Modelling of Alfvén waves in JET plasmas with the CASTOR-K code*

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION Nucl. Fuion 42 (22) PII: S29-555(2) Modelling of Alfvén wave in JET plama with the CASTOR-K code* D. Borba,2, H.L. Berk 3, B.N. Breizman 3, A. Faoli 4,7, F. Nabai, S.D. Pinche 5, S.E. Sharapov 6, D. Teta 4 and contributor to the EFDA-JET Work Programme a Aociacão EURATOM/IST, Av. Rovico Pai, 49- Liboa, Portugal 2 EDFA Cloe Support Unit, Culham Science Centre, Abingdon OX4 3DB, UK 3 Intitute for Fuion Studie, Univerity of Texa at Autin, Autin, TX 7872, USA 4 Plama Science and Fuion Centre, MIT, Cambridge, Maachuett, MA 239, USA 5 Max-Planck Intitut für Plamaphyik, Euratom Aociation, D85748 Garching, Germany 6 Euratom/UKAEA Fuion Aociation, Culham Science Center, Abingdon OX4 3DB, UK 7 CRPP-EPFL, Aociation EURATOM-Swi Confederation, CH-5 Lauanne, Switzerland Duarte.Borba@jet.efda.org Received 22 January 22, accepted for publication June 22 Publihed 7 Augut 22 Online at tack.iop.org/nf/42/29 Abtract A hybrid magnetohydrodynamic (MHD)-gyro-kinetic model CASTOR-K developed for the tudy of Alfvén eigenmode (AE) tability in the preence of energetic ion ha been applied to the interpretation of recent meaurement of Alfvén wave in JET. Thee include the detailed AE damping meaurement performed uing the AE antenna excitation ytem and alo the obervation of Alfvén cacade in trongly revered hear cenario at JET. The mode converion between the AE and kinetic Alfvén wave and the relation to the Alfvén continuum i tudied and the calculated damping i compared with the experimental data. The contribution of ion cyclotron reonant heating driven minority ion to the growth rate of the novel-type mode localized around the point of zero magnetic hear i calculated. Thi mode i hown to be clearly linked to the ideal MHD Alfvén continuum, computed with the CSCAS code and conitent with the obervation of a quai-periodic pattern of upward frequency weeping Alfvén cacade in JET. PACS number: Fa, Pi, Tn. Introduction The undertanding of plama intabilitie i of great importance for the optimization of the deign and future operation of a fuion tokamak reactor []. Alfvén intabilitie are particularly important, due to the fact that the charged fuion product (α particle ) birth velocity i larger than the Alfvén velocity V A = B / 4πρ, where B repreent the equilibrium magnetic field and ρ the plama ma denity [2, 3]. Alfvén eigenmode (AE), eigenmode of the ideal magnetohydrodynamic (MHD) incompreible equation uch a toroidicity induced Alfvén eigenmode (TAE) [4 6] and ellipticity induced Alfvén eigenmode (EAE) [7], detabilized by energetic ion may reditribute α and reduce * Thi work wa performed under the European Fuion Development Agreement. a See appendix of the paper by Pamela J. 2 Overview of recent JET reult Proc. IAEA Conf. on Fuion Energy (Sorrento, 2). the ignition margin of a tokamak reactor and/or caue damage to the firt wall. The detabilization of AE [4 6] by the fuionborn α particle in tokamak wa firt theoretically analyed in [8, 9] and experimentally oberved in TFTR []. The tudy of AE detabilization by energetic ion produced by auxiliary heating, uch a ion cyclotron reonant heating (ICRH) [ 3], i a valuable tool in undertanding the phyic iue related to Alfvén intabilitie. Thee experiment offer the poibility of validating the model ued in the extrapolation to reactor condition []. The ytematic paive meaurement of intabilitie in the Alfvén frequency range were carried out at JET, tarting from 997 [4]. The new high frequency digital recorder were able to tore 4 of magnetic fluctuation data at a ampling frequency of MHz. With thi ytem, detailed meaurement of Alfvén intabilitie have been done. It wa oberved that the ICRH driven energetic ion were able to detabilize AE at JET [5], for a wide range of plama parameter. Under thee /2/829+$3. 22 IAEA, Vienna Printed in the UK 29

