The toroidicity-induced Alfven eigenmode structure in DIII-D: Implications of soft x-ray and beam-ion loss data

The internal structure of the toroidicity-induced Alfve´n eigenmode ~ TAE ! is studied by comparing soft x-ray proﬁle and beam ion loss data taken during TAE activity in the DIII-D tokamak @ W. W. Heidbrink et al. , Nucl. Fusion 37 , 1411 ~ 1997 !# with predictions from theories based on ideal magnetohydrodynamic ~ MHD ! , gyroﬂuid, and gyrokinetic models. The soft x-ray measurements indicate a centrally peaked eigenfunction, a feature which is closest to the gyrokinetic model’s prediction. The beam ion losses are simulated using a guiding center code. In the simulations, the TAE eigenfunction calculated using the ideal MHD model acts as a perturbation to the equilibrium ﬁeld. The predicted beam ion losses are an order of magnitude less than the observed ; 6%–8% losses at the peak experimental amplitude of d B r / B 0 . 2–5 3 10 2 4 . © 2001 American Institute of


I. INTRODUCTION
Toroidicity-induced Alfvén eigenmodes ͑TAEs͒ are fastion driven instabilities that can cause energetic particle losses in tokamaks. These losses are a concern for future devices with reactor-like plasmas since losses significantly degrade performance and escaping particles may cause damage to the vacuum vessel and to plasma-facing components. 1 Theoretical work on the TAE is extensive. Initial papers noted the existence of a spectral gap in the ideal magnetohydrodynamic ͑MHD͒ continuum 2 that supported an eigenmode 3 and pointed out that fast ions could destabilize the mode. 3,4 After the instability was observed experimentally 5,6 work intensified on understanding the stability properties of the mode. Inclusion of damping effects associated with the coupling to kinetic Alfvén waves [7][8][9] gave better agreement with the experimentally observed stability thresholds. These treatments of ''continuum'' and ''radiative'' damping employed the global MHD eigenfunction ͑outside of narrow damping layers͒. Subsequent papers stressed modifications of the eigenfunction by kinetic effects. The eigenfunction is altered by coupling to other plasma waves, 10-12 by energetic particles 11,[13][14][15][16][17] and by sheared rotation profiles. 18 Experimentally, there have been many studies of various properties of the TAE ͑Ref. 19͒ but accurate measurements of the TAE eigenfunction are rare. At the Tokamak Fusion Test Reactor ͑TFTR͒, beam emission spectroscopy and reflectometry were used to make fluctuation measurements. In the former measurement, the poloidal mode number derived from beam emission spectroscopy was consistent with predictions of ideal MHD to within ϳ50% uncertainties. 20 In the reflectometry measurements, Nazikian et al. measured the radial structure of alpha particle driven TAEs over several shots with similar Mirnov activity. 21 The structure of an nϭ4 TAE had qualitative agreement with a prediction from an ideal MHD model but an nϭ2 TAE which coexisted with the other mode had a core-localized antiballooning structure inconsistent with theory. ͑n is the toroidal mode number and m is the poloidal mode number.͒ Reflectometry measurements of an nϭ1 TAE on JT-60U found qualitative agree-a͒ Electronic mail: wwheidbr@uci.edu ment with the expected ballooning structure. 22 In DIII-D, a poloidal array of magnetic probes was used to measure fluctuations during TAE activity. 23 Although the amplitude of the fluctuations did not agree with predictions from ideal and resistive MHD models, it compared fairly well with the predictions of a model retaining Landau damping and finite Larmor radius ͑FLR͒ effects. For this model, the comparison yielded reduced 2 's of 1.2 and 1.6, respectively, for two different toroidal mode numbers. On the W7-AS stellarator, an x-ray imaging system consisting of 200-400 sightlines was used to construct a three-dimensional rendering of the TAE, 24 yielding the most detailed picture of the internal TAE mode structure to date. So far, the data have not been compared with theory.
An alternative method of testing theoretically predicted eigenfunctions is to use particle simulations. By following fast ion trajectories in the presence of a perturbation to the equilibrium field, particle transport with different TAE eigenfunctions can be calculated and those losses compared with observations. A study of this type was done in the Poloidal Divertor Experiment for fishbones, an MHD mode with a simpler structure. The model eigenfunction in that case successfully predicted anomalous 3 He losses over a range of mode amplitudes. 25 Similar comparisons have been performed for the sawtooth instability 26,27 and for tearing modes. 28,29 For the TAE, particle simulation studies have focused on the mode's interaction with alpha particles. Particle simulations have been used to investigate the effect of TAE modes on alpha particle transport in the International Tokamak Engineering Reactor ͑ITER͒, 30 the nonlinear saturation of the instability due to alpha particle redistribution, 31 the expected alpha particle diffusion from observed TAEs in the Joint European Torus ͑JET͒, 32 and the effect of the TAE on fast particles in spherical tokamaks. 17 Unfortunately, in deuterium-tritium ͑D-T͒ experiments, no anomalous alpha particle losses were observed in TFTR ͑Ref. 33͒ and no alpha-driven TAEs were detected in JET ͑Ref. 34͒ D-T experiments, so comparisons between particle simulations and experiment were impossible.
