Kinetic calculation of the resistive wall mode and fishbone-like

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1 inetic calculation of the resistive wall mode and fishbone-like mode instability in tokamak G.Z. Hao 1,2, S.X. Yang 3, Y.Q. Liu 4,2, Z.X. Wang 3, a, A.. Wang 2, and H.D. He 2 1 University of California, Irvine, California, Southwestern Institute of Physics, P. O. Box 432 Chengdu 6141, China 3 ey Laboratory of materials Modification by Beams of the Ministry of Education, School of Physics and Optoelectronic Technology 4 CCFE, Culham Science Centre, Abingdon OX14 3DB, United ingdom a ABSTRACT inetic ects of both trapped thermal and energetic particles on the resistive wall mode (RWM and on the fishbone-like mode (FLM are investigated in theory. Here, the trapped thermal particles include both ions and electrons. The FLM is driven by trapped energetic particles. The results demonstrate that thermal particle collisions can either stabilize or destabilize the RWM, depending on the energetic particle pressure. Furthermore, the critical value of h h for triggering the FLM is increased when the thermal particle contribution is taken into account. The critical value sensitively depends on the plasma collision frequency. In addition, the plasma inertia is found to have a negligible influence on the FLM. I. INTRODUCTION In magnetic confinement fusion devices, the achievable plasma pressure is limited by several types of magneto-hydrodynamic (MHD instabilities, such as the tearing mode and the resistive wall mode (RWM. The RWM is a global MHD instability with external kink mode structure. Its growth time is comparable with the eddy current decay time of the resistive wall surrounding the plasma. The RWM is unstable when the plasma pressure is in the range of no wall N and ideal wall N. Here, nowall ideal wall N N N are the normalized pressure limits, without and with an ideal conducting wall, respectively. 1 An unstable RWM can cause plasma disruption and discharge termination. For the steady state operation of tokamak in the high pressure regime, such as that in the future ITER advanced scenarios, it is necessary to suppress the RWM instability. Experimental results indicate that plasma toroidal rotation, and/or its shear, has a significant stabilizing ect on the RWM. 2, 3 Theoretical models have shown that the 1

2 combination of the plasma flow with certain energy dissipation mechanisms can completely stabilize the RWM instability In order to explain the suppression of the RWM by a very low critical toroidal rotation speed, drift kinetic models have been developed, which predict that the mode-particle resonance has a significant contribution to suppression of the RWM. Recent observations in DIII-D tokamak have shown that the resonant interaction between the RWM and the thermal particles is the likely cause of the improved plasma stability at very slow plasma rotation. 14 Some instability scenarios of the RWM, including thermal and alpha particles, have been discussed in Ref. 15, though without taking account into the thermal particle collision ects. Following a perturbative approach, it has been shown that the plasma collisionality has a significant destabilizing ect on the RWM, when the plasma is in the resonance phase. 16 The kinetic RWM theory model is applied to include the kinetic contribution from trapped energetic particles as well. 17 It shows that trapped energetic particles (EPs play a stabilizing role on the RWM, due to the mode resonance with the precessional drift motion of EPs. On the other hand, when the EPs pressure exceeds a critical value, an instability, with global mode structure and labeled fishbone-like mode (FLM, can be triggered Since in real experiments, kinetic contributions from both thermal and energetic particles co-exist, it is therefore worthwhile to investigate combined influence of both types of particles on the RWM and on the FLM. This is the purpose of the present study. In particular, the ect of thermal particle collisions on these instabilities is non-perturbatively (to be specified later on investigated, in the presence of trapped EPs. The remainder of the paper is organized as follows. Section II describes the extended RWM dispersion relation including kinetic contributions from both trapped thermal and energetic particles. Based on this dispersion relation, kinetic ects of trapped particles on the RWM stability are studied in Section III. Section IV reports results for the FLM. Section V draws the main conclusions. II. INETIC RWM DISPERSION RELATION INCLUDING BOTH THERMAL AND ENERGETIC PARTICLES The extended dispersion relation of the RWM, with the kinetic contributions of 1, 7, both trapped thermal particles and EPs, can be written as i in E W W D i w, 2 i in W W 2 E b where r i is the eigenvalue (complex frequency of the mode in the laboratory frame, with 2m 2m respectively. w bd 1 a / b r and being the real frequency and the growth rate, is the wall time, with a, b, d, m,, and being the plasma minor radius, the wall position, the wall thickness, the (1 2

