Spheromak research from 1979 to the present is reviewed including over I60 references. Emphasis is on understanding and interpretation of results. In addition to summarizing results some new interpretations are presented. An introduction and brief history is followed by a discussion of generalized helicity and its time derivative. Formation and sustainment are discussed including five different methods, flux core, theta -pinch z-pinch, coaxial source, conical theta -pinch, and kinked z-pinch. All methods use helicity injections. Steady-state methods and rules for designing spheromak experiments are covered, followed by equilibrium and stability. Methods of stabilizing the tilt and shift modes are discussed as well as their impact on the reactor designs. Current-driven and pressure-driven instabilities as well as relaxation in general are covered. Energy confinement is discussed in terms of helicity decay time and and beta s limits. The confinement in high and low open-flux geometries are compared and the reactor implications discussed.

https://iopscience.iop.org/article/10.1088/0741-3335/36/6/002/pdf

Review of spheromak research

Ball lightning as an example of a magnetohydrodynamic equilibrium.

The Grad-Shafranov equation for a prolate spheroidal plasma is solved by using the prolate spheroidal coordinates. The magnetic field in the vacuum region surrounding the plasma is determined by use of the Maxwell equation and the pressure-balance condition on the plasma-vacuum interface. The result obtained in this letter is an exact solution to the magnetohydrodynamic equilibrium equation. The magnetic-field configuration of the spheroidal plasma is found to be considerably different from that of the spherical plasnla.

Magnetohydrodynamic Equilibrium with Prolate Spheroidal Plasma-Vacuum Interface

The analytic solutions to the Grad-Shafranov equation are obtained for a prolate and an oblate spheroidal plasma by using Hill's vortex model. Effects of a toroidal magnetic field B ϕ on the MHD equilibrium configurations are investigated by using these analytic solutions. When B ϕ is stronger than that of the force-free configuration, the spheroidal plasmas in a vacuum magnetic field are shown to be unable in the MHD equilibrium. The several physical quantities on the equilibrium configuration are evaluated. The spheromak plasma is proved to be unstable if d p /dψ≠0 and d 2 F /dψ 2 ≥0 on the magnetic axis. Here p is the pressure and V (ψ) the volume surrounded by a magnetic surface of ψ=const. The equilibrium configurations of the spheroidal plasmas by using Hill's vortex model are shown to satisfy the above conditions, i.e., to be unstable.

Magnetohydrodynamic Equilibrium and Stability of Spheromak with Spheroidal Plasma-Vacuum Interface

The Grad-Shafranov equations for an oblate and a prolate spheroidal plasmas are solved analytically under the assumptions, B ϕ =0 and d p /dψ=constant. Here B ϕ is the toroidal magnetic field, p is the kinetic pressure, and ψ is the magnetic flux function. The plasmas in magnetohydrodynamic equilibrium are shown to be toroidal . The equilibrium magnetic-field configurations outside the spheroidal plasmas are considerably different from that of a spherical plasma. A line cusp or two point cusps appear outside the oblate or the prolate spheroidal plasma, respectively.

Magnetohydrodynamic Equilibrium with Spheroidal Plasma-Vacuum Interface : Compact Toroid without Toroidal Magnetic Field

Solutions representing axially symmetric hydromagnetic equilibria with spherical plasma‐vacuum interfaces are extended from a low‐pressure paramagnetic regime to a high‐pressure diamagnetic regime.

G. K. Morikawa

Papers

Axially Symmetric Hydromagnetic Equilibria with Spherical Plasma‐Vacuum Interfaces