A Study of the Nonlinear Landau Damping in the Fourier Transformed Velocity Space

Abstract

The collisionless plasma oscillations in the nonlinear regime (the nonlinear Landau damping) are investigated numerically in the Fourier transformed velocity space. A scattering process of a Maxwellian initial disturbance is again observed, as in the linear case. It is found that at the moment the initially excited Landau damped modes stop damping they are converted into Van Kampen‐Case eigenmodes. These modes create a quasi‐stationary wave pattern in the Fourier transformed phase plane, characteristic for a BGK mode. The scattered disturbance preserves its identity but exercises, on the background of the wave pattern, a complicated motion and undergoes a strong dispersion, up to complete disappearance.

Keywords:

• Landau damping
• Van Kampen‐Case eigenmodes
• BGK modes
• free streaming

Notes

¹The precise meaning of this term is given in section 6. The problem of the relation between the Van Kampen‐Case eigenmodes and the BGK modes has recently been raised by Schamel (2000) who obtained an expansion of the BGK mode in terms of Van Kampen‐Case eigenmodes. Here we show that these modes are in a sense hidden in the BGK mode and become visible in the Fourier transformed velocity space.

²No static pictures and verbal description can substitute seeing the results of numerical experiments in action. The author has therefore prepared a CD‐ROM containing many animated solutions of both the linear and the nonlinear Vlasov‐Poisson equation in the Fourier transformed velocity space together with a guide and other related material. The author will send with pleasure this CD‐ROM to anyone who asks for it.

³This is the approach used originally by Knorr (1963). Some authors like Neunzert (1968) and Nührenberg (1971) and most recently Eliasson (2002) do not transform with respect to the spatial coordinate, but with this approach most of the phenomena described in this article are lost.

⁴For the solution of the nonlinear Vlasov‐Poisson equation we used a Runge‐Kutta method of 5th order with an adaptive step algorithm. With the linear Vlasov Poisson equation an “exact” method was used, based on calculation of the exponential function of the matrix of the discretized Vlasov‐Poisson operator (Sedláček 1998).

⁵The reason for the best performance of nonsymmetrical upwind‐biased schemes is obviously the fact that they realize the zero boundary condition at the upwind infinity and thus generate the discretized advection operator with a non‐Hermitian matrix which has complex eigenvalues in the lower half plane. A symmetrical, nonbiased scheme realizes the nonzero boundary condition at the upwind infinity and thus generates a Hermitian matrix with only real eigenvalues which approximate the real continuous spectrum of the advection operator for that boundary condition. Carver and Hinds (1978) examined only the frequency characteristics of the schemes but not the eigenvalues of the corresponding matrices.

⁶The symbolic capability was used to calculate “on the fly” the exact coefficients of differentiation schemes by the method published by Ogˇuztöreli, Şuhubi, and Leung (1980).

⁷A possible alternative to this procedure would be to use directly the Fourier transform of the Van Kampen‐Case eigenmode as an initial condition, similarly as in the case of the eigenfunctions of discrete spectrum, but in this way many interesting transient dynamical phenomena would be completely missed.

⁸Some results on weakly nonlinear phenomena like the so‐called second order waves and the temporal echo were published in Sedláček and Nocera (1992).

⁹Interestingly enough, a similar pattern can be seen in Figure 4 of Knorr's paper (1963) or in Figure 3 of Armstrong et al. (1970), but this author failed to recognize it as the Van Kampen‐Case eigenmode. Since Knorr uses only two harmonics in total, the wave trains look quite regular. At the moment the graph in Figure 3(a) is plotted the outgoing disturbance should be located at w=20 (w corresponds to our q). The large amplitude wave packet visible in the graph is just at this place. In Figure 3(b) the outgoing disturbance is located at w=35, already off the limit of the relatively short calculational interval.

¹⁰Under‐resonant modes are encountered if some higher harmonic is initially excited with a nonzero amplitude. This is typical for the side‐band instability.

¹¹In recent simulations of Brunetti, Califano, and Pegoraro (2000) it was found that, in agreement with a theory of Lancellotti and Dorning (2003), there is a critical initial amplitude of the perturbation such that if the perturbation is excited with an amplitude lower than some threshold the electric field is permanently damped. This would mean that for such small amplitude waves the quasi‐stationary wave pattern does not develop, and the wave field that is created in the Fourier transformed phase plane escapes into infinity, similarly as in the linear regime.

¹²This is what we call the “finite” Van Kampen‐Case eigenmode to distinguish it from the genuine Van Kampen‐Case eigenmode. The “finite” Van Kampen‐Case eigenmode has an inverse Fourier transform in an ordinary sense whereas the inverse Fourier transform of the genuine Van Kampen‐Case eigenmode is a singular generalized function because the incoming and outgoing wave trains extend to infinity.

¹³This is the reason why, asymptotically, the frequency of the electric field is proportional to the mode number m (cf. Equation (18) in Knorr 1963).

¹⁴As a matter of fact, the Fourier transformed Van Kampen‐Case eigenmode (21) has also this form; ψ˜ _kσ(q) in (21) is a solution of the linearized Equation (33).

¹⁵When animated, this wave pattern creates a (false) impression of a fan rotating around the origin of the Fourier transformed phase plane.