Travelling Breaking Waves

We study a mathematical model of coastal waves in the shallow water approximation. The model contains two empirical parameters. The first one controls turbulent dissipation. The second one is responsible for the turbulent viscosity and is determined by the turbulent Reynolds number. We study travelling waves solutions to this model. The existence of an analytical and numerical solution to the problem in the form of a traveling wave is shown. The singular points of the system are described. It is shown that there exists a critical value of the Reylnols number corresponding to the transition from a monotonic profile to an oscillatory one. The paper is organized as follows. First, we present the governing system of ordinary differential equations (ODE) for travelling waves. Second, the Lyapunov function for the corresponding ODE system is derived. Finally, the behavior of the solution to the ODE system is discussed.

Keywords

shallow-water equation; Lyapunov function; Reynolds number; travelling wave solution.

References

1. Salmon R. Lectures on Geophysical Fluid Mechanics. New York, Oxford, Oxford University Press, 1998.
2. Stoker J.J. Water Waves: the Mathematical Theory with Applications. New York, Interscience, 1957.
3. Lannes D. The Water Waves Problem: Mathematical Analysis and Asymptotics. Providence, American Mathematical Society, 2013. DOI: 10.1090/surv/188
4. Whitham G.B. Linear and Nonlinear Waves. New York, John Wiley and Sons, 1974.
5. Cienfuegos R., Barthelemy E., Bonneton P. Wave-Breaking Model for Boussinesq-Type Equations Including Roller Effects in the Mass Conservation Equation. Journal of Waterway, Port, Coastal and Ocean Engineering, 2010, vol. 136, no. 1, pp. 10-26. DOI: 10.1061/AASCEWW.1943-5460.0000022
6. Freeman J.C., Lemehaute B. Wave Breakers on a Beach and Surges in a Dry Bed. Journal of Hydraulic Engineering, 1964, vol. 90, pp. 187-216.
7. Karambas T.V., Tozer N.P. Breaking Waves in the Surf and Swash Zone. Journal of Coastal Research, 2003, vol. 19, pp. 514-528. DOI: 10.2307/4299194
8. Kazakova M., Richard G.L. A New Model of Shoaling and Breaking Waves: One-Dimensional Solitary Wave on a Mild Sloping Beach. Journal of Fluid Mechanics, 2019, vol. 862, pp. 552-591. DOI: 10.1017/jfm.2018.947
9. Richard G.L., Duran A., Fabr`eges B. A New Model of Shoaling and Breaking Waves: Part 2. Run-up and Two-Dimensional Waves. Journal of Fluid Mechanics, 2019, vol. 867, pp. 146-194. DOI: 10.1017/jfm.2019.125
10. Richard G.L., Gavrilyuk S.L. A New Model of Roll Waves: Comparison with Brock's Experiments. Journal of Fluid Mechanics, 2012, vol. 698, pp. 374-405. DOI: 10.1017/jfm.2012.96
11. Richard G.L., Gavrilyuk S.L. The Classical Hydraulic Jump in a Model of Shear Shallow-Water Flows. Journal of Fluid Mechanics, 2013, vol. 725, pp. 492-521. DOI: 10.1017/jfm.2013.174
12. Richard G.L., Gavrilyuk S.L. Modelling Turbulence Generation in Solitary Waves on Shear Shallow Water Flows. Journal of Fluid Mechanics, 2015, vol. 773, pp. 49-74. DOI: 10.1017/jfm.2015.236
13. Gavrilyuk S.L., Liapidevskii V.Yu., Chesnokov A.A. Spilling Breakers in Shallow Water: Applications to Favre Waves and to the Shoaling and Breaking of Solitary Wave. Journal of Fluid Mechanics, 2016, vol. 808, pp. 441-468. DOI: 10.1017/jfm.2016.662
14. Chesnokov A.A., Gavrilyuk S.L., Liapidevskii V.Yu. Mixing and Nonlinear Internal Waves in a Shallow Flow of a Three-Layer Stratified Fluid. Physics of Fluids, 2022, vol. 34, no. 7, article ID: 075104, 16 p. DOI: 10.1063/5.0093754