Prediction of Alfvén eigenmode dampings in the Joint European Torus

Predictions from a gyrokinetic toroidal plasma model reproduce for the ﬁrst time the evolution of Alfve´n eigenmode (cid:126) AE (cid:33) dampings over a range of discharges. The coupling between shear-and kinetic-Alfve´n waves is responsible for the main source of damping through Landau interactions and can be an order of magnitude larger than ﬂuid predictions neglecting global kinetic effects. Strong stabilization occurs when the wave ﬁeld gets localized radially by a rise in the edge magnetic shear, explaining why global AEs have never been detected in the Joint European Torus (cid:64) Rebut, Bickerton, and Keen, Nucl. Fusion 25 , 1011 (cid:126) 1985 (cid:33)(cid:35) in the presence of an X point and suggesting how global Alfve´n instabilities could be avoided in future reactors. © 1998 American Institute of Physics. (cid:64) S1070-664X (cid:126) 98 (cid:33) 02508-7 (cid:35)


I. INTRODUCTION
Whether global modes of the Alfve ´n wave driven by fusion-born ␣ particles are stable is a critical issue for fusion devices such as the International Thermonuclear Experimental Reactor ͑ITER͒. 1 The growth rate of the instability depends on the strength of the ␣-particle pressure gradient drive, which has to remain smaller than the overall damping from the collisions and the resonant Landau interactions between the particles and the wave field.3][4][5][6][7] Growth rates may then be evaluated approximately from the shear-Alfve ´n wave field with simplified models including direct Landau damping, 8,7 continuum damping, [9][10][11] and radiative damping. 12Even if the theoretical calculations 7,13 performed in this manner could only occasionally explain the observed instability thresholds when the mode coupling to kinetic waves did not affect the eigenmode structure, it is remarkable how the theoretical understanding of the beam Landau damping 14 paved the way for the first observation of ␣-particle driven toroidicity Alfve ´n eigenmodes ͑TAEs͒. 15

II. MODELING
A proper evaluation of the growth rate requires taking into account the finite ion gyroradius responsible for the mode conversion from fast to slow waves such as the kinetic-Alfve ´n16 and the drift waves.This generally occurs when the spatial scale of two waves match at a resonance, but can also be induced through mode coupling by the magnetic field curvature 17,18 and, as will be shown below, by the weak magnetic shear in the plasma core.In the studies below, the gyrokinetic predictions are found to agree with the measured frequencies and damping rates 19 over a range of discharges and times, this in contrast with computed damping rates that are an order of magnitude smaller when global kinetic effects are neglected.Apart from quantifying for the first time the relative importance of what could be associated in simplified models with continuum and radiative damping, the results illustrate that when the central magnetic shear s ϭ(/q)(‫ץ‬q/‫)ץ‬ is smaller than ⑀ϭ/R the inverse aspect ratio, there can be a dramatic change in the AE structure due to mode conversion to the kinetic-Alfve ´n wave, which strongly enhances the Landau absorption in the plasma core.The calculations also show the stabilizing effect of a high magnetic shear at the plasma edge, giving a plausible explanation why antenna excited ͑global͒ Alfve ´n eigenmodes ͑AEs͒ have never been observed in Joint European Torus ͑JET͒ X-point configurations.To our knowledge, this study is the most detailed comparison which has been undertaken between a nonideal MHD spectrum and experimental measurements; it also provides a strong validity check for the ITER predictions, showing that in a reference equilibrium, AEs with low to intermediate toroidal mode numbers n ϭ1 -12 are stable for a large variety of burn conditions. 20aving chosen a discharge with good damping measurements, the equilibrium magnetic field, the density, and temperature profiles have been reconstructed with the best possible fit to the experimental diagnostics. 21,22The gyrokinetic toroidal PENN code 23 is then used to predict the spectrum from the equilibrium data only, monitoring the response peaks in the same manner as in the experiment ͑Ref.19͒ to determine the frequency and damping of AEs that are sufficiently global to reach the saddle coil antenna at the bottom of the plasma.The plasma model 24 is based on a finite Lar-mor radius expansion for the passing bulk particles and takes into account the magnetic and diamagnetic drifts induced by the equilibrium inhomogeneities.Resonant wave-particle interactions are modeled with an approximate functional dependence for the parallel wave vector k ʈ ϭn/R, where n is the toroidal mode number and R the major radius, justified by an iterative evaluation of k ʈ from the wave field along the same lines as in Ref. 23.The PENN code has been successfully tested for heating scenarios from the ion-Bernstein 25 down to the Alfve ´n range of frequencies, 23 and has been validated for lower frequencies with measurements of the AE spectrum 26 and eigenmode structures 27 in the JET tokamak.