Verification of the theoretical models for the TAE with experimental evidence is necessary in order to have confidence in our ability to use these models to predict highfrequency Alfvén activity in future devices. The investigation reported in this paper is part of the effort to understand the details of the TAE mode structure. In this study, we take two sets of data acquired during a DIII-D discharge with TAE activity ͑shot 71524͒ and compare them with predictions based on several theoretical models.
The first set of data consists of soft x-ray measurements from two arrays ͑one array has a vertical view of the plasma while the other has a horizontal view͒ on DIII-D during TAE activity. These data depend directly on the mode's spatial structure. We compare them with the expected soft x-ray emission from the nϭ5 linear eigenfunctions generated by codes based on ideal MHD, gyrofluid and gyrokinetic models. The models and the discharge are described in detail in Sec. II while specifics of the comparison and the results are given in Sec. III. The data show an eigenfunction with a strongly peaked amplitude in the center, similar to the struc-ture implied by measurements from a poloidal array of Mirnov coils. 23 The gyrokinetic prediction compares well with the data qualitatively but quantitative discrepancies exist, although these may by accounted for by uncertainties in the modeling.
The second set of data is the 2.5 MeV neutron signal during TAE bursts. Drops in this signal imply significant losses of beam ions. To properly make this comparison, we use a guiding center code to simulate fast ion motion in the presence of TAE activity, similar to the work done for fishbones on PDX. 25 Here, the TAE eigenfunction is generated using the ideal MHD model and introduced in the code as a time dependent perturbation to the equilibrium. Particle losses over a range of amplitudes are calculated and compared with losses in the range of experimental mode amplitudes ͑Sec. IV͒. The particle simulations predict only a fraction of the losses measured in the experiment.
A discussion on possible reasons why the data do not compare well with theoretical predictions, and the consequences, is presented in Sec. V. Conclusions from this study are summarized in Sec. VI.

II. THEORETICAL MODELING
For this study we select a well-documented DIII-D discharge, shot 71524. This discharge is a low field ͑0.8 T͒, inner-wall limited deuterium plasma of moderate elongation ͑1.6͒ that is heated by 5 MW of deuterium neutral beams. The flux surfaces and kinetic profiles appear in Figs. 3-5 of Ref. 35 and the time evolution in Refs. 36 and 37. The TAE activity occurs in ϳ1 ms bursts separated by ϳ8 ms; the cycle resembles those shown in Ref. 38 and is caused by beam-ion losses. ͑The escaping particles are observed to reach the vessel wall.͒ Each burst contains a ''cluster'' of several toroidal modes with mode numbers n ranging from 1 to 9. 35 The Fourier spectrum and the Alfven gap structure 39 for one of the bursts are shown in Fig. 1. The splitting of the peaks in the spectrum is caused by the Doppler shift although for some of the bursts each peak in the spectrum contains more than one toroidal mode. 36 If one assumes that different toroidal modes are excited in the same region in the plasma, the spectral data indicate that the mode is excited near the qϭ1 surface ͓Fig. 1͑b͔͒. The inferred frequency in the plasma frame is near the bottom of the toroidicityinduced gap and may actually lie in the continuum. The dominant toroidal mode numbers in the TAE feature are usually nϭ3 -5. Although fishbone activity is also present ͓Fig. 1͑a͔͒, the TAEs are primarily responsible for beam-ion transport. 40 The drop in the neutron emission at the burst analyzed in Sec. IV indicates that an ϳ8% loss of the injected beam particles occurs.
Three separate wave codes that feature different physical models are used to calculate the linear response of the plasma. ͑Only one of the codes is used for the particle simulation analysis, see Sec. IV.͒ None of the models takes into account the effect of the energetic ions on the mode structure. In the calculations, all three codes use a numerical equilibrium that is based on the experimental equilibrium fitting code EFIT. 41 The NOVA code 42 solves the linearized ideal MHD equations using cubic B spline finite elements and a Fourier expansion in a general flux coordinate system. The code was updated to include Landau and radiative damping in its stability model ͑the NOVA-K code͒ though the updated version still uses the same ''ideal MHD'' model to solve for the linear eigenfunction. A search for core localized modes for this discharge was conducted using this updated code but none were found, most likely because the core TAE does not exist in plasmas with a large pressure gradient 43 and the experimental pressure gradients are a factor of 2 larger than the critical value. We solve for three unstable eigenfunctions with toroidal mode numbers 3, 4, and 5, all with frequencies of 64 kHz in the plasma frame. The poloidal components ranging from mϭ1 through 15 for the nϭ3 TAE calculated using NOVA are shown in Fig. 2͑a͒. Though not shown, the solutions for the nϭ4 and 5 modes are qualitatively similar to the nϭ3, peaking around p / p,edge Ӎ0.93 but with the mϭ10 and mϭ13 poloidal components being strongest for each n, respectively.
