Stable ellipticity-induced Alfven eigenmodes in the Joint European Torus

An external antenna excites stable eigenmodes in elongated Ohmically heated plasmas in the Joint European Torus (cid:126) JET (cid:33) (cid:64) P.-H. Rebut, R. J. Bickerton, and B. E. Keen, Nucl. Fusion 25 , 1011 (cid:126) 1985 (cid:33)(cid:35) . The frequency of the modes (240 (cid:50) 290 kHz (cid:33) falls in the gap in the magnetohydrodynamic (cid:126) MHD (cid:33) continuum that is produced by ellipticity. Some modes are very weakly damped ( (cid:103) / (cid:118) (cid:44) 10 (cid:50) 3 ). © 1997 Physics. (cid:64) S1070-664X (cid:126) 97 (cid:33) 01210-X (cid:35)


I. INTRODUCTION
In a cylinder, the spectrum of Alfvén waves is continuous in the ideal magnetohydrodynamic ͑MHD͒ model. In tokamaks, departures from cylindrical symmetry create gaps in the continuum. MHD theory predicts Alfvén eigenmodes ͑AE͒ in the gaps produced by beta ͑BAE͒, 1 toroidicity ͑TAE͒, 2 and ellipticity ͑EAE͒. 3 Unstable modes with frequencies similar to the expected BAE 4 and TAE 4,5 frequencies were first observed during neutral-beam heating. A possible EAE driven by beam ions was also reported. 1 In recent months, tail ions that are accelerated by ion cyclotron waves have destabilized EAE in the Joint European Torus ͑JET͒ and in the Japan Atomic Energy Research Institute Tokamak-60 Upgrade ͑JT-60U͒. 6 Measurements of unstable, fast-ion driven instabilities are complemented by studies of stable modes. In JET, an external antenna excited stable TAE 7 and kinetic AE. 8 The first observation of stable EAE was also briefly reported. 9 This paper further documents the identification of the mode as an EAE. In addition, the first systematic measurements of the damping rate are presented and initial comparisons with theoretical models are given.

II. EXPERIMENT
The modes are excited by passing ϳ5 A through two saddle coils on the bottom of JET. 9 For the experiments reported here, the antenna phasing is adjusted to excite predominantly modes with toroidal mode numbers n of Ϯ2. The excitation frequency is swept ͑typically between 150 and 300 kHz͒ and the driven response of the plasma is extracted from background noise using a set of synchronous detectors that provide the real and imaginary components of the signal. For these experiments, data from twelve electron cyclotron emission ͑ECE͒ radiometer signals, 10 four ordinary-mode reflectometer signals, 11 and a toroidal array of nine magnetic probes are archived.
The detector response to the antenna current can be described as a transfer function H(). An EAE resonance in the transfer function is shown in Fig. 1, corresponding to a mode at the frequency of 262 kHz. In the complex plane, the magnetic probe signal encircles the pole at pϭi 0 ϩ␥, where 0 ϭ2 f exp is the ͑real͒ resonant frequency and ␥ is the ͑imaginary͒ damping rate. The data from a complete set of diagnostic signals ͕x i ͖ are analyzed by simultaneously fitting the measured transfer functions H(,x i ) to a rational fraction, HϭB/A, where B and A are complex polynomials. 9 The denominator A is assumed the same for all the signals and determines the characteristics of the resonance. Here A is chosen to be of the second order to describe a single resonance. The numerators B(,x i ) ͑chosen of 5th order in this case to account for direct coupling between the antenna and the detectors͒ are proportional to the strength of the response. In particular, for the ECE measurements, the residues B are related to the wave amplitude as a function of space, i.e., the radial eigenfunction. For the measurements presented here, the ECE and reflectometer signals are relatively weak, so the magnetic probe data govern the determination of f exp and ␥.
Determination of the MHD gap structure requires knowledge of the profiles of safety factor q and mass density . The q profile is calculated by the equilibrium reconstruction code EFIT, 12 using magnetics data and the sawtooth inversion radius ͑from ECE͒ as input to the code. The mass density is inferred from measurements of the electron density by six interferometer chords 13  in these deuterium plasmas with few high-Z impurities and little hydrogen the mass density is approximately Ӎ2m p n e , where m p is the proton mass. Systematic uncertainties in the data contribute more to the uncertainty in the calculated gap structure than random errors. At the plasma edge, the density inferred from interferometric measurements can differ by as much as 50% from Thomson scattering measurements, yielding a ϳ20% variation in the predicted frequency. At the center, uncertainty in the q profile typically generates ϳ10% uncertainty in the continuum frequency. The uncertainty in the local magnetic shear is particularly large (ϳ50% in the plasma interior͒. Corrections associated with the Doppler shift are negligible (ϳ1Ϫ2 kHz͒. The measured frequency f exp generally lies in the computed ellipticity-induced gap in the Alfvén continuum ͑Fig. 2͒. The center of the EAE gap occurs at a frequency of f EAE ϭv A /2qR, 3 where v A is the Alfvén speed. In Fig. 3, all of the measurements of f exp are compared with f EAE at sӍ0.95, f edge . In 80% of the cases, the measured frequency lies in the computed EAE gap; in the remaining cases, f exp is from 1-9% higher than the calculated continuum at the upper edge of the EAE gap, but this is within the estimated uncertainty of the calculated value. The correlation of f exp with f edge (rϭ0.53) is stronger than the correlation with the EAE frequency at sϭ0.5, f middle . Averaging over the data, the ratio f exp / f edge ϭ1.06Ϯ0.  Fig. 1 to the nϭ2 Alfvén continuum as calculated by the CSCAS code. 15 The frequency falls in the gap associated with ellipticity ͑EAE͒; the toroidicity-induced gap is also shown ͑TAE͒. The radial coordinate is the square root of the normalized poloidal flux sϭͱ(⌿Ϫ⌿ 0 )/(⌿ 1 Ϫ⌿ 0 ). ͑Fig. 3͒. This lower-frequency mode falls in the gap created by toroidicity and is therefore identified as a TAE. 7 Further confirmation that the observed resonances are global Alfvén eigenmodes is obtained from the ECE measurements ͑Fig. 4͒. Although the signals are too weak to obtain an accurate profile of the radial eigenfunction, the observation of measurable residues on several detectors confirms that the eigenfunction is globally extended, as expected for an EAE. For the case shown in Fig. 4, the ECE signal is largest where the frequency of the eigenmode intersects the mϭ3 Alfvén continuum in the middle of the plasma. Calcu-lations of the expected eigenfunction with the CASTOR 16 and PENN 17 codes also predict a large amplitude near this intersection point ͑Fig. 4͒; however, the radius of the largest ECE signal does not coincide with the radius of the continuum crossing for all the modes in our EAE database.
The measured damping rates ͑Fig. 5͒ vary considerably, from values as low as ␥/ϭ8ϫ10 Ϫ4 to values as large as 3%. The dependence of ␥/ on plasma parameters is complicated. Even during nominally steady-state conditions in the same discharge, ␥/ can double on successive frequency sweeps. For our dataset, the correlation of ␥/ with v A , n e (0), a, , ␦, q 95 , I p , and T e is weak (r 2 Ͻ0.22); the correlation with the magnetic shear at sϭ0.50, s 50 , and with the shear at the edge, s 95 , is also weak. For the weakly damped modes (␥/Ͻ1%), the strongest correlation in the dataset is with the toroidal field (rϭϪ0.72). This dependence may reflect an underlying dependence of the damping rate on the gyroradius. ͓The correlation with ͱT e (0)/B 2 is weaker, however.͔ No correlation with the nonideal parameter 18 ϰs 95 q 95 ͱT e /B T is observed.