III. ANALYSIS
The first discharge being analyzed ͑No.31561 at 10 s͒ is an ohmic fat pear-shaped plasma with a weak magnetic shear s ˆϽ⑀ from the core out to a normalized radius sϭͱ Ӎ0. 18, where stands for the poloidal magnetic flux.The safety factor q(s) rises rapidly toward the edge so that a multitude of gaps appears in Fig. 1͑a͒ by the coupling of neighboring poloidal Fourier harmonics mϭϪ2,...,Ϫ10.The misalignment of individual gap frequencies ͑which decrease with the deuterium density n D as 1/qͱn D when moving radially outward͒ formally closes the global gap in the fluid spectrum, so that an Alfve ´n resonance remains in the plasma for all frequencies.Scanning the interval ͓120;180͔ kHz, an nϭ2 AE is predicted at 161 kHz with a damping rate of ␥/ pred ϭ0.013Ϯ0.003,where the uncertainty refers to oscillations in the numerical convergence.A single nϭ2 eigenmode is also observed in the experiment with a frequency of 131 kHz which is somewhat lower, probably because of the uncertainty on the safety factor in the core which is larger when no sawtooth inversion radius can be used as a diagnostic; the damping rate ␥/ exp ϭ0.0146 is, however, in very good agreement with the predicted value.Figure 1͑b͒ shows that the mode has a global radial extension; the toroidicity induced variation of the shear-Alfve ´n wave field matches the kinetic-Alfve ´n wavelength in the ͑Ϫ2,Ϫ3͒ TAE gap and creates standing wave between the mode coupling region around sϭ0.35 and the plasma core. 17The electric field component parallel to the magnetic field E ʈ gives rise to an electron Landau damping which can be partitioned with the integrated power P(s)ϭ͐ 0 s P(sЈ)dsЈ in Fig. 1 ͑c͒ as follows: 70% absorbed by the kinetic-Alfve ´n wave induced in the core where no resonance is present, 5%-10% by mode conversion at Alfve ´n resonances (sϭ0.62,0.68, 0.75͒, and the remaining 20%-25% by direct Landau damping of the shear-Alfve ´n wave field in the edge region, where the magnetic shear localizes the mode radially and thereby increases the absorption.From the wave field, it is clear that simplified models such as radiative damping and continuum damping are not applicable, the former because a standing kinetic wave is created in the core and the latter because the kinetic wavelength is comparable with the gap size.The global character of wave fields such as the one in Fig. 1͑b͒ also explains why no correlation is observed in Fig. 5 of Ref. 27 between the high-n radiative damping model 12 and the AE damping measurements in JET. The second discharge ͑No.33273 at 16.35 s͒ is an elongated pear-shaped plasma which has been examined experimentally in Ref. 27.Two nϭ2 global AEs are predicted in the interval ͓100;300͔ kHz, one around 140 kHz with a weak antenna coupling and relatively large damping and a second with a 30 times better coupling at 266 kHz and a weak damping ␥/ th EAE ϭ0.0026Ϯ0.0004depending rather sensitively on the magnetic shear.Two modes are also found in the experiment at 142 and 277 kHz; the first disappears in the sweep 1 s later, in agreement with the theoretical prediction of weak coupling for the lower frequency mode, while the damping measured for the second ␥/ expt.EAE ϭ0.0014 is a factor of 2 smaller than the experimental value.Figure 2 shows that 60% of the electron Landau absorption occurs through FIG. 1. AE mode nϭ2 at 161 kHz in the discharge No. 31561 at 10 s with parameters B T ϭ1.65 T, q 0 ϭ1.13, q a ϭ5.49, n e (0)ϭn D (0)ϭ1.6 ϫ10 19 m Ϫ3 , T e (0)ϭT D (0)ϭ2 keV.The plots show a sketch of the shear Alfve ´n spectrum with a dashed line for the global AE frequency ͑a͒, the Fourier components of the radial electric field Re(E n ) in ͑b͒ and P(s) the power absorption integrated from the center ͑c͒, with circles identifying the radial discretization as a function of the normalized radius s.FIG. 2. AE mode nϭ2 at 266 kHz in the discharge No. 33273 at 16.35 s with parameters B T ϭ3.1 T, q 0 ϭ0.86, q a ϭ4.46, n e (0)ϭn D (0)ϭ4.48 ϫ10 19 m Ϫ3 , T e (0)ϭT D (0)ϭ2 keV.The same type of plots as in Fig. 1.
mode conversion at the Alfve ´n resonances (sϭ0.52,0.83͒, another 20% by mode coupling in the weak magnetic shear region (sϽ0.17),and the remaining 20% by direct damping of the shear-Alfve ´n wave field in the edge where the magnetic shear is stronger.This global wave field with kinetic Alfve ´n waves damped in the vicinity of the conversion layers provides an example of experimental interest where the continuum damping ␥/ fluid EAE ϳ0.01 calculated in Ref. 27 using a fluid plasma model can be very misleading. 11A reason which may explain the overestimation of the electron Landau damping in our gyrokinetic calculation could be that we have neglected the reduction of the number of resonant electrons by the trapping in the toroidal magnetic field.This has been examined within a large aspect ratio expansion for the shear-Alfve ´n wave field only in Refs.28 and 29, but would here require a fully toroidal ͑nonlocal͒ gyrokinetic calculation to take into account the coupling to the kinetic Alfve ´n wave.Studies carried out by varying the equilibrium parameters moreover show that when the damping is as small as ␥/ Ӎ0.001, the theoretical result depends sensitively on the equilibrium profiles, so that a ϳ5% uncertainty in the safety factor can also account for a factor of 2 discrepancy in the damping.