The FAR code 44 is a program that solves initial value problems in full toroidal geometry. A modified version ͑TAE/FL͒ of this code 45 was used to calculate the linear TAE eigenfunction for this burst. This version uses a ''gyrofluid'' model that retains ion and electron Landau damping and thermal ion FLR effects but ignores the fast Alfvén wave. The background plasma is modeled using the reduced MHD equations with a consistent Landau closure in slab geometry to account for the energetic species moment equations ͑retaining the shear Alfvén spectrum and continuum damping͒ FIG. 1. ͑a͒ Cross-power spectrum from a pair of magnetic probes at 1883-1886 ms in discharge 71524. The numbers by the spectral peaks represent the toroidal mode numbers obtained from a toroidal array of eight unequally spaced probes. The low frequency feature with the dominant nϭ1 peak is a fishbone oscillation; the cluster of spectral peaks around 100 kHz is the TAE feature. ͑b͒ Radial profiles of Doppler-shift-corrected frequencies f lab Ϫn f rot for the TAE feature. Here f lab is the measured frequency in the laboratory frame and f rot is the measured toroidal rotation frequency. If one assumes that the different toroidal mode numbers have the same frequency in the plasma frame f pl , the intersection of the curves ͑indicated by the solid dot͒ is f pl . Also shown are the frequency of the center of the TAE gap v A /4qR ͑dashed line͒, the radial location of the qϭ1 surface ͑dashed-dotted line͒, and the envelopes of the Alfvén continuum as calculated by the CONT code ͑Ref. 39͒ ͑dotted lines͒. The CONT code includes plasma pressure terms that create a beta-induced ͑BAE͒ gap and raise the frequencies of the TAE and ellipticity-induced ͑EAE͒ gaps. The harmonics are given as m␣ m since ␦B r ϳm␣ m /qr. ͑a͒ For the NOVA case, solutions for the mϭ1-15 harmonics are provided ͑components with amplitudes Ͻ10% of the peak amplitude are not shown͒ evaluated in a straight-field line coordinate system with a Jacobian JϳB Ϫ2 . ͑b͒ For TAE/FL case, solutions for the mϭ1-15 harmonics are provided ͑components with amplitudes Ͻ5% of the peak amplitude not shown͒ in a coordinate system with JϳR Ϫ2 . Not pictured is the imaginary part of this eigenfunction. ͑c͒-͑d͒ For the PENN case, pictured in ͑c͒ are the mϭ1-4 ͑heavy solid, heavy dashed, light solid, and light dashed trace, respectively͒ while shown in ͑d͒ are the mϭ5-9 harmonics ͑heavy solid, heavy dashed, light solid, light dashed, and very light solid trace, respectively͒. This eigenfunction also has radially dependent phases, nm ϭ nm ( p ), which are not pictured, and was evaluated in the same coordinate system as the NOVA case. and the assumption that the fast ion distribution is Maxwellian. Linearly unstable modes are found by calculating growth rates of possible modes using an expression derived from the gyrofluid equations. The Fourier components of the real part of the eigenfunction from this code are given in Fig. 2͑b͒.
The PENN code 46 solves the Maxwell equations using cubic finite elements in the radial and poloidal directions. An oscillating helical source current is distributed inside the plasma to excite bulk modes. ͑In the experiment, these modes are driven by the energetic beam-ions.͒ Only eigenmodes with sufficiently small damping and wavefields independent of the excitation are regarded as physically meaningful. The linear response of the plasma is defined in terms of a dielectric tensor featuring a resistive plasma model and a kinetic FLR model that is appropriate for higher temperatures. By taking into account the FLR excursions, the drifts of thermal particles and the resonant Landau interactions, this ''gyrokinetic'' model not only is well-suited for the propagation of the fast waves but also models the linear mode conversion to the slower waves such as the kinetic Alfvén, the ion acoustic, and drift waves. Several unstable modes with sufficiently small damping were found; the linear response of the plasma in this discharge was calculated for a 62.5 kHz, nϭ3 mode ͓as shown in Figs. 2͑c͒ and 2͑d͔͒ and for a 71 kHz, nϭ5 mode.
The differences in the predicted mode structure ͑Figs. 2-3͒ are striking. ͑Remember, these predictions are for the same TAE mode.͒ Some of these differences are attributed to numerics ͑both different computational techniques and equilibrium processing͒ and some have their origin in different physical approximations. For example, the ideal MHD eigenfunction calculated by NOVA peaks at Ӎ0.8, while the ideal MHD eigenfunction calculated by the GATO code ͑Fig. 3 of Ref. 47͒ peaks at Ӎ0.6. Despite these differences, the ideal MHD eigenfunctions are qualitatively similar and both lack the large central wavefields predicted by the gyrokinetic code. The dramatic differences in Figs. 2-3 indicate that comparison with internal measurements can provide a sensitive test of the validity of the theoretical treatments.

