.. _osc-rot: 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 :ads_citealp:`unno: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. .. _osc-rot-doppler: 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 :ref:`separated equations `, all instances of the inertial-frame frequency :math:`\sigma` are replaced by the co-rotating frequency .. math:: :label: e:sigmac \sigmac \equiv \sigma - m \Orot, where :math:`m` is the azimuthal order of the mode and :math:`\Orot` is the rotation angular frequency. GYRE assumes shellular rotation (see, e.g., :ads_citealp:`meynet:1997`), and so the latter can in principle be a function of radial coordinate :math:`r`. The corresponding modifications to the :ref:`dimensionless formulation ` involve replacing the dimensionless inertial-frame frequency :math:`\omega` with the dimensionless co-rotating frequency .. math:: :label: e:omegac \omegac \equiv \omega - m \Orot \sqrt{\frac{\Rstar^{3}}{G\Mstar}}. .. _osc-rot-coriolis-p: 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 :ads_citealp:`unno:1989`). To first order in :math:`\Orot`, the frequency of a mode is shifted by the amount .. math:: \Delta \sigma = m \int_{0}^{\Rstar} \Orot \, \deriv{\beta}{r} \diff{r}, where the rotation splitting kernel is .. math:: \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 :math:`\txir` and :math:`\txih` are evaluated from solutions to the oscillation equations without rotation. Therefore, the expression above for :math:`\Delta \sigma` can be applied as a post-calculation correction to non-rotating eigenfrequencies. .. _osc-rot-coriolis-np: Non-Perturbative Coriolis Force Treatment ----------------------------------------- The perturbation expansion technique above breaks down when :math:`\Orot/\sigmac \gtrsim 1`. To deal with such cases, the :program:`gyre` frontend\ [#gyre-tides]_ 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., :ads_citealp:`bildsten:1996`; :ads_citealp:`lee:1997`; :ads_citealp:`townsend:2003a`; :ads_citealp:`bouabid:2013`; :ads_citealp:`townsend:2020`). Within the TAR, the solution forms given in equation (:eq:`e:osc-sol-forms`) are replaced by .. math:: :label: e:osc-sol-forms-hough \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} Here, the Hough functions :math:`\houghr`, :math:`\hought` and :math:`\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 (:math:`\theta`) coordinate (see :ads_citealt:`townsend:2020`). Together with their associated eigenvalue :math:`\lambda`, they depend on the harmonic degree :math:`\ell`\ [#harmonic-deg]_, azimuthal order :math:`m`, and spin parameter .. math:: q \equiv \frac{2 \Orot}{\sigmac}. .. _osc-rot-solfam: Solution Families ^^^^^^^^^^^^^^^^^ Solutions to the TEs can be grouped into two families based on the behavior of the eigenfunctions and eigenvalue in the limit :math:`\Orot \rightarrow 0`. For the gravito-acoustic family, .. math:: :label: e:hough-lim-ga \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, and :math:`\lambda \rightarrow \ell(\ell+1)`. With these expressions, the solution forms (:eq:`e:osc-sol-forms-hough`) reduce to those given in equation (:eq:`e:osc-sol-forms`). Conversely, for the Rossby family .. math:: :label: e:hough-lim-ross \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, and :math:`\lambda \rightarrow 0`. The solution forms (:eq:`e:osc-sol-forms-hough`) then reduce to toroidal-mode solutions of the oscillation equations, as described in section 13.3 of :ads_citet:`unno:1989`. In the approach to the non-rotating limit, the frequencies of Rossby modes behave as .. math:: :label: e:ross-freq \sigmac \approx \frac{2 m \Orot}{\ell(\ell+1)} + \mathcal{O}(\Orot^2) :ads_citep:`saio:1980`, meaning that the spin parameter :math:`q` remains finite. Implementing the TAR ^^^^^^^^^^^^^^^^^^^^ To implement the TAR in the :ref:`separated equations ` the angular eigenvalue :math:`\lambda` is evaluated from the TE rather than using equation (:eq:`e:lambda-norot`). Moreover, all instances of the harmonic degree :math:`\ell` in the :ref:`boundary conditions ` are replaced by :math:`\elle`, an effective harmonic degree found as the positive root of the equation .. math:: \elle(\elle+1) = \lambda. Similar steps are taken in the :ref:`dimensionless formulation `, but in the definitions of the dependent variables :math:`\{y_{1},y_{2},\ldots,y_{6}\}`, :math:`\ell` is replaced by :math:`\elli`, the effective harmonic degree evaluated at the inner boundary. .. rubric:: Footnotes .. [#gyre-tides] Currently the TAR cannot be used with the :program:`gyre_tides` frontend, because it doesn't play well with forcing by the tidal potential :math:`\PhiT`. .. [#harmonic-deg] 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 :math:`\Orot \rightarrow 0`, as given in eqns. (:eq:`e:hough-lim-ga`) and (:eq:`e:hough-lim-ross`).