Combined effects of topography and bottom friction on shoaling internal solitary waves in the South China Sea

doi:10.1007/s10483-019-2465-8

Abstract

Abstract:

A numerical study to a generalized Korteweg-de Vries (KdV) equation is adopted to model the propagation and disintegration of large-amplitude internal solitary waves (ISWs) in the South China Sea (SCS). Based on theoretical analysis and in situ measurements, the drag coefficient of the Chezy friction is regarded as inversely proportional to the initial amplitude of an ISW, rather than a constant as assumed in the previous studies. Numerical simulations of ISWs propagating from a deep basin to a continental shelf are performed with the generalized KdV model. It is found that the depression waves are disintegrated into several solitons on the continental shelf due to the variable topography. It turns out that the amplitude of the leading ISW reaches a maximum at the shelf break, which is consistent with the field observation in the SCS. Moreover, a dimensionless parameter defining the relative importance of the variable topography and friction is presented.

Key words: probabilistic metric space, Caristi’s coincidence theorem, Ekeland’s variational principle, partial ordering set., internal solitary wave (ISW), Chezy friction, South China Sea (SCS), variable topography

2010 MSC Number:

35Q53
76B55

Dalin TAN, Jifu ZHOU, Xu WANG, Zhan WANG. Combined effects of topography and bottom friction on shoaling internal solitary waves in the South China Sea. Applied Mathematics and Mechanics (English Edition), 2019, 40(4): 421-434.