III. SOFT X-RAY PROFILES
The soft x-ray emissivity S depends on the electron density n e , the density of impurity ions n I , and the electron temperature T e . Since Alfvén modes are low frequency waves, the electron and ion fluids are expected to oscillate with the field, so oscillations in the soft x-ray emissivity s are related to the radial displacement of the field by sϭ•ٌs.
At the time of the experiment, the poloidal soft x-ray array at DIII-D ͑Ref. 48͒ consisted of two sets of silicon diodes. The horizontal camera had 32 channels, while the vertical camera had 16 channels. All of the channels measured the slowly varying ''dc'' soft x-ray emission, but only half of the channels were instrumented to measure highfrequency fluctuations. The frequency response of the instrumented channels was 250 kHz.
The dc emissivity profile s is inferred from inversion of the chordal measurements. Because the data are too sparse for a complete tomographic inversion, s is assumed to be an axisymmetric function of the magnetic flux ͑Fig. 4͒. ͑Actually, in rotating plasmas, centrifugal effects cause an outward shift of the impurity density n I , 49 but we ignore this effect in these plasmas with subsonic carbon impurities.͒ The geometry of the flux surfaces is obtained from the EFIT equilibrium reconstruction. The plasma is first partitioned into a finite number of flux zones then the path length of each soft x-ray chord through each zone is computed. The results are assembled into a response matrix. A vector of the measured chord-averaged signals S x is also constructed. The emissivity profile is then obtained by matrix inversion using singular value decomposition. To check the inversion, the expected chord-averaged signals S are compared with the measured values S x ͑Fig. 5͒. The agreement between S and S x is optimized for 9-11 flux zones ͑with subsequent smoothing of the s profile͒. The biggest discrepancy between the measured profiles and the simulated profiles is in the horizontal camera. For this symmetric plasma that was limited on the inner wall, the simulated profiles are vertically symmetric, but the actual emission from the upper half of the plasma is somewhat larger than the emission from the lower half.
To obtain the amplitude of the fluctuations S x , the fast Fourier transform of the cross power is computed for an n ϭ5 TAE mode with a frequency of 118 kHz in the laboratory frame. ͑The toroidal mode number n is inferred from measurements with a toroidal array of eight unequally spaced magnetic probes.͒ The data with the largest signal to noise ratio are from later in the discharge ͑2.17 s͒ than the time of the theoretical calculations ͑1.875 s͒ but the differ- ences in plasma parameters are small ͑Ͻ10% changes in density, ␤, ␤ f , and central q͒. The uncertainty in the amplitude is estimated from the coherency and from the amplitude at 110 kHz ͑in a ''valley'' between the TAE peaks in spec-trum͒; the two methods yield comparable error estimates. For both the horizontal and vertical cameras, the signals are an order of magnitude larger for the central channels than for the edge channels.
To compare with the theoretical predictions, the calculated perturbation for the nϭ5 mode is converted to an image on a uniform grid ͑Fig. 3͒ and multiplied by the gra-dient of the dc emissivity ٌs. The expected signals are obtained by integrating •ٌs over the detector sightlines. The resulting comparisons are shown in Figs. 6-8. The profile simulated using the NOVA eigenfunction has less variation between edge channels and central channels than the experimental data; also, the predicted signals for the horizontal camera are larger than the predicted signals for the vertical camera, in contradiction to the data. The NOVA eigenfunction peaks strongly near the inner and outer midplane, but the soft x-ray data are inconsistent with strong peaking at the periphery. ͓Use of the ideal MHD eigenfunction computed by GATO ͑Ref. 47͒ yields a similar conclusion.͔ The contrast between edge channels and central channels is greater for the eigenfunction calculated by the gyrofluid code, but is still less than the measured contrast. The contrast between edge and central channels predicted by the PENN code is comparable to the experimental observations. This reflects the relatively large central displacement of the PENN eigenfunction.  Qualitatively, the data agree best with the PENN prediction, but the quantitative agreement is poor for all of the calculated eigenfunctions. The experimental uncertainties are known accurately, but the uncertainties in the theoretical predictions are difficult to quantify. The sensitivity of the predictions to the inversion of the dc emissivity s was studied by using different EFIT equilibria and by varying the order of the inversion matrix. Because the predicted signals are dominated by the edge, the NOVA predictions were sensitive to these variations, but the PENN and gyrofluid predictions were scarcely affected. The sensitivity of the eigenfunction itself was studied three ways. For NOVA, pressure, density, and q profiles at the extremes of the experimental errors were generated and employed in a series of runs, but all of the calculated eigenfunctions peaked strongly at the edge. For the gyrofluid code, the sensitivity to uncertainties in the gap structure was artificially explored by varying the assumed value of v ʈ /v A ; this shifts the location of the peak of the eigenfunction. The differences in simulated signals for v ʈ /v A between 0.3 and 0.85 are comparable to the symbol sizes in Fig. 7. The PENN prediction is sensitive to the position of the qϭ1 surface and to the pressure gradient in this region, which have experimental uncertainties of approximately 10% and 20%, respectively. Estimates of the likely effect of these errors were used to distort the calculated eigenfunction ͑Fig. 3͒, and the effect on the simulated soft x-ray signals was explored. ͑The PENN prediction is too computationally expensive for a rigorous sensitivity study.͒ This sensitivity study suggests that, with judicious adjustment of the input parameters within experimental uncertainties, the PENN eigenfunction could be made consistent with the measured profile.

