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PRL 102, 135001 (2009)

PHYSICAL REVIEW LETTERS

week ending 3 APRIL 2009

Kinetic Effects of Energetic Particles on Resistive MHD Stability R. Takahashi,1 D. P. Brennan,1 and C. C. Kim2 1

Department of Physics and Engineering Physics, University of Tulsa, 800 South Tucker Drive, Tulsa, Oklahoma 74104, USA 2 Plasma Science and Innovation Center, University of Washington, Seattle, Washington 98195, USA (Received 16 September 2008; published 30 March 2009) We show that the kinetic effects of energetic particles can play a crucial role in the stability of the m=n ¼ 2=1 tearing mode in tokamaks (e.g., JET, JT-60U, and DIII-D), where the fraction of energetic particle frac is high. Using model equilibria based on DIII-D experimental reconstructions, the nonideal MHD linear stability of cases unstable to the 2=1 mode is investigated including a f particle-incell model for the energetic particles coupled to the nonlinear 3D resistive MHD code NIMROD [C. C. Kim et al., Phys. Plasmas 15, 072507 (2008)]. It is observed that energetic particles have significant damping and stabilizing effects at experimentally relevant , frac , and S, and excite a real frequency of the 2=1 mode. Extrapolation of the results is discussed for implications to JET and ITER, where the effects are projected to be significant. DOI: 10.1103/PhysRevLett.102.135001

PACS numbers: 52.30.Cv, 52.55.Fa

Introduction.—Understanding the physics of the onset and evolution of tearing modes in toroidal confinement plasmas is an important issue for the achievement of a burning plasma state in future experiments. Tearing modes are instabilities causing the topological change of field lines through reconnection to form magnetic islands, resulting in enhanced cross-field transport that can significantly reduce confinement. Experimental attempts to access the highest ¼ 0 P=B2 in tokamak discharges are typically terminated by the growth of a large m=n ¼ 2=1 tearing mode which often leads to disruption. In these experiments a large fraction of plasma particles can have large gyro-orbits and banana orbits due to their large kinetic energy. These high energy populations arise from various heating mechanisms, and can be accurately predicted [1]. The ensuing non-Maxwellian distribution function can strongly affect the equilibrium, stability, and transport in the plasma. The energetic particle stabilization of the internal kink mode has been studied extensively. For example, this model fits into the Porcelli formulation [2] which predicts the frequency and intensity of onset of the sawtooth mode. High energy particles can stabilize this mode and cause larger, more infrequent sawtooth oscillations. However, the effect these high energy particles have on linear and nonlinear resistive stability of m > 1 modes has only begun to be investigated. We examine the energetic particle effects on the nonideal MHD linear stability of the 2=1 mode using model equilibria based on DIII-D experimental reconstructions as initial value states in a f particle-in-cell (PIC) model coupled to the nonlinear 3D resistive MHD code NIMROD [3]. This hybrid kinetic-MHD model has recently been used to study kinetic effects on the 1=1 ideal internal kink mode [4,5]. In the present work, it is observed that the energetic particles have a significant effect on the stability of the 2=1 0031-9007=09=102(13)=135001(4)

mode. The growth rates of the 2=1 mode are calculated as a function of three quantities: Lundquist number S ¼ R =A > 104 , the fraction of attributed to the energetic particles frac h = (h is the energetic particle ), and the plasma normalized to the toroidal current N =ðI=aB Þ. In the section on our methods, the theoretical and computational methods used in this study are briefly reviewed. The results of the analysis are described in the next section. We also discuss the implication of the results, with particular attention to recent results from JET [6]. Furthermore, implications of energetic particle effects are discussed based on extrapolation to the ITER operational parameter regime, where S and frac will exceed current experiments. Our methods.—A detailed description of our methods is written in Refs. [4,5,7], while here we describe only the key elements. NIMROD [3] solves the linear and nonlinear MHD equations as initial value computations with a mesh of finite elements for the poloidal (R Z) plane and a finite Fourier series for the toroidal direction. In the hybrid kinetic-MHD model, it is assumed that the density of energetic particles (nh ) is negligible compared to the bulk MHD density (n0 ): (nh n0 ) but energetic particle h is on the order of the bulk plasma 0 : (h 0 ), where ¼ 0 þ h . In this approximation, any energetic particle species contribution to the center of mass velocity is neglected. However, the energetic species modifies the momentum equation with an additional energetic ion pressure tensor ph as @V þ V rV ¼ J B þ r rV rpb r ph ; @t (1) where pb is the background pressure tensor. ph is calculated from the moment equation,

