NONADIABATIC TIDAL FORCING OF A MASSIVE, UNIFORMLY ROTATING STAR

We study the fully non-adiabatic tidal response of a uniformly rotating 20-M. ZAMS star to the dominant I = m = 2 component of the companion's perturbing potential. This is done numerically with a 2D implicit finite difference scheme. We assume the star is rotating slowly with angular speed Omega(s) much less than Omega(c), so that the centrifugal force can be neglected, but we take the Coriolis force fully into account. It is found that the l = m = 2 forcing can be resonant not only with predominantly l = 2 gravity modes but, as expected, also with gravity modes with predominantly I = 4, 2 = 6, etc. because of the rotational coupling between different I-components. We have used our results for the non-adiabatic response to calculate the tidal spin-up rate of the slowly rotating massive star. Because of the additional resonances the tidal spin-up rate of a rotating star is a considerably more erratic function of orbital frequency than that of a non-rotating star. We compare the rotational frequency shift of resonances with m = 2 modified g-modes with the values obtained from first-order perturbation theory. By extrapolating our numerical results to low rotation speeds we obtain frequency shifts consistent with the first-order approximation. However, even for the moderately small rotation speeds considered in this paper, the calculated frequency shifts deviate substantially from the values predicted by first-order perturbation theory. In the inertial regime, in which the relative forcing frequency is less than 2 Omega(s), it is found that the response contains large-amplitude, very short-wavelength components which cannot be resolved on the numerical grid (effectively 400 x 256 for the full meridional cross-section), unless the so-called 'traditional approximation' is used, in which the theta-component of the rotational angular velocity is ignored. Then tidal resonances with rotationally modified gravity modes continue into the inertial regime. Outside the inertial regime the traditional approximation gives results similar to the full code calculations. Inside the inertial regime this is only true for the strongly stratified layers where the Brunt-Vaisalla frequency is larger than about three times the stellar break-up speed Omega(c). We find evidence that the singular response in the inertial regime, obtained with the full code, may be real and due to resonant excitation of rotationally controlled inertial modes in the convective core and the adjacent, weakly stratified, radiative layers. Because the spectrum of the rotationally controlled inertial modes is dense, resonant excitation of these modes may give rise to significant tidal effects. Further progress requires higher resolution calculations incorporating viscosity to deal with the very short-wavelength components in the inertial regime.