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apply a mature, benchmarked turbulence simulation, the Gyrokinetic Toroidal Code (GTC)41,42 to the FRC with real- istic parameters from the C-2 experiment. GTC has been successfully applied to simulate microturbulence,43 energetic particle transport,44 Alfv en eigenmodes,45,46 and magnetohy- drodynamic (MHD) instabilities, including the kink mode47 and tearing mode48 in fusion plasmas. This paper details the necessary upgrades which have been made to GTC to carry out simulations of an FRC.
In this study, a formulation for gyrokinetic particle simulation of the FRC is presented, with emphasis on the construction of a field-aligned mesh using magnetic Boozer coordinates49,50 and the implementation of a field solver in the FRC magnetic geometry. As a first step in developing FRC simulation capabilities, this study is confined to elec- trostatic effects. Electromagnetic instabilities, demonstrated to be relevant in FRC plasmas,25,27,51 will be investigated in detail in a future study, and the authors note that electro- magnetic capabilities are already present in the GTC framework.41,52–54 Verifications carried out on the newly implemented features are noted and results from first simu- lations are presented. Initial simulations find a pressure- gradient-driven driftwave instability on the ion scale in the SOL. Most interestingly, the ion scale driftwave is stabi- lized in the core by the large ion gyroradius.55
In Section II, we develop a coordinate mapping algo- rithm to produce magnetic Boozer coordinates from an MHD equilibrium in cylindrical coordinates. We also detail an extension of these Boozer coordinates into the SOL, con- stituting the first open field line simulation domain for GTC. Section III explains modifications to the field solver to accommodate the FRC’s poloidal-only magnetic field. We demonstrate some initial simulation results in Section IV, in both the FRC core and SOL. Finally, conclusions, discussion, and direction of future efforts are presented in Section V.
II. MAPPING TO MAGNETIC BOOZER COORDINATES IN AN FRC MAGNETIC FIELD GEOMETRY
There are a number of advantages to using magnetic- flux coordinates in a particle-in-cell simulation of magneti- cally confined plasma. Magnetic-flux coordinates are obtained by applying the constraint that the constant coordi- nate surfaces of one coordinate, typically referred to as the “radial coordinate,” correspond to the surfaces traced out by magnetic field lines. Such a coordinate system greatly sim- plifies expressions for physical quantities, such as the mag- netic field, B, and electric current, J. Plasma equilibrium quantities, including density, temperature, and pressure, are uniform on a magnetic flux surface and thus can be expressed as one-dimensional profiles using flux coordinates. Applying a second constraint on a magnetic-flux coordinate system, namely, that magnetic field lines are straight in the plane of the remaining two coordinates, results in straight field line coordinates. Advantages to using straight field line coordinates are manyfold. One such advantage is the ability to represent the magnetic field with scalar functions. Another advantage is having a field-aligned computational mesh,
Phys. Plasmas 23, 012509 (2016) which simplifies the implementation of the field solver and
particle equations of motion and minimizes the grid number required to resolve unstable modes, which are typically elon- gated along magnetic field lines. For any such flux coordi- nates, the magnetic field may be expressed in terms of coordinate gradients, B 1⁄4 $wtor $h þ $wrpol $f, where wtor is the toroidal flux and wrpol is the poloidal flux. Represented in this form, the magnetic field is automatically divergence-free and the existence of flux surfaces is guaranteed.
A third constraint is still required to establish a unique coordinate system. One common choice is to require that the periodic part of the magnetic scalar potential goes to zero, resulting in a simplified expression for the coordinate Jacobian. This particular choice, established by Boozer in the early 1980s, is referred to as magnetic Boozer coordi- nates.56 The simplified form of the Jacobian appears as J 1⁄4 ðqIpol þ ItorÞ=B2, where Ipol is the poloidal current, Itor is the toroidal current, and q is the safety factor. These terms will be explained in greater depth in Section II B.
GTC is formulated using Boozer coordinates, so the first step in implementing FRC simulation capabilities is produc- ing a mapping algorithm to translate FRC equilibria from cy- lindrical coordinates into magnetic Boozer coordinates. The mapping algorithm is most easily formulated by division into three stages. First, we map from the cylindrical coordinates, given as input from an MHD equilibrium code, to any set of magnetic-flux coordinates. Second, we establish a system of straight field line coordinates from the initial magnetic-flux coordinates. Third, we develop a transformation from the straight field line coordinates to the unique Boozer coordi- nates. With this three-stage coordinate mapping, we may interpolate values of the needed physical quantities onto the new grid in GTC.
One important note in establishing this algorithm is that the coordinate transformation only needs to be performed once for a given equilibrium, and the output may be saved. This work flow makes run-time efficiency of the coordinate mapping a relatively low priority. Numerical fidelity, by comparison, is critical, since the self-consistency of the equi- librium magnetic field will affect the numerical accuracy of the particle push and field solver in every time step of the simulation.
In the remainder of this section, we present an imple- mentation of coordinate mapping from cylindrical coordi- nates to Boozer magnetic coordinates, including simplifications specific to FRC magnetic field geometry and verifications.
A. Input to the coordinate mapping
The input magnetic equilibrium to the coordinate map- ping algorithm is generated by a 2-D equilibrium code, LR_eqMI, developed at TAE.57 The code LR_eqMI is designed to model equilibria in plasmas with multiple ion species. It allows for realistic boundary conditions with any combination of electrically conducting or insulating walls in an arbitrary geometry.