Self-similarity and nonlinear dynamics of thermally unstable media

Abstract

We apply the theory of continuous group transformations to equations governing optically thin gaseous media in the isobaric limit in order to find the class of self-similar solutions admitted of such equations. Depending on the character of the diffusivity (linear or nonlinear) three types of self-similar variables are found generating (i) the well-known root-mean-square law of heat transfer, (ii) a power-law enhanced or de-pressed by nonlinearity of diffusivity or radiation losses, and (iii) progressive wave type of solutions, respectively. The constraints on the form of the source (i.e., cooling) function have been determined for each solution. These results have been applied to the study of the nonlinear evolution of a thermal instability of a plasma as an example of astrophysical interest. We find that the nonlinear phase of the thermal instability leads to the formation of conductive/cooling fronts in the medium if a thermally unstable equilibrium point does exist. The specific dependence of diffusivity on temperature was shown to determine the shape of cooling/conductive fronts, while radiative losses influence the temporal behavior of perturbations. The transition between different types of self-similar solutions are discussed along with their stability.