The effects of magnetic island on toroidal ion temperature gradient mode instability

In this work, we have investigated the influences of magnetic island (MI) on electrostatic toroidal ion temperature gradient (ITG) mode, where the ions are described by gyro-kinetic equations including MI, and adiabatic approximation is used for electrons. The eigen-equation for short-wavelength toroidal ITG mode in Fourier-ballooning representation is derived, and the corresponding eigen-value as well as mode structure are solved. Both the flattening effects of MI on plasma pressure and MI-scale shear flow are considered. It is found that when only considering the flattening effects of MI, ITG mode can be stabilized as compared to the case without MI. While, the effective drive of toroidal ITG mode could be enhanced by including MI-scale flow, which indicates the dominant destabilizing by MI-scale flow over the stabilizing by flattening profile and results in higher growth rate than the case without MI. It is also found that the total flow shearing may prevent the ITG turbulence spreading from X-point of MI but not strong enough to prevent spreading from the seperatrix across O-point of larger MI via comparison between the flow shearing rate and the linear growth rate. Furthermore, the corresponding width of lowest-order mode structure in ballooning angle is slightly widened (narrowed) for the case without (with) MI-scale flow, as compared to the case without MI. Besides, the shifted even symmetry in ballooning angle is not qualitatively influenced by the presence of MI. The mode structure is radially asymmetric, but is symmetric with respect to the phase of MI at the O-point.