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Observations of toroidicity-induced Alfvén eigenmodes in a reversed fieldpinch plasma

G. RegnoliConsorzio RFX, Associazione Euratom-ENEA sulla Fusione, Corso Stati Uniti 4, 35127 Padova, Italyand Associazione Euratom-ENEA sulla fusione, Centro Riccerche Frascati, Cassella Postale 65, 00044Frascati, Italy

H. Bergsåker and E. TennforsAlfvén Laboratory, Royal Institute of Technology (Association EURATOM/VR), 100 44 Stockholm, Sweden

F. ZoncaAssociazione Euratom-ENEA sulla Fusione, Centro Riccerche Frascati, Cassella Postale 65, 00044Frascati, Italy

E. Martines, G. Serianni, M. Spolaore, and N. VianelloConsorzio RFX, Associazione Euratom-ENEA sulla Fusione, Corso Stati Uniti 4, 35127 Padova, Italy

M. CecconelloAlfvén Laboratory, Royal Institute of Technology (Association EURATOM/VR), 100 44 Stockholm, Sweden

V. Antoni and R. CavazzanaConsorzio RFX, Associazione Euratom-ENEA sulla Fusione, Corso Stati Uniti 4, 35127 Padova, Italy

J.-A. MalmbergAlfvén Laboratory, Royal Institute of Technology (Association EURATOM/VR), 100 44 Stockholm, Sweden

sReceived 6 July 2004; accepted 23 December 2004; published online 9 March 2005d

High frequency peaks in the spectra of magnetic field signals have been detected at the edge ofExtrap-T2R fP. R. Brunsell, H. Bergsåker, M. Cecconello, J. R. Drake, R. M. Gravestijn, A.Hedqvist, and J.-A. Malmberg, Plasma Phys. Controlled Fusion,43, 1457s2001dg. The measuredfluctuation is found to be mainly polarized along the toroidal direction, with high toroidalperiodicity n and Alfvénic scalingsf ~B/Îminid. Calculations for a reversed field pinch plasmapredict the existence of an edge resonant, high frequency, high-n number toroidicity-induced Alfvéneigenmode with the observed frequency scaling. In addition, gas puffing experiments show that edgedensity fluctuations are responsible for the rapid changes of mode frequency. Finally a coupling withthe electron drift turbulence is proposed as drive mechanism for the eigenmode. ©2005 AmericanInstitute of Physics. fDOI: 10.1063/1.1861896g

I. INTRODUCTION

Plasmas confined in reversed field pinchsRFPd configu-rations are characterized by relatively high amplitude ofmagnetic fluctuationsstypically of the order of 1% of theaverage fieldd with a broad spectrum of frequencies andwave vectors. Most of the fluctuation power is concentratedin the low frequency range dominated by several tearingmodes resonating in the core region.1,2 Historically, most ofthe effort has been devoted to the study of these modes, asthey are responsible for the transport in the core region andfor the dynamo process which sustains the configurationagainst resistive diffusion. The higher frequency of the spec-trum has received much less attention as it was thought togive a smaller contribution to the confinement properties ofthe configuration.3

Among the possible causes for the magnetic fluctuationsin this frequency region, Alfvén waves and eigenmodes de-serve a special mention. Indeed a generation of an Alfvénwave cascade up to the ion-cyclotron frequency of the mainatomic species is expected. Moreover turbulence induced byAlfvén waves has been indicated as spontaneous mechanism

for anomalous ion heating4 or as possible current drivemethod in RFPs.5

Up to now Alfvén eigenmodessAEd and especiallytoroidicity-induced Alfvén eigenmodessTAEd have been in-tensively studied in tokamak experiments since they are ex-pected to strongly interact with resonanta particlessprod-ucts of D-T reactionsd via direct or inverse Landau dampingthus influencing the performance of future reactors. Inpresent tokamaks TAE modes can be driven unstable by ion-cyclotron resonant heating6 sICRHd and by ions produced byneutral beam injectionsNBId.6 However, they have also beenobserved during purely ohmic discharges.7

So far no clear evidence of TAE modes in RFP deviceshas been reported, although it is worth mentioning that peaksin the spectra of magnetic field signals have been detected atthe edge of the previous Extrap-T2 device, for short timeintervals of the dischargesDt=1 msd.8,9 Their frequencieswere near the TAE frequency expected for a two-ion speciesRFP plasma made up of hydrogen and C+ carbon impuritiessreleased by the old graphite first walld.4,9,10 However, RFPshave always been theoretically studied in the cylindrical ap-proximation and it is generally believed that toroidicity ef-

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http://dx.doi.org/10.1063/1.1861896

fects should not play an important role for the edge plasmadynamic in RFPs since the edge toroidal magnetic field issmall and because an intense electron Landau damping ofAlfvén waves is expected to occur.

