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China Ocean Engineering

, Volume 33, Issue 1, pp 65–72 | Cite as

Two-Layer Non-Hydrostatic Model for Generation and Propagation of Interfacial Waves

  • S. R. PudjaprasetyaEmail author
  • I. Magdalena
Article
  • 5 Downloads

Abstract

When pycnocline thickness of ocean density is relatively small, density stratification can be well represented as a two-layer system. In this article, a depth integrated model of the two-layer fluid with constant density is considered, and a variant of the edge-based non-hydrostatic numerical scheme is formulated. The resulting scheme is very efficient since it resolves the vertical fluid depth only in two layers. Despite using just two layers, the numerical dispersion is shown to agree with the analytical dispersion curves over a wide range of kd, where k is the wave number and d the water depth. The scheme was tested by simulating an interfacial solitary wave propagating over a flat bottom, as well as over a bottom step. On a laboratory scale, the formation of an interfacial wave is simulated, which also shows the interaction of wave with a triangular bathymetry. Then, a case study using the Lombok Strait topography is discussed, and the results show the development of an interfacial wave due to a strong current passing through a sill.

Key words

interfacial waves two-layer non-hydrostatic model dispersion relation 

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References

  1. Bai Y.F., Yamazaki Y. and Cheung K.F., 2018. Convergence of multilayer nonhydrostatic models in relation to Boussinesq-type equations, Journal of Waterway, Port, Coastal, and Ocean Engineering, 144(2), 06018001-1–06018001-10.CrossRefGoogle Scholar
  2. Bai Y.F. and Cheung K.F., 2012. Depth-integrated free-surface flow with a two-layer non-hydrostatic formulation, International Journal for Numerical Methods in Fluids, 69(2), 411–429.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bijvelds M., 2003. A Non-Hydrostatic Module for Delft3D-Flow, Technical Report, WL-Delft Hydraulics.Google Scholar
  4. Casulli V. and Stelling G.S., 1998. Numerical simulation of 3D quasi-hydrostatic. free-surface flows, Journal of Hydraulic Engineering, 124(7), 678–686.CrossRefGoogle Scholar
  5. Choi W. and Camassa R., 1999. Fully nonlinear internal waves in a two-fluid system, Journal of Fluid Mechanics, 396, 1–36.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fang K.Z., Liu Z.B. and Zou Z.L., 2015. Efficient computation of coastal waves using a depth-integrated. non-hydrostatic model, Coastal Engineering, 97, 21–36.CrossRefGoogle Scholar
  7. Grimshaw R., Pelinovsky E. and Talipova T., 2007. Modelling internal solitary waves in the coastal ocean, Surveys in Geophysics, 28(4), 273–298.CrossRefGoogle Scholar
  8. Grimshaw R., Pelinovsky E. and Talipova T., 2008. Fission of a weakly nonlinear interfacial solitary wave at a step, Geophysical & Astrophysical Fluid Dynamics, 102(2), 179–194.MathSciNetCrossRefGoogle Scholar
  9. Helfrich K.R. and Melville W.K., 1986. On long nonlinear internal waves over slope-shelf topography, Journal of Fluid Mechanics, 167, 285–308.CrossRefGoogle Scholar
  10. Hibiya, T., 2004. Internal wave generation by tidal flow over a continental shelf slope, Journal of Oceanography, 60(3), 637–643.CrossRefGoogle Scholar
  11. Kanarska Y., Shchepetkin A. and McWilliams J.C., 2007. Algorithm for non-hydrostatic dynamics in the regional oceanic modeling system, Ocean Modelling, 18(3–4), 143–174.CrossRefGoogle Scholar
  12. Lamb K.G., 1994. Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge, Journal of Geophysical Research: Oceans, 99(C10), 843–864.CrossRefGoogle Scholar
  13. Lawrence C., Adytia D. and Van Groesen E., 2018. Variational Boussinesq model for strongly nonlinear dispersive waves, Wave Motion, 76, 78–102.MathSciNetCrossRefGoogle Scholar
  14. Lindzen, R.S., 1974. Stability of a Helmholtz velocity profile in a continuously stratified. infinite Boussinesq fluid-applications to clear air turbulence, Journal of Atmospheric Sciences, 31(6), 1507–1514.CrossRefGoogle Scholar
  15. Liu Z.B., Fang K.Z. and Cheng Y.Z., 2018. A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed, Journal of Fluid Mechanics, 842(2), 323–353.MathSciNetCrossRefGoogle Scholar
