Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T14:18:59.478Z Has data issue: false hasContentIssue false
  • English
  • Français

Consistent nonlinear stochastic evolution equations for deep to shallow water wave shoaling

Published online by Cambridge University Press:  04 April 2016

Teodor Vrecica  and
Yaron Toledo
Teodor Vrecica
Affiliation:
School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv 6997801, Israel
Yaron Toledo*
Affiliation:
School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv 6997801, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear interactions between sea waves and the sea bottom are a major mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave forecasting models due to its mathematical complexity, which mostly consists of computing either the bispectral evolution or non-local shoaling coefficients. In this work, quasi-two-dimensional stochastic energy evolution equations are derived for dispersive water waves up to quadratic nonlinearity. The bispectral evolution equations are formulated using stochastic closure. They are solved analytically and substituted into the energy evolution equations to construct a stochastic model with non-local shoaling coefficients, which includes nonlinear dissipative effects and slow time evolution. The nonlinear shoaling mechanism is investigated and shown to present two different behaviour types. The first consists of a rapidly oscillating behaviour transferring energy back and forth between wave harmonics in deep water. Owing to the contribution of bottom components for closing the class III Bragg resonance conditions, this behaviour includes mean energy transfer when waves reach intermediate water depths. The second behaviour relates to one-dimensional shoaling effects in shallow water depths. In contrast to the behaviour in intermediate water depths, it is shown that the nonlinear shoaling coefficients refrain from their oscillatory nature while presenting an exponential energy transfer. This is explained through the one-dimensional satisfaction of the Bragg resonance conditions by wave triads due to the non-dispersive propagation in this region even without depth changes. The energy evolution model is localized using a matching approach to account for both these behaviour types. The model is evaluated with respect to deterministic ensembles, field measurements and laboratory experiments while performing well in modelling monochromatic superharmonic self-interactions and infra-gravity wave generation from bichromatic waves and a realistic wave spectrum evolution. This lays physical and mathematical grounds for the validity of unexplained simplifications in former works and the capability to construct a formulation that consistently accounts for nonlinear energy transfers from deep to shallow water.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 794 , 10 May 2016 , pp. 310 - 342
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agnon, Y. 1999 Linear and nonlinear refraction and Bragg scattering of water waves. Phys. Rev. E 59, 1319–1322.CrossRefGoogle Scholar
Agnon, Y. & Sheremet, A. 1997 Stochastic nonlinear shoaling of directional spectra. J. Fluid Mech. 345, 79–99.CrossRefGoogle Scholar
Agnon, Y. & Sheremet, A. 2000 Stochastic evolution models for nonlinear gravity waves over uneven topography. In Advances in Coastal and Ocean Engineering (ed. Liu, P. L.-F.), vol. 6, pp. 103–133. World Scientific.CrossRefGoogle Scholar
Agnon, Y., Sheremet, A., Gonsalves, J. & Stiassnie, M. 1993 A unidirectional model for shoaling gravity waves. Coast. Engng 20, 29–58.CrossRefGoogle Scholar
Becq-Girard, F., Forget, P. & Benoit, M. 1999 Non-linear propagation of unidirectional wave fields over varying topography. Coast. Engng 38, 91–113.CrossRefGoogle Scholar
Benney, D. J. & Saffman, P. G. 1966 Nonlinear interactions of random waves. Proc. R. Soc. Lond. A 289, 301–321.Google Scholar
Bredmose, H., Agnon, Y., Madsen, P. & Schaffer, H. 2005 Wave transformation models with exact second-order transfer. Eur. J. Mech. (B/Fluids) 24 (6), 659–682.CrossRefGoogle Scholar
Eldeberky, Y. & Battjes, J. A. 1995 Parameterization of triad interactions in wave energy models. In Coastal Dynamics ’95 (ed. Dally, W. R. & Zeidler, R. B.), pp. 140–148. ASCE.Google Scholar
Eldeberky, Y. & Madsen, P. A. 1999 Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves. Coast. Engng 38, 1–24.CrossRefGoogle Scholar
Freilich, M. H. & Guza, R. T. 1984 Nonlinear effects on shoaling surface gravity waves. Phil. Trans. R. Soc. Lond. A 311, 1–41.Google Scholar
Herbers, T. & Burton, M. 1997 Nonlinear shoaling of directionally spread waves on a beach. J. Geophys. Res. 102, 21101–21114.CrossRefGoogle Scholar
Janssen, P. A. 2009 On some consequences of the canonical transformation in the hamiltonian theory of water waves. J. Fluid Mech. 637, 1–44.CrossRefGoogle Scholar
Janssen, T.2006 Nonlinear surface waves over topography. PhD thesis, University of Delft.Google Scholar
Janssen, T. T., Herbers, T. H. C. & Battjes, J. A. 2008 Evolution of ocean wave statistics in shallow water: refraction and diffraction over seafloor topography. J. Geophys. Res. 113, C03024.Google Scholar
Kaihatu, J. M. & Kirby, J. T. 1995 Nonlinear transformation of waves in finite water depth. Phys. Fluids 8, 175–188.Google Scholar
Kofoed-Hansen, H. & Rasmussen, J. H. 1998 Modeling of nonlinear shoaling based on stochastic evolution equations. Coast. Engng 33, 203–232.CrossRefGoogle Scholar
Liu, Y. & Yue, D. K. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326.CrossRefGoogle Scholar
Luth, H. R., Klopman, G. & Kitou, N.1994 Kinematics of waves breaking partially on an offshore bar, LVD measurements for waves without a net onshore current. Tech. Rep. No. H1573, Delft Hydraulics, Delft, The Netherlands.Google Scholar
Madsen, P. A., Fuhrman, D. R. & Wang, B. 2006 A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Engng 53, 487–504.CrossRefGoogle Scholar
Polnikov, V. 1997 Nonlinear energy transfer through the spectrum of gravity waves for the finite depth case. J. Phys. Oceonogr. 27, 1481–1491.2.0.CO;2>CrossRefGoogle Scholar
Sharma, A., Panchang, V. & Kaihatu, J. 2014 Modeling nonlinear wave–wave interactions with the elliptic mild slope equation. Appl. Ocean Res. 48, 114–125.CrossRefGoogle Scholar
Stiassnie, M. & Drimer, N. 2006 Prediction of long forcing waves for harbor agitation studies. J. Waterway Port Coast. Ocean Engng 132 (3), 166–171.CrossRefGoogle Scholar
Tang, Y. & Ouellet, Y. 1997 A new kind of nonlinear mild-slope equation for combined refraction–diffraction of multifrequency waves. Coast. Engng 31, 3–36.CrossRefGoogle Scholar
Toledo, Y. 2013 The oblique parabolic equation model for linear and nonlinear wave shoaling. J. Fluid Mech. 715, 103–133.CrossRefGoogle Scholar
Toledo, Y. & Agnon, Y. 2009 Nonlinear refraction–diffraction of water waves: the complementary mild-slope equations. J. Fluid Mech. 641, 509–520.CrossRefGoogle Scholar
Toledo, Y. & Agnon, Y. 2012 Stochastic evolution equations with localized nonlinear shoaling coefficients. Eur. J. Mech. (B/Fluids) 34, 13–18.CrossRefGoogle Scholar
Willebrand, J. 1975 Energy transport in a nonlinear and inhomogeneous random gravity wave field. J. Fluid Mech. 70, 113–126.CrossRefGoogle Scholar