headerpos: 12198
 
 
 

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
Publisher
Journal Information
» Editorial Board
» Editorial Policy
» Archival Policy
» Article Publication Charges
» Copyright and Licensing Policy
Guidelines for Authors
» For Authors
» Instructions to Authors
» LaTex style files
Guidelines for Reviewers
» For Reviewers
» Review Form
Open Access
List of Issues
» 2019
» 2018
» 2017
» 2016
Vol. 65, Issue 4
Vol. 65, Issue 3
Vol. 65, Issue 2
Vol. 65, Issue 1
» 2015
» 2014
» 2013
» 2012
» 2011
» 2010
» 2009
» 2008
» Back Issues Phys. Math.
» Back Issues Chemistry
» Back issues (full texts)
  in Google. Phys. Math.
» Back issues (full texts)
  in Google. Chemistry
» Back issues (full texts)
  in Google Engineering
» Back issues (full texts)
  in Google Ecology
» Back issues in ETERA Füüsika, Matemaatika jt
Subscription Information
» Prices
Internet Links
Support & Contact
Publisher
» Staff
» Other journals

On the influence of wave reflection on shoaling and breaking solitary Waves; pp. 414–430

(Full article in PDF format) doi: 10.3176/proc.2016.4.06


Authors

Amutha Senthilkumar

Abstract

A coupled BBM system of equations is studied in the situation of water waves propagating over a decreasing fluid depth. A conservation equation for mass and also a wave breaking criterion, both valid in the Boussinesq approximation, are found. A Fourier collocation method coupled with a 4-stage Runge–Kutta time integration scheme is employed to approximate solutions of the BBM system. The mass conservation equation is used to quantify the role of reflection in the shoaling of solitary waves on a sloping bottom. Shoaling results based on an adiabatic approximation are analysed. Wave shoaling and the criterion of the breaking of solitary waves on a sloping bottom are studied. To validate the numerical model the simulation results are compared with reference results and a good agreement between them can be observed. Shoaling of solitary waves is calculated for two different types of mild slope model systems. Comparison with reference solutions shows that both of these models work well in their respective regimes of applicability.

Keywords

coupled BBM system, shoaling rates, mass conservation law.

References

1. Ali , A. and Kalisch , H. Mechanical balance laws for Boussinesq models of surface water waves. J. Nonlinear Sci. , 2012 , 22 , 371–398.
https:/doi.org/10.1007/s00332-011-9121-2

2. Ali , A. and Kalisch , H. On the formation of mass , momentum and energy conservation in the KdV equation. Acta Appl. Math. , 2014 , 133 , 113–131.
https:/doi.org/10.1007/s10440-013-9861-0

3. Benjamin , T. B. , Bona , J. L. , and Mahony , J. J. Model equations for long waves in nonlinear dispersive systems. Philos. T. Roy. Soc. A , 1972 , 272 , 47–78.
https:/doi.org/10.1098/rsta.1972.0032

4. Bjørkavåg , M. and Kalisch , H.Wave breaking in Boussinesq models for undular bores. Phys. Lett. A , 2011 , 375(14) , 1570–1578.
https:/doi.org/10.1016/j.physleta.2011.02.060

5. Bona , J. L. , Chen , M. , and Saut , J. C. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. , 2002 , 12(4) , 283–318.
https:/doi.org/10.1007/s00332-002-0466-4

6. Camfield , F. E. and Street , R. L. Shoaling of solitary waves on small slopes. J. Waterw. Harb. Coast. Eng. , 1969 , 95(1) , 1–22.

7. Chen , M. Exact solutions of various Boussinesq systems. Appl. Math. Lett. , 1998 , 11(5) , 45–49.
https:/doi.org/10.1016/S0893-9659(98)00078-0

8. Chen , M. Equations for bi-directional waves over an uneven bottom. Math. Comput. Simulat. , 2003 , 62(1-2) , 3–9.
https:/doi.org/10.1016/S0378-4754(02)00193-3

9. Chou , C. R. and Quyang , K. The deformation of solitary waves on steep slopes. J. Chin. Inst. Eng. , 1999 , 22(6) , 805–812.
https:/doi.org/10.1080/02533839.1999.9670516

10. Chou , C. R. and Quyang , K. Breaking of solitary waves on uniform slopes. China Ocean Eng. , 1999 , 13(4) , 429–442.

11. Chou , C. R. , Shih , R. S. , and Yim , J. Z. Numerical study on breaking criteria for solitary waves. China Ocean Eng. , 2003 , 17(4) , 589–604.

12. Duncan , J. H. Spilling breakers. Annu. Rev. Fluid Mech. , 2001 , 33 , 519–547.
https:/doi.org/10.1146/annurev.fluid.33.1.519

13. Filippini , A. G. , Bellec , S. , Colin , M. , and Ricchiuto , M. On the nonlinear behaviour of Boussinesq type models: Amplitudevelocity vs amplitude-flux forms. Coast. Eng. , 2015 , 99 , 109–123.
https:/doi.org/10.1016/j.coastaleng.2015.02.003

14. Gottlieb , D. and Orszag , S. A. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM , Philadelphia , 1977.
https:/doi.org/10.1137/1.9781611970425

15. Green , A. E. and Naghdi , P. M. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. , 1976 , 78(2) , 237–246.
https:/doi.org/10.1017/S0022112076002425

16. Grilli , S. T. , Subramanya , R. , Svendsen , I. A. , and Veeramony , J. Shoaling of solitary waves on plane beaches. J. Waterw. Port C-ASCE , 1994 , 120 , 609–628.
https:/doi.org/10.1061/(ASCE)0733-950X(1994)120:6(609)

17. Grilli , S. T. , Svendsen , I. A. , and Subramanya , R. Breaking criterion and characteristics for solitary waves on slopes. J.Waterw. Port C-ASCE , 1997 , 123 , 102–112.
https:/doi.org/10.1061/(ASCE)0733-950X(1997)123:3(102)

18. Ippen , A. and Kulin , G. The shoaling and breaking of the solitary wave. Coast. Eng. Proc. , 1954 , No 5 , Chapter 4 , 27–47.

