Benchmark: disc-load response

Reference: Spada et al. (2011), disc-load test, earth model M3–L70–V01 (Spada et al. 2011). Test: test_benchmark_disc (in make check). Status: ✅ displacement \(\sim 1\%\) near-field; geoid \(\sim 1\%\) for \(\ell\ge 2\); transient to \(\sim 0.5\%\).

This benchmark integrates the per-degree response over a spatial load and adds the time relaxation, testing the spatial synthesis and the Maxwell stepper together against a spatial reference.

Load and setup

  • A 10° radius disc of 1000 m ice (\(\rho_{\text{ice}} = 931\ \mathrm{kg\,m^{-3}}\)) on elevated bedrock — no ocean load.
  • Earth model M3–L70–V01 (as for the Love numbers).
  • The disc is expanded in Legendre coefficients \(\sigma_n = \tfrac{\sigma_0}{2}\,[P_{n-1}-P_{n+1}]\) and the spatial response is synthesized as \(u(\theta) = \sum_n U_n(a)\,\sigma_n\,P_n(\cos\theta)\).

Reference data and metrics

The reference is the spatial response at colatitudes \(\theta = 0\):\(0.1\):\(20°\) (201 points) and times \(t = [0,1,2,5,10,100]\) kyr (column 1, \(t=0\), is the elastic response). Three fields are compared: vertical displacement \(u\), geoid \(n\), and uplift rate \(\mathrm{d}u/\mathrm{d}t\).

The reference uses a mixed reference frame: displacement in a CE-like gauge (\(h_1\approx 0\), geocenter), geoid in the CM frame (\(N_1 = 0\)). FastEarth3D reproduces both by referencing the degree-1 geoid to CM (fe_response).

All three reference files in this repository (u_*, n_*, dudt_*) are stored in ascending \(\theta\) order and match the ordering used by the model, so they can be paired with the \(\theta\) grid directly.

Result

  • Elastic uplift (\(t=0\)): centre \(-19.81\) m vs. reference \(-20.57\) m (\(3.7\%\), dominated by the Gibbs ripple at the sharp 10° disc edge); the profile is \(<2\%\) across \(\theta=1\)–\(9°\).
  • Viscoelastic transient: the centre uplift matches to \(\sim 0.5\%\) at \(t=0,1,2,5,10\) kyr (ratio 0.997–1.005).
  • Geoid: matches to \(\sim 1\%\) for degrees \(\ge 2\) (centre 32.57 vs. 33.05; far field 2.88 vs. 2.92).
  • The far-field forebulge (\(\theta\gtrsim 12°\)) is small-amplitude and low-degree-truncation sensitive, so it shows larger relative differences.

Figures

Figures to be added: \(u(\theta)\) at \(t=0,1,2,5,10,100\) kyr; geoid \(n(\theta)\); centre uplift time series; FastEarth3D vs. Spada disc reference.

Provenance

Reference: data/benchmarks/disc_spada2011/{u,n,dudt}_disc.txt, from the isostasy_data repository [J. Jereczek], model_outputs/Spada-2011/, originating in the Spada et al. (2011) benchmark. Full details in data/benchmarks/PROVENANCE.md.

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References

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.