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.
1. Ali, A. and Kalisch, H. Mechanical balance laws for Boussinesq models of surface water waves. J. Nonlinear Sci., 2012, 22, 371–398.
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.
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.
4. Bjørkavåg, M. and Kalisch, H.Wave breaking in Boussinesq models for undular bores. Phys. Lett. A, 2011, 375(14), 1570–1578.
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.
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.
8. Chen, M. Equations for bi-directional waves over an uneven bottom. Math. Comput. Simulat., 2003, 62(1-2), 3–9.
9. Chou, C. R. and Quyang, K. The deformation of solitary waves on steep slopes. J. Chin. Inst. Eng., 1999, 22(6), 805–812.
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.
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.
14. Gottlieb, D. and Orszag, S. A. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia, 1977.
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.
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.
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.
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.
20. Kalisch, H. and Senthilkumar, A. Derivation of Boussinesq’s shoaling law using a coupled BBM system. Nonlin. Processes Geophys., 2013, 20, 213–219.
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.
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.
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.
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.
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.
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.
29. McCowan, J. On the highest wave of permanent type. Philos. Mag., 1894, 38, 351–357.
30. Miles, J. W. On the Korteweg–de Vries equation for a gradually varying channel. J. Fluid Mech., 1979, 91, 181–190.
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.
32. Nwogu, O. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port C-ASCE., 1993, 119, 618–638.
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.
36. Peregrine, D. H. Long waves on a beach. J. Fluid Mech., 1967, 27(4), 815–827.
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.
40. Teng, M. H. and Wu, T. Y. Evolution of long water waves in variable channels. J. Fluid Mech., 1994, 266, 303–317.
41. Trefethen, L. N. Spectral Methods in Matlab. SIAM, Philadelphia, 2000.
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.