2 D. Borba et al condition, a very complex Alfvén pectrum emerged and it ha been a challenge to undertand all the phyic iue in detail. Neverthele, a comprehenive picture of AE detabilization by energetic ion ha been etablihed, with the aid of the AE active diagnotic [6, 7]. The preent phyic undertanding can be ummarized within the following topic: plama equilibrium, Alfvén wave propagation, particle orbit, wave particle interaction, energetic particle mode (EPM), nonlinear behaviour, aturation amplitude and energetic particle reditribution and loe. The recontruction of the plama equilibrium i the firt tep and an important one in the detailed modelling of AE. Uually, an accurate recontruction of the afety factor (q)-profile can be achieved uing information from the awteeth inverion radiu or other MHD intabilitie. However, the accurate equilibrium recontruction of revered hear cenario at JET, obtained with the ue of lower hybrid current drive, require ome further reearch, in particular cenario with zero or negative current cloe to the axi [8]. However, uing motion Stark effect (MSE) meaurement [9] together with the information from MHD intabilitie, reaonably accurate inverted q-profile equilibrium can be obtained. The propagation of the Alfvén wave in toroidal plama i well etablihed within the MHD framework, while both particle orbit and the wave particle interaction can be decribed uccefully within the gyro-averaged approximation. In deep revered hear cenario, the orbit of energetic ion with energie in the MeV range are motly in the non-tandard regime, where the preceion drift frequency i comparable with the poloidal bounce/tranit frequency. In thi cae, full orbit effect need to be retained, including non-tandard potato orbit. The tudy of the occurrence of EPM in plama with a ignificant fraction of energetic ion i alo crucial. EPM doe not exit a an eigenmode of the plama in the abence of energetic ion [2, 2], but can become untable above a critical value of the fat particle preure gradient [22, 23]. Thee mode have, in general, a higher fat ion intability threhold but once untable can alo have a larger impact on the confinement of energetic ion. Thi paper will focu on the calculation of the AE pectrum and the mode detabilization by energetic ion. AE non-linear behaviour [24] and the effect on particle confinement will not be addreed. The modelling of AE intabilitie in JET can be carried out uing the CASTOR-K code [25]. The CASTOR-K code i a hybrid model, containing a fluid part for the propagation of AE and mode converion to kinetic Alfvén wave, and a gyro-kinetic part for the interaction of the energetic ion with AE. After the decription of the CASTOR-K model and ome of the recent enhancement (ection 2 and 3), the code will be ued to analye the damping of AE in Ohmic low temperature plama (ection 4) and the AE pectrum in deep revered hear plama (ection 5). Thee reult extend ignificantly the analyi of AE in JET plama reported previouly [4], which focued on monotonic q-profile and high temperature plama. The analyi in ection 5 focue on MHD mode localized around the point of zero magnetic hear. They are referred to a zero hear Alfvén eigenmode (ZAE). Thee mode are cloely linked to multiple toroidal AE dicued in [27] and to the EPM aociated with Alfvén cacade (ee [26]). It i noteworthy that ZAE can exit in the abence of energetic particle provided that the mode frequency i ufficiently cloe to the TAE gap frequency. The contribution of ICRH driven minority ion to the growth rate of the mode with non-tandard particle orbit, can be decribed uing the CASTOR-K model. The calculation ue the aumption that the eigenfrequency i cloe to the TAE gap frequency and that the eigenfunction i not ignificantly modified by fat particle even for the low frequency range. 2. CASTOR-K model (fluid part) The hybrid MHD-kinetic CASTOR model [28, 29] olve the linearized reitive MHD equation in toroidal geometry, where the finite Larmor radiu effect and the effect of the parallel electric field are included in the model within the complex reitivity approximation [3], and it i unchanged in relation to the previou verion of the code [25]. In the ideal MHD framework, the damping of AE i dominated by continuum damping, caued by the reonant aborption of the AE energy at the Alfvén reonance [3, 6]. Thi i due to the fact that the propagation of MHD wave in a non-uniform plama become ingular at the Alfvén reonance, yielding a continuou pectrum for a bounded plama [32]. The continuum mode wave energy i aborbed at the Alfvén reonance layer due to phae mixing of the ingular wave. In hot plama, with the introduction of additional phyic including non-ideal effect, uch a finite perturbed electric field and finite Larmor radiu, the Alfvén ingularity i reolved. Non-ideal effect raie the order of the ytem of differential equation for AE introducing new olution, which decribe hort wavelength ocillation. The analyi of the higher order ytem of equation how that the energy aborbed by phae mixing in ideal MHD i now converted into a new et of hort wavelength mode with non-zero parallel electric field. The perturbed parallel electric field and firt order finite Larmor radiu of core ion give the following correction to the vorticity equation [33]: ( ω 2 ) ( b )( 2 ( b )φ) + VA 2 φ } {{ } Ideal MHD part ω 2 V 2 A ρ 2 i 4 φ } {{ } FLR + ( iδ e ) ω2 VA 2 ρs 2 4 φ =, }{{} E term with diipation δ e due to the electron. The vorticity equation ha the following form in reitive MHD: ( ω ( b )( 2 2 ) ( b )φ) + VA 2 φ }{{} Ideal MHD part + ( b ) i η 4πω ( 4 ( b )φ) =. }{{} Reitive term The direct comparion of the previou equation give u the complex reitivity, where ω A = V A R, ρ S = T e T i ρ i 3

