Energy principle for 2D electromagnetic, relativistic, interpenetrating, counter streaming plasma flows

A relativistic electron beam propagating through plasma induces a return current in the system. Such a system of counterstreaming forward and return current is susceptible to host of instabilities out of which Weibel remains a dominant mode for destabilizing the system. Weibel instability has been widely investigated in simulations, experiments, as well as, analytically using fluid and kinetic treatments. A purely growing mode like Weibel instability can also be understood by using conservation theorems and energy principle analysis. An electrostatic analog to the Weibel instability, two stream instability in a beam plasma system has already been investigated using energy principle analysis [Lashmore-Davies 2007]. A detailed analytical description of conservation theorem for 2D Weibel instability in a beam plasma system has been carried out in this manuscript.