Page 2 - Compact Toroid Injection Fueling in a Large Field-Reversed Configuration

082512-2 Lau et al.
inspiration from these observations, we study the stability properties of the microscopic drift-wave instabilities in the FRC plasma, both in the core and in the SOL.
By identifying and studying transport mechanisms, a suit- able transport scaling may be found and applied toward predict- ing confinement performance in larger, hotter, and denser FRC plasmas. To our knowledge, first-principles simulation of turbu- lent transport in FRC geometry has not been previously carried out. To fill this void in theoretical understanding, and, in sup- port of ongoing experiments at TAE, we extend a mature, well- benchmarked turbulence simulation code, the Gyrokinetic Toroidal Code (GTC),38,39 to a system with C-2-like geometry and parameters.35,40,41 The focus of this study is to find and characterize the linear properties of drift-waves in the FRC core and SOL separately using gyro-kinetic simulation.
In early work in slab geometry, drift-waves were shown to be always unstable without particular thresholds, thus called “universal instability.”42,43 In the same slab geometry with the addition of finite magnetic shear, however, drift- waves were then found to become completely stabilized.44–47 The inclusion of electron non-linearity, however, can de- stabilize the drift wave.48 Investigations of toroidal coupling, in toroidal geometries such as the tokamak, then lead to the de-stabilization of drift-waves yet again.49–51 In toroidal geometry, the addition of shear flow was then found to be partially stabilizing for the drift-wave instabilities.52,53 In the last two decades, the paradigm has been dominated by the understanding of zonal flow generation as a non-linear mech- anism of regulation for the drift-wave instabilities.39 With the work of this paper, we address the new aspect of drift- wave stability in the FRC geometry which we are finding quite distinct from those in tokamaks.
In our simulations of the FRC SOL, which has mirror- like and slab-like geometry, the collisionless electrostatic drift-wave in the ion-to-electron-scale is destabilized by the electron temperature gradient due to resonance with locally barely trapped electrons. Collisions suppress this instability, but a collisional drift-wave instability can still exist at realis- tic pressure gradients.
We find that drift-waves in the FRC core geometry, on the other hand, are robustly stable in simulations (our simu- lations cover wavelengths up to kfqe < 0.3 so far). We should note here that, unlike tokamak geometry, there is no toroidal coupling which destabilizes the tokamak drift-waves49–51 in FRC geometry because the FRC lacks the toroidal fields and magnetic shear. Our study in simulations of limiting cases of FRC show this stability to be due to the features of the FRC core: (1) the field-lines of the ideal FRC core geometry are not toroidally coupled, similar to the early slab geometry but with closed field-lines, leading to extremely short connection lengths to shield electronic charge separation; (2) while the curvature of the field-lines is always bad, aligning with the direction of decreasing pressure, but the magnetic field is always increasing radially outward, leading to rB drift stabi- lization; and (3) the high temperature and low magnetic field lead to a large stabilizing finite Larmor radius (FLR) effect.2,8
The characteristics of these two regions have been com- pared to recent TAE experiments35 and found to be in
Phys. Plasmas 24, 082512 (2017) agreement. In particular, the lack of ion-scale instability in
the core is consistent with experimental measurements of a fluctuation spectrum showing a depression in the ion-scale.35 In addition, linear pressure gradient thresholds for stability found in simulations are consistent with thresholds observed in experiments. The survey of these modes will guide nonlin- ear and cross-separatrix simulations, as well as simulations including the effects of fast ions and ion cyclotron motion.
The remainder of this paper is organized as follows. Section II briefly details the simulation model which is more explicitly discussed in previous papers.40,41 Sections III and IV present the results for the core and SOL regions respec- tively. Section V discusses the interpretation of these results, including comparisons to recent experimental data.35
II. SIMULATION MODEL
Electrostatic simulations presented in this paper have been conducted with the Gyrokinetic Toroidal Code (GTC) using gyro-kinetic ions and gyro-kinetic electrons. GTC is a well- benchmarked, first principles code which has been extensively applied to study microturbulence and transport, including ion and electron temperature gradient driven modes,54 collisionless trapped electron modes,55 energetic particle transport,56 Alfv en eigenmodes,57,58 kink,59 and tearing modes.60
While the FRC core does contain a magnetic null-point, the simulation domains used for this paper do not include the null-point allowing gyro-kinetics to remain valid. This is dis- cussed in Sec. II B and graphically shown in Fig. 1.
Recently, GTC has been extended to study instabilities in the core and scrape-off layer (SOL) regions of the FRC.40,41 In this work, electrostatic perturbative df simula- tions38,43,61–63 are confined either in the core or SOL region separately with no cross-separatrix coupling. The domain is reduced to a toroidal wedge and localized to a single flux surface as described in Subsection IIB. The equilibrium parameters of the simulations are detailed in Subsection II A.
A. Equilibrium
Simulations are initialized with a FRC equilibrium which is representative of typical FRC plasmas realized in the C-2 experiment. The equilibrium is calculated using the LR_eqMI code, which is an axisymmetric force balance solver including realistic wall and coil geometry and the pos- sibility of multiple ion species, arbitrary rotation profiles, and arbitrary temperature profiles.64 These quantities are then transformed65 from cylindrical coordinates ðR; Z; /Þ to magnetic Boozer coordinates (w, h, f) for use in GTC40 as shown in Fig. 1. The origin of the Boozer coordinate system is located at the magnetic null-point. The magnetic field points in the poloidal direction, ~h, and the guiding-center drifts are in the toroidal direction, ~f, as shown in Fig. 1. Here, the ion diamagnetic direction is positive ð~fÞ, and the electron diamagnetic direction is negative ð ~fÞ. The major radius R0 1⁄4 27 cm is the distance from the machine cylindri- cal axis (geometry center) to the null-point (magnetic axis) as indicated by the blue dashed line. The minor radius a 1⁄4 11 cm is the distance from the null-point to the separa- trix. The separatrix radius Rs 1⁄4 38 cm is the distance from