TH/P6-08. CompX, Del Mar, CA 92014, USA; 2Xcel Engineering, Oak Ridge, TN 37830, USA; 3PSFC-MIT,Cambridge, MA 02139

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1 1 Differences Between QL and Exact Ion Cyclotron Resonant Diffusion R.W. Harvey1, Yu. V. Petrov1, E.F. Jaeger2, P.T. Bonoli3, A. Bader3, and RF-SciDAC Group 1 CompX, Del Mar, CA 92014, USA; 2Xcel Engineering, Oak Ridge, TN 37830, USA; 3PSFC-MIT,Cambridge, MA contact of main author: rwharvey@compxco.com Abstract. This work reports some differences between ICRF quasi-linear (QL) diffusion theory and RF diffusion coefficients obtained with an "exact" orbit calculation using the DC (Diffusion Coefficient) code. These calculations are performed within the context of modeling vertical viewing neutral particle analyzer (NPA) experimental results for an Ion Cyclotron Resonant Heating (ICRH) CMod minority heating experiment. Time-dependent simulations are performed by successive execution of the AORSA full-wave code, the DC code, and the time-dependent CQL3D Fokker-Planck code. Comparisons are made with simulation using AORSA-generated quasilinear coefficients and time-dependent CQL3D. The simulation results obtained with the exact RF diffusion coefficients from DC indicate a shorter ramp up time for NPA fluxes than that obtained using QL coefficients, in general agreement with the experiment. However, NPA ramp down time after turn off of the RF remains slower than the experimental observations in both simulations. 1. Introduction This work reports differences in results obtained with ICRF quasi-linear (QL) diffusion theory and with RF diffusion coefficients from an "exact" orbit calculation using the DC (Diffusion Coefficient) code. The latter preserves correlations between subsequent resonance crossings, and naturally includes finite Larmor radius effects. These calculations are performed within the context of modeling time-dependent vertical viewing NPA experimental results for an Ion Cyclotron Resonant Heating (ICRH) C-Mod minority heating experiment, as shown in Fig. 1. We will show that in combination with the AORSA [1] full wave and CQL3D FokkerPlanck [2] codes, the DC results indicate a shorter ramp up time for NPA fluxes than that obtained using QL coefficients, in general agreement with the experiment. By "exact" orbit calculation of ICRF diffusion, we refer to the case that the DC code obtains diffusion coefficients by numerically integrating the Lorentz equation trajectories of ions in the combined tokamak magnetic equilibrium and AORSA full-wave electromagnetic fields. The particles are launched from tokamak midplane points, initially equispaced in gyro-phase (with constant gyroradius) about the given gyro-centers Fig. 1. Simulated NPA view chords on the C- and also equispaced in toroidal length along a Mod experiment, at 65, 67.5 and 70cms. The given RF mode wavelength, and averages the heavy dashed line gives the location of minority H+ cyclotron resonance. The lighter dashed resulting square of the velocity changes after one lines are at the Doppler shifted resonances for (or more) poloidal circuits, to obtain the ICRF parallel energy 50 kev ions, with Nφ =+/- 10. bounce-averaged, gyro-phase and wave-phase

2 2 averaged diffusion tensor. This is carried out for a 3D array (u, u, R) of initial conditions, giving the six independent RF diffusion coefficients in 3D constant-of-motion space. The method follows the formalism of Refs. [3,4]. For comparison, we have the zero-banana-width quasilinear (QL) diffusion coefficients obtained in [1] and calculated in the AORSA code. These coefficients are directly compared with DC Lorentz-orbit-based coefficients. Selfconsistent time-evolution of the ion distributions and radial power deposition are obtained by importing the RF diffusion coefficients into the time-dependent CQL3D bounce-averaged Fokker-Planck code. Ten FP time steps are taken for each AORSA/DC execution. The physical configuration in the C-Mod NPA experiment in Fig. 1 includes a range of vertical viewing sightlines[5,6], shown here superimposed on the poloidal magnetic flux surfaces for the experimental shot being examined. ICRF wave heating is applied at 80 MHz, with balanced toroidal modes peaked at Nφ =+/-10. As shown in the heavy dashed line, the H+ cyclotron resonance passes 2 cms outside the magnetic axis. The lighter lines give the location of the Doppler shifted resonance for particles with 50 kev parallel energy. Figure 2 shows the left-hand polarized electric field strength given by AORSA for the subject C-Mod shot, with localization of the mode-converted ion cyclotron and ion Bernstein waves near the H+ resonance layer. Since this spatial region of large E+ is narrowly peaked relative to the spatial spread of the 50 kev parallel resonances in Fig. 1, it implies that the resonance regions are narrow in pitch angle when measured in velocity space at the midplane, as will be shown below. Although the machine geometry is nearly up-down symmetric, and the ICRF antenna is also symmetric, the left-hand field component is highly localized and non-symmetric [7]. Comparison with zero-banana-width QL theory is more directly achieved in the present work by subtracting off the perpendicular drifts in DC code Fig. 2. Left-hand polarized E+ electric using a fictitious force in the Lorentz equation, F = field, for Nφ = +/-10 excitation, using ugc B. This removes the finite banana width the AORSA code[1]. effects, but leaves correlation, finite gyro-radius, and other effects. The integration of (32 radii) (128 u ) (256 u ) (8 gyro-phase) (8 toroidal angle) starting positions (134M Lorentz orbits) for a single toroidal mode is wellparallelized and takes 0.25 hour on 4096 cores, and 8 times as long summing over 101 complex toroidal modes; these global calculations are enabled by recent advances in supercomputing[8]. The effect of the zero-orbit-width approximation on the simulated vertical viewing NPA diagnostic is not expected to be substantial, in that the detected particles are at points near their banana tips. The DC code is similar to the MOKA code[9], but has been coupled to the CQL3D Fokker-Planck and AORSA full-wave codes to obtain a time-dependent, noise-free solution to

