Kinetic damping of resistive wall modes in ITER

Transcription

1 PHYSICS OF PLASMAS 9, 55 () Kinetic damping of resistive wall modes in ITER I. T. Chapman, Y. Q. Liu, O. Asunta, J. P. Graves, T. Johnson, and M. Jucker 5 EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX DB, United Kingdom Department of Applied Physics, Association EURATOM-Tekes, Aalto University, P.O. Box FI-76 AALTO, Finland CRPP, Association EURATOM/Confédération Suisse, EPFL, 5 Lausanne, Switzerland EURATOM-VR Association, EES, KTH, Stockholm, Sweden 5 GFDL/Princeton University, AOS Program, Princeton, New Jersey 85, USA (Received February ; accepted 8 April ; published online 5 May ) Full drift kinetic modelling including finite orbit width effects has been used to assess the passive stabilisation of the resistive wall mode (RWM) that can be expected in the ITER advanced scenario. At realistic plasma rotation frequency, the thermal ions have a stabilising effect on the RWM, but the stability limit remains below the target plasma pressure to achieve Q ¼ 5. However, the inclusion of damping arising from the fusion-born alpha particles, the NBI ions, and ICRH fast ions extends the RWM stability limit above the target b for the advanced scenario. The fast ion damping arises primarily from finite orbit width effects and is not due to resonance between the particle frequencies and the instability. [ I. INTRODUCTION In the steady-state 9 MA scenario planned for ITER,, prolonged plasma operation is facilitated by high pressure and low plasma current leading to high self-generated noninductive bootstrap current fraction. However, such plasma parameters make the discharges more susceptible to deleterious magnetohydrodynamic (MHD) instabilities, which would not be unstable with conventional H-mode profiles. The ITER steady-state scenario is designed to operate slightly above the no-wall beta limit, meaning that the stability and control of the resistive wall mode (RWM) is a significant concern. The RWM is a macroscopic pressure-driven kink mode, whose stability is mainly determined by damping arising from the relative rotation between the fast rotating plasma and the slowly rotating wall mode. In the absence of a surrounding wall, the plasma is stable to kink modes until the normalised plasma pressure, b ¼ l hpi=b, exceeds a critical value, b, the no-wall b-limit. In the presence of an ideally conducting wall, the plasma is stable to a critical value, b b, with the range b < b < b b called the wallstabilised region. In practice, the vessel wall has a finite resistivity. Thus, on the time scale required for eddy currents to decay resistively, the magnetic perturbation of the external kink mode can penetrate the wall and so wall-stabilisation is lost. Reversed shear discharges have low plasma inductance (l i < :8), as the current density peaks off axis, and have predominantly peaked pressure profile due to the presence of (weak) internal transport barriers. Both of these facets make such advanced scenarios more prone to external kink instabilities manifest as RWMs. It has been shown in a number of machines that the plasma can operate above the no-wall b-limit, even with very low rotation. Recent experiments with nearly balanced neutral beam injection (NBI), have found a critical velocity for RWM stabilisation well below that found in previous magnetic braking experiments. Consequently, in order to make reliable extrapolation to ITER, it is crucial to understand the passive stabilisation allowing operation above the predicted no-wall stability limits. Various models have been presented to explain the RWM damping due to kinetic effects, such as sound-wave damping, 5 ion Landau damping, 6 or damping arising from resonance with the precessional drift of thermal ions. 7,8 Numerical simulation has shown that the damping from resonance with the precession frequency of thermal ions 9 or fast ions, can significantly raise the RWM stability limit. Furthermore, experiments designed to deliberately vary the fast ion distribution have also shown a change in the damping of the RWM as measured by the resonant field amplification,,5 which can be (at least qualitatively) explained by numerical simulation of the kinetic damping of the mode. There has been significant effort to model the passive stabilisation of the RWM that might be expected in ITER in recent years. It was shown in Ref. 7 that the kinetic effects associated with the RWM resonance with thermal particle precession drifts could partially stabilise the RWM provided the rotation frequency was below the ion diamagnetic frequency, though this analysis was for a simplified up-down symmetric plasma boundary. The self-consistent inclusion of thermal kinetic effects on RWM stability in generalised toroidal geometry was then assessed in Refs. 6 and 7, before the effects of alpha particles were added in Refs. and. All of these studies, either with a perturbative or selfconsistent treatment of the fast ion interaction with the RWM displacement, find some passive stabilisation of the RWM above the ideal no-wall limit due to the presence of thermal and alpha particles. None of these studies, however, take into account the finite orbit width effects of the energetic particles. 