Critical Points and Separatrix Characteristics in Solar and Astrophysical MHD Flows

Abstract

One of the main difficulties in solving the MHD equations is due to the fact that a physically accepted solution is determined by the requirement that it passes through some critical points which are not known a priori, but are only determined simultaneously with the complete solution. in this chapter the relation of such critical points to separatrix characteristics is established, through examples of several classes of analytical, axisymmetric and self-similar solutions for flows in a uniform or spherical gravitational field. Besides the well known polytropic cases with constant polytropic index, more general flows with a variable polytropic index are also examined. These examples are the only available classes of exact 2-D dynamical equilibria which give us a chance to understand some novel and subtle properties of the nonlinear set of the MHD equations encountered in the modelling of various astrophysical flows, such as in winds, jets, coronal loops, etc.

In the reviewed examples of exact solutions, the various families of characteristics of the MHD partial differential equations are plotted; it is then found that one member of each family of characteristics is asymptotically tangent to the corresponding separatrix. Furthermore, we also find that these separatrices coincide with the location of novel X-type critical points which control the topology of the solutions and wherein nevertheless the flow speed does not correspond to any of the known speeds for MHD wave propagation in a plasma (cusp/tube speed, or fast/slow and sound/Alfven speeds). Instead, it is shown that at the critical points of all axisymmetric and self-similar solutions examined, the component of the flow velocity perpendicular to the directions of axisymmetry and self-similarity, equals the characteristic slow/fast MHD wave speed in that direction. As a byproduct of this understanding of the nature of the critical points, the sound speed can be calculated locally even in flows where no polytropic relation exists.