Finite difference modeling of coherent wave amplification in the Earth's radiation belts

Modeling of gyroresonant wave-particle interactions in the radiation belts requires solving the Vlasov-Maxwell system of equations in an inhomogenous background geomagnetic field. Previous works have employed particle-in-cell methods or Eulerian solvers (such as the Vlasov Hybrid Simulation code) to provide numerical solutions to the problem. In this report, we provide an alternative numerical approach by utilizing a first order finite difference upwind scheme. When coupled to the narrowband Maxwell's equations, the model reproduces linear as well as nonlinear wave growth of coherent signals. Wave growth is nonlinear growth when the wave amplitude exceeds the minimum value for phase trapping of counterstreaming particles and is linear otherwise. The model also demonstrates free-running frequency variation for a case with a high linear growth rate. In addition, the model confirms the theoretical prediction of a stable “phase-space hole” during the nonlinear growth process. The plasma parameters and L shell used in this study are typical of those associated with the Siple Station wave injection experiment.