A. Simulation details
The Hamiltonian guiding-center code ORBIT ͑Ref. 50͒ is used to simulate the effect of TAE modes on beam ion confinement. In ORBIT, an ensemble of particles are ''born'' according to a user-defined probability distribution then the code follows the trajectories of the particles' guiding centers. Particles whose guiding centers cross the plasma boundary before the end of the run are judged lost. The same EFIT equilibrium used with the other codes is adapted for these simulations, with both axisymmetry and up-down symmetry assumed.
Establishing the length for a run to simulate the burst is nontrivial. Data from a typical burst appear in Fig. 9. The TAE mode amplitude evolves rapidly, with the entire burst persisting for longer than a millisecond. However, as in the case shown in Fig. 9, the bulk of the losses generally occur in the 0.1-0.3 ms range when the mode is near its maximum amplitude. In the simulations, the mode amplitude is assumed constant in time and the total run duration is set at ϳ430 s, or about 30 toroidal transits for a deeply passing 75 keV deuteron. ͑The nominal transit time is trans Ӎ4.3 s.͒ The equations of motion are solved at intervals of ϳ trans /300, and the energy change per particle is limited to less than 20%. The run time is sufficiently short compared to any characteristic collision time that no collisional effects are included in the simulations. Beam fueling is also ignored.
For the beam ion distribution function, the pitch ϭv ʈ /v, energy E, poloidal angle , and minor radius along the midplane r of individual particles are randomly generated using a Monte Carlo prescription that ''fits'' the population to a specified probability distribution. The code TRANSP ͑Ref. 51͒ is used to determine the initial distribution of beam ions for this discharge ͑with the additional assumption that the particles are uniformly distributed in ͒. In the TRANSP calculation, profiles of the electron density and temperature were used to calculate the beam deposition. Because we are inferring beam ion confinement from the 2.5 MeV neutron measurement, only particles above the injection half-energy (EϾ38 keV) are considered. The initial particle distribution used in this study is shown in Fig. 10 and is in essence a classical slowing down distribution. Most ions are on cogoing passing orbits. The particle population is calculated in the lab frame, while the simulations are done the plasma frame, but the Doppler shift in particle velocity is neglected since the plasma rotation of ϳ10 kHz is small compared to typical transit frequencies. For all simulations, 10 000 birth particles are initially generated.
In general, an instability can cause spatial transport through both its magnetic field and its electric field. For the magnetic field, the TAE eigenfunction is introduced as an oscillating perturbation to the equilibrium field. In this formalism, the coordinate system is defined by three position variables ͑the poloidal flux p , the poloidal angle , and toroidal angle ͒ and a Jacobian of the form Jϳ1/B 0 2 . 52 ͑B 0 represents the unperturbed equilibrium field.͒ The perturbation ␦B is introduced into the equations of motion using the scalar function ␣ which is related to the perturbation through ␦Bϭٌϫ␣B 0 . The code expects coefficients ␣ nm ( p ) for a Fourier decomposed scalar perturbation ␣( p ,,) ϭ ͚␣ mn ( p )sin(nϪmϩt), where is the frequency of the mode, and t is the time. In the ideal MHD case, NOVA's solution is given as the radial component of the displacement vector r ϭ•(ٌ p /͉ p ͉). To interface properly with ORBIT, r is converted into ␣ using 31 which assumes that r has been Fourier-decomposed in and with Fourier coefficients nm ( p ).
In the ideal MHD model, electrons rapidly short out the parallel electric field, so the electric field is induced by the time-dependent magnetic perturbation. In the plasma frame, the equilibrium electric field is negligible. The perturbed potential 1 is Fourier analyzed and the coefficients are given by The singularity when qϭm/n is a by-product of the ideal MHD model. To smooth the potential inside the magnetic island at the rational surface, the replacement To evaluate the effect of the instability on the neutron rate, we write out the confined particle distribution at the end of each run and calculate the beam-plasma reaction probability n d ͗v͘ b-p of each particle using profiles of deuterium density n d and temperature T d inferred from experimental data. ͑From TRANSP, the neutron signal is dominated by beam-plasma reactions with Ͻ5% contributions from beambeam and thermonuclear reactions.͒ The beam-plasma reactivity is approximately 53 ͗v͘ b-p Ӎ͓1ϩc 1 T d /E͔v, where v is evaluated at the injection energy E and c 1 is a constant that depends on E.

