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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 42, NO. 10, OCTOBER 2014 3137
Hybrid MHD Model for a Driven, Ion-Current FRC Hafiz U. Rahman, Frank J. Wessel, Norman Rostoker, Michl W. Binderbauer, and Paul Ney
Abstract— A standard magnetohydrodynamic code, MACH2 [1], is modified in 1-D to account for two-fluid behavior, to include the effects of a finite-electric field during the formation of a driven, field-reversed configuration (FRC). The simulation is run for a period of 150 μs. Initially an azimuthal ion current is produced, then the FRC forms, compressing radially and axially, and finally the FRC decays. Previous experiments [3] agree with the simulations, specifically in the magnitude of the total current produced, including ion and electron flows, and the radial profile for the FRC, which appears to be rigid rotor [2].
self-consistent plasma rotation that leads naturally to a rigid- rotor equilibrium.
In the following sections, this paper is organized as follows. Section II describes the derivation of the hybrid MHD model, starting from a two-fluid plasma model, Section III describes the results from the simulation, using a modified version of the MACH2 code [1], [9] to model the approximate experimen- tal configuration, Section IV compares the simulations with experimental results, and Section V concludes this paper.
II. HYBRID MODEL
The model is 2-D, in r and z, and assumes azimuthal symmetry in the third dimension, θ, and charge neutrality, ni = ne. The standard, two-fluid momentum equations are
∂ v⃗ i
Index Terms— 2-fluid, field-reversed magnetohydrodynamic (MHD).
I. INTRODUCTION
configuration
(FRC),
THE field-reversed configuration (FRC) is distinguished from other fusion concepts by its high-beta plasma,
2
β = 8πnkT/B ≥ 1 nature. The plasma is formed in
cylindricalconfigurationwithalarge,ion-gyro-orbits,ρi ∼R, where R is the size of the chamber, and there is a presence of amagnetic-fieldnull.Theseeffectsleadnaturallytophysics that is not captured by standard magnetohydrodynamic (MHD) models. Thus, hybrid models, and even better, kinetic models, are needed to study the formation, and long-term stability, of FRCs.
The equilibrium of a rotating FRC has been studied analyti- cally in the context of the rigid-rotor model [2], [4]–[6] where the entire plasma rotates with a constant angular frequency. Experiments at the University of California, Irvine report that a driven FRC, formed and sustained between coaxial solenoids, contains an azimuthal current that is due primarily to ion rotation [3]. The FRCs formed by a reversed-field θ-pinch [7] also report toroidal spin-up, e.g., in the C-2 device, formed by merging two θ-pinch plasmas [8]. The origin of the spin-up is suggested to be due to end-shorting of the radial-electric field, that is found on open, magnetic-field lines, and particle-loss jetting from the FRC, inside the separatrix [7].
This paper employs the use of a hybrid, two-fluid code to study the formation and equilibration of the FRC, driven by an external, inductive-electric field. Standard MHD descriptions are used for the radial and axial components, while the azimuthal component is two-fluid. This simulation produces a
Manuscript received October 10, 2013; revised January 2, 2014; accepted February 18, 2014. Date of publication September 17, 2014; date of current version October 21, 2014.
H. U. Rahman, F. J. Wessel, N. Rostoker, and M. W. Binderbauer are with Tri Alpha Energy, Inc., Rancho Santa Margarita, CA 92688 USA (e-mail: hrahman@trialphaenergy.com; fwessel@uci.edu; norman.rostoker@uci.edu; michl@trialphaenergy.com).
P. Ney is with Mount San Jacinto College, Menifee, CA 92584 USA (e-mail: pney@trialphaenergy.com).
+ v⃗ · ∇ v⃗ = e n E⃗ + v⃗ × B⃗ − ∇ p + P i∂tii i iie
m n
wherePie=−Pei=−ηenJ⃗andηistheresistivityof the plasma. In cylindrical coordinates, with B⃗ = ( Br , 0, Bz ), (1) and (2) can be split into the radial components
dvir ⃗ ⃗ ∂pi
mindt =eE+v⃗i×Br−∂r−ηenJr (3)
dver ⃗ ⃗ ∂pe
mendt =−eE+v⃗e×Br−∂r+ηenJr (4)
and axial components
dviz ⃗ ⃗ ∂pi
mindt =eE+v⃗i×Bz−∂z−ηenJz (5)
dvez ⃗ ⃗ ∂pe
mendt =−eE+v⃗e×Bz−∂z+ηenJz. (6)
( 1 ) men ∂v⃗e +v⃗e·∇ v⃗e =−en E⃗+v⃗e×B⃗ −∇pe+Pei (2)
∂t
Using Ampere’s Law
∇ × B⃗ = μ J⃗ + J⃗ . (7)
0ei
Equations (3)–(6) may be combined into single MHD fluid
equations of motion
ρ∂vr +ρ v ∂vr +v ∂vr −ρvθ2
∂t r∂r z∂z r
∂ B2 1 ∂Br ∂Br
=−∂r p+2μ +μ Br ∂r +Bz ∂z (8)
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org. ∂z
Digital Object Identifier 10.1109/TPS.2014.2320407
−
∂r
=μ0Jθ =μ0en(viθ −veθ)
= μ0ern(ωi − ωe) ≈ μ0envθ (10)
00
ρ∂vz +ρ v ∂vz +v ∂vz
∂t
r ∂r z ∂z
∂ B2 1 ∂Bz ∂Bz =− p+ + Br +Bz
(9)
∂z 2μ0 μ0 ∂r ∂z
where p = pe + pi = n(Te + Ti ). The azimuthal component of Ampere’s law can be written as
∂Br ∂Bz
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