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Ó 2009 The American Physical Society

PRL 102, 135001 (2009)

ðv vh Þ2 fh ðr; vÞd3 v;

(2)

where vh is the center of mass velocity and mh is the mass of energetic species. The f PIC model [8] evolves along the phase space characteristics (or trajectories) of the Vlasov equation @f @f þ z_ ¼ 0; @z @t

(4)

Then, we substitute f ¼ f0 þ f (where f is the time evolving phase space distribution function, and f0 is the equilibrium or steady state distribution) into the pressure moment Eq. (2). We only explicitly calculate ph . ph0 is implicitly assumed to be a part of the steady state initial conditions. Z p h ¼ ph0 þ mh ðv vh Þ2 fh d3 v; (5) where ph0 is the equilibrium energetic pressure tensor. The condition for steady state fields within the computation is given by J 0 B0 ¼ rp0 þ rph0 ;

(6)

where the assumption is that the equilibrium anisotropic energetic pressure component is 0 and the tensorial ph0 reduces to a scalar ph0 . Notice that the steady state fields satisfy a scalar pressure force balance, which limits the form of the equilibrium energetic particle distribution to isotropic distributions in velocity. The hybrid kinetic-MHD model is applied to the drift kinetic regime for the energetic particles. The drift kinetic equations [Eqs. (20) and (21) in Ref. [5] ] are used, and the linear response of the energetic minority ions is entirely contained in the evolution of the weight equation. The drift kinetic approximation leads to the CGL like pressure tensor 0 1 0 0 p? 0 A; p? ph ¼ @ 0 (7) 0 0 pk R R where p? ¼ Bfd3 v, and pk ¼ v2k fd3 v. The ‘‘slowing down’’ distribution function is used for the energetic minority ions, f0 ¼

P P0 expð c n Þ "3=2 þ "3=2 c

;

_ ¼ v rf0 ev0 E@" f0 ; f

(3)

where f ¼ fðz; tÞ is the time dependent phase space distribution function spanned by the 6D phase space variable z ¼ ðx; vÞ. These characteristics are chosen to be the drift _ kinetic equations of motion. Along the characteristics z, there is an evolution equation for f, _ ¼ z _ @f0 : f @z

where P0 is a normalization constant, " is the particle energy, "c is the critical energy, P is the canonical toroidal momentum, c n ¼ C c 0 , where c 0 is the total poloidal magnetic flux and C is a constant matching the equilibrium pressure profile [5]. Then, the linearized evolution equation for Eq. (4) is written in an explicit form [5] as

(8)

(9) 2

2

0 mvk v? m 2 where v0 ¼ vk b^ 0 þ eB 3 ðvk þ 2 ÞðB rBÞ þ eB2 J? and v ¼ EB þ vk B B . B2 Equilibria are constructed as described by Ref. [7]. Figure 1 shows the pressure and q profiles (qmin 1:5, q95 4:4) of a typical equilibrium as a function of normalized c . Our equilibria have C ¼ 1=4 with p0 / expð c =CÞ, which is a reasonable approximation to experiment that allows for lower local gradients in the edge region, while still reaching large values of . The gradient in P approaches zero near the edge, while the q profile and zero-dimensional parameters approach those of the experiment with N near the (m=n ¼ 2=1) ideal kink limit. By comparison pressure profiles in Ref. [7] have p / expð6 c Þ as seen in Fig. 1. Results.—In general, the linear growth rates reduce with frac , with the strongest reduction at the highest S. A real frequency of the 2=1 mode is found to increase with frac in qualitative agreement with studies of ideal m ¼ 1 modes [5]. However, the far lower (than ideal) growth rates of the resistive mode are accompanied by far lower real frequencies of the mode. In Fig. 2 the MHD only growth rates normalized to A ð4:7 107 sÞ of the 2=1 mode are shown as a function of N =4li for varying S while holding Prandtl number Pr visc = diff fixed at 100 (the ratio of kinetic viscosity to electric diffusivity) for all calculations. A stability boundary can be found at low S, N =4li , where this region is essentially damped by viscosity and resistivity. At higher S and N =4li < 0:9 the mode is in the experimentally relevant asymptotic regime, with low viscosity, in which the linear growth rate decreases with S. For N =4li > 0:9

8 q

exp(−4ψ) exp(−6ψ)

0.25

6

0.2 0.15

q

Z

u0 P

p h ¼ mh



4

0.1 2

0.05 0

0

0.2

0.4

ψ

0.6

0.8

1

0

0

0.2 0.4 0.6 0.8

1

ψ

FIG. 1. The equilibrium pressure and safety factor profiles as a function of normalized c . The pressure profile function / expð4 c Þ and produces experimentally relevant tearing instabilities near the ideal limit [7].

