Page 3 - Gyrokinetic simulation of driftwave instability in field-reversed configuration

056111-3 Fulton et al.
III. SIMULATION MODEL
To the authors’ knowledge, first-principles simulation of turbulent transport using realistic FRC magnetic geometry has never been previously carried out. In this study, we apply the Gyrokinetic Toroidal Code (GTC)19 to driftwave turbu- lence using numerical geometry based on the TAE advanced beam-driven FRC experiment, C-2. GTC is a full-featured, well-benchmarked, kinetic PIC code with gyrokinetic or fully kinetic ion models, kinetic electron effects,20 electro- static or fully electromagnetic field solvers, a Fokker–Planck collision operator,21 perturbative df or full-f resolution of the particle distribution functions, and an efficient, fully 3D magnetic coordinate grid which follows either analytically or numerically specified magnetic equilibrium geometry. GTC has been applied to simulations of microturbulence,22 ener- getic particle transport,23 Alfv en eigenmodes,24,25 and to MHD modes include tearing26 and kink27 instabilities in tokamak plasmas. A number of upgrades were recently implemented in GTC for the purpose of FRC simulation. These include (1) a coordinate transformation pipeline to produce Boozer magnetic coordinates from numerical FRC geometry, and (2) reimplementation of the electrostatic field- solver in order to optimize parallelism for an equilibrium magnetic field line orientation in the poloidal direction.28
For the initial effort presented in this paper, only linear mode growth of electrostatic instabilities is considered. Electromagnetic turbulence may play a crucial role in trans- port and will be investigated in the future. The ion model is gyrokinetic and electrons are drift kinetic using a realistic mass ratio. Boundary conditions for the particles are reflec- tive on the radial boundaries and periodic on the axial boundary. The field solver assumes a fixed value field on the edges of the grid. Simulations are initialized with par- ticles distributed uniformly across computational cells, with random spatial placement within each cell. The veloc- ity distribution of particles is assumed to be Gaussian about the thermal temperature for that species. Both fields and particle distributions are computed with the perturbative df model. Early simulations indicate the effect of collisions on linear growth rate and mode frequency is small, consistent with a low effective collisionality (see Section IV). For simplicity, collisions are excluded in the following results, but are under investigation and will be reincorporated in future simulations.
The magnetic coordinate system in GTC provides nu- merical advantages in the required grid size for the field solver, however is unable to simulate the separatrix, where there is a coordinate singularity. To circumvent this problem, the core and SOL regions have been simulated separately
Phys. Plasmas 23, 056111 (2016)
with the separatrix excluded. The magnetic axis is also excluded, since the gyrokinetic model is invalid at the field null in the center. A further approximation is that, in linear simulations, the toroidal wavelength is much shorter than the radial wavelength. It may be expressed in terms of the toroi- dal and radial wavenumbers as kr kf, and allows single flux surfaces to be simulated in isolation. The result is signif- icant savings in simulation time due to reduced domain and simplified Laplacian operator. The approximation is accom- plished by modifying particle gather and scatter operations to only sample one flux surface radially. Both core and SOL regions, with the simulated flux surfaces indicated, are shown in Fig. 1.
While the end goal of this work is to produce global do- main simulations of an FRC, using a significantly smaller do- main for initial simulations results in a rapid turnaround time for individual simulations. This rapid turnaround is critical to enable careful evaluation of all aspects of the physics model before more-expensive global simulations are carried out. Other approximations made in this paper are motivated similarly. Rapid iteration of simulations is an absolute neces- sity in the development of a complete integrated physics model for FRC transport.
IV. SIMULATION RESULTS
In this section, we report on initial results of driftwave instability simulations in FRC geometry. The simulation in the FRC core is performed on a flux surface with r=a 0:906. In the SOL, a flux surface with r=a 2:244 is used. The radial coordinate, r, is the position of the flux sur- face, measured outward on the midplane from the magnetic axis, and a, the minor radius of the FRC, is the value of r at the magnetic separatrix. These locations were chosen by per- forming radial scans to determine flux surfaces with maximal growth rates.
Driftwave instabilities are driven by density and temper-
ature gradients. These gradients may be characterized by
their length scales, where, for a plasma parameter, g, the
inverse length scale is defined as 1=Lg @ lnðgÞ. Both tem- @r
perature and density gradients are included. The ratio of the scale lengths of these two gradients, g, indicates their relative importance. For ions and electrons, gi 1⁄4 LTi =Ln and ge 1⁄4 LTe =Ln, respectively. In all of the following simulations g gi 1⁄4 ge 1⁄4 1. The simulated ion species is deuterium. The major radius, measured from the machine axis to the magnetic axis, is R0 1⁄4 26:96 cm, and the minor radius
is a 1⁄4 11:15 cm.
Simulation parameters and derived
FIG. 1. Magnetic geometry in both the FRC core and SOL used in GTC simu- lations. Black solid lines indicate the simulated flux surfaces in each region, and the dashed line indicates the separatrix.
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