Self-similar magnetohydrodynamics. I. The γ = 4/3 polytrope and the coronal transient

The full set of ideal magnetohydrodynamic (MHD) equations for a γ = 4/3 polytrope admits self-similar solutions which can be derived by analytic methods. An axisymmetric magnetic field in a stratified stellar atmosphere is assumed. The general properties of these self-similar solutions can be demonstrated without obtaining the solutions explicitly and are discussed in connection with the coronal transient as an illustration. The solutions admit a large variety of magnetic structures, including those in the form of loops, moving through the corona with large scale coherence and sharp small scale features. The Lagrangian velocity of an individual transient feature is found to accelerate or decelerate according to a positive or negative gain in momentum by the plasma in a self-similar distribution of the Lorentz force, the pressure gradient, and the gravitational force. In both cases, the net force decreases with time so that, at large radial distances, the motion becomes inertial. The physical implications are discussed, arguing in favor of the transient beginning as a fully nonlinear MHD motion that ejects both magnetic field and plasma out of the gravitational bond of the Sun. Boundary conditions and construction of solutions are discussed. Illustration with explicit solutions and application to the coronal transient as well as stellar explosions will be taken up in papers to follow