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Vr

1.0

(b)

0.5 0.0

Z (m)

5

0.6

0.0

0.65

0.5

9 7

0.7

Z (m)

βN/4li

0.75

Vr

(a)

1.0

21 17 13

0.8

1.5

1.5

0.9 0.85



6

6.5

-1.0

5.5 Log10 S

-1.5

5

-1.0

4.5

-0.5

3

1

-1.5

0.5

-0.5

0.55

FIG. 2. The curved lines show the linear growth rate A ð4:7 107 sÞ contours of the 2=1 mode for equilibria in this study. Above S 1 105:5 the asymptotic regime is apparent, and near N =4li 1 the ideal limit is approached.

1.0

1.5

x 10 6

2.0

1.0

2.5

1.5

2.0

1

|δ |

(c)

v⊥ (m/s)

0.8

the mode approaches the ideal MHD stability limit. Note that when N =4li is larger than 0.95, the n ¼ 2 is the most unstable mode for this equilibrium specification. Thus the most interesting experimentally relevant regime, for example, for nonlinear study, is N =4li 0:8 and S > 106 . Our study of the energetic particle effects focuses on the N =4li ¼ 0:83 cases, which are close to the ideal limit. Figure 3 shows the eigenfunction and the n ¼ 1 spatial projection of the phase space for the S ¼ 1 105 MHD only and energetic particle inclusive cases. The effect of real frequency is evident in the poloidal phase lag between certain minor radial locations. In Fig. 3(c) the jfj for this result is shown, where the mode activity is dominant in the trapped cone, and asymmetry is evident in vk . Figure 4 shows growth rates for the equilibrium with N =4li ¼ 0:83 for various frac as a function of S. Also, in this equilibrium, the MHD only growth rate has a peak near S 105:5 . The MHD only cases have been temporally and spatially converged in growth rates to within few percentage points. Since the growth rates are noisy with the energetic particle cases in general, especially in the kinetic energy, the growth rate is calculated from the average of the mean value of magnetic and kinetic energies, averaging over times much shorter than the growth time. The convergence test is then posed by varying total particle numbers (4.8, 19.2, 38:4 106 ). The growth rates of the magnetic part are converged, and the standard deviation of the growth rate of the kinetic part is reduced to less than =3 for the largest numbers of particles in our simulation, except for specifically noted marginal cases. The real frequencies are calculated from a poloidal Fourier transform at a series of times in the calculation, selecting out and tracking the phase of the m ¼ 2 component of the mode at the q ¼ 2 surface in time. These results show that with the inclusion of energetic particles there are significant and increasing damping effects, especially at high frac and S. The real frequencies

2.5

R (m)

R (m)

0.6 0.4 0.2 0 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

v|| (m/s)

0.4

0.6

0.8

1 6

x 10

FIG. 3. The eigenfunction of Vr for the MHD only results (S ¼ 105 ) (a), the energetic particles (frac ¼ 12:5%) inclusive results (b), and the n ¼ 1 spatial projection of jfj in phase space for this case (c). The effect of real frequency of the mode is evident as is a reduction of the radial extent of the mode.

generally increase with frac , in qualitative agreement with results for the 1=1 mode [5]. Varying "c from 0.2 to 0.6 in Eq. (8) produced changes in growth rate of a few percent. Varying the maximum velocity of the initial distribution produced less of an effect than those reported in Ref. [5], but can quantitatively change the results. A specific point of interest is the S 106:5 case, where marginality is found at frac 20%. For cases with S > 105 and frac > 50%, the mode is generally damped to a marginal state or stabilized. Circled points are marginal cases. The mode is stabilized at or near the marginal points, but it is a prohibitively demanding and subtle task to accurately and thoroughly resolve the marginal region with initial value codes. The real frequencies of some marginal cases are not shown, because the 2=1 eigenfunction never emerges in the computation. Also, when the mode approaches marginality, the real frequencies generally increase dramatically. It is expected, and interpreted from observation, that this is a consequence of the Alfve´n continuum beginning to have a significant effect on the mode structure, thus changing the nature of the mode and its response to the energetic par-