3 poloidal mode number, the wall conductivity, and the vacuum permeability, respectively. E is the EB rotation frequency. is the inertial term, which is normally neglected, since the RWM frequency is normally very low. In this work, the ect of on the FLM will be investigated, and the results will show that this inertial term is also not important for the FLM. W and Wb represent the perturbed fluid potential energies without and with an ideal conducting wall, respectively. W Wkt Wkh is the perturbed drift kinetic energy, where Wkh and Wkt are contributions from the trapped thermal particles and EPs, respectively. In this work, for simplicity, we assume a cylindrical equilibrium with a parabolic 2 pressure profile Pr P 1 rˆ, where P. J is the (constant plasma J 2 /4 current density, yielding a flat safety factor profile qr q 2 B / R J. In addition, we choose the external kink mode eigenfunction to calculate the perturbed energies in Eq. (1. Following Ref. 2, the perpendicular displacement of the instability is taken as m1 ( ˆ ( i mn r / ξ amr e ie e F, with n being the toroidal mode number and m the poloidal mode number. We choose m/ n 2 /1 in this work. and are the poloidal and toroidal angles, respectively; F ( m nq a / ( Rq and rˆ r / a. We also assume an anisotropic slowing down equilibrium distribution function F n B B E 3/2 t ( for EPs, induced by neutral beam injection, where B B / Ek is defined as the pitch angle of the trapped EPs. and m v 2 /2B E m v 2 /2 m E are the magnetic momentum and the kinetic energy of EPs, k h h respectively. v and v denote the particle velocity and its component perpendicular to the magnetic field, respectively. r and R are the radial variable and major radius of the torus, respectively; B B 1 cos ò c denotes the equilibrium magnetic field to the lowest order in ò r/ R. The kinetic contribution from EPs is obtained as, 17 h ˆ B R ˆ ˆ 2 1 Wh A B 2 a 7 2 h 12 (1 {( ln(1 2 ˆ 5 ˆ ( A B [2( ln( ]} M where ( i r / ds i / d s ra / ds,, (2 ˆ 1 (2 2(2E 1 A m 2 (2E q q, (3 1 d E 1 E kt 1 1 d (2E ( [ ] (2E B dk B q 2 B 2q dk t t 3

4 ˆ 1 2kt 1 E d E (2E B [ (2 1] (, (4 q B dk q t M 3 R 1 2kt q 1 d h 2a 2m 3/ 2 q dk 2 4(1.5 B ( (2E t. ( Here, Wˆh is the normalized form following Wˆ 2 R F W / B a m. h is the normalized energetic particle pressure. 17 ( /[ ( ] ds 2 a Emq b a mhac R denotes the magnetic precession frequency of the trapped EPs at the plasma edge, with E being the birth energy of the trapped EPs. m 2 1/2 ò t t and b(a 2 2 / ( B ( kt / (a 2 2 / B [2E( k ( k ]/ ra where k ( t and ( t kinds, respectively, with the argument kt B E( kt E and ( kt. ò, Ek are the complete elliptic integrals of the first and second ra (1/ ò1 / (2 ò. In the following, Following the same normalization as for Wˆkh, the kinetic contribution from trapped thermal particles, calculated in Ref. 2, can be rewritten as ˆ J Wkt drr 1 r d G 4k G Here, / 1 2 m1 ò /2 2 1/ 1 1 t B ò ei, 2 5/2 ˆ k ˆ a n k e d ˆ ˆ k bk nb b n b d n 2 2ò / k. (6 B is defined as the pitch angle of the thermal particle, with and k being the magnetic momentum and the particle kinetic energy, respectively. i and e represent the ion and electron species, respectively. The factor 1/ d n represents the mode-particle resonance condition, with 3 n / 2 i /, / a N T ds b T ds, n n i / ds, d d / ds. T and N are the diamagnetic drift frequencies due to the plasma 2q k 2E 1 temperature and density gradients, respectively. d d aer rˆ 2 is the bounce averaged toroidal precession frequency of trapped particle with 1 for ions and 1 for electrons. e is electron charge. ef f d We assume that the equilibrium plasma density is constant, yielding N. 2mm ò with ij i 1 3/2 ˆ r k m j n Z e ln 4 4 i i i 3/2 2 3/2 12 ò mt i i represents the ective collisionality frequency between the test particles j and the thermal ions i, where d 4

5 m m m / m m is the reduced mass, Z the particle charge number, ln the ij i j i j Coulomb logarithm, ò the vacuum permittivity, and ˆ / T. 16 The thermal k k j electron collision frequency is 2 m / m 1 times larger than that of thermal ions. In the following calculation, we scan parameter collisionality on the instability. The function G p where p, 1. /2 i e to investigate the ect of G p is defined as 2 m p p 2 12 tsin sin 1 tsin p 1 d k k Based on the chosen equilibrium and the form of the plasma displacement, the normalized form of other perturbed energies can be analytically obtained as follows. 2 ˆ 2 2, (8 A m m nq 2 ˆ 4 1 W 1, 2 m q m nq, (7 (9 m nq 2 ˆ Wb. 2 2m m q m nq 1b (1 where / is the Alfvén time at the magnetic axis. 1 A A R B It is worthwhile to point out that the kinetic contribution in our model includes both thermal and energetic particles. In addition, the collisionality of the thermal particle is also taken into account. Therefore, our present model includes a rather complete set of kinetic physics, compared to the previous analytic work. We note that the above fluid potential energies, Wˆ and Wˆb, do not explicitly include the equilibrium pressure of the plasma. This is because a current driven RWM is assumed in our model. On the other hand, the eventual drift kinetic ects, that we shall study in the following, depend only on the values of the fluid potential energy perturbations, not on the origin of the mode drive. In other words, our final qualitative conclusions do not depend on the choice of the fluid model for the RWM. For convenience, we shall neglect the over-hat on the normalized perturbed energies in further analysis. III. INETIC CALCULATION ON THE RWM A. Collisionless case Usually, the plasma inertia can be neglected for the RWM, due to its small frequency. Under this approximation, the RWM dispersion is written as: 5

