We have performed two-dimensional hybrid simulations of non-relativistic collisionless shocks in the presence of pre-existing energetic particles (‘seeds’); such a study applies, for instance, to the re-acceleration of galactic cosmic rays (CRs) in supernova remnant (SNR) shocks and solar wind energetic particles in heliospheric shocks. Energetic particles can be effectively reflected and accelerated regardless of shock inclination via a process that we call diffusive shock re-acceleration. We find that re-accelerated seeds can drive the streaming instability in the shock upstream and produce effective magnetic field amplification. This can eventually trigger the injection of thermal protons even at oblique shocks that ordinarily cannot inject thermal particles. We characterize the current in reflected seeds, finding that it tends to a universal value $J\simeq en_{\text{CR}}v_{\text{sh}}$ , where $en_{\text{CR}}$ is the seed charge density and $v_{\text{sh}}$ is the shock velocity. When applying our results to SNRs, we find that the re-acceleration of galactic CRs can excite the Bell instability to nonlinear levels in less than ${\sim}10~\text{yr}$ , thereby providing a minimum level of magnetic field amplification for any SNR shock. Finally, we discuss the relevance of diffusive shock re-acceleration also for other environments, such as heliospheric shocks, galactic superbubbles and clusters of galaxies.

I review the theoretical understanding of the global structure of pulsar magnetospheres concentrating on recent progress in force-free electrodynamics and first-principles simulations of magnetospheres.

We present results of time-dependent numerical modeling of the internal structure of the collisionless shock terminating the pulsar wind in Crab Nebula. We treat the equatorial relativistic wind as composed of ions and electron-positron plasma with an embedded toroidal magnetic field. Relativistic cyclotron instability of the ion ring downstream from the shock is found to launch outward propagating magnetosonic waves. Due to the fresh supply of ions crossing the shock, the time-dependent process achieves a limit-cycle pattern, in which the waves are launched with periodicity on the order of the ion Larmor time. Compressions in magnetic field and pair density associated with these waves as well as their propagation speed qualitatively reproduce the features observed in the wisps.

Current models of pulsar magnetospheres typically assume either a complete absence of plasma or abundant ideal plasma filling the magnetosphere in order to compute the field structure. The latter condition is thought to be closer to reality, but we know of a number of pulsars in which the ideal conditions break down, resulting in dissipation and high-energy emission. In this work we formulate a resistive force-free scheme that allows us to consider the effects of resistive plasma and accelerating fields on the magnetospheric structure. We run numerical simulations to construct a family of resistive solutions that smoothly bridges the gap between the vacuum and the force-free magnetosphere solutions. We further provide a self-consistent model for the spin-down of intermittent pulsars, pulsars which appear to transition between radio-loud and radio-quiet states with different spin-down rates. Finally, we present models for high-energy emission from reconnecting current sheets in Gamma-ray pulsars.

The inner workings of pulsar magnetospheres have fascinated and confused researchers since the discovery of pulsars. I will review the status of magnetospheric models, including vacuum, space-charge-limited and resistive force-free MHD. I will highlight model predictions for the integrated pulsar quantities (such as spin down and torques) and the observational consequences of calculated magnetic field structure. Particularly, high-energy emission from pulsars allows putting new constraints on the geometry of the emission region and the physics of particle acceleration in the magnetosphere.