max

Te;5%.

We can now estimate a value for the normalized flucating electric field energy density,E/nTi , where E'Emax

2 /8p5kmax2 ^f2&max/8p. As was discussed in Sec

IV A 3, a statistically significant value for the mode wavlength was not measured, however we can estimate the wnumber in these fluctuations from the linear theory,kmax

;re21 . Using this estimate, we find that the peak fluctuati

electric field value, based onf;0.40 V and k;re21

;1700 m21, is E;700 V/m. Using this estimate andn;2.531013 cm23 and Ti;Te;8 eV, we find that in thesemeasurements

Emax

nTi;731028.

Now we can compute the predictions, based on msured plasma parameters, of the electron resonance broaing and nonlinear mode coupling saturation models for coparison. The electron resonance broadening model predusing n;2.531013 cm23, Ti;Te;8 eV, B;100 G ~at r'0.36 cm!,

S EnTi

D'2

5

me

M

Ve2

vp,e2 S Ti

TeD 1/4 V2

v th,i2 5531028.

This value is quite comparable to the computed valueE/nTi for the measurements reported here (;731028).However, it should be pointed out that we might expectelectron resonance broadening mechanism to be hampby electron collisions in MRX, so it might be surprising tfind agreement with this prediction. The nonlinear Landdamping saturation mechanism predicts

ef

Ti'2.4S 2me

M D 1/2 V

v th,i'20%.

This prediction is larger than the normalized amplitude dduced from the measurements~5%!, but is quite close con-sidering the limitations of the theoretical model used in tcalculation. The theory used to make this estimate ignocoupling of wave energy in unstable long parallel wavlength modes to damped shorter parallel wavelenmodes.26 For this reason, it may overpredict the saturatamplitude in these experiments.

It should be noted that the electron resonance broadeand mode coupling saturation predictions have the sascaling with plasma parameters, and differ only by a costant. This coupled with the fact that both are within a facof 2 of the measurement makes it difficult to argue whicheither, is the correct model for saturation of the LHDIMRX.

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5. Magnetic measurements of LHDI

A brief discussion of electromagnetic modificationsthe LHDI was offered in Sec. II, suggesting that magnefluctuations should be expected along with electrostLHDI fluctuations in high-beta current sheets. While the dtailed study performed using electrostatic diagnostics~as pre-sented previously! has not yet been reproduced with manetic diagnostics, initial evidence for electromagnetic LHfluctuations has been found. Magnetic pickup loops wused to study magnetic fluctuations in low-collisionality curent sheets~12/10 kV, 4 mT, hydrogen!. These studies revealed high frequency (f & f LH) magnetic fluctuations on thinner edge of the current sheet. The magnetic fluctuationsobserved concomitantly with the electrostatic LHDI fluctutions measured using floating probes. These signals aretatively identified as magnetic LHDI fluctuations, whicshould be expected to appear due to electromagnetic cotions to the LHDI in high beta current sheets. The ampliturange of these fluctuations isdB;1 – 10 G or dB/B;1 – 10%, similar to the normalized amplitude in the eletrostatic fluctuations. Currently, a detailed study of magnefluctuations in MRX is under way which should shed molight on the source of these signals.

B. Role of the LHDI in reconnection in MRX

One of the primary motivations for studying fluctuatioin MRX is to determine the role of any observed fluctuatioin the reconnection process. Of particular interest is wheor not the LHDI can generate anomalous resistivity in MRcurrent sheet or play some other role in establishing theserved enhanced resistivity and fast reconnection rateMRX.50 The data and analyses presented in Sec. IV sugthat the LHDI is not essential in determining the reconntion rate in MRX. This conclusion stems largely from cosideration of the radial profile of the fluctuation amplitudthe time behavior of the fluctuation amplitude and the scaof the fluctuation amplitude and effective collision rate wCoulomb collisionality.