19. Johnson , R. S. On the development of a solitary wave moving over uneven bottom. Proc. Cambridge Philos. Soc. , 1973 , 73 , 183–203.
https:/doi.org/10.1017/S0305004100047605

20. Kalisch , H. and Senthilkumar , A. Derivation of Boussinesq’s shoaling law using a coupled BBM system. Nonlin. Processes Geophys. , 2013 , 20 , 213–219.
https:/doi.org/10.5194/npg-20-213-2013

21. Kennedy , A. B. , Chen , Q. , Kirby , J. T. , and Dalrymple , R. A. Boussinesq modeling of wave transformation , breaking , and runup. I: 1D. J. Waterw. Port C-ASCE. , 2000 , 126 , 39–47.
https:/doi.org/10.1061/(ASCE)0733-950X(2000)126:1(39)

22. Khorsand , Z. and Kalisch , H. On the shoaling of solitary waves in the KdV equation. Coast. Eng. Proc. , 2014 , 34 , waves 44.

23. Kishi , T. and Saeki , H. The shoaling , breaking and runup of the solitary wave on impermeable rough slopes. Coast. Eng. Proc. , 2011 , 10 , 322–347.

24. Longuet-Higgins , M. S. On the mass , momentum , energy and circulation of a solitary wave. Proc. Roy. Soc. A. , 1974 , 337 , 1–13.
https:/doi.org/10.1098/rspa.1974.0035

25. Longuet-Higgins , M. S. and Fenton , J. D. On the mass , momentum , energy and circulation of a solitary wave II. Proc. Roy. Soc. A. , 1974 , 340 , 471–493.
https:/doi.org/10.1098/rspa.1974.0166

26. Madsen , O. S. and Mei , C. C. The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. , 1969 , 39 , 781–791.
https:/doi.org/10.1017/S0022112069002461

27. Madsen , P. A. , Murray , R. , and Sørensen , O. R. A new form of the Boussinesq equations with improved linear dispersioon characteristics. Coast. Eng. , 1991 , 15(4) , 371–388.
https:/doi.org/10.1016/0378-3839(91)90017-B

28. Madsen , P. A. and Schäffer , H. A. Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Philos. T. Roy. Soc. A. , 1998 , 356 , 3123–3184.
https:/doi.org/10.1098/rsta.1998.0309

29. McCowan , J. On the highest wave of permanent type. Philos. Mag. , 1894 , 38 , 351–357.
https:/doi.org/10.1080/14786449408620643

30. Miles , J. W. On the Korteweg–de Vries equation for a gradually varying channel. J. Fluid Mech. , 1979 , 91 , 181–190.
https:/doi.org/10.1017/S0022112079000100

31. Mitsotakis , D. E. Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves. Math. Comput. Simulat. , 2009 , 80(4) , 860–873.
https:/doi.org/10.1016/j.matcom.2009.08.029

32. Nwogu , O. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port C-ASCE. , 1993 , 119 , 618–638.
https:/doi.org/10.1061/(ASCE)0733-950X(1993)119:6(618)

33. Ostrovsky , L. A. and Pelinovsky , E. N. Wave transformation on the surface of a fluid of variable depth. Atmos. Oceanic Phys. , 1970 , 6 , 552–555.

34. Pelinovsky , E. N. and Talipova , T. G. Height variations of large amplitude solitary waves in the near-shore zone. Oceanology , 1977 , 17(1) , 1–3.

35. Pelinovsky , E. N. and Talipova , T. G. Change of height of the solitary wave of large amplitude in the beach zone. Mar. Geodesy , 1979 , 2(4) , 313–321.
https:/doi.org/10.1080/15210607909379359

36. Peregrine , D. H. Long waves on a beach. J. Fluid Mech. , 1967 , 27(4) , 815–827.
https:/doi.org/10.1017/S0022112067002605

37. Senthilkumar , A. BBM equation with non-constant coefficients. Turk. J. Math. , 2013 , 37 , 652–664.

38. Svendsen , I. A. Introduction to Nearshore Hydrodynamics. World Scientific , Singapore , 2006 , 24.

39. Synolakis , C. E. The runup of solitary waves. J. Fluid Mech. , 1987 , 185 , 523–545.
https:/doi.org/10.1017/S002211208700329X

40. Teng , M. H. and Wu , T. Y. Evolution of long water waves in variable channels. J. Fluid Mech. , 1994 , 266 , 303–317.
https:/doi.org/10.1017/S0022112094001011

41. Trefethen , L. N. Spectral Methods in Matlab. SIAM , Philadelphia , 2000.
https:/doi.org/10.1137/1.9780898719598

42. Wei , G. , Kirby , J. T. , Grilli , S. T. , and Subramanya , R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. , 1995 , 294 , 71–92.
https:/doi.org/10.1017/S0022112095002813

43. Whitham , G. Linear and Nonlinear Waves. Wiley , New York , 1974.

 
Back

Current Issue: Vol. 68, Issue 2, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December