Studies Of Nonlinear Heating By Multiple Waves And Alfvén Eigenmodes In Tokamak Geometry

The nonlinear interaction between waves and particles is important in controlled nu-clear fusion research as well as in astrophysics because it is associated with ion heating in tokamak fusion experiments using lower-hybrid wave, transverse energization of ions in the ionosphere, and heating of ions in the solar corona by Alfven waves. It is interesting to examine whether multiple waves with frequency at a fraction of the cyclotron frequency and small amplitude can still lead to effective heating or coherent acceleration of ions, since they are much easier to be achieved than a single wave with high frequency and large am-plitude in experiments and astrophysics, and whether there is a general nonlinear resonance condition for the multiple wave case.It is well known that shear Alfven eigenmodes(AEs) in tokamaks can be resonantly destabilized by energetic particles 1 and the Alfven instability will degrade the confinement of energetic particles. In addition, the AEs are useful for plasma diagnostic such as mea-surement of safety factor profile.Therefore it is important to study the properties of AEs such as the existence and stability of the modes.We report the mechanisms of the interaction of waves and particles which can lead to heating in plasma and the properties of shear Alfven waves in tokamak. In the the-sis, we adopt multi-scale expansion method and Lie perturbation method respectively to study in particular acceleration and heating ions by multiple electrostatic waves and shear Alfven waves with small amplitude and low frequency, and then apply a nonperturbative kinetic/magnetohydrodynamics eigenvalue code to investigate the kinetic effect of shear Alfven eigenmodes in general tokamak geometry with reversed shear profile of safety fac-tor.In Chapter 1, a review on the importance of the interaction of waves and particles which can lead to heating in plasma and shear Alfven wave in tokamak is introduced.In Chapter 2, the nonlinear interaction of ions with multiple electrostatic waves propa- gating perpendicularly across a uniform magnetic field is investigated both analytically and numerically. Applying a multi-scale expansion method with the wave amplitude as the per-turbation parameter, a general nonlinear resonance condition is analytically derived. Under this condition, it is confirmed that multiple waves even below the cyclotron frequency and small amplitude are capable of effectively producing acceleration or stochastic heating by numerical simulation. Compared to the single wave situation, the stochastic threshold for heating by multiple waves with frequencies satisfied with a nonlinear resonance condi-tion is significantly reduced because the nonlinear interaction of ions with multiple waves leads more easily to overlapping of islands and spreading of the stochastic layer in phase space.The above result is helpful to understand the energization mechanism of ions in the solar corona.In Chapter 3, the nonlinear interaction of ions with multiple Alfven waves is stud-ied both analytically and numerically. Adopting a Lie-perturbation method with the wave amplitude as the perturbation parameter, a general nonlinear resonance condition is ana-lytically derived. Under this condition, it is confirmed that multiple waves even below the cyclotron frequency and small amplitude can effectively producing acceleration or stochas-tic heating by numerical simulation. Compared to the single wave situation, the stochastic threshold for heating by multiple waves with frequencies satisfied with a nonlinear reso-nance condition is significantly reduced due to the nonlinear interaction of ions with mul-tiple waves leading more easily to overlapping of islands and spreading of the stochastic layer in phase space.In Chapter 4, a nonperturbative kinetic/magnetohydrodynamics eigenvalue code has been constructed for calculation of kinetic damping of shear Alfven eigenmodes. The model generalizes the previous work [G. Y. Fu et al., Phys. Plamsas 12,082505 (2005)] to general tokamak equilibria with finite aspect ratio, finite beta and non-circular shape. The model describes shear Alfven waves with kinetic effects from thermal species including thermal ion finite Larmor radius effects and parallel electric field. An analytic formula for the radiative damping of reversed shear Alfven eigenmodes is obtained for tokamak plas- mas with reversed shear q profile. Numerical calculations reveal the existence of multiple kinetic reversed shear Alfven eigenmodes (KRSAEs). It is found that the damping rate of the KRSAEs scales linearly with the thermal ion gyroradius. The damping rates are larger for modes with more peaks in the radial structures. These results are consistent with ana-lytic expectation. The KRSAEs found here can be used to interpret the RSAEs frequency sweeping down observed sometime in the tokamak experiments.In the final Chapter, conclusions are given.