The aim of this work is to report clear experimental evi-dence of Alfvénic fluctuations in the Extrap-T2RsRef. 11dreversed field pinch plasma and to offer an interpretation ofthem in the framework of TAE theory.

The remainder of this work is organized as follows: areview of the theory basics is given in Sec. II; experimentalsetup and the observations of high frequency fluctuations arereported in Sec. III; the Alfvénic scalingsf ~B/Îminid offluctuations is shown to be consistent with the TAE gap inSec. IV shints about the drive mechanism are also given inthis sectiond; conclusions are drawn in Sec. V; and contraryto common wisdom the existence of a finite TAE gap for theRFP configuration is analytically derived in the Appendix.

II. TAE THEORY BASICS

In an axisymmetric cylindrical single-ion species plasmaof length 2pR0 and radiusa, shear Alfvén waves have acontinuous spectrum given by the solutions of the dispersionrelation v2srd=ki

2srdvA2srd, where vA=B/Îm0r is the local

Alfvén velocity, ki=sm+nqdBu / rB is the component of thewave vector parallel to the local magnetic field,q=rBf /RBu is the safety factor,m andn are the poloidal andtoroidal mode numbers, andr=mini is the mass density.12 Allperturbations belonging to the Alfvén continuous suffer astrong damping due to wave phase mixing.13 The cylindricalcontinuous spectrum of Alfvén waves is characterized by thepresence of degeneracies, namely, radial positions wheremodes localized on different rational surfacessm,nd have thesameki and consequently the samev.

A TAE mode is a normal mode of a toroidal plasma.Where the Alfvén cylindrical continua of two rational sur-faces with subsequent poloidal mode numberssuDmu=1d andsame toroidal mode numbern have a degeneracy, toroidicitycorrections, expressed in terms of the inverse aspect ratioa/R0, remove the degeneracy by coupling the two modes andcreating a gap in the spectrum.12 Inside the gap Alfvén wavescannot exist except for a discrete frequency which constitutesthe TAE mode. Such a mode does not experience anycon-tinuum dampingand can grow if there is enough free energyto drive it unstable with an eigenfunction localized at theradial positionr =rgap.

14,13 The dispersion relation near thetoroidicity-induced gap takes the form given by the follow-ing equation:7,15

v2 = ki,m2 srdvA

2srdgsrd, s1d

with

g =1

f2 − e02/2g

h1 + s ± Îs1 − sd2 + e02sj

and

s = ki,m+12 /ki,m

2 .

The two branches, solutions of the above equation, de-scribe the gap whose width depends on the value ofe0

,Osr /R0d.14,16 In the limit of e0→0 the gap amplitude iszero and the cylindrical dispersion relation mentioned aboveis recovered.

Equation s1d is configuration independent, but the ex-pression fore0 changes for different magnetic configurations.In the Appendix the expression fore0 is analytically derivedfor any axisymmetric toroidal configuration with shiftedmagnetic surfaces characterized by large aspect ratiosa/R0

!1d. In particular, it is shown there thate0 is small but finiteat the edge of RFP plasma and therefore gaps in the Alfvéncontinuous spectrum are opened by toroidicity in that region.

The TAE-mode frequency is approximately equal to thedegenerate frequency of the cylindrical spectrum and can beestimated writing down the degeneracy condition:4

ki,m = − ki,m−1 ⇒ m+ nqsrgapd < ± 12 , s2d

which is solved forkigap. ±Bu /2Brgap and the TAE-modefrequency isfTAE.Bu / s4pÎm0rrgapd. From conditions2d itis evident that a TAE mode is the result of the toroidicity-induced coupling of two modes traveling in opposite direc-tions with same phase velocityvA. Plasma inhomogeneitieslocalize the resulting beating at the gap positionr =rgap.

