Head-on Collision of Internal Waves of Large Amplitudes
Abstract
The dynamics and energetics of a frontal collision of internal solitary waves of high amplitude propagating in a two-layer stratified fluid are studied numerically. The computations are carried out within the framework of the Navier—Stokes equations in the Boussinesq approximation. It was shown that the frontal collision of internal solitary waves of moderate amplitude leads to a small phase shift and to the generation of dispersive wavetrain trailing behind transmitted solitary wave. The phase shift grows with increasing amplitudes of the interacting waves and approaches the limiting value at large amplitudes of the waves. The deviation of the maximum wave height during collision from the twice the amplitude does not grow with increasing amplitude in the case of interaction of wave of large amplitude in contrast to the moderate amplitude waves. It was shown that the interaction of waves of large amplitude leads to the shear instability and the formation of Kelvin—Helmholtz vortices in the interface layer, however, subsequently waves again become stable.
About the Authors
V. S. Maderich
The Institute of Mathematical Machines and Systems Problems of the Ukraine National Academy of Science (IMMSP NASU)
Russian Federation
Russian Federation
K. V. Terletska
The Institute of Mathematical Machines and Systems Problems of the Ukraine National Academy of Science (IMMSP NASU)
Russian Federation
Russian Federation
I. A. Brovchenko
The Institute of Mathematical Machines and Systems Problems of the Ukraine National Academy of Science (IMMSP NASU)
Russian Federation
Russian Federation
References
1. Grimshaw R., Pelinovsky E., Talipova T. Modeling internal solitary waves in the coastal ocean. Survey in Geophysics. 2007, 28, 273—298.
2. Helfrich K. R., Melville W. K. Long nonlinear internal waves. Ann. Review Fluid Mech. 2006, 38, 395—425.
3. Grimshaw R., Zhu Y. Oblique interactions between internal solitary waves. Stud. Appl. Math. 1994, 92, 249—270.
4. Wang C., Pawlowicz R. Oblique wave-wave interactions of nonlinear near-surface internal waves in the Strait of Georgia. J. Geophys. Res. 2012, 117, C06031.
5. Mirie R., Su C. H. Internal solitary waves and their head-on collision. Part 1. J. Fluid Mech. 1984, 147, 213—231.
6. Mirie R. M., Su C. H. Internal solitary waves and their head-on collision. II. Phys. Fluids. 1986, 29, 31—37.
7. Nguyen H. Y., Dias F. A Boussinesq system for two-way propagation of interfacial waves. Physica D. 2008, 237, 2365—2389.
8. Jo T.-C., Choi W. On stabilizing the strongly nonlinear internal wave model. Stud. Appl. Math. 2008, 120, 65—85.
9. Mellor G. L. An equation of state for numerical models of ocean and estuaries. J Atmos. Ocean. Tech. 1991, 8, 609—611.
10. Kanarska Y., Maderich V. A non-hydrostatic numerical model for calculating free-surface stratified flows. Ocean Dynamics. 2003, 53, 176—185.
11. Kao T. W., Pan F. S., Renouard D. Internal solitions on the pycnocline: generation, propagation, shoaling and breaking over a slope. J. Fluid Mech. 1985, 159, 19—53.
12. Hsu R.-C., M. H. Cheng, Chen C.-Y. Potential hazards and dynamical analysis of interfacial solitary wave interactions. Nat Hazards. 2013, 65:255–278, DOI 0.1007/s11069-012-0360-9.
13. Miles J. W., Howard L. N. Note on a heterogeneous shear flow. J. Fluid Mech. 1964, 20, 331—336.
14. Miyata M. An internal solitary wave of large amplitude. La Mer. 1985, 23, 43—48.
15. Choi W., Camassa R. Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 1999, 396, 1—36.
16. Troy C. D., Koseff J. R. The instability and breaking of long internal waves. J. Fluid Mech. 2005, 543, 107—336.
17. Fructus D., Carr M., Grue J., Jensen A., Davies P. A. Shear-induced breaking of large internal solitary waves. J. Fluid Mech. 2006, 620, 1—29.
18. Maderich V., Talipova T., Grimshaw R., Terletska K., Brovchenko I., Pelinovsky E., Choi B. H. Interaction of a large amplitude interfacial solitary wave of depression with a bottom step. Physics of Fluids. 2010, 22, doi:10.1063/1.3455984.
19. Barad M. F., Fringer O. B. Simulations of shear instabilities in interfacial gravity waves. J. Fluid Mech. 2010, 644, 61—95.
20. Maxworthy T. Experiments on collisions between solitary waves. J. Fluid Mech. 1976, 76, 177—185.
21. Chambarel J., Kharif C., Touboul J. Head-on collision of two solitary waves and residual falling jet formation. Nonlinear Proc. Geoph. 2009, 16, 111—122.
Review
For citations:
Maderich V.S., Terletska K.V., Brovchenko I.A. Head-on Collision of Internal Waves of Large Amplitudes. Fundamental and Applied Hydrophysics. 2015;8(3):44-52. (In Russ.)
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