6 D i. w W W b W W As shown in Section II, the perturbed drift kinetic energy is eventually a non-linear function of the mode s eigenfrequency. In this work, we numerically solve the non-linear equation (11 with respect to the eigenfrequency. In this sense, we call the solution non-perturbatively. We choose an equilibrium with a 1m, R 3m, B 2.3T, 6 1 1, m (11 q, E 85eV and the pitch angle of trapped energetic particles B.95. A resistive wall with thickness d.1a is located at b 1.2a. The m/ n 2 /1 perturbation is considered. These operating conditions are chosen such that, (i the fluid RWM is unstable without the drift kinetic damping, with W.76 and.22 ; (ii the fluid potential energy is W b comparable to the drift kinetic energy. Non-perturbative computations from MARS- also indicate that the kinetic contribution is generally comparable with the fluid potential energy. 24 Figures 1(a and (b show the mode s normalized growth rate and real frequency, respectively, versus EP pressure h / m. Here, is the total equilibrium pressure of the plasma. For comparison, three cases are considered: (a includes trapped EP contribution; (b EPs and trapped thermal ions; (c W only W includes contributions from both trapped W includes full kinetic contributions (including trapped EPs, thermal ions, and thermal electrons. The ion-electron temperature ratio is assumed to be 1, yielding C T T T /.5. Particle collisions are neglected p i i e in this subsection. Figure 1(a indicates that the trapped thermal particles have a stabilizing ect on the RWM, in the region of.13, compared with a case of including the EP kinetic contribution alone. On the other hand, the thermal particles destabilize the mode when.13, increasing the critical value for complete suppression of the instability. Figure 1(b shows that the mode s real frequency / is r r A reduced when thermal particles are considered. The main reason for these results is the modification of Re W by thermal particles (shown in Fig.2. This can be qualitatively understood byre-writing the dispersion relation (11 into real and imaginary parts 25 Wb W Wb ReW w W ReW Im W r w 2 2 b Wb W ImW Re Im 2 2 Wb W W 1,. (12 (13 6

7 Equation (12 implies that the growth rate decreases with increasing ReW under the condition ImW W ReW and W W Im W W ReW b b b Re. When, the growth rate of the instability significantly increases with decreasing Im W frequency decreases with increasing Re W W b. Eq. (13 indicates that the mode s real (shown in Fig.2(a, when ReW. However, when ImW W ReW b eigenvalue sensitively depends on both ImW and Re W nonlinearly depends on the mode s eigenfrequency. The latter affects the resonance factor, which in turn determines the mode s eigenvalue., the mode. In fact, Eq. (11 W through FIG.1 (color online (a the normalized growth rate, and (b the real frequency of the RWM, versus the trapped EP pressure /. The trapped EP pitch angle is chosen as B.95. h 3 / , p.5 Plasma rotation frequency A C,.. Black curve: trapped EPs only; Blue curve: both EPs and thermal ions; Red curve: trapped EPs, thermal ions and thermal electrons. Figure2 shows the corresponding perturbed drift kinetic energy W versus for the case shown in Fig. 1. It is evident that the real part, Re( W, is enhanced by the thermal particle contribution when.13 imaginary part, Im( W, hardly changes when,. On the other hand, the.13 for different cases. However, the imaginary part slightly reduces for the cases including thermal particles in the region of.13. In the presence of plasma rotation of 3 / A , the mode resonance with the trapped thermal particles is very weak. Hence, the imaginary part Im( W mainly comes from the mode resonance with the precession motion of trapped EPs in this case. We point out that, at 7

8 , thermal particles yield a finite Im( W Fig.6, which is much smaller than W ReW rate is determined by the real part Re W w corresponding real frequency r w at b, in the order of. Hence, at 3 1 (shown in, the growth, as can be seen from Eq.(12. The is very small, due to ImW as shown by Eq.(13. This is consistent with the red curve shown in r w Fig.1(b. Despite the fact that the direct resonance between the mode and the trapped thermal particles is weak, the latter can affect the instability by enhancing the real part of the kinetic energy Re( W. The increase of Re( W by the thermal particle contribution results in a decrease of w in the region.13. In the relatively high region (.13, additional thermal particle contribution yields Im W W ReW. The Re( and Im( have a comparable b influence on the mode s complex eigenvalue. As a result, in W W.13, the combination of Im( W and Re( W enhances the instability, when the thermal particle contribution is included. FIG.2 (color online (a the real and (b the imaginary part of kinetic contribution, versus /, are calculated for the three cases as in Fig.1. Here, W.76 and.22 h W b 8

9 FIG.3 (color online Im( W and Re( W in the complex plane for the same cases shown in Fig.1. Dashed curves indicate constant values of the growth/damping rate of the mode. Arrows on solid curves indicate the increase of. Figure 3 plots the perturbed drift kinetic energy in the complex plane, for three cases as shown in Fig. 1. We re-write Eq. (12 as follows 26 W 2 W 2 2 Re Im =, (14 Wb W w Wb W 2 21 W b W 21 w w, (15. (16 The marginal stability curve is obtained by setting w in Eq. (14. The RWM is unstable when Re W and Im W curve. It is interesting to note that the real frequency RWM can be obtained as are located below the marginal r of the marginally stable r w Wb W W Re W b 1. (17 Equation (17 indicates that r w of a marginally stable RWM does not depend on 9

