New bootstrap current formula valid for steep edge pedestal, and its implication to pedestal stability

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1 1 TH/P4-12 New bootstrap current formula valid for steep edge pedestal, and its implication to pedestal stability C.S. Chang 1,2, Sehoon Koh 2,*, T. Osborne 3, R. Maingi 4, J. Menard 1, S. Ku 1, Scott Parker 5, W. Wan 5, Yang Chen 5, A. Pankin 6 and P.B. Snyder 3 1 Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA 2 Korea Advanced Institute of Science and Technology, Daejeon , Korea 3 General Atomics, San Diego, CA , USA 4 Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA 5 University of Colorado, Boulder, CO 80309, USA 5 Tech-X Corporation, Boulder, CO 80303, USA * Present address, WCI, National Fusion Research Institute, Daejeon, Korea Corresponding Author: Abstract: A drift-kinetic neoclassical particle code XGC0, equipped with a mass-momentum-energy conserving collision operator, is used to study the edge bootstrap current in a realistic diverted magnetic field geometry with a self-consistent radial electric field. A new analytic fitting formula, as a simple modification to the formula, is obtained based upon the XGC0 simulation results. The new bootstrap current formula brings the linear peelingballooning stability boundary to the right ballpark with the experimental observation in NSTX edge, as also observed in the conventional aspect ratio tokamaks. The new bootstrap current formula also makes the linear growth rate of the collisionless trapped electron modes to be higher in NSTX edge pedestal. 1 Introduction While most other transport phenomena in tokamak plasma are dominated by turbulence physics, the parallel plasma current has been experimentally validated to obey neoclassical physics. Existing studies of the bootstrap current and the analytic formulas being used in the equilibrium and stability analyses so far have been mainly focused on the core plasma (represented by the formula [1]). It is necessary to improve the bootstrap current formula for application to edge pedestal. The neoclassical physics in the edge plasma is unconventional and difficult, compared to the core plasma. Specifically, the edge pedestal plasma has a narrow passing particle (current carrier) region in velocity space, which can be easily modified or destroyed by Coulomb collisions, while the

2 TH/P formulation of the conventional bootstrap current has been focused on the core plasma and is based upon the collisional modification of the effective trapped particle fraction. The edge pedestal plasma has a steep gradient, whose scale length is similar to the ion banana width, so that one must include the nonlinear interaction between the ion radial excursion and the plasma pressure gradient. The existing formulas are based upon linearized approximation under the assumption that the radial banana excursion width is much smaller than the plasma pressure gradient scale length. Moreover, the pedestal plasma contains a magnetic separatrix surface, across which the bootstrap-current generating topological property of particle orbits change abruptly. Another physics effect not recognized in the previous theories is the extreme largeness of the toroidal magnetic field component compared to the poloidal component at the high field side, as the toroidal aspect ratio becomes small. Many trapped particles then execute multiple toroidal rotations at the high field side before they recognize that they are in the trapped particle regime in a tight aspect ratio tokamak such as NSTX. Under reasonable pitch-angle collisions, these particles are virtually indistinguishable with the passing particles. As a result, the effective passing particle fraction is higher in a tight aspect ratio tokamak edge than the conventional estimates. In this study, the drift-kinetic particle code XGC0, equipped with a mass-momentumenergy conserving collision operator, is used to obtain the neoclassical bootstrap current J b. In order to include the edge effect faithfully, the simulation is performed in realistic diverted geometry with the self-consistent radial electric field solution, unlike in the previous studies. We solve the drift-kinetic equation without linearization in banana width. The Coulomb collision operator used in the present study is, however, linearized, and is similar to that used in [1]. A simple modification to the formula has been obtained to bring the analytic fitting formula to a better agreement with the XGC0 results in the edge pedestal. More details can be found in [2]. NSTX edge plasma had a long standing issue [4] that the experimental ELM boundary [γ/(ω /2) 0.1] is far away from the theoretical peeling-ballooning instability boundary [γ/(ω /2) 1] [5], while other higher aspect ratio tokamaks have shown agreement between them. When the new XGC0-based formula is applied to a few representative NSTX edge plasmas, it is found that the agreement between the linear peeling-ballooning boundary and the experimental instability boundary is recovered. A gyrokinetic simulation of micro-instabilities using the delta-f electromagnetic gyrokinetic code GEM shows that the linear growth rate of the collisionless trapped electron modes become higher in NSTX pedestal. Stabilizing effect of Coulomb collisions on these modes is under investigation and to be reported in a subsequent publication. 2 Numerical Results from XGC0 The magnetic field is given as B = B P + B T = φ ψ + I φ, with φ being the toroidal angle and I = RB, and the plasma current density in steady state is given by J = RdP/dψ ˆφ + K(ψ) B, where K(ψ) =< J B > / < B 2 > +I/ < B 2 > dp/dψ. It can be easily shown that the flux surface averaged toroidal bootstrap current can be