III. THEORY
Possible EAE damping mechanisms include trapped electron collisional absorption, 19,20 continuum damping, 21,22 radiative damping, 18 and, more generally, Landau damping through mode conversion. 23 ͑Ion Landau damping 24 should be negligible in these Ohmically-heated discharges.͒ A formalism for calculating the expected damping rate associated with electron collisional and radiative damping in realistic geometry was developed by Mett et al. 18 The theory only treats the interaction of a single pair of poloidal harmonics. This ''high-n'' assumption is of dubious validity for the n ϭ2 modes considered here, 25 although the theory did successfully predict the stability threshold of nϭ4 TAE modes in DIII-D ͑to within a factor of two͒. 18 We have applied this theory to our data. The frequencies at the top and bottom of . The error bars are derived from the covariance matrix of the fitting routine. 9 ͑b͒ Poloidal decomposition of the radial magnetic field Ќ calculated by CASTOR. 16 The eigenfunction is multiplied by the derivative of the electron temperature since the expected ECE fluctuation is -ٌT e . ͑c͒ Binormal electric field calculated by the kinetic version of PENN 17 multiplied by ٌT e . ͑The binormal component of E is approximately proportional to Ќ .) In ͑b͒ and ͑c͒, only the largest amplitude harmonics are shown: mϭ1 ͑solid͒, 2 ͑dash͒, 3 ͑dash-dotted͒, 4 ͑long dash͒, 5 ͑solid͒, 6 ͑dot͒, 7 ͑dash͒. the gap top and bottom are obtained from the calculations of the gap structure. Two different radial locations are selected for this evaluation: near sϭ0.95 and at the gap adjacent to the interior continuum crossing ͑for example, for the case shown in Fig. 2, the continuum frequencies are measured at sϭ0.66). The results of this analysis are shown in Fig. 5 for the interior gaps. The results for sϭ0.95 are similar. Clearly, this simple theory cannot explain the observations. On the other hand, the predictions are of the right order of magnitude, so it is possible that electron collisional and radiative damping are important damping mechanisms.
Comparisons that properly treat the mode structure are computationally expensive, so only a single discharge with both an EAE and a TAE resonance is analyzed in this study. Initially, the PENN code 17 found eigenmodes at frequencies that are consistent with the experimental values, but the predicted damping of the EAE exceeded the experimental value (␥/ϭ0.14Ϯ0.06%) by a factor of 5-10. A numerical convergence study performed a posteriori showed that higher numerical resolution was in fact required to represent correctly the mode coupling occurring in the plasma core (s Ͻ0.2); using a densified mesh with 96, 128, or 192 radial mesh points finally yielded a theoretically converged value (0.26Ϯ0.04%) which is in acceptable agreement with the experimental measurement. Initial calculations with the CASTOR code 16 correctly predicted the frequency of the TAE, but the frequency of the computed EAE was only ϳ80% of the experimental value. Judicious reduction of the density near the edge by 20% ͑which is within experimental uncertainties͒ yielded satisfactory agreement with the measured frequencies; however, the predicted damping (ϳ1%) still exceeded the experimental value. The CASTOR damping prediction is large because the computed EAE singularity occurs within a dominant poloidal harmonic. Further tailoring of the profile to shift the location of the singularity could yield a smaller damping rate.

IV. CONCLUSION
Eigenmodes with frequencies that lie in the ellipticityinduced gap in the Alfvén continuum are observed in JET. The damping of these EAE span from ␥/Շ10 Ϫ3 to values Շ0.1, i.e., the same range as the TAE. 8 Although the predicted destabilizing term produced by energetic ions is a factor of two smaller for the EAE, 24 the measured damping rates vary by orders of magnitude, so the EAE could prove dangerous in a reactor. The damping seems to depend sensitively on subtle details of the plasma profiles, thus making stability projections problematic.