  Plasma outflows from gamma-ray bursts, supernovae, and relativistic jets, in
general, interact with the surrounding medium through collisionless shocks. The
microphysics of such shocks are still poorly understood, which, potentially,
can introduce uncertainties in the interpretation of observations. It is now
well established that the Weibel two-stream instability is capable of
generating strong electromagnetic fields in the transition region between the
jet and the ambient plasma. However, the parameter space of collisionless
shocks is vast and still remains unexplored. In this Letter, we focus on how an
ambient magnetic field affects the evolution of the electron Weibel instability
and the associated shock. Using a particle-in-cell code, we have performed
three-dimensional numerical experiments on such shocks. We compare simulations
in which a jet is injected into an unmagnetized plasma with simulations in
which the jet is injected into a plasma with an ambient magnetic field both
parallel and perpendicular to the jet flow. We find that there exists a
threshold of the magnetic field strength below which the Weibel two-stream
instability dominates, and we note that the interstellar medium magnetic field
strength lies well below this value. In the case of a strong magnetic field
parallel to the jet, the Weibel instability is quenched. In the strong
perpendicular case, ambient and jet electrons are strongly accelerated because
of the charge separation between deflected jet electrons and less deflected jet
ions. Also, the electromagnetic topologies become highly non-linear and complex
with the appearance of anti-parallel field configurations.