1. Radial profiles

The radial profiles shown in Fig. 7 seem to suggest tsome penetration of the LHDI into the magnetic null is oserved in these measurements. However, it is importannote that the peak of the current density is slightly offfrom the magnetic null early in the reconnection processMRX, due to the asymmetries inherent in the toroidal geoetry in MRX. This is demonstrated in Fig. 12, where profilof fitted magnetic field, fit-derived current density, and mesured fluctuation amplitude are plotted. From Fig. 12,fluctuation amplitude is not seen to penetrate into the nconsistent with linear theoretical predictions of finite-bestabilization. The simplest mechanism of anomalous resisity generation by LHDI turbulence is by effective scatteriof the current carrying particles by the wave electric fielThe measured amplitude profile of the LHDI makes it qudifficult to apply this model to the MRX current sheet, as tturbulence is not present at the magnetic null, where it wobe needed to provide dissipation.


cic-

Ie

re-n-

ec-e

-c

ser

b-inst-

,g

t-totn-

-ell,

v-

.

ld

It is also interesting to discuss the relationship betwethe current profile and the fluctuation amplitude profile. Tcurrent density in MRX is generally observed to be symmric, while the fluctuation amplitude profile is markedly asymmetric. In addition, the thickness of the current sheet~see,e.g., Fig. 8! seems insensitive to the time history of the aplitude of the LHDI fluctuations. Both of these observatiosuggest that the fluctuations are not the primary mechanby which the shape of the current profile is established.

2. Time behavior of the LHDI amplitude

An observation which provides further support for a coclusion that the observed fluctuations are not essentialfast reconnection in MRX is the measured time behaviorthe fluctuation amplitude. The time behavior of the fluctution amplitude is compared to that of the average reconntion electric field (Eu) and the average central current desity (Ju) in Fig. 13. The reconnection electric field is thtime derivative of calculated poloidal flux value in the cenof the current sheet, and represents the rate of reconne~rate of destruction of poloidal flux interior to the curresheet in radius!. From Fig. 13, the quasi-steady reconnectiphase can be identified as the time period over whichreconnection electric field is steady, roughly fromt5260 ms to t5280 ms. The fluctuation amplitude shown iFig. 13 ( df f&max) is the peak value in space at each pointtime. As was discussed in Sec. IV A 2, the fluctuation amptude grows as the current sheet forms and reconnectiongins, but is seen to decrease rapidly with time before theof the quasi-steady reconnection phase~note that the radiallyintegrated or averaged fluctuation power would exhibiteven steeper decline, as is evident when inspecting Fig!.Both the reconnection electric field and the peak current dsity seem rather insensitive to the fairly extreme time behior of the peak fluctuation amplitude neart5265 ms. This

FIG. 12. Profiles of magnetic field, current density, and fluctuation amtude at t5264 ms, demonstrating that no significant penetration of tLHDI into the magnetic null is observed.


ruhen

me

n

Dilarere

aigt

ioa

gghafo

oc

, i

Droghenk

ernell

eoh

con-atein

idetua-ioneren-earl-a-dlli-ts,

r of

ce-heea-de-te

li-

ns

l-olt-

ofnog.

o-m-

teythe

an.


observation suggests that the LHDI fluctuations are not ccial in determining the reconnection rate in MRX, since treconnection rate is essentially unphased by a rapid chain the amplitude of the fluctuations. In fact, there is soevidence that the reconnection electric field and current dsity actuallyincreaseslightly following the rapid decrease ithe fluctuation amplitude neart5275 ms. Although this ob-servation is not conclusive, this might suggest that the LHactually impedes the reconnection process in MRX. A simconclusion has been made with respect to recent thdimensional Hall MHD simulations of reconnection whethe LHDI is seen to arise.14

The measurements reported here are taken only nez50, and it is possible that the fluctuations persist at hamplitude elsewhere in the current sheet even thoughamplitude drops dramatically at the measurement locatHowever, measurements of plasma profiles downstre(uzu.0) have been made, and these measurements sushallower density gradients and lower current densities tat z50. Therefore it is expected that the strongest drivethe LHDI should be located atz50. Even if the fluctuationsdid persist elsewhere, the simplest theoretical pictureanomalous resistivity generation by the LHDI, through effetive scattering of the current carrying particles at the nullunlikely to be valid in light of the observations.