3 Modelling of Alfvén wave in JET plama and ρ i the ion Larmor radiu. ( ) 2ω 2 ( 3 η = 4πωρS 2 δ e +i4πω ω A 4 + T ) ( ) e 2ω 2 ρi 2. T i ω A In order to undertand the behaviour of Alfvén wave in the preence of FLR and finite parallel electric field, the Alfvén pectrum can be olved within the Wentzel Kramer Brillouin Jeffrey (WKBJ) approximation. The tranition from the reitive Alfvén pectrum into the kinetic Alfvén pectrum can be tudied by changing the uual reitivity into a general complex parameter η. The fat varying olution in the radial direction can be decribed accurately by the phae integral in the WKBJ approximation [34] φ e ±iφf, where dφ f = ( k b) + ω 2. dr ηω To obtain the WKBJ pectrum the following eigenvalue condition for ω mut be atified: (a) no anti-stoke line croing φ f (b) φ f (a) = pπ Im φ f (a) < Im φ f (b) <, (b) one at (a) and half at (b) anti-stoke line croing φ f (a) = pπ Im φ f (b) >, (c) one at (b) and half at (a) anti-stoke line croing φ f (b) = pπ Im φ f (a) >, where a,b are the boundarie. In the generalized cae of complex η the pectrum rotate and the third branch diappear with the two remaining branche forming the kinetic Alfvén pectrum. The kinetic Alfvén wave correponding to the econd branch have a turning point at the Alfvén reonance giving rie to the kinetic toroidicity induced Alfvén eigenmode (KTAE) in toroidal geometry [35]. The firt branch doe not have a turning point inide the plama and play an important role in radiative damping of Alfvén wave. In the low hear ( ) limit the diperion relation for each branch take the form +2(+ω 2 ) 2(+ω 2 ) ω2 + ηω = pπ, ω +2i(+ω 2 ) 2 η (ω 2 +)/ ηω = pπ, ( + ) 2 +2iω ω2 + ηω = pπ. For very large value of p the diperion relation can be implified ω = 2 ( ηp2 π 2 ± η 2 p 4 π 4 4). Since p i the number of radial wave period, p k, one obtain γ = Im(ω) ηk 2, howing that within the CASTOR-K complex reitivity approximation the hort wavelength wave are damped proportionally to the perpendicular radial wave number quare k 2. The CASTOR-K code calculate the non-ideal Alfvén pectrum uing two ditinct numerical algorithm. In the firt procedure the linearized non-ideal MHD equation are olved a an eigenvalue problem uing invere vector interaction. In the econd method the plama repone to an external antenna excitation i calculated uing linear olver. The damping of the eigenmode i determined by the width of the reonance or directly from the eigenvalue. The ideal and non-ideal plama repone for a imple benchmark cylindrical equilibrium [36] i hown in figure. A the non-ideal complex parameter η increae, the damping of the TAE [4] located inide the toroidicity induced gap in the continuum increae, cauing the reonance oberved in the plama repone to broaden. The break up of the continuum in a dicrete et of kinetic Alfvén wave i alo clearly viible in figure a the non-ideal complex parameter η i increaed. Numerical convergence for a JET limiter Ohmic dicharge require around 5 radial finite cubic element and 7 poloidal Fourier harmonic, depending on edge q and toroidal mode number of the eigenmode. Computational reource conitent with thee requirement and with a ytematic comparion with the experimental data were made available in the lat few year, therefore allowing the analyi preented in thi paper. Figure 2 how the damping of n = TAE mode a a function of the underlying plama diipation Re( η) δ e. Calculation of the damping of thi mode how that above a certain value of the underlying plama diipation δ e, the damping become independent of the plama diipation δ e and depend only on the amount of energy converted from the AE to the runaway kinetic Alfvén wave. It i, therefore, hown that in thi parameter regime the CASTOR-K model reproduce the radiative damping limit. The analyi of the dependence of the damping of the TAE on the non-ideal parameter η how the exitence of three eparate cae that need to be conidered. The damping of the TAE mode located at the bottom of the TAE gap tend to be dominated by radiative damping, which increae with η. The TAE mode located at the top of the gap uually croe the Alfvén continuum and i continuum damped. Thi mode ha a finite damping in ideal MHD and the damping ha a weak dependence on η. KTAE mode do not exit in the ideal MHD and the damping decreae with η a hown in figure 3. Antenna Repone (x -7 ) Frequency normalized to the Alfven Frequency Figure. Ideal (red-dahed) and non-ideal (black) plama repone for a imple benchmark cylindrical equilibrium, howing the increae in damping of the TAE located inide the continuum gap and the break up of the continuum in a dicrete et of kinetic Alfvén wave a the complex parameter i increaed. JG.28-c 3

