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

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

of partial tidal potentials defined by

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

(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,

\[\epsT = \left( \frac{R}{a} \right)^{3} = \frac{\Oorb R^{3}}{GM} \frac{q}{1+q}\]

quantifies the overall strength of the tidal forcing, in terms of the companion’s mass \(q M\), 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).

Separated Equations

Because the tidal potential (19) superposes many different spherical harmonics, the solution forms (8) must be replaced by the more-general expressions

(20)\[\begin{split}\begin{aligned} \xir(r,\theta,\phi;t) &= \sum_{\ell,m,k} \txirlmk(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t), \\ \xit(r,\theta,\phi;t) &= \sum_{\ell,m,k} \txihlmk(r) \, \pderiv{}{\theta} Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t), \\ \xip(r,\theta,\phi;t) &= \sum_{\ell,m,k} \txihlmk(r) \, \frac{\ii m}{\sin\theta} Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t), \\ f'(r,\theta,\phi;t) &= \sum_{\ell,m,k} \tflmk'(r) \, Y^{m}_{\ell}(\theta,\phi) \, \exp(-\ii k \Oorb t) \end{aligned}\end{split}\]

(the notation for the sums has been abbreviated). Substituting these solution forms into the linearized equations, 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, with just a couple changes:

  • The perturbation \(\tPhi'\) is replaced by \(\tPsi' \equiv \tPhi' + \tPhiT\), representing the total (self + tidal) gravitational potential perturbation.

  • Rather than being an eigenvalue parameter, the oscillation frequency is set by \(\sigma = k \Oorb\), representing the forcing frequency of the partial tidal potential in an inertial frame.

The latter change means that the dimensionless frequency (13) becomes

\[\omega = \alphafrq \, k \Oorb \sqrt{\frac{R^{3}}{GM}},\]

where \(\alphafrq\) is an additional term introduced to allow tuning of the tidal forcing frequency (see the alpha_frq parameter in the Tidal Parameters section).

Boundary Conditions

The boundary conditions accompanying the separated equations for a given \(\{\ell,m,k\}\) combination resemble those presented previously, except that the outer potential boundary condition becomes

\[(\ell + 1) \tPsi' + r \deriv{\tPsi'}{r} = (2 \ell + 1) \tPhiTlmk,\]


(21)\[\tPhiTlmk \equiv - \epsT \, \frac{GM}{R} \, \cbar_{\ell,m,k} \left( \frac{r}{R} \right)^{\ell}.\]

describes the radial dependence of the partial tidal potential.