# Particle Simulation of Lower Hybrid Waves and Electron-Ion Hybrid Instability

## Abstract

Lower hybrid wave (LHW) has been of great interest to laboratory plasma physics for decades due to its important applications in particles heating and current drive in plasmas devices. There are two fundamental characteristics of LHWs, Landau damping and parametric instability (PI), both of which play key roles in particles heating and current drive problems. Linear physics of LHWs has been studied in great details in analytical theories, while nonlinear physics is usually too complicated to be resolved analytically. Computer kinetic simulation technique has developed to be one of the best tools for the investigation of kinetic physics, especially in the nonlinear stage. Although a great amount of theoretical work has been done in the investigation of LHWs, little particle simulation work can be found in published literatures. In this thesis, an electrostatic gyro-kinetic electron and fully kinetic ion (GeFi) particle simulation scheme is utilized to study the linear and nonlinear physics of LHWs. GeFi model is particularly suitable for plasma dynamics with wave frequencies lower than the electron gyrofrequency, and for problems in which the wave modes ranging from Alfv\'en waves to lower-hybrid/whistler waves that need to be handled on an equal footing with realistic electron-to-ion mass ratio. Firstly, Interactions of LHWs with both electrons and ions through nonlinear Landau damping, as well as linear Landau damping, is investigated by utilization of GeFi particle simulation in electrostatic limit. Landau damping, a wave-particle interaction process, provides a way for LHWs to exchange energy and momentum with electrons and ions, thus to heat plasmas and generate electric currents in the plasmas. Unlike most other wave modes, LHWs can resonantly interact with both electrons and ions, with the former being highly magnetized and the latter being nearly unmagnetized around lower hybrid frequency. Direct interactions of LHWs with electrons and/or ions are investigated for cases with various $k_\parallel/k$, $T_i/T_e$, and wave amplitudes. Here, $k$ is wave vector, $k_\parallel$ is parallel (to static magnetic field) wave vector, $T_i$ and $T_e$ are ion and electron temperatures, respectively. In the linear electron Landau damping (ELD), real frequencies and damping rates obtained from our kinetic simulations have excellent agreement with the analytical linear dispersion relation. As wave amplitude increases, the nonlinear Landau effects are present, and a transition from strong decay at smaller amplitudes to weak decay at larger amplitudes is observed. In the nonlinear stage, LHWs in a long time evolution finally exhibit a steady BGK mode, in which the wave amplitude is saturated above noise level. While resonant electrons are trapped in the wave electric field in the nonlinear ELD, resonant ions are untrapped in LHWs time scales. Ion Landau damping is thus predominantly in a linear fashion, leading to a wave saturation level significantly lower than that in the ELD. On long time scales, however, ions are still weakly trapped. Simulation results show a coupling between LHW frequency and ion cyclotron frequency during the long-time LHW evolution. Secondly, our investigation extends to a transverse sheared flow driven instability, electron-ion hybrid (EIH) instability, whose frequency is in the lower hybrid frequency range. EIH instability is studied by the GeFi model in a magnetized plasma with a localized electron cross-field flow. Macroscopic flows are commonly encountered in various plasmas, such as plasmas in tokamak devices, laser-produced plasmas, Earth's magnetopause, plasma sheet boundary layer, and the Earth's magnetotail. As a benchmark, linear simulations of EIH are firstly performed in both slab geometry and cylindrical geometry with $k_z=0$ in either uniform plasmas or nonuniform plasmas, and the results are compared with linear theories in a slab geometry. Here for the slab geometry, static magnetic field is at $z$ axis, and electron shear flow as a function of $x$ is put at $y$ axis. And in the cylindrical geometry, static magnetic field is also at $z$ direction, while electron shear flow as a function of radial position $r$ is in $\theta$ (poloidal) direction. Linear eigen mode structures and growth rates of EIH instability are calculated for various $k_yL$, $\alpha_1=V_E^0/L\Omega_e$, and $L/L_n$. Here $k_y$ ($k_z$) is the wave vector in $y$ ($z$) direction, $L$ the shear length of electric field, $V_E^0$ the peak value of the $\mathbf{E}\times\mathbf{B}$ drift velocity, $\Omega_e$ the electron gyro frequency, $L_n$ the scale of density gradients. The results have very good agreement with the theoretical predications. Nonlinear simulations are performed to investigate the nonlinear evolution of EIH instabitly. It is found that the EIH instability nonlinearly evolves from a short wave length ($k_x\rho_i\sim 12$) mode to a long wave length ($k_x\rho_i\sim 3$) mode with frequency $\sim \omega_{LH}$. Simulation results under realistic plasma conditions of the Auburn Linear Experiment for Instability Studies (ALEXIS) device are discussed and compared with ALEXIS experimental results as well. Finally, parametric instability of LHWs is investigated using the GeFi model. Parametric instability is a nonlinear process that involves wave-wave interactions, and is of great interest. In the propagation of LHWs, a pump LHW $(\omega_0, \mathbf{k_0})$ decays into a low frequency wave mode $(\omega, \mathbf{k})$ and two high frequency sidebands $(\omega\pm\omega_0, \mathbf{k}\pm\mathbf{k_0})$, where $\omega$ represent the wave frequency, $\mathbf{k}$ represent the wave vector, and the subscript "0" indicates the pump wave. Two different cases with parametric instability are discussed in great details. The parametric decay process is found to occur very fast within several lower hybrid wave periods. Growth rates of the excited modes are estimated as well, and results are compared with the analytical theory. The simulation shows that the parametric instability process is complicated due to multiple decay channels. These multiple parametric instability processes usually occur simultaneously. The corresponding electron and ion particle distributions are investigated in the decay process. Finally, a discussion is presented for the future study of electron and ion nonlinear physics of the PI instability.