Benchmark: Love numbers

Reference: Spada et al. (2011), the Charles University GIA benchmark, earth model M3–L70–V01 (Spada et al. 2011). Test: test_benchmark_love (in make check). Status: ✅ elastic and fluid \(h,\ell,k\) match to \(\sim 0.1\%\) at every degree 2–256.

This benchmark tests the per-degree radial solver in isolation: for a unit surface load at degree \(j\), do the elastic (instantaneous) and fluid (fully relaxed) loading Love numbers match the reference?

Earth model

M3–L70–V01 — incompressible, self-gravitating, layered Maxwell:

  • a 70 km elastic lithosphere (\(\eta\to\infty\)),
  • three viscous mantle layers,
  • an inviscid fluid core (\(\mu=0\)).

The layer table (radius, density, shear modulus, viscosity) is the incompressible model of the Spada et al. (2011) benchmark. FastEarth3D’s fe_earth_structure reproduces the benchmark’s interface gravities, total mass, and moment of inertia as an independent check before any Love number is computed.

Definitions

For a degree-\(j\) surface load of coefficient \(\sigma\), with the load’s own surface potential \(\varphi^{L} = 4\pi G\,a\,\sigma/(2j+1)\), the loading Love numbers are extracted from the surface radial functions (Farrell 1972 normalization, (Farrell 1972)):

\[ h_j = \frac{g_0(a)\,U(a)}{\varphi^{L}}, \qquad \ell_j = \frac{g_0(a)\,V(a)}{\varphi^{L}}, \qquad k_j = -\frac{F(a)}{\varphi^{L}} - 1. \]

The \(-1\) in \(k_j\) follows from Martinec’s \(\varphi_1 = F\) being the total perturbation potential, which carries the load’s direct potential with the opposite sign to \(\varphi^{L}\) (\(F\to-\varphi^{L}\) for a rigid sphere).

Analytic anchors

Two limits are exact and are checked first (test_love, to \(\sim 10^{-5}\)):

  • Fluid (\(\mu\to 0\), homogeneous): \(h_j\to-(2j+1)/3\), \(k_j\to-1\).
  • Rigid (\(\mu\to\infty\)): \(h_j,\ell_j,k_j\to 0\).

The fluid limit independently exercises self-gravity, incompressibility, the Poisson coupling, and the load forcing; the rigid limit checks the shear block and the \(F\) sign.

Result

The elastic and fluid \(h,\ell,k\) match the benchmark table to \(\sim 0.1\%\) (the P1 discretization level) at every degree from 2 to 256. The fluid limit was always exact (\(<0.5\%\)); the elastic numbers are the discriminating test. Correct discretization of the self-gravity \(U\)–\(F\) block is essential for the low-degree elastic numbers — see the typo in Martinec (2000) eq. (81). TABOO (NV=3, CODE=7) served as an independent reference to regenerate the M3–L70–V01 table.

Figures

Figures to be added: \(h_j,\ell_j,k_j\) versus degree (elastic and fluid), FastEarth3D vs. the M3–L70–V01 table; relative-error spectra.

Provenance

Reference table: data/benchmarks/love_M3-L70-V01/mod_M3-L70-V01, from the giapy package (MIT) [S. B. Kachuck], originating in the Charles University GIA benchmark; independently regenerable with TABOO (GPLv3). Full details in data/benchmarks/PROVENANCE.md.

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References

Farrell, W. E. 1972. “Deformation of the Earth by Surface Loads.” Reviews of Geophysics 10 (3): 761–97. https://doi.org/10.1029/RG010i003p00761.
Martinec, Z. 2000. “Spectral–Finite Element Approach to Three-Dimensional Viscoelastic Relaxation in a Spherical Earth.” Geophysical Journal International 142 (1): 117–41. https://doi.org/10.1046/j.1365-246x.2000.00138.x.
Spada, G., V. R. Barletta, V. Klemann, et al. 2011. “A Benchmark Study for Glacial Isostatic Adjustment Codes.” Geophysical Journal International 185 (1): 106–32. https://doi.org/10.1111/j.1365-246X.2011.04952.x.