Global MHD GAMs in toroidal plasmas with reversed magnetic shear

A formalism is developed for axisymmetric (n = 0) perturbations of toroidal plasmas in the framework of ideal magnetohydrodynamics, and is applied successfully to the explanation of the global structures of geodesic acoustic modes (GAMs) that have been revealed by the numerical simulations of tokamak plasmas with an off-axis maximum of local GAM frequency. It turns out that the global structures of GAM can emerge as the consequence of balancing the momenta associated with the second poloidal harmonics introduced by the coupling between the curvatures of equilibrium magnetic field lines and the pressure perturbations via Alfvenic perturbations. The global GAM frequency is allowed to reside in the narrow gap between the maxima of the modified local GAM frequencies due to Alfvenic perturbations. This mode features a distinct 'singular' layer centered around the local GAM maxima, where both the acoustic/geodesic acoustic and Alfvenic perturbations are equally important and the eigenfunction resembles the symbol integral. This layer is sandwiched between two wider regions where the Alfvenic activities dominate. The negative magnetic shear plays an indispensable role in establishing the global mode structures, in that it shapes the eigenfunctions in the right way in the outer regions and allows them to match with the inner segment smoothly. Many features of the global GAM predicted by our theory are confirmed by running our shooting code for both weakly and strongly reversed magnetic shear profiles.