Cylindrical and Spherical Pistons as Drivers of MHD Shocks

We consider an expanding three-dimensional (3-D) piston as a driver of an MHD shock wave. It is assumed that the source-region surface accelerates over a certain time interval to achieve a particular maximum velocity. Such an expansion creates a large-amplitude wave in the ambient plasma. Owing to the nonlinear evolution of the wavefront, its profile steepens and after a certain time and distance a discontinuity forms, marking the onset of the shock formation. We investigate how the formation time and distance depend on the acceleration phase duration, the maximum expansion velocity (defining also acceleration), the Alfvén velocity (defining also Mach number), and the initial size of the piston. The model differs from the 1-D case, since in the 3-D evolution, a decrease of the wave amplitude with distance must be taken into account. We present basic results, focusing on the timing of the shock formation in the low- and high-plasma-beta environment. We find that the shock-formation time and the shock-formation distance are (1) approximately proportional to the acceleration phase duration; (2) shorter for a higher expansion velocity; (3) larger in a higher Alfvén speed environment; (4) only weakly dependent on the initial source size; (5) shorter for a stronger acceleration; and (6) shorter for a larger Alfvén Mach number of the source surface expansion. To create a shock causing a high-frequency type II burst and the Moreton wave, the source region expansion should, according to our results, achieve a velocity on the order of 1000 km?s^?1 within a few minutes, in a low Alfvén velocity environment.