# Rotation Effects

The oscillation equations and boundary conditions laid out in the
Dimensionless Formulation section are formulated for a non-rotating
star. Solving the corresponding equations for a rotating star is a
challenging task, and a complete treatment lies beyond the scope of
**gyre**. However, **gyre** does include two important
modifications arising from rotation.

## Doppler Shift

The lowest-order effect of rotation appears in the Doppler shift that arises when transforming between the inertial reference frame and the local co-rotating reference frame. To incorporate this effect in the oscillation equations, all instances of the inertial-frame frequency \(\sigma\) are replaced by the co-rotating frequency

where \(m\) is the azimuthal order of the mode and \(\Omega\) is the rotation angular frequency. Because GYRE assumes so-called shellular rotation (see, e.g., Meynet & Maeder, 1997), both \(\Omega\) and \(\sigmac\) are functions of radial coordinate \(r\).

## Coriolis Force

### The Traditional Approximation of Rotation

Higher-order effects of rotation arise through the Coriolis force,
which appears in the linearized momentum equation to correct for the
non-inertial nature of the co-rotating reference
frame. **gyre** incorporates an approximate treatment of the
Coriolis force based on the traditional approximation of rotation
(TAR), which was first introduced by Eckart (1960; Hydrodynamics of
Oceans and Atmospheres) and has since then been used extensively
within the pulsation community (see, e.g.,
Bildsten et al., 1996; Lee & Saio, 1997;
Townsend, 2003a; Bouabid et al., 2013;
Townsend, 2020).

Within the TAR, the solution forms given in eqn. (7) are replaced by

(cf. equations 1-3 of Townsend, 2020). Here, the Hough functions \(\houghr\), \(\hought\) and \(\houghp\) are the eigenfunctions obtained by solving Laplace’s tidal equations (TEs), a second-order system of differential equations and boundary conditions in the polar (\(\theta\)) coordinate. Together with the associated eigenvalue \(\lambda\), they depend on the harmonic degree \(\ell\)1 and azimuthal order \(m\), and the spin parameter

### Solution Families

Solutions to the TEs can be grouped into two families based on the behavior of the eigenfunctions and eigenvalue in the limit \(\Omega \rightarrow 0\). For the gravito-acoustic family,

and \(\lambda \rightarrow \ell(\ell+1)\). With these expressions, the solution forms (10) reduce to those given in eqn. (7).

Conversely, for the Rossby family

and \(\lambda \rightarrow 0\). Moreover, Rossby-mode eigenfrequencies also show the limiting behavior

which is independent of the stellar structure.

### Incorporating the TAR

To incorporate the TAR in the oscillation equations, all instances of the term \(\ell(\ell+1)\) are replaced by the TE eigenvalue \(\lambda\). Then, all instances of the harmonic degree \(\ell\) are replaced by \(\elli\), an effective harmonic degree found by solving

at the inner boundary (remember, because \(\sigmac\) is a function of radial coordinate, so too are \(q\) and \(\lambda\)).

Footnotes