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Non-linear excitation of low frequency modes by overstable convective modes in rotating stars

We discuss non-linear excitation and amplitude saturation of $g$-modes, $r$-modes and overstable convective (OsC) modes in early type main sequence stars, taking account of the effects of three-mode couplings on amplitude evolutions. OsC modes are rotationally stabilized convective modes in the convective core and they resonantly excite low frequency $g$-modes to obtain large amplitudes in the envelope when the rotation rate of the core is larger than critical rates. We use, for a network of three-mode couplings, amplitude equations governing the time evolution of the mode amplitudes where each of three-mode couplings is assumed to occur between two stable modes and one unstable mode. Assuming that the unstable modes in the couplings are OsC modes in the core and the stable modes are $g$- and $r$-modes in the envelope, we integrate the amplitude equations to see how the $g$- and $r$-modes are non-linearly excited by the OsC modes and whether or not the amplitude evolutions tend toward a state of finite amplitudes. We find that the non-linear three-mode couplings do excite low frequency $g$- and $r$-modes but they are not necessarily effective to achieve amplitude saturation since the three-mode couplings between the OsC modes with large growth rates and $g$- and $r$-modes with small damping rates tend to destabilize amplitude evolutions.