A new gyro-Landau fluid model for the Landau damping of circulating particles is presented. Since trapped particles can bounce average low frequency waves, they are not affected by Landau damping. The loss of bounce averaging when the mode frequency exceeds the trapped particle bounce frequency is included in the model by changing the phase space integration boundary. An excellent fit to the kinetic response for the circulating particle density and the parallel and perpendicular pressure is obtained with six-moment equations.