A further feature of a pure shear Alfvén eigenmode isthat it does not compress the plasma. Hence the parallel com-ponent of perturbed magnetic field is given by the perpen-dicular pressure balancem0dp+BdBi=0. Moreover the per-pendicular fluctuationdB' is by definition

dB' = sB/Bd 3 = dAi . SuBf

B− f

Bu

BD ]dAi

]r+ i rSnBu

RB

−mBf

rBDdAi, s3d

wheredAi is the parallel component of the vector potential.13

Equations3d, as well as the previous ones, is valid for anytoroidal plasma configuration. Nevertheless, for the edge re-gion sr .ad of RFP plasmas, some simplifications can bemade. Indeed in such a case, since the gradient term]dAi /]ris expected to be the dominant one at the edge, and theaverage toroidal field is smallfBfsad /Bsad!Busad /Bsadg,the perpendicular perturbationdB' is almost toroidal.

III. RESULTS

A. Experimental setup

Extrap-T2R sRef. 11d is a torus of major radiusR=1.24 m and minor radiusa=0.183 m. The results presentedherein have been obtained in low current hydrogen dis-charges withIp ranging from 60 to 80 kA, core densityne

.131019 m−3, and pinch parameterQ=Busad / kBfl rangingbetween 1.6 and 2.4. As the T2R first wall is metallicsmo-lybdenumd, the impurities concentration is lower than in theprevious T2 device and is expected not to modify the elec-tromagnetic fluctuations spectrum in the studied frequencyranges0.1ø f ø fhi.1.3 MHzd.10 Hence the plasma can beconsidered as a single-ion species plasma for the purpose ofthis work.

Magnetic signals have been collected by two groups ofthree Mirnovsmagneticd coils measuring the time derivative

042502-2 Regnoli et al. Phys. Plasmas 12, 042502 ~2005!


of magnetic fieldsBr ,Bu ,Bfd. The two groups have beenplaced inside a boron nitride case at the same radial andpoloidal position but toroidally spaced byDx=1.3 cm. Atriple Langmuir probe, located in between, allowed measure-ments of the ion saturation currentIs. The signals have beensampled at a frequency of 2 MHz with a measured band-width of 1 MHz for the Mirnov coils and 700 kHz for theLangmuir probes. The probes have been inserted through anoutboard horizontal port in the equatorial plane of the ma-chine and located outside the limitersrprobe/a=1.06d.

A third group of three Mirnov coils has been placed atthe same toroidal and radial position but inserted through avertical port from the top of the machine. The signals havebeen sampled at 2 MHz but the probe bandwidth was higherthan 1 MHz. The possible aliasing effects have been checkedby performing measurements at 4 and 8 MHz during repro-ducible discharges and found to be negligible by using thesame method described elsewhere.17

B. Experimental results

In Fig. 1sad the Fourier spectra of the three components

of the magnetic field time derivativesBr ,Bu ,Bfd for the dis-charge 13 915 have been plotted.18 The spectra are charac-terized by two peaks. The lower frequency activitysf,70 kHzd has already been observed in the past and is dueto toroidal rotation of tearing modes in the plasma core.1 Thesecond peak at higher frequencysf ,400 kHzd is present on

all the components, though the amplitude of theBu spectrumis smaller than that of the other two components. This resultis consistent with previous findings on the Extrap-T2 devicein which a higher amplitude of the high frequency part of the

Bf spectrum with respect to the other two components wasobserved.9

In order to get clear insights about the wave field polar-ization the high frequency fluctuation has been analyzedalong the so-calledmaximum variancedirectionsMVD d. The

MVD technique is advantageous whenever the axis of one ofthe magnetic Mirnov coils is not exactly aligned with theaverage magnetic fieldB0, as in the case of astrophysicalplasmas19,20 and often in laboratory plasmas as well. Such amisalignment causes spurious couplings between magneticsignals locally measured by the perpendicular coils. As aconsequence the matrix of fluctuations, defined as

dAi,j = kBiBjl − kBilkBjl, i, j = r,u,f,

is not diagonal.19,20 The k¯l indicates the time average.The MVD analysis consists in projecting the measured

field componentssBr ,Bu ,Bfd on the orthogonal referenceframe in whichdAi,j is diagonal. By definition the unit vec-tors of the new reference framese1,e2,e3d are eigenvectorsof the matrixdAi,j.

