Collisionless shocks are one of the most ubiquitous phenomena in space plasmas. The directed flow energy is converted into other forms at the shock front: ion and electron heating, particle acceleration, and magnetic field enhancement. One of the most important problems of the collisionless shock physics is prediction of the post-shock (downstream) state of the plasma and fields given the corresponding plasma and magnetic field state before the shock (upstream). In the absence of dissipative processes the relations between the upstream and downstream parameters (Rankine–Hugoniot relations, boundary conditions, and jump conditions) are just conservation laws of particle number, momentum, and energy. There is a complete hierarchy of scales in collisionless shocks and, accordingly, the boundary conditions depend on the scale at which the boundaries are placed. The standard Rankine–Hugoniot relations (RH) are formulated on the largest, magnetohydrodynamic (MHD) scale, at which the distributions thermalize (Kennel 1994). In many space plasma environments the ambient conditions change on spatial or temporal scales, which are too short to achieve such thermalization. Observational comparison of the upstream and downstream plasma is often done in various regions that do not satisfy the conditions for establishing the MHD RH. In the close vicinity of the shock front, ion distributions are significantly gyrophase dependent, and the jump conditions at the very transition should take this into account (Gedalin & Balikhin 2008; Gedalin 2016a). On larger scales or for measurements invoking substantial temporal and/or spatial averaging, the distributions become gyrotropic but can still remain anisotropic, since isotropization may be slow. The corresponding RH should take into account this anisotropy (Lyu & Kan 1986; Gedalin 2017; Gedalin et al. 2020). Additional complications arise when there are different populations of ions that undergo gyrotropization and isotropization on different scales. Eventual equilibration of temperatures may never happen. Even for different populations of ions of the same kind eventual merging into one thermal population may never be observed. The latter situation is expected to be typical when pickup ions (PUIs) are important constituents, as at the termination shock (Zank et al. 1996; Li et al. 2008; Richardson et al. 2008; Burrows et al. 2010; Matsukiyo & Scholer 2011, 2014; Ariad & Gedalin 2013; Jokipii 2013; Mostafavi et al. 2017, 2018; Kumar et al. 2018). In this case, MHD has to be replaced with a multispecies model, where each population obeys the corresponding conservation laws separately, while the magnetic field jump is obtained by combining all species together. In this approach one has to know the equations of state for each species. Typically, some form of thermodynamics motivated state equations are assumed (Florinski et al. 2009; Borovikov et al. 2011; Pogorelov et al. 2013; Zank et al. 2014; Mostafavi et al. 2018). This approach implicitly assumes that the population is in a kind of local thermodynamic equilibrium. However, observations show that this is not the case. In fact, the downstream ion distribution is determined primarily by ion dynamics in the macroscopic fields of the shock front. Interaction with the electromagnetic fluctuations plays a secondary role causing slow relaxation toward equilibrium, while binary collisions are negligible. The resolution of particle observations in the vicinity of the heliospheric termination shock corresponds to the scale on which the solar wind protons are possibly (but not certainly) isotropic but PUI are still anisotropic. Yet most studies of RH at the termination shock aim at the determination of jump conditions on the MHD scale, assuming isotropy of the distributions. Some previous analyses within a two-fluid model (solar wind and PUI) introduced heat flux and collisionless viscosity of PUI (Zank et al. 2014; Mostafavi et al. 2016, 2017, 2018; Zank 2016). This approach explicitly assumes that ion scattering within the shock transition is substantial so that the focused transport equation is valid throughout the shock (Zank et al. 2014). However, the shock transition is scatter-free (Toptyghin 1980; Drury 1983), so that the approach loses the physics of relation between the kinetic and MHD scales. Accordingly, the obtained magnetic profiles are monotonic and cannot explain observed overshoots (Mostafavi et al. 2018). Previous attempts to take into account the ion kinetics at a shock front assumed magnetic moment conservation (Fahr & Chalov 2008; Fahr et al. 2012; Fahr & Siewert 2013, 2015). However, it has been shown that magnetic moment is, in general, not conserved (Terasawa 1979; Gedalin 2020), and such approximation may be satisfactory only in the perpendicular regime. Proper establishment of the boundary conditions on MHD scales requires establishing a relation of the ultimately isotropic distributions to the collisionless ion dynamics within a shock front. This objective requires proper determination of downstream gyrophase averaged distributions, which are gyrotropic but anisotropic, with further relating these to the isotropic pressure (Gedalin et al. 2020). In the present paper, we numerically determine the parameters of these distributions, which will be used for modifying the Rankine–Hugoniot relation.
Citation: Gedalin 2017
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