Recent theoretical work has suggested that the LHmay provide a trigger for reconnection, through either pviding an initial resistivity or through nonlinearly steepeninthe current and density profiles at the edges of a current sand triggering additional instabilities such as the drift-kiinstability ~see, e.g., Refs. 13, 42, and 54!. It is not clear thatthere is an onset problem in MRX, as reconnection is drivthrough boundary perturbations imposed by the extecoils ~note that this is not unlike the initial tearing modperturbation imposed on some recent simulations of cosionless reconnection15!. However, it is possible that thLHDI plays an important role early in the reconnection prcess in MRX, when its amplitude is strongest, even thoug

FIG. 13. Time traces of reconnection electric field, peak current density,peak rms fluctuation amplitude in 12/10 kV 4 mT hydrogen discharges


-

geen-

Ire-

rhhen.mestnr

f-s

I-

et

nal

i-

-it

does not seem to influence the eventual quasi-steady renection rate. Therefore, future experiments will investigthe influence of the LHDI on the onset of reconnectionMRX current sheets.

3. Scaling of fluctuation amplitude and quasilinearresistivity with collisionality

The discussions already presented in Sec. IV provevidence supporting the argument that the measured fluctions are not of crucial importance in setting the reconnectrate in MRX. An additional data set which provides furthsupport for this conclusion was taken to explore the depdence of the fluctuation amplitude and computed quasilinresistivity on the collisionality in MRX current sheets. Colisionality in MRX current sheets is characterized by the prameterlmfp /d, whered is the width of the current sheet anlmfp is the electron mean free path against Coulomb cosions. As the collisionality is lowered in MRX current sheethe measured toroidal reconnection electric field,Eu , is nolonger balanced by classical collisional drag at the centethe current sheet,Eu /hSpj u@1 ~where hSp is the classicalSpitzer perpendicular resistivity!.49 The size of this discrep-ancy, which could be characterized as a resistivity enhanment, increases rapidly with decreasing collisionality. If tmeasured LHDI fluctuations were responsible for this msured resistivity enhancement, one might expect a strongpendence of the LHDI amplitude and effective collision raon the Coulomb collisionality in MRX.

Figure 14~a! shows the measured peak fluctuation amptude ~peak amplitude in both space and time! versuslmfp /dfrom a scan of fill pressure. The amplitude of the fluctuatiodoes tend to increase with decreasing collisionality~increas-ing lmfp /d!. However, if the fluctuation amplitude is normaized to the measured electron temperature, which from Bzmann’s equation might be considered as an estimatedn/n in the turbulence, we find that there is essentiallychange in this quantity with collisionality, as shown in Fi14~b!.

Theoretical estimates of effective collision rates prduced by LHDI fluctuations depend on the normalized aplitude of the fluctuations,Ek /nT;(dn/n)2 @see Eq.~8!#.Figure 14~b! then suggests that the effective collision raprovided by the LHDI fluctuations in MRX should be fairlconstant as the collisionality is drastically changed in

d

FIG. 14. ~a! Fluctuation amplitude and~b! normalized fluctuation amplitudevs collisionality from a scan in fill pressure.


iar

I

thle

on-w

he

nitin

c

D

debbDw

yas

tialeseca--of

orting

oftialtouredre-ce

ve-ithea-d aec-ityredf theoad-

onsuredblely a

sti-

uredter-olex-r ofen-col-isDIua-re-

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ur-

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ivi


current sheet. However, as the collisionality is raisedMRX ne,i increases dramatically, and therefore the normized LHDI resistivity,nLHDI /ne,i might behave in a manneconsistent with the observed resistivity enhancementMRX. We can now compute the normalized effective LHDcollision rate for this set of data, using Eq.~8! along with themeasured amplitude, plasma parameters and the linearretical estimates for the LHDI shown in Fig. 9. For exampfor the lowest collisionality data point in Fig. 14,

nLHDI5ImS k'

4vp,i2

k'2 v th,i

2 z iZ~z i ! Dk',max

Ti

meV

EnTi

5vp,i

2

V i2

v th,i

V

EnTi

ImS 4

k'rez iZ~z i ! D

k',max

vLH

'0.6vLH526 MHz.

This estimate is actually lower than the Coulomb collisirate for that data point,ne,i'35 MHz, suggesting a resistivity enhancement of less than a factor of 2. Figure 15 shothe computed LHDI resistivity enhancement along with tmeasured resistivity enhancement (E/hSpj ) as a function ofcollisionality for all the data points in the pressure scaWhile the LHDI resistivity enhancement does increase wdecreasing collisionality, it is clearly insufficient to explathe observed value ofE/hSpj . It should be noted that theeffective collision rate is computed using the maximum flutuation amplitude~maximum in both time and space!, andtherefore provides a very generous estimate of the LHresistivity. The amplitude at the null point, whereE/hSpj ismeasured, is significantly lower than this peak amplituand an estimate of the effective collisionality there shouldmore than an order of magnitude lower. It should alsonoted that the theory used in computing the effective LHcollisionality is a collisionless theory, and it is not clear hothese estimates change when the Coulomb collisionalitclose to the linear frequency of the instability, as is the chere.

