On internal wave breaking and tidal dissipation near the centre of a solar-type star

Abstract

We study the fate of internal gravity waves approaching the centre of an initially non-rotating solar-type star, primarily using two-dimensional numerical simulations based on a cylindrical model. A train of internal gravity waves is excited by tidal forcing at the interface between the convection and radiation zones of such a star. We derive a Boussinesq-type model of the central region of a star and find a non-linear wave solution that is steady in the frame rotating with the angular pattern speed of the tidal forcing. We then use spectral methods to integrate the equations numerically, with the aim of studying at what amplitude the wave is subject to instabilities. These instabilities are found to lead to wave breaking whenever the amplitude exceeds a critical value. Below this critical value, the wave reflects perfectly from the centre of the star. Wave breaking leads to mean flow acceleration, which corresponds to a spin-up of the central region of the star, and the formation of a critical layer, which acts as an absorbing barrier for subsequent ingoing waves. As these waves continue to be absorbed near the critical layer, the star is spun up from the inside out.Our results point to an important amplitude dependence of the (modified) tidal quality factor Q′, since non-linear effects are responsible for dissipation at the centre of the star. If the amplitude of the tidal forcing exceeds the critical amplitude for wave breaking to occur, then this mechanism produces efficient dissipation over a continuous range of tidal frequencies. This requires , for a planet of mass mp in an orbit of period P around the current Sun, neglecting stellar rotation. However, this criterion depends strongly on the strength of the stable stratification at the centre of the star, and so it depends on stellar mass and main-sequence age. If breaking occurs, we find , for the current Sun. This varies by no more than a factor of 5 throughout the range of solar-type stars with masses between 0.5 and 1.1 M⊙, for fixed orbital parameters. This estimate of Q′ is therefore quite robust and can be reasonably considered to apply to all solar-type main-sequence stars, if this mechanism operates. The strong frequency dependence of the resulting dissipation means that this effect could be very important in determining the fate of close-in giant planets around G and K stars. We predict fewer giant planets with orbital periods of less than about 2 d around such stars if they cause breaking at the centre, due to the efficiency of this process.Even if the waves are of too low amplitude to initiate breaking, radiative damping could, in principle, lead to a gradual spin-up of the stellar centre and to the formation of a critical layer. This process could provide efficient tidal dissipation in solar-type stars perturbed by less massive companions, but it may be prevented by effects that resist the development of differential rotation.These mechanisms would, however, be ineffective in stars with a convective core, such as WASP-18, WASP-12 and OGLE-TR-56, perhaps partly explaining the survival of their close planetary companions. © 2010 The Authors. Journal compilation © 2010 RAS.