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Sum-frequency triad interactions among surface waves propagating through an ice sheet

Published online by Cambridge University Press:  06 February 2024

Max W. Pierce Open the ORCID record for Max W. Pierce [Opens in a new window] ,
Yuming Liu Open the ORCID record for Yuming Liu [Opens in a new window]  and
Dick K.P. Yue Open the ORCID record for Dick K.P. Yue [Opens in a new window]
Max W. Pierce
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Yuming Liu
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Dick K.P. Yue*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: yue@mit.edu
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Abstract

We study nonlinear resonant wave–wave interactions which occur when ocean waves propagate into a thin floating ice sheet. Using multiple-scale perturbation analysis, we obtain theoretical predictions of the wave amplitude evolution as a function of distance travelled past the ice edge for a semi-infinite ice sheet. The theoretical predictions are supported by a high-order spectral (HOS) method capable of simulating nonlinear interactions in both open water and the ice sheet. Using the HOS method, the amplitude evolution predictions are extended to multiple (coupled) triad interactions and a single ice sheet of finite length. We relate the amplitude evolution to mechanisms with strong frequency dependence – ice bending strain, related to ice breakup, as well as wave reflection and transmission. We show that, due to sum-frequency interactions, the maximum strain in the ice sheet can be more than twice that predicted by linearised theory. For an ice sheet of finite length, we show that nonlinear wave reflection and transmission coefficients depend on a parameter in terms of wave steepness and ice length, and can have values significantly different than those from linear theory. In particular, we show that nonlinear sum-frequency interactions can appreciably decrease the total wave energy transmitted past the ice sheet. This work has implications for understanding the occurrence of ice breakup, wave attenuation due to scattering in the marginal ice zone and the resulting ice floe size distribution.

