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

Abstract

We perform a stability analysis of a tidally excited non-linear internal gravity wave near the centre of a solar-type star in two-dimensional cylindrical geometry. The motivation is to understand the tidal interaction between short-period planets and their slowly rotating solar-type host stars, which involves the launching of internal gravity waves at the top of the radiation zone that propagate towards the centre of the star. Studying the instabilities of these waves near the centre, where non-linearities are most important, is essential, since it may have implications for the survival of short-period planets orbiting solar-type stars. When these 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, in previous simulations the waves have not been observed to undergo instability for smaller amplitudes. Here we perform a stability analysis of a non-linear standing internal gravity wave in the central regions of a solar-type star. This work has two aims: to determine any instabilities that set in for small-amplitude waves, and to further understand the breaking process for large-amplitude waves that overturn the stratification. Our results are compared with the stability of a plane internal gravity wave in a uniform stratification, and with previous work by Kumar & Goodman on a similar problem to our own. Our main result is that the waves undergo parametric instabilities for any amplitude (in the absence of viscosity and thermal conduction). However, because the non-linearity is spatially localized in the innermost wavelengths, the growth rates of these instabilities tend to be sufficiently small that they do not result in astrophysically important tidal dissipation. Indeed, we estimate that the modified tidal quality factors of the star that result are Q′★≳ 107, and possibly much greater, which implies that the resulting dissipation is at least two orders of magnitude 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 Wide Area Search for Planets and Kepler.