4 D. Borba et al Damping (%) Colliional parameter (x -7 ) Figure 2. Damping of n = TAE mode a a function of the underlying plama diipation. Above a certain value of diipation the overall damping become independent of the underlying plama diipation, reproducing the radiative damping limit. Damping (%).5..5 n = TAE mode Upper TAE mode Lower TAE mode n = KTAE mode Complex parameter (x -7 ) Figure 3. Damping of n = TAE and KTAE mode a a function of the non-ideal complex parameter. The damping of the TAE mode increae with plama preure, directly related non-ideal complex parameter, while the damping of KTAE mode decreae with plama preure. 3. CASTOR-K model (gyro-kinetic part) The gyro-kinetic part of the CASTOR-K code calculate the tranfer of energy between the fat particle and the mode. Thi code wa originally developed in order to tudy TAE detabilization by α particle, and later it wa extended to analye the influence of ICRH heated particle on the tability of lower frequency MHD mode. The CASTOR-K code then calculate the contribution of the fat particle to the energy of the mode uing a perturbative approach. It compute the firt order perturbation on the eigenvalue due to the interaction between the wave JG.28-2c JG.28-3c Table. Variable Integration proce α Gyro-angle Analytical Gyro-average θ Poloidal angle Numerical Fourier tranform φ Toroidal angle Analytical Fourier decompoition E Energy Analytical Pole integration µ/e Magnetic moment Numerical Binary earch P φ Toroidal momentum Numerical Binary earch Table 2. Variable Integration proce α Gyro-angle Analytical Gyro-average θ Poloidal angle Numerical Fourier tranform φ Toroidal angle Analytical Fourier decompoition E Energy Numerical Binary earch µ Magnetic moment Analytical ICRH ditribution P φ Toroidal momentum Numerical Binary earch and the energetic ion population, uing the eigenfunction determined by the fluid part of the code. CASTOR-K decompoe the hot particle energy functional into poloidal bounce harmonic and integrate the contribution over the particle phae pace, a een in equation (): δw HOT = 2π 2 dp m 2 φ de dµ σ f τ b Y p 2 (ω n ω ), p= E ω + n ω D + pω b dτ Y p = L () e ipωbτ. () τ b P φ repreent the toroidal canonical momentum, µ the magnetic momentum, E the energy, L () the perturbed orbit Lagrangian, ω the diamagnetic frequency of the fat ion, ω the perturbation frequency, ω D toroidal preceion drift frequency and ω b the poloidal bounce frequency, the cyclotron frequency and m the ma of the fat ion. The imaginary part of δw HOT, which repreent the contribution of the energetic ion to the linear MHD eigenvalue, i related to the growth rate of the mode (γ/ω)by γ ω = 2ω 2 Im[δW HOT ] E k, (2) where E k i proportional to the mode energy and γ i the growth rate of the unperturbed mode, a hown in equation (2). The ix-dimenional integration in phae pace i performed uing both numerical and analytic method. In the original CASTOR-K verion the integration wa performed uing the procedure repreented in table. The new verion of the CASTOR-K code ue a modified procedure a hown in table 2. The new procedure [37] aume that ICRH tranfer energy to the fat ion only in the perpendicular direction, generating ditribution with a ingle value of the magnetic momentum divided by energy. Thi allow the energy integration to be performed numerically uing the two-dimenional binary earch algorithm and the conequent calculation of both the real and imaginary part of the quadratic form δw HOT. 32

