Theory of nonlinear diffusion of plasma across a magnetic field. 1: Solution of a class of quasilinear parabolic equations

The mathematical theory of a class of equations arising in studies of cross-field diffusion in toroidal multipole plasmas is presented. This class of quasilinear parabolic equations has a separable solution (the 'normal mode') which decays in time. It is shown that the normal mode is stable against infinitesimal perturbations; all small perturbations decay more rapidly than the normal mode. Numerical experiments show that large perturbations also decay more rapidly than the normal mode. It is demonstrated analytically that large perturbations decay exponentially whereas small perturbations decay as the fourth power (or higher) of the normal mode time dependence. Thus, the normal mode will rapidly evolve out of an initial distribution of particles, in agreement with the experiments. The asymptotic behavior of the system can be predicted approximately from knowledge of the initial particle distribution. The class of equations studied is also of interest in a variety of other physical, chemical, and engineering problems.