10 Im. That is, only the Re W W marginal stability point. Here, the thermal particles enhance W ReW frequency near the marginal point (shown in Fig.1(b. b can determine the mode real frequency at the W and Wb are fixed. As shown in Fig.3,, which in turn reduces the mode s real B. Collisional plasmas In the absence of trapped EPs, it has been shown that the kinetic damping of the thermal particle collision on the RWM depends on the plasma rotation and the mode s real frequency. 27 In this subsection, the results with the inclusion of trapped EPs are presented. For the chosen plasma rotation 3 / in this work, the A mode resonance with trapped EPs is dominant over thermal particles. In the calculation, both the ion-ion and the electron-ion collisions are considered. Since the collision between energetic particles and thermal ions is relatively low, the energetic particle collisionality is neglected here. FIG.4 (color online (a the mode growth rate and (b the mode frequency versus kinetic case, with B.95, denote the different choices of collision frequency in full , and Cp.5. The curves with different color Figure 4(a shows that the plasma collisionality has a destabilizing ect on the mode in the region increasing.1, and a slight stabilizing ect with further. In the calculation, we assume that the collisionality depends on the particle kinetic energy as 3/2 ˆk the mode s real frequency in the whole.the plasma collisionality leads to slight increase of region. The corresponding Re W and Im W are plotted in Fig.5. Since the resonance between the mode and thermal particles is weak for the chosen rotation amplitude, the thermal particles are in the off-resonance phase, and Re W from kt 1

11 thermal particles decreases with collision frequency. 25 The reduction of ReW due to collision results in the reduction of Re W in the whole kt region. Collision slightly increases the imaginary part Im W. However, Im W mainly comes from the mode resonance with trapped EPs. The decrease of Re W still results in the reduction of the mode real frequency, based on Eq.(13. In addition, in the lower Im w W W ReW b W b region where, the growth rate can be approximately written as b W W Re W determining in the lower Im w w 1. Hence, W ReW. Reduction of Re W b is the dominant factor yields a destabilizing ect on the mode region, due to the collision ect. In the higher W W ReW b region where, the growth rate can be approximately written as Wb W Wb W 1, and reduction of Re W ImW 2 Re decrease of the mode s growth rate, as shown in Fig.4. results in a FIG.5 (color online The (a real and (b imaginary parts of kinetic contribution, versus the trapped EP, The other parameters are the same as those presented in Fig.4. Here, W.76 and.22. W b 11

12 FIG.6 (color online The (a real and (b imaginary parts of ion kinetic contribution, and (c real and(d imaginary parts of electron kinetic contribution versus the trapped EP pressure, respectively. The other parameters are the same as those used in Fig.4. versus Figure 6 shows the perturbed kinetic energies of trapped ions and electrons, for different values of, with a fixed plasma rotation frequency The plasma is on the off-resonance phase with thermal particles. In addition, the finite rotation prevents the cancellation between Im W Im Wkte and, which is discussed in Ref.2. Figure 6 demonstrates that the real part of the kinetic contribution is much larger than the imaginary part for thermal particles. Moreover, collisions tend to increase Im W while decreasing Re W fixed kti kti at a, which is consistent with the analysis discussed in Ref.16. At a given collisionality, the thermal ion contribution depends on the trapped EP pressure especially in the region.1.15 kti,. This is because the trapped EPs introduce significant changes to the mode s growth rate and real frequency in region, which in turn affects Wkti For trapped thermal electrons, Re W is also much larger than Im W Similar to thermal ions, the collisionality has an ect of reducing Re W fixed kte kte. kte at a. Since the electron collisionality is much larger compared to the ion-ion collision, for the case of , the electron collision frequency is dominant over other frequencies in the kinetic resonances for electrons, resulting in both Re and Im W Wkte being independent of the mode s eigenvalue. In other kte words, the electron kinetic contribution does not depend on EP pressure sufficiently high collision regime. We also point out that, as, in a approaches to, 12

13 both Wkti and kte W recover that values of the collisionless case. C. Effect of plasma rotation on the kinetic contribution At certain values of plasma rotation frequency,when thermal particles are in resonance with the mode, the kinetic contribution from bulk particles should have a large contribution. We thus calculate the kinetic contribution of thermal ions while scanning the plasma rotation frequency. of Figure 7 plots the components of. We find that Im Wkti than Im W kh peaks near W as a function of, for different values. Im W at small rotation frequency.2, at fixed is larger kti. Im W is nearly symmetric with respect to the sign of, due to the Maxwell distribution of the thermal ions. This shows that, at slow flow, the dominant kinetic damping comes from the bulk plasma. On the other hand, at faster flow (e.g..5 in our case, the kinetic damping from trapped EPs becomes stronger. The results qualitatively imply that the mode resonance with the thermal particles and with EPs occur in the lower and higher region, respectively. Moreover, the amplitude of Re( W kti is comparable to that of Re( W kh for the considered cases. For trapped EPs, as expected, Re( W kh and Im W are clearly proportional to the EPs pressure kh at fixed. We point out that the dependence of thermal electrons kinetic contribution is qualitatively similar to that of thermal ions. kti, FIG.7 (color online The (a real and (b imaginary parts of Wkti (solid curves and trapped EP W kh (dashed curves, versus the toroidal rotation frequency. The other parameters are given as B.95, and. IV. INETIC FISHBONE-LIE MODE 13

