The shear Alfvén continuum spectrum is studied for a tokamak with a single island chain using the ideal magnetohydrodynamics theory. We have taken into account the toroidal geometry and toroidal mode coupling with the island considered as a highly-shaped stellarator. Various new frequency gaps open up inside the island due to its asymmetry both poloidally and toroidally, such as the mirror-induced Alfvén eigenmode (MAE) gap and the helicity-induced Alfvén eigenmode (HAE) gap. We have shown that the MAE gap acts as the continuation of the outside toroidal Alfvén eigenmode (TAE) gap into the island. However, the combined TAE/MAE gap is getting narrower as the island grows, leaving only half of its original width with a moderate island size as much as 3.2% of the minor radius. In addition, the two-dimensional eigenfunction of the continuum mode on the lower tip of the MAE gap now has highly localised structures around the island's long axis, contrary to the usual oscillatory global solutions found with no or a low level of toroidal asymmetry—an indication of the continuous spectrum becoming discrete and dense. These results have implications for the frequency, mode structure and continuum damping of global TAEs residing in the gap.
Calculation of toroidal Alfvén eigenmode mode structure in general axisymmetric toroidal geometry