How really transverse is the filamentation instability?

It is generally considered that the linear filamentation instability encountered when two counter streaming electron beams interpenetrate is purely transverse. Exact and approximated results are derived in the relativistic fluid approximation showing that within some parameter range, filamentation can be indeed almost longitudinal with cos(k,Ê)≲1−3.1∕γb, where γb is the relativistic factor of the beam. Temperature effects are then evaluated through relativistic kinetic theory and yield even fewer transverse filamentation modes. In the cold case, the transverse approximation overestimates the growth rate by a factor ∝γb.