Singular global components and frequency shift of the geodesic acoustic continuum modes in shaped tokamaks

Within the ideal magnetohydrodynamic (MHD) model, the geodesic acoustic modes (GAMs) in tokamaks derived by Winsor et al (1968 Phys. Fluids 11 2448) belong to the continuous spectrum, characterised by unbounded non-square integrable eigenfunctions (delta functions) at the singular surfaces (Goedbloed 1975 Phys. Fluids 18 1258). The eigenfunctions of the MHD continua in cylindrical as well as toroidal plasmas include, in addition, components that exist outside the singular surfaces and have singularities of type (psi - psi(0))(-1) or ln, vertical bar psi - psi(0)vertical bar where psi is a flux function that labels the magnetic surfaces, and psi = psi(0) defines the singular surface (Pao 1975 Nucl. Fusion 15 631). Using a large aspect ratio approximation of tokamak plasmas it is shown in this paper that the GAMs indeed include such singular components. Hence, in addition to the non-square integrable m = 0 and m = 1 components of the plasma flow and of the density and pressure perturbations at the GAM surface, the GAM continua also include accompanying m = 0 and m = 1 singular components varying as (psi - psi(0))(-1) This gives the m = 0 and m = 1 components of each GAM in the continuum radially extended profiles and a global character also within ideal theory. To the same order in the expansion, effects of a finite aspect ratio and a non-circular plasma cross section on the GAM frequency are also calculated, and we recover the dependence on inverse aspect ratio and Shafranov shift of the real GAM frequency previously calculated within gyrokinetic theory by Gao (2010 Phys. Plasmas 17 092503). Furthermore, while the dominating shaping effect on the GAM frequency comes from plasma elongation, as shown previously, it is shown in this paper that there is a higher-order triangularity effect that can also be significant. The calculated triangularity effect predicts a nearly linearly increasing GAM frequency with increasing triangularity, a phenomenon observed also in the TCV tokamak.