Nonlinear evolution of the dynamical instability in a Keplerian disk - Resonant excitation of higher modes

The second order perturbations induced by linearly unstable perturbations were computed in order to study the nonlinear evolution of the dynamical instabilities. When the linearly most unstable mode with azimuthal wavenumber, m, is excited, the second order perturbation with azimuthal wavenumber, 2m, is also excited with a large amplitude. Even when the first order density fluctuation is 0.01, the second order fluctuation has an amplitude of the same order. This large amplitude is ascribed to the resonance of the radial wavenumber of the second order perturbation to that of the cross term of the first-order perturbations. The third and higher modes are also expected to be highly excited. The excitation of these shorter wavelength modes makes the wave profile steep and may lead to a shock wave. The angular momentum transfer associated with the dynamical instability is discussed, based on the second-order change in the angular momentum.