Practical criteria for the Weibel instability and its saturation

摘要

We consider the Weibel, purely aperiodic instability in a collisionless plasma, relativistic or not, for an important in practice case when the particle distribution functions of all species exhibit mirror symmetry with respect to a certain plane xy and a wave vector of an ordinary wave perturbation is parallel to this plane, i.e., k_z= 0 and f0α(px,Py, Pz)= foa(Px,Py, -p_z), where p= (Px,Py,Pz) is the particle momentum. In this case, we obtain a novel analytical criterion for the Weibel instability using its analogy with a long-wavelength soft-mode instability which is well known in the solid state physics. It facilitates an analysis of the Weibel instability and agrees with the results which have been known for the certain particle distributions, including the bi-Maxwellian, power-law, and parallelepiped ones as well as various variants of the so-called waterbag distributions. Also, for a series of special cylindrically-symmetric particle distributions we find the analytical dependence of the Weibel-instability growth rate on the wavenumber of perturbation and show that it agrees well with the criterion presented [1].