TrendMD

References

[1] DUDA, T. F., LYNCH, J. F., and IRISH, J. D. Internal tide and nonlinear internal wave behavior at the continental slope in the northern South China Sea. IEEE Journal of Oceanic Engineering, 29(4), 1105-1130(2004)
[2] RAMP, S. R., YANG, Y. J., and BAHR, F. L. Characterizing the nonlinear internal wave climate in the northeastern South China Sea. Nonlinear Processes in Geophysics, 17(5), 481-498(2010)
[3] ALFORD, M. H., LIEN, R. C., and SIMMONS, H. Speed and evolution of nonlinear internal waves transiting the South China Sea. Journal of Physical Oceanography, 40(6), 1338-1355(2010)
[4] CAI, S. Q., XIE, J. S., and HE, J. L. An overview of internal solitary waves in the South China Sea. Surveys in Geophysics, 33(5), 927-943(2012)
[5] GRIMSHAW, R., GUO, C., and HELFRICH, K. Combined effect of rotation and topography on shoaling oceanic internal solitary waves. Journal of Physical Oceanography, 44(4), 1116-1132(2014)
[6] LAMB, K. G. Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annual Review of Fluid Mechanics, 46(1), 231-254(2014)
[7] ALFORD, M. H., PEACOK, T., and MACKINNON, J. A. The formation and fate of internal waves in the South China Sea. nature, 528(7580), 65-69(2015)
[8] FETT, R. and RABE, K. Satellite observation of internal wave refraction in the South China Sea. Geophysical Research Letters, 4(5), 189-191(1977)
[9] ZHAO, Z., KLEMAS, V., and ZHENG, Q. Remote sensing evidence for baroclinic tide origin of internal solitary waves in the northeastern South China Sea. Geophysical Research Letters, 31(6), L06302(2004)
[10] JACKSON, C. R. An empirical model for estimating the geographic location of nonlinear internal solitary waves. Journal of Atmospheric and Oceanic Technology, 26(10), 2243-2255(2009)
[11] WEI, G., LI, J. C., and DAI, S. Q. Surface effects of internal wave generated by a moving source in a two-layer fluid of finite depth. Applied Mathematics and Mechanics (English Edition), 24(9), 1025-1040(2003) https://doi.org/10.1007/BF02437635
[12] CAI, S. Q. and XIE, J. S. A propagation model for the internal solitary waves in the northern South China Sea. Journal of Geophysical Research, 115(C12), C12074(2010)
[13] QU, F. L. and WANG, W. Q. Alternating segment explicit-implicit scheme for nonlinear thirdorder KdV equation. Applied Mathematics and Mechanics (English Edition), 28(7), 973-980(2007) https://doi.org/10.1007/s10483-007-0714-y
[14] HELFRICH, K. R. and MELVILLE, W. K. Long nonlinear internal waves. Annual Review of Fluid Mechanics, 38(38), 395-425(2006)
[15] GRIMSHAW, R. Nonlinear wave equations for oceanic internal solitary waves. Studies in Applied Mathematics, 136(2), 214-237(2016)
[16] ZHOU, H. Y. and PIAO, D. X. Hamiltonian long wave expansions for internal waves over a periodically varying bottom. Applied Mathematics and Mechanics (English Edition), 29(6), 745-756(2008) https://doi.org/10.1007/s10483-008-0606-x
[17] CAI, S. Q., GAN, Z. J., and LONG, X. M. Some characteristics and evolution of the internal soliton in the northern South China Sea. Chinese Science Bulletin, 47(1), 21-27(2002)
[18] DJORDJEVIC, V. D. and REDEKOPP, L. G. The fission and disintegration of internal solitary waves moving over two-dimensional topography. Journal of Physical Oceanography, 8(6), 1016-1024(1978)
[19] KNICKERBOCKER, C. J. and NEWELL, A. C. Shelves and the Korteweg-de Vries equation. Journal of Fluid Mechanics, 98, 803-818(1980)
[20] GRIMSHAW, R. and MITSUDERA, H. Slowly varying solitary wave solutions of the perturbed Korteweg-de Vries equation revisited. Studies in Applied Mathematics, 90(1), 75-86(1993)
[21] GRIMSHAW, R., HELFRICH, K. R., and JOHNSON, E. R. Experimental study of the effect of rotation on nonlinear internal waves. Physics of Fluids, 25(5), 1-27(2013)
[22] OSTROVSKY, L. A. Nonlinear internal waves in a rotating ocean. Oceanology, 18(2), 181-191(1978)
[23] HOLLOWAY, P. E., PELINOVSKY, E., and TALIPOVA, T. A nonlinear model of internal tide transformation on the Australian North West Shelf. Journal of Physical Oceanography, 27(6), 871-896(1997)
[24] HOLLOWAY, P. E., PELINOVSKY, E., and TALIPOVA, T. A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone. Journal of Geophysical Research, 104(C8), 18333-18350(1999)
[25] DENG, X. D. and CAI, S. Q. A numerical study of rotation effect on the propagation of nonlinear internal solitary waves in the northern South China Sea. Applied Mathematical Modelling, 46, 581-590(2017)
[26] MILES, J. W. Wave evolution over a gradual slope with turbulent friction. Journal of Fluid Mechanics, 133, 207-216(1983)