3 3 TH/P4-12 measured as J bφ B/B 0 = Jb B/B 0 B φ B / B 2, (1) where J is the net current density vector, and B 0 is the magnetic field magnitude at the magnetic axis. The inductive Ohmic loop voltage, which could be used to determine the neoclassical electrical conductivity, is set to zero in the present study. Since the fidelity of the electron physics is important in the evaluation of the bootstrap current, the real electron mass is used instead of an artificially enhanced mass. The simulation normally used about 100 millions particles and takes about 6 hours on 70,000 Hopper cores at NERSC. In XGC0, the radial electric field and the ion toroidal/poloidal flows are generated consistently with the edge effects: i.e., steep pedestal plasma profile, magnetic separatrix geometry and the X-transport phenomenon [3]. The peak statistical error (1/ N) is estimated to be less than 1%. The XGC0 simulation has been verified against the existing formula of Ref. [1] in its confidence regime, x 10 4 i.e., tokamak core plasma with high toroidal aspect ratio 6 XGC0 r/r < 0.2. As the bootstrap current becomes large Modified in the steep edge pedestal (r/r > 0.3, corresponding 4 to trapped fraction of 0.75 or higher), surprisingly, the 2 XGC0 results still traces formula reasonably closely for ψ N < 0.99 if the effective collisionality of the passing electrons is low in both DIII-D and NSTX Current Density (A/m 2 ) Current Density (A/m 2 ) x XGC0 2.5 Modified Normalized Psi FIG. 1: Collisional bootstrap current results in edge pedestal for (a) DIII-D with ν e 7.7 at the peak of bootstrap current, and (b) NSTX with ν e 5. geometries (corresponding to ν e 1). Here, ψ N is the poloidal magnetic flux normalized to be unity at the separatrix and zero at the magnetic axis. It is found that the agreement with the formula begins to deteriorate as the effective passing particle collision frequency is raised to > 1. We present here a case with the electron collisionality raised to ν e 5, thus pushing the effective collisionality of the pedestal passing particles ν e,p = ˆν e /( θ p ) 2 into highly collisional regime. For the DIII-D pedestal, the plasma density has been raised and the temperature has been lowered from plasma discharge number in order to enhance the collisionality. For the NSTX pedestal, the natural plasma profile from experimental discharge number has been modeled after. Figure 1 shows comparison of the simulation results with formula for (a) DIII-D at ν e 7.7 (ˆν e 1.6 and ν e,p 28) and (b) NSTX at ν e 5 (ˆν e 2.3 and ν e,p 117) at the radial positions where the bootstrap current peaks. The pedestal bootstrap current is significantly reduced from the value in a conventional aspect ratio tokamak (by 35% in FIG. 1 (a)) and significantly enhanced in a tight aspect ratio tokamak (by 50% in FIG. 1 (b)).

4 TH/P We would like to note here that the present kinetic simulation shows that the ion contribution to the bootstrap current is still insignificant ( < 10%) even in the edge pedestal. Thus, some change in the ion contribution does not affect the total bootstrap current as significantly as the electrons contribution does. The success of the formula in the weakly collisional edge pedestal region, even though it neglects the radial orbit excursion and the ExB flow effects, is found to be largely due to the smallness of the ion contribution to the total bootstrap current. For the same reason, a modification of the radial electric field solution by turbulence would have little effect on the edge bootstrap current. The only place where the radial excursion effect makes a significant correction to the formula in the wearkly collisional regime is in the thin radial boundary layer (ψ N > 0.99, several electron banana widths from ψ N = 1) in contact with the magnetic separatrix surface. Even this is the electron contribution effect. 3 New XGC0-based bootstrap current formula The existing formula as given in [1] is in the form < J b B P d ln P >= Ip e (L 31 p e dψ + L d ln T e 32 dψ + L 34α T i d ln T i ZT e dψ ), where I(ψ) = RB φ and Z is the ion charge number. Because the formula gives reasonably good agreement with simulation results in the deep banana regime, we maintain the functional form of the transport coefficients L 31, L 32, and L 34 to be unchanged and modify the effective trapped particle fractions fteff ee 32 ei 34, fteff, fteff, and fteff in these transfort coefficients. Following Ref. [1], L 31, L 32, and L 34 are given as L 31 = F 31 (X = fteff) 31 = ( )X Z + 1 f 31 teff = Z + 1 X Z + 1 X Z + 1 X4, f t 1 + (1 0.1f t ) ν e + 0.5(1 f t )ν e /Z, 32 ee L 32 = F 32 ee (X = f F 32 ee (X) = teff ) + F 32 ei (Y = f Z Z( Z) (X X4 ) + 32 ei teff ), Z [X2 X 4 1.2(X 3 X 4 )] Z X4, F 32 ei (Y ) = Z Z( Z) (Y Y ) Z [Y 2 Y (Y 3 Y 4 )] 32 ee fteff = 32 ei fteff = Z Y 4, f t (1 f t ) ν e (1 0.37f t )ν e / Z, f t 1 + ( f t ) ν e (1 0.37f t )ν e (1 + Z),

