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:
| State Function | External Potential ($U$) | Mass Load ($\zeta_n$) | Traction ($\tau$) | Pressure ($P$) |
|---|---|---|---|---|
| $y_3(R)$ (Normal Stress) | $0$ | $-g_e \zeta_n$ | $0$ | $-P_n$ |
| $y_4(R)$ (Tangential Stress) | $0$ | $0$ | $\tau_n$ | $0$ |
| $\frac{n+1}{R} y_5(R) + y_6(R)$ | $\frac{2n+1}{R} U_n$ | $4\pi G \zeta_n$ | $0$ | $0$ |
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:
\[\begin{aligned} y_{3}(R) &= - \frac{(2n + 1)g_e}{4 \pi G R} U'_n - P_n \\ y_{4}(R) &= \tau_n \\ \frac{n+1}{R} y_5(R) + y_6(R) &= \frac{2n+1}{R} (U_n + U'_n) \end{aligned}\]
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:
- $k^T_n$ and $k^L_n$ are the tidal and load Love numbers of the solid mantle only.
- $k_n^{T, \text{(fluid)}}$ represents the response of the fluid layer.
- 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.