14 A. Collisionless model Trapped EPs can stabilize the RWM, but can also trigger a fishbone-like mode (FLM instability as has been shown in both experiments and theory It is found that when the EPs pressure exceeds a critical value c, a FLM instability can be excited. In this section, we consider the role of thermal species on the triggering of the FLM. FIG.8 (color online The (a normalized growth rate and (b real frequency of the FLM versus the, with , B.95,, and Cp.5. Black curve: only includes trapped EPs ; Blue curve: both EPs and thermal ions; Red curve: EPs, thermal ions and thermal electrons. Figure 8 plots the FLM growth rate and real frequency versus, with and without thermal particle species. We find that c is substantially enhanced in the presence of the kinetic contribution from thermal particles (either trapped ions or both trapped ions and electrons. The real frequency of the FLM, at the triggering point, is also increased. These results are qualitatively consistent with those from the full toroidal, non-perturbative MHD-kinetic hybrid computations by the MARS- code. 33 At the marginal stability point (the triggering point with w, we have the following condition satisfied, based on Eqs. (12 and (13, b b Re W W W Im W W W. 2 2 Since the drift motions of the bulk plasma particles are off-resonance for the chosen parameters in this section, Im W Im W ImW. However, Re W with Re W Re kh kh kt (18 from thermal species is small (i.e. from thermal particles is comparable kt, being close to that for the RWM (shown in Fig.6. Furthermore, W, and Re. 16 Whist, kh W kh Re. As a result, in order to W kt 14

15 keep the balance of Eq. (18 in the presence of thermal species, a larger value of is required. On the other hand, as c c changes, the real frequency of the FLM at the triggering point also changes, which in turn affects the value of Figure 9 plots the perturbed W. 17 W as a function of, corresponding to the cases shown in fig.8. It shows that the amplitude of Re W is slightly increased with the inclusion of the thermal particle contributions. However, for all cases, Re W is close to W b.22. Figure 9 also shows that, at the marginal stability point, for all the three cases considered here, Im W Re Wb of magnitude larger than W EPs, trapped ions and electrons, Im.4, W based on Eq. (13, 1/ ImW the amplitude of Im W r w W is about one order. For example, in the case of including Re W.6. Hence, at the marginal point. Figure 9(b shows that, at the marginal stability point, is reduced in the presence of thermal particle contributions. Hence, the real frequency of the FLM (at the triggering point, with the inclusion of thermal species, is larger than the case of only trapped EPs. b FIG.9 (color online The (a real and (b imaginary parts of kinetic contribution versus the EP The other parameters are the same as those presented in Fig.8., B. Collisonal model This section reports the ects of the plasma collisionality FLM. Figure 1 shows that c decreases with increasing on the triggered, for the case with trapped EPs and with thermal ions. For the case of including trapped EPs and both thermal species, c is strongly enhanced at very low collision frequency. As 15

16 increases, reaches its minimum at c with further increasing of collisionality, the required , followed by a slight increase. This suggests that, for a plasma regime with lower is higher. In fact, Re( collisionality 16, resulting in a reduction of the required c c W kt decreases with with increasing collision frequency, when the thermal species kinetic damping is included. On the other hand, collisions of thermal species generally increase the required case of including the contribution of trapped EPs alone. c, compared with the FIG. 1 (color online The critical value c for triggering FLM versus the thermal species collisionality, with , Cp.5. Both rotation frequency and collision frequency are normalized by the Alfven frequency. Figure 11 shows the growth rate and real frequency of the FLM as functions of, for different choices of and. The chosen and values yield unstable FLMs near the marginal stability point. Figure 11(a shows that generally destabilizes the FLM. Moreover, frequency. Figure 12 plots Re W and Im W slightly increases the mode real in the complex plane. It shows that the closer the curve is to the point ( Wb,, the more sensitive the mode eigenvalue becomes, with respect to the change of the collisionality. Generally, along the solid curves, W ReW ImW, and Im W b roughly decreases with increasing. As a result, the approximation 16

17 Wb W Wb W ImW Re 1 w 1 and 2 r w based on Im W Eqs.(12-(13, predicts that both the growth rate and the real frequency of the mode increase with increasing. The collisionality dependences of the components of the perturbed kinetic energy are plotted in Fig.13. It is evident that the value of Re( W Re( W tends to be close to W b.22 with varying collisionality. In addition, in the low region, Im W is dominant over Im W kh lower rotation (i.e. kt. In the higher, Im W is comparable with Im W kt kh kt region with kh. That is, the mode resonance contribution from thermal particles can be comparable with that from trapped EPs in the slowly rotating plasma. FIG. 11 (color online The (a modes growth rate and (b real frequency, versus collision frequency. Different color curves label different choose of and. Here, B.95, and Cp.5. 17

