Damping of toroidal ion temperature gradient modes

The temporal evolution of linear toroidal ion temperature gradient (ITG) modes is studied based on a kinetic integral equation including an initial condition. It is shown how to evaluate the analytic continuation of the integral kernel as a function of a complex-valued frequency, which is useful for investigating the asymptotic damping behavior of the ITG mode. In the presence of the toroidal magnetic drift, the potential perturbation consists of normal modes and a continuum mode, which correspond to contributions from poles and from an integral along a branch cut, respectively, of the Laplace-transformed potential function of the frequency. The normal modes have exponential time dependence while the continuum mode, which has a ballooning structure, shows a power law decay [proportional] t^?2, where t is the time variable. Therefore, the continuum mode dominantly describes the long-time asymptotic behavior of the perturbation for the stable system. By performing proper analytic continuation for the dispersion relation, the normal modes\u27 growth rate, real frequency, and eigenfunction are numerically obtained for both stable and unstable cases