A pressure tensor description for the time-resonant Weibel instability

We discuss a fluid model with inclusion of the complete pressure tensor dynamics for the description of Weibel type instabilities in a counterstreaming beams configuration. Differently from the case recently studied in Sarrat et al. 2016, where perturbations perpendicular to the beams were considered, here we focus only on modes propagating along the beams. Such a configuration is responsible for the growth of two kind of instabilities, the Two-Stream Instability and the Weibel instability, which in this geometry becomes "time-resonant", i.e. propagative. This fluid description agrees with the kinetic one and makes it possible e.g. to identify the transition between non-propagative and propagative Weibel modes, already evidenced by Lazar et al. 2009 as a "slope-breaking" of the growth rate, in terms of a merger of two non propagative Weibel modes.