Stability analysis of a tidally excited internal gravity wave near the centre of a solar-type star

We perform a stability analysis of a tidally excited nonlinear internal gravity wave near the centre of a solar-type star in two-dimensions. The motivation is to understand the tidal interaction between short-period planets and their solar-type host stars, which involves the launching of gravity waves at the top of the radiation zone that propagate towards the stellar centre. Studying the instabilities of these waves near the centre, where nonlinearities are most important, is essential, since it may have implications for the survival of these planets. When the waves have sufficient amplitude to overturn the stratification, they break and form a critical layer, which efficiently absorbs subsequent ingoing wave angular momentum, and can result in the planet spiralling into the star. However, previous simulations do not find the waves to undergo instability for smaller amplitudes. This work has two aims: to determine any instabilities that set in for small-amplitude waves, and to further understand the breaking of large-amplitude waves. Our main result is that the waves undergo parametric instabilities for any amplitude. However, because the nonlinearity is spatially localised in the innermost wavelengths, their growth rates are sufficiently small that they do not result in astrophysically important tidal dissipation. The resulting modified tidal quality factors are estimated to be Q'_star>10^7, and possibly much greater, so the dissipation is much weaker than that which results from critical-layer absorption. These results support our explanation for the survival of all currently observed short-period planets around solar-type main-sequence stars: that planets unable to cause wave breaking at the centre of their host stars are likely to survive against tidal decay. This hypothesis will be tested by ongoing and future observations of transiting planets, such as WASP and Kepler.