NONLINEAR OBLIQUE INTERACTION OF LARGE AMPLITUDE INTERNAL SOLITARY WAVES

Keisuke Nakayama, Taro Kakinuma, Hidekazu Tsuji, Masayuki Oikawa

Abstract


Solitary waves are typical nonlinear long waves in the ocean. The two-dimensional interaction of solitary waves has been shown to be essentially different from the one-dimensional case and can be related to generation of large amplitude waves (including ‘freak waves’). Concerning surface-water waves, Miles (1977) theoretically analyzed interaction of three solitary waves, which is called “resonant interaction” because of the relation among parameters of each wave. Weakly-nonlinear numerical study (Funakoshi, 1980) and fully-nonlinear one (Tanaka, 1993) both clarified the formation of large amplitude wave due to the interaction (“stem” wave) at the wall and its dependency of incident angle. For the case of internal waves, analyses using weakly nonlinear model equation (ex. Tsuji and Oikawa, 2006) suggest also qualitatively similar result. Therefore, the aim of this study is to investigate the strongly nonlinear interaction of internal solitary waves; especially whether the resonant behavior is found or not. As a result, it is found that the amplified internal wave amplitude becomes about three times as much as the original amplitude. In contrast, a "stem" was not found to occur when the incident wave angle was more than the critical angle, which has been demonstrated in the previous studies.

Keywords


variational principle; internal wave; two-layer system; soliton resonance; fully nonlinear; strong dispersive

References


Aghsaee P., L. Boegman and K. G. Lamb. 2010. Breaking of shoaling internal solitary waves, Journal of Fluid Mechanics, 659, 289-317.http://dx.doi.org/10.1017/S002211201000248X

Antenucci, J.P. and J. Imberger. 2001. Energetics of long internal gravity waves in large lakes, Limnology and Oceanography, 46, 1760-1773.http://dx.doi.org/10.4319/lo.2001.46.7.1760

Boegman, L., G. N. Ivey and J. Imberger 2005. The degeneration of internal waves in lakes with sloping topography, Limnology and Oceanography, 50, 1620–1637.http://dx.doi.org/10.4319/lo.2005.50.5.1620

Boegman, L. and G. N. Ivey. 2009. Flow separation and resuspension beneath shoaling nonlinear internal waves, Journal of Geophysical Research, 114, C02018.http://dx.doi.org/10.1029/2007JC004411

Choi, W. and R. Camassa. 1999. Fully nonlinear internal waves in a two-fluid system, Journal of Fluid Mechanics, 396, 1–36.http://dx.doi.org/10.1017/S0022112099005820

Funakoshi, M. 1980. Reflection of obliquely incident solitary waves, Journal of the Physical Society of Japan, 49(6), 2371-2379.http://dx.doi.org/10.1143/JPSJ.49.2371

Helfrich, K.R. 1992. Internal solitary wave breaking and run-up on a uniform slope, Journal of Fluid Mechanics, 243, 133-154.http://dx.doi.org/10.1017/S0022112092002660

Horn, D.A., J. Imberger and G.N. Ivey. 2001. The degeneration of large-scale interfacial gravity waves in lakes, Journal of Fluid Mechanics, 434, 181-207.http://dx.doi.org/10.1017/S0022112001003536

Kakinuma, T. 2001. A set of fully nonlinear equations for surface and internal gravity waves, Proceedings of 5th International Conference on Computer Modelling of Seas and Coastal Regions, WIT Press, 225-234.

Kakinuma T. and K. Nakayama. 2007. Numerical simulation of internal waves using a set of fully nonlinear internal wave equations, Annual Journal of Hydraulic Engineering, JSCE, 51, 169-174.

Kodama, Y. 2010. KP solitons in shallow water, Journal of Physics A: Mathematical and Theoretical, 43, 434004-434057.http://dx.doi.org/10.1088/1751-8113/43/43/434004

Li, W., H. Yeh and Y. Kodama. 2011. On the Mach reflection of a solitary wave: revisited, Journal of Fluid Mechanics, 672, 326-357.http://dx.doi.org/10.1017/S0022112010006014

Luke, J.C. 1967. A variational principle for a fluid with a free surface, Journal of Fluid Mechanics, 27, 395–397.http://dx.doi.org/10.1017/S0022112067000412

Miles, J.W. 1977. Resonantly interacting solitary waves. Journal of Fluid Mechanics, 79, 171-179.http://dx.doi.org/10.1017/S0022112077000093

Mirie, R.M. and S.A. Pennell. 1989. Internal solitary waves in a two-fluid system, Physics of Fluids, A1(6), pp.986-991.

Mirie, R.M. and S.A. Pennell. 1989. Internal solitary waves in a two-fluid system, Physics of Fluids, A1(6), 986-991.

Nakayama K. and J. Imberger. 2010. Residual circulation due to internal waves shoaling on a slope, Limnology and Oceanography, 55, 1009-1023.http://dx.doi.org/10.4319/lo.2010.55.3.1009

Nakayama K. and T. Kakinuma. 2010. Internal waves in a two-layer system using fully nonlinear internal-wave equations, International Journal for Numerical Methods in Fluids, 62(5), 574-590.

Nakayama K., T. Shintani, K. Kokubo, Y. Maruya, T. Kakinuma, K. Komai and T. Okada. 2012. Residual current over a uniform slope due to breaking of internal waves in a two-layer system, Journal of Geophysical Research, accepted.http://dx.doi.org/10.1029/2012JC008155

Pierson, D.C. and G. A. Weyhenmeyer. 1994. High resolution measurements of sediment resuspension above an accumulation bottom in a stratified lake, Hydrobiologia, 284, 43-57.http://dx.doi.org/10.1007/BF00005730

Tanaka, M. 1993. Mach reflection of a large-amplitude solitary wave, Journal of Fluid Mechanics, 248, 637-661.http://dx.doi.org/10.1017/S0022112093000941

Tsuji, H. and M. Oikawa. 2007. Oblique interaction of solitons in an extended Kadomtsev–Petviashvili equation, Journal of Physical Society of Japan, 76, 84401–84408.http://dx.doi.org/10.1143/JPSJ.76.084401

Wallace, B.C., and D. L. Wilkinson. 1988. Run-up of internal waves on a gentle slope in a two-layered system, Journal of Fluid Mechanics, 191, 419-442.http://dx.doi.org/10.1017/S0022112088001636


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