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Appendix A: Dynamical equations for tidal flows in convective envelopes

The solutions of the system of dynamical equations for the envelope written in the co-rotating frame are separated into a non-wavelike part (with subscripts _nw), which corresponds to the immediate hydrostatic adjustment to the external tidal potential (U), and a wavelike part (with subscript _w) driven by the action of the Coriolis acceleration on the non-wavelike part, (A.1)

where s is the displacement, e_z the unit vector along the rotation axis, h the specific enthalpy, and Φ is the self-gravitational potential of A. Primed variables denote an Eulerian perturbation in relation to the unperturbed state with unprimed variables. We note that U being of the first order in tidal amplitude, it appears in perturbation equations. Finally, W ≡ W_nw + W_w = h′ + Φ′ + U, while is the acceleration driving the wavelike part of the solution.

© ESO, 2015