Page 2 - Rotational stability a long field-reversed configuration

032507-2
Rahman et al.
Phys. Plasmas 21, 032507 (2014) The azimuthal components can be written as
m dvih 1⁄4 eE þ ev B ev B geJ ; (11) idt h izr irz h
m dveh 1⁄4 eE ev B þ ev B þ geJ : (12) edt h ezr erz h
Initially, when B~ 1⁄4 B0z^, the h-components reduce to
dvih 1⁄4 e Eh Xivir geJh; (13)
FIG. 1. Flux-coil (driven), field reversed configuration.
@ ~v dveh 1⁄4 e Eh Xever þgeJh; (14)
dt mi mi
e ~~ dtme me
men @t þ~ve r ~ve 1⁄4 enðEþ~ve BÞ rpe þPei; (2)
where Pie 1⁄4 Pei 1⁄4 genJ~ and g is the resistivity of the plasma. The above equations assume charged neutrality ni 1⁄4 ne. In cylindrical coordinates the equations can be sim- plified, assuming azimuthal symmetry and B~ 1⁄4 ðBr ; 0; Bz Þ. Equations (1) and (2) can be split into the radial components
where Xe 1⁄4 eB0/me and Xi 1⁄4 eB0/mi.
If we ignore electron inertia and assume x Xe, then
the azimuthal component of the electron equation of motion reduces to ver 1⁄4 Eh/B0, which is a radial drift. Any radial imbalance in the electric field can be compensated by the free motion of electrons along the field lines.
In the azimuthal direction, only the ion equation of motion is applicable. Initially, ions carry most of the plasma current in the azimuthal direction, because of their large Larmor radius, at least until the closed-field structure of the FRC develops.
Assuming axial symmetry, the one dimensional equilib- rium solution can be written as
v2h @p Bz@Bz minr1⁄4 @rþl0 @r; (15)
@Bz 1⁄4 l0envh: (16) @r
dvir ~ ~ @pi
min dt 1⁄4enðEþ~vi BÞr @r genJr;
dver ~~@pe
men dt 1⁄4 enðEþ~ve BÞr @r þgenJr;
and axial components
dviz ~ ~ @pi
min dt 1⁄4enðEþ~vi BÞz @z genJz;
dvez ~~@pe
men dt 1⁄4 enðEþ~ve BÞz @z þgenJz:
(3) (4)
(5) (6)
2
Using N1⁄4n/n , r2Dr2 1⁄4 4e2n0l0 vh, and n 1⁄4 r2 , we obtain
0 0 ðTeþTiÞr2 r0Dr d2 ln N 1⁄4 2N;
Using Ampere’s Law
r B~ 1⁄4 l J~ 1⁄4 l ð J~ þ J~ Þ ; (7) 2
00ei dn
the radial and axial components may be combined into single MHD fluid equations of motion
@vr @vr @vr v2 q@tþqvr@rþvz@z qrh
(17)
with the standard 1-D Rigid-Rotor Model (RRM) solutions17 N 1⁄4 cosh 2ðn n0Þ; (18) Bz 1⁄4 tanhðn n0Þ: (19)
2-D equilibrium solutions are also available.11–14
The above model accounts for the finite gyro-radius and gyro-period of the ions in the azimuthal direction but not that of the electrons. The electrons are assumed to be a back- ground fluid, in order to maintain charge neutrality and to cancel any ion current in the radial and axial directions. However, the small gyroradius of the electrons prevents their canceling of the ion currents in the azimuthal direction.
Therefore, the current density is Jh 1⁄4 nevih.
Note also that the plasma may shield the pulsed,
induction-electric field at the radius where the FRC equili- brates. In the quasi-equilibrium, the field-reversed magnetic topology is maintained due to the total current, while the ions keep rotating due to conservation of angular momen- tum. Dissipative effects, due to drag or instabilities, will
1⁄4 @ pþB2 þ1 B@BrþB@Br ; (8) @r 2l l r@r z@z
q @vz þ q v @vz þ v @vz @t r @r z @z
00
@ B2 1 @Bz @Bz 1⁄4 @z pþ2l þl Br @r þBz @z ; (9)
00
where p 1⁄4 pe þ pi 1⁄4 nðTe þ TiÞ. The azimuthal component of Ampere’s law can be written as
@Br @Bz 1⁄4 l0Jh 1⁄4 l0enðvih vehÞ @z @r
where Jh envh.
1⁄4 l0ernðxi xeÞ l0envh; (10)