Fully Ionized Pinch Collapse

Abstract

A fully ionized plasma is assumed. To this plasma cylindrically-symmetric magnetic fields are applied, thus causing a pinch collapse. The plasma is treated in hydromagnetic approximation, including electric and thermal conductivity. Separate temperatures are assigned to the electrons and ions.Two schemes are developed for solving numerically the resulting system of six partial differential equations: the explicit scheme for rather fast pinches, where a numerical stability requirement causes the timestep to be bounded by the characteristics given by the ALFVEN speed, and an implicit scheme, which consists essentially in converting the momentum equation into a second order difference equation with coefficients determined by iteration; here there is no such restriction on the timestep. These schemes were made to work on the U.K.A.E.A. IBM 704 and IBM 709.A run is described in which the initial state was one with uniform density, temperature and B z field. The boundary temperatures were assumed to remain constant, while the magnetic fields at the boundary were determined by the circuits for the j z and currents. The results of the computations are in good agreement with experimental results obtained at the Technische Hochschule Mün-chen by one of the authors (KÖPPENDÖRFER) .The whole program is a joint effort between A.E.R.E. Harwell and the Max-Planck-Institut, intended to discover by comparison with experiments how good the hydromagnetic approximations are. If the agreement is satisfactory (eventually using a generalised program which includes neutral gas) it should be possible to design experiments so that specified field configurations are set up. This paper treats the simplest possible model for the dynamical problem of the pinch collapse, namely a fully ionized plasma. It is the first of a series of papers dealing with calculations on the pinch. The series will describe a joint attempt by A.E.R.E. Harwell and the Max-Planck-Institut für Physik und Astrophysik, to set up quite general programs for big computers to calculate the pinch effect in hydromagnetic approximation. Within this restriction the equations should be as close as possible to the physics of the actual experiments. The comparison between experiments and these computations should give evidence on how good the hydromagnetic approximation is. It is planned to generalize this program in some respects, e. g. by including neutral gas, anisotropic pressure, toroidal geometry. Eventually one would hope to be able to design circuits to achieve field distributions specified beforehand, e. g. for reasons of stability.The paper consists mainly of three parts. In the first part the set of equations is derived and the boundary conditions are discussed. In the second part the numerical methods are described. The third part gives the comparison between experiments and computations. I. The hydromagnetic equations (two fluid model)A fully ionized plasma is assumed to fill an infinitely long cylinder. Only z and Q magnetic field components are allowed. All quantiti...

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