American Institute of Mathematical Sciences

Advanced
June  2010, 13(4): 773-782. doi: 10.3934/dcdsb.2010.13.773

Focusing wave group in shallow water in the presence of wind

Julien Chambarel 1, , Christian Kharif 1, and  Olivier Kimmoun 1,

1. 

Institut de Recherche sur les Phénomènes Hors Equilibre, Technopôle de Château-Gombert 49, rue Joliot Curie - B.P. 146, 13384 Marseille Cedex 13, France, France, France

Received  March 2009 Revised  July 2009 Published  March 2010

The influence of wind on extreme wave events in shallow water is investigated numerically. A series of numerical simulations using a pressure distribution over the steep crests given by the modified Jeffreys' sheltering theory shows that wind blowing over a strongly modulated wave group due to the dispersive focusing of a chirped long wave packet increases the time duration and maximal amplitude of the extreme wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water.
Keywords: wind., shallow water, steep wave group.
Mathematics Subject Classification: 76B07, 76B15, 76B25, 76D33, 76M2.
Citation: Julien Chambarel, Christian Kharif, Olivier Kimmoun. Focusing wave group in shallow water in the presence of wind. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 773-782. doi: 10.3934/dcdsb.2010.13.773
[1]

Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089
[2]

Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065
[3]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
[4]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
[5]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 239-269. doi: 10.3934/dcdss.2014.7.239
[6]

Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186
[7]

Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144
[8]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[9]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[10]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[11]

Issam S. Strub, Julie Percelay, Olli-Pekka Tossavainen, Alexandre M. Bayen. Comparison of two data assimilation algorithms for shallow water flows. Networks & Heterogeneous Media, 2009, 4 (2) : 409-430. doi: 10.3934/nhm.2009.4.409
[12]

Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799
[13]

Olivier Delestre, Arthur R. Ghigo, José-Maria Fullana, Pierre-Yves Lagrée. A shallow water with variable pressure model for blood flow simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 69-87. doi: 10.3934/nhm.2016.11.69
[14]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[15]

Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115
[16]

Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
[17]

Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375
[18]

Marcel Oliver, Sergiy Vasylkevych. Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 827-846. doi: 10.3934/dcds.2011.31.827
[19]

Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
[20]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences