Quick ion injection and acceleration at quasi‐parallel shocks

Particle acceleration processes at quasi‐parallel shocks have been widely discussed; however, the very initial injection from thermal to suprathermal energies is still controversial. Here we show that a nonlinear wave‐particle interaction at quasi‐parallel shocks results in quick injection and quick further acceleration of ions to nonthermal (NT) energies. Instead of an ensemble of small‐amplitude random waves, a large‐amplitude, monochromatic, upstream wave is set to propagate into the shock transition layer and test‐particle orbits are deterministically calculated. First, we superimpose a purely right‐handed, circularly polarized, monochromatic upstream wave whose amplitude is as large as those observed in the upstream of Earth's bow shock. The conversion of the wave at the shock front brings about quick acceleration of selected ions into NT energies. Although the observed orbit is similar to previous results, we propose that the process can be better understood in terms of a nonlinear wave‐particle interaction in which the phase angle between an ion's velocity and the upstream wave field is playing a key role. Next, we add a left‐handed, circularly polarized wave of the same wavelength to make up an elliptically/linearly polarized upstream wave, which is also observed in the bow shock upstream. Some of the NT injected ions that are leaving the shock front are seen to be quickly scattered back to the shock by the wave. These ions experience repeated acceleration within the limited time available for acceleration in the upstream of Earth's bow shock. The resultant energy spectrum has the exponential slope extending up to ∼ 70 E 0 (where E 0 is the upstream bulk flow energy), with the characteristic energy ∼ 7 E 0 . We have compared this energy spectrum with self‐consistent hybrid simulation results and with Geotail satellite observations in the upstream region to find reasonably good agreement.