5 5 TH/P4-12 L 34 = F 31 (X = f 34 teff), f 34 teff = f t 1 + (1 0.1f t ) ν e + 0.5(1 0.5f t )ν e /Z. Here fteff ee 32 ei 34, fteff, fteff, and fteff modify the collisionless trapped particle fraction f t by collisions: f t = B 2 1/B max 0 λdλ < 1 λb >. It is found that the separatrix effect can be accounted for by multiplying a numerical fitting factor H(ψ) to the collisionless trapped particle fraction f t,new = f t H(ψ), (2) where, if the electron magnetic drift is into the X-point in a single null diverted geometry, ( ) H(ψ) = 1 (0.2/Z 4 ) exp ψ s ψ 2.7 log(ɛ 1.5 ν e / ) ψe. Otherwise (including double null), H(ψ) is fitted to ( ) H(ψ) = 1 (0.6/Z 4 ) exp ψ s ψ 3.3 log(ɛ 1.5 ν e + 2) ψe, where ψ s is the value of ψ( ψ N ) at the magnetic separatrix surface, ψe is the electron banana width in ψ space, ψe = (dψ/dr) be = RB p ɛ me v th,e /eb p is the electron banana width in the ψ space measured at the outside midplane. This effect is confined to a thin radial layer (ψ N > 0.99) in contact with the magnetic separatrix surface, corresponding to several electron banana widths. The small passing particle region effect is found to be modeled by modifying f 3j f 3j teff,new as follows: teff into f 3j teff,new = f 3j teff [1 + δ(ɛ, ν e, Z)], (3) δ(ɛ, ν e, Z) = 0.55Z 0.2 ( tanh ( 3.2β(ɛ)(ɛ 3/2 ν e ) 1.4 /Z α(z)) where j=1, 2, 4, and + (1 exp( ν e /0.1)) tanh ( 2.2β(ɛ)ɛ 2.8 νe 0.1 /Z α(z))), β(ɛ) = Re((ɛ 0.44) 0.7 ), α(z) = ( Z Z 4.981)/(4.294Z Z ) for 1 Z 5 and α(z) = 0 for Z > 5.

6 TH/P The modified L new 31, L new 32, and L new 34 then become L new 31 = F 31 (X = fteff,new) 31 = ( ) 1.9 X Z + 1 f 31 teff,new = L new F 32 ee (X) = Z + 1 X Z + 1 X Z + 1 X4, f t,new [1 + δ(ɛ, ν e )] 1 + (1 0.1f t,new ) ν e + 0.5(1 f t,new )ν e /Z, 32 ee 32 = F 32 ee (X = fteff,new) + F 32 ei (X = f Z Z( Z) (X X4 ) + 32 ei teff,new), Z [X2 X 4 1.2(X 3 X 4 )] Z X4, F 32 ei (Y ) = Z Z( Z) (Y Y ) Z [Y 2 Y (Y 3 Y 4 )] 32 ee fteff,new = 32 ei fteff,new = Z Y 4, f t,new [1 + δ(ɛ, ν e )] (1 f t,new ) ν e (1 0.37f t,new )ν e / Z, f t,new [1 + δ(ɛ, ν e )] 1 + ( f t,new ) ν e (1 0.37f t,new )ν e (1 + Z), L new 34 = F 31 (X = f 34 teff,new), f 34 teff,new = f t,new [1 + δ(ɛ, ν e )] 1 + (1 0.1f t,new ) ν e + 0.5(1 0.5f t,new )ν e /Z. Since the above correction factors are for electrons, the ion charge number Z in the δ formula is equal to Z eff. It can be easily seen that in the weakly collisional limit νe or in the large aspect ratio limit ɛ (3/2) (and several electron banana-width away from the magnetic separatrix surface), the modified formula reduces to the formula. 4 Application to NSTX edge stability NSTX edge plasma has been having a long standing issue [4] that the experimentally inferred ELM boundary [γ/(ω /2) 0.1] was far away from the theoretical peelingballooning instability boundary [γ/(ω /2) 1] [5], while other tokamaks with conventional aspect ratio have shown agreement between them. One suspicion leading to this disagreement was that the s bootstrap current formula was not applicable to NSTX edge pedestal. We have re-examined the problem. When the new XGC0-based formula is applied to a set of representative NSTX edge plasmas as chosen in [4], it is found that the experimentally inferred ELM boundary agree with the peeling (kink) instability boundary within error bar. Figure 2 shows the γ/(ω /2) contour plot from the Elite code [5] and the experimental pedestal parameters around the ELM onset, evaluated from both the and the XGC0-based bootstrap current. NSTX discharge has been used.

