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Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory

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Nonlinear Water Waves

Abstract

We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schrödinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about the stability of open ocean spectra to inhomogeneous disturbances, as well as new stochastic equations for the nearshore.

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Notes

  1. 1.

    However, the ω i satisfy the dispersion relation of the NLS rather than the linear deep-water dispersion relation in (3.3).

  2. 2.

    “Over 2000 wave spectra were measured; about […] 121 corresponded to “ideal” stationary and homogeneous wind conditions.” p. 10.

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Acknowledgements

RS is grateful for the hospitality and support of the Erwin Schrödinger Institute for Mathematics and Physics (ESI), Vienna, Austria, as well as support from a Small Grant from the IMA.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, University of Plymouth, Plymouth, UK

    Raphael Stuhlmeier

  2. School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv, Israel

    Teodor Vrecica & Yaron Toledo

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  1. Raphael Stuhlmeier
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  2. Teodor Vrecica
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  3. Yaron Toledo
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Correspondence to Raphael Stuhlmeier .

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Editors and Affiliations

  1. School of Mathematical Sciences, University College Cork, Cork, Ireland

    David Henry

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

    Konstantinos Kalimeris

  3. School of Mathematics, University of East Anglia, Norwich, UK

    Emilian I. Părău

  4. Department of Mathematics, University College London, London, UK

    Jean-Marc Vanden-Broeck

  5. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Erik Wahlén

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Stuhlmeier, R., Vrecica, T., Toledo, Y. (2019). Nonlinear Wave Interaction in Coastal and Open Seas: Deterministic and Stochastic Theory. In: Henry, D., Kalimeris, K., Părău, E., Vanden-Broeck, JM., Wahlén, E. (eds) Nonlinear Water Waves . Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-33536-6_10

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