  The Weibel instability driven by two symmetric counter-streaming relativistic
electron plasmas, also referred to as current-filamentation instability, is
studied in a constant and uniform external magnetic field aligned with the
plasma flows. Both the linear and non linear stages of the instability are
investigated using analytical modeling and Particle-In-Cell (PIC) simulations.
While previous studies have already described the stabilizing effect of the
magnetic field, we show here that the saturation stage is only weakly affected.
The different mechanisms responsible for the saturation are discussed in detail
in the relativistic cold fluid framework considering a single unstable mode.
The application of an external field leads to a slighlt increase of the
saturation level for large wavelengths, while it does not affect the small
wavelengths. Multi-mode and temperature effects are then investigated. While at
large temperature the saturation level is independent of the external magnetic
field, at small but finite temperature the competition between different modes
in the presence of an external magnetic field leads to a saturation level lower
with respect to the unmagnetized case.