B. Determining the perturbation strength
The available experimental measurements of the mode amplitude are the magnetic perturbation ␦B near the vacuum vessel and the amplitude of the soft x-ray measurements. In this subsection, these measurements are related to the Fourier components ␣ mn employed in the simulation. Our conclusions are summarized in Table I.
The maximum amplitude of ␦B measured at a coil along the outer midplane is ϳ0.6 G; the toroidal field is ϳ0.8 T. The magnetics signal is affected by vessel currents so the equivalent vacuum field is about half as large. To estimate ͉␦B͉ for the NOVA eigenfunction, we use a formula derived from a multipole expansion about the magnetic axis, ͉␦B r,m ͉Ӎ͉␦B ,wall ͉(b/r) mϩ1 , where b is the distance from the magnetic axis to the coil, and rӍsa is the distance from the magnetic axis to the radius of maximum amplitude for the mth poloidal harmonic. For the nϭ3 case, the largest contributions to the field at the coil are from the mϭ6 and mϭ7 poloidal harmonics. The inferred maximum value of FIG. 10. Birth distribution of 10 000 deuterium beam ions generated from a probability distribution calculated by TRANSP using a Monte Carlo prescription. Shown are histograms for the population in terms of ͑a͒ pitch and ͑b͒ energy, as well as the ͑c͒ radial profile of the particle density from TRANSP ͑solid͒ and using a radially flat profile ͑dashed͒. Pitch is defined as vʈ /v. All particles are launched along the outer midplane, ϭ0. the total perturbation is ␦B r /B 0 ϳ2ϫ10 Ϫ4 . In the experiment, several toroidal harmonics are simultaneously excited.
For the NOVA simulations, the maximum values of ␦B r /B 0 inferred for the nϭ3,4,5 toroidal components are in the ratio 1.00:1.11:0.74. The measured mode frequencies in the plasma frame are nearly identical, so a value of 64 kHz is employed for all three toroidal harmonics. An independent estimate of the internal perturbation amplitude is available from the soft x-ray data. The measured fluctuation in signal at the TAE frequency is S x . The displacement r of the flux surface intersected by the soft x-ray chord is r ӍS x /ٌS x , where ٌS x is the gradient of the dc soft x-ray emissivity at the innermost portion of the soft x-ray chord. If the perturbed field is ''tied'' to the fluid, as expected for a shear Alfvén wave, the perturbed field is ␦B r ϭB 0 k ʈ r , where the parallel component of the wave vector k ʈ ϭ1/2qR for a TAE. Thus, ␦B r /B 0 Ӎ r /2qR. The maximum measured value of r for the available chords is ϳ0.4 mm, which implies a maximum perturbation ␦B r /B 0 Ӎ0.7ϫ10 Ϫ4 . Since the available chords may miss the maximum perturbation and the measurements are chord averaged, this value is an approximate lower bound.
To relate the amplitude of the mode used in the code to the measured value of ␦B r , this amplitude needs to be converted into physical units by relating the scalar function ␣ to the perturbation strength ␦B r /B 0 . The equilibrium magnetic field in the ( p ,,) coordinate system can be expressed as 54 B 0 ϭg( p )ٌϩI( p )ٌϩ␦( p ,)ٌ p . In this coordinate system, the Jacobian is given by Jϭ(Iϩgq)/B 0 2 . Assuming that the minor radius rӍͱ2 p /B 0 , we can express the radial component of the perturbation ␦B r in terms of the Fourier decomposed scalar perturbation, ␣ nm , as with F nm ϵ(r/R 0 )͓(Iϩgq)/(nIϩmq)͔ and R 0 as the plasma major radius. Only the peak value is required to make the conversion from code to physical units. For the nϭ3 mode, we evaluate Eq. ͑2͒ for , , and t set to 0 and find the ratio between the peak values of ␦B r /B 0 and ␣ is 7.4ϫ10 Ϫ4 .

C. Simulation results
The simulation results using all three (nϭ3 -5) toroidal components from NOVA are given in Fig. 11. Near the measured peak mode amplitude (␦B r /B 0 Ӎ2ϫ10 Ϫ4 ) and using the TRANSP distribution, particle losses are nonexistent and neutron losses are minuscule. Even at a significantly stronger amplitude (␦B r /B 0 Ӎ5ϫ10 Ϫ3 ), neutron losses only reach ϳ4%, still below the observed 6%-8% drop in the neutron emission. Additional runs are conducted at frequencies close to the observed TAE frequency, i.e., f TAE Ϯ50 kHz at ϳ10 kHz intervals, with the perturbation strength set at the experimental mode amplitude to check for the presence of a transport resonance, but no significant particle losses are seen.
