Solution of initial value problem of gyro-kinetic equation

Applying the Laplace transform technique, the initial value problem of gyro-kinetic equation is solved for the electrostatic ion temperature gradient instability situation in slab and toroidal systems. The transformed perturbed scalar potential, when evaluated on the real frequency axis, tends to small as the growth rate increases, and shows a singularity only at the marginal stability state. The time dependent solution for perturbed scalar potential is expressed in terms of the discrete mode and the continuum contribution. At the marginal stability state, the discrete eigenvalue attains the continuous eigenvalue spectrum. For the subcritical state the discrete eigenvalue moves into the next Riemann surface and tends to a damping mode, which has been numerically examined making use of the analytically continued dispersion relation. The time-dependent perturbed distribution is expressed in terms of the discrete mode, the velocity dependent beam mode, and the continuum contribution. Some characteristics of the discrete and continuum contribution are examined, and application to anomalous transport theory is suggested. (author)