Ocean Surface Gravity Wave Excitation of Flexural Gravity and Extensional Lamb Waves in Ice Shelves


  • Lauren Abrahams Department of Geophysics, Stanford University, Stanford, CA, USA, now at Lawrence Livermore National Laboratory, Livermore, CA, USA
  • Jose Mierzejewski Department of Geophysics, Stanford University, Stanford, CA, USA, now ow at California Polytechnic State University, San Luis Obispo, CA, USA
  • Eric Dunham Department of Geophysics, Stanford University, Stanford, CA, USA, & Institute of Computational and Mathematical Engineering, StanfordUniversity, Stanford, CA, USA https://orcid.org/0000-0003-0804-7746
  • Peter D. Bromirski Scripps Institution of Oceanography, University of California San Diego, La Jolla, California, USA https://orcid.org/0000-0002-5342-9252




Flexure and extension of ice shelves in response to incident ocean surface gravity waves have been linked to iceberg calving, rift growth, and even disintegration of ice shelves. Most modeling studies utilize a plate bending model for the ice, focusing exclusively on flexural gravity waves. Ross Ice shelf seismic data shows not only flexural gravity waves, with dominantly vertical displacements, but also extensional Lamb waves, which propagate much faster with dominantly horizontal displacements. Our objective is to model the full-wave response of ice shelves, including ocean compressibility, ice elasticity, and gravity. Our model is a 2D vertical cross-section of the ice shelf and sub-shelf ocean cavity. We quantify the frequency-dependent excitation of flexural gravity and extensional Lamb waves and provide a quantitative theory for extensional Lamb wave generation by the horizontal force imparted by pressure changes on the vertical ice shelf edge exerted by gravity waves. Our model predicts a horizontal to vertical displacement ratio that increases with decreasing frequency, with ratio equal to unity at ~0.001 Hz. Furthermore, in the very long period band (<0.003 Hz), tilt from flexural gravity waves provides an order of magnitude larger contribution to seismometer horizontal components than horizontal displacements from extensional Lamb waves.


Achenbach, J. D. (1973). Wave propagation in elastic solids. In ISBN-13. Elsevier.

Achenbach, J. D. (2003). Reciprocity in elastodynamics. Cambridge University Press.

Achenbach, Jan D. (2003). Laser excitation of surface wave motion. Journal of the Mechanics and Physics of Solids, 51(11–12), 1885–1902. https://doi.org/https://doi.org/10.1016/j.jmps.2003.09.021

Aki, K., & Richards, P. G. (2002). Quantitative Seismology. University Science Books.

Aster, R. C., Lipovsky, B. P., Cole, H. M., Bromirski, P. D., Gerstoft, P., Nyblade, A., Wiens, D. A., & Stephen, R. (2021). Swell-triggered seismicity at the near-front damage zone of the Ross Ice Shelf. Seismological Research Letters, 92(5), 2768–2792. https://doi.org/https://doi.org/10.1785/0220200478

Banwell, A. F., MacAyeal, D. R., & Sergienko, O. V. (2013). Breakup of the Larsen B Ice Shelf triggered by chain reaction drainage of supraglacial lakes. Geophysical Research Letters, 40(22), 5872–5876. https://doi.org/https://doi.org/10.1002/2013GL057694

Banwell, A. F., Willis, I. C., Macdonald, G. J., Goodsell, B., Mayer, D. P., Powell, A., & Macayeal, D. R. (2017). Calving and rifting on the McMurdo ice shelf, Antarctica. Annals of Glaciology, 58(75pt1), 78–87. https://doi.org/https://doi.org/10.1017/aog.2017.12

Biot, M. A. (1952). The interaction of Rayleigh and Stoneley waves in the ocean bottom. Bulletin of the Seismological Society of America, 42(1), 81–93. https://doi.org/https://doi.org/10.1785/BSSA0420010081

Bromirski, P. D., Chen, Z., Stephen, R. A., Gerstoft, P., Arcas, D., Diez, A., Aster, R. C., Wiens, D. A., & Nyblade, A. (2017). Tsunami and infragravity waves impacting Antarctic ice shelves. Journal of Geophysical Research: Oceans, 122(7), 5786–5801. https://doi.org/https://doi.org/10.1002/2017JC012913

Bromirski, P. D., Sergienko, O. V., & MacAyeal, D. R. (2010). Transoceanic infragravity waves impacting Antarctic ice shelves. Geophysical Research Letters, 37(2). https://doi.org/https://doi.org/10.1029/2009GL041488

