Application of a Consistent Nonlinear Mild-Slope Equation Model to Random Wave Propagation and Dissipation
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 150, Issue 6
Abstract
In the context of actual surface wave conditions, the wave field is represented as a set of complex fluctuations that randomly change in both time and space, commonly known as “random waves.” These random waves can be expressed mathematically as a combination of multiple monochromatic waves, each having unique phases, directions, and amplitudes. Frequency-domain phase-resolving wave models have been shown to be robust predictors of random wave propagation provided the dispersive characteristics are valid for the range of water depths considered. Recently, a new dispersive nonlinear mild-slope equation model was developed by establishing a closer correspondence between the scaling of nonlinearity, horizontal depth variation, and modulation scale during the derivation process. In this work, this new model is augmented with a wave-breaking dissipation model using frequency-squared dissipation weighting over the wave spectrum. The new model and previous models are compared with laboratory data for accuracy in modeling the evolution of the random wave spectrum. Overall, the new model demonstrates improved agreement with results compared with the previously derived models. The additional nonlinear terms of the model, indicating the interaction effects between amplitude and amplitude change, correct the overprediction of wave spectral energy from prior models, especially at the lower frequencies of the shallowest gauges. Furthermore, the predictions of free surface elevation by the newly derived model are in excellent agreement with the observations at the shallowest gauge, primarily due to the alleviation of phase mismatch caused by the additional terms. Lastly, we provide the nonlinear modification to linear wavenumber on the basis of the additional nonlinearity.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The first author was supported by a Doctoral Fellowship from the Zachry Department of Civil & Environmental Engineering at Texas A&M University. The work was also partly supported by grant P42 ES027704 from the National Institute of Environmental Health Sciences. We are thankful to Dr. Serdar Beji for providing the laboratory data of Beji and Battjes (1994), and to Drs. Anouk de Bakker and Gerben Ruessink for the laboratory data of Ruessink et al. (2013).
References
Agnon, Y., A. Sheremet, J. Gonsalves, and M. Stiassnie. 1993. “Nonlinear evolution of a unidirectional shoaling wave field.” Coastal Eng. 20 (1–2): 29–58. https://doi.org/10.1016/0378-3839(93)90054-C.
Akrish, G., A. Reniers, M. Zijlema, and P. Smit. 2024. “The impact of modulational instability on coastal wave forecasting using quadratic models.” Coastal Eng. 190: 104502. https://doi.org/10.1016/j.coastaleng.2024.104502.
Alsina, J. M., and T. E. Baldock. 2007. “Improved representation of breaking wave energy dissipation in parametric wave transformation models.” Coastal Eng. 54 (10): 765–769. https://doi.org/10.1016/j.coastaleng.2007.05.005.
Ardani, S., and J. M. Kaihatu. 2019. “Evolution of high frequency waves in shoaling and breaking wave spectra.” Phys. Fluids 31 (8): 087102. https://doi.org/10.1063/1.5096179.
Baldock, T. E., P. Holmes, S. Bunker, and P. Van Weert. 1998. “Cross-shore hydrodynamics within an unsaturated surf zone.” Coastal Eng. 34 (3–4): 173–196. https://doi.org/10.1016/S0378-3839(98)00017-9.
Battjes, J. A., and J. P. F. M. Janssen. 1978. “Energy loss and set-up due to breaking of random waves.” Coastal Eng. Proc. 1: 569–587. https://doi.org/10.1061/9780872621909.034.
Beji, S., and J. A. Battjes. 1994. “Numerical simulation of nonlinear wave propagation over a bar.” Coastal Eng. 23 (1–2): 1–16. https://doi.org/10.1016/0378-3839(94)90012-4.
Bouws, E., H. Günther, W. Rosenthal, and C. L. Vincent. 1985. “Similarity of the wind wave spectrum in finite depth water: 1. Spectral form.” J. Geophys. Res. Oceans 90 (C1): 975–986. https://doi.org/10.1029/JC090iC01p00975.
Bowen, G. D., and J. T. Kirby. 1994. Shoaling and breaking random waves on a 1: 35 laboratory beach. Newark, DE: Department of Civil Engineering, Center for Applied Coastal Research, Univ. of Delaware.
Chapalain, G., R. Cointe, and A. Temperville. 1992. “Observed and modeled resonantly interacting progressive water-waves.” Coastal Eng. 16 (3): 267–300. https://doi.org/10.1016/0378-3839(92)90045-V.