A third series of predictions has been carried out for the discharge No. 38573 in which the evolution of an nϭ1 AE has been measured from its birth around 3 s when a ͑Ϫ1,Ϫ2͒ TAE gap is formed around q 0 ϭ1.5 on the plasma axis, drifting first rapidly outward to sӍ0.55 at 4.7 s, the AE going through a nonresonant 80 keV beam heating phase between 5.5 and 7.5 s, until it again disappears after 10 s when an X point is formed at the bottom of the elliptical plasma and the safety factor in the core reaches a minimum of q 0 ϭ0.92.Two global kinetic Alfve ´n eigenmodes KAEs 17,26 are predicted in the interval ͓160;210͔ kHz, with wave fields that reflect the lϭ0,1 radial oscillations of the kinetic-Alfve ´n wave modulating the TAE wave field inside the ͑Ϫ1,Ϫ2͒ gap.
Because the interval scanned experimentally is very small ⌬Ӎ3␥ expt.and the antenna coupling to the higher frequency lϭ0 mode is predicted to be better until 4.7 s, only the high frequency KAE appears on the measurements.Figure 3 shows that the predicted AE frequencies fall within ϳ3% of the experimental measurements, reproducing well the variation from the beam fueling.The agreement achieved for the damping is around 30% for most of the discharge except in the beginning when the gap position varies very rapidly, making the predictions very sensitive to the mode conversion parameters in the plasma core.This is clearly apparent in the wave fields of Fig. 4, where the kinetic-Alfve ´n wave dominates the mode structure until 4.7 s.Most important however is that the largest fraction of the Landau damping from 4.0 to 8.4 s is here induced by short wavelength oscillations in the plasma core ͑Fig.5͒: removing the kinetic effects artificially from the model for small radii (s Ͻ0.2) yields a fluid-like electron Landau damping rate from mainly the shear Alfve ´n wave ␥/ art Ӎ0.003 which is an order of magnitude smaller than observed in the experiment.Since the gap opens up monotonically from sӍ0.65 to the center, this core-localized mode conversion has to be of dif-ferent nature than the mode coupling induced inside the toroidicity gap. 17Studies carried out by varying locally the central magnetic shear show that the oscillations are caused by an increase of the kinetic-Alfve ´n wavelength in the weak shear region, resulting in the mode conversion once the spatial scale of the kinetic-Alfve ´n wave matches the shear-Alfve ´n wave field scale length.Because of the complicated nature of this mechanism, we believe that it is not possible to derive a simplified model that fits well the global AE measurements.Instead, we use here the predictions from the comprehensive PENN model to illustrate the phenomenon and show not only that it is important, but that it is also in agreement with the measurements.Another effect apparent in FIG. 3. Comparison between the nϭ1 AE frequencies ͑top͒ and dampings ͑bottom͒ predicted ͑*͒ and measured ͑ϩ͒ during the evolution in the discharge No. 38573.Experimentally, the systematic error from the choice of the relevant diagnostic channels to fit is the dominant source of uncertainty ͑ϳ10%-30% for ␥/͒; for the theory the uncertainty comes mainly from the reconstruction of the safety factor profile ͑ϳ10%-20% for q 0 ).Parameters at 4.0 s are B T ϭ2.56 T, q 0 ϭ1.36, q a ϭ4.

IV. CONCLUSION
In summary, gyrokinetic calculations of global AE dampings predicted for JET plasmas are in good agreement with the measurements and the uncertainties obtained from these studies show to which extent predictions are possible for ITER.Resonant Landau interaction with the global shearand kinetic-Alfve ´n wave field provide for the dominant damping mechanism, with mode conversion induced by Alfve ´n resonances, toroidal mode coupling, and the weak magnetic shear in the plasma core.

Figs. 3
Figs. 3-5 which becomes dominant after ϳ8 s, is the enhanced global damping rate due to the radial localization of the shear-Alfve ´n wave field in the edge region.It is caused by the rise of the edge magnetic shear when the plasma gets diverted, explaining why antenna excited AEs have never been observed in JET in the presence of X points.