Four Applications of Lagrangian Mass Variables in Plasma Physics.

Four separate one-dimensional nonlinear fluid plasma problems are mathematically investigated in the frame of a Langrangian representation that uses the Lagrangian mass variables (LMV). The four problems are: (1) the space-charge-limited current (SCLC) problem, (2) three examples of exact nonlinear electrostatic motion of a plasma, (3) the self-similar expansion of a magnetized plasma slab, and (4) the self-similar implosion in a two-temperature plasma. The SCLC problem is concerned with the decay of a steady-state current in a dielectric placed between two parallel electrodes held at a constant potential difference after the ion supply has been interrupted. It is shown that the solution using LMV is equivalent to the traditional approach using the method of characteristics yet is simpler to manipulate and yields an exact integral of an earlier calculation. The exact solutions of the motion of cold, single - and two-species plasmas are derived in terms of arbitrary functions to be specified by initial and boundary conditions. Natural mathematical restrictions are interpreted in terms of the plasma frequency. A nonlinear wave equation is derived for a warm two-species plasma whose linearized form does not require any conditions on the initial profiles as has been in previous calculations. An alternate derivation of the self-similar expansion of a semi-infinite quasi-neutral slab of magnetized plasma is performed. In this frame, the magnetic field is trivially shown to be frozen into the plasma. An additional calculation is presented corresponding to the requirement of total particle conservation for the system. The self-similar-stationary profile of a two-temperature plasma shock implosion with zero thermal flux back boundary is solved numerically. It is found that a small supersonic precursor dominated by the electron thermal flux is followed by a shock compression that heralds a larger subsonic region dominated by heat exchange between the electrons and hot ions. The shock profile is virtually identical for planar, cylindrical, and spherical geometries. An approximate analytic solution is also generated which is in excellent agreement with the numerical solution.