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Generation of mode-2 internal waves in a two-dimensional stratification by a mode-1 internal wave. (English) Zbl 07214088
Summary: The generation of mode-2 nonlinear internal waves (IWs) by the evolution of a mode-1 IW in a two-dimensional stratification is investigated. A generation model accounting for intermodal interaction is derived based on a multi-modal approach in a weakly nonlinear and non-hydrostatic configuration. The generation model is numerically solved to simulate the evolution of mode-1 and mode-2 IWs in an inhomogeneous pycnocline. The numerical experiments confirm that mode-2 IWs are generated due to linear and nonlinear intermodal interaction. The mode-2 IW continues growing and gradually separates with the mode-1 IW during the generation process. The numerical results suggest that the pycnocline strength or thickness prominently affects the generation of mode-2 IWs, followed by pycnocline depth. A weakening or thinning pycnocline favors the generation of mode-2 IWs by evidently enhancing linear and nonlinear intermodal interaction, whereas a shoaling pycnocline favors a rapid growth rate mainly by enhancing linear intermodal interaction. The wave amplitude of an initial mode-1 IW strongly affects the generation of mode-2 IWs and increasing it can noticeably enlarge mode-2 IWs.
76 Fluid mechanics
35 Partial differential equations
Full Text: DOI
[1] Helfrich, K. R.; Melville, W. K., Long nonlinear internal waves, Annu. Rev. Fluid Mech., 38, 395-425 (2006) · Zbl 1098.76018
[2] Jackson, C., An Atlas of Internal Solitary-Like Waves and their Properties (2004), http://www.internalwaveatlas.com
[3] Cai, S.; Long, X.; Gan, Z., A method to estimate the forces exerted by internal solitons on cylindrical piles, Ocean Eng., 30, 673-689 (2003)
[4] Zhou, J. X.; Zhang, X. Z.; Roger, P. H., Resonant interaction of sound wave with internal solitons in the coastal zone, J. Acoust. Soc. Am., 90, 2042-2054 (1991)
[5] Bogucki, D. J.; Redekopp, L. G.; Barth, J., Internal solitary waves in the coastal mixing and optics 1996 experiment: Multimodal structure and resuspension, J. Geophys. Res., 110, 0 (2005)
[6] Benney, D. J., Long non-linear waves in fluid flows, J. Math. Phys., 45, 52-63 (1966) · Zbl 0151.42501
[7] Yang, Y. J.; Fang, Y. C.; Tang, T. Y.; Ramp, S. R., Convex and concave types of second baroclinic mode internal solitary waves, Nonlinear Processes Geophys., 17, 605-614 (2010)
[8] Davis, R. E.; Acrivos, A., Solitary internal waves in deep water, J. Fluid Mech., 29, 593-607 (1967) · Zbl 0147.46503
[9] Kao, T. W.; Pao, H.-P., Wake collapse in the thermocline and internal solitary waves, J. Fluid Mech., 97, 115-127 (1980)
[10] Mehta, A. P.; Sutherland, B. R.; Kyba, P. J., Interfacial gravity currents. II. wave excitation, Phys. Fluids, 14, 3558-3569 (2002) · Zbl 1185.76252
[11] Helfrich, K. R.; Melville, W. K., On long nonlinear internal waves over slope-shelf topography, J. Fluid Mech., 167, 285-308 (1986)
[12] New, A. L.; Pingree, R. D., Large-amplitude internal soliton packets in the central Bay of Biscay, Deep-Sea Res. Part A, 37, 513-524 (1990)
[13] New, A. L.; Pingree, R. D., Local generation of internal soliton packets in the central bay of Biscay, Deep-Sea Res. Part A, 39, 1521-1534 (1992)
[14] Mercier, M. J.; Mathur, M.; Gostiaux, L.; Gerkema, T.; Magalh√£es, J. M.; Da Silva, J. C.B.; Dauxois, T., Soliton generation by internal tidal beams impinging on a pycnocline: Laboratory experiments, J. Fluid Mech., 704, 37-60 (2012) · Zbl 1246.76007
[15] Stastna, M.; Peltier, W. R., On the resonant generation of large-amplitude internal solitary and solitary-like waves, J. Fluid Mech., 543, 267-292 (2005) · Zbl 1137.76328
[16] Vlasenko, V. I.; Hutter, K., Generation of second mode solitary waves by the interaction of a first mode soliton with a sill, Nonlinear Processes Geophys., 8, 223-239 (2001)
[17] Lamb, K. G., A numerical investigation of solitary internal waves with trapped cores formed via shoaling, J. Fluid Mech., 451, 109-144 (2002) · Zbl 1009.76010
