Scalar transport by propagation of an internal solitary wave over a slope-shelf


Internal solitary waves (ISWs) of depression are commonly found in the coastal environment and are believed to re-suspend sediments in coastal regions where the waves break. In this research, the direct numerical simulation is used to study the scalar transport induced by the ISWs of depression propagating over a slope-shelf topography. The scalar in this paper is considered to represent the concentrations of very fine suspended solids or pollutants. Vortices are observed from the numerical results at the bottom boundary layer on the slope during the ISW shoaling process, resulting in a scalar transport. All incident ISWs of depression are observed to produce a waveform inversion on the shelf. The scalar transport from a slope to a shelf is the consequence of the combined vortices at the bottom boundary layer and the overturning of the ISWs of depression, and the latter was commonly ignored in previous studies. This study shows that the ISW-induced scalar transport consists of the following four stages: the slip transport, the wash transport, the vortex transport, and the secondary transport. A dimensionless time scales of the four stages are calculated, and the beginning times of the wash transport and the secondary transport are found to be uncorrelated with the slope gradients, taking values of 1.26, 4, respectively.

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  1. [1]

    Perry M. J., Sackmann B. S., Eriksen C. C. et al. Seaglider observations of blooms and subsurface chlorophyll maxima off the Washington coast [J]. Limnology and Oceanography, 2008, 53(5 Part 2): 2169–2179.

    Article  Google Scholar 

  2. [2]

    Woodson C. B. The fate and impact of internal waves in nearshore ecosystems [J]. Annual Review of Marine Science, 2018, 10: 421–441.

    Article  Google Scholar 

  3. [3]

    Jackson J. F. E., Elliott A. J. Internal waves in the Clyde Sea [J]. Estuarine, Coastal and Shelf Science, 2002, 54(1): 51–64.

    Article  Google Scholar 

  4. [4]

    Alpers W., Huang W. On the discrimination of radar signatures of atmospheric gravity waves and oceanic internal waves on synthetic aperture radar images of the sea surface [J]. IEEE Transactions on Geoscience and Remote Sensing, 2011, 49(3): 1114–1126.

    Article  Google Scholar 

  5. [5]

    Susanto R., Mitnik L., Zheng Q. Ocean internal waves observes [J]. Oceanography, 2005, 18(4): 80.

    Article  Google Scholar 

  6. [6]

    Vlasenko V., Hutter K. Numerical experiments on the breaking of solitary internal wavesover a slope shelf topography [J]. Journal of Physical Oceanography, 2002, 32(6): 1779–1793.

    MathSciNet  Article  Google Scholar 

  7. [7]

    Boegman L., Ivey G. N., Imberger J. The degeneration of internal waves in lakes with sloping topography [J]. Limnol Oceanogr, 2005, 50(5): 1620–1637.

    Article  Google Scholar 

  8. [8]

    Cheng M. H., Hsu R. C., Chen C. Y. Laboratory experiments on waveform inversion of an internal solitary wave over a slope-shelf [J]. Environmental Fluid Mechanics, 2011, 11(4): 353–384.

    Article  Google Scholar 

  9. [9]

    Butman B., Alexander P. S., Scotti A. et al. Large internal waves in Massachusetts Bay transport sediments offshore [J]. Continental Shelf Research, 2006, 26(17-18): 2029–2049.

    Article  Google Scholar 

  10. [10]

    Stastna M., Lamb K. G. Sediment resuspension mechanisms associated with internal waves in coastal waters [J]. Journal of Geophysical Research, 2008, 113: 193–199.

    Article  Google Scholar 

  11. [11]

    Puig P., Palanques A., Guillén J. Near-bottom suspended sediment variability caused by storms and near-inertial internal waves on the Ebro mid continental shelf