18 FIG. 12 (color online Re W and ImW in the complex plane, corresponding to the cases shown in Fig. 11. Dashed curves indicate constant values for the growth/damping rate of mode. Arrows along the curves indicate the direction of increasing collision frequency. FIG. 13 (color online The profile of (a real and (b imaginary parts of Wkh, and the profile of (c real and (d imaginary parts of Wkt versus collision frequency, corresponding to the case in Fig.11 Figure 14 plots the radial profiles of the drift kinetic energy perturbation, 18

19 associated with thermal particle species, for different choices of Different values of radial profile of at.24. result in different eigenvalues, which in turn affect the. At low collisionality, e.g. or Wkt , Re( W kti is comparable with Re( W kte. At, Im( W kti and Im( W kte are comparable, but with opposite signs, thus tending to cancel each other. At high collisionality, e.g , Re( W kti Im( W kte is also much larger than Im( W kti is much larger than Re(, and W kte. Both of them tend to reduce the imaginary part of the total kinetic energy, since Im( W kh (shown in Fig.13(b. Here, the profile of Wkt finally depends on the competition among various drift frequencies in the kinetic resonance Eq.(6. FIG. 14 (color online Radial profiles of (a real and (b imaginary parts of the perturbed drift kinetic energy of thermal ions and electrons for various values of, with , C.5, p.24. The critical value c also depends on the toroidal rotationfrequency. 34 Hence, it is worthwhile to investigate the combined ect of collisionality and plasma rotation on c. Figure 15(a shows that fixed rotation frequency. In addition, c generally decreases with collisionality c in the lower range is qualitatively larger than that in the higher range. Figure 15 reveals that there are two boundaries, where rotation. The minimum c is sensitive to either the collisionality or the plasma is obtained near the point ( 8 1 3, c at Furthermore, Fig. 15(b shows that the dependence of the real frequency of the FLM on and, at the triggering point, is similar to that for c. Basically, this 19

20 frequency c region. Re( W r, in the higher c region (red color, is larger than that in the smaller For the same case reported in Fig.15, Figs. 16(a and (b show the corresponding and Im W shows that Re( W in the and b domain, respectively. Figure 16(a W in the whole 2D domain. Equation (18 suggests that, at the critical point for the FLM, the change of Im W is synchronized with that of Re( W. This is demonstrated in Figs. 16(a and (b. In addition, at the marginal point, proportional to Im W r w Wb W ImW, the mode s real frequency is inversely, which is also demonstrated in the and by Figs. 15(b and 16(b. Figure 16 reveals two boundaries, where to either or. domain, W is sensitive FIG. 15 (color online The (a c and (b the initial mode frequency r of the FLM, vary with toroidal rotation and the thermal particle collisionality, with Cp.5. 2

21 Fig. 16 (color online The (a real and (b imaginary part of c. C. EFFECT OF INERTIAL TERM ON THE FLM W at the marginal stable point In this section, the ect of plasma inertia on the FLM is investigated. Figure 17 shows that the inertial term has little influence on the mode real frequency in the whole in the high region. In fact the inertia only has a slight destabilizing ect on the mode, region where the mode real frequency is relatively large. However, for the high frequency energetic driven mode (frequency > 1kHz observed in sperical tokamaks, the plasma inertia may contributeto the mode growth rate. Fig. 17 (color online The normalized (a mode growth rate and (b real frequency r of the FLM versus for the case with (dash curves and without (solid curves plasma inertia term. Different color curves denote the cases with different thermal collisionality. The other parameters are given as , B.95, and C.5. p 21

22 V. CONCLUSION inetic ects of both trapped thermal particles and energetic ions on the RWM and the FLM are investigated, using the drift kinetic RWM dispersion relation. We find that the kinetic damping/destabilization the instability significantly depends on the plasma rotation frequency. At a fixed energetic particle pressure, the thermal particle resonance contribution can be dominant over the EP contribution in the slow rotation regime. At fast plasma flow, the EP s resonant contribution becomes dominant. In addition, the non-resonant contribution of thermal particles plays a stabilizing (destabilizing role on the mode in the low (high region, at fixed plasma rotation. The thermal particle collisionality has a destabilizing (stabilizing ect on the mode at the low (high. Thermal particles can strongly increase the EP pressure threshold value c for driving the fishbone-like mode. This is because Re( W kt of thermal particles cancels that of EPs, and a larger EP Re( W kt required is required to balance Re(. However, c decreases with plasma collisionality, resulting in the reduction of the c, especially for the case of including the thermal electron-ion collisionality. In addition, the EPs kinetic contribution W kt Wkh is also modified by the thermal particle collisionality, through the nonlinear dependence of mode s eigenvalue. We show that the minimum of localizes near the point ( 8 1 3, Wkh on the c, in the 2D rotation-collisionality domain, 1 1 4, as a result of synchronization ect of rotation and collisionality on the FLM. Qualitatively, in the low collisionality plasma, it is more difficult to drive the fishbone-like mode. In addition, the plasma inertia term slightly destabilizes the FLM instability. In this work, the theoretical model is applied with the large aspect ratio. We have also assumed flat safety factor and density profile and a single pitch angle for all EPs. In the future, the self-consistent simulations with realistic profile, in a full toroidal geometry, will be carried out, especially to investigate the combined ect of plasma flow and collisionality on the RWM instability as well as on the threshold value for triggering the FLM. c ACNOWLEDGMENTS S.X.Yang thanks the assistance from X. X. Zhang. This work was supported by the U.S. Department of Energy under DE-FG3-2ER54681 and DE-AC2-9CH This work was partly supported by National Magnetic confinement Fusion Science Program (No. 214GB124, 215GB144 and No.213GB1129 and by National Natural Science Foundation of China 22