In Fig. 1sbd the Fourier power spectra of the components

sB1,B2,B3d, resulting from the projection of the measuredmagnetic signals on the new reference frame, are plotted. Itis evident that the high frequency fluctuationsf ,400 kHzdis linearly polarized along the directione3 which constitutesMVD for it. The matrix of the anglesakl, which the eigen-vectors se1,e2,e3d share with the measurement referenceframe ser ,eu ,efd, is the following:

akl = 126.7° 69.6° 73.5°

73.8° 22.3° 75.2°

69.4° 81.4° 22.4°2 ,

whereakl=arccossek ·eld, with k=1,2,3 andl =r ,u ,f. Themode polarization observed with the MVD technique is con-sistent with that of an edge resonant AE mode, discussed inthe preceding section of the paper. Indeed, the unit vectore3is oriented predominantly along the toroidal direction, asa3,f!a3,u ,a3,r.

Furthermore the MVD analysis has allowed to divide offthe time evolution of the tearing mode rotation frequencyfrom that of the high frequency fluctuation. This is shown in

FIG. 1. sColor onlined. sDischarge 13 915.d Graphsad, the Fourier power spectra of the measured magnetic field components. Graphsbd, the Fourier spectraof the magnetic field components after projection on the MVD eigenvectors frame.

042502-3 Observations of toroidicity-induced Alfvén eigenmodes… Phys. Plasmas 12, 042502 ~2005!


Fig. 2 where the Fourier spectrograms of theB3 andB2 com-ponents for the considered discharge 13 915 are plotted to-gether with the time evolution of the plasma currentIp onpanelsscd, sbd, andsad, respectively. The spectrogram of the

B2 component is dominated by the tearing mode rotation

whereas theB3 spectrogram is characterized by the high fre-quency fluctuation. The latter is found to change its fre-quency with time around an average value off .400 kHz.As mentioned in the Introduction of the paper, in previousexperimental campaigns in Extrap-T2, similar high fre-quency magnetic activity could be detected but for only afraction of the discharge durationsDt=1 msd. Hence it wasnot possible to state how long the high frequency modecould last.9 The spectrogram presented in Fig. 2scd showsthat the high frequency fluctuation can be excited for thewhole discharge duration.

The edge ion-cyclotron frequencyfhi is proportional tothe plasma currentIp as Busad.Bsad for RFP plasmas. Inparticular, for the considered hydrogen dischargefhisad.1.3 MHz. By comparing graphsscd andsad of Fig. 2 it canbe noted that during the plasma current ramp phase the modefrequency increases followed by a sort of plateau and a finaldecreasing phase at the end of the discharge. However, ascaling like fstd~ Ipstd~Bstd appears to be incomplete andnot accurate enough to account for the rapid changes of thefrequency observed in the spectrogram. Further insights onthe frequency scaling will be given in the following sectionof the paper.

For a detailed characterization of the mode it has been

necessary to investigate its spatial structure. The two groupsof magnetic probes, located at two closely spaced toroidalpositions, allowed to study the phase properties of the fluc-tuations and to compute the toroidal wave numbersn=kfRby the two-point correlation technique.21 In Fig. 3 the spectraSskf , fd are shown as a function of toroidal wave numberkf

and frequencyf for the B2 componentfgraphsadg and for the

B3 componentfgraphsbdg. As expected the two spectra arecharacterized by different frequency ranges. The wide spread

of frequencies in theB3 fluctuation spectrum is due to thechange of the mode frequency with time, see Fig. 2scd. Thehigh frequencies 0.2 MHz, f ,0.6 MHz of the Sskf , fdspectrum are found to correspond approximately to the samekf>−20 m−1. Considering the radial position where mea-surements have been performed, the correspondent toroidalperiodicity of the moden>kfsR+rprobed>−30 has been de-duced. In particular, the mode has been found to rotate in thesame direction as the edge electrons diamagnetic drift. The

Sskf , fd spectrum of theB2 component shows only the tear-ing mode rotation frequency, characterized by a lower toroi-dal numbern.10 and traveling in the opposite direction.However, in Extrap-T2R, the internally resonant tearingmodes are known to form a “slinky” mode which rotates inthe electron diamagnetic drift direction as well. More detailsabout tearing mode and slinky mode rotation can be found inRefs. 1 and 2.

The poloidal periodicitym of the fluctuations could notbe measured but the analysis of correlations between mag-

FIG. 2. sColor onlined. sDischarge13 915.d Graphsad, the time evolutionof plasma currentIp proportional tothe edge ion-cyclotron frequencyfhi.Graphsbd, Fourier spectrogram of the

B2 component. Graphscd, Fourier

spectrogram of theB3 component.



netic signals measured on the outboard side and on the up-board side gives a negative time delay ofdt.−3 ms for thehigh frequency range of fluctuations. This result suggeststhat the high frequency mode is characterized by a negativepoloidal periodicitym.

