On the Bernstein–Landau paradox

Abstract

The essence of the Bernstein–Landau paradox is that in a stable unmagnetized plasma electrostatic waves exhibit collisionless Landau damping, while in a magnetized plasma the Bernstein modes, perpendicular to the magnetic field, are exactly undamped, independent of the strength of the magnetic field. This problem is the subject of the present study. An analytical solution of the initial value problem for perturbations perpendicular to the magnetic field is given, which is a generalization of the well-known Landau work to magnetized plasmas. By introducing, according to Plemelj’s prescription, plus- and minus-functions, having unique analytical properties, the character of the short-term and long-term plasma response is revealed, showing in the small magnetic field limit Landau damping in the first gyroperiod, followed by recurrence, and exhibiting irregular behavior with no damping at large times. The initial damping rate is seen to be close to the commonly used Landau damping rate for unmagnetized plasmas, however with a significant systematic deviation. A corrected expression for the Landau damping rate is found which yields a perfect description of the initial damping of oscillations perpendicular to a weak magnetic field. An alternative approach, expansion over Bernstein modes, is also employed. It is found that a zero-frequency (convective) mode, revealed earlier in particle simulations, is included in the complete linear treatment.

Numerical Simulations of the Fourier-Transformed Vlasov-Maxwell System in Higher Dimensions—Theory and Applications

“…This problem was investigated theoretically by Sukhorukov and Stubbe (1997) who derived analytical solutions of the problem. In Fig.…”

“…In Fig. 12, we have compared a numerical solution of the Vlasov equation with the analytic solution of Sukhorukov and Stubbe (1997) for a wave with the wavenumber k x = 0.4 r −1 D and with different values on the magnetic field, such that ω ce /ω pe = 0.4/10, 0.4/7 and 0.4/4, where ω ce = eB ext /m e . The numerical result obtained from the Vlasov simulation can be seen in Fig.…”

A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case

“…11 shows the effect of an increasing magnetic field on the time oscillations. According to the Shukorukov and Stubbe theory, the effect of the Landau damping is visible in the first gyroperiod, but each cyclotron period the magnetic field raises the electric oscillations and recurrence peaks are strongly visible [23]. In correspondence to the strongest value of the magnetic field (B = 0.3), in Fig.…”

“…Another analytical and numerical study has been performed by Shukorukov and Stubbe [23], who solved numerically the dispersion relation of the Bernstein waves, and showed that, in the magnetized case, the effect of the Landau damping is visible in the first gyroperiod, for very brief time transients, but the waves are not damped at large times and the amplitude of the oscillations grows with increasing values of the external field. In this way, they solved the problem in linear approximation, paying attention to the time evolution of the density perturbation, under the effects of the magnetic field.…”

Laser-pulse compression using magnetized plasmas

“…While linear collisionless damping vanishes when the wave propagates exactly perpendicular to the magnetic field [8,10], nonlinear collisionless damping persists due to surfatron acceleration [11] and stochastic heating [12]. To give a conservative estimation, note that the UH frequency is typically comparable to the gyrofrequency.…”