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2 55- Chapman et al. Phys. Plasmas 9, 55 () passing ions is essentially neglected, though such effects can be important. Furthermore, the finite orbit width affects the stabilisation arising from energetic trapped ions as well. 8 In this work, we use a guiding-centre drift kinetic code to analyse the effect of the thermal ions, the alpha particles, and also the contribution of energetic ions arising from neutral beam injection (NBI) or ion cyclotron resonance heating (ICRH) on the stability of the RWM, whilst retaining finite orbit width effects throughout. In Sec. II, the modelling tools used to assess the energetic ion populations are described, with the details of the resultant fast ion populations given in Sec. III. The tools used to model RWM stability are outlined in Sec. IV together with a brief discussion of the pertinent physics determining kinetic damping of RWMs. The effect of thermal particles on the RWM in ITER is assessed in Sec. V, whilst the effects of energetic particles are described in Sec. VI. The implications for operating ITER advanced scenarios are discussed in Sec. VII. II. ENERGETIC PARTICLE POPULATION MODELLING TOOLS In order to model the alphas and the neutral beam fast ion distribution, the ASCOT code 9, has been used whilst the SCENIC code has been used to simulate the ICRH distribution. Once the fast ion distributions are simulated, the HAGIS drift kinetic code has been employed to study the effect of the various fast ion populations on external kink stability perturbatively. ASCOT is a guiding-centre orbit following Monte Carlo code, which integrates the particles equation of motion in time over a five-dimensional space. Collisions with the background plasma are modelled using Monte Carlo operators allowing an acceleration of collisional time scales and reduced computational time. The alpha particle markers are initialised by the local hrvi DT, whereas the beam ions are followed starting from the injector taking into account the beamlet position, direction, beam species, energy, total power, and its bi-gaussian dispersion. The ionization crosssection is calculated at each step using the local temperature and density, and analytic fits from Ref.. In addition to thermal fusion reactions, also fusion reactions between the fast NBI particles and thermal plasma particles are included in the ASCOT code using the model described in Ref.. The ICRH fast ion populations are simulated using SCENIC. The SCENIC integrated code package,5 takes an equilibrium from VMEC, 6 the wave fields and wave numbers from LEMan 7 and iterates with the distribution function evolved by VENUS. 8,9 These codes are iterated to form a self-consistent solution, which can incorporate the anisotropic equilibrium in full d geometry. For the equilibrium and wave field computations, a bi-maxwellian distribution is used for the hot minority, allowing for pressure anisotropy and stronger poloidal dependence of the pressure and dielectric tensor. Whereas LEMan is limited to leading order finite Larmor radius effects, and thus to fundamental harmonic without mode conversion, it computes the wave vectors with the help of an iterative scheme and can, therefore, treat correctly upshifted wave numbers without the use of a local dispersion relation. The guiding centre equations of motion employed in the Boozer coordinate system of VENUS are exact in axisymmetric plasmas and incorporate all finite orbit width effects. After running VENUS, the moments of the distribution function are fitted to the bi-maxwellian on which the equilibrium and wave-field computations are based, allowing them to be iterated to convergence. SCENIC has been used for an assessment of the (de)stabilisation of the internal kink mode and corresponding effect on sawtooth oscillations in ITER. III. ENERGETIC PARTICLE DISTRIBUTIONS IN ITER ASCOT follows a limited number of test particles, each representing a group of real particles. Each test particle is given a weight factor to indicate the number of real particles it represents. When simulating alpha particles born in thermonuclear reactions