5 Modelling of Alfvén wave in JET plama The binary earch algorithm i important, becaue mall reonance area in phae pace dominate the wave particle interaction. Thi algorithm allow conecutive meh refinement in area where the wave particle interaction i tronget. In an initial tep the algorithm urvey the entire phae pace, and in the following tep the meh i refined uing evaluation, ordering and toring procedure. The convergence i found to be linear in the number of tep and adequate in mot practical application. 4. Alfvén pectrum excited by external antenna in Ohmic plama The calculated continuum and radiative damping were found to be very enitive to the experimental profile, in particular the denity and afety factor profile. However, uing the detabilization of the awtooth a evidence of the appearance of the q = urface in the centre of the plama, a relatively accurate q-profile can be recontructed for (hot #558 at t = 6 ) uing the code EFIT [38] and HELENA [39]. The denity profile can alo be recontructed accurately uing the data from the LIDAR Thomon cattering and microwave interferometer diagnotic. The calculated frequency and damping i compared with the meaurement uing the AE active excitation diagnotic. In thi dicharge, the probing frequency i canned with the frequency range calculated to be in the vicinity of the toroidicity induced gap. Once the ytem detect a reonance, it follow the mode meauring the frequency and damping a a function of time. In thi dicharge two n = mode are found in the TAE gap. The MHD model alo how the exitence of two n = TAE mode in the gap in agreement with the experiment, and the model i able to reproduce the frequency of the AE with high accuracy a hown in figure 4. The Pule No: Alfvén continuou pectrum, and frequency of the eigemode found in the TAE gap for dicharge #558 for the time lice t = 6.5 i hown in figure 5. The eigenfunction calculated by the CASTOR for thi dicharge are hown in figure 6 and 7. The radial plama diplacement i repreented a a function of the normalized magnetic poloidal flux = ψ/ψ, which i approximately equivalent to the normalized plama minor radiu (r/a). The upper frequency (f = 5 7 khz) TAE mode, hown in figure 6, ha no parity inverion, in contrat with the lower frequency (f = 2 3 khz) TAE mode which ha a parity inverion at =.75 a hown in figure 7. The calculated damping for the lower frequency (f = 2 3 khz) TAE mode for thi dicharge i found Frequency (khz) TAE gap Upper TAE Lower TAE EAE gap Figure 5. Alfvén continuou pectrum, and frequency of the eigenmode with toroidal mode number n = found in the TAE gap for dicharge #558 for the time lice t = 6.5. JG2.468-c Frequency (khz) Modelling Experiment ξ r m =. m = 2 m = 4 m = 5 m = 3 m = Time () Figure 4. Frequency of two n = TAE mode (lower TAE, upper TAE) meaured uing the active AE excitation diagnotic at JET (continuou line), compared with the frequency calculated uing the CASTOR code (ymbol ) for dicharge #558. The MHD model reproduce the experimentally oberved frequency with high accuracy. JG.468-7c -., Figure 6. Radial plama diplacement a a function of the quare root of the normalized poloidal magnetic flux of the upper frequency (f = 5 7 khz) n = TAE mode calculated uing the CASTOR code for dicharge #558. JG.468-8c 33

6 D. Borba et al to be.5 %, a factor of 2 maller than the meaured damping of 2% a hown in figure 8. Thi tudy how that the Alfvén wave mode converion model implemented in the CASTOR-K code underetimate the meaured damping by a factor 2. Thi dicrepancy cannot be attributed to uncertaintie in the denity and q-profile. Thi i confirmed by the enitivity tudie and the fact that the meaured frequency for two eigenmode provide a trong contraint in the enitivity analyi. Thi diagreement i not unexpected taking into account that the model i baed on the implifying aumption of the complex reitive approximation decribed previouly ξ r m = m = 2, m = 6 m = 3 m = 5 m = Figure 7. Radial plama diplacement a a function of the quare root of the normalized poloidal magnetic flux of the lower frequency (f = 2 3 khz) n = TAE mode calculated uing the CASTOR code for dicharge #558. JG.468-9c [3]. However, if mode converion were not included in the CASTOR-K model the dicrepancy would be much larger, therefore, the importance of mode converion in the analyi of the damping of TAE. Thi reult emphaize the importance of mode converion in addition to continuum damping in the overall damping of AE in JET Ohmic plama, a hown in figure Alfvén pectrum in deep revered hear plama A detailed analyi of the Alfvén pectrum in deep revered hear plama i carried out uing an equilibrium recontruction obtained with the MSE diagnotic at JET for dicharge # The recontructed q-profile i trongly revered with q on axi q 6. and q min 2.5 located around r(q min ).62 minor radiu. The Alfvén pectrum a calculated with the fluid part of the CASTOR-K code i quite complex and a number of eigenmode have been found in the vicinity of the TAE gap. Among thee, only