Upstream structure of collisionless perpendicular shocks

Abstract

A full set of Vlasov–Maxwell equations describing a stationary perpendicular collisionless shock has been solved in the asymptotic case x→±∞, where x is the spatial coordinate of the one dimensional (1-D) shock geometry. The perpendicular magnetic field in the upstream region has been found to vary asymptotically as Bz(x)−Bz(−∞)∼exp(−bx/Rp), where b(MA,βp,βe)<0 is a function of the Alfvén Mach number, MA, and the proton and electron betas; Rp is the proton Larmor radius. The electron and proton distribution functions are expressed as one-dimensional integrals to be evaluated numerically. Two general results are derived: (i) there occurs an instability, in the sense that the distribution functions are not unique, for −b/MA=(mp/me)1/2, where mp and me are the proton and the electron mass, respectively. Numerical computations show that the lowest value of MA fulfilling this instability condition is Mc≊6.5. (ii) The perpendicular shock cannot develop if MA≤(1+βp+βe)1/2. Moreover, it was found that for MA, βp, and βe from a certain range, the proton distribution function exhibits a peak of reflected particles, with vx≊0 and vy≊2.33v0, i.e., with an energy of about 5.4 times that of the incident flux.