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

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

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 (16) superposes many different spherical harmonics, the solution forms (7) must be replaced by the more-general expressions

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

(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 (10) 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

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

where

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

describes the radial dependence of the partial tidal potential.