three AE with damping rate γ/ω le than 3% were found. The Alfvén continuou pectrum, the frequency and approximate location of the eigenmode found in the vicinity of the TAE gap are hown in figure 9. A core localized EAE wa found with frequency jut above the TAE gap. The radial plama diplacement of the core localized EAE eigenfunction, repreented a a function of the quare root of the normalized magnetic poloidal flux, i hown in figure. A global TAE exit in the middle of the TAE gap with mall damping. The radial plama diplacement of the global TAE, repreented a a function of the quare root of the normalized magnetic poloidal flux, i hown in figure. In addition, a localized eigenmode wa found located in radiu at the poition of the minimum of q with frequency below the TAE mode frequency EAE gap Damping (%).5. Experimental Reult CASTOR-K Calculation Frequency (khz) 2 EAE TAE gap TAE Zero Shear Alfven Eigenmode (ZAE) Time () Figure 8. Comparion between the TAE damping meaured by the active AE antenna ytem and the damping calculated uing the CASTOR-K code for hot #558, for the lower frequency (f = 2 3 khz) TAE mode. The calculated damping for the lower TAE mode for thi dicharge i found to be.5 %, a factor of 2 maller than the meaured damping of 2%. JG.28-4c Figure 9. Alfvén continuou pectrum, the frequency and approximate location of the eigemode, with toroidal mode number n =, found in the vicinity of the TAE gap for the deep revered hear dicharge # A global TAE exit in the middle of the TAE gap and a core localized EAE i found with frequency jut above the TAE gap. In addition, a localized eigenmode referred to in the paper a ZAE i found located in radiu at the poition of the minimum of q with frequency below the TAE mode frequency. JG.468-c 34

7 Modelling of Alfvén wave in JET plama.5.8. m = 6.6 m = m = 2 ξ r m = m = 2 m = 3 m = 4, m = Figure. Radial plama diplacement a a function of the quare root of the normalized poloidal magnetic flux of a n = core localized EAE with frequency jut above the TAE gap for the deep revered hear dicharge # ξ r m = m = 6 m = 5 m = 4 m = 3, m = Figure. Radial plama diplacement a a function of the quare root of the normalized poloidal magnetic flux of a global n = TAE that exit in the middle of the TAE gap for the deep revered hear dicharge # The radial plama diplacement of thi localized eigenmode, repreented a a function of the quare root of the normalized magnetic poloidal flux, i hown in figure 2. The analyi of the frequency dependence on the evolution of q for thee mode wa carried out by varying the value of the minimum of q from 2. to 3.. It wa found that the frequency of the mode located at the minimum of q wa clearly linked with the tip of the Alfvén continuum pectrum: d dr (k V A ) = at the ame poition a hown in figure 3. The tranition from the low hear TAE mode [27] compoed of two dominant JG.468-2c JG.468-3c ξ r m = 6 m = 4 m =, m = Figure 2. The radial plama diplacement a a function of the quare root of the normalized poloidal magnetic flux of a localized n = eigenmode (ZAE) found localized in radiu at the poition of the minimum of q, for the value of q min = 2.5 with mode frequency f mode = 2 khz, with frequency below the TAE mode frequency for the deep revered hear dicharge # Frequency (khz) Core localized EAE γ~% Other mode γ>3% ZAE γ~% TAE γ<% Alfvén Continuum q (min) Figure 3. Analyi of the frequency dependence on the evolution of q of the Alfvén pectrum by varying the value of the minimum of q from 2. to 3.. The n = Alfvén continuum at the poition of q min i repreented by the red line. The eigenmode with the calculated damping maller than % are alo indicated in the figure. The frequency of the mode located at the minimum of q (ZAE) i clearly linked with the continuum pectrum at the ame poition. poloidal harmonic (m = 2,m = 3), into a cylindrical mode [26] dominated by a ingle poloidal harmonic i clearly viible in figure 4, a the mode frequency decreae and move further away from the TAE gap. We conclude therefore that the Alfvén cacade oberved in JET plama with the deep revered q-profile can be interpreted a a ZAE located at the poition of q min and whoe frequency i cloe to the Alfvén continuou pectrum at the poition of q min. Thi JG.468-4c JG.468-5c 35