In the following section an interpretation of the observedhigh frequency–high-n fluctuation as TAE is given. A modelbased on the computation of the Alfvén continuous spectrumis built up and tested.

IV. THE MODEL: ALFVÉN CONTINUOUS SPECTRUMFOR EXTRAP-T2R

The Alfvén continuous spectrum of modes with a toroi-dal periodicity n=−30 has been calculated for the Extrap-T2R RFP configuration. Solving Eq.s1d for the T2R mag-netic field profiles, estimated by the polynomial funtionmodel sPFMd,22 and for density profiles, obtained by en-semble averaging single chord interferometer measurementscollected along different lines of sight during highly repro-ducible shots,23 produces the graph shown in Fig. 4.

In particular, the Alfvén continua of thesm=−1, . . . ,5 ;n=−30d, rational surfaces in the cylindrical limitof Eq. s1d have been plotted as dashed lines whereas thetoroidicity corrected continua for the surfacessm=−1,0;n=−30d have been superimposed as full lines. The hydrogenion–cyclotron frequency profilefhisrd, which establishes anupper limit to the shear Alfvén spectrum, has been plottedtoo.24

One peculiarity of the Alfvén continuous spectrum ofRFP magnetic configurations is that the number of rational

surfaces with the same toroidal mode numbern and Dm= ±1 that can be coupled by toroidicity effects is limitedbecause the safety factorqsrd is a decreasing function ofr.Indeed from the degeneracy condition, Eq.s2d, it follows that

m± 0.5 =nqsrd ⇒ unqminu ø um8u ø unqmaxu,

with m8=m±0.5. For a RFP configuration one has that 0, um8u, unuqs0d as q is maximum on the axis and goes tozero near the edge. In Fig. 4 it can be noted that only sur-faces with umu,3 have crossing pointssi.e., degeneraciesdwith the subsequent surfacessuDmu=1d that are suited to besolved by toroidicity effects, generating a gap.

As the m number of the observed fluctuation has notbeen measured it is not possible to address which couples ofresonant surfaces are involved in a TAE generation. How-ever, considering that the observed polarization points to-wards an edge resonant Alfvénic mode and that the poloidalcorrelations give a negative value form, the most suitedcouple that can give rise to TAE consistent with the obser-vations is the couplesm=0,−1;n=−30d.

Toroidicity corrections open a gap in the spectrumaround a frequencyfTAE=vTAE/2p.450 kHz, a value nearthe observed high frequency peak. The gap amplitude isfound to bedfTAE.100 kHz which implicitly gives a theo-retical uncertainty in the TAE-mode frequency.

Nevertheless it is worth noting that Extrap-T2R mag-netic field and density profiles are such that the gap fre-quency fTAE.Bu /4prÎm0r is almost independent from thechosen couple of rational surfaces. That is different couplesof rational surfaces withsuDmu=1d have similar gap frequen-cies but localized at different radial positions.

A. Frequency scaling

In order to test the Alfvénic nature of the high frequencypeak and the validity of the model, the time evolution ofTAE-mode frequency has been computed using the modeland compared with the one observed in the data.

FIG. 3. sDischarge 13 915.d Graphsad, Sskf , fd spectrum for theB2 compo-

nent. Graphsbd Sskf , fd spectrum for theB3 component.

FIG. 4. The Alfvén continuous spectrum for the discharge 13 915. Dashedlines are solutions for the cylindrical approximation. Full lines are obtainedincluding toroidicity corrections. Hydrogen ion-cyclotronfhi has been super-imposed as full line too.



The local densityn0 has been estimated from ion satura-tion current signalIs which, according to the Langmuir probetheory,25 is given by

Is = An0vse,

whereA is a constant related to the ions collecting area of theprobe andvs=ÎsTe+Ti /mid is the sound speed velocity. Inparticular, quasineutralityne<ni <n0 has been assumed andan average temperature valueTisad<Tesad<kTesadl=10 eV, measured in previous works,26 has been used.

The time evolution of the TAE-mode frequency has beenestimated by inserting in Eq.s1d the density signaln0 low-pass filteredsf* =50 kHz being the cutoff frequencyd, themagnetic profiles from the PFM modelsRef. 22d and bylooking for them=0,−1 degeneracy at each time step.