A lower frequency fishbone is present during the burst along with the TAE ͑Fig. 1͒, which may result in a radial flattening of the beam ion distribution profile and lead to enhanced transport. To test this hypothesis, a distribution with an artificially flat profile in minor radius ͓ f beam (r) ϳͱ1Ϫr͔ was used as the birth distribution. Simulations with the flattened radial distribution resulted in a larger particle loss as compared with the TRANSP case ͓Fig. 11͑a͔͒ but little change in the predicted drop in neutron emission ͓Fig. 11͑b͔͒. Therefore, at least for this ideal MHD case, any prior redistribution of the energetic particle population by preceding TAE and fishbone bursts has little effect on the predicted neutron reduction. As before, additional runs about the TAE frequency were conducted with a similarly null result. For the record, particle losses with both distributions scaled linearly in ␦B r over the range of mode amplitudes given in Fig. 11, with correlation coefficients for the least squares fit of 0.999 and 0.996 for the TRANSP and radially flattened cases, respectively.

A. Origin of the discrepancy
Neither the soft x-ray data nor the beam-ion loss measurements agree quantitatively with the predictions made using the ideal MHD, gyrofluid, and gyrokinetic models. This suggests that either the experimental data, the comparisons, or the theoretical models are erroneous. The data are unlikely culprits. The soft x-ray measurements are valid. At other frequencies in the same discharge, normal mϭ3, nϭ2 tearing modes are observed; also, the profile for ''BAEs'' ͑Ref. 55͒ differs considerably from the TAE profile. There is also no question that beam ions are really lost under these conditions, as the neutron data are corroborated by stored energy, foil bolometer, and active charge exchange diagnostics, 40,55 as well as by observations of internal damage to vacuum vessel components. 40 The observation that the mode amplitude is relatively small is well documented as well: when ''chirping modes'' ͑modes with frequencies that change by a factor of 2 in a millisecond͒ cause 8% drops in neutron emission, the magnetic perturbation ␦B is an order of magnitude larger than during TAE activity. 56 In our analyzed discharge ͑71524͒, the soft x-ray fluctuation from the nϭ2 tearing mode was two orders of magnitude larger than the TAE signal.
Systematic errors in the comparisons are a more plausible culprit. There are two main caveats in the soft x-ray comparison. First, because of the coarseness of the array, the dc emissivity s was assumed to be a flux function. In reality, the emissivity, which depends linearly on the distribution of impurities, may exhibit significant poloidal asymmetries. In the comparison, the predicted signals are multiplied by the gradient of the dc emissivity, so the sensitivity to errors in s is appreciable. Second, the reported soft x-ray measurements are from later in the discharge ͑2.17 s͒ than the beam-ion loss data and theoretical calculations ͑1.875 s͒. ͑Because the beam-ion losses caused an influx of impurities, the signal to noise ratio of the soft x-ray data is superior later in the discharge.͒ The plasma parameters at these two times are similar, however, so this is probably not a major source of error.
There are three major caveats in the comparison with the beam-ion loss data. First, it was not feasible to retain all of the poloidal and toroidal harmonics that are excited in the experiment. The largest harmonics were employed in the reported simulations and tests indicate only weak sensitivity ͑Շ50% enhancement of total losses͒ to the inclusion of additional modes of comparable amplitude, but this is a potential source of systematic error. The second caveat is associated with the choice of the beam-ion distribution function. Most of the simulations used the classical beam-ion distribution calculated by TRANSP. In the experiment, the losses occur in repetitive bursts, so prior bursts almost certainly altered the distribution function prior to the analyzed burst. Available measurements suggest anomalous pitch-angle scattering is relatively weak, so these modifications likely consist of a broadening of the profile ͑due to internal transport͒ and a reduction in low-energy beam ions ͑due to prior losses͒. Both effects are expected to increase the calculated losses, although artificial flattening of the profile resulted in only a small change ͑Fig. 11͒. The third caveat is that the bursts had concurrent fishbone activity. Experimentally in DIII-D, 40 for a given amplitude TAE mode, the measured losses are generally higher during combined fishbone and TAE activity than during TAE activity alone, the losses depend much more strongly on the TAE amplitude than on the fishbone amplitude, and losses are negligible ͑Ͻ1%͒ for pure fishbone modes. The simulation with the flat beam-ion profile was intended to mimic the effect of fishbone activity, but conceivably there is a synergistic effect that was not included in our simulations.
The third possibility is that the experimental eigenfunction really differs from the theoretical eigenfunctions. This hypothesis is compatible with prior studies. Although rough consistency between theoretical eigenfunctions and measurements have been reported, [20][21][22]57 the most detailed comparisons found discrepancies. 21,23 An earlier study of discharge 71524 concluded that the predictions of the MHD model disagree in both amplitude and phase with the poloidal distribution of edge magnetic fluctuations, while the gyrokinetic model gives satisfactory agreement for the amplitude but not the phase. 23 Thus, three sets of data-soft x-ray, beam-ion loss, and poloidal magnetic-are all inconsistent with the theoretical predictions for this discharge.
