Trapped ion temperature gradient driven instabilities with bounce resonance

A kinetic analysis of ion temperature gradient modes with the trapped ion bounce resonance in tokamaks is presented under the long-wavelength limit. The dispersion relations of the toroidal and slab branches are derived by employing two-scale expansion, and their eigenfunctions are obtained analytically. The growth rates and stability thresholds of the more unstable toroidal and slab ion modes are evaluated in the cases with and without trapped electron dynamics. It is found that the minority trapped ion bounce resonances possess approximately the same effects on the toroidal and slab ion modes as the majority ion transit resonances. The nonadiabatic trapped electron dynamics does also strongly affect the toroidal ion mode while it affects the slab branch, the trapped electrons have a destabilizing effect on these ion modes. In addition, the effects of trapped electron temperature gradient on the trapped ion temperature gradient modes are considered. For high collisionality, the trapped electron temperature gradient destabilizes toroidal and slab ion modes. On the contrary, it plays a role of stabilizing to the toroidal and slab ion modes in the case of very low collisionality.