Theory of phase-space hydrodynamics of electron and ion holes in collisionless plasmas

Phase-space holes are well-known Bernstein–Greene–Kruskal (B.G.K.) modes and are formed by particle-trapping in solitary potential waveforms. They exhibit orbital particle trajectories in the phase-space, due to which they are also referred to as phase-space vortices. In this article, we develop the theory of phase-space hydrodynamics for electron and ion phase-space in collisionless plasmas. The analogy between ordinary two-dimensional fluids and 1D−1V phase-space has been explored by introducing a momentum equation and a phase-space vorticity field, which enable the fluid-like analyses of the plasma phase-space. The developed kinetic-hydrodynamic equations are then employed to address the vortical nature of phase-space holes by exploring their fluid-analogous vortex-like characteristics, an identification technique of phase-space vortices, an exact derivation of the Schamel-df equations, and a measurable definition of the particle-trapping β parameter. This article introduces a new technique to the study of phase-space holes which focuses on the fluid-analogous vortical nature of the phase-space holes and prevents the need for an initial assumption of the trapped and free particle phase-space densities, thus presenting itself as a precursor to the Schamel-pseudopotential method.