Bromwich, D. H., & Nicolas, J. P. (2010). Ice-sheet uncertainty. Nature Geoscience, 3(9), 596–597. https://doi.org/https://doi.org/10.1038/ngeo946

Brunt, K. M., Okal, E. A., & MacAYEAL, D. R. (2011). Antarctic ice-shelf calving triggered by the Honshu (Japan) earthquake and tsunami, March 2011. Journal of Glaciology, 57(205), 785–788. https://doi.org/DOI: https://doi.org/10.3189/002214311798043681

Chen, Z, Bromirski, P., Gerstoft, P., Stephen, R., Lee, W. S., Yun, S., Olinger, S., Aster, R., Wiens, D., & Nyblade, A. (2019). Ross Ice Shelf icequakes associated with ocean gravity wave activity. Geophysical Research Letters, 46(15), 8893–8902. https://doi.org/https://doi.org/10.1029/2019GL084123

Chen, Zhao, Bromirski, P. D., Gerstoft, P., Stephen, R. A., Wiens, D. A., Aster, R. C., & Nyblade, A. A. (2018). Ocean-excited plate waves in the Ross and Pine Island Glacier ice shelves. Journal of Glaciology, 64(247), 730–744. https://doi.org/https://doi.org/10.1017/jog.2018.66

De Angelis, H., & Skvarca, P. (2003). Glacier surge after ice shelf collapse. Science, 299(5612), 1560–1562. https://doi.org/10.1126/science.1077987

Diez, A., Bromirski, P., Gerstoft, P., Stephen, R., Anthony, R., Aster, R., Cai, C., Nyblade, A., & Wiens, D. (2016). Ice shelf structure derived from dispersion curve analysis of ambient seismic noise, Ross Ice Shelf, Antarctica. Geophysical Journal International, 205(2), 785–795. https://doi.org/10.1093/gji/ggw036

Dingemans, M. W. (1997). Water wave propagation over uneven bottoms: Part 1 – Linear wave propagation. World Scientific.

Dunham, E. M., Belanger, D., Cong, L., & Kozdon, J. E. (2011). Earthquake ruptures with strongly rate-weakening friction and off-fault plasticity, Part 1: Planar faults. Bulletin of the Seismological Society of America, 101(5), 2296–2307. https://doi.org/https://doi.org/10.1785/0120100075

Dupont, T., & Alley, R. (2005). Assessment of the importance of ice-shelf buttressing to ice-sheet flow. Geophysical Research Letters, 32(4). https://doi.org/https://doi.org/10.1029/2004GL022024

Ewing, M., & Crary, A. (1934). Propagation of elastic waves in ice. Part II. Physics, 5(7), 181–184.

Fox, C., & Squire, V. A. (1990). Reflection and transmission characteristics at the edge of shore fast sea ice. Journal of Geophysical Research: Oceans, 95(C7), 11629–11639. https://doi.org/https://doi.org/10.1029/JC095iC07p11629

Fox, C., & Squire, V. A. (1991). Coupling between the ocean and an ice shelf. Annals of Glaciology, 15, 101–108. https://doi.org/https://doi.org/10.3189/1991AoG15-1-101-108

Holdsworth, G., & Glynn, J. (1978). Iceberg calving from floating glaciers by a vibrating mechanism. Nature, 274(5670), 464–466. https://doi.org/https://doi.org/10.1038/274464a0

Ilyas, M., Meylan, M. H., Lamichhane, B., & Bennetts, L. G. (2018). Time-domain and modal response of ice shelves to wave forcing using the finite element method. Journal of Fluids and Structures, 80, 113–131. https://doi.org/https://doi.org/10.1016/j.jfluidstructs.2018.03.010

Kalyanaraman, B., Bennetts, L. G., Lamichhane, B., & Meylan, M. H. (2019). On the shallow-water limit for modelling ocean-wave induced ice-shelf vibrations. Wave Motion, 90, 1–16. https://doi.org/https://doi.org/10.1016/j.wavemoti.2019.04.004

Kalyanaraman, B., Meylan, M. H., Bennetts, L. G., & Lamichhane, B. P. (2020). A coupled fluid-elasticity model for the wave forcing of an ice-shelf. Journal of Fluids and Structures, 97, 103074. https://doi.org/https://doi.org/10.1016/j.jfluidstructs.2020.103074

Kozdon, J. E., Dunham, E. M., & Nordström, J. (2012). Interaction of waves with frictional interfaces using summation-by-parts difference operators: Weak enforcement of nonlinear boundary conditions. Journal of Scientific Computing, 50(2), 341–367. https://doi.org/https://doi.org/10.1007/s10915-011-9485-3

Kozdon, J. E., Dunham, E. M., & Nordström, J. (2013). Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods. Journal of Scientific Computing, 55(1), 92–124. https://doi.org/https://doi.org/10.1007/s10915-012-9624-5

Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2015). Fluid Mechanics. Academic Press.