Chen, Y., R. T. Guza, and S. Elgar. 1997. “Modeling spectra of breaking surface waves in shallow water.” J. Geophys. Res. Oceans 102 (C11): 25035–25046. https://doi.org/10.1029/97JC01565.
Chen, Y., and P. L.-F. Liu. 1995. “Modified Boussinesq equations and associated parabolic models for water wave propagation.” J. Fluid Mech. 288: 351–381. https://doi.org/10.1017/S0022112095001170.
Eldeberky, Y., and J. A. Battjes. 1996. “Spectral modeling of wave breaking: Application to Boussinesq equations.” J. Geophys. Res. Oceans 101 (C1): 1253–1264. https://doi.org/10.1029/95JC03219.
Eldeberky, Y., and P. A. Madsen. 1999. “Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves.” Coastal Eng. 38 (1): 1–24. https://doi.org/10.1016/S0378-3839(99)00021-6.
Elgar, S., R. T. Guza, B. Raubenheimer, T. H. C. Herbers, and E. L. Gallagher. 1997. “Spectral evolution of shoaling and breaking waves on a barred beach.” J. Geophys. Res. Oceans 102 (C7): 15797–15805. https://doi.org/10.1029/97JC01010.
Freilich, M., and R. Guza. 1984. “Nonlinear effects on shoaling surface gravity waves.” Philos. Trans. R. Soc. London, Ser. A 311 (1515): 1–41.
Gobbi, M. F., J. T. Kirby, and G. Wei. 2000. “A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to o(kh)4.” J. Fluid Mech. 405: 181–210. https://doi.org/10.1017/S0022112099007247.
Hasselmann, K., et al. 1973. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Hamburg, Germany: Duetches Hydrographisches Institute.
Janssen, T. T., and J. A. Battjes. 2007. “A note on wave energy dissipation over steep beaches.” Coastal Eng. 54 (9): 711–716. https://doi.org/10.1016/j.coastaleng.2007.05.006.
Kaihatu, J. M. 2001. “Improvement of parabolic nonlinear dispersive wave model.” J. Waterw. Port Coastal Ocean Eng.127 (2): 113–121. https://doi.org/10.1061/(ASCE)0733-950X(2001)127:2(113).
Kaihatu, J. M., and J. T. Kirby. 1995. “Nonlinear transformation of waves in finite water depth.” Phys. Fluids 7 (8): 1903–1914. https://doi.org/10.1063/1.868504.
Kaihatu, J. M., and J. T. Kirby. 1997. “Effects of mode truncation and dissipation on predictions of higher order statistics.” Coastal Eng. 1996: 123–136. https://doi.org/10.1061/9780784402429.010.
Kaihatu, J. M., and J. T. Kirby. 1998. “Two-dimensional parabolic modeling of extended Boussinesq equations.” J. Waterw. Port Coastal Ocean Eng. 124 (2): 57–67. https://doi.org/10.1061/(ASCE)0733-950X(1998)124:2(57).
Kaihatu, J. M., J. Veeramony, K. L. Edwards, and J. T. Kirby. 2007. “Asymptotic behavior of frequency and wave number spectra of nearshore shoaling and breaking waves.” J. Geophys. Res.: Oceans 112 (C6): C06016.
Kim, I.-C. 2022. “A consistent nonlinear frequency domain model for finite depth ocean wave propagation.” Ph.D. thesis, Dept. of Civil and Environmental Engineering, Texas A&M Univ.
Kim, I.-C., and J. M. Kaihatu. 2021. “A consistent nonlinear mild-slope equation model.” Coastal Eng. 170: 104006. https://doi.org/10.1016/j.coastaleng.2021.104006.
Kim, I.-C., and J. M. Kaihatu. 2022. “A modified frequency distribution function of wave breaking-induced energy dissipation.” J. Geophys. Res. Oceans 127 (12): e2022JC018792. https://doi.org/10.1029/2022JC018792.
Kim, I.-C., and J. M. Kaihatu. 2024. “A simplified consistent nonlinear mild-slope equation model for random waves propagation and dissipation.” Coastal Eng. 189: 104449. https://doi.org/10.1016/j.coastaleng.2023.104449.
Kirby, J. T. 1991. “Modelling shoaling directional wave spectra in shallow water.” Coastal Eng. 1990: 109–122.
