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The filamentation (Weibel) instability plays a key role in the formation ofcollisionless shocks which are thought to produce Gamma-Ray-Bursts andHigh-Energy-Cosmic-Rays in astrophysical environments. While it has been knownfor long that a flow-aligned magnetic field can completely quench theinstability, it was recently proved in 2D that in the cold regime, suchcancelation is possible if and only if the field is perfectly aligned. Here,this result is finally extended to a 3D geometry. Calculations are conductedfor symmetric and asymmetric counter-streaming relativistic plasma shells. 2Dresults are retrieved in 3D: the instability can never be completely canceledfor an oblique magnetic field. In addition, the maximum growth-rate is alwayslarger for wave vectors lying in the plan defined by the flow and the obliquefield. On the one hand, this bears consequences on the orientation of thegenerated filaments. On the other hand, it certifies 2D simulations of theproblem can be performed without missing the most unstable filamentation modes.

Robustness of the filamentation instability in arbitrarily oriented magnetic field: Full 3D calculation