FIG. 15. Measured resistivity enhancement and computed LHDI resistenhancement as a function of collisionality.


nl-

in

eo-,

s

.h

-

I

,eeI

ise

V. SUMMARY AND DISCUSSION

In this paper, detailed measurements of floating potenfluctuations in the MRX current sheet were presented. Thmeasurements have led to the first experimental identifition of the lower-hybrid drift instability in a laboratory current sheet, and to the first opportunity for a detailed studythe role of this instability in magnetic reconnection. Suppfor identifying the measured potential fluctuations as bedue to the LHDI was provided by detailed measurementsthe frequency spectrum, radial amplitude profiles, and spacorrelations. A local linear theory of the LHDI was usedsuccessfully explain asymmetries observed in the measradial fluctuation amplitude profile. Correlation measuments indicated a decorrelation length in the turbulenwhich was comparable to the theoretically predicted walength of the LHDI, an observation which is consistent wa theoretically predicted strong peak linear growth rate. Msurements of phase velocity in the fluctuations suggestepreference for propagation in the electron diamagnetic dirtion, but a statistically significant value for the phase velocwas not found due to a large variations in the measuphase at each frequency in the turbulence. Estimates oexpected saturation amplitude by electron resonance brening and nonlinear mode coupling were made basedmeasured plasma parameters. The estimate for the meapotential fluctuation amplitude was found to be comparato the electron resonance broadening estimate, but roughfactor of 4 lower than the nonlinear Landau damping emate.

The observations presented suggest that the measpotential fluctuations do not play an essential role in demining the quasi-steady reconnection rate in MRX. The rof the LHDI in the reconnection process in MRX was eplored through studying the spatial and temporal behaviothe fluctuation amplitude and through studying the depdence of the fluctuation amplitude on the current sheetlisionality. The observed radial profile of the fluctuationsconsistent with several theoretical predictions that the LHshould not penetrate to the high-beta null point. The flucttion amplitude was observed to drop dramatically duringconnection while the reconnection rate~electric field! wassteady. The mechanism for the drop in amplitude is still nfully understood, but this observation makes it difficultclaim that the LHDI is providing anomalous dissipation duing reconnection in MRX. Finally, a study of the effect ocollisionality in the current sheet on the fluctuation amptude and computed effective collisionality was performeThe normalized fluctuation amplitude was found to be faiinsensitive to the collisionality in MRX current sheets. Thquasilinear estimate of the LHDI collisionality was foundfall short of the Coulomb collision rate in low collisionalitdischarges, even when the peak fluctuation amplitude is uin the computation, further suggesting that the fluctuatioare not responsible for enhancing the resistivity in MRX crent sheets.

Magnetic fluctuations have also been observed in MRand any relationship between these fluctuations and theconnection rate is currently the subject of intense investi

ty


ntin

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enla

thnrrfon

nlysntifo

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on

nsa

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ity,


tions. While fluctuation studies will continue, with curredata it is difficult to construct a theory of fast reconnectionMRX based solely on extending MHD with an anomaloresistivity generated by the observed electrostatic potenfluctuations. Therefore future experimental campaignsMRX will also focus on exploring alternative fast reconnetion mechanisms, including looking for signatures of fastconnection mediated by the Hall term in the generalizOhm’s law. There is some consistency between the datathe Hall-dominated models in the observation of a currsheet thickness proportional to the ion skin depth; simutions have shown that an ion current layer can exist atscale.16 However, the simulations predict that the curresheet can have two scales, and that an inner, electron cuscale can be as small asc/vp,e ~depending on the source odissipation!. Such a current layer could solve the electrforce balance problem in MRX through an increased~andcurrently unresolved! current density,j * , such thatE5h j *~dissipation could be provided by collisions!. The real ex-perimental tests of these simulations will therefore ocome through attempting to resolve smaller scale featurethe MRX current sheet—features that are potentially matimes smaller than the current size of individual magnedetectors in MRX. The development of new diagnosticsthis purpose is already underway.

