Kinetic simulations of ladder climbing by electron plasma waves

The energy of plasma waves can be moved up and down the spectrum using chirped modulations of plasma parameters, which can be driven by external fields. Depending on whether the wave spectrum is discrete (bounded plasma) or continuous (boundless plasma), this phenomenon is called ladder climbing (LC) or autoresonant acceleration of plasmons. It was first proposed by Barth et al. [Phys. Rev. Lett. 115, 075001 (2015)] based on a linear fluid model. In this paper, LC of electron plasma waves is investigated using fully nonlinear Vlasov-Poisson simulations of collisionless bounded plasma. It is shown that, in agreement with the basic theory, plasmons survive substantial transformations of the spectrum and are destroyed only when their wave numbers become large enough to trigger Landau damping. Since nonlinear effects decrease the damping rate, LC is even more efficient when practiced on structures like quasiperiodic Bernstein-Greene-Kruskal (BGK) waves rather than on Langmuir waves per se.

Figure 1

Evolution of plasma wave parameters during LC according to the linear theory: (a) wave frequency calculated using the fluid model, ${\omega }_m/{\omega }_{pe}=\sqrt{1+\tilde{\beta }{m}^2}$; (b) phase velocity, $v_{\varphi }=\omega /k$ within the same model; and (c) linear Landau damping rate, ${\gamma }_L$, from Eq. (5).

Figure 2

The evolution of the field-energy spectrum. Also shown is the sum of the field energy for $m\ge 2$.The contribution of mode with $m=1$ is excluded in order to eliminate the interference with $E_{\mathrm{drive}}$, which has the wave number equal to that of the first mode.

Figure 3

Snapshots of a plasma wave undergoing LC: (a) electric field and (b) electron distribution at $\tau =$ 21.1, 33.9, 43.1, 55.9, 70.6, and 85.2, when transitions occur to modes with $m=4,5,6,7,8$, and 9 (see Fig. 1). The same log-scale color map is used in all six subfigures in (b). The blue dashed lines in (b) show the linear phase velocities of the relevant modes.

Figure 4

Spatially averaged electron velocity distribution functions (VDFs). The time steps correspond to the ones shown in Fig. 3. Plateaus of the VDFs are found around the phase velocities predicted by the analytic theory. The insert, Fig. 4(b), is a zoomed-in view of dash-dotted box in Fig. 4(a) showing the evolution of the modes with $m=6$ and $m=7$.

Figure 5

An illustration of a transition between two modes, specifically $m=6$ to $m=7$: (a) wave energy, zoomed in from Fig. 2. The zoomed-in electron distribution shown in (b), (c), and (d) correspond to the moments of time marked in (a) with pink dashed lines. The blue dashed lines in (b)–(d) show the phase velocities of the modes with $m=6$ and $m=7$ correspondingly.

Figure 6

Comparison between our numerical simulation and the analytic theory for Landau damping. The dash-dotted, dashed, dotted lines are theoretical prediction of Landau damping for $m=6,m=7$, and $m=8$, respectively.

Figure 7

Simulations of a single mode plasma wave with reflecting-wall boundary conditions. Panels (a), (b) correspond to the mode with $m=4$, and (c), (d) correspond to the mode with $m=9$. Particle trapping occurs around $v_{\varphi }=\pm 4.64v_{\mathrm{th},e}$.