Collisionless Trapped Ion Temperature Gradient Instabilities

Under the fluid limit, the collisionless electrostatic trapped ion temperature gradient instabilities are investigated by retaining the trapped ion parallel compressibility in the long wavelength limit. The two-scale analysis should be employed in view of the fact that the eigenfunctions vary over the connection width scale with an envelope varying over a secular scale. The results show that the slab branch of trapped ion modes, corresponding to the aperiodic trapped ion instability discovered by Kadomtzev, propagates along ion diamagnetic direction. The toroidd branch is also obtained in low-frequency and long wavelength limit. In this case, besides the unstable one propagating in ion diamagnetic direction, there exists a marginally stable toroidal odd mode propagating in electron diamagnetic direction. The eigenfrequencies and perturbative structures of these unstable modes are given analytically and numerically. The numerical results are in good agreement with the analytical ones.