Geodesic acoustic modes with poloidal mode couplings ad infinitum

Geodesic acoustic modes (GAMs) are studied including all poloidal mode (m) couplings within a drift reduced Braginskii framework. An exact analytical formula for GAM frequency is given within the toroidal Hasegawa Mima model with the full finite larmor radius effect and poloidal mode couplings ad infinitum using a scalar continued fraction formulation, which results from reduction of the semi-infinite chain of interactions that is obtained from the nearest neighbor coupling pattern due to geodesic curvature. This pattern can be described by a semi-infinite chain model of the GAM with the mode-mode coupling matrix elements proportional to the radial wave number k_r. In the more general case of multi-field description of the GAM, the infinite chain can be reduced to a renormalized bi-nodal chain with a matrix continued fraction formulation. The convergence study of the linear GAM dispersion with respect to k_r and the m-spectra confirms that the coupling beyond m = 1 is sustained only when k_r ≠ 0 and the higher m couplings become important with increasing k_r and increasing ion to electron temperature ratio τ_i.