Faraday resonance in dynamically bar unstable stars

We investigate the nonlinear behaviour of the dynamically unstable rotating star for the bar mode by both three-dimensional hydrodynamics in Newtonian gravity and our simplified mathematical model. We find that an oscillation along the rotation axis is induced throughout the growth of the unstable bar mode, and that its characteristic frequency is twice as that of the bar mode, which oscillates mainly along the equatorial plane. We also find that, by examine several azimuthal modes, mode coupling to even modes, i.e., the bar mode and higher harmonics, significantly enhances the amplitudes of odd modes, unless they are exactly zero initially. Therefore, non-axisymmetric azimuthal modes cannot be neglected at late times in the growth of the unstable bar-mode even when starting from an almost axially symmetric state.