Kinetic theory of Geodesic Acoustic Modes: radial structures and nonlinear excitations

Geodesic Acoustic Modes (GAM) are shown 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 radii; analogous to kinetic Alfvén waves (KAW). KGAM are shown to propagate radially outward; consistent with experimental observations and numerical simulation results. The degeneracy of GAM/KGAM with Beta induced Alfvén Eigenmodes (BAE) is demonstrated and discussed. We show that energetic particle driven oscillations can be excited from the GAM continuum, similarly to the Energetic Particle Mode (EPM) case. Furthermore, it is shown that KGAM can be nonlinearly excited by drift-wave (DW) turbulence via 3-wave parametric interactions, and the resultant driven-dissipative nonlinear system exhibits typical prey-predator self-regulatory dynamics. KGAM are preferentially excited with respect to GAM because of the radial wave-number dependence of the parametric excitation process. Plasma non-uniformity effects on nonlinear KGAM excitations are discussed. In this work, we show that Geodesic Acoustic Modes (GAM) [1] constitute a continuous spectrum due to radial inhomogeneities. The existence of a singular layer, thus, suggests GAM collisionless damping due to absorption at the GAM continuum resonance [2] and linear mode conversion to short-wavelength kinetic GAM (KGAM) via finite ion Larmor radii (FLR) and finite magnetic drift orbit widths (FOW) [3]. This result is demonstrated by derivations of the GAM/KGAM mode structure and dispersion relation in the singular layer, indicating that, typically, KGAM propagate radially outward [3]. The formal identity of the GAM/KGAM wave equation to that describing shear Alfvén wave (SAW) mode conversion to kinetic Alfvén Wave (KAW) near the SAW resonance [4], suggests a complete analogy of GAM/KGAM ⇔ SAW/KAW. Our analyses also confirm that GAM and Beta induced Alfvén Eigenmodes (BAE) [5, 6] are degenerate in the long wavelength limit, where diamagnetic effects are ignored, even when FLR and FOW corrections are accounted for. Besides the importance of its physics implications, the usefulness of this result on the BAE/GAM degeneracy is that we may straightforwardly derive the governing equations for GAM using the kinetic theory results on BAE developed earlier [7, 8, 9]. We show that dynamics of GAM/KGAM excitations should be addressed as initial value problem in the presence of a radial nonuniform source. Details of these analyses are reported in a separate work. Since GAM/KGAM are toroidally and (nearly) poloidally symmetric structures, the source term can be associated with either an anisotropic distribution of (fast) particles in velocity space [10, 11] or by nonlinear excitations due to drift-wave (DW) turbulence [3, 12, 13, 14]. In this work, we show that, while GAM/KGAM are linearly stable due to ion Landau damping, they can be excited from the GAM continuum by energetic particles (Section 2), in analogy to Energetic Particle Modes (EPM) [15], and by DW turbulence via 3-wave resonant parametric interactions (Section 3) [3]. GAM are important to turbulence transport studies, since their low frequency radial structures can scatter DW fluctuations to sta-