Simulations of Ion Acceleration at Non-relativistic Shocks. II. Magnetic
  Field Amplification
  We use large hybrid simulations to study ion acceleration and generation of
magnetic turbulence due to the streaming of particles that are
self-consistently accelerated at non-relativistic shocks. When acceleration is
efficient, we find that the upstream magnetic field is significantly amplified.
The total amplification factor is larger than 10 for shocks with Alfv\'enic
Mach number $M=100$, and scales with the square root of $M$. The spectral
energy density of excited magnetic turbulence is determined by the energy
distribution of accelerated particles, and for moderately-strong shocks
($M\lesssim30$) agrees well with the prediction of resonant streaming
instability, in the framework of quasilinear theory of diffusive shock
acceleration. For $M\gtrsim30$, instead, Bell's non-resonant hybrid (NRH)
instability is predicted and found to grow faster than resonant instability.
NRH modes are excited far upstream by escaping particles, and initially grow
without disrupting the current, their typical wavelengths being much shorter
than the current ions' gyroradii. Then, in the nonlinear stage, most unstable
modes migrate to larger and larger wavelengths, eventually becoming resonant in
wavelength with the driving ions, which start diffuse. Ahead of strong shocks
we distinguish two regions, separated by the free-escape boundary: the far
upstream, where field amplification is provided by the current of escaping ions
via NRH instability, and the shock precursor, where energetic particles are
effectively magnetized, and field amplification is provided by the current in
diffusing ions. The presented scalings of magnetic field amplification enable
the inclusion of self-consistent microphysics into phenomenological models of
ion acceleration at non-relativistic shocks.