3 3 the ICRF heating problem across the whole plasma width. (a) 2. Comparison of DC Exact Coefficients and AORSA QL Coefficients In this section, focused on an ITER relevant discharge in the Alcator C-Mod tokamak, we compare the ICRF diffusion coefficients calculated by DC, which includes correlations, with the AORSA QL coefficients derived using the randomphase approximation and neglecting all interresonance correlations. The toroidal variation of (b) the RF fields is accounted for by Fourier decomposition into 101 modes with Nφ = [-50,+50], described in [10]. C-Mod is in an intermediate toroidal damping regime with the waves propagating toroidally, being damped in ~0.5 toroidal turns. An ITER case gives single pass absorption with little toroidal propagation, whereas NSTX HHFW waves extend around the tokamak from the antenna with damping length of (c) several toroidal turns[10]. Fig. 3(a-c) compares the velocity space Duu diffusion coefficient calculated by DC for 1,2, and 4 complete turns of ion orbit in the poloidal plane. The fig 3(b) coefficient for 2 turns shows significantly more variations of Duu in pitch angle than the single turn results in 3(a); fig. 3(c) for 4 turns shows little additional correlation Fig. 3. DC Duu diffusion coefficients: (a) 1 effects. Peaks of the Fig. 3 coefficients are 1.46(1 poloidal orbit turn; (b) 2 turns; (c) 4 turns. These coefficients, for C-Mod minority H, are turn), 1.66 (2 turns), 1.77(4 turns), in accord with at the same radii near the peak of the radial heuristic expectations for correlations which are absorption profile. reaching maximum effect. All three DC coefficient radial sets show remarkable agreement in radial power absorption, shown in Fig. 4. The overall conclusion is that correlation effects in this case are nearly fully developed after two poloidal orbit turns, and one poloidal turn has given the major features for the power deposition. Noteworthy in Fig. 3(b-c) and Fig. 4, is the remarkable accuracy that can be obtained for the DC Duu coefficient: the integrated over pitch angle diffusion coefficients are largely unchanged when calculated at the end of 1, 2 or 4 poloidal turns, and the power absorption profiles, Fig. 4, do not significantly change Fig. 4. Radial profile of ICRF power deposition derived from DC with 1,2, and 4 poloidal orbit turns, and from AORSA QL coefficients. Next question to address is how sensitive the final result is to number of included toroidal modes. In

