Weibel Instability in the Presence of an External Magnetic Field: Analytical Results

We perform dispersion analysis of the Weibel-type electron instability in collisionless bi-Maxwellian plasma with an external magnetic field aligned with the anisotropy axis, which corresponds to the highest temperature. The range of the wave numbers of unstable modes, their growth rates, maximum growth rates, and the corresponding optimal wave number as a function of the density and anisotropy of the plasma and the external magnetic field are found for ordinary mode perturbations with the wave vector and the vector of the electric field being perpendicular and parallel to the external field, respectively. The external magnetic field, which suppresses the instability of these modes, and the r.m.s. generated magnetic field, which saturates it at the nonlinear stage, are also estimated.