JFM classification

Geophysical and Geological Flows: Sea ice Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 980 , 10 February 2024 , A45
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Alam, M.-R. 2013 Dromions of flexural-gravity waves. J. Fluid Mech. 719, 1–13.CrossRefGoogle Scholar
Alam, M.-R., Liu, Y. & Yue, D.K.P. 2009 Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation. J. Fluid Mech. 624, 225–253.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D.K.P. 2010 Oblique sub- and super-harmonic Bragg resonance of surface waves by bottom ripples. J. Fluid Mech. 643, 437–447.Google Scholar
Armstrong, J.A., Bloembergen, N., Ducuing, J. & Pershan, P.S. 1962 Interactions between light waves in a nonlinear dielectric. Phys. Rev. Lett. 127, 1918–1939.Google Scholar
Bisht, N., Boral, S., Sahoo, T. & Meylan, M.H. 2022 Triad resonance of flexural gravity waves in a two-layer fluid within the framework of blocking dynamics. Phys. Fluids 34, 116606.Google Scholar
Bonnefoy, F., Meylan, M.H. & Ferrant, P. 2009 Nonlinear higher-order spectral solution for a two-dimensional moving load on ice. J. Fluid Mech. 621, 215–242.Google Scholar
Das, S., Sahoo, T. & Meylan, M.H. 2018 Dynamics of flexural gravity waves: from sea ice to Hawking radiation and analogue gravity. Proc. R. Soc. A 474 (2209), 20170223.Google Scholar
Dommermuth, D.G. & Yue, D.K.P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.CrossRefGoogle Scholar
Forbes, L.K. 1986 Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409–428.Google Scholar
Fox, C. & Squire, V.A. 1990 Reflection and transmission characteristics at the edge of shore fast sea ice. J. Geophys. Res. 95 (C7), 11629–11639.CrossRefGoogle Scholar
Fox, C. & Squire, V.A. 1991 Strain in shore fast ice due to incoming ocean waves and swell. J. Geophys. Res. 96 (C3), 4531–4547.Google Scholar
Guyenne, P. & Părău, E.I. 2012 Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713, 307–329.CrossRefGoogle Scholar
Guyenne, P. & Părău, E.I. 2017 a Numerical simulation of solitary-wave scattering and damping in fragmented sea ice. In Proceedings of the 27th International Offshore and Polar Engineering Conference, ISOPE-I-17-510. International Society of Offshore and Polar Engineers.Google Scholar
Guyenne, P. & Părău, E.I. 2017 b Numerical study of solitary wave attenuation in a fragmented ice sheet. Phys. Rev. Fluids 2, 034002.Google Scholar
Herman, A., Evers, K.-U. & Reimer, N. 2018 Floe-size distributions in laboratory ice broken by waves. Cryosphere 12, 685–699.CrossRefGoogle Scholar
Hunkins, K. 1966 Ekman drift current in the Arctic Ocean. Deep-Sea Res. 13 (4), 607–620.Google Scholar
Kirby, J.T. 1992 Water waves in variable depth under continuous sea ice. In Proceedings of the 2nd International Offshore and Polar Engineering Conference, ISOPE-I-92-227. International Society of Offshore and Polar Engineers.Google Scholar
Kohout, A.L. & Meylan, M.H. 2008 An elastic plate model for wave attenuation and ice floe breaking in the marginal ice zone. J. Geophys. Res. 113, C09016.Google Scholar
Kohout, A.L., Smith, M., Roach, L.A., Williams, G., Montiel, F. & Williams, M.J. 2020 Observations of exponential wave attenuation in Antarctic Sea ice during the PIPERS campaign. Ann. Glaciol. 61 (82), 196–209.CrossRefGoogle Scholar
Kohout, A.L., Williams, M.J.M., Dean, S.M. & Meylan, M.H. 2014 Storm-induced sea-ice breakup and the implications for ice extent. Nature 509, 604–607.Google Scholar
Liu, A.K., Holt, B. & Vachon, P.W. 1991 Wave propagation in the marginal ice zone: model predictions and comparisons with buoy and synthetic aperture radar data. J. Geophys. Res. 96 (C3), 4605–4621.Google Scholar
Liu, A.K. & Mollo-Christensen, E. 1988 Wave propagation in a solid ice pack. J. Phys. Oceanogr. 18, 1702–1712.2.0.CO;2>CrossRefGoogle Scholar
Liu, Q., Rogers, W.E., Babanin, A., Li, J. & Guan, C. 2020 Spectral modeling of ice-induced wave decay. J. Phys. Oceanogr. 50 (6), 1583–1604.CrossRefGoogle Scholar
Marchenko, A.V. 1999 Stability of flexural-gravity waves and quadratic interactions. Fluid Dyn. 34 (1), 78–86.Google Scholar
Marchenko, A.V. & Shrira, V.I. 1991 Theory of two-dimensional nonlinear waves in liquid covered by ice. Fluid Dyn. 26, 580–587.Google Scholar
McGoldrick, L.F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305–331.Google Scholar
Mei, C.C., Stiassnie, M. & Yue, D.K.P. 2005 Theory and Application of Ocean Surface Waves, chap. 13.2. World Scientific.Google Scholar
Meylan, M.H. & Squire, V.A. 1993 Finite-floe wave reflection and transmission coefficients from a semi-infinite model. J. Geophys. Res. 98 (C7), 12537–12542.Google Scholar
Pan, Y. 2020 High-order spectral method for simulation of capillary waves with complete order consistency. J. Comput. Phys. 408, 109299.CrossRefGoogle Scholar
Părău, E.I. & Dias, F. 2002 Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech. 460, 281–305.CrossRefGoogle Scholar
Plotnikov, P.I. & Toland, J.F. 2011 Modelling nonlinear hydroelastic waves. Phil. Trans. R. Soc. A 369, 2942–2956.Google Scholar
Simmons, W.F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. A 309 (1499), 551–575.Google Scholar
Vanden-Broeck, J. & Părău, E.I. 2011 Two-dimensional generalized solitary waves and periodic waves under an ice sheet. Phil. Trans. R. Soc. A 369 (1947), 2957–2972.Google Scholar
Wadhams, P., Squire, V.A., Goodman, D.J., Cowan, A.M. & Moore, S.C. 1988 The attenuation rates of ocean waves in the marginal ice zone. J. Geophys. Res. 93 (C6), 6799–6818.CrossRefGoogle Scholar
Waseda, T., Alberello, A., Nose, T., Toyota, T., Kodaira, T. & Fujiwara, Y. 2022 Observation of anomalous spectral downshifting of waves in the Okhotsk Sea marginal ice zone. Phil. Trans. R. Soc. A 3809, 20210256.Google Scholar
Xiao, W., Liu, Y., Wu, G. & Yue, D.K.P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357–392.Google Scholar
Xu, B. & Guyenne, P. 2023 Nonlinear simulation of wave group attenuation due to scattering in broken floe fields. Ocean Model. 181, 102139.CrossRefGoogle Scholar