In broad astrophysical contexts of large-scale gravitational collapses and outflows and as a basis for various further astrophysical applications, we formulate and investigate a theoretical problem of self-similar MHD for a non-rotating polytropic ga
s of quasi-spherical symmetry permeated by a completely random magnetic field. We derive two coupled nonlinear MHD ordinary differential equations (ODEs), examine properties of the magnetosonic critical curve, obtain various asymptotic and global semi-complete similarity MHD solutions, and qualify the applicability of our results. Unique to a magnetized gas cloud, a novel asymptotic MHD solution for a collapsing core is established. Physically, the similarity MHD inflow towards the central dense core proceeds in characteristic manners before the gas material eventually encounters a strong radiating MHD shock upon impact onto the central compact object. Sufficiently far away from the central core region enshrouded by such an MHD shock, we derive regular asymptotic behaviours. We study asymptotic solution behaviours in the vicinity of the magnetosonic critical curve. Numerically, we construct global semi-complete similarity MHD solutions that cross the magnetosonic critical curve zero, one, and two times. For comparison, counterpart solutions in the case of an isothermal unmagnetized and magnetized gas flows are demonstrated in the present MHD framework at nearly isothermal and weakly magnetized conditions. For a polytropic index $gamma=1.25$ or a strong magnetic field, different solution behaviours emerge. In these cases, there exist semi-complete similarity solutions crossing the magnetosonic critical curve only once, and the MHD counterpart of expansion-wave collapse solution disappears.
We construct magnetohydrodynamic (MHD) similarity rebound shocks joining `quasi-static asymptotic solutions around the central degenerate core to explore an MHD model for the evolution of random magnetic field in supernova explosions. This provides a
theoretical basis for further studying synchrotron diagnostics, MHD shock acceleration of cosmic rays, and the nature of intense magnetic field in compact objects. The magnetic field strength in space approaches a limiting ratio, that is comparable to the ratio of the ejecta mass driven out versus the progenitor mass, during this self-similar rebound MHD shock evolution. The intense magnetic field of the remnant compact star as compared to that of the progenitor star is mainly attributed to both the gravitational core collapse and the radial distribution of magnetic field.