between the fuel tritons and deuterons, the weight factors are assigned according to the local fusion rate, i.e., the birth rate of the fusion alphas, R ¼ n D n T hrvi DT where hrvi DT is the fusion reactivity in ½m =sš. Thevalueofthefusionreactioncrosssectiondepends only on the local plasma temperature and was obtained from Ref.. Thisisachievedbymeansofhighspatialdensityof the test particles. A test particle with a random pitch k ¼ v k =v ½ ; Š is initialized in a random (R, z) locationwithin the given lower and upper limits q min and q max.thepitchhasa strong effect on the behavior of the particle and, therefore, the entire range [, ] should be covered as uniformly as possible. This is achieved by means of high spatial density of the test particles. The test particle weights were calculated from the local fusion density. The D density of the alpha particles is shown in Figure. Thealphasarewellconfinedwithinq :6 andareapproximatelyisotropic,asillustratedinfigure. z (m) R (m) x n (/m ) FIG.. The particle density of the slowing-down distribution of the fusion alpha population in ITER advanced scenario on an (R, Z) grid as calculated by ASCOT. Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

3 55- Chapman et al. Phys. Plasmas 9, 55 () vpe (m/s) x vpar (m/s) s /m x 7 x 6 (a) vpe (m/s) x s /m x FIG.. Contours of hot particle density of the a particles in terms of their perpendicular velocity (on the ordinate) and parallel velocity (on the abscissa) as predicted by the ASCOT code. It is clear that the a particles are approximately isotropic in velocity space. In order to penetrate the hot, dense plasmas in ITER, neutral deuterium beam energies of the order of.5. MeV are necessary. In this study, the N-NBI is assumed to consist of MeV (D) neutrals from a negative ion-beam system injected in the co-current direction, at a tangency radius of 6 m. The beam can be aimed at two extreme (on-axis and off-axis) positions by tilting the beam source around a horizontal axis on its support flange, resulting in N-NBI injection in the range of Z ¼.56 m to.7 m at the tangency radius of R ¼ 5. m. Simulations have been carried out to predict the fast ion distribution function due to the N-NBI when it is aimed both on- and off-axis, as well as the corresponding pressure and current density profiles. The on- and off-axis distributions are shown in (R, Z) space in Figure. The off-axis fast ion population is peaked at approximately r/a ¼.6. The distribution of beam particles predicted by ASCOT in velocity space and radial space is illustrated in Figures and 5, respectively. Ion cyclotron range of frequency waves will be strongly damped in ITER, leading to various possible heating schemes: second harmonic tritium or fundamental He heating; fundamental D heating (although a low deuterium concentration is required for efficient wave damping making this (b) vpe (m/s) 6 8 vpar (m/s) x x vpar (m/s) x 6 s /m x 8 FIG.. Contours of hot particle density of the (left) on-axis and (right) offaxis NNBI particles in terms of their perpendicular velocity (on the ordinate) and parallel velocity (on the abscissa) as predicted by the ASCOT code. It is clear that the NNBI particles are anisotropic in velocity space. low relevance); electron Landau damping (ELD)/transit time magnetic pumping (TTMP), both of which are fast wave current drive scenarios leading to insignificant fast ion populations; or off-axis He heating, which has moderate absorbed power density but can result in a significant population of ICRH energetic ions near the q ¼ surface. The SCENIC code has been applied to simulate the application of He minority heating in the reversed shear scenario with the resonance.5.5 (a) x 8.5 (b) x 7 z (m).5.5 n (/m ) z (m) 8 6 n (/m ) FIG.. The particle density of the slowing-down distribution of the (left) on-axis and (right) off-axis NNBI hot ions in ITER on an (R, Z) grid as predicted by the ASCOT code R (m) R (m) Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

4 55- Chapman et al. Phys. Plasmas 9, 55 () n (/m ).5 x ρ pol slightly off-axis for different minority concentrations. The off-axis resonance has been simulated in order to generate a strong radial gradient in the passing fast ion pressure nearer to the point of maximum displacement due to the RWM, as indicated by the radial distribution shown in Figure 6. Whilst the far-off-axis heating only gives rise to a low power per particle and no highly energetic tails in the distribution, it does nonetheless incur fast ion distributions capable of affecting kink stability. These simulations only treated the acceleration of He, whereas tritium may also be accelerated. Since tritium damping is nd harmonic, the acceleration