Predictions from a gyrokinetic toroidal plasma model reproduce for the first time the evolution of Alfven eigenmode (AE) dampings over a range of discharges. The coupling between shear-and kinetic-Alfven waves is responsible for the main source of damping through Landau interactions and can be an order of magnitude larger than fluid predictions neglecting global kinetic effects. Strong stabilization occurs when the wave field gets localized radially by a rise in the edge magnetic shear, explaining why global AEs have never been detected in the Joint European Torus [Rebut, Bickerton, and Keen, Nucl. Fusion 25, 1011 (1985)] in the presence of an X point and suggesting how global Alfven instabilities could be avoided in future reactors. (C) 1998 American Institute of Physics.