Kinetic Theories of Geodesic Acoustic Mode in Toroidal Plasmas

Geodesic Acoustic Modes (GAM) are known to constitute a continuous spectrum due to radial inhomogeneities. The existence of a singular layer causes GAM to mode convert to short wavelength kinetic GAM (KGAM) via finite ion Larmor radius (FLR) and finite guiding-center drift-orbit-width (FOW) effects. The dispersion relation of GAM/KGAM with FLR/FOW as well as parallel electric field contributions is derived to demonstrate the mode conversion to KGAM and propagation in the lower-temperature and/or higher-q region. Corresponding collisionless damping of GAM/KGAM excited in the large q region, including higher-order harmonics of ion transit resonances, has been investigated and the analytical expression for the damping rate agrees well with numerical results in its validity regime. Excitation of energetic-particle-induced GAM (EGAM) by velocity space anisotropy is also investigated taking into account the coupling to the GAM continuous spectrum. The response of energetic particles is studied nonperturbatively, and both local and nonlocal EGAM dispersion relations are derived assuming a single pitch-angle slowing-down energetic particle equilibrium distribution function. For a sharply localized energetic particle (EP) source, it is shown that the EGAMmode is self-trapped where the EP drive is strongest, with an exponentially small damping due to tunneling coupling to outward propagating KGAM. While for a broadly distributed EP source, it is shown that the EGAM will be heavily continuum damped due to the strong coupling to GAM continuous spectrum. 1. GAM continuous spectrum and collisionless damping Geodesic AcousticModes (GAMs) [1] are toroidally symmetric normal modes unique to toroidal plasmas, and the mode structure is also nearly poloidally symmetric. They exist since the charge separation effect, due to ion radial magnetic drift associated with geodesic curvature, causes a finite parallel a.c. electric field (∝ Te/Ti) and a perturbed ion diamagnetic current to ensure quasi-neutrality via electron and ion dynamic responses, respectively. 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. GAM can be described by the magnetic flux surface averaged quasineutrality condition, which reads ∂r ( δJr ) = 0, (1) where δJr is the fluctuating radial current and (· · · ) denotes magnetic flux surface averaging. Here, we consider a large aspect-ratio axisymmetric Tokamak with straight field line flux coordinates (r, θ, ξ), and the equilibrium magnetic filed is given by B0 = B0(eξ/(1 + ε cos θ) + (ε/q)eθ), where, ξ and θ are respectively, toroidal and poloidal angle-like flux coordinates of the torus. δJr is made up of polarization current and the perturbed diamagnetic current due to density accumulation in poloidal direction. The wave equation of GAM, at the lowest order, is ∂r [ n0(r)ω ( 1− ω G(r)/ω )] ∂rδφ = 0, (2) with ωG the lowest order GAM frequency. Equation (2) is identical to that describing shear Alfvén wave (SAW) resonance [2] and, thus, it demonstrates that GAM also constitutes a con-