of a tail may strongly increase the tritium absorption. IV. NUMERICAL TOOLS FOR MODELLING THE KINETIC DAMPING OF RWMS alphas on axis NBI off axis NBI FIG. 5. The fast particle density resultant from the on-axis and off-axis NNBI injection as well as the fusion born alphas as a function of the poloidal magnetic flux as predicted by the ASCOT code. In order to assess the rôle of these kinetic effects in damping the RWM, the drift-kinetic particle-following HAGIS code has been used to calculate the change in the potential energy of the kink mode in the presence of both trapped and passing ions. HAGIS is a particle-orbit code following the guiding centre motion of the particles, meaning that all the relevant resonances with particle motion are included rigorously. This modelling includes finite drift orbit width effects that have been neglected in previous studies. 8,, The code has been used in a perturbative manner to study kinetic resonance damping effects by diagnosing which particles cause the strongest wave-particle interactions. Whilst this gives an exact treatment of the kinetic effects by calculating the orbit frequencies of a distribution of markers, it does not self-consistently evolve the eigenmode structure in the presence of the particles, which has been shown to affect the RWM stability. It should be noted that both the plasma rotation and the resistive wall geometry will also provide a damping to the RWM (see for instance Refs., 5,, and 57) though these effects are not included here. The MISHKA MHD linear stability code does not include a resistive wall, and the effect of toroidal rotation on the eigenfunction is neglected. In order to quantitatively assess the damping from the wall, full three dimensional geometry of the wall is required, 57 though this is not possible using the numerical tools used here. Further, previous studies have shown that the rotation in ITER is likely to be too low to suppress the RWM passively, 5 so here we focus on passive kinetic damping. Finally, non-ideal MHD effects such as resistive layer damping or viscous boundary layer damping are also neglected. The change in the potential energy of the unstable mode due to kinetic effects is expressed as 6 dw K ¼ ð n? ðr P e K ÞdV; () where n is the plasma displacement eigenfunction, P e K is the perturbed pressure tensor which can be found by taking moments of the perturbed distribution function, which itself can be found by solving the linearised drift kinetic equation, and V is phase space including both velocity and volume integrals. By solving the drift kinetic equation and substituting in for P e K then performing phase space integrals over particle energy, E, pitch angle, k ¼ v k =v, and flux, w, it can be shown that the kinetic potential energy of the mode incorporates a frequency resonance condition, 7,, dw K X l¼ n ðx nx E Þ@f j =@E ez j =@w ; () x nx E þ i eff;j nhxi d;j lx b;j FIG. 6. The fast ion density resultant from off-axis ion cyclotron resonance heating using He minority at 9.8 MHz as a function of the normalised minor radius as predicted by the SCENIC code. The vertical line indicates the average resonance layer averaged over all wave-particle interactions and time. where x is the complex eigenmode frequency, the E B frequency is x E ¼ x / x i ; x / is the toroidal rotation frequency, x i is the ion diamagnetic frequency, Z is the effective plasma charge, e is the electron charge, the particle energy is E ¼ v =, hxi d is the orbit-averaged drift precession frequency, x b is the trapped-particle bounce frequency, and eff is the collision frequency. Using the HAGIS code, all of the toroidal and poloidal bounce (transit)-averaged frequencies for any species of particles can be calculated, though collisions are neglected. It is clear from Eq. () that a resonance between the Doppler shifted mode frequency and these particle frequencies can occur when the denominator vanishes. Whilst it has been shown that the low-frequency Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