4 4 fig. 5, the RF minority H+ velocity-space Duu diffusion coefficients are shown (a) directly from the AORSA QL coefficient generator, (b) from one-poloidal-turn with DC, both of these with a complete finite-antenna-lenth 101 mode representation of the launched spectrum, and (c) with only the two central modes, Nφ = +/-10. The top row is for the full energy range of the calculation, to 2 MeV. The bottom row zooms in to the lower velocities to 125 kev, where most of the power absorption occurs. Of particular note is the filling in of the diffusion coefficient for pitch angles inside the trapped-passing region (bounded by the magenta lines). This has substantial ramifications for diffusion of the fast ions observed by the NPA, up to 0.8 MeV. 101 Mode 101 Mode 2 Mode DC Coeffs Filled in Fig. 5. Velocity-space plots of Duu Diffusion coefficients at radius near the peak of the ICRF power deposition profiles from (a) 101 mode spectrum for C-Mod minority heating from AORSA using quasilinear theory, (b) 101 mode spectrum from DC, and (c) the dominant Nφ =+/-10 two modes also from DC. The top row shows results to 2000 kev, whereas the bottom row plots are zoomed in to maximum energy 125 kev. The DC diffusion coefficients, even the 2 mode case, are more filled in, in pitch angle, near the trapped-passing boundary than the AORSA coefficients in (a). The t-p boundary is indicated by the magenta lines. From these comparisons between correlated DC and uncorrelated QL AORSA ICRF coefficients in Figs. 3 and 5, we conclude that the broad toroidal spectrum resulting from damping in cases in C-Mod, and particularly as occurs in ITER[10], leads to saturation of correlation effects after one or two poloidal orbital turns of the ions. If the toroidal damping length exceeds the machine circumference, then more poloidal turns are required for an accurate description of correlation effects. But even in single (or two) toroidal mode simulations with just one poloidal turn to obtain DC rf coefficients, simply neglecting further correlations which are particularly important for a single toroidal mode, still leads to reasonably accurate C-Mod simulation. Consequently, to reduce required computer resources for the time-dependent C-Mod simulations, we use just the two main modes, N φ =+/- 10.

5 5 3. Time-Dependent Evolution of Distributions and of NPA Diagnostic Signals The time-dependence of the minority H+ distributions (nh/nd ~ 9%) is simulated over a 60 msec ICRF on-period, and 30 msec off-period. (This pulse sequence was repeated in the experiment, to build up NPA statistics.) The simulations were conducted with the combination of the AORSA full-wave code with the two main toroidal modes (N φ =+/- 10), the DC diffusion code, and the CQL3D Fokker-Planck code. Simulation sequencing of the codes, and variation of some of the parameters on the simulation, was controlled by a python script. Simulations were carried out on NERSC parallel computers[8] typically using 1152 core. Three types of simulations of the full 90 msec period were carried out: (1) AORSA-DC-CQL3D, where DC computes the Lorentz-orbit-based RF diffusion coefficients for a single poloidal turn and the results are passed in to CQL3D. CQL3D is restarted for ten sub-time steps until the next step of the codes. The maximum energy and time-steps are suitably varied over the course of the calculation, with 40 steps sufficient for the 90 msec simulation. (Simulation wall-clock time ~20 hours.) (2) AORSA-CQL3D, where QL RF diffusion coefficients are calculated internally in AORSA and passed to CQL3D for each cycle of the codes. The power deposited in this simulation typically agrees to within 10% with the prior three-code simulation. (Simulation wall-clock time ~10 hours.) In order to more closely compare the results of these two types of calculation, a third simulation is performed, to obtain the same total powers. (3) Renormalized DC coefficients with CQL3D are used in which a Newton iteration is applied in order to scale the DC diffusion coefficients from (1), above, to give the same power as in the AORSA-CQL3D QL calculation, (2) above. (Typically three iterations per timestep, 4 hours wall-clock time for the complete re-simulation.) The purpose of this step is so that QL and DC-based results can be compared at closely the same powers, and differences will not be due to power. The comparison of the total powers is shown in Fig. 6, and shows close agreement between aorsa-cql3d and the renormalized aorsa-dc-cql3d throughout the simulation. As shown in Fig. 7, the diffusion is broader in pitch angle for aorsa-dc-cql3d. Fig. 7 shows the resulting H+ ion distribution functions at radius near the peak of the power deposition, at two different times: early (t=0.1 msec) and intermediate (t=10 msec). Compared to the aorsa-cql3d QL case, the results of the simulation with DC show marked inflation of the ions in the perpendicular direction. Particles viewed by the perpendicular NPA, as seen in the outer equatorial plane velocity space, are approximately at an pitch angle Fig. 6. Total powers versus time in the δ θ0= ϵ(1 cos θ pol ) from perpendicular, simulations. The blue curve is from aorsa-dcwhere ϵ=ρ/r. At poloidal angle cql3d, and shows quite good agreement θ pol =π /2, that is, vertically viewing through (~10% greater) than the aorsa-cql3d quasilinear results in red. The green curve is the magnetic axis, the viewed pitch angle in the for the renormalized aorsa-dc-cql3d. equatorial plane is at 0.71 of the trapped passing

