Reference (6)

Surface Loading

Total $k_{2}$

To calculate the response of a planet to various external forcings, we define general boundary conditions at the surface ($r = R$). These conditions account for external gravitational potentials, mass loading, tangential tractions, and surface pressure.

General Boundary Conditions

For a given degree $n$, the boundary conditions for the state vector components $y_i$ are summarized in the following table:

No $y_5$ term appears in the third row: in Obliqua's convention $y_6$ is already Takeuchi & Saito's (1972) combined "potential stress" variable, whose own radial equation has no coupling back to $y_5$, so the surface condition is a direct statement about $y_6(R)$ alone (see Solid-Phase for the underlying motion matrix).

By expressing a surface mass load $\zeta_n$ as an equivalent external potential $U'$, where $\zeta_n = \frac{2n + 1}{4 \pi G R} U'_n$, the system simplifies to:

\[\begin{aligned} y_{3}(R) &= - \frac{(2n + 1)g_e}{4 \pi G R} \left[\frac{G}{R} U'_n \right] - P_n \\ y_{4}(R) &= \tau_n \\ y_{6}(R) &= \frac{2n+1}{R} \left(U_n + \left[\frac{G}{R} U'_n \right] \right) \end{aligned}\]

Note that Obliqua's actual get_surface_bc! (src/common.jl) does not literally apply this $\zeta_n \leftrightarrow U'_n$ conversion; it sets $(U,U',\tau,P)$ directly as dimensionless $0$/$1$ selector flags, which for the load case numerically works out to $y_3(R) = -(2n+1)g(R)/(4\pi R^2)$ and $y_6(R) = (2n+1)G/R^2$ — see Solid-Phase - solid1d for the concrete tidal/load values the code actually produces.

Calculation of Love Numbers

In Obliqua, Love numbers are non-dimensionalized by setting the forcing terms to either $1$ (present) or $0$ (absent).

• Tidal Love Number (TLN): Calculated by setting $(U, U', \tau, P) = (1, 0, 0, 0)$.
• Load Love Number (LLN): Calculated by setting $(U, U', \tau, P) = (0, 1, 0, 0)$.

Setting $U=1$ allows us to determine the intrinsic response of the planet; the final physical solution is obtained by scaling these non-dimensional Love numbers by the actual tidal potential $U_{n,m,k}$ of the system.

Global Potential and Layer Coupling

The total distortion potential of the planet, $U_n^D$, can be constructed by combining the responses to different forcings:

\[U_n^D = k_n^T + (1+k_n^L)U_n^L + K_n^P U_n^P\]

Where:

• $U_n^L$: Potential due to surface mass-loading (e.g., a magma ocean or ice sheet).
• $U_n^P$: Potential due to surface pressure (e.g., atmospheric loading).

Current Implementation

Currently, Obliqua couples the solid and fluid responses to calculate a Global Tidal Love Number $k_n$ using the following recursive form:

\[k_n = k^T_n + (1 + k^L_n) k_n^{T, \text{(fluid)}}\]

In this formulation:

1. $k^T_n$ and $k^L_n$ are the tidal and load Love numbers of the solid mantle only.
2. $k_n^{T, \text{(fluid)}}$ represents the response of the fluid layer.
3. The term $(1 + k^L_n)$ accounts for how the solid interior deforms under the weight of the fluid's own tidal response.

While additional corrections for atmospheric pressure or specialized crustal rheologies (Andrade/Maxwell) are not yet active, the framework is designed to incorporate these by calculating the respective pressure and loading Love numbers which are already implemented in the solver.

Function Documentation

get_surface_bc!(R, g, n, U, U_prime, tau, P; G0=1, Y=[1,2,3,4,5,6])