Page 4 - Feasibility study of microwave electron heating on the C-2 field-reversed configuration device

012509-4 Fulton et al.
derivative of the poloidal flux with respect to wf, and U~ is
the periodic part of the magnetic scalar potential. Here, we haver chosen the radial flux coordinate wrpol 1⁄4 wf so the value of w_ pol 1⁄4 1. The toroidal flux, wtor, on a flux surface labelled wf is the toroidal magnetic field component integrated over a surface,St,inthewf hfplanðecontainedwithinthefluxsur- face of interest,
wtor1⁄4 B dS: St
The magnetic scalar potential, U, is the homogeneous solu- tion to Ampere’s law, that is
l0J 1⁄4 $ B 1⁄4$ ð$UþBjÞ where
0 1⁄4 $ $U l0J 1⁄4 $ Bj:
It may be composed of both constant and periodic parts, U 1⁄4 U0 þ U~ , of which we are only interested in the periodic portion, U~ .
In the case of FRC magnetic geometry, simplifications to Eq. (4) may be considered. First, since there is no toroidal magnetic field, the toroidal flux and its derivatives are zero, wtor 1⁄4 w_ tor 1⁄4 0. Correspondingly, there may only be equilib- rium current in the toroidal direction and thus the poloidal current is also zero, Ipol 1⁄4 0. Making these substitutions, (4) becomes
w1⁄4w Bf
h B 1⁄4 h f þ 2 p U~ l0 Itor
fB1⁄4ff: (5)
Combining (5) and (3), we can write our fully simplified Boozer coordinate transformation in an FRC magnetic field geometry
wB 1⁄4wrpolðR;ZÞ
Phys. Plasmas 23, 012509 (2016) @ U~ !
B1⁄4 l0~gþ@w rwf f!
þ l 0 I þ @ U~ r h 2ptor @h f
f!
þ l0Ipolþ@U~ rff: (7)
2p @ff
Since there is no toroidal field in FRC geometry, the
rff component of B is zero. This gives ~
l0 Ipol þ @U 1⁄4 0: (8) 2p @ff
As noted, Ipol 1⁄4 0, and therefore @U~ =@ff 1⁄4 0 must also be true. The magnetic scalar potential, U~ , has no ff dependence, which is expected since we still have axisymmetry in the straight field line coordinates.
The rhf component of B is non-zero
Bhf jB rwfj 1⁄4l0Itorþ@U: (9)
~
jrwf rhfj 2p @hf
term on the left-hand side of (9) may be computed
The Bhf
directly by expressing the terms in cylindrical coordinates. Doing so yields
@w
BRR^þBZZ^þBf^f fR^þ fZ^þ f^f
Bhf @w @w @w @h @h @h f R^ þ f Z^ þ f ^f f R^ þ f Z^ þ f ^f
@w @w @R @Z @f
@R @Z @f @R @Z @f 1⁄4 l0 Itor þ @U~ :
Using the relationship expressed in (3), partial derivatives may be numerically computed everywhere.
Because U~ is periodic in hf, the derivative, @U~ =@hf , can- not have a constant component. By contrast, Itor is a flux function and is constant with respect to hf. By applying a dis- crete Fourier transform to the total term, Bhf , the constant and oscillatory parts can be separated as Fourier components, giving values for Itor and @U~ =@hf , respectively.
In our implementation, the forward and inverse Fourier transforms of Bhf take the form
XN ð 2pıÞð j 1Þðk 1Þ=N Bk1⁄4 Bje
Z 2pU~ wrpolðR;ZÞ;hf;ff hB 1⁄4 arctan þ
2p @hf
(10)
R R0 l0 Itor
fB1⁄4f: (6)
Notably, the periodic magnetic scalar potential, U~ , is left in terms of hf and ff since we are considering the periodicity in the straight field line coordinates. All of the quantities on the right hand side of this transformation are physical and may be determined from the information given on the input cylin- drical mesh from LR_eqMI.
To implement the coordinate transformation, numeri- cal values for wrpol, U~, and Itor must be determined. The poloidal flux, wrpol, is provided directly in the LR_eqMI input, but the others must be computed. If we consider the covariant expression of the magnetic field, B, the directional components of B contain both U~ and Itor terms:
j1⁄4X1
1 N ð2pıÞð j 1Þðk 1Þ=N
the
real-space
Bj 1⁄4N Bke : (11) k1⁄41
function is Bj 1⁄4 Bhf ðhf ðjÞÞ, where
Here,
hf ðjÞ 1⁄4 2pðj 1Þ=N.
by Bk, where k 1⁄4 1 represents the constant component and k 1⁄4 2; 3; 4; ... represent higher oscillatory harmonics. To ensure good numerical resolution, the number of grid points, N, of the intermediary straight field line grid, Bhf , is at least a
The k-space
function is represented