Two-dimensional Vlasov simulation of electron plasma wave trapping, wavefront bowing, self-focusing, and sideloss

Two-dimensional Vlasov simulations of nonlinear electron plasma waves are presented, in which the interplay of linear and nonlinear kinetic effects is evident. The plasma wave is created with an external traveling wave potential with a transverse envelope of width Dy such that thermal electrons transit the wave in a “sideloss” time, tsl Dy=ve. Here, ve is the electron thermal velocity. The quasisteady distribution of trapped electrons and its self-consistent plasma wave are studied after the external field is turned off. In cases of particular interest, the bounce frequency, xbe 1⁄4 k ffiffiffiffiffiffiffiffiffiffiffiffiffi e/=me p , satisfies the trapping condition xbetsl > 2p such that the wave frequency is nonlinearly downshifted by an amount proportional to the number of trapped electrons. Here, k is the wavenumber of the plasma wave and / is its electric potential. For sufficiently short times, the magnitude of the negative frequency shift is a local function of /. Because the trapping frequency shift is negative, the phase of the wave on axis lags the off-axis phase if the trapping nonlinearity dominates linear wave diffraction. In this case, the phasefronts are curved in a focusing sense. In the opposite limit, the phasefronts are curved in a defocusing sense. Analysis and simulations in which the wave amplitude and transverse width are varied establish criteria for the development of each type of wavefront. The damping and trapped-electron-induced focusing of the finite-amplitude electron plasma wave are also simulated. The damping rate of the field energy of the wave is found to be about the sideloss rate, e t 1 sl . For large wave amplitudes or widths Dy, a trapping-induced self-focusing of the wave is demonstrated. VC 2011 American Institute of Physics. [doi:10.1063/1.3577784]