5 55-5 Chapman et al. Phys. Plasmas 9, 55 () RWM can resonate with both the precession drift frequency and bounce frequencies of thermal ions (depending on the plasma rotation), 9 for energetic particles typically x d ; x b x x E. Consequently, it is likely that the contribution from the energetic particles is not a resonant process, but related to () an increase in the resistance to changes in the magnetic flux by the kink mode through an inhibition of toroidal coupling of harmonics and () finite orbit width effects of the energetic ions. The effect of orbit widths on external kink stability is likely to be similar to the effect on the stability of the internal kink mode, which has been examined recently: Orbit width effects have been discussed analytically, 7,8 studied numerically using the HAGIS code,,9,5 and then demonstrated experimentally. 6 For internal kink stability, the mechanism responsible was found to be due to the contribution of passing particles intersecting the q ¼ rational surface. An important fast ion effect is obtained when the distribution function of passing ions is asymmetrically distributed in the velocity parallel to the magnetic field. This effect is exacerbated when the orbit widths of the energetic ions are larger, and the parallel asymmetry of the distribution function is more strongly radially sheared. Whilst the analysis of Refs. 7 and 8 is for a top-hat displacement associated with the n ¼ internal kink mode, it is probable that the physical mechanism also applies to the more complex external kink eigenfunction, provided that there is a parallel asymmetry in the distribution function and that the ions are sufficiently energetic that their orbit widths cut across the region of maximal plasma displacement. In Ref. 8, the importance of finite orbit width effects on RWM stability was simulated by changing the toroidal field for a typical MAST plasma and calculating the change in the mode potential energy due to the fast ion kinetic contribution, dw K. The radial excursion of energetic ion orbits can be expressed as 5 D r ¼ qm rezb ðjv k R lr qx bjþ; () where M is the ion mass, e is the electron charge, Z is the effective plasma charge, v k is the velocity of the ions parallel to the magnetic field, RðR Þ is the major radius (at the magnetic axis), l ¼ for trapped ions and l ¼ for passing ions, and x b is the bounce frequency of the ions. As the radial excursion of the fast ion orbits was decreased (with increasing magnetic field), the effect of the fast ions on external kink stability was diminished. Finally, having calculated the change in the potential energy of the external kink mode due to the presence of the energetic particles, the growth rate of the RWM can be formulated in terms of the MHD perturbed energy and the change due to kinetic corrections as 8 cs W ¼ dw þ dw K dw b þ dw K ; () where dw b; represents the sum of the plasma and vacuum energy with and without a wall respectively and where dw < and dw b >, indicating that the plasma is in the FIG. 7. The pressure and safety factor profiles as a function of the normalised minor radius for the ITER advanced scenario plasma. These profiles are derived from transport simulations using the ASTRA code. region where the ideal external kink mode is stable, but the RWM is MHD unstable. Here, s W is the characteristic resistive decay time for currents in the wall induced by the RWM. V. EFFECT OF THERMAL PARTICLES ON RWM STABILITY The ITER advanced scenario reversed-shear plasma studied in this paper has a peaked pressure profile, a safetyfactor on-axis of q ¼ :5, and a minimum safety-factor of q min ¼ :57, as illustrated in Figure 7. The profiles are simulated using the ASTRA transport code. 5 ASTRA solves a basic set of four one-dimensional flux-surface-averaged diffusive equations, along with a variety of modules describing additional heating, current drive, and other non-diffusive processes in tokamak plasmas. The toroidal field and plasma current are taken as B T ¼ 5: T and I p ¼ 9 MA, respectively, the bootstrap fraction is calculated as f BS ¼ 5% giving rise to a fusion yield of Q ¼ 5. The current drive is provided by 6.5 MW of off-axis NBI and MW of electron cyclotron current drive (ECCD) (split between upper and equatorial launchers) at r/a ¼.. In other transport simulations, FIG. 8. The growth rate of the external kink mode normalised to the Alfven frequency as a function of the normalised pressure, b N as calculated by the MISHKA code when there is assumed to be no wall and an ideally conducting wall at the edge of the plasma. This stability analysis used the equilibrium for the ITER advanced scenario, using the profiles illustrated in Figure 7. Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

