Coupling, degeneracy breaking and isolation of Weibel modes in relativistic plasmas: I. General theory

A general proof is given that for an asymmetric 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. Starting from the linearized relativistic Vlasov equation it is shown that, unless the asymmetry in the distribution function is precisely zero, the broad ranges of unstable wavenumbers occurring for symmetric distribution functions are reduced to discrete, isolated wavenumbers for which unstable modes can exist. For asymmetric plasmas, electrostatic and electromagnetic wave modes are coupled to each other and the degeneracy of the two electromagnetic wave modes ( that holds for symmetric distributions) is therefore broken.