Reference (4)

Liquid-Phase

When the melt fraction of a magma layer exceeds a critical value $F_m \gtrsim F_{m,c}$, its response to tidal forcing becomes fluid-like rather than solid-like. In this regime, tidal deformation is governed by the linearized Laplace tidal equations (LTEs), which describe the dynamics of a thin, global fluid shell of uniform thickness $H$ on a planet of radius $R_p$.

Neglecting nonlinear advection and mean flows, momentum and mass conservation for the magma ocean read

\[\partial_t \vec{u} + \sigma_R \vec{u} + \vec{f} \times \vec{u} + g \nabla \zeta = g \nabla \zeta_{\mathrm{eq}}, \qquad \partial_t \zeta + \nabla \cdot (H \vec{u}) = 0,\]

where $\vec{u}$ is the horizontal velocity, $\zeta$ the tidally induced surface displacement, $\zeta_{\mathrm{eq}} = U_T/g$ the equilibrium tide associated with the tidal potential $U_T$, $g$ the surface gravity, and $\sigma_R$ a Rayleigh drag frequency parameterizing dissipation in the magma ocean.

We focus on the strongly damped, highly viscous limit ($\sigma_R \gg 2\Omega$), appropriate for magma oceans, in which dissipative processes dominate over rotational effects. In this high-Ekman-number regime, the Coriolis term is negligible and the tidal response becomes overdamped. Dissipation is encapsulated by $\sigma_R$, which should be interpreted as an effective damping rate accounting for viscous resistance, boundary-layer friction, form drag, and porous-flow dissipation near the rheological transition.

Transforming the governing equations to the frequency domain and expanding tidal quantities in spherical harmonics yields a closed-form expression for the complex, frequency-dependent tidal Love number $k_\ell(\sigma)$.

\[k_{\ell}(\sigma) = -\frac{\varrho_\ell \bar{\sigma}_\ell^2}{\sigma\tilde{\sigma} - \bar{\sigma}_\ell^2},\]

where $\varrho_\ell = 3/(2n+1)(\rho_f/\rho_b)$ is the degree-$\ell$ density ratio between the fluid and the bulk part of the planet. The characteristic frequency is given by

\[\bar{\sigma}_\ell = \sqrt{\frac{\mu_\ell g H_\mathrm{magma}}{R^2}},\]

where $H_\mathrm{magma}$ the ocean depth, $R$ the radius of the planet, and with

\[\mu_\ell = \ell(\ell+1),\]

the eigenvalue of the nth degree spherical harmonic, and

\[\tilde{\sigma} = \sigma - i \sigma_R.\]