CHARACTERISTIC VELOCITY OF MHD SHOCK

According to the similarity between the boundary conditions on the MHD discontinuity and the equations of the MHD simple wave, a physical quantity U_* called the characteristic velocity of MHD shock has been introduced. U_* is defined by Eq. (4) as the ratio of the jump of total presure p^* to that of density p, and equal to the geometric mean value of the shock velocities Wrelative to the up and down streams (Eq. (5)), approaching to the MHD wave velocity Uin the weak shock limit.The system of equations (7) express the shock relations with U_* as the strengthparameter, where u, v =H/√4πρ and U± are respectively the velocity of fluid relativeto the shock front, and of the Alfven and the magnetosonic waves. These relations simplify the usual shock calculations and degenerate into the formulae for the MHD simple wave (Eq. (11)) in the infinitesimal jump limit.The fast, slow, and intermediate shock can be distinguished by Eq. (12-a, b, c) respectively. It is interesting to note that the jump formulae (7-a) and (13) show explicitly the transmission relations among the various modes of MHD shock and wave (Eq. (16-a, b, c) for the above three modes respectively), which are nothing but the immediate consequence of the compressive nature of shocks together with the conservation of mass (Eq. (14) and Eq. (15)). However, it needs a lengthy derivation to prove Eq. (16) without introducing U_*.