6 55-6 Chapman et al. Phys. Plasmas 9, 55 () (a)... Thermal ions Alphas. -. Re(δW k ). Re(γτ W ) C β =.5 C β =. C β =..... v /v A v /v A (b).5. Thermal ions Alphas FIG.. The growth rate of the RWM found using expression with dw k calculated using HAGIS as a function of the rotation speed normalised to the Alfven speed for different values of C b when the damping arising from thermal particles is included. Im(δW k ) v /v A FIG. 9. (a) The real and (b) imaginary parts of dw k of the RWM as calculated by HAGIS due to the presence of both thermal and a particles for a range of different rotation speeds normalised to the Alfven speed in ITER advanced scenario at C b ¼ :5. core ICRH is also assumed to mediate the safety factor onaxis. 5,5 The target normalised pressure is, b N bab =I p ¼ :9 where a is the minor radius, which is above the calculated no-wall limit, b N ¼ :. Here, the ideal n ¼ stability limits are calculated using the MISHKA-F code, 55 which takes a fixed-boundary static equilibrium calculated by the Helena code. 56 The no-wall b-limit is found to be b N ¼ :, whereas the limit with an ideally conducting realistic ITER wall geometry is b b N ¼ :6, as shown in Figure 8, in good agreement with the ideal fluid limits found in Ref. 57. Taking the external kink eigenfunction in the absence of a conducting wall from MISHKA and an isotropic, Maxwellian low-energy distribution, the change in the potential energy of the RWM due to kinetic effects of the thermal population is calculated using HAGIS to find dw K and then assess the damping rate using Eq. (). Figure 9 shows the real and imaginary parts of the drift kinetic energy perturbation with respect to the plasma rotation speed as a fraction of the Alfvén speed. Here, the Alfvén rotation frequency is v A ¼ B =l q R ; B is the magnetic field, q is the mass density, and R is the major radius at the magnetic axis, respectively. Transport predictions suggest that the rotation speed at the magnetic axis in ITER is likely to be approximately v ¼ :v A At the expected rotation in ITER, the thermal particles and a particles give a similar, but small, stabilising contribution. As the rotation frequency drops, the thermal contribution increases significantly. This occurs as the plasma frequency approaches the toroidal precession frequency of the thermal population, at which point a resonance occurs, seen when the denominator of Eq. () tends to zero as x nx E x d;th. In contrast, the damping arising from the presence of the a particles is relatively constant for this range of rotation. This damping is a non-resonant effect and arises due to the inclusion of orbit width effects, as will be discussed in Sec. VI. The a stabilisation does increase slightly at the highest rotation speeds, as the plasma rotation frequency begins to approach the fast ion precession frequency and so a small minority of the ensemble of particles approach resonance. Figure shows the growth or damping rate of the RWM calculated from Eq. () as a function of plasma rotation when the damping arising from the thermal population is included. The growth rate is shown for different plasma pressures, quantified here by the parameter C b ¼ ðb b nowall Þ=ðb wall b nowall Þ, where the target for ITER advanced scenario is C b ¼ :. It is evident that the RWM is, at best, marginally unstable at the pressure and rotation speed expected in ITER when only thermal damping effects are considered. The implication is that passive stabilisation of the RWM by rotation and thermal damping alone cannot be relied upon, and that active control of the RWM, most probably using externally applied magnetic fields from in-vessel control coils, would be required for stable operation at the target plasma pressure required to meet the goal of fusion yield Q ¼ 5. VI. EFFECT OF ENERGETIC PARTICLES ON RWM STABILITY Since the damping arising from thermal particles appears insufficient to passively stabilise the RWM at the expected flow and pressure in ITER, considering the passive damping that might be expected in the presence of energetic particles is consequently of great interest. In this section, we again take the external kink eigenfunction from MISHKA, but now use the fast ion distributions as markers from the Monte Carlo ASCOT and SCENIC codes to find the change in the potential energy of the RWM due to kinetic effects. Figure shows the contributions from both trapped and passing energetic ions for the NBI, a, and ICRH Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

7 55-7 Chapman et al. Phys. Plasmas 9, 55 () FIG.. The contributions to the real part of dw k from both passing and trapped particles for the thermal, a, NBI and ICRH distributions as calculated by HAGIS at v ¼ :v A and C b ¼ :5. populations for v ¼ :v A and C b ¼ :5 compared to the contribution for the thermal particle distribution as discussed in Sec. V. Here, the neutral beam injection is oriented in its most off-axis position, and MW of injected power is assumed. The ICRH distribution assumes MW injected power at a frequency of 9.8 MHz equating to a resonance at R ¼ 6.8 m (when R ¼ 6. m) with % He minority concentration in a deuterium plasma. As will be discussed later, it should be noted that the contribution from both NBI and ICRH is diminished if the fast ion distribution is peaked nearer