[18] Lamb, K. G.; Warn-Varnas, A., Two-dimensional numerical simulations of shoaling internal solitary waves at the ASIAEX site in the South China Sea, Nonlinear Processes Geophys., 22, 289-312 (2015)
[19] Grisouard, N.; Staquet, C.; Gerkema, T., Generation of internal solitary waves in a pycnocline by an internal wave beam: A numerical study, J. Fluid Mech., 676, 491-513 (2011) · Zbl 1241.76101
[20] Chen, Z.; Xie, J.; Wang, D.; Zhan, J.; Xu, J.; Cai, S., Density stratification influences on generation of different modes internal solitary waves, J. Geophys. Res., 119, 7029-7046 (2014)
[21] Guo, C.; Chen, X., Numerical investigation of large amplitude second mode internal solitary waves over a slope-shelf topography, Ocean Model., 42, 80-91 (2012)
[22] Belogortsev, A. S.; Rybak, S. A.; Serebryanyi, A. N., Second-mode nonlinear internal waves over a sloping bottom, Acoust. Phys., 59, 62-67 (2013)
[23] Xie, J. S.; Pan, J. Y.; Jay, D. A., Multimodal internal waves generated over a subcritical ridge: Impact of the upper-ocean stratification, J. Phys. Oceanogr., 45, 904-926 (2015)
[24] Yang, Y. J.; Fang, Y. C.; Chang, M.-H.; Ramp, S. R.; Kao, C.-C.; Tang, T. Y., Observations of second baroclinic mode internal solitary waves on the continental slope of the northern South China Sea, J. Geophys. Res., 114, 10 (2009)
[25] Liu, A. K.; Su, F.-C.; Hsu, M.-K.; Kuo, N.-J.; Ho, C.-R., Generation and evolution of mode-two internal waves in the South China Sea, Cont. Shelf Res., 59, 18-27 (2013)
[26] Ramp, S. R.; Yang, Y. J.; Reeder, D. B.; Bahr, F. L., Observations of a mode-2 nonlinear internal wave on the northern Heng-Chun Ridge south of Taiwan, J. Geophys. Res., 117, C03043 (2012)
[27] Shroyer, E. L.; Moum, J. N.; Nash, J. D., Mode 2 waves on the continental shelf: Ephemeral components of the nonlinear internal wavefield, J. Geophys. Res., 115, 0 (2010)
[28] Konyaev, K. V.; Sabinin, K. D.; Serebryany, A. N., Large-amplitude internal waves at the Mascarene Ridge in the Indian Ocean, Deep-Sea Res. Part I, 42, 2075-2091 (1995)
[29] da Silva, J. C.B.; New, A. L.; Magalhaes, J. M., On the structure and propagation of internal solitary waves generated at the Mascarene Plateau in the Indian Ocean, Deep-Sea Res. Part I, 58, 229-240 (2011)
[30] Dong, D.; Yang, X. F.; Li, X. F.; Li, Z. W., SAR observation of eddy-Induced mode-2 internal solitary waves in the South China Sea, IEEE Trans. Geosci. Remote Sens., 54, 6674-6686 (2016)
[31] Vasiliy, V.; Stashchuk, N.; Hutter, K., Baroclinic Tides Theoretical Modelling and Observational Evidence (2005), Cambridge University Press: Cambridge University Press New York · Zbl 1432.86001
[32] Gerkema, T., Development of internal solitary waves in various thermocline regimes - a multi-modal approach, Nonlinear Processes Geophys., 10, 397-405 (2003)
[33] Griffiths, S. D.; Grimshaw, R. H.J., Internal tide generation at the continental shelf modeled using a modal decomposition: Two-Dimensional results, J. Phys. Oceanogr., 37, 428-451 (2007)
[34] Sakai, T.; Redekopp, L. G., A weakly nonlinear model for multi-modal evolution of wind-generated long internal waves in a closed basin, Nonlinear Processes Geophys., 16, 487-502 (2009)
[35] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16-42 (1992) · Zbl 0759.65006
[36] Tyliszczak, A., Influence of the compact explicit filtering method on the perturbations growth in temporal shear-layer flow, J. Theoret. Appl. Mech., 44, 19-32 (2003)
[37] Wei, G.; Kirby, J. T., Time-dependent numerical code for extended boussinesq equations, J. Waterw. Port Coast Ocean Eng., 121, 251-261 (1995)
[38] Buijsman, M. C.; McWilliams, J. C.; Jackson, C. R., East-west asymmetry in nonlinear internal waves from Luzon Strait, J. Geophys. Res., 115 (2010)
[39] Gerkema, T., Internal and interfacial tides: Beam scattering and local generation of solitary waves, J. Mar. Res., 59, 227-255 (2001)
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