Geodesic acoustic modes in a fluid model of tokamak plasma: The effects of finite beta and collisionality

Starting from the Braginskii equations, relevant for the tokamak edge region, a complete set of nonlinear equations for the geodesic acoustic modes (GAM) has been derived which includes collisionality, plasma beta and external sources of particle, momentum and heat. Local linear analysis shows that the GAM frequency increases with collisionality at low radial wave number k_r and decreases at high k_r. GAM frequency also decreases with plasma beta. Radial profiles of GAM frequency for two Tore Supra shots, which were part of a collisionality scan, are compared with these calculations. A discrepancy between experiment and theory is observed, which seems to be explained by a finite k_r for the GAM when flux surface averaged density 〈n〉 and temperature 〈T〉 are assumed to vanish. It is shown that this agreement is incidental and self-consistent inclusion of 〈n〉 and 〈T〉 responses enhances the disagreement more with k_r at high k_r. So the discrepancy between the linear GAM calculation and experiment, (which also persist for more complete linear models such as gyrokinetics) can probably not be resolved by simply adding a finite k_r.