Evolution of the Weibel instability in relativistically hot electron–positron plasmas

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Abstract
Analytical and numerical studies of the evolution of the Weibel instability in relativistically hot electron–positron plasmas are presented. Appropriate perturbations on the electromagnetic fields and the particle orbits, corresponding to a single unstable mode, are determined analytically and used as initial conditions in the numerical simulations to excite a single unstable mode. A simple estimate of the saturation amplitude is also obtained analytically. Numerical simulations are carried out when a single unstable mode is favorably excited. Comparisons of the simulation results with the analytical ones show very good agreement. Also observed in the simulations are mode competition, mode suppression, and the difference in the long‐term evolution between the magnetized and unmagnetized plasmas. For relativistic unmagnetized plasmas, energy‐like global constraints, which are conservation laws in addition to the conservation of energy and momentum, are derived. Numerical simulations of the multimode evolution are described. Simulation results show growth in electromagnetic energy in the early stage, a narrowing in the bandwidth and a shift in the peak of the spectrum to longer wavelength in the subsequent evolution, and a decrease in the temperatureanisotropy. In a simulation for an unmagnetized plasma, it is observed that the system reaches a steady state halfway through the simulation. In contrast, the peak of the spectrum continues to shift to lower wave number k, and the temperatureanisotropy continues to decrease during the entire simulation for a magnetized plasma.