Theories Of Geodesic Acoustic Modes In Tokamak Plasmas

Geodesic Acoustic Modes (GAMs) are toroidally symmetric normal modes unique to toroidal plasmas, and the mode structure is also nearly poloidally symmetric. Their existence is associated with the charge separation effect, due to ion radial geodesic magnetic curvature drift and polarization drift. GAMs have received much attention in magnetic fusion plasma due to their potentially important roles in regulating drift waves, and, hence, transports via nonlinear interactions. In this dissertation, we employ the kinetic-theoretic approaches and study various properties of GAM.First, we derive the dispersion relation of GAM with fluid equations, and show that GAM constitutes a continuous spectrum due to inherent radial inhomogeneity of fusion plasmas. When kinetic effects due to, e.g., finite ion Larmor radii(FLR) and finite guiding-center drift-orbit-width(FOW) are included, the singularity in GAM con-tinuous spectrum can be removed. We study analytically and numerically various phe-nomenons associated with GAM continuous spectrum. We show that, in the absence of an external source, initial wave packet of GAM will spontaneously generate short-wavelength structures, and decay in time as t-1 due to phase-mixing. When kinetic ef-fects are considered, the initial pulse can be mode-converted to short wavelength kinetic GAM(KGAM), and propagate radially outward. When there is an external source, we show that, at the radial location where GAM continuum frequency matches the driving frequency of the source, the mode structure of GAM becomes singular and the external source can be resonantly absorbed by the plasma. Including kinetic effects, the singu-larity is removed and the external source is then resonantly mode-converted to outgoing KGAM.Second, we employ gyrokinetic equation and systematically study the dispersion relation of GAM for a wide range of wavelengths; including effects of FLR/FOW and parallel electric field. In the limit when the wavelength of GAM is smaller than the drift orbit width of resonant ions, we study the collisionless damping of GAM due to reso-nant interaction with the ion radial magnetic drift. Our analytical formula, combined with previous analytical works in the small drift orbit limit, thus, provides collision-less damping rate of GAM over a broad range of parameters. Our analytical formula, furthermore, agrees well with TEMPEST simulation in its validity regime.Third, we investigate nonperturbatively the nonlocal theory of energetic-particle-induced GAM(EGAM) excited via transit resonances with energetic particles, taking into account the coupling to GAM continuous spectrum and the nonlocal dispersion relation of EGAM excited by the transit resonance of energetic particles is derived. We first derive the local dispersion relation of EGAM assuming a single pitch-angle slowing down energetic particle distribution function, and give the critical pitch angle for the local EGAM instability. Including the FLR/FOW terms of both bulk thermal ions and energetic particles, then leads to the eigenmode equation of EGAM. Con-sidering an energetic particle beam radially localized away from the singular layer of GAM continuous spectrum, the FLR/FOW term is then dominated by energetic parti-cles inside the localization domain of energetic particles, and we obtain the bounded EGAM eigenmode. Away from the localized energetic particles, EGAM then couples to the background GAM continuous spectrum and mode-converts to radially outgoing KGAM. Asymptotically matching the solutions in the inner and outer regions, we then drive the corresponding global dispersion relation of EGAM. We show both analyti-cally and numerically that, when the energetic particle beam is localized away from the singular layer of GAM continuous spectrum, the mode is radially self-trapped where the energetic-particle drive is strongest. There exist, moreover, an exponentially small tunneling coupling to outgoing KGAM, which leads to convective damping of EGAM and hence, a finite threshold for EGAM excitation. We show numerically that, as the-oretically predicted, the threshold value increases with enhance coupling to the GAM continuous spectrum.The corresponding global mode structure of EGAM is also given numerically.Finally, conclusions and suggestions for possible future works on this important topic are also given.