COMPX^SM
Computational Modeling and Software Development

DC

Description | Purpose | Algorithms | Results | Publications

Short Description

Diffusion Coefficient Calculator code. Forms gyro- and bounce- averaged RF diffusion coefficients by direct integration of particle orbits in combined equilibrium and full-wave RF.

Date/Active Use

2000.

Authors

R.W. Harvey, Yu.V. Petrov

Language

F77/F90 + MPI

Purpose/Function/Special Features

The DC diffusion coefficient calculator [1-3] numerically integrates the trajectories of ions launched from tokamak midplane points in the combined equilibrium and AORSA full-wave RF fields. Particles are launched equispaced in initial gyro-phase about a given gyro-center and also equispaced in toroidal length. Diffusion coefficients are obtained by averaging the resulting square of the velocity changes after one (or more) ion poloidal circuits, to obtain the ICRF bounce-averaged diffusion tensor. This is carried out for a 3D array (u_par, u_perp, R) of initial conditions, giving the six independent RF diffusion coefficients in 3D constant-of-motion space. The method follows the formalism of Refs. [4,5]. For comparison, we have the zero-banana-width (ZOW) RF diffusion coefficients calculated in the AORSA code[6]. Comparison is more directly achieved with DC by subtracting off the perpendicular guiding center drifts using a fictitious force in the Lorentz equation, F_perp = u_gc X B. This removes the finite banana width effects, but leaves correlation, finite gyro-radius, and other effects. The DC code is similar to the MOKA code[7], but has been coupled to the CQL3D Fokker-Planck code[6] and AORSA to obtain a time-dependent, noise-free solution to the ICRF heating problem across the whole plasma width. The integration of (32 radii) X (128 u_perp) X (256 u_par) X (8 gyro-phase) X (1 toroidal angle) starting positions (8M Lorentz orbits) is well-parallelized and takes 5 minutes on 1152 cores; these calculations are implemented on the Cray XC30 supercomputer. For the time-dependent calculation, each aorsa-dc-cql3d cycle takes 22 minutes using 1192 core. The DC portion is 6 minutes; the 10 subcycle steps of CQL3D-HYBRID-FOW require 4 minutes. Forty time-steps are used.

Basic Algorithms

Orbit integration is by 4th order Runge-Kutta. Sufficiently accurate orbits are determined by varying the time step, to get converged solutions.

Coupled Diagnostics

The usual suite of diagnostics is available through the coupling to CQL3D.

Key Results

The principal results are that resonable agreement is obtained between the numerical integration determination of RF diffusion coefficients and the quasilinear theory. Details of the pitch angle dependence of the Duu diffusion coefficient, evidently due to correlations between resonance crossings in the numerical integration case are visible, but the overall effect on the radial damping profile is not large, at least for the C-Mod tokamak case examined.

Fig. 1(a-c) compares the velocity space Duu diffusion coefficient calculated by DC for 1, 2, and 4 complete turns in the poloidal plane, for a symmetric 101 toroidal mode case modeling the finite length C-Mod antenna. The Fig. 1(b) coefficient for 2 turn shows significantly greater pitch angle dependence than the single turn results in 1(a); Fig. 1(c) for 4 turns shows little additional correlation effects. The close similarity of (b) and (c) support the accuracy of the code. Peaks of the Fig. 1 coefficients are 1.46(1 turn), 1.66(2 turns), 1.77(4 turns), and 0.55(AORSA QL), Fig 1(d), in accord with heuristic expectations for correlations which are reaching maximum effect. The AORSA-QL coefficients are calculated directly from the full-wave fields using quasilinear theory which incorporates spatial delta function wave-particle interactions at resonances, with no correlations[1]. All three DC coefficient radial sets show remarkable agreement in radial power absorption, shown in Fig. 2.

Selected Publications, and References

R.W. Harvey, Yu. V. Petrov, D.L. Green, E.F. Jaeger, P.T. Bonoli, D.B. Batchelor, L.A. Berry, J.C. Wright, A. Bader, and RF-SciDAC Group, "Comparison of Quasi-linear and Exact Ion Cyclotron Resonant Heating Diffusion, With and Without Finite Width Ion Orbits", paper THW/P7-09, Proceedings of 23rd IAEA FEC 2010, Daejon, Korea, 11-16 October (2010); Copy Here.
R.W. Harvey, Yu. Petrov, E.F. Jaeger, and RF-SciDAC Group, Validation Studies Of Quasilinear Theory Of Resonant Diffusion In The Ion Cyclotron Range Of Frequencies By Comparison With Exact Integration Results'', 19th Topical Conference on Radio Frequency Power in Plasmas,Newport, Rhode Island, June 1-3 (2011); https://doi.org/10.1063/1.3664995 Copy Here.
R.W. Harvey, Yu. Petrov, E.F. Jaeger, L.A. Berry, P.T. Bonoli, and A. Bader, "Time-Dependent Distribution Functions and Resulting Synthetic NPA Spectra in C-Mod Calculated with the CQL3D-Hybrid-FOW, AORSA Full-Wave, and DC Lorentz Codes" 21st Topical Conference on Radiofrequency Power in Plasmas, Lake Arrowhead, California, April 27 - 29, 2015 AIP Conference Proceedings 1689, 060005 (2015); https://doi.org/10.1063/1.4936503 Copy Here.
A. Kaufman,"Quasilinear Diffusion of an Axisymmetric Toroidal Plasma ", Phys. Fluids 15, 1063 (1972). https://doi.org/10.1063/1.1694031
K. Kupfer, "Fokker-Planck Formulation for RF Current Drive, Including Wave Driven Radial Transport", Proc. of IAEA TCM on FWCD in Reactor Scale Tokamaks, Arles(1991). Copy Here.
E.F. Jaeger, R.W. Harvey, L.A. Berry, et al., " Global-wave solutions with self-consistent velocity distributions in ion cyclotron heated plasmas", Nucl. Fusion 46, S397 (2006). https://doi.org/10.1088/0029-5515/46/7/S02
V. Bergeaud, F. Nguyen, A. Becoulet and L-G. Eriksson, "Superadiabatic behavior of fast particles accelerated by ion cyclotron resonance heating", Physics of Plasmas 8, 139 (2001). https://doi.org/10.1063/1.1332988