A comparative study of the filamentation and Weibel instabilities and their cumulative effect. I. Non-relativistic theory

A comparative study of the electromagnetic instabilities in anisotropic unmagnetized plasmas is undertaken. The instabilities considered are the filamentation and Weibel instability, and their cumulative effect. Dispersion relations are derived and the growth rates are plotted systematically for the representative cases of non-relativistic counterstreaming plasmas with isotropic or anisotropic velocity distributions functions of Maxwellian type. The pure filamentation mode is attenuated by including an isotropic Maxwellian distribution function. Moreover, it is observed that counterstreaming plasmas can be fully stabilized by including bi-Maxellian distributions with a negative thermal anisotropy. This effect is relevant for fusion plasma experiments. Otherwise, for plasma streams with a positive anisotropy the filamentation and Weibel instabilities cumulate leading to a growth rate by order of magnitude larger than that of a simple filamentation mode. This is noticeable for the quasistatic magnetic field generated in a astrophysical sources, and which is expected to saturate at higher values and explain the non-thermal emission observed.