to the magnetic axis. It is clear that the trapped ion effects dominate for the thermal species, since these ions influence the RWM through the resonance with the thermal trapped ion precession drift frequency, whilst the passing ion transit frequency is significantly larger than the Dopplershifted mode frequency at this modest rotation. In contrast, the passing energetic ions play a significant role, and for the ICRH population actually have a more stabilising contribution than the trapped particles. When the passing fast ion population is asymmetric in velocity space, there is an important finite orbit contribution to the mode stability. This strong contribution of the circulating particles comes from the ions close to the trapped-passing boundary where their orbit widths, D b are large, dw passing k D b. The damping effects of all energetic species are relatively unaffected by the plasma rotation frequency because, in all cases, the damping arises from orbit width effects rather than the resonance effects, which dominate the thermal species contribution. In order to explicate the importance of finite orbit width effects on RWM stability, the effect of changing fast ion birth energy on the change in the mode potential energy due to the fast ion kinetic contribution, dw K, has been considered. The radial excursion of energetic ion orbits can be expressed by Eq. (). Evidently, increasing v k will increase the orbit width of the particles. Figure shows <eðdw k Þ calculated using Hagis for different birth energy of an analytic a particle distribution function, given by an isotropic slowing-down distribution function. It is clear that as the energy of the a particles increases and thus the orbit widths FIG.. The real part of dw k as found by HAGIS resulting from the a particles as a function of different assumed birth energy when the distribution function is assumed to be isotropic and slowing down. increase, the number of particles that cross the radial region encompassed by the external kink eigenfunction increases and thus the damping increases. Further, the damping is enhanced approximately quadratically with energy, since the number of ions, which overlap the RWM eigenfunction increases nonlinearly. At full-field in ITER, MeV ions have an orbit width D r =a ¼ :6, whereas for comparison, MeV ions in JET with He minority at B T ¼ :75 T (as per experiments in Ref. 6) have an orbit width of D r =a ¼ :5. Despite this relatively small orbit width, these ions can still have a significant non-resonant damping effect on the RWM, which is amplified at higher energy. The damping from the energetic ions is also enhanced when the distribution of the fast ions is located further offaxis, allowing more interaction between the fast ion population and the external kink displacement, which is predominantly in the outer half of the plasma minor radius. This can be seen by radially moving the peak of an analytic distribution function of the NBI and ICRH fast ions. The effects of the fast ions are taken into account by making approximations for both the NBI and ICRH populations. The ICRH distribution function is assumed to be bi-maxwellian in form, as in Refs. 6 and 6 " m = fh ICRH n c ðrþ ¼ p T? ðrþt = k ðrþ exp lb c T? ðrþ je lb c j T k ðrþ where the particle energy E ¼ mv =, the magnetic moment l ¼ mv? =B; k and? represent the components parallel and perpendicular to the magnetic field respectively, B c is the critical field strength at the resonance, and n c is the local density evaluated at B ¼ B c. Similarly, the passing NBI ions can be approximately represented by a non-symmetric distribution, which is slowing down with respect to energy and Gaussian with respect to radius and pitch angle, 6 such that h f p w h ¼ C exp w i h k k exp Dw Dk i E = þ E = c # ; (5) h E E i Erf c ; DE (6) Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

8 55-8 Chapman et al. Phys. Plasmas 9, 55 () Re(δW k ).... NBI ICRH ψ (NBI), ψ(b c ) (ICRH) FIG.. The real part of dw k as calculated using HAGIS resulting from the NBI and ICRH particles as a function of different radial positions of the peak of the distribution functions when the NBI population is assumed to be given by Eq. (5) and the ICRH distribution is assumed to be given by Eq. (6). where Dw ¼ :; k ¼ :5; Dk ¼ :5; E ¼ MeV; E c ¼ kev; and DE ¼ 5 kev. As these model fast ion distributions are varied in radial space, the variation in dw k is illustrated in Figure. The stabilising contribution to the kink mode potential energy arising from the fast ions is increased as the distribution function is moved further off axis. This is synergistic in the case of neutral beam injection, since the off-axis beam driven