Though it is not possible to predict the exact value of theTAE gap frequency in this way, its time evolution can becompared with the one of the measured high frequency fluc-tuation in order to verify whether the mode displays anAlfvénic-like scalingffstd~Bstd /Îministdg with time or not.In Fig. 5 the model’s prediction for the discharge 13 915 has

been superimposed on theB3 spectrogram and a good agree-ment between the two is found.

The important role played by local density fluctuationsin establishing the mode frequency evolution with time hasbeen confirmed by gas puffing experiments. In Fig. 6scd asudden decrease of mode frequencysdischarge 13 978d, con-sistent with the model’s prediction obtained by including lo-cal density fluctuations, can be observed. The frequency drop

FIG. 5. sColor onlined. sDischarge 13 915.d The modelprediction for the time evolution of the TAE frequencyis plotted as white full line over the spectrogram of the

B3 component.

FIG. 6. sColor onlined. sDischarge13 978.d Graphsad, the time evolutionof the edge density measured by theLangmuir probe. Graphsbd, Ha emis-sivity signal measured at the edge nearthe magnetic probe. Graphscd, spec-

trogram of the B3 component andmodel prediction for TAE frequencyevolution superimposed as white fullline.



occurs when the gas puffing starts. In Fig. 6sbd theHa signal,measured in a toroidal position near to the magnetic probes,shows the time occurrence and the duration of gas puffing,whereas the local density measured by the Langmuir probeincreases when gas puffing starts, Fig. 6sad.

B. Drive mechanism: Hints

In the following some qualitative considerations aboutthe drive and damping of the mode are given. The commonunderstanding about the excitation of TAE modes is that theyare driven by fast ions with velocities in the range of theAlfvén velocity.

TAEs can tap free energy from particle source via theparticle diamagnetic drifts ifv* =sk' /eB0ds1/ndusd/drdnkTuùvTAE. In tokamaks such a condition is never satisfied bythermal plasma particles because of the strength of the mag-netic field; thus fast ions and are needed. The modes rotatethen in the ion diamagnetic drift direction. However, therehas also been observations of TAE modes in tokamaks whichwere not driven by fast ions and rotate in the electron dia-magnetic drift direction.27–29These modes appear to occur inthe edge region and the excitation can be explained by cou-pling to short wavelength electromagnetic drift Alfvén turbu-lence but inverse energy cascade process has been claimed inthat case.7,30

In RFPs the situation is rather different than in tokamaksas the magnetic field is lower and the linear coupling condi-tion with the drift wave above mentioned can be satisfied atthe edge where the pressure gradients are higher.

In our experiment the observed high frequency Alfvénicfluctuation has a toroidal phase velocity of

vf = vTAER

n,

which is negative forn,0 and in the same direction of theedge electron diamagnetic drift. The drift wave which movesin the electron diamagnetic direction has a frequency givenby

v* =k'

eB0

1

neU d

drnekTeU ,

and a toroidal phase velocity of

vD,f =B0

Bu

1

eB0neS d

drnekTeD .

For coupling, the frequencies and toroidal phase veloci-ties should be equal for the two waves, which yields a rela-tion betweenn andk',

n

R= k'

B0

Bu

,

and givesk'<kf at the edge asBusad.B0. Substituting forExtrap-T2R electron density and temperature profiles mea-sured during low current discharges in previous works23,26 ithas been found that the electron drift frequency and the tor-oidal drift velocity can be of the same order of magnitude orhigher than the mode’s ones at the edgesr /aù0.95d. Alter-

natively it can be said that the modes which are suited to bedestabilized by the electron pressure profile at the edge ofExtrap-T2R are high-n modes withunuù30 consistently withthe high-n numbers observed in the experiment.

Regarding the possible damping mechanisms it is worthreminding that the gap amplitude obtained from the model isdfTAE.100 kHz. Thus the large aspect ratio of T2R deviceprevents the gap from being too large, avoiding the strongdamping due to finite frequencyfTAE/ fhi effects. Neverthe-less the electron landau damping is expected to be strong asvA.vTe

.ÎTe/me at Extrap-T2R edge.However, other drive and damping mechanisms should

be taken into account to compute the right stability thresholdfor the TAE mode. For instance, it has to be considered thepossibility for the mode to be driven by an anisotropic ion orelectron distribution function, sideband excitation mecha-nisms and nonlinear transfer of energy to the relevant spa-tiotemporal scales, the latter being very attractive as directlylinked to the well-known intermittency phenomena in theRFP edge turbulence.31,32Among the damping mechanisms,the ion Landau damping, the coupling of the mode with thecontinuous spectrum,14,13 and also the possibleE3B sup-pression mechanism must be estimated at the edge of anRFP. Analytical and numerical work is in progress to com-pute the mode eigenfunction and the right stability thresholdof the mode.

