Field-reversed Configuration Equilibrium Research Articles

A new tool GSEQ-FRC for two-dimensional field-reversed configuration equilibrium

A new tool (GSEQ-FRC) is developed for solving two-dimensional equilibrium of the field-reversed configuration (FRC) based on fixed boundary and free boundary conditions that includes external coils. Benefiting from the two-parameter modified rigid rotor radial equilibrium model and the numerical approaches presented (Ma et al 2021 Nucl. Fusion 61 036046) in this work, the GSEQ-FRC code is used to study the equilibrium properties of FRC quantitatively and will be used for fast FRC equilibrium reconstruction. In GSEQ-FRC, the FRC equilibrium is determined by two parameters, i.e. the ratio between thermal pressure and magnetic pressure at the separatrix β s , and the normalized scrape of layer width δ s . Examples with fixed and free boundary conditions are given to demonstrate the capability of GSEQ-FRC for equilibrium calculations. This new tool is used to quantitatively study the factors affecting the shape of the FRC separatrix, which reveals how the FRC changes from racetrack-like to ellipse-like.

Two-parameter modified rigid rotor radial equilibrium model for field-reversed configurations

A new equilibrium pressure profile is proposed that extends the rigid rotor (RR) model with a simple unified expression P = P(ψ; β s, α, σ) for both inside and outside the separatrix. A new method is described for solving the field-reversed configuration (FRC) one-dimensional (1D) equilibrium with only two parameters. The radial normalized FRC equilibrium profiles for pressure, magnetic field, and current density can be uniquely determined by the dimensionless parameters of and δ s ≡ L ps/R s. This modified rigid rotor (MRR) model has sufficient flexibility to accommodate narrow scrape-off layer (SOL) widths and hollow current density profiles, and can be used to fit experimental measurements. A database of 1D plasma equilibria in the (β s, δ s) space is provided for the convenience of real-time approximate FRC equilibrium reconstruction. The detailed one-dimensional (1D) characteristics of the equilibrium models are investigated analytically and numerically. The 1D results from the MRR models are also validated using two-dimensional (2D) Grad–Shafranov (G–S) equilibrium solutions.

Theory of ion dynamics and heating by magnetic pumping in FRC plasma

A new mathematical framework is developed for efficient evaluation of ion heating during magnetic pumping of field-reversed configuration (FRC) plasmas. The topology of ion trajectories is examined through the use of action variables applied to a typical FRC equilibrium. Orbit bifurcations are identified and characterized analytically. These bifurcations are associated with the breakdown of the action variables as adiabatic invariants. The associated pitch-angle-mixing is important to the efficiency of magnetic pumping and is included in a new model for the kinetic behavior of the ions during the heating process.

Effects of Neutral Particle Density on Equilibrium of Field-Reversed Configuration Sustained by Rotating Magnetic Field

The effects of neutral particle density on equilibrium of field-reversed configuration (FRC) plasmas sustained by rotating magnetic fields (RMF) were investigated in the FRC Injection Experiment apparatus. Two different gas feeding methods were used to realize different neutral particle density conditions. The experimental results show that a higher current drive efficiency with full penetration of the RMF was achieved in the lower neutral particle density case. On the other hand, the plasma current in a higher neutral particle density case flowed in a narrower region near the geometric axis than in a lower neutral particle density case. The FRC equilibrium with higher neutral particle density resulted in lower current drive efficiency, possibly due to the shorter penetration length of the RMF. No significant ion spin-up was observed in the present conditions.

Two-fluid physics and field-reversed configurations

In this paper, algorithms for the solution of two-fluid plasma equations are presented and applied to the study of field-reversed configurations (FRCs). The two-fluid model is more general than the often used magnetohydrodynamic (MHD) model. The model takes into account electron inertia, charge separation, and the full electromagnetic field equations, and it allows for separate electron and ion motion. The algorithm presented is the high-resolution wave propagation scheme. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. The algorithms are benchmarked against the Geospace Environmental Modeling Reconnection Challenge problem. Equilibrium of FRC is studied. It is shown that starting from a MHD equilibrium produces a relaxed two-fluid equilibrium with strong flows at the FRC edges due to diamagnetic drift. The azimuthal electron flow causes lower-hybrid drift instabilities (LHDI), which can be captured if the ion gyroradius is well resolved. The LHDI is known to be a possible source of anomalous resistivity in many plasma configurations. LHDI simulations are performed in slab geometries and are compared to recent experimental results.