23 (No , No , No , and No This work was part-funded by European Union s Horizon 22 research and innovation program under grant agreement number and from the RCU Energy Program [grant EP/I5145]. The views and opinions expressed herein do not necessarily reflect those of the European Commission. References B. Hu and R. Betti, Phys. Rev. Lett 93, 152 (24. E. J. Strait, T. S. Taylor, A. D. Turnbull, J. R. Ferron, L. L. Lao, B. Rice, O. Sauter, S. J. Thompson, and D. Wróblewski, Phys. Rev. Lett 74, 2483 (1995. S. A. Sabbagh, R. E. Bell, M. G. Bell, J. Bialek, A. H. Glasser, B. LeBlanc, J. E. Menard, F. Paoletti, D. Stutman, E. Fredrickson, A. M. Garofalo, D. Gates, S. M. aye, L. L. Lao, R. Maingi, D. Mueller, G. Navratil, M. Ono, M. Peng, E. Synakowski, W. Zhu, and N. R. Team, Phys. Plasmas 9, 285 (22. A. Bondeson and D. Ward, Phys. Rev. Lett 72, 279 (1994. R. Betti and J. Freidberg, Phys. Rev. Lett 74, 2949 (1995. R. Fitzpatrick and Y. A. A, Nucl. Fusion 36, 11 (1996. M. S. Chu, J. M. Greene, T. H. Jensen, R. L. Miller, A. Bondeson, R. W. Johnson, and M. E. Mauel, Phys. Plasmas 2, 2236 (1995. Y. He, Y.Q. Liu, Y. Liu, C. Liu, G. Xia, A. Wang, G. Hao, L. Li, and S. Cui, Phys. Plasmas 23, 1256 (216. G. Xia, Y. Q. Liu, Y. Liu, G. Z. Hao, and L. Li, Nucl. Fusion 55, 937 (215. C. Liu, Y. Q. Liu, Y. Liu, G. Z. Hao, L. Li, and Z. R. Wang, Nucl. Fusion 55, 6322 (215. I. T. Chapman, C. G. Gimblett, M. P. Gryaznevich, T. C. Hender, D. F. Howell, Y. Q. Liu, S. D. Pinches, and J. E. Contributors, Plasma Phy. Control. Fusion 51, 5515 (29. Y.Q. Liu, M. S. Chu, I. T. Chapman, and T. C. Hender, Nucl. Fusion 49, 354 (29. Y. Q. Liu, M. S. Chu, W. F. Guo, F. Villone, R. Albanese, G. Ambrosino, M. Baruzzo, T. Bolzonella, I. T. Chapman, A. M. Garofalo, C. G. Gimblett, R. J. Hastie, T. C. Hender, G. L. Jackson, R. J. L. Haye, M. J. Lanctot, Y. In, G. Marchiori, M. Okabayashi, R. Paccagnella, M. F. Palumbo, A. Pironti, H. Reimerdes, G. Rubinacci, A. Soppelsa, E. J. Strait, S. Ventre, and D. Yadykin, Plasma Phys. Control. Fusion 52, 142 (21. H. Reimerdes, J. W. Berkery, M. J. Lanctot, A. M. Garofalo, J. M. Hanson, Y. In, M. Okabayashi, S. A. Sabbagh, and E. J. Strait, Phys. Rev. Lett. 16, 2152 (211. B. Hu, R. Betti, and J. Manickam, Phys. Plasmas 12, 5731 (25. J. W. Berkery, S. A. Sabbagh, R. Betti, R. E. Bell, S. P. Gerhardt, B. P. LeBlanc, and H. Yuh, Phys. Rev. Lett. 16, 754 (211. G. Z. Hao, A.. Wang, Y. Q. Liu, and X. M. Qiu, Phys. Rev. Lett. 17, 151 (211. G. Matsunaga, N. Aiba,. Shinohara, Y. Sakamoto, A. Isayama, M. Takechi, T. Suzuki, N. Oyama, N. Asakura, Y. amada, T. Ozeki, and JT-6 Team, Phys. Rev. Lett. 13, 451 (29. W. W. Heidbrink, M. E. Austin, R.. Fisher, M. Garcia-Munoz, G. Matsunaga, G. R. Mcee, R. A. Moyer, C. M. Muscatello, M. Okabayashi, D. C. Pace,. Shinohara, W. M. Solomon, E. J. Strait, M. A. Van Zeeland, and Y. B. Zhu, Plasma Phys. Controlled Fusion 53, 8528 (

