Specular refection at a non-stationary shock: A simple model

Analytic treatments of a particle encountering a collisionless shock have commonly been based on the assumption that the shock surface is quasi-planar with length scales larger than the particle gyroradius. Within this framework, the particle distribution function width is supposed to be conserved in any shock reflection process. It is well known, however, that the thermal energy associated with backstreaming ions upstream of Earth's bow shock is significantly larger than the incident solar wind thermal energy. In a previous study, we found that non-thermal features of ions reflected quasi-adiabatically can be accounted for by considering the effect of small, normally distributed fluctuations of the shock normal over short temporal or spatial scales. The strong dependence of the particle acceleration on shock geometry leads to an increase in the temperature and to a non-thermal tail. Here, we conduct a similar analysis to investigate the effects of small, normally distributed fluctuations in the shock normal direction for specularly reflected ions. This later mechanism is considered of first importance in the dissipation process occurring at quasi-perpendicular shocks. We have derived the probability distribution functions f(^v∥) and f(^v⊥) of ions issued from a specular reflection of incident solar wind in the presence of normal direction fluctuations. These distributions deviate weakly from a Maxwellian, in agreement with the observations. In particular, a qualitative agreement with the ion thermal energy is obtained for fluctuations of the normal orientation in the 58° range about the nominal direction. Also, we have found that the shock ^θBn has a weak effect on the shape of the distribution. While, not a strong determinant of the reflected distribution characteristics, the dynamical shock structure at ion scales cannot be ignored when accounting for the shock-accelerated particle thermal energy.