6 6 angle, as measured from the perpendicular direction. Thus, from Fig. 7, this lies within the vicinity of the inflated distributions both for the QL and the DC based diffusion results, and modeling must be particularly accurate to distinguish the two cases. In steady state, the AORSA-CQL3D calculated NPA energy using QL theory generally agrees with experiment [6]. Similar results are obtained with DC. However, as shown in the experiment/simulation comparison in Fig. 8, the experimental flux both increases and decreases in time at about twice faster rate than in the simulation. Aorsa-cql3d (QL) t=0.1 msec t=10 msec Aorsa-dc-cql3d t=0.1 msec t=10 msec Fig. 7. Contours of minority H+ ion distributions at rho/a=0.143 for early and intermediate times in the RF simulations. Results on the left are from aorsa QL coefficients, and on the right from the Lorentz-orbit based DC diffusion coefficients. Velocity normalization is for 2 MeV at t=0.1 msec, 5 MeV at 10 msec. Contour levels are chosen to be equispaced for the initial Maxwellian distribution. Fig. 8. Comparison of experimental total flux in the energy range kev, and that in aorsa-cql3d QL simulation (From [6]).

7 7 DC DC QL DC QL QL Fig. 9. Simulated total NPA fluxes in the energy range keV, versus time, for three vertical viewlines effectively at (a) 67, (b) 69.5, and (c) 72 cms. View-chord (b) corresponds most closely to the experimental case shown in Fig. 8. The blue curve using DC Lorentz-orbit-based coefficients gives flux growth rate approximately twice as fast as the simulations with QL coefficients. Figure 9 compares simulated keV signal for three viewing chords of Fig. 1, using DC ICRF diffusion coefficients and using QL coefficients. Effectively, these spectra are for points shifted outwards by the gyro-radius (1.6 cms at 300 kev, 2.6 cms at 800 kev). Figure 9(b) at effectively 69.5 cms best compares with the 70.2 cm experimental data given in Fig. 8. The modification of the ICRF diffusion coefficients obtained from the DC Lorentz-orbit calculation has approximately reproduced the halving of NPA flux ramp-up time compared to quasilinear theory, and thus supports the differences in diffusion obtained by DC, relative to QL theory. However, DC did not show evidence of the shorter decay time of the total flux signal as observed experimentally. In summary, the effects of the Lorentz-orbit DC calculation of diffusion have substantial effects on formation of minority ion tail distributions, and on the time-dependent signal of vertically viewing NPA spectra. But, this exact type of diffusion calculation has not explained the full time-history. In a subsequent publication[11], results of direct inclusion of the gyro-orbit shift as a function of energy on the NPA spectra calculation, and also direct orbit losses to the wall, will be given. Additionally, we will investigate use of full guiding-center orbits on the distributions and NPA spectra using the new Hybrid finite-orbit-width option in CQL3D[12]. Research supported by USDOE Contracts DE-FC02-01ER54649, DE-FC02-01ER54648, DE-FG02-04ER54744, and DE-AC05-00R22725 with UT-Battelle, LLC.

8 8 REFERENCES 1. E.F. Jaeger, L.A. Berry, et al., Nucl. Fusion 46, (2006) S397-S R.W. Harvey and M.G. McCoy, The CQL3D Fokker-Planck Code, IAEA TCM, on Advances in Simulation and Modeling of Thermonuclear Plasmas, Montreal (1992). 3. A. Kaufman, Phys. Fluids 15, 1063 (1972). 4. L-G. Eriksson and P. Helander, Phys. Plasmas 1, 308 (1994). 5. A. Bader, R.S. Granetz, R.R. Parker et al., Nucl. Fusion (2012). 6. A. Bader, Experimental Measurements and Numerical Modeling of Fast-ion Distributions in the Alcator C-Mod Tokamak, Ph.D. Thesis, MIT (2012). 7. E.F. Jaeger, M.D. Carter, L.A. Berry, D.B. Batchelor, C.B. Forest, H. Weitzner, Nucl. Fusion 38, 1 (1998). 8. This research used resources of the National Energy Research Scientific Computing Center, supported by the Office of Science of the U.S. Dept. of Energy, Contract DEAC02-05CH V. Bergeaud, F. Nguyen, A. Becoulet and L-G. Eriksson, Phys. of Plasmas 8, 139 (2001). 10. E.F. Jaeger, L.A. Berry, et al., Phys. of Plasmas 15, (2008). 11. R.W. Harvey, Yu. V. Petrov, E.F. Jaeger et al., in preparation (2012). 12. Yu.V. Petrov, R.W. Harvey, Finite Orbit Width Features in the CQL3D Code, 24th IAEA Fusion Energy Conference, San Diego (this meeting), TH/P6-02 (2012).