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3.0 2.5 βfrac =0.0

γ τA

2.0

βfrac =6.25%

1.5

βfrac =12.5% βfrac =25% βfrac =50%

1.0 0.5

βfrac =75%

0.0 0.6

4

4.5

5

4

4.5

5

5.5

6

6.5

7

5.5

6

6.5

7

0.5 0.4

ω τA

0.3 0.2 0.1 0.0

log10S

FIG. 4. The growth rates (upper panel) and the real frequency (lower panel) of the equilibrium N =4li ¼ 0:83 for various frac as a function of S. Note that due to the adjustment of 0 , the equilibrium N =4li does not change for various frac .

ticles. However, the details of such an investigation are beyond the scope of this study. Conclusions and discussion.—We have presented a linear resistive MHD stability analysis, including energetic particle effects, for the n ¼ 1 mode in a series of equilibria based on experimental equilibrium reconstructions in the DIII-D tokamak. These equilibria are most unstable to the m=n ¼ 2=1 mode as the ideal MHD stability limit in N is approached. The analyses show the growth rates reduce with increasing energetic particle fractions, most significantly at high S, accompanied by an increasing real frequency of the mode. Thus, these effects are expected to be stronger in ITER, where frac and S are even larger. Hybrid kinetic-MHD studies of the 1=1 internal kink mode [4,5,9] have shown that the mode interaction with the trapped particles and ‘‘barely passing’’ particles can be stabilizing and drive a real frequency. A similar effect is found in this work. This is an energetic particle effect driven by the trapped and barely passing particles in the bulk of the plasma and not a direct effect in the tearing layer. The free energy available to, and hence the growth rates of, resistive modes are in general far smaller than that of ideal modes. The effect the energetic particles have on the resistive mode is strongest in the asymptotic regime with increasing S, where the growth rates of the MHD only solutions decrease with the decreasing resistivity and viscosity. The inner layer physics can be thought of as independent of the ideal outer region physics at high S, where the inner layer width is much smaller than the gradient scale length of the mode. It is apparent from the results that the energetic particles are affecting the stability of the mode through interaction with the ideal outer region, and the effect is stronger with S because of the weakening drive


to the mode growth from the inner layer. An analysis of the scaling of growth rates with S will appear in further publication, but it should be noted that for S 106 the growth rates for unstable modes with varying frac differ mainly by an additive constant, scaling approximately by the analytic form of / S3=5 . It is conjectured that energetic particle effects may play a role in recent results from the JET experiment, where the 2=1 mode is shown to be stable in parameter regimes where other experiments have been found unstable [6]. In these cases, as the ideal limit is approached the experimental data clearly indicate that JET is stable for cases well above N values where DIII-D and JT-60U are found to be unstable. The most significant difference in the dimensionless parameters of these otherwise similar discharges is frac , where JET exceeds 30% while DIII-D and JT-60U are typically below 20% [10,11]. These results suggest that a qualitative extrapolation is reasonable for what to expect from energetic particle effects on resistive MHD modes in ITER. In general, the stabilizing effects should be extremely significant. Accurately computationally calculating the inner layer physics for resistive MHD mode onset at the huge S values expected in ITER is not a trivial task. The details of what to expect from the equilibrium state itself are not yet clear. However, the results here suggest that for the large frac and S expected in ITER, as the ideal and resistive MHD limits are approached, energetic particle effects will cause real frequency, damping and possibly stabilization of resistive MHD modes. This work is supported by US DOE Grant No. DEFG02-07ER54931. The authors would like to thank the NIMROD team, R. Buttery (UKAEA), R. J. La Haye, and A. D. Turnbull (GA) for numerous helpful discussions regarding this work.

[1] M. Choi et al., Phys. Plasmas 14, 112517 (2007). [2] F. Porcelli, Plasma Phys. Controlled Fusion 33, 1601 (1991). [3] C. Sovinec et al., J. Comput. Phys. 195, 355 (2004). [4] C. C. Kim, C. Sovinec, and S. Parker, Comput. Phys. Commun. 164, 448 (2004). [5] C. C. Kim, Phys. Plasmas 15, 072507 (2008). [6] R. J. Buttery et al., in Proceedings of the 22nd IAEA Fusion Energy Conference, Geneva, Switzerland, IT/P68 (IAEA, Geneva, Switzerland, 2008). [7] D. P. Brennan, S. E. Kruger, T. A. Gianakon, and D. D. Schnack, Nucl. Fusion 45, 1178 (2005). [8] S. Parker and W. Lee, Phys. Fluids B 5, 77 (1993). [9] G. Y. Fu et al., Phys. Plasmas 13, 052517 (2006). [10] A. Fasoli et al., Nucl. Fusion 47, S264 (2007). [11] T. Hender et al., Nucl. Fusion 44, 788 (2004).

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