Critical Alfven-Mach numbers for transverse field MDH shocks

Just as ordinary gas dynamic shocks in a gas of zero viscosity, but finite thermal conductivity do not possess a continuous structure above a certain critical Mach number, MHD shocks in a two-fluid plasma also exhibit discontinuities when viscosity is neglected. These discontinuities occur when the Alfven-Mach number of the shock, M A , exceeds a certain critical value M A *; the precise value of M A * depends on which of the three dissipative mechanisms of thermal conductivity for ions and electrons and electrical resistivity are retained and is independent of their actual values. In the practically important case of strong narrow shocks in which ion collisions can be neglected, but the electron gas still exhibits fluid behaviour, one finds that M A *=3.46. This is the case tau i / tau s approximately infinity and tau e / tau s <1, where tau i , tau e are collision times in the ion and electron gases respectively and tau s is the characteristic shock time (shock thickness divided by shock speed). The ion viscosity tends to zero in both of the limits tau i / tau s to 0, infinity and while the resulting discontinuity at M A >M A * is known to be stable for the first of these limits, in the second limit ion-ion streaming instabilities are likely to occur. The addition of collisionless electron viscosity due to finite Larmor radius results no change in the value of M A *.