Lamb, H. (1905). On deep-water waves. Proceedings of the London Mathematical Society, 2(1), 371–400.

Lamb, H. (1917). On waves in an elastic plate. Proceedings of the Royal Society of London. Series A, 93(648), 114–128. https://doi.org/https://doi.org/10.1098/rspa.1917.0008

Lipovsky, B. P. (2018). Ice shelf rift propagation and the mechanics of wave-induced fracture. Journal of Geophysical Research: Oceans, 123(6), 4014–4033. https://doi.org/https://doi.org/10.1029/2017JC013664

Lotto, G. C., & Dunham, E. M. (2015). High-order finite difference modeling of tsunami generation in a compressible ocean from offshore earthquakes. Computational Geosciences, 19(2), 327–340. https://doi.org/10.1007/s10596-015-9472-0

MacAyeal, D. R., Okal, E. A., Aster, R. C., Bassis, J. N., Brunt, K. M., Cathles, L. M., Drucker, R., Fricker, H. A., Kim, Y.-J., Martin, S., & others. (2006). Transoceanic wave propagation links iceberg calving margins of Antarctica with storms in tropics and Northern Hemisphere. Geophysical Research Letters, 33(17). https://doi.org/https://doi.org/10.1029/2006GL027235

Massom, R. A., Scambos, T. A., Bennetts, L. G., Reid, P., Squire, V. A., & Stammerjohn, S. E. (2018). Antarctic ice shelf disintegration triggered by sea ice loss and ocean swell. Nature, 558(7710), 383–389. https://doi.org/https://doi.org/10.1038/s41586-018-0212-1

Mattsson, K., Dunham, E. M., & Werpers, J. (2018). Simulation of acoustic and flexural-gravity waves in ice-covered oceans. Journal of Computational Physics, 373, 230–252. https://doi.org/https://doi.org/10.1016/j.jcp.2018.06.060

Metropolis, N., & Ulam, S. (1949). The Monte Carlo Method. Journal of the American Statistical Association, 44(247), 335–341. https://doi.org/10.1080/01621459.1949.10483310

Meylan, M. H., Ilyas, M., Lamichhane, B. P., & Bennetts, L. G. (2021). Swell-induced flexural vibrations of a thickening ice shelf over a shoaling seabed. Proceedings of the Royal Society A, 477(2254), 20210173. https://doi.org/https://doi.org/10.1098/rspa.2021.0173

Miles, J. W. (1967). Surface-wave scattering matrix for a shelf. Journal of Fluid Mechanics, 28(4), 755–767. https://doi.org/https://doi.org/10.1017/S0022112067002423

Newman, J. (1965). Propagation of water waves over an infinite step. Journal of Fluid Mechanics, 23(2), 399–415. https://doi.org/https://doi.org/10.1017/S0022112065001453

Olinger, S., Lipovsky, B. P., Denolle, M., & Crowell, B. W. (2022). Tracking the cracking: a holistic analysis of rapid ice shelf fracture using seismology, geodesy, and satellite imagery on the Pine Island Glacier ice shelf, West Antarctica. Geophysical Research Letters, e2021GL097604. https://doi.org/https://doi.org/10.1029/2021GL097604

Olinger, S., Lipovsky, B., Wiens, D., Aster, R., Bromirski, P., Chen, Z., Gerstoft, P., Nyblade, A., & Stephen, R. (2019). Tidal and thermal stresses drive seismicity along a major Ross Ice Shelf rift. Geophysical Research Letters, 46(12), 6644–6652. https://doi.org/https://doi.org/10.1029/2019GL082842

Paolo, F. S., Fricker, H. A., & Padman, L. (2015). Volume loss from Antarctic ice shelves is accelerating. Science, 348(6232), 327–331. https://doi.org/10.1126/science.aaa0940

Press, F., & Ewing, M. (1951). Propagation of elastic waves in a floating ice sheet. Eos, Transactions American Geophysical Union, 32(5), 673–678. https://doi.org/https://doi.org/10.1029/TR032i005p00673

