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 temperature anisotropy. 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 temperature anisotropy continues to decrease during the entire simulation for a magnetized plasma.
A linear stability analysis is carried out for the Weibel instability in relativistic magnetized electron–positron-pair plasmas, with the propagation direction parallel to the background magnetic field. The instability in the ultrarelativistic regime, with the typical Lorentz factor γ much greater than unity, is emphasized for its relevance to astrophysical sources of synchrotron radiation. Detailed stability properties are examined, in the ultrarelativistic regime, for two model distribution functions, the water-bag distribution function, and a smooth distribution function. The dispersion relations are obtained in closed analytic forms for both distribution functions. The necessary and sufficient conditions for instability are determined when the temperature along the background magnetic field is cold (T∥=0). The dispersion relations are solved numerically with T∥≠0 over a wide range of system parameters to determine the detailed dependence of the instability on the strength of the background magnetic field and the temperature anisotropy. The present analysis shows that both a decrease in temperature anisotropy and an increase in the background magnetic field can cause a significant decrease in growth rate. For the smooth distribution function, it is found that, for a given plasma density, the system stabilizes completely when the background magnetic field is stronger than the moderate threshold value [(ωp±/ωc±)2≤2/π], corresponding to T∥=0. As the temperature anisotropy decreases, the threshold magnetic field decreases.
The original sheath inverse bremsstrahlung model [P. J. Catto and R. M. More, Phys. Fluids 20, 704 (1977)] is modified by including the v×B term in the equation of motion, as the evanescent magnetic field in an overdense plasma is greater than the corresponding electric field. It is shown that the present results are significantly different from those derived without the v×B term. The v×B term is also important in interpreting the absorption mechanism. If the v×B term were neglected, the absorption of the light would be incorrectly interpreted as an increase in the transverse components of the canonical momentum, in the case of a normally incident laser light. It is also shown that both the sheath inverse bremsstrahlung and the anomalous skin effect are limiting cases of the same collisionless absorption mechanism. Results from particle-in-cell (PIC) plasma simulations are compared with the absorption coefficient calculated from the linear theory. Finally, the effects of finite density gradients are investigated by PIC simulations.
For p-polarized laser light obliquely incident on overdense plasmas with steep density gradients, a new collisionless absorption mechanism (sheath-transit absorption) is studied analytically and numerically. Complementary to Brunel’s ‘‘not-so-resonant’’ resonant absorption, and to the conventional resonant absorption, the sheath-transit absorption is most effective for steep density gradients and when the light pressure is less than the plasma pressure. It is also shown that the assumption of instantaneous particle reflection, usually a reasonable assumption for the normal incidence case, is invalid for the p-polarized oblique incident case. A test-particle model which provides a simple physical picture of the sheath-transit absorption is presented. Absorption coefficients obtained from the test-particle model agree reasonably well with those from particle-in-cell (PIC) simulations. The transition from the resonant absorption to the sheath-transit absorption as the density gradient steepens is demonstrated by PIC simulations with a wide range of density gradients.
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