current is necessary to broaden the current profile and achieve the reversed shear q-profile symptomatic of the advanced scenario. The strongest contribution towards the damping arising from ICRH comes from the highly energetic ions with largest orbit widths that traverse the external kink eigenfunction. This means that as the radial deposition is shifted outwards, there is a preferential resonance position, beyond which these extremely energetic ions are no longer confined. Finally, it is possible to use the full marker populations for all the energetic species to assess the increase in the RWM stability limit due to kinetic damping. Figure shows the growth/damping rate of the RWM for different C b. Clearly, in the absence of any damping, the RWM is unstable at the no-wall limit. The thermal particles alone Re(γτ W ) Fluid Fluid + Thermal Fluid + Thermal + α Fluid + Thermal + α + NBI + ICRH c β FIG.. The growth or damping rate of the RWM found using expression with dw k calculated using HAGIS as a function of C b when only ideal fluid stability is considered or when additional damping from thermal and fast ion populations are considered for v ¼ :v A. The marginal stability point is increased significantly by including all these damping effects, only made possible by including finite orbit width effects. account for a significant increase in the stability limit, % into the wall-stabilised region. The additional damping arising from non-resonant finite orbit width effects of the a particles, and the NBI and ICRH fast ions further extends the RWM stability limit to more than the target value for ITER advanced scenario of C b ¼ :5 equating to b N ¼ :95. This means that assuming off-axis NBI and ICRH fast ion populations, finite orbit-width drift kinetic modelling predicts that the RWM will be passively damped by kinetic effects at the target pressure in ITER. It should, however, be noted that whilst the fast ion effects are treated rigorously, the eigenfunction is considered as fixed and this perturbative approach has been shown to over-estimate kinetic damping. VII. DISCUSSION The ITER advanced scenario, with peaked pressure profile and reversed shear safety-factor, is expected to be unstable to RWMs according to fluid theory, with the target plasma pressure required to achieve the goal of Q ¼ 5 well above the ideal no-wall b -limit. Therefore, it is of considerable interest to consider mechanisms to stabilise the RWM, especially when such stabilisation is achieved passively just from the foreseen heating and current drive actuators. In recent years, it has been demonstrated numerically, and then experimentally, that tailoring both the thermal and fast ion distributions can lead to damping of the RWM and consequent enhancement of the stability limits. Here, for the first time, the damping that may be expected in ITER has been studied including finite orbit width effects. We employ the HAGIS guiding centre drift kinetic code to calculate dw K, meaning that all the relevant resonances with particle motion are included rigorously. The code has been used in a perturbative manner to study kinetic resonance damping effects by diagnosing, which particles cause the strongest wave-particle interactions. Whilst this gives an exact treatment of the kinetic effects by calculating the orbit frequencies of a distribution of markers, it does not self-consistently evolve the eigenmode structure in the presence of the particles, which has been shown to affect the RWM stability. The inclusion of kinetic damping from the thermal ions results in an increase in the stability limit for RWMs. This thermal ion damping depends sensitively upon the plasma rotation, since the kinetic stabilisation occurs due to a resonance between the thermal ion precession frequency and the Doppler-shifted mode frequency. For the low plasma rotation anticipated in ITER, the thermal damping contributes more than the non-resonant fast ion effects. If the rotation were to be even smaller than expected, the thermal stabilisation is increased as the mode frequency approaches the ensemble-averaged precession frequency. At the expected mode frequency, the thermal ion effects do not passively stabilise the RWM at the target plasma pressure for the advanced scenario in ITER. However, when the non-resonant damping arising from finite orbit width effects of very energetic alpha, NBI, and ICRH particles is included, the RWM is predicted to be stable at the target b N ¼ :95 in ITER advanced scenario. This effect is only captured by full drift kinetic modelling, which Downloaded 9 Jul to Redistribution subject to AIP license or copyright; see

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