  16. Maderich, V., Talipova T., Grimshaw R., Pelinovsky E., Choi B.H., Brovchenko I., Terletska K. and Kim D.C., 2009. The transformation of an interfacial solitary wave of elevation at a bottom step, Nonlinear Processes in Geophysics, 16, 33–42.CrossRefzbMATHGoogle Scholar
  17. Maderich V., Talipova T., Grimshaw R., Terletska K., Brovchenko I., Pelinovsky E. and Choi B.H., 2010. Interaction of a large amplitude interfacial solitary wave of depression with a bottom step, Physics of Fluids, 22(7), 076602.CrossRefzbMATHGoogle Scholar
  18. Magdalena I., 2015. An Efficient Two-Layer Non-hydrostatic Numerical Model for Waves Propagation, Ph.D. Thesis, ITB.Google Scholar
  19. Marshall J., Hill C., Perelman L. and Adcroft A., 1997. Hydrostatic. quasi-hydrostatic, and nonhydrostatic ocean modeling, Journal of Geophysical Research: Oceans, 102(C3), 5733–5752.CrossRefGoogle Scholar
  20. Maxworthy T., 1979. A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge, Journal of Geophysical Research: Oceans, 84(C1), 338–346.CrossRefGoogle Scholar
  21. Murray, S.P., Arief D., Kindle J.C. and Hurlburt H.E., 1990. Characteristics of circulation in an Indonesian Archipelago strait from hydrography, current measurements and modeling results, in: Pratt L J. (ed.), The Physical Oceanography of Sea Strait, Springer, Dordrecht, 318, 3–23.CrossRefGoogle Scholar
  22. Nwogu O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal, and Ocean Engineering, 119(6), 618–638.CrossRefGoogle Scholar
  23. Osborne A.R. and Burch T.L., 1980. Internal solitons in the Andaman Sea, Science, 208(4443), 451–460.CrossRefGoogle Scholar
  24. Pudjaprasetya S.R., 2009. A study on internal solitary waves in lombok strait using the KdV Two-Layer Model, Far East J. of Oce. Res., 2(2), 83–91.Google Scholar
  25. Pudjaprasetya S.R. and Tjandra S.S., 2014. A hydrodynamic model for dispersive waves generated by bottom motion, in: Fuhrmann J. Ohlberger M, Rohde C. (eds.), Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, Springer, Cham.Google Scholar
  26. Pudjaprasetya S.R., Magdalena I. and Tjandra S.S., 2017. A nonhydrostatic two-layer staggered scheme for transient waves due to antisymmetric seabed thrust, Journal of Earthquake and Tsunami, 11(1), 1740002.CrossRefGoogle Scholar
  27. Stelling G. and Zijlema M., 2003. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation, International Journal for Numerical Methods in Fluids, 43(1), 1–23.MathSciNetCrossRefzbMATHGoogle Scholar
  28. Stelling G.S. and Duinmeijer S.P.A., 2003. A staggered conservative scheme for every Froude number in rapidly varied shallow water flows, International Journal for Numerical Methods in Fluids, 43(12), 1329–1354.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Susanto R.D., Mitnik L. and Zheng Q.N., 2005. Ocean internal waves observed in the Lombok Strait, Oceanography, 18(4), 80–87.CrossRefGoogle Scholar
  30. Sutherland B.R., Keating S. and Shrivastava I., 2015. Transmission and reflection of internal solitary waves incident upon a triangular barrier, Journal of Fluid Mechanics, 775, 304–327.MathSciNetCrossRefGoogle Scholar
  31. Tjandra S.S. and Pudjaprasetya S.R., 2015. A non-hydrostatic numerical scheme for dispersive waves generated by bottom motion, Wave Motion, 57, 245–256.MathSciNetCrossRefGoogle Scholar
  32. Visser W.P., 2004. On the Generation of Internal Waves in Lombok Strait through Kelvin-Helmholtz Instability, MSc Thesis, University of Twente, Twente.Google Scholar
  33. Wessels, F. and Hutter K., 1996. Interaction of internal waves with a topographic sill in a two-layered fluid, Journal of Physical Oceanography, 26(1), 5–20.CrossRefGoogle Scholar
  34. Yamazaki Y., Kowalik Z. and Cheung K.F., 2008. Depth-integrated. non-hydrostatic model for wave breaking and run-up, International Journal for Numerical Methods in Fluids, 61(5), 473–497.MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zhou J.G. and Stansby P.K., 1999. An arbitrary Lagrangian-Eulerian s (ALES) model with non-hydrostatic pressure for shallow water flows, Computer Methods in Applied Mechanics and Engineering, 178(1–2), 199–214.CrossRefzbMATHGoogle Scholar
  36. Zijlema M., Stelling G. and Smit P., 2011. SWASH: an operational public domain code for simulating wave fields and rapidly varied flows in coastal waters, Coastal Engineering, 58(10), 992–1012.CrossRefGoogle Scholar

Copyright information

© Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural SciencesInstitut Teknologi BandungBandungIndonesia

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