Rotation of a Field-Reversed Configuration due to Resistive Flux Decay

Rotation of a Field-Reversed Configuration (FRC) plasma due to a resistive flux decay is numerically studied. When the anomaly factor is 10, the flux lifetime is found to be about 60 μsec in a case that the external magnetic field is O.4 T and the wall radius is 0.17 m. Single-particle motions in a quasi-steady resistively decaying FRC equilibrium are calculated, and a local flow velocity is estimated by a particle-in-cell method. An electric acceleration of a betatron particle near the field-null is shown; this can cause a plasma rotation. From a comparison of the toroidal ion flow velocity profile between with and without the flux decay, it is found that the ion rotation begins at the field-null.

Advances in the numerical modeling of field-reversed configurations

The field-reversed configuration (FRC) is a compact torus with little or no toroidal magnetic field. A theoretical understanding of the observed FRC equilibrium and stability properties presents significant challenges due to the high plasma beta, plasma flows, large ion gyroradius, and the stochasticity of the particle orbits. Advanced numerical simulations are generally required to describe and understand the detailed behavior of FRC plasmas. Results of such simulations are presented in this paper. It is shown that 3D nonlinear hybrid simulations using the HYM code [E. V. Belova et al., Phys. Plasmas 7, 4996 (2000)] reproduce all major experimentally observed stability properties of elongated (theta-pinch-formed) FRCs. Namely, the scaling of the growth rate of the n=1 tilt mode with the S*∕E parameter (S* is the FRC kinetic parameter, E is elongation, and n is toroidal mode number), the nonlinear saturation of the tilt mode, ion toroidal spin-up, and the growth of the n=2 rotational mode have been demonstrated and studied in detail. The HYM code has also been used to study stability properties of FRCs formed by the counterhelicity spheromak merging method. A new stability regime has been found for FRCs with elongation E∼1, which requires a close-fitting conducting shell and energetic beam ion stabilization.

Profile consistency of an elongated field-reversed configuration. I. Asymptotic theory

An asymptotic theory of field-reversed configuration (FRC) equilibrium is developed, where the small expansion parameter is the square of the inverse elongation of the separatrix. It is shown that equilibrium alone completely determines the closed-field pressure profile of an elongated FRC in terms of the open-field profile. Examples show that the closed profile is insensitive to details of the open profile. A surprising result is that the open outflow plasma (axially beyond closed region) is always totally diamagnetic on the axis (β=1, where β is measured in the θ-pinch sense). The separatrix shape (axial variation) depends uniquely on the first-order pressure profile, and any separatrix shape may be realized within the limitations of the asymptotic theory. This sensitive dependence of shape on pressure profile explains extreme stiffness of the FRC equilibrium problem which was reported earlier. These results are compared favorably with experimental observations.

Analytic, two fluid, field reversed configuration equilibrium with sheared rotation

A two fluid model is used to derive an analytical equilibrium for elongated field reversed configurations containing shear in both the electron and ion velocity profiles. Like some semiempirical models used previously, the analytical expressions obtained provide a satisfactory fit to the experimental results for all radii with a few key parameters. The present results reduce to the rigid rotor model and the infinite conductivity case for a specific choice of the parameters.

Plasma stability of a nearly spherical field-reversed configuration

A nearly spherical field-reversed configuration (FRC) is created by employing a combination of a double-cusp bias magnetic field and laser-produced plasma, and its plasma stability is studied. The FRC equilibrium configuration is found to last for 20–25 μsec, which is not determined by any plasma instability in the present experiment and is estimated to be longer than the Alfvén time by a factor of more than 40. That is, the occurrence of neither the n=2 rotational instability nor the active magnetohydrodynamic (MHD) (tilting) mode is observed during this time period. The stabilization of the rotational mode may be attributed to the effect of magnetic field line bending. Although the exact reason for stabilizing the tilting mode is not known, it has become clear that the geometry of an equilibrium FRC has no significant effect on its stability, based upon comparison with the other experimental results obtained thus far from theta-pinch-based FRCs with a very elongated cross section. It is suggested that the ion kinetic, nonadiabatic effect may be largely responsible for this stability.