Page 5 - Feasibility study of microwave electron heating on the C-2 field-reversed configuration device

012509-5 Fulton et al.
Phys. Plasmas 23, 012509 (2016)
factor of five greater than the desired number of grid points in the final Boozer coordinate mesh.
To obtain the toroidal current, Itor, only the constant component is kept from the transform
Notably, there is no j dependence in the final expression since we keep only the k 1⁄4 1 term. As expected, Itor is a flux surface constant.
We need to integrate the partial derivative term from (10) to obtain U~ . Integrating around the flux surface in real space aggregates numerical error, so the total error in U~ near hf 1⁄4 2p would become large. By taking advantage of the Fourier transform and integrating in k-space instead, these numerical issues are avoided. Applying the Fourier trans- form, the integration operator becomes
X1 2p N k1⁄41
l0 Itorðwf;hfðjÞÞ 1⁄4 1 1
Bk eð2pıÞð j 1Þðk 1Þ=N 1⁄4NBk1⁄41
1 XN 1⁄4N Bje0
Bhf ðhfðjÞÞ: U~ ðwf ; hf ð jÞÞ 1⁄4 1 XN ı
0
U~ðwf;hfðjÞÞ1⁄41XN XN ı Bhf hf j0 eð2pıÞðj jÞðk 1Þ=N : (13)
j1⁄41X 2p N
ð
ı dhf !k 1:
ItorðwfÞ 1⁄4 l0N
(12)
Keeping only the oscillatory components, the complete mag- netic scalar potential looks like
j1⁄41
Bk eð2pıÞð j 1Þðk 1Þ=N
1⁄4 1 XN 0@ ı XN Bj0 eð 2pıÞ ðj0 1Þ ðk 1Þ=N eð2pıÞ ð j 1Þ ðk 1Þ=N 1A
N k1⁄42 k 1
N k1⁄42 k 1 j01⁄41
Nk1⁄42j01⁄41 k 1
Now the results in (12) and (13) are simply plugged into transformation (6), and the cylindrical to Boozer coordinate transformation algorithm is established.
D. Output requirements from the coordinate mapping
The code which we are modifying for the FRC simula- tion, GTC, is formulated to push particles and solve electric and magnetic fields in Boozer coordinates. The desired out- put of the mapping algorithm is a coordinate grid system, regularly spaced in Boozer coordinates, with the cylindrical coordinate location of each of these new grid points. This constitutes an inverse coordinate mapping, RðwB; hB; fBÞ; ZðwB; hB; fBÞ, and fðwB; hB; fBÞ, from Boozer to cylindrical coordinates. These inverse coordinate transformations are used for gyroaveraging and computation of the metric tensor in GTC. Using splines, they also provide a quick method to determine a cylindrical position from a given Boozer coordi- nate location, which is practical for plotting diagnostics and comparison to experiments.
A series of one-dimensional high-resolution splines are used to implement the inverse transformation. For a given flux surface, a set of regularly spaced h is generated, and cor- responding hB are produced using (3) and (5). From these val- ues, a one-dimensional spline of hðhBÞ is generated on the flux surface, and this spline is used to determine h values at the desired, regularly spaced hB positions. This must be done for every flux surface of interest, so regularly spaced values of wB are selected to be sampled from the LR_eqMI input from
the beginning of the mapping algorithm. Looking at Eq. (6) makes it apparent that most of the complexity of the coordi- nate mapping algorithm is contained in the relation between h and hB. Fig. 1 shows a sample of the relationship between h and hB on a reference flux surface. A similar one-dimensional spline may be formed on a given straight radial ray in toroidal coordinates which corresponds to a single h. First, a set of reg- ularly spaced r is generated. With h fixed, each r corresponds to a single flux value, giving the spline, rðwBÞ.
The physical values of magnetic field magnitude, den- sity, and temperature are also needed on the new grid. Once the (R, Z) coordinate locations are obtained for the regularly spaced Boozer grid, these physical quantities are simply interpolated from the original LR_eqMI input file using a standard 2-D interpolation algorithm.
E. Extension of Boozer coordinates in the SOL
Understanding transport in the scrape-off-layer is criti- cal to understanding global FRC transport. Because the SOL acts as a divertor, transport behaviour inside and outside of the separatrix is qualitatively different.
A major obstacle to using GTC for simulations of the SOL is that magnetic Boozer coordinates are not defined for open magnetic field lines. Since the field lines in the region are not closed, the periodic magnetic scalar potential becomes meaningless. For equilibria that are symmetric over the Z 1⁄4 0 axis, which is approximately true for C-2, one solu- tion is to enforce periodicity across the Z boundary when