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.