Nonlinear wave scattering and electron beam relaxation

Abstract

The role played by nonlinear scattering during the relaxation of a warm electron beam is investigated with the help of a numerical code. The code is based on kinetic equations and includes the quasilinear wave–electron interaction as well as wave–wave scattering off ion clouds. Both mechanisms have been observed to play key roles in a recent particle simulation with a large number of modes. It is found that (1) ions with velocity 2vi (vi being the ion thermal velocity) are the most efficient to scatter the Langmuir waves off their polarization clouds. As a result, the transfer rate of the spectrum out of resonance with the beam is larger by a factor 3 compared to the usual estimates in the literature, which assume a static ion response. The predicted wave number k of the secondary spectrum differs also substantially. (2) If the beam density nb, drift Ub, and width vb satisfy the condition nb/n0>4.2(ve/Ub)2 ×(vb/Ub)3, the changes brought to the dispersion relation by the presence of the beam electrons dramatically alter the characteristics of the secondary spectrum. Forward propagating waves may grow where the conventional picture expects backward propagating waves. Most strikingly, in a late phase the classic condensate about k∼0 is depleted with the formation of a new condensate in resonance with the flat-topped beam distribution. This contradicts the commonly assumed cascade in wave numbers, but follows simply from the fact that the mere presence of the beam electrons creates a minimum in the frequency–wave-number relation. There is no contradiction with a cascade toward lower frequencies driven by an isotropic ion distribution. For strong and slow beams (nb/n0∼10−2, Ub∼10ve) the predictions of this code can be compared with the results obtained in the particle simulation. The agreement is excellent if one uses a dispersion relation that includes the beam. Complete plateau formation by resonant diffusion and late formation of a secondary spectrum are observed. Time scales and spectral characteristics compare well. For faster and weaker beams, it is demonstrated that the nonlinear wave scattering may intervene before complete quasilinear relaxation. Once the beam top has been erased by diffusion, a wave condensate forms, which inhibits further relaxation toward lower velocities. Modes in resonance with the positive slope at the low-velocity front of the flat-topped beam are stabilized by a fast transfer of their energy into the condensate.