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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 42, NO. 10, OCTOBER 2014
where Jθ ≈ envθ . Assuming axial symmetry, the 1-D equilib- rium solution can be written as
III. HYBRID MHD SIMULATION
The simulation tracks ion acceleration, FRC formation and equilibration, using a modified version of the 2-1/2-D, hybrid MHD code, MACH2 [9]; 2-1/2-D code has a 2-D grid, but computes all three components for the velocity and magnetic field.
MACH2 is a time-dependent, arbitrary Lagrangian–Eulerian simulation code that solves for the continuity, momentum, energy, and magnetic field equations on a grid composed of quadrilateral cells in the r- and z-plane. In the azimuthal direction, it solves the ion equation of motion. The code uses a finite-volume-differencing technique, and either analytic mod- els, or tabular values (http://t1web.lanl.gov/doc/SESAME_ 3Ddatabase_1992.html), for the equation-of-state and transport variables. In modeling the formation of rotating FRC, when the plasma is optically thin, a simple radiation emission model employing nonlocal thermodynamic equilibrium opacities was used.
One of our interests is to account for observations in the Irvine FRC experiment. The experimental geometry is cylindrical, with plasma injectors located on both ends of the central-confinement region, an outer (flux-conserving) limiter coil, and a central flux-coil. The injected plasma is inductively accelerated by the concentric coils, which are basically pulsed LC circuits with the rise-times matched.
The published reports are for an azimuthal-ion flow of, Viθ ∼ 7 km/s, measured by Doppler spectroscopy of multiple impurity lines [3]. Initially, an axial (background) magnetic field is applied, around 150–250 G, by pulsing the outer (limiter) coil. The azimuthal-electric field is generated at the location of the plasma fill, by pulsing the inner (flux) coil. This produces a diamagnetic-plasma current that reverses the applied-magnetic field at the plasma’s inner radius.
The azimuthal, electric-field generator was modeled by an equivalent solenoid circuit connected to the coaxial-flux coil. The circuit-series resistance was 0.05 with a coil radius of 10 cm and 7 turns connected to a capacitor of 1200 μFd, charged to 5 kV. The quarter-period rise time of the flux-coil current is 60 μs with a maximum of 5 kA. After this initial rise the flux-coil current is crowbarred. The background-magnetic field provided by the limiter coil is held fixed at 100 G, to suppress electron acceleration. The radial scale for the simulation is from, r = 10 − 45 cm and for the axial scale, z = −90 to 90 cm. The simulation grid is comprised of 128 × 48 cells and the initial (start-up) plasma density and temperature are, n0 = 5 × 1012 cm−3 andT0=5eV.
The modified MACH2 code solves the hybrid set of equa- tions described previously. Assuming, ω ≫ i, (16) simpli- fies to
dviθ e
dt = m Eθ −(ηe/mi)Jθ (18)
i
where Eθ is calculated from Faraday’s Law
vθ2 ∂p Bz ∂Bz −min r = −∂r + μ0 ∂r
∂Bz
∂r = −μ0envθ.
(11) (12)
Using N = n/n0, r02 r2 = 4e2n0μ0/(Te + Ti)vθ2/r2, and ξ = r2/r0 r, we obtain
d2lnN =−2N dξ2
2-D RR equilibrium solutions have been derived using a Green’s function approach [2], [4]–[6]. The azimuthal com- ponents can be written as
(13)
with the following 1-D rigid-rotor solutions: −2
N=cosh (ξ−ξ0)andBz=tanh(ξ−ξ0).
dviθ
mi dt =eEθ+evizBr−evirBz−ηeJθ
dveθ
me dt =−eEθ−evezBr+everBz+ηeJθ.
(14) (15)
Initially, when B⃗ = B0zˆ, the θ-components reduce to dviθ = e Eθ − ivir −(ηe/mi)Jθ
(16) dveθ = − e Eθ − ever + (ηe/me)Jθ (17)
dt mi dt me
where e = eB0/me and i = eB0/mi.
Ignoring electron inertia and assuming ω ≪ e, then
the azimuthal component of the electron equation of motion reduces to, ver = Eθ/B0, which is a radial drift of the elec- trons; any radial field imbalance 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. Ions will carry most of the plasma current in the azimuthal direction because of their large Larmor radius, at least initially, until the FRC closed-field structure develops.
The above model accounts for the finite gyroradius and gyroperiod of the ions in the azimuthal direction, but not that of the electrons. The electrons are assumed to be a background fluid, to maintain charge neutrality and to cancel any ion current in the radial and axial direction. The small gyroradius of the electrons, however, prevents their canceling the ion currents in the azimuthal direction. Therefore, the current will be, Jθ =nevθ.
Note that the pulsed, induction-electric field will be either reduced, or become very weak, at the large radius where the FRC exists, due to plasma shielding. In equilibrium the ion rotation is preserved, due to conservation of angular momentum, provided the the field-reversed magnetic topology is maintained by the total current. Dissipative effects will result in particle loss, particularly by the electrons, whereas instabilities may lead to larger-scale losses.
1 ∂ BZ
Eθ=−2πr ∂t (19)
and B Z
is the magnetic flux inside the central coil.