Pritchard, Hd., Ligtenberg, S. R., Fricker, H. A., Vaughan, D. G., van den Broeke, M. R., & Padman, L. (2012). Antarctic ice-sheet loss driven by basal melting of ice shelves. Nature, 484(7395), 502–505. https://doi.org/https://doi.org/10.1038/nature10968

Rignot, E., Jacobs, S., Mouginot, J., & Scheuchl, B. (2013). Ice-shelf melting around Antarctica. Science, 341(6143), 266–270. https://doi.org/10.1126/science.1235798

Rodgers, P. (1968). The response of the horizontal pendulum seismometer to Rayleigh and Love waves, tilt, and free oscillations of the Earth. Bulletin of the Seismological Society of America, 58(5), 1385–1406. https://doi.org/https://doi.org/10.1785/BSSA0580051385

Rott, H., Skvarca, P., & Nagler, T. (1996). Rapid collapse of northern Larsen ice shelf, Antarctica. Science, 271(5250), 788–792. https://doi.org/10.1126/science.271.5250.788

Scambos, T. A., Bohlander, J., Shuman, C. A., & Skvarca, P. (2004). Glacier acceleration and thinning after ice shelf collapse in the Larsen B embayment, Antarctica. Geophysical Research Letters, 31(18). https://doi.org/10.1029/2004GL020670

Sells, C. L. (1965). The effect of a sudden change of shape of the bottom of a slightly compressible ocean. Philosophical Transactions of the Royal Society of London. Series A, 258(1092), 495–528. https://doi.org/https://doi.org/10.1098/rsta.1965.0049

Sergienko, O V. (2017). Behavior of flexural gravity waves on ice shelves: Application to the Ross Ice Shelf. Journal of Geophysical Research: Oceans, 122(8), 6147–6164. https://doi.org/https://doi.org/10.1002/2017JC012947

Sergienko, Olga V. (2010). Elastic response of floating glacier ice to impact of long-period ocean waves. Journal of Geophysical Research: Earth Surface, 115(F4). https://doi.org/https://doi.org/10.1029/2010JF001721

Sergienko, Olga V. (2013). Normal modes of a coupled ice-shelf/sub-ice-shelf cavity system. Journal of Glaciology, 59(213), 76–80. https://doi.org/https://doi.org/10.3189/2013JoG12J096

Squire, V. A. (2007). Of ocean waves and sea-ice revisited. Cold Regions Science and Technology, 49(2), 110–133. https://doi.org/https://doi.org/10.1016/j.coldregions.2007.04.007

Squire, V. A., Dugan, J. P., Wadhams, P., Rottier, P. J., & Liu, A. K. (1995). Of ocean waves and sea ice. Annual Review of Fluid Mechanics, 27(1), 115–168. https://doi.org/https://doi.org/10.1146/annurev.fl.27.010195.000555

Tanimoto, T., & Wang, J. (2018). Low-frequency seismic noise characteristics from the analysis of co-located seismic and pressure data. Journal of Geophysical Research: Solid Earth, 123(7), 5853–5885. https://doi.org/https://doi.org/10.1029/2018JB015519

Tazhimbetov, N., Almquist, M., Werpers, J., & Dunham, E. (2022). Simulation of flexural-gravity wave propagation for elastic plates in shallow water using energy-stable finite difference method with weakly enforced boundary and interface conditions. Available at SSRN 4147169. https://doi.org/http://dx.doi.org/10.2139/ssrn.4147169

Timoshenko, S. P., & Goodier, J. N. (1970). Theory of Elasticity. McGraw Hill.

Walker, R., Dupont, T., Parizek, B., & Alley, R. (2008). Effects of basal-melting distribution on the retreat of ice-shelf grounding lines. Geophysical Research Letters, 35(17). https://doi.org/https://doi.org/10.1029/2008GL034947

Yamamoto, T. (1982). Gravity waves and acoustic waves generated by submarine earthquakes. International Journal of Soil Dynamics and Earthquake Engineering, 1(2), 75–82. https://doi.org/https://doi.org/10.1016/0261-7277(82)90016-X


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How to Cite

Abrahams, L., Mierzejewski , J., Dunham, E., & Bromirski, P. D. (2023). Ocean Surface Gravity Wave Excitation of Flexural Gravity and Extensional Lamb Waves in Ice Shelves. Seismica, 2(1). https://doi.org/10.26443/seismica.v2i1.213