Dynamic tides in rotating objects: a numerical investigation of inertial waves in fully convective or barotropic stars and planets

Abstract

We perform direct numerical simulations of the tidal encounter of a rotating planet on a highly eccentric or parabolic orbit about a central star formulated as an initial value problem. This approach enables us to extend previous work of Ivanov & Papaloizou to consider planet models with solid cores and to avoid making an anelastic approximation. We obtain a power spectrum of the tidal response of coreless models which enables global inertial modes to be identified. Their frequencies are found to be in good agreement with those obtained using either a WKBJ approach or the anelastic spectral approach adopted in previous work for small planet rotation rates. We also find that the dependence of the normal mode frequencies on the planet angular velocity in case of higher rotation rates can for the most part be understood by applying first order perturbation theory to the anelastic modes. We calculate the energy and angular momentum exchanged as a result of the tidal encounter and for coreless models again find good agreement with results obtained using either the anelastic spectral method. Models with a solid core showed evidence of the emission of shear layers at critical latitudes and possibly wave attractors after the encounter but the total energy exchanged during the encounter did not differ dramatically from the coreless case as long as the ratio of the core radius to the total radius was less than 50%, there being hardly any difference at all when this ratio was less than 25% of the total radius. We give a physical and mathematical interpretation of this result. Finally we are able to validate the use of the anelastic approximation for both the work presented here and our previous work which led to estimates of circularisation rates for planets in highly eccentric orbits.