Page 8 - Feasibility study of microwave electron heating on the C-2 field-reversed configuration device

012509-8 Fulton et al.
Phys. Plasmas 23, 012509 (2016)
the simulation runtime. In future simulations, with increased number of particles, if the current implementation becomes a bottleneck, the Bessel functions may be implemented more efficiently by table look-up.
C. Semi-spectral operator
Taking advantage of the toroidal periodicity, we Fourier-decompose in the ^f direction to obtain the semi- spectral Laplacian operator
FIG. 3. Simulation domain used in core region.
matched, before undertaking more intensive simulations with the new implementation. Additionally, energy and mo- mentum were recorded over the timescale of the simulation to ensure conservation. The divergence of the magnetic field was computed numerically in the new equilibrium and con- firmed to be zero everywhere. The following results come af- ter these verifications.
Linear simulations assume that the toroidal wavelength is much shorter than the radial wavelength of the instabil- ities, or equivalently, kr kf. Thus, we can approximate that the radial simulation domain is localized to a single flux surface. This is achieved by modifying the scatter and gather operations such that all dynamics of the system are on the same flux surface. For these single flux surface simulations, the Laplacian operator is further reduced from its spectral form in (21) to r2n 1⁄4 n2 gff.
Initial linear simulations are localized to the flux surface where r=a 0:906 in the core and r=a 2:244 in the SOL, where r is the position of the flux surface relative to the mag- netic axis and a is the distance from the separatrix to the magnetic axis, as measured along the mid-plane axis. Radial scans show these locations to be in the region where growth- rates are largest. The domain of the core simulations also excludes the magnetic axis where gyro-kinetic approxima- tions are poor due to the null magnetic field. The simulation domains are shown in Figs. 3 and 4, for the core and SOL, respectively. In future simulations, where the domain is a fi- nite annulus, the radial derivative in the Laplacian operator will be computed via finite differencing.
In the simulations presented in this section, both tempera-
ture and density gradient drives are included. The scale length
of the gradient of a plasma parameter, f, is 1=Lf 1⁄4 @ lnðfÞ. @r
Here, g characterizes the relative strength of the temperature and density gradients, and is the ratio between the two scale lengths, where gi 1⁄4 LTi =Ln and ge 1⁄4 LTe =Ln correspond to ions and electrons, respectively. In these simulations, gi 1⁄4 ge 1⁄4 g 1⁄4 1. The simulated ion species is deuterium. Collisional effects were evaluated using the Fokker-Planck model67 and found to have only small effect on linear growth rates, chang- ing them by less than 10%. For simplicity, collisions are excluded in the following results. The distance from the
FIG. 4. Simulation domain used in scrape-off layer region.
^ ınf fðw;h;fÞ ! fðw;hÞe
@ ! ın @f
2 2 ww@2 ff2 r?!rn1⁄4g @w2 gn:
(21)
Numerically, the derivative in the w^ direction is still found by central finite differencing, where i is the grid index corresponding to w
r2 1⁄4gwwð/ 2/ þ/ Þ gffn2/ n iþ1 i i 1 i
1⁄4 gwwð/iþ1 þ /i 1Þ ð2 gww þ n2 gffÞ/i: (22)
Because one of the perpendicular directions is ^f, this results in a tri-diagonal matrix with total number of non-zero elements equal to 3 Nw and a total matrix size of Nw2 which is solved using the Krylov method implemented in PETSc.65 Here, Nw is the number of radial grid points, Nh is the num- ber of poloidal grid points, and Nf is the number of toroidal grid points. The general process of solving the Poisson equa- tion is summarized in the following:
(1) Density is split into w f planes of size Nw Nf to pro- cessors based on h value assigned to individual processor;
(2) Transform density planes of size Nw Nf by FFT to semi-spectral density on w n planes of size ðNw NfÞ=2þ1;
(3) Semi-spectral potential is found on each w n plane of sizeðN N Þ=2þ1byPETSc;
wf
(4) Semi-spectral potential on w n plane of size ðNw NfÞ=
2 þ 1 is transformed back by FFT 1 to real-space potential in w f plane of size (mpsi mtoroidal), keeping only n of interest non-zero;
(5) Potential on w f plane of size Nw Nf is passed back to every processor to construct 3D potential of size Nw Nf Nh;
IV. INITIAL SIMULATION RESULTS
In this section, we report on initial simulation of drift- wave instabilities in the FRC, using the newly implemented code features described in Sections II and III. More detailed physics analysis will be reported in a forthcoming publica- tion.66 Prior to these simulations, simple test cases were run using both the new implementation and old less efficient (for FRC geometry) algorithms in GTC. We verified that the real frequency and growth rate from both sets of simulations