Electron and Ion Runaway in a Fully Ionized Gas. I

Hydrodynamic equations are used to describe the flow of the electrons and ions of a fully ionized gas under the action of an electric field, E, of arbitrary magnitude. The dynamical friction force exerted by the electrons and ions upon each other through the agency of two-body Coulomb encounters is evaluated. In this connection the electrons and ions have been assigned Maxwellian velocity distributions which are displaced from each other by their relative drift velocity. This treatment yields a dynamical friction force which maximizes when the relative drift velocity is equal to the sum of the most probable random electron and ion speeds. For relative drift velocities in excess of this value the friction force decreases rapidly. As a consequence, it is found that a fully ionized gas cannot exhibit the steady-state behavior characterized by time independent drift velocities which has previously been accredited to it by other authors. Instead, it is shown that the electron and ion currents flowing parallel to the existing magnetic fields increase steadily in time (i.e., runaway) as long as a component of the electric field persists along the magnetic field. Drift velocities which greatly exceed the random speeds of the plasma particles can be created in this manner.

The theory yields a critical electric field parameter, $E_c$, which is proportional to the plasma density and inversely proportional to its temperature. It is a measure of the electric field which is required if the velocities are to increase and exceed the most probable random speeds in the gas in one mean free collision time. For electric fields in excess of $E_c$ runaway proceeds even faster. In smaller fields runaway occurs when Joule heating has depressed $E_c$ sufficiently. Several interpretations of $E_c$ are given in terms of the collisional phenomenon involved.

Within the framework of the hydrodynamic equations it is shown that the well-known ${\left(\mathrm{temperature}\right)}^{\frac{3}{2}}$ electrical conductivity law can be recovered, provided $E\ll E_c$ and the electron temperature is held constant.

Numerical solutions giving electron temperature and drift velocity as a function of time are presented for a range of the ratio $\frac{E}{E_c}$. The assumption of the displaced Maxwellian distribution is justified on the basis of a comparison between the rate of Joule heating and the rate of equipartition of random speeds. Moreover, it is found that the use of an anisotropic velocity distribution does not affect the runaway phenomenon in any important way.

The possibility of runaway induced across magnetic fields by steep pressure gradients and its relation to diffusion across magnetic fields is examined and discussed in detail.