[27] EL, G. A., GRIMSHAW, R., and KAMCHATNOV, A. M. Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction. Journal of Fluid Mechanics, 585, 213-244(2007)
[28] GRIMSHAW, R., PELINOVSKY, E., and TALIPOVA, T. Damping of large-amplitude solitary waves. Wave Motion, 37(4), 351-364(2003)
[29] HENDERSON, S. M. Turbulent production in an internal wave bottom boundary layer maintained by a vertically propagating seiche. Journal of Geophysical Research, 121(4), 2481-2498(2016)
[30] WUEST, A. and LORKE, A. Small-scale hydrodynamics in lakes. Annual Review of Fluid Mechanics, 35(35), 373-412(2003)
[31] LIU, A. K., CHANG, Y. S., and HSU, M. K. Evolution of nonlinear internal waves in the East and South China Seas. Journal of Geophysical Research, 103(C4), 7995-8008(1998)
[32] ZHENG, Q., YUAN, Y., and KLEMAS, V. Theoretical expression for an ocean internal soliton synthetic aperture radar image and determination of the soliton characteristic half width. Journal of Geophysical Research, 106(C12), 31415-31424(2001)
[33] XIE, J. S., HE, Y. H., LU, H. B., CHEN, Z. W., XU, J. X., and CAI, S. Q. Distortion and broadening of internal solitary wave front in the northeastern South China Sea deep basin. Geophysical Research Letters, 43(14), 7617-7624(2016)
[34] APEL, J. R., OSTROVSKY, L. A., and STEPANYANTS, Y. A. Internal solitons in the ocean. Journal of the Acoustical Society of America, 98(5), 2863-2864(2006)
[35] LAMB, K. G. and YAN, L. The evolution of internal wave undular bores:comparisons of a fully nonlinear numerical model with weakly nonlinear theory. Journal of Physical Oceanography, 26(12), 2712-2734(1996)
[36] ZHOU, X. and GRIMSHAW, R. The effect of variable currents on internal solitary waves. Dynamics of Atmospheres and Oceans, 14(1), 17-39(1989)
[37] GRIMSHAW, R., PELINOVSKY, E., and TALIPOVA, T. Internal solitary waves:propagation, deformation and disintegration. Nonlinear Processes in Geophysics, 17(6), 633-649(2010)
[38] GRIMSHAW, R. Internal solitary waves. Environmental Stratified Flows, Springer, United States, 1-30(2003)
[39] LI, Q. and FARMER, D. The generation and evolution of nonlinear internal waves in the deep basin of the South China Sea. Journal of Physical Oceanography, 41(7), 1345-1363(2011)
[40] HUANG, X., CHEN, Z., and ZHAO, W. An extreme internal solitary wave event observed in the northern South China Sea. Scientific Reports, 6, 30041(2016)
[41] AMANTE, C. and EAKINS, B. W. ETOPO11 Arc-minute global relief model:procedures, data sources and analysis. Psychologist, 16(3), 20-25(2009)
[42] BOYER, T. P., ANTONOV, J. I., BARANOVA, O. K., COLEMAN, C., GARCIA, H. E., GRODSKY, A., JOHNSON, D. R., LOCARNINI, R. A., MISHONOV, A. V., O'BRIEN, T. D., PAVER, C. R., REAGAN, J. R., SEIDOV, D., SMOLYAR, I. V., and ZWENG, M. M. World Ocean Database 2013, National Oceanographic Data Center, Silver Springer, Maryland (2013)
[43] WU, Z. R., CHENG, Y. L., WANG, S. L., and LYU, Y. K. Effects of varying bottom on nonlinear surface waves. Applied Mathematics and Mechanics (English Edition), 27(3), 409-416(2006) https://doi.org/10.1007/s10483-006-0318-y
[44] LIU, A. K., RAMP, S. R., and ZHAO, Y. A case study of internal solitary wave propagation during ASIAEX 2001. IEEE Journal of Oceanic Engineering, 29(4), 1144-1156(2005)
[45] FARMER, D., LI, Q., and PARK, J. H. Internal wave observations in the South China Sea:the role of rotation and non-linearity. Atmosphere-Ocean, 47(4), 267-280(2009)
[46] HOLLOWAY, P., PELINOVSKY, E., and TALIPOVA, T. Internal tide transformation and oceanic internal solitary waves. Environmental Stratified Flows, Springer, Boston§29-60(2003)
[47] LIEN, R. C., D'ASARO, E. A., and HENYEY, F. Trapped core formation within a shoaling nonlinear internal wave. Journal of Physical Oceanography, 42(4), 511-525(2012)
[48] LIEN, R. C., HENYEY, F., and MA, B. Large-amplitude internal solitary waves observed in the northern South China Sea:properties and energetics. Journal of Physical Oceanography, 44(4), 1095-1115(2013)
[49] STASTNA, M. and LAMB, K. G. Large fully nonlinear internal solitary waves:the effect of background current. Physics of Fluids, 14(9), 2987-2999(2002)
[50] LAMB, K. G. and WARNVARNAS, A. Two-dimensional numerical simulations of shoaling internal solitary waves at the ASIAEX site in the South China Sea. Nonlinear Processes in Geophysics, 22(3), 289-312(2015)
[51] LAMB, K. G. A numerical investigation of solitary internal waves with trapped cores formed via shoaling. Journal of Fluid Mechanics, 451, 109-144(2002)