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, 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

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

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

\[\omegac \equiv \omega - m \Orot \sqrt{\frac{R^{3}}{GM}}.\]

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

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

where the rotation splitting kernel is

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

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 eqn. (7) are replaced by

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

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 the associated eigenvalue \(\lambda\), they depend on the harmonic degree \(\ell\)[2] and azimuthal order \(m\), and the 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,

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

and \(\lambda \rightarrow \ell(\ell+1)\). With these expressions, the solution forms (12) reduce to those given in eqn. (7).

Conversely, for the Rossby family

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

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

(15)\[\sigmac = \frac{2 m \Orot}{\ell(\ell+1)} \quad \text{as } \Orot \rightarrow 0,\]

which is independent of the stellar structure.

Implementing the TAR

To implement the TAR in the separated equations and boundary conditions, 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 \(\elle\), an effective harmonic degree found by solving

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

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

[1]

Currently the TAR cannot be used with the gyre_tides frontend, because it doesn’t play well with forcing by the tidal potential \(\PhiT\).

[2]

The harmonic degree isn’t formally a ‘good’ quantum number in the TAR; however, it can still be used to identify Hough functions by considering their behavior in the limit \(\Orot \rightarrow 0\), as given in eqns. (13) and (14).