Electron–Langmuir wave resonance in three dimensions

Abstract

In the one-dimensional (1D) treatment of Langmuir wave generation by a particle distribution via the kinetic beam instability, there is a one-to-one resonance between the electron speed v and the phase speed vϕ of the wave. The 1D condition for wave growth is ∂f/∂v>0 with v=vϕ, and f(v) evolves due to quasilinear relaxation toward a plateau distribution ∂f/∂v=0. We show here that none of these results apply in a 3D treatment of the problem. For a wave with wavevector k and phase speed vϕ, there is a many-to-one resonance with all electrons with v>vϕ moving obliquely to k. Although growth requires a region with ∂f/∂v>0 below a peak in f(v), the growth can be driven primarily by particles above the peak where ∂f/∂v<0, and growth can even occur at phase velocities vϕ where ∂f/∂v|v=vϕ<0. Resonance at v≫vϕ favors diffusion of the particle distribution in angle, rather than plateau formation. These properties imply that intuition based on the 1D model can be seriously misleading, with far-reaching implications for modeling phenomena such as type III solar radio bursts.