Page 3 - Feasibility study of microwave electron heating on the C-2 field-reversed configuration device

012509-3 Fulton et al.
Phys. Plasmas 23, 012509 (2016)
This input includes the magnitude and vector direction of the magnetic field on a cylindrical coordinate grid, ðR; Z; fÞ, where
fR 2 R j 0 < R < 1g
fZ 2 R j 1 < Z < 1g ff 2 R j 0 f < 2pg:
Cartesian coordinates may be related back to these cylindri- cal coordinates via the expression
x 1⁄4 Rcosf
y1⁄4Z
z1⁄4Rsinf: (1)
Note that this definition is different from the conventional cylindrical coordinates, ðr; h; zÞ. To maintain right- handedness, the direction of the angular coordinate is reversed relative to convention. A number of physical quantities may be optionally included in the LR_eqMI equilibrium. The mapping algorithm requires only the magnetic flux, wrpol and the vector magnetic field in cylin- drical coordinates, B 1⁄4 ðBR;BZ;BfÞ. Plasma parameters, including ion density, electron density, ion temperature, and electron temperature, are taken from the LR_eqMI equilibrium grid to produce one-dimensional plasma pro- files which are used directly as GTC input.
The top panel of Fig. 2 shows the cylindrical input grid, which is symmetric in f, in the R–Z plane, along with color contours of the flux surfaces.
B. Establishing flux coordinates and straight field line coordinates
A magnetic flux surface is a smooth surface such that B n 1⁄4 0 everywhere, where n is a perpendicular to the sur- face. In a toroidal field geometry, these surfaces are nested and donut shaped. The innermost surface, as the donut nar- rows to a ring, is designated as the magnetic axis. Flux coor- dinates use these surfaces to represent one position in three- dimensional coordinate space.
The first step in developing the desired mapping algo- rithm is establishing some set of flux coordinates. As desig- nated in Section IIA, our initial cylindrical coordinate system is ðR;Z;fÞ. These cylindrical coordinates may be simply transformed into toroidal coordinates, a donut-shaped coordinate system conventionally used for toroidal geome- tries, denoted ðr; h; fÞ, where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r1⁄4 ðR R0Þ2þZ2
h 1⁄4 arctanðZ=ðR R0ÞÞ
f1⁄4f: (2)
The R0 in (2) is the R location of the magnetic axis, and is easily obtained from the LR_eqMI input equilibrium by locating the minimum flux value. The r^; ^h, and ^f compo- nents of vector quantities are, respectively, referred to as radial, poloidal, and toroidal, and this nomenclature carries
over to topologically similar coordinate systems, such as those discussed in the remainder of this paper.
Obtaining magnetic flux coordinates requires replacing the radial coordinate, r, with a flux surface label, w. As men- tioned in Section II A, this magnetic flux is supplied on the cy- lindrical grid in the input to the mapping algorithm. Specifically, we have the poloidal flux as a function of cylin- drical coordinates, wrpolðR;ZÞ. Here, “poloidal” refers to the poloidal component of magnetic field, which is integrated over a surface, Sp, a ring-shaped ribbon in the r f plane, stretched from the magnetic axis to the flux surface of interest.
ð
wrpol 1⁄4 B dS: Sp
This flux may be used in place of r in the toroidal coordi- nates expressed in (2). One feature of the FRC geometry is symmetry in the coordinate f which means establishing the radial flux coordinate is a matter of simple 2-D interpolation on the R–Z coordinate plane.
The second step in developing a mapping algorithm is obtaining straight field line coordinates, which we denote with a subscript “f.” Straight field line coordinates are a set of mag- netic flux coordinates, ðwf;hf;ffÞ, such that magnetic field lines drawn in the hf ff plane are straight lines. In the gen- eral case, establishing a set of straight field line coordinates from arbitrary flux coordinates may be quite involved. FRCs, however, contain only poloidal magnetic field, that is to say, Bf 1⁄4 0, so magnetic field lines drawn in the h f plane are al- ready straight, horizontal lines. Therefore, the set of magnetic flux coordinates which we have already established is also a set of straight field line coordinates in an FRC geometry. Our straight field line flux coordinate now looks like
wf 1⁄4wrpolðR;ZÞ
hf 1⁄4 h
ff 1⁄4 f: (3)
C. Determining Boozer coordinates from straight field line coordinates
The formal derivation of Boozer coordinates is well established, and we leave it to the reader to procure the details from a previous publication. We recommend the text by D’haeseleer et al., which provides a general treatment for the derivation of Boozer coordinates along with a myriad of other flux coordinate choices.50 Here, we summarize the result for the general case
; (4)
where l0 is the permeability of free space, w_ tor is the deriva- tive of the toroidal flux with respect to wf, w_rpol is the
wB 1⁄4 wf
h 1⁄4 h þ w_ r 2 p U~
B f pol l0 w_ rpolItor þ w_ torIpol f1⁄4fþw_ 2p U~
B f tor l0 w_ rpolItor þ w_ torIpol