A relativistic electron beam propagating through plasma induces a return electron current in the system. Such a system of interpenetrating forward and return electron current is susceptible to a host of instabilities. The physics of such instabilities underlies the conversion of the flow kinetic energy to the electromagnetic field energy. Keeping this in view, an energy principle analysis has been enunciated in this paper. Such analyses have been widely utilized earlier in the context of conducting fluids described by MHD model [I. B. Bernstein et al., Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 244(1236), 17-40 (1958)]. Lately, such an approach has been employed for the electrostatic two stream instability for the electron beam plasma system [C. N. Lashmore-Davies, Physics of Plasmas 14(9), 092101 (2007)]. In contrast, it has been shown here that even purely growing mode like Weibel/current filamentation instability for the electron beam plasma system is amenable to such a treatment. The treatment provides an understanding of the energetics associated with the growing mode. The growth rate expression has also been obtained from it. Furthermore, it has been conclusively demonstrated in this paper that for identical values of S4=σαn0αv0α2/n0γ0α, the growth rate is higher when the counterstreaming beams are symmetric (i.e. S3 = σαn0αv0α/n0γ0α = 0) compared to the case when the two beams are asymmetric (i.e. when S3 is finite). Here, v0α, n0α and γ0α are the equilibrium velocity, electron density and the relativistic factor for the electron species 'α' respectively and n0 = σαn0α is the total electron density. Particle - In - Cell simulations have been employed to show that the saturated amplitude of the field energy is also higher in the symmetric case.
Kumar, A., Shukla, C., Das, A., & Kaw, P. (2018). Energy principle for excitations in plasmas with counterstreaming electron flows. AIP Advances, 8(5). https://doi.org/10.1063/1.5008254