Why might the eigenfunctions be incorrect? It is known from stability studies of discharge 71524 that radiative damping ͑or related effects͒ is needed to account for the observed power threshold. 58,59 Also, to obtain agreement with JET measurements of mode damping it was necessary to include mode coupling to kinetic Alfvén waves. [60][61][62] This effect is not included in the ideal MHD or gyrofluid models and may need further refinements in the PENN treatment. Another possibility is that energetic-particle effects associated with the beam ions modify the eigenfunction. ͑The beam beta is comparable to the thermal beta in this discharge.͒ A final possibility is that the use of linear eigenfunctions does not properly model the experiment. Experimentally, the different toroidal modes tend to lock and rotate together, even though their expected linear frequencies differ. 36

B. Mechanism of beam-ion loss
A TAE mode can transport beam ions out of a tokamak plasma in several ways: ͑a͒ A class of passing particles can circulate with the mode and resonantly drift out of the plasma via an EϫB drift, similar to mode-particle pumping for fishbones. 63 The losses would scale as ␦B r /B 0 and require only a few transits for losses to occur. ͑b͒ Countergoing ions can be resonantly converted into large banana orbits that intersect the wall as the particle interacts with the wave over a short period of time. 30,31 The losses scale as (␦B r /B 0 ) 2 . For significant ͑Ͼ1%͒ losses, however, the time of the run needs to be long compared to the transit time ͑the time it takes for a passing particle to complete one circuit around the torus͒ as particles need to slow down or pitch-angle scatter into this resonant region of phase space. ͑c͒ The particles are born near a loss boundary in phase space and their resonant interaction with the wave causes them to cross that boundary into a loss orbit; 30 alternatively, some particles may be sent into a ripple-trapped orbit before being lost. 64 The overall losses can occur within a few transits of the particle's birth and are significant only for a short period of time. ͑d͒ Finally, multiple harmonics can lead to islands in the particle's phase space which, at sufficiently large perturbations, can overlap and the particle's motion can become stochastic. 65 This mechanism is a threshold effect and occurs once the mode strength creates large enough islands.
Considering the short time in which the losses occur, we can eliminate mechanism ͑b͒ as a possible explanation of the dominant loss mechanism during this burst. In the simulations, all the other mechanisms are explored, except for particles getting nudged into a ripple-trapped orbit ͓as in mechanism ͑c͔͒ since ripple is not included in the calculation. ͑DIII-D has 24 coils so the ripple-trapping region is relatively small.͒ The calculated losses for the MHD eigenfunction scale approximately linearly with ␦B r and the pitch of lost particles tends to be Ͼ0.7, suggesting that resonant convective losses ͓mechanism ͑a͔͒ predominate in these simulations. Since the calculated increase in losses with increasing ␦B r is modest, we can conclude that the simulations never reached a stochastic threshold, as would be the case for mechanism ͑d͒.
Based on linear scaling of the losses with ␦B r and on the ϳ10 cm radial step size of the final loss orbit, it was previously concluded 40 that mechanism ͑a͒ predominates in the experiment. Because the calculated losses are inconsistent with the measurements, the present simulations do not definitively establish the mechanism of beam-ion transport.

VI. CONCLUSIONS
All three codes compute a mode frequency close to the experimentally measured frequency but, due to the different physics assumptions, the predicted eigenfunctions are radically different ͑Figs. 2-3͒. Eigenfunction measurements provide a more stringent test of theoretical models than frequency measurements alone.
Predictions based on the linear MHD eigenfunction are inconsistent with soft x-ray, poloidal magnetic, and beam-ion loss data for this discharge. Apparently, the actual eigenfunction is not as large near the periphery as predicted. The gyrokinetic eigenfunction is in better qualitative agreement with the soft x-ray measurements than the ideal MHD model, but the quantitative agreement is only fair. It is possible that systematic uncertainties account for the discrepancy.
Particle simulations based on the MHD eigenfunction predict several times smaller losses than observed experimentally. Possible explanations include the synergistic effect of fishbone activity and nonlinear couplings between different toroidal modes. The most likely explanation is that the inductive electric field predicted by MHD fails to model the actual electric field. The amplitude of the magnetic perturbation (␦B/Bϳ10 Ϫ4 ) is very small; typically, a value of ␦B/Bϳ10 Ϫ3 is required for appreciable transport in simulations. In the experiment, the likely transport mechanism is the resonant EϫB drift. Kinetic modifications of the eigenfunction may produce appreciable electric fields, facilitating radial transport. Future work should concentrate on simulations employing the full magnetic and electric fields predicted by codes that retain kinetic effects.

ACKNOWLEDGMENTS
Guido Huysmans generously shared his Interactive Display Language ͑IDL͒ programs for analysis of JET soft xray data, which served as a model for the analysis in Sec. III. The support of the DIII-D team and insightful suggestions by Liu Chen and Ted Strait are also gratefully acknowledged.
This work was supported by General Atomics subcontract No. SC-G903402 to United States Department of Energy Contract No. DE-AC03-99ER54463.