8 D. Borba et al.8.6 q min =2.45 Frequency=23 khz m= 3 m= q min =2.49 Frequency=2 khz m= m= m=. m= S S.8.6 q min =2.53 Frequency=7 khz m= q min =2.57 Frequency= khz m= m= m= 2. m=. m= S S Figure 4. Detail of the radial plama diplacement a a function of the quare root of the normalized poloidal magnetic flux ( =.4.8) of a localized n = eigenmode (ZAE) found localized in radiu at the poition of the minimum of q, for the value of q min = 2.45, 2.49, 2.53 and 2.57 with mode frequency f mode = 23, 2, 7 and khz. The tranition from the low hear TAE mode compoed of two dominant poloidal harmonic (m = 2,m= 3) into a cylindrical mode dominated by a ingle poloidal harmonic i clearly viible a the mode frequency decreae and move further away from the TAE gap. mode exit in the abence of energetic ion, provided that the mode frequency i ufficiently cloe to the TAE gap. In the cae of the frequency being far away from the TAE gap the ZAE mode exitence require finite preure of energetic ion. However, the frequency of the mode remain linked with the Alfvén continuum and the frequency change within a timecale conitent with the q-profile evolution [4], in contrat to the EPM where the frequency change i et by the characteritic invere growth rate timecale [22] ( τ A ). In order to evaluate the tability of thee mode in the preence of ICRH accelerated minority ion, it i neceary to take into account the orbit of high energy ion in deep revered hear plama in ome detail. In the mall banana limit the particle orbit can be claified into paing and trapped particle. In the cae of high energy ion, where the banana width i comparable with the plama minor radiu the topology of the orbit i more complicated. Conidering only the region in phae pace relevant to ICRH accelerated minority ion the orbit can be claified in two group, banana orbit and potato orbit. Due to the mall current in the plama core, a large fraction of ion with energie in the MeV range are in the potato regime and the toroidal preceion drift frequency i comparable with the poloidal bounce frequency. Therefore, in the calculation of the detabilization of AE, it i crucial to take into account the contribution of non-tandard orbit. For the configuration conidered, it wa found that the main detabilizing influence i dominated by the trapped banana orbit, while mot particle in the potato regime have a tabilizing influence. Figure 5 repreent the energy exchange between the energetic particle and the eigenmode localized around the poition of q min a a function of the energy and the toroidal canonical momentum. The tabilizing influence of mainly non-tandard potato orbit i repreented in red, while the detabilizing influence of the trapped banana orbit i repreented in blue. It i alo clearly een in figure 5 that the relatively narrow reonant region in phae pace dominate the interaction between the mode and the energetic ion. Thee are the region at which the CASTOR-K code ue the meh refinement decribed in ection 3 in order to improve the accuracy of the calculation. A tability diagram for the leat damped eigenmode calculated by the CASTOR-K code for the equilibrium of the deep revered hear dicharge #49382 i hown in figure 6. The TAE mode require lower fat ion β for intability becaue it ha lower continuum damping. The ZAE 36

9 Modelling of Alfvén wave in JET plama Normalized toroidal canonical momentum Energy in (kev) Figure 5. Energy exchange between the energetic particle and the eigenmode localized around the poition of q min (ZAE) a a function of the energy and the toroidal canonical momentum. The tabilizing influence of mainly non-tandard orbit i repreented in red, while the detabilizing influence of the trapped banana orbit i repreented in blue.. JG.53-c Frequency (khz) n= n= n=2 q min = Time () Figure 7. Spectrogram of the magnetic fluctuation howing the preence of Alfvén cacade common in deep revered hear dicharge with ICRH. The calculated frequency of the Alfvén continuum at the poition of q min modified by f corrected = f continuum (.2/n), where n i the toroidal mode number, i hown for comparion. JG.468-6c Fat ion θ.. Zero Shear Alfvén Eigenmode (ZAE) TAE EAE q min ha been performed. The Alfvén continuum at q min behaviour i qualitatively the ame a the Alfvén cacade if only cae where the Alfvén continuum ha a maximum at the q min urface are conidered. Thi i conitent with the exitence of the mode near the TAE gap calculated by the CASTOR-K code. Furthermore, better agreement i obtained by including a reduction of the mode frequency by a factor inverely proportional to the toroidal mode number (n), f corrected = f continuum (.2/n), a hown in figure Concluion ICRH ion tail energy (kev) Figure 6. Stability diagram for the leat damped eigenmode calculated by the CASTOR-K code for the equilibrium of the deep revered hear dicharge # The TAE mode require