Tidal Effects

To simulate the effects of tidal forcing by a companion, the gyre_tides frontend solves a modified form of the linearized momentum equation (6), namely

\[\rho \pderiv{\vv'}{t} = - \nabla P' - \rho' \nabla P - \rho \nabla \Phi' - \rho \nabla \PhiT.\]

The final term on the right-hand side represents the external force arising from the tidal gravitational potential \(\PhiT\).

Tidal Potential

The tidal potential can be expressed via the superposition

(27)\[\PhiT = \sum_{\ell=2}^{\infty} \sum_{m=-\ell}^{\ell} \sum_{k=-\infty}^{\infty} \PhiTlmk.\]

of partial tidal potentials defined by

(28)\[\PhiTlmk \equiv - \epsT \, \frac{G\Mstar}{\Rstar} \, \cbar_{\ell,m,k} \left( \frac{r}{\Rstar} \right)^{\ell} Y^{m}_{\ell}(\theta, \phi) \, \exp(- \ii k \Oorb t).\]

(the summation over \(\ell\) and \(m\) comes from a multipolar space expansion of the potential, and the summation over \(k\) from a Fourier time expansion). Here,

\[\epsT = \left( \frac{\Rstar}{a} \right)^{3} q_{2} = \frac{\Oorb^{2} \Rstar^{3}}{G\Mstar} \frac{q_{2}}{1+q_{2}}\]

quantifies the overall strength of the tidal forcing, in terms of the companion’s mass \(q_{2} \Mstar\), semi-major axis \(a\) and orbital angular frequency \(\Oorb\). These expressions, and the definition of the tidal expansion coefficients \(\cbar_{\ell,m,k}\), are presented in greater detail in Sun et al. (2023).

Tidal Equations

Because the tidal potential (27) superposes many different spherical harmonics, the solution forms given in equations (8) must be replaced by

(29)\[\begin{split}\begin{aligned} f'(r,\theta,\phi;t) &= \sum_{\ell,m,k} \tflmk'(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t) \\ \va'(r,\theta,\phi;t) &= \sum_{\ell,m,k} \left[ \tarlmk'(r) \, \ver + \tahlmk'(r) \, r \, \nablah \right] Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t), \end{aligned}\end{split}\]

and those in equation (10) by

(30)\[\vxi(r,\theta,\phi;t) = \sum_{\ell,m,k} \left[ \txirlmk(r) \, \ver + \txihlmk(r) \, r \, \nablah \right] Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii \sigma t)\]

(the notation for the sums has been abbreviated). Substituting these solution forms into the linearized equations (with the additional \(\rho \nabla \PhiT\) term, as above), and taking advantage of the orthonormality of the spherical harmonics, leads to a fully separated set of differential equations for each combination of \(\ell\), \(m\) and \(k\). A given set resembles the regular separated equations, but with the radial momentum equation (11) replaced by

\[-\sigmac^{2} \rho \txir = - \deriv{\tP'}{r} - \trho' \deriv{\Phi}{r} - \rho \deriv{\tPhi'}{r} - \rho \deriv{\tPhiT}{r}\]

and the horizontal momentum equation (12) by

\[-\sigmac^{2} \rho \txih = - \frac{1}{r} \left( - \tP' - \rho \tPhi' - \rho \tPhiT \right).\]

(in the interests of brevity, the \(\ell,m,k\) are now dropped). Here,

\[\tPhiT = - \epsT \, \frac{G\Mstar}{\Rstar} \, \cbar_{\ell,m,k} \left( \frac{r}{\Rstar} \right)^{\ell}\]

describes the radial dependence of the partial tidal potential defined by eqn. (28), while

\[\sigmac = k \Oorb - m \Orot\]

is the forcing frequency associated with the potential, corrected for the Doppler shift due to rotation.

In the dimensionless oscillation equations, the tidal modifications consist of additional inhomogeneous terms on the right-hand sides of the differential equations for \(y_1\), \(y_2\), \(y_5\) and \(y_6\), as follows:

\[\begin{split}\begin{aligned} x \deriv{y_{1}}{x} &= [ \ldots ] + \frac{\lambda}{c_{1} \omegac^{2}} \yT \\ x \deriv{y_{2}}{x} &= [ \ldots ] - \ell \yT \\ x \deriv{y_{5}}{x} &= [ \ldots ] + \frac{V}{\frht} \left[ \frac{\lambda}{c_{1} \omegac^{2}} (\nabad - \nabla) + \ell \nabad \right] \yT \\ x \deriv{y_{6}}{x} &= [ \ldots ] + \left[ \lambda \crad \frac{3 + \dcrad}{c_{1}\omegac^{2}} \right] \yT \end{aligned}\end{split}\]

where \([\ldots]\) represents the right-hand side in the absence of tidal forcing, and

\[\yT \equiv x^{2-\ell}\, \frac{\tPhiT}{gr} = - \epsT \, c_{1} \cbar_{\ell,m,k}.\]

(The differential equations for \(y_{3}\) and \(y_{4}\) remain unchanged). Moreover, the first of the regularity-enforcing equations (18) becomes

\[c_{1} \omegac^{2} y_{1} - \ell y_{2} - \alphagrv \ell y_{3} = \ell \yT.\]

In these expressions,

\[\omegac = \alphafrq \sigmac \sqrt{\frac{\Rstar^{3}}{G\Mstar}}\]

is the dimensionless tidal forcing frequency in the co-rotating reference frame. The \(\alphafrq\) term is included here to allow the forcing frequency to be adjusted independently of \(m\) and \(k\) (see the alpha_frq option). All occurrences of \(\omega\) in the other dimensionless equations are likewise replaced by \(\omegac\).