Plasma produced by a laser in a medium with convection and free surface satisfying a Hamilton-Jacobi equation

Abstract

A nonlinear diffusion equation is considered which models the temperature distribution in a laser-sustained plasma subject to wind. As certain parameters are small a singular perturbation problem arises and the method of matched asymptotic expansions is applied to approximate the solution. An essential role in this problem is played by the plasma front. This is a free surface separating the plasma and non-plasma phases. One of the main results is that we derive a 1st order nonlinear P.D.E. of Hamilton-Jacobi type, which describes certain parts of the free surface in the stationary case. This equation is analysed for various wind directions. It appears that for the initial conditions for these parts of this free surface there are different possibilities depending on the wind direction.

We show further that the solution of the Hamilton-Jacobi equation can contain singularities of corner type. Furthermore, the effect of wind on the stability region of the stationary full plasma solution is analysed. The method of analysis presented in this paper is not restricted to the cone-like geometry or the specific form of the non-linearity of the problem considered here, but has potentially a much wider scope. However, the case under study in this paper is certainly representative for the effects that have to be taken into account.