24 M. Okabayashi, G. Matsunaga, J. S. degrassie, W. W. Heidbrink, Y. In, Y. Q. Liu, H. Reimerdes, W. M. Solomon, E. J. Strait, M. Takechi, N. Asakura, R. V. Budny, G. L. Jackson, J. M. Hanson, R. J. La Haye, M. J. Lanctot, J. Manickam,. Shinohara, and Y. B. Zhu, Phys. Plasmas 18, (211. Y.Q. Liu, M. S. Chu, C. G. Gimblett, and R. J. Hastie, Phys. Plasmas 15, 9255 (28. Y.Q. Liu, M. S. Chu, I. T. Chapman, and T. C. Hender, Phys. Plasmas 15, (28. S. W. Haney and J. P. Freidberg, Phys. Fluids 1, 1637 (1989. Z. R. Wang, S. C. Guo, and Y. Q. Liu, Phys. Plasmas 19, (212. G. Z. Hao, Y. Q. Liu, A.. Wang, and X. M. Qiu, Phys. Plasmas 19, 3257 (212. J. W. Berkery, S. A. Sabbagh, R. Betti, B. Hu, R. E. Bell, S. P. Gerhardt, J. Manickam, and. Tritz, Phys. Rev. Lett 14, 353 (21. S. X. Yang, S. Wang, Y. Q. Liu, G. Z. Hao, Z. X. Wang, X. M. Song, and A.. Wang, Phys. Plasmas 22, 975 (215. M. Okabayashi, G. Matsunaga, J. S. degrassie, W. W. Heidbrink, Y. In, Y. Q. Liu, H. Reimerdes, W. M. Solomon, E. J. Strait, M. Takechi, N. Asakura, R. V. Budny, G. L. Jackson, J. M. Hanson, R. J. La Haye, M. J. Lanctot, J. Manickam,. Shinohara, and Y. B. Zhu, Phys. Plasmas 18, (211. M. Okabayashi, I. N. Bogatu, M. S. Chance, M. S. Chu, A. M. Garofalo, Y. In, G. L. Jackson, R. J. La Haye, M. J. Lanctot, J. Manickam, L. Marrelli, P. Martin, G. A. Navratil, H. Reimerdes, E. J. Strait, H. Takahashi, A. S. Welander, T. Bolzonella, R. V. Budny, J. S. im, R. Hatcher, Y. Q. Liu, and T. C. Luce, Nucl. Fusion 49, 1253 (29. G. Matsunaga, N. Aiba,. Shinohara, Y. Sakamoto, A. Isayama, M. Takechi, T. Suzuki, N. Oyama, N. Asakura, Y. amada, T. Ozeki, and J. T. Team, Phys. Rev. Lett 13, 451 (29. Y.Q. Liu, Nucl. Fusion 5, 958 (21. G. Z. Hao, A.. Wang, Y. Q. Liu, and X. M. Qiu, Phys. Rev. Lett 17, 151 (211. S.C.Guo, X.Y.Xu, Y.Q.Liu, and Z.R.Wang, Nucl. Fusion 56,566 (216. G. Z. Hao, Y. Q. Liu, A.. Wang, and X. M. Qiu, Phys. Plasmas 19, 3257 (

25 4 3.5 (a 4 x 1 3 (b γτ w Ω r T i +T i +T e β 1 +T i +T i +T e β

26 .5.35 Re(δ W.5 +T i +T i +T e Im(δ W T i +T i +T e.1 (a β.1.5 (b β

27 .5 +T i γτ w =.3 γτ w =. Im(δ W T i +T e γτ w = Re(δ W

28 3 (a 2.5 x 1 3 (b γτ w ν =. ν = ν = ν = β Ω r ν =. ν = β

29 Re(δW ν =. ν = ν = Im(δW.35 ν =..3 ν = ν = (a (b β β

30 Re(δW kti Re(δW kte (a β (c ν =. ν = ν = Im(δW kti Im(δW kte 12 x (b β x 1 3 (d β β

31 Re(δW β =.1 β =.1 β =.2 β =.2 β =.3 β =.3 (a.4 (b.3 Im(δW Ω Ω

32 1 8 (a +T i +T i +T e.5 (b.4 γτ w Ω r β β

33 +T i +T i +T e.26 (a.1 (b Re(δW Im(δW β β

34 β c T i +T i +T e ν

35 γτ w Ω =.,β =.258 Ω = ,β =.2435 Ω = ,β =.233 Ω = ,β =.26 (a Ω r ν (b ν

36 Im(δW γτ w =.3 γτ w =. γτ w =2.2 γτ w =5. δw b Re(δW Ω =.,β=.258 Ω =4.9e 3,β=.2435 Ω =7.2e 3,β=.233 Ω =9.4e 3,β=.26

37 Ω =.,β =.258 Ω =4.9e 3,β =.2435 Ω =7.2e 3,β =.233 Ω =9.4e 3,β =.26 Re(δW kt Re(δW kh (a Im(δW kh ν (c ν Im(δW kt.7 (b ν x (d ν

38 Re(δW kt ν =. ion ν =. electron ν =1.e 4 ion ν =1.e 4 electron ν =1.e 3 ion ν =1.e 3 electron.5 1 r (a Im(δW kt (b r

39

40

41 15 ν = (a.35 (b ν = γτ w 1 5 ν = ν = ν = ν = Ω r β β

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