Rotation Effects

The oscillation equations presented in the preceding sections are formulated for a non-rotating star. The corresponding equations for a rotating star are significantly more complicated (see Unno et al., 1989 for details), and a complete treatment of rotation lies beyond the scope of GYRE. However, GYRE can include two important effects arising from rotation.

Doppler Shift

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

(16)\[\sigmac \equiv \sigma - m \Orot,\]

where \(m\) is the azimuthal order of the mode and \(\Orot\) is the rotation angular frequency. GYRE assumes shellular rotation (see, e.g., Meynet & Maeder, 1997), and so the latter can in principle be a function of radial coordinate \(r\). The corresponding modifications to the dimensionless formulation involve replacing the dimensionless inertial-frame frequency \(\omega\) with the dimensionless co-rotating frequency

(17)\[\omegac \equiv \omega - m \Orot \sqrt{\frac{\Rstar^{3}}{G\Mstar}}.\]

Perturbative Coriolis Force Treatment

Another lowest-order effect of rotation arises from the Coriolis force. For slow rotation, this effect can be determined through a perturbation expansion technique (see, e.g., section 19.2 of Unno et al., 1989). To first order in \(\Orot\), the frequency of a mode is shifted by the amount

\[\Delta \sigma = m \int_{0}^{\Rstar} \Orot \, \deriv{\beta}{r} \diff{r},\]

where the rotation splitting kernel is

\[\deriv{\beta}{r} = \frac{\left\{ \txir^{2} + [\ell(\ell+1) - 1] \txih^{2} - 2 \txir \txih \right\} \rho r^{2}} {\int_{0}^{\Rstar} \left\{ \txir^{2} + \ell(\ell+1) \txih^{2} \right\} \rho r^{2} \diff{r}}\]

In this latter expression, the eigenfunctions \(\txir\) and \(\txih\) are evaluated from solutions to the oscillation equations without rotation. Therefore, the expression above for \(\Delta \sigma\) can be applied as a post-calculation correction to non-rotating eigenfrequencies.

Non-Perturbative Coriolis Force Treatment

The perturbation expansion technique above breaks down when \(\Orot/\sigmac \gtrsim 1\). To deal with such cases, the gyre frontend[1] can incorporate a non-perturbative treatment of the Coriolis force based on the ‘traditional approximation of rotation’ (TAR). The TAR was first introduced by Eckart (1960; Hydrodynamics of Oceans and Atmospheres) and has since 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 equation (8) are replaced by

(18)\[\begin{split}\begin{aligned} \xir(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txir(r) \, \houghr(\theta) \, \exp(\ii m \phi -\ii \sigma t) \right], \\ \xit(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, \frac{\hought(\theta)}{\sin\theta} \, \exp(\ii m \phi -\ii \sigma t) \right], \\ \xip(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, \frac{\houghp(\theta)}{\ii \sin\theta} \, \exp(\ii m \phi -\ii \sigma t) \right], \\ f'(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \tf'(r) \, \houghr(\theta) \, \exp(\ii m \phi -\ii \sigma t) \right] \end{aligned}\end{split}\]

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 (see Townsend 2020). Together with their associated eigenvalue \(\lambda\), they depend on the harmonic degree \(\ell\)[2], azimuthal order \(m\), and spin parameter

\[q \equiv \frac{2 \Orot}{\sigmac}.\]

Solution Families

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

(19)\[\begin{split}\left. \begin{aligned} \houghr(\theta) \ \rightarrow & \ Y^{m}_{\ell}(\theta,0) \\ \hought(\theta) \ \rightarrow & \ \sin\theta \pderiv{}{\theta} Y^{m}_{\ell}(\theta,0) \\ \houghp(\theta) \ \rightarrow & \ - m Y^{m}_{\ell}(\theta,0) \end{aligned} \right\} \quad \text{as } \Orot \rightarrow 0,\end{split}\]

and \(\lambda \rightarrow \ell(\ell+1)\). With these expressions, the solution forms (18) reduce to those given in equation (8).

Conversely, for the Rossby family

(20)\[\begin{split}\left. \begin{aligned} \houghr(\theta) \ \rightarrow & \ 0 \\ \hought(\theta) \ \rightarrow & \ m Y^{m}_{\ell}(\theta,0) \\ \houghp(\theta) \ \rightarrow & \ - \sin\theta \pderiv{}{\theta} Y^{m}_{\ell}(\theta,0) \end{aligned} \right\} \quad \text{as } \Orot \rightarrow 0,\end{split}\]

and \(\lambda \rightarrow 0\). The solution forms (18) then reduce to toroidal-mode solutions of the oscillation equations, as described in section 13.3 of Unno et al. (1989). In the approach to the non-rotating limit, the frequencies of Rossby modes behave as

(21)\[\sigmac \approx \frac{2 m \Orot}{\ell(\ell+1)} + \mathcal{O}(\Orot^2)\]

(Saio, 1980), meaning that the spin parameter \(q\) remains finite.

Implementing the TAR

To implement the TAR in the separated equations the angular eigenvalue \(\lambda\) is evaluated from the TE rather than using equation (11). Moreover, all instances of the harmonic degree \(\ell\) in the boundary conditions are replaced by \(\elle\), an effective harmonic degree found as the positive root of the equation

\[\elle(\elle+1) = \lambda.\]

Similar steps are taken in the dimensionless formulation, but in the definitions of the dependent variables \(\{y_{1},y_{2},\ldots,y_{6}\}\), \(\ell\) is replaced by \(\elli\), the effective harmonic degree evaluated at the inner boundary.

Footnotes