Nonlinear Landau Damping

Abstract

We present a new description of nonlinear Landau damping, applicable to wave propagation in a weakly collisional as well as a collisionless plasma. We derive a set of equations with a simple energy conservation law which is useful for understanding the evolution of the system. A numerical investigation shows significant differences as compared to previous studies. The reasons for the deviations from the earlier results are explained. [S0031-9007 (97)02399-5] PACS numbers: 52.35.Mw, 52.25.Dg, 52.35.Sb Wave-particle interaction processes are of fundamental interest in plasma physics. Besides the purely theoretical interest, they are important for many applications-for example, for the supplementary heating of magnetically confined fusion plasmas [1], for the absorption of laser radiation in inertial fusion experiments [2], and for particle acceleration by means of plasma accelerators [3]. In particular, both linear (e.g., Refs. [4-6]) and nonlinear (e.g., Refs. [7-16]) Landau damping have been discussed extensively in the literature. The problem of nonlinear Landau damping has been investigated for more than two decades, experimentally [7], analytically [8], with computer aided calculations [9], and with intermediate approaches [10]. Furthermore, the presentation given in recent textbooks (e.g., ) agrees with the early results. Basically, the picture of nonlinear Landau damping is the following: If the bounce frequency v B of the particles trapped in the wave field is comparable to the linear damping rate g L , the initial decay of the wave amplitude will soon turn into nonlinear oscillations around a value somewhat lower than the initial value. After a few oscillations the amplitude will approach a steady value, and the wave becomes a Bernstein-GreekKruskal (BGK) mode (e.g., Refs. [12,14]). However, our picture differs significantly: According to our calculations the amplitude never settles to a steady value, although the nonlinearities obstruct the linear decay. Instead the energy of the resonant particles diffuses slowly into higher harmonics. Furthermore, a flat distribution of the trapped particles corresponding to the BGK modes never develops. The discrepancy with previous analytical work [8] is due to a lack of self-consistency in that paper. In particular, the neglect of the dynamics associated with the trappinguntrapping transitions of particles is crucial. The calculations made in Ref.[10] are self-consistent on the other hand, but use the assumption that the amplitude of the electric field deviates only slightly from a final steady value. This is fulfilled neither in experiments [7] nor in our calculations for any value of g L ͞v B . Furthermore, previous numerical work starting from the basic equations [9] has used a too low resolution [17] to describe the excitation of high harmonics of resonant particle oscillations. Finally, experimental results [7] show an average decay of the large amplitude waves not predicted by previous theories [18], but in agreement with the results of the present Letter.From ...

Equilibrium statistical mechanics for single waves and wave spectra in Langmuir wave-particle interaction

“…The existence of BGK steady states is supported by some theoretical analysis and numerical results [21,22,23]. However, Brodin's numerical simulations [19] suggested that the wave amplitude never settles to a steady value as the energy of the resonant particles diffuses slowly into higher harmonics. Moreover Isichenko [20] argued that nonlinearities should not stop the damping of the electric field towards a vanishing time-asymptotic amplitude.…”

“…However, this result is qualitative. Actually, in spite of major recent advances in the analytical and computational fields [19,20,21,22,23,24,25,26], mostly in the full Vlasov-Poisson framework, as well as in experiments [27], no definite answer presently exists to this intrinsically nonlinear and time-asymptotic issue, in the sense that no quantitative threshold separating nonzero time-asymptotic wave amplitude states from zero ones is known. This is basically due to the facts that analytical reductions of the problem should e.g.…”

Detailed dynamics of electron beams self-trapped and accelerated in a self-modulated laser wakefield

“…This is referred to as electron beam loading or nonlinear Landau damping. [27][28][29] In fact, untrapped electrons can also gain energy ͑as seen in Fig. 1͒ and damp the wave, a process referred to as Landau damping.…”

Covariant Constitutive Relations, Landau Damping and Non-Stationary Inhomogeneous Plasmas

“…There has been much debate in the literature [16] as to whether Landau damping leads to a stationary BGK solution [17]. This has been challenged by new simulations [18,19]. Looking at the perturbations of the BGK solutions may give new insight into this debate.…”