Theory and simulation of collisionless parallel shocks

A one-dimensional theory of shocks propagating parallel to the ambient magnetic field in a collisionless plasma is presented. We show that shock formation and plasma heating can result from parallel propagating electromagnetic ion beam-driven instabilities for a wide range of Mach numbers and upstream plasma conditions. The plasma upstream from a parallel shock overtakes the slower, downstream plasma in the absence of any electrostatic or electromagnetic wave fields. An interface region is created that is unstable to electromagnetic ion beam-driven growth. The waves grow until they can scatter and couple the upstream and downstream plasma populations and can provide the necessary entropy increase for the shock. The marginal firehose state is reached downstream from the shock if the Alfven Mach number MA is high enough (MA ≡ Uu/CA, where Uu is the upstream shock velocity and CA is the upstream Alfven speed). A one-dimensional compression of the ions along the magnetic field line direction results for weaker shocks. We solve a set of jump conditions and show that the marginal firehose state is first reached at a small magnetosonic Mach number (less than 2). We derive various conditions for creating and maintaining the shock via the ion beam-driven instability and show that these conditions are roughly equal. The results from many one-dimensional hybrid simulation runs are compared with the predictions of the theory and are shown to be in good agreement. At low Mach numbers, resonant group-standing waves are the most strongly amplified and scatter the ions over several wavelengths. The transition in the ion pressure tensor is smooth, going from an isotropic upstream state to an anisotropic firehose state downstream in several wavelengths. The ions are compressed in one dimension at small Mach numbers, and there is little change in the perpendicular ion temperature across the shock. At high Mach numbers a small fraction of the upstream ions are backscattered by large-amplitude magnetic waves at the shock front. The backscattered ions resonantly generate waves in the foreshock region, which are then convected downstream and are strongly compressed and amplified at the shock front. The bulk of the ion population passes through the shock and is nonresonantly scattered and heated by the amplified downstream waves. The change in the pressure tensor across the shock is abrupt; most of the entropy increase occurs within one or two wavelengths beyond where the waves are compressed and amplified. We compare the theoretical predictions with recently published observations of quasi-parallel interplanetary shocks and the Earth's quasi-parallel bow shock and find good agreement. Some theoretical limits on the fraction of ions that can be scattered back upstream are derived, and the implications for cosmic ray injection, particle acceleration, and astrophysical shocks are discussed.

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