Coupling, degeneracy breaking and isolation of Weibel modes in relativistic plasmas: II. Specific examples

Recently, a general proof was given (Tautz et al 2006 J. Phys. A: Math. Gen. 39 13831) that for an asymmetric relativistic particle phase-space distribution function, and in the absence of a homogeneous background magnetic field, any unstable linear Weibel modes are isolated, i.e. restricted to discrete wavenumbers. In this paper, for a specific distribution function consisting of mono-energetic counterstreaming electron and positron beams, growth rates and associated wavenumbers for the isolated modes are calculated, proving the existence of discrete values for unstable wavenumbers. Furthermore, electrostatic and electromagnetic Weibel modes are investigated for monoenergetic counterstreaming plasmas, yielding constraints to the momentum components that have to be fulfilled in order to have unstable wave modes.