Skip to main content Accessibility help
×
Home

Transformation of a shoaling undular bore

  • G. A. El (a1), R. H. J. Grimshaw (a1) and W. K. Tiong (a1)

Abstract

We consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coefficient Korteweg–de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons – an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments.

Copyright

Corresponding author

Email address for correspondence: G.El@lboro.ac.uk

References

Hide All
1. Ablowitz, M. J., Baldwin, D. E. & Hoefer, M. A. 2009 Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction. Phys. Rev. E 80, 016603.
2. Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.
3. Boussinesq, J. 1872 Théorie des ondes des remous qui se propagent le long d’un canal rectangulaire, en communuuant au liquide contenu dans ce canal des vitesses sensblemnt pareilles de la surface au fond. J. Math. Pures Appl. 17, 55108.
4. Claeys, T. & Grava, T. 2010 Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit. SIAM J. Math. Anal. 42, 21322154.
5. El, G. A. 2005 Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15, 037103.
6. El, G. A. & Grimshaw, R. H. J. 2002 Generation of undular bores in the shelves of slowly-varying solitary waves. Chaos 12, 10151026.
7. El, G. A., Grimshaw, R. H. J. & Kamchatnov, A. M. 2007 Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction. J. Fluid Mech. 585, 213244.
8. El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2006 Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104.
9. El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2009 Transcritical shallow-water flow past topography: finite-amplitude theory. J. Fluid Mech. 640, 187215.
10. El, G. A., Khodorovskii, V. V. & Leszczyszyn, A. M. 2012 Refraction of dispersive shock waves. Physica D 241, 15671587.
11. Esler, J. G. & Pearce, J. D. 2011 Dispersive dam-break and lock-exchange flows in a two-layer fluid. J. Fluid Mech. 667, 555585.
12. Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A 289, 373404.
13. Grimshaw, R. 1979 Slowly varying solitary waves. I Korteweg-de Vries equation. Proc. R. Soc. Lond. A 368, 359375.
14. Grimshaw, R. 1981 Evolution equations for long nonlinear internal waves in stratified shear flows. Stud. Appl. Maths 65, 159188.
15. Grimshaw, R. 2007a Internal solitary waves in a variable medium. Gesellsch. Angew. Math. 30, 96109.
16. Grimshaw, R. 2007b Solitary waves propagating over variable topography. In Tsunami and Nonlinear Waves (ed. Kundu, A. ), pp. 4962. Springer.
17. Grimshaw, R., Pelinovsky, E., Talipova, T. & Kurkin, A. 2004 Simulation of the transformation of internal solitary waves on oceanic shelves. J. Phys. Oceanogr. 34, 27742779.
18. Grimshaw, R. H. J. & Smyth, N. F. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429464.
19. Gurevich, A. V. & Pitaevskii, L. P. 1974 Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38, 291297.
20. Johnson, R. S. 1973 On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183203.
21. Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.
22. Kakutani, T. 1971 Effect of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272276.
23. Kamchatnov, A. M. 2004 On Whitham theory for perturbed integrable equations. Physica D 188, 247261.
24. Kaup, D. J. & Newell, A. C. 1979 Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory. Proc. R. Soc. Lond. A 361, 413446.
25. Kawahara, T. 1975 Derivative-expansion method for nonlinear waves on a liquid layer of slowly varying depth. J. Phys. Soc. Japan 38, 12001206.
26. Khruslov, E. A. 1976 Asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with step-like initial data. Math. USSR-Sb. 28, 229248.
27. Kivshar, Yu. S. & Malomed, B. A. 1989 Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61, 763907.
28. Madsen, O. S. & Mei, C. C. 1969 The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 39, 781891.
29. Madsen, P. A., Fuhrman, D. R. & Schäffer, H. A. 2008 On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113, C12012.
30. Malomed, B. A. & Shrira, V. I. 1991 Soliton caustics. Physica D 53, 112.
31. Miles, J. W. 1983a Solitary wave evolution over a gradual slope with turbulent friction. J. Phys. Oceanogr. 13, 551553.
32. Miles, J. W. 1983b Wave evolution over a gradual slope with turbulent friction. J. Fluid Mech. 133, 207216.
33. Newell, A. 1985 Solitons in Mathematics and Physics. SIAM.
34. Ostrovsky, L. A. & Pelinovsky, E. N. 1975 Refraction of nonlinear sea waves in a coastal zone. Izv. Akad. Nauk SSSR Atmos. Ocean. Phys. 11, 3741.
35. Schiesser, W. E. 1991 The Numerical Method of Lines: Integration of Partial Differential Equations. Academic.
36. Scotti, A., Beardsley, R. C., Butman, B. & Pineda, J. 2008 Shoaling of nonlinear internal waves in Massachusetts bay. J. Geophys. Res. 113, C08031.
37. Smyth, N. F. & Holloway, P. E. 1988 Hydraulic jump and undular bore formation on a shelf break. J. Phys. Oceanogr. 18, 947962.
38. Tissier, M., Bonneton, P., Marche, F., Chazel, F. & Lannes, D. 2011 Nearshore dynamics of tsunami-like undular bores using a fully nonlinear Boussinesq model. J. Coastal Res. 603607 (Special Issue 64).
39. Whitham, G. B. 1965 Non-linear dispersive waves. Proc. R. Soc. Lond. A 283, 238261.
40. Whitham, G. B. 1974 Linear and Nonlinear Waves. J. Wiley and Sons.
41. Whitham, G. B. 1984 Comments on periodic waves and solitons. IMA J. Appl. Maths 32, 353366.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Transformation of a shoaling undular bore

  • G. A. El (a1), R. H. J. Grimshaw (a1) and W. K. Tiong (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed