Beam plasma electromagnetic instabilities in a smooth density gradient:Applications to ICF fast ignition

We detail a calculation of the integrated growth rate(GR) of an instability in a weakly varying plasma density gradient using a WKB-like approximation.We justify such an assumption in the fast ignitor scenario context.Our formalism includes 2- stream, filamentation and 2-stream/filamentation instabilities.The latter features an appropriate mixture of the separate two former ones.It is also the fastest.We demonstrate that filamentation is damped through the density gradient whereas 2-stream and 2-stream/filamentation saturate even before being submitted to the density gradient.