ACKNOWLEDGMENTS

The authors would like to thank D. Cylinder and R. Culer for their excellent technical support.

MRX is jointly funded by DOE, NASA, and NSFT.A.C. acknowledges support from NSF Graduate Reseaand NASA GSRP Fellowships.

APPENDIX: DERIVATION OF LHDI ELECTRONDENSITY PERTURBATION

In the following we use the electrostatic approximatiand introducek'5Akx

21ky2 and ki5kz . The Vlasov equa-

tion is used to calculate perturbed distribution functiofrom which the perturbed charge densities are calculatedused in Poisson’s equation,

S dd f e

dt D0

52q

mE"

] f e0

]v.

We use the method of characteristics to solve ford f e , inte-grating along the zero-order orbits of the particles:

d f e52q

m E2`

t

dt8S E"]

]vf e

0Dv8,r 8,t8

5q

m E2`

t

dt8F2v"“f2FM,e

v th,e2 1

en

VeFM,e~“f!yG

522q

m

FM,e

v th,e2 f t85t1

2q

m

FM,e

v th,e2 E

2`

t

dt8]f

]t8

1ikyqen

mVeFM,eE

2`

t

dt8 f.


aln

-dndt-istent

inycr

ch

,nd

The final step is accomplished by usingv"“f5df/dt2]f/]t, and by assumingf t8→2`50.

In order to complete the time integrals, we must fisolve the single particle equations of motion for the eletrons. These are

dv

dt5

q

me

vÃB~x!

c,

dr

dt5v.

Assuming that the gradient scale length in the magnetic fiis much longer than the electron gyroradius, we can useguiding center expansion to obtain the electron orbit. Intducing the variablet5t8– t, we find

y8'v'

Vecos~w1Vet!2

v'

Vecosw2

1

2eb

v'2

Vet,

x8'v'

Vesin~w1Vet!2

v'

Vesinw,

z852v it,

where eb5(1/B)]B/]x and ebv'2 /2Ve5V

“B , the electron“B drift speed. Here we are ignoring oscillating termsorder ebv'

2 /Ve . If we assumef5f exp(ik"r2 ivt), thenthe equation ford f e becomes

d f e522qf

m

FM,e

v th,e2 F11 i ~v2kyvD,e!

3E0

`

dt expS 2 i S k'v'

Ve~cos~w1Vet!2cosw! D

1~v2kiv i2kyV“B!t D G .Using the fact that

exp~ iz sinw!5 (n52`

`

exp~ inw!Jn~z!

and

exp~ iz sin~w1Vet!!5 (n52`

`

exp~ im~w1Vet!!Jn~z!,

d f e becomes

d f e522q

m

FM,e

v th,e2 fF12~v2kyvD,e!

3(n,m

S Jn~z!Jm~z!exp~ i ~m2n!~w2p/2!!

v2kiv i2kyV“B2mVeD G ,

where we have introduced the electron diamagnetic velocvD,e5env th,e

2 /2Ve andz5k'v' /Ve . The perturbed electrondensity can now be calculated by integratingd f e over veloc-ity space:


I

e

res

da

ins,

s.

ys.

,

ys.

-

a A.

ys.

ev.


dne522qn0

mv th,e2 f2

2q


3(m,n

E v'dv'E dv i

Jn~z!Jm~z!FM,e

v2kiv i2kyV“B2mVe

3E dw exp~ i ~m2n!~w2p/2!!.

The velocity phase integral is nonzero only form5n. Theintegral overv i evaluates to a plasma dispersion function,Z.The perturbed electron density then becomes

dne522qn0

mv th,e2 f2

2qn0


2

kiv th,e

3(nE xdxexp~2x2!Jn

2~k'rex!

3ZS v2kyV“Bx22nVe

kiv th,eD ,

where we have introducedx5v' /v th,e , and V“B

5ebv th,e2 /2Ve . The frequency range of interest for the LHD

is v;vLH!Ve , and we will therefore keep only then50term in the sum. So the final expression for the perturbelectron density is then

dne522qn0

mv th,e2 f2

2qn0


2

kiv th,e


kiv th,eD .

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