Kirby, J. T., and R. A. Dalrymple. 1983. “A parabolic equation for the combined refraction–diffraction of stokes waves by mildly varying topography.” J. Fluid Mech. 136: 453–466. https://doi.org/10.1017/S0022112083002232.
Kirby, J. T., and J. M. Kaihatu. 1997. “Structure of frequency domain models for random wave breaking.” Coastal Eng. 1996: 1144–1155. https://doi.org/10.1061/9780784402429.089.
Liu, P. L.-F., S. B. Yoon, and J. T. Kirby. 1985. “Nonlinear refraction–diffraction of waves in shallow water.” J. Fluid Mech. 153: 185–201. https://doi.org/10.1017/S0022112085001203.
Longuet-Higgins, M. S., and R.w. Stewart. 1964. “Radiation stresses in water waves; a physical discussion, with applications.” Deep Sea Res. Oceanogr. Abstr. 11 (4): 529–562. https://doi.org/10.1016/0011-7471(64)90001-4.
Lozano, C., and P. L. F. Liu. 1980. “Refraction-diffraction model for linear surface-water waves.” J. Fluid Mech. 101: 705–720.
Madsen, P. A., and O. R. Sørensen. 1993. “Bound waves and triad interactions in shallow water.” Ocean Eng. 20 (4): 359–388. https://doi.org/10.1016/0029-8018(93)90002-Y.
Mase, H., and J. T. Kirby. 1993. “Hybrid frequency-domain kdv equation for random wave transformation.” Coastal Eng. 1992: 474–487.
Mase, H., and T. Kitano. 2000. “Spectrum-based prediction model for random wave transformation over arbitrary bottom topography.” Coastal Eng. J. 42 (1): 111–151. https://doi.org/10.1142/S0578563400000067.
Peregrine, D. H. 1967. “Long waves on a beach.” J. Fluid Mech. 27 (4): 815–827. https://doi.org/10.1017/S0022112067002605.
Ruessink, B. G., H. Michallet, P. Bonneton, D. Mouazé, J. L. Lara, P. A. Silva, and P. Wellens. 2013. “GLOBEX: Wave dynamics on a gently sloping laboratory beach.” In Proc., 7th Int. Conf. on Coastal Dynamics, 1351–1362. Utrecht, The Netherlands: Utrecht University.
Smith, J. M., and C. L. Vincent. 1992. “Shoaling and decay of two wave trains on beach.” J. Waterw. Port Coastal Ocean Eng. 118 (5): 517–533. https://doi.org/10.1061/(ASCE)0733-950X(1992)118:5(517).
Smith, R., and T. Sprinks. 1975. “Scattering of surface waves by a conical island.” J. Fluid Mech. 72 (2): 373–384. https://doi.org/10.1017/S0022112075003424.
Tang, Y., and Y. Ouellet. 1997. “A new kind of nonlinear mild-slope equation for combined refraction-diffraction of multifrequency waves.” Coastal Eng. 31 (1-4): 3–36. https://doi.org/10.1016/S0378-3839(96)00050-6.
Thornton, E. B., and R. T. Guza. 1983. “Transformation of wave height distribution.” J. Geophys. Res. Oceans 88 (C10): 5925–5938. https://doi.org/10.1029/JC088iC10p05925.
Vrecica, T., and Y. Toledo. 2019. “Consistent nonlinear deterministic and stochastic wave evolution equations from deep water to the breaking region.” J. Fluid Mech. 877: 373–404. https://doi.org/10.1017/jfm.2019.525.
Yue, D. K., and C. C. Mei. 1980. “Forward diffraction of stokes waves by a thin wedge.” J. Fluid Mech. 99 (1): 33–52. https://doi.org/10.1017/S0022112080000481.
Information & Authors
Information
Published In
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Jan 29, 2024
Accepted: Jul 3, 2024
Published online: Aug 27, 2024
Published in print: Nov 1, 2024
Discussion open until: Jan 27, 2025
ASCE Technical Topics:
- Coasts, oceans, ports, and waterways engineering
- Continuum mechanics
- Dynamics (solid mechanics)
- Engineering mechanics
- Fluid mechanics
- Frequency response
- Gravity waves
- Hydraulic engineering
- Hydrologic engineering
- Motion (dynamics)
- Natural frequency
- Nonlinear waves
- Ocean engineering
- Oscillations
- Random waves
- Solid mechanics
- Surface waves
- Water and water resources
- Water waves
- Wave equations
- Wave propagation
- Waves (fluid mechanics)
- Waves (mechanics)
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.