Page 3 - Tomographic imaging system for measuring impurity line emission in a field-reversed configuration

022506-3 D. C. Barnes and L. C. Steinhauer
Phys. Plasmas 21, 022506 (2014)
with
There remains to compute the effects of finite, but small k. WewillorderksothatkVA x x Xsothattermsof order ðkVAÞ2 are comparable to those retained in our previous analysis. This ordering implies that kdi 1 because of the assumed elongation of the FRC. Thus, kdi < ka a=L 1. Because two-fluid corrections are of order ðkdiÞ2 these may be ignored and we find from the single-fluid Ohm’s law
ix^ B rX0Brh^ 1⁄4 ikB u: (10) Using Eq. (10) the line bending contribution to the right
There are additional GV terms which are perturbations of those written previously by finite k. It is easy to show that these corrections are of order ðkVA=x^Þkqi, compared with those included. so far, where qi is the thermal ion gyroradius in the external magnetic field. Because the first factor is of order unity and the second is small, these may be neglected.
Equations (9) and (12) comprise a non-standard eigen- value problem (because x^ enters in a complicated manner) once homogeneous boundary conditions are given. One of these is that there is an outer conducting wall at which U 1⁄4 0. The second is that of regularity near r 1⁄4 0, where it iseasytoshowthatU/r‘.
We find eigenvalues by a shooting method. Beginning with an appropriate multiple of r‘ at a small radius, we inte- grate the ODE Eq. (9) outward to the radius at which the outer boundary condition is to be satisfied (wall radius rw). The complex value of U at this radius is a residual, which vanishes at normal mode frequencies. Mode frequencies are complex and require non-standard shooting methods. We apply the intersecting contour method previously devel- oped.17 A specified portion of the complex plane is scanned for possible roots. The eigenvalue/eigenfunction solution is then found by applying the secant method to make the resid- ual smaller than some specified tolerance, with an initial guess determined from examination of the contour plot.
One additional modification which greatly simplifies the numerics is to rewrite the ODE in a form which eliminates some of the high order radial derivatives on equilibrium quantities. This considerably simplifies the problem of obtaining sufficiently smooth and accurate equilibrium solu- tions and allows an adaptive step ODE intergrator to operate efficiently. One can show that the following system is equiv- alent to Eqs. (9) and (12)
2‘ g0 A1⁄4rMn þ x^ ;
F 1⁄4 ‘M d n ð2X þ rX0Þ ‘2Mn x^ þ ‘2MX2n 0 r d r r 2 r x^
2‘ d rX0 g0 ð‘2 1Þ g0 00 þr2‘drx^ r g: (9)
hand side of Eq. (1) is
B • rB ikB
l1⁄4lB: (11) 00
After applying the curl operator, we find additional k2 cor- rections to our previous theory. Equation (8) remains as writ- ten, with the extended coefficients
A 1⁄4 r M n þ

0 ;
2‘ g 0 x^
rk2 B 2 =l
‘ x^ 2
F 1⁄4 ‘M d n ð2X þ rX0Þ ‘2Mn x^ þ ‘2MX2n 0
r d r r 2 r x^ þ2‘ ‘ d rX0 g0 ð‘2 1Þ g0 g00
r2"drx^ r # 2 ‘B 2=l0 1 d r B 2=l0 X0
0 V ZU U1⁄4 W ;
(13)
X0 ;
(14)
V0 1⁄4XUþYU0; !
r r2 x^ x^2 r r2 x^ ‘x^2 W 1⁄4 Mn x^ þ 2‘ g0 k2B 2=l0 ; Z 1⁄4 ‘Mn ð2X þ rX0Þ þ 2‘ g0 ‘rX0 1 k2 B 2=l0 X0 :
þkr2x^ rdrx^2 : (12)
X 1⁄4 ‘2Mn x^ ‘2MX2n 0 þ 2‘ð‘2 1Þ g0 k2‘B 2=l0 ‘Mn ð2X þ rX0Þ 4‘ g0
where
B 2=l0 r2rx^r3r2x^r2 r3x^rx^2
‘rX0 1 þ k2
Y 1⁄4 ‘Mn ð2X þ rX0Þ þ 2‘ g0 ‘rX0 1 k2 B 2=l0 X0 ‘X0 þ x^ rMn þ 2‘ g0 rk2B 2=l0 ;
r ‘x^ r r2 x^ x^2
It is obvious that the ODE in this form only requires first derivatives of equilibrium quantities to evaluate all coeffi- cients, while the previous form required derivatives up to second order. We integrate this system with the previously described boundary conditions to obtain normal modes of our rotating system.
III. NON-REVERSED THETA PINCH
We apply our incompressible model to the family of rigid-rotor equilibria considered in Ref. 7. As mentioned ear- lier, we use the rigid rotor equilibrium for electrons and rotating ions, including the centrifugal shift of the ion