Even and odd modes driven by the trapped ion temperature gradient over the secular scale

In the long wavelength limit, the trapped ion temperature gradient driven modes on the secular scale are investigated by considering the lowest order effects of the finite banana orbit width and using the two-scale expansion. Attention is focused on the effects of the wave–particle interaction over the secular scale. Also, the effects of the collisionless trapped electron dynamics are considered. The eigenmode equation derived is in the form of a modified Bessel equation. The even modes are almost not influenced by the effects of the wave–particle interaction. But for odd modes, the wave–particle interaction can give rise to instabilities of the ion branches. These odd modes are similar to the trapped ion scattering modes pointed out first by Coppi and Rewoldt [Advances in Plasma Physics, edited by A. Simon and W. Thompson (Wiley, New York, 1976), Vol. 6, p. 421]. The dependence of the growth rates on the ion temperature gradient is analyzed numerically. The typical eigenfunction curves of both the even and odd modes driven by the trapped ion temperature gradient over the secular scale are displayed.