7 7 TH/P x x8099 XGC FIG. 2: Comparison between the peelingballooning mode stability and the XGCenhanced experimental ELM boundary in NSTX discharge shall be pursued in the near future. It can be seen that the experimental point inferred from the new XGC0-based bootstrap current sits near the theoretical peeling (kink) instability boundary, similarly to the observations made in high aspect ratio tokamaks, while the experimental point inferred from formula is far away. The ballooning mode instability boundary exits to the right and can not be seen in the figure box. JET also observes a significant modification of the pedestal bootstrap current (in the opposite direction to NSTX modification) by the new formula, improving their stability understandings[7]. On the other hand, DIII-D edge gets an insignificant influence by the new formula. Effect on the ITER pedestal stability will be of interest, and 5 Gyrokinetic investigation of the new bootstrap current effect on mocro instabilities r/a q Modified 2.2 x γ [rad/s] Modified n FIG. 3: The new bootstrap current makes the collisionless trapped electron modes more unstable in the edge pedestal of NSTX discharge Edge localized modes may be from MHDtype instabilities as shown above. But, the transport in the pedestal is most likely influenced by micro instabilities. The electromagnetic delta-f gyrokinetic code GEM has been used to study the effect of the new bootstrap current formula on micro instabilities in the same NSTX pedestal equilibria as those used for the above MHD analysis. Coulomb collision is turned off in the present study. Unstable modes are found and identified to be the the collisionless trapped electron modes. NSTX plasma edge is heavily populated by trapped particles. Linear growth rate of these modes become stronger with the higher pedestal bootstrap current as found from the new formula (FIG. 3). The q-profile is less steep (left figure) with the higher bootstrap cur-

8 TH/P rent, leading to weaker magnetic shear in the pedestal. Collisions usually weaken the trapped electron mode growth rate. Electron collisionality in NSTX edge pedestal is rather high. Effect of Coulomb collisions on these modes is under investigation and to be reported in a subsequent publication. We note here that GEM has also identified the kinetic version of the peeling-ballooning modes at intermediate toroidal mode number n and the kinetic ballooning modes at high n in DIII-D edge pedestal[6]. Near threshold conditions exist. Present study shows a strong sensitivity of these modes to the pedestal bootstrap current. Thus, even though the modification of the bootstrap current by the new formula is small in DIII-D edge pedestal 1%, its effect on theses modes are large 10%. Kinetic version peelingballooning and kinetic ballooning modes are not yet found in NSTX pedestal from the GEM study. 6 Conclusion and Discussion The full-f drift-kinetic particle code XGC0, equipped with a mass-momentum-energy conserving collision operator, has been used to study the bootstrap current in steep edge pedestal in realistic magnetic separatrix geometry under self-consistent radial electric field development. A simple modification to the formula is obtained to bring the analytic fitting formula to a better agreement (within several percent accuracy compared to the peak value) with the XGC0 results in the edge pedestal. When the new XGC-based formula is applied to the representative NSTX edge plasmas, the long-standing discrepancy between the peeling-ballooning stability boundary and the experimental observation appears to be resolved. The new formula may also help other tokamak edge stability analysis. Linear growth rate of micro-instabilities, identified as the collisionless trapped electron modes, becomes higher in NSTX edge by enhanced bootstrap current from the new formula. A realistic collision effect is under investigation and to be presented in a subsequent publication. Work supported by US Department of Energy and the Korean National R & D Program through the National Research Foundation of Korea ( ). We acknowledge helpful collaboration with Drs. Samuli Sarrelma and Andrew Kirk on the JET and MAST bootsrap current and stability evaluation using the new formula. References [1] O., C. Angioni et al., Phys. Plasmas 6, 2834(1999); 9, 5140 (2002) [2] S. Koh, C. S. Chang et al., Phys. Plasmas 19, (2012) [3] C. S. Chang, Seunghoe Ku et al., Phys. Plasmas 11, 2649 (2004) [4] R. Maingi, T.H. Osborne et al., Phys. Rev. Lett. 103, (2009) [5] P.B. Snyder et al., Nucl. Fusion 49, (2009) [6] W. Wan, S. Parker et al., Phys. Rev. Lett. (submitted) [7] S. Sarrelma and A. Kirk, private communication

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