Quasi-linear Analysis of Proton-cyclotron Instability

The proton-cyclotron (PC) instability operates in various space plasma environments. In the literature, the so-called velocity moment-based quasi-linear theory is employed to investigate the physical process of PC instability that takes place after the onset of early linear exponential growth. In this method, the proton velocity distribution function (VDF) is assumed to maintain a bi-Maxwellian form for all time, which substantially simplifies the analysis, but its validity has not been rigorously examined by comparing against the actual solution of the kinetic equation. The present paper relaxes the assumption of the velocity moment-based quasi-linear theory by actually solving for the velocity space diffusion equation under the assumption of separable perpendicular and parallel VDFs, and upon comparison with the simplified velocity moment theory, it demonstrates that the simplified method is largely valid, despite the fact that the method slightly overemphasizes the relaxation of temperature anisotropy when the system is close to the marginally stable state. The overall validation is further confirmed with the results of particle-in-cell and hybrid-code simulations. The present paper thus provides a justification for making use of the velocity moment-based quasi-linear theory as an efficient first-cut theoretical tool for the PC instability.