Nonlinear development of the plasma-beam instability in a magnetized plasma

The nonlinear development of a plasma-mean system immersed in a static magnetic field B₀ is investigated. The beam electrons are assumed to be streaming parallel to the direction of B₀ and have a spread in the perpendicular component of the velocity. We restrict our consideration to the P instabilities which result in growing whistler waves aligned in a direction parallel to that of B₀. Scaled equations describing the time evolution of these instabilities are derived under the assumptions that the beam density is small and that the resulting whistler wave is nearly sinusoidal. The perturbation of the perpendicular component of the electron velocity during the interaction can be excluded from the equation determining the wave dynamics, and then these scaled equations can be reduced to the simpler ones amenable to numerical solution which depends analytically on all the basic parameters of the problem. This solution shows that the amplitude of an unstable whistler wave first increases at the linear growth rate until the electrons are trapped and then, after overshooting, it approaches a steady state. The time evolution of the amplitude does not show an oscillation with a constant period; this results from the phase mixing enhanced by the spread in perpendicular velocities of the beam electrons.