  Magnetic-field generation by a relativistic ion beam propagating through an
electron-ion plasma along a homogeneous magnetic field is investigated with
2.5D high-resolution particle-in-cell (PIC) simulations. The studies test
predictions of a strong amplification of short-wavelength modes of magnetic
turbulence upstream of nonrelativistic and relativistic parallel shocks
associated with supernova remnants, jets of active galactic nuclei, and
gamma-ray bursts. We find good agreement in the properties of the turbulence
observed in our simulations compared with the dispersion relation calculated
for linear waves with arbitrary orientation of ${\vec k}$. Depending on the
parameters, the backreaction on the ion beam leads to filamentation of the
ambient plasma and the beam, which in turn influences the properties of the
magnetic turbulence. For mildly- and ultra-relativistic beams, the instability
saturates at field amplitudes a few times larger than the homogeneous magnetic
field strength. This result matches our recent studies of nonrelativistically
drifting, hot cosmic-ray particles upstream of supernova-remnant shocks which
indicated only a moderate magnetic-field amplification by nonresonant
instabilities. We also demonstrate that the aperiodic turbulence generated by
the beam can provide efficient particle scattering with a rate compatible with
Bohm diffusion. Representing the ion beam as a constant external current, i.e.
excluding a backreaction of the magnetic turbulence on the beam, we observe
non-resonant parallel modes with wavelength and growth rate as predicted by
analytic calculations. In this unrealistic setup the magnetic field is
amplified to amplitudes far exceeding the homogeneous field, as observed in
recent MHD and PIC simulations.