lower fat ion β for intability but the mode localized at the minimum of q (ZAE) i the mot untable for a wide range of parameter. mode localized at the minimum of q i the mot untable mode for a wider range of parameter, becaue the TAE mode ha a tronger damping from the high energy generated ICRF ion in the potato regime. The overall tability analyi performed by the CASTOR-K code, where the ZAE mode i hown to be the mot likely untable mode, i conitent with the experimental obervation that Alfvén cacade are the dominant Alfvén intability in plama with deep revered q-profile. Detailed comparion of the ZAE mode frequency pattern of the Alfvén cacade and Alfvén continuum located at JG.53-3c The CASTOR-K model for the radiative damping of TAE mode ha been compared with the damping meaurement performed at JET, uing the AE active eigenmode excitation ytem. The mode frequency in Ohmic plama i reproduced by the ideal MHD model with great accuracy. The calculated damping wa found to be around a factor of 2 maller than the meaured damping. Thi diagreement i conitent with the implifying aumption of the complex reitive approximation implemented in the CASTOR-K code. However, thee reult how the importance of mode converion and continuum damping in the overall damping of AE in JET Ohmic plama. In deep revered hear cenario, a mode localized at the poition of the minimum of the afety factor q, referred to a ZAE, wa calculated by the CASTOR-K code. The ZAE wa found to exit in the ideal framework only if the frequency of the mode i cloe to the frequency of the TAE gap. Stability calculation how that the ZAE i the mot untable eigenmode for a wide range for parameter conitent with the experimental obervation of Alfvén cacade. The frequency pattern of the Alfvén cacade i qualitatively in 37

10 D. Borba et al agreement with the time evolution of the Alfvén continuum at the location of q min which i linked to the frequency of the mode calculated by the CASTOR-K code. Meaurement of thee Alfvén intabilitie are ued to olve the invere problem of identifying the plama parameter, epecially the minimum of the afety factor (q min ) hown in figure 7. Acknowledgment Thi work ha been carried out within the framework of the contract of aociation between the European Atomic Energy Community and Intituto Superior Técnico and ha alo received financial upport from Fundação para a Ciencia e a Tecnologia (FCT). The content of the publication i the ole reponibility of the author and it doe not necearily repreent the view of the Commiion of the European Union or their ervice. Reference [] ITER Phyic Bai (chapter 5) 999 Nucl. Fuion [2] Roenbluth M.N. and Rutherford P.H. 975 Phy. Rev. Lett [3] Mikhailovkii A.B. 975 Sov. Phy. JETP 4 89 [4] Cheng C.Z., Chen L. and Chance M.S. 985 Ann. Phy. 6 2 [5] Dewar et al 974 Phy. Fluid 9 93 [6] Kiera and Tataroni 982 J. Plama Phy [7] Betti R. et al 99 Phy. Fluid B [8] Chen L. 988 Proc. Joint Varenna-Lauane Int Workhop on Theory of Fuion Plama (Chexbre, Switzerland, 3 7 October 988) ed J. Vaclavik, F. Troyon and E. Sindoni, p 327 [9] Fu G.Y. and Van Dam J.W. 989 Phy. Fluid B 949 [] Wong K.L. et al 99 Phy. Rev. Lett [] Ali-Arhad S. and Campbell D. 995 Plama Phy. Control. Fuion [2] Wilon et al 993 Int. Conf. Plama Phyic Cont. Fuion Reearch 992: Proc. 4th Int. Conf. (Wurzburg, 992) vol (Vienna: IAEA) p 66 [3] Saigua et al 995 Plama Phy. Control. Fuion [4] Faoli A. et al 997 Plama Phy. Control. Fuion 39 B287 [5] Kerner W. et al 998 Nucl. Fuion [6] Faoli A. et al 995 Nucl. Fuion [7] Faoli A. et al 996 Nucl. Fuion [8] Hawke N. et al 2 Phy. Rev. Lett [9] Levinton F. et al 989 Phy. Rev. Lett [2] Chen L. 994 Phy. Plama 59 [2] Zonca F. and Chen L. 996 Phy. Plama [22] Chen et al 995 Nucl. Fuion [23] Bernabei et al 999 Phy. Plama 6 88 [24] Faoli A. et al 998 Phy. Rev. Lett [25] Borba D. and Kerner W. 999 J. Comput. Phy. 53 [26] Berk H. et al 2 Phy. Rev. Lett [27] Candy J. et al 996 Phy. Lett. A [28] Kerner W. et al 998 J. Comput. Phy [29] Huyman G.T.A. et al 993 Phy. Fluid B [3] Connor J. et al 994 2th EPS Proc. vol 8B (part II), p 66 [3] Vaclavik J. and Appert K. 99 Nucl. Fuion [32] Goedbloed J.P. 975 Phy. Fluid [33] Candy J. and Roenbluth M.N. 994 Phy. Plama 356 [34] Borba D. et al 994 Phy. Plama 35 [35] Mett R. and Mahajan S. 992 Phy. Fluid B [36] Jaun A. et al 997 Plama Phy. Control. Fuion [37] Nabai F. et al 2 Proc. European Conf. on Plama Phy. and Control. Fuion (Funchal, 8 22 June 2) p 557 [38] Lao L. et al 99 Nucl. Fuion 3 35 [39] Huyman G.T.A et al 992 Phy. Fluid B 3 56 [4] Sharapov S. et al 2 Phy. Lett. A