Mathematical Theory of Cylindrical Isothermal Blast Waves in a Magnetic Field

Abstract

An investigation is made of the self-similar flow behind a cylindrical blast wave from a line explosion (situated on r = 0, using conventional cylindrical coordinates r, 4>, z) in a medium whose density and magnetic field both vary as r -w ahead of the blast front, with the assumption that the flow is isothermal. The magnetic field can have components in both the azimuthal B(jJ and longitudinal B, directions. It is found that: (i) For B(jJ =f:. 0 =f:. B, a continuous single-valued solution with a velocity field representing outflow of material away from the line of explosion does not exist for OJ < 0, but only for OJ ~ o. (ii) For Bz = 0, but B(jJ i= 0, there is no ontinuous single-valued solution with a velocity field representing outflow of material away from the line of the explosion for any OJ value. (iii) For B(jJ = 0, but Bz =f:. 0, the behaviour is as follows: (a) for OJ < 0 a continuous singlevalued solution with outflow of gas everywhere ehind the shock does not exist ; (b) for OJ = 0 the solution is singular and piecewise continuous with an inner region where no fluid flow occurs and an outer region where the fluid flow gradually increases; (c) for 1 > OJ > 0 the governing equation possesses a set of movable critical points. In this case it is shown that the fluid flow velocity is bracketed between two curves and that the asymptotes of the velocity curve on the shock are intersected by, or are tangent to, the two curves. Thus a solution always exists in the physical domain r ~ o. The overall conclusion from the investigation is that the behaviour of isothermal blast waves in the presence of an ambient magnetic field differs substantially from the behaviour calculated for no magnetic field. These results have an impact upon previous applications of the theory of self-similar flows to evolving supernova remnants without allowance for the dynamical influence of magnetic pressure and magnetic tension.