Self-induced transparency of long water waves over bathymetry: the dispersive shock mechanism

Sheremet, Alex; Qayyum, Rizwan; Shrira, V.I.

Abstract

The shoreline hazard posed by ocean long waves such as tsunamis and meteotsunamis critically depends on the fraction of energy transmitted across the shallow near-shore shelf. In linear setting, bathymetric inhomogeneities of length comparable to the incident wavelength act as a protective high-pass filter, reflecting long waves and allowing only shorter waves to pass through. Here, we show that, for weakly nonlinear waves, the transmitted energy flux fraction can significantly depend on the amplitude of the incoming wave. The basis of this mechanism is the formation of dispersive shock waves (DSWs), a salient feature of nonlinear evolution of long water waves, often observed in tidal bores and tsunami/meteotsunami evolution. Within the framework of the Boussinesq equations, we show that the DSWs efficiently transfer wave energy into the high wavenumber band, where reflection is negligible. This is phenomenologically similar to self-induced transparency in nonlinear optics: small amplitude long waves are reflected by the bathymetric inhomogeneity, while larger amplitude waves that develop DSWs blueshift into the transparency regime and pass through. We investigate this mechanism in a simplified setting that retains only the key processes of DSW disintegration and reflection, while the effects such as bottom dissipation and breaking are ignored. The results suggests that the phenomenon is a robust, order-one effect. In contrast, the increased transmission due to the growth of bound harmonics associated with the steepening of the wave is weak. The results of the simplified modelling are validated by simulations with the FUNWAVE-TVD Boussinesq model.

Citation

Sheremet, A., Qayyum, R., & Shrira, V. (2024). Self-induced transparency of long water waves over bathymetry: the dispersive shock mechanism. Journal of Fluid Mechanics, 997, https://doi.org/10.1017/jfm.2024.850

• Journal Article Type: Article
• Acceptance Date: Aug 26, 2024
• Online Publication Date: Oct 8, 2024
• Publication Date: Oct 25, 2024
• Deposit Date: Oct 25, 2024
• Journal: Journal of Fluid Mechanics
• Print ISSN: 0022-1120
• Electronic ISSN: 1469-7645
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 997
• DOI: https://doi.org/10.1017/jfm.2024.850
• Public URL: https://keele-repository.worktribe.com/output/955278
• Publisher URL: https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/selfinduced-transparency-of-long-water-waves-over-bathymetry-the-dispersive-shock-mechanism/10FF6BD0548D9395E52913346E537250
• Additional Information: Copyright: © The Author(s), 2024. Published by Cambridge University Press.; License: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.; Free to read: This content has been made available to all.