References

Primary method

The spectral–finite-element, time-domain method and its rotational extension:

  • Martinec (2000) — the spectral–finite-element approach to 3D viscoelastic relaxation in a spherical Earth. The core method.
  • Martinec and Hagedoorn (2014) — time-domain rotational feedback.
  • Kendall et al. (2005) — the sea-level equation with moving shorelines (pseudo-spectral fixed point).

Benchmarks

  • Spada et al. (2011) — the GIA-code benchmark (Love numbers, disc-load response, polar motion; Charles University benchmark, earth model M3–L70–V01).
  • Martinec et al. (2018) — the sea-level-equation intercomparison.

VILMA physics and coupling

  • Albrecht et al. (2024) — coupled PISM–VILMA; documents VILMA’s physics.
  • Bagge et al. (2021) — VILMA with 3D earth structure.
  • Willeit et al. (2022) — the CLIMBER-X Earth system model (the host).

Supporting theory

  • Farrell (1972) — load Love numbers and the deformation of the Earth by surface loads.
  • Mitrovica et al. (2005) — revised rotational stability theory (fixing \(k_f\) from the observed flattening).
  • Blewitt (2003) — reference frames, geocenter, and degree-1 surface loading.
  • Hanýk (1999) — initial-value (time-domain) viscoelastic relaxation.

Numerical libraries

  • Schaeffer (2013) — SHTns, efficient spherical-harmonic transforms.
  • Frigo and Johnson (2005) — FFTW3.

Full bibliography

Albrecht, T., M. Bagge, and V. Klemann. 2024. “Feedback Mechanisms Controlling Antarctic Glacial-Cycle Dynamics Simulated with a Coupled Ice Sheet–Solid Earth Model.” The Cryosphere 18 (9): 4233–55. https://doi.org/10.5194/tc-18-4233-2024.
Bagge, M., V. Klemann, B. Steinberger, M. Latinović, and M. Thomas. 2021. “Glacial-Isostatic Adjustment Models Using Geodynamically Constrained 3D Earth Structures.” Geochemistry, Geophysics, Geosystems 22 (11): e2021GC009853. https://doi.org/10.1029/2021GC009853.
Blewitt, G. 2003. “Self-Consistency in Reference Frames, Geocenter Definition, and Surface Loading of the Solid Earth.” Journal of Geophysical Research: Solid Earth 108 (B2): 2103. https://doi.org/10.1029/2002JB002082.
Farrell, W. E. 1972. “Deformation of the Earth by Surface Loads.” Reviews of Geophysics 10 (3): 761–97. https://doi.org/10.1029/RG010i003p00761.
Frigo, M., and S. G. Johnson. 2005. “The Design and Implementation of FFTW3.” Proceedings of the IEEE 93 (2): 216–31. https://doi.org/10.1109/JPROC.2004.840301.
Hanýk, L. 1999. “Viscoelastic Response of the Earth: Initial-Value Approach.” PhD thesis, Charles University, Prague.
Kendall, R. A., J. X. Mitrovica, and G. A. Milne. 2005. “On Post-Glacial Sea Level – II. Numerical Formulation and Comparative Results on Spherically Symmetric Models.” Geophysical Journal International 161 (3): 679–706. https://doi.org/10.1111/j.1365-246X.2005.02553.x.
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.
Martinec, Z., and J. Hagedoorn. 2014. “The Rotational Feedback on Linear-Momentum Balance in Glacial Isostatic Adjustment.” Geophysical Journal International 199 (3): 1823–46. https://doi.org/10.1093/gji/ggu369.
Martinec, Z., V. Klemann, W. van der Wal, et al. 2018. “A Benchmark Study of Numerical Implementations of the Sea Level Equation in GIA Modelling.” Geophysical Journal International 215 (1): 389–414. https://doi.org/10.1093/gji/ggy280.
Mitrovica, J. X., J. Wahr, I. Matsuyama, and A. Paulson. 2005. “The Rotational Stability of an Ice-Age Earth.” Geophysical Journal International 161 (2): 491–506. https://doi.org/10.1111/j.1365-246X.2005.02609.x.
Schaeffer, N. 2013. “Efficient Spherical Harmonic Transforms Aimed at Pseudospectral Numerical Simulations.” Geochemistry, Geophysics, Geosystems 14 (3): 751–58. https://doi.org/10.1002/ggge.20071.
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.
Willeit, M., A. Ganopolski, A. Robinson, and N. R. Edwards. 2022. “The Earth System Model CLIMBER-X V1.0 – Part 1: Climate Model Description and Validation.” Geoscientific Model Development 15 (14): 5905–48. https://doi.org/10.5194/gmd-15-5905-2022.
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