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Benchmark: low-viscosity zone (lateral viscosity)

Reference: Weerdesteijn et al. (2023), section 5.2, the short-timescale low-viscosity-region case computed with ASPECT, Abaqus and TABOO (Weerdesteijn et al. 2023). Test: test_benchmark_lvz (in make check-slow). Asserted: vertical displacement at the load center after 200 yr within 8 % of the published values, without and with the low-viscosity zone. Measured (ifx, albedo): 2.6 % and 1.0 %.

This benchmark tests the laterally varying viscosity against published results of three-dimensional finite-element codes. It is the only test of the 3-D path against codes other than VILMA1.

Load and setup

  • Earth model M3–L70–V01.
  • An axisymmetric ice disc of 100 km radius (\(\rho_\text{ice} = 931\ \mathrm{kg\,m^{-3}}\)). Its height grows linearly from 0 to 100 m over 100 yr and is then held to 200 yr.
  • A cylindrical low-viscosity zone (LVZ) under the load: radius 100 km, depth 70–170 km, viscosity \(10^{19}\) Pa s in an upper mantle of \(10^{21}\) Pa s. The shear modulus is unchanged.
  • Two runs: the homogeneous model (no LVZ) and the model with the LVZ.

The load and the LVZ are axisymmetric, so the problem has order \(m = 0\) only. The test runs with mmax = 0 at \(\ell_\max = 512\) (a shortest half-wavelength of about 40 km, against the 100 km radius of the disc) with a step of 2.5 yr. It drives the viscoelastic response with the ice load directly. There is no ocean and no sea-level equation.

Reference data and metrics

The reference values are the vertical surface displacement at the load center at \(t = 200\) yr: \(-0.75\) m for the homogeneous model and \(-1.23\) m with the LVZ. The published codes differ among themselves. ASPECT and Abaqus agree to 1–3 %. ASPECT, which has no self-gravity, and TABOO, which has, agree to 0.3 % at the load center of the homogeneous model. Self-gravity is thus a sub-percent effect in this case, and the published numbers are a valid reference for a self-gravitating spherical model to within a few percent. The tolerance of the test is 8 %.

Result

The numbers below are for model 1790f86 built with ifx on albedo.

Table 1: Vertical displacement at the load center at 200 yr.
Case VILMA2 Reference Difference Tolerance
homogeneous \(-0.731\) m \(-0.75\) m 2.6 % 8 %
with LVZ \(-1.218\) m \(-1.23\) m 1.0 % 8 %
amplification by the LVZ 1.67 1.64 1.6 % not asserted

VILMA2 reproduces the displacement with the LVZ to 1 % and the homogeneous displacement to 2.6 %. Both differences are within the 1–3 % spread of the published codes. The low-viscosity zone increases the displacement by a factor of 1.67, against 1.64 in the reference.

The scope of this benchmark is limited. It compares one number per run, the displacement at the load center, on a time scale of 200 yr. It has a single order (\(m = 0\)), so it does not exercise the coupling between orders or the toroidal flow, which vanishes for an axisymmetric problem. The rotational-invariance test test_rotinv complements it: the same kind of load and LVZ placed away from the pole gives the peak displacement of the on-pole case to 0.5 % at \(\ell_\max = 128\) (tolerance 1 %, in make check-slow). A comparison of the 3-D path at glacial time scales exists only against VILMA1.

Figures

Figure 1: Low-viscosity-zone benchmark of Weerdesteijn et al. (2023): vertical displacement at the load center against time and the displacement profile at 200 yr, for the homogeneous model and the model with the low-viscosity zone, with the published values at the load center.

With VILMA_BENCH_DUMP set, the test writes the center displacement against time and the profile at 200 yr of both runs (lvz_homog_series.txt, lvz_homog_profile.txt, lvz_lvz_series.txt, lvz_lvz_profile.txt); see Benchmarks.

Provenance

The reference values are quoted from Weerdesteijn et al. (2023) (their section 5.2 and Table 4). No reference data file is needed.

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References

Weerdesteijn, M. F. M., J. B. Naliboff, C. P. Conrad, et al. 2023. “Modeling Viscoelastic Solid Earth Deformation Due to Ice Age and Contemporary Glacial Mass Changes in ASPECT.” Geochemistry, Geophysics, Geosystems 24 (3): e2022GC010813. https://doi.org/10.1029/2022GC010813.