V. CONCLUSIONS

Concluding, a high frequency–high-n mode whichshows the proper time scaling, polarization, and phase prop-erties of an edge resonant TAE eigenmode has been observedin the Extrap-T2R reversed field pinch device. The modeldeveloped here has a general validity and the same kind ofphenomenon is expected to be present in the other RFP de-vices such as RFX or MST depending on the discharge pa-rameters. An important role is played by the local mass den-sity. According to its value the TAE frequency can be verylow and as a consequence be hidden by tearing mode rota-tion frequency or vice versa can be very high and near to theion-cyclotron thus suffering a strong kinetic damping. Gaspuffing experiments can certainly give a help in resolvingthese problems as they are supposed to change both massdensity and plasma profiles,23 opening the possibility to ob-serve TAE modes in a clear undamped way. Furthermore theobservation of an Alfvénic nature of high frequency mag-netic fluctuations certainly gives insights to the investigationof RFP edge turbulence and related transport phenomena.

ACKNOWLEDGMENT

The authors would like to thank Dr. P. Brunsell for thehelpful discussions on this subject.

APPENDIX: THE TAE GAP STRUCTURE

The formal expression of the toroidal coupling param-eter e0 is derived here for an axisymmetric shifted toroidalequilibrium with nested flux surfaces and large aspect ratio



sa/R0!1d. According to Ref. 16, the basic equation forshear Alfvén waves in such an equilibrium can be derivedfrom the quasineutrality condition

= · dJ = = ·J' + B · =dJi

B= 0, sA1d

wheredJ is the perturbed density current.In particular, the gap structure is given by balancing field

line bending and inertia near the Alfvén resonance, definedby Eq. s2d.

For n@1 each harmonic of the TAE wave is stronglylocalized around theq=s2m±1d /2n surface and the radialwidth of the TAE eigenfunction isDrTAE<e0/nq8.16 Thisconsideration allows to get a simplified version of the Alfvénwave equation in the gap region. Indeed near the Alfvénresonance,¹'

2 df.u¹ r u2]r2df and the parallel Ampere’s law

gives the following expression fordJi:

dJi = −1

m0='

2 dAi. sA2d

Here dAi is the parallel component of the perturbed vectorpotential and can be expressed, in terms of the scalar poten-tial perturbation by the use of the parallel Ohm’s lawdEi

=0,

b · = df +]dAi

]t= 0. sA3d

The perpendicular current perturbation is given by the per-pendicular force balance and, in the gap region, it reduces tothe polarization current,16,13

dJ' = −nemi

B2 ='

]df

]t. sA4d

Substituting Eqs.sA2d–sA4d into ]t= ·dJ=0 and elimi-natingdAi for df, the shear Alfvén wave equation near theresonance is obtained,

Bb · = F u = r u2

Bb · =

]2df

]r2 G +v2

vA2 u = r u2

]2df

]r2 = 0, sA5d

where vA=B/Îm0nemi and a time dependence of the kinddf,exp−ivt has been assumed. EquationsA5d is generallyvalid and assumes only the existence of nested flux surfacesfor the definition of r as radial-like flux variable, i.e.,B ·= r =0. In order to compute the TAE gap structure it isnecessary to choose a coordinate system.

A toroidal coordinate systemscp,q ,wd can be adoptedand the magnetic field written as

Bscp,q,wd = Fscpd = w + = w 3 = cp, sA6d

where cp is the poloidal flux which is a function of theradial-like flux variabler, q is the poloidal angle, andw isthe ignorable toroidal angle on which the axisymmetric tor-oidal equilibrium does not depend. Hence the variablessr ,q ,wd are defined as

R= R0 + r cosq − Dsrd, sA7d

Z = r sinq, sA8d

w = w, sA9d

whereR0 is the major radius andDsrd is the Shafranov shiftwhich, in the large aspect ratio approximation, is of orderOsbd<Osr2/R0

