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Geodesic acoustic modes (GAMs) are studied, for the first time, including all poloidal mode $(m)$ couplings using drift reduced fluid equations. The nearest neighbor coupling pattern, due to geodesic curvature, leads to a semi-infinite chain model of the GAM with the mode-mode coupling matrix elements proportional to the radial wave number $k_{r}$. The infinite chain can be reduced to a renormalized bi-nodal chain with a matrix continued fractions. Convergence study of linear GAM dispersion with respect to $k_{r}$ and the $m$-spectra confirms that high m couplings become increasingly important with $k_{r}$. The radially sorted roots overlap with experimentally measured GAM frequency profile in low collisionality shots in Tore Supra thus explaining the reduced frequency of GAM in Tore Supra.

Geodesic acoustic modes with poloidal mode couplings ad infinitum