2d.33 Expanding terms up to orderOsr /R0d itis straightforward to show that

=r . s1 + D8 cosqdsR cosq + Z sinqd, sA10d

r = q . s1 + D8 cosqdfZscosq − D8d − R sinqg,

sA11d

=w .1

Rw. sA12d

Here D8 is the radial derivative of the Shafranov shift

whereassR ,Z ,wd are the unit vectors of thesR,Z,wd cylin-drical coordinates. Nevertheless Eq.sA5d and, in particular,

the operatorb ·= has a complex expression if such a coordi-nate system is chosen. It is found to be more convenient forour purpose to transform to the coordinatesscp,x ,jd withx=q+ xsq ,cpd andj=w−nsq ,cpd in which the equilibriumfield lines are straight. Bothxsq ,cpd and nsq ,cpd are as-sumed to be periodic functions ofq and of orderOsr /R0d.The Jacobian of the transformation is easily verified to be, upto the first order inr /R0,

J = s=cp 3 = x · = jd−1 =s=cp 3 = q · = wd−1

S1 +]x

]qD

<rR0

dcp/dr

S1 +r

R0cosq − D8 cosqDS1 +

]x

]qD . sA13d

The condition for straight field lines requiresq=sB ·=jd / sB ·=xd to be a flux functionq=qsrd, that is,

B · = j

B · = x= qsrd = −

]n

]x+

B · = w

B · = x=

FJ

R2 −]n

]x, sA14d

and such a condition is satisfied by choosing

]n

]x= − qS r

R0+ D8Dcosq − q

]x

]q. sA15d

Considering that]n /]x<]n /]qf1−s]x /]qdg to the firstorder in r /R0, integration of Eq.sA15d gives a relation be-tween the anglesnsq ,cpd and xsq ,cpd,

n = − qS r

R0+ D8Dsinq − qx. sA16d

Therefore the new coordinates are clearly defined oncexsq ,cpd is chosen. In this coordinate system the operator

b ·= is



b · = =1

JBS ]

]x+ q

]

]jD . sA17d

Substituting this expression in Eq.sA5d it is found that anappropriate choice ofxsq ,cpd is the one that makes the ratiou¹ r u2/JB2 to be a flux function itself.

By the use of Eqs.sA6d–sA13d it is derived that such acondition is satisfied for

x = −r

R0sinq − D8 sinq

1 + 3q2R02

r

1 + q2R02

r

, sA18d

and in such a case the Alfvén wave equationsA5d is writtenas

S ]

]x+ q

]

]jD2S ]2df

]r2 D +v2

vA2 J2B2S ]2df

]r2 D = 0. sA19d

Substituting forJ the expression given by Eq.sA13d it iseasily found that

v

vAJB=

v

vA0

JB0

=v

vA0

SqR0

2

dB0r2

dcD

3S1 −r

R0cosq + D8 cosq +

]x

]qD , sA20d

whereB0 is the magnetic field on axis,vA0=B0/Îm0nimi, and

c is the total magnetic flux. Hence Eq.sA19d can be formallyreduced to the Mathieu equation16

U ]2F

]x2 Uj

+ V2s1 + 2e0 cosxdF = 0 sA21d

by recognizing that, at constantz=j−qx,

S ]

]x+ q

]

]jD2

=U ]2

]x2Uj

sA22d

and by setting F=s]2dfd /]r2, V=sv /vA0dfsqR0/

2dsdB0r2/dcdg and

S1 −r

R0cosq + D8 cosq +

]x

]qD2

= 1 + 2e0 cosx.

sA23d

The Mathieu equationsA21d admits infinite pairs of realvalues V which correspond to exponentially decaying/growing solutions and define gaps between the continuumbands.16 The gap amplitude is finite fore0Þ0. Assuminge0

to be of orderr /R0 and cosx<cosq, the expression fore0 isfound to be

e0 =r

R0− D8 −

1

cosq

]x

]q= 2

r

R0+ 2D81 q2

q2 +r2

R022 . sA24d

In particular, in the limit ofq2,1@ r2/R02 which corre-

sponds to the tokamak core approximation the standard ex-pressione0,2fsr /R0d+D8g is obtained. Nevertheless whenq2& r2/R0

2, which is the case of the RFPs edge, the limitexpression fore0 is

e0 , 2r

R0. sA25d

This demonstrates that torodicity opens finite amplitudegaps in the Alfvén continuous spectrum of RFP plasmas. TheexpressionsA25d for e0 has to be inserted into Eq.s1d inorder to account for the right toroidicity corrections on theAlfvén continuous spectrum in RFPs edge.

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quencyf / fhi effects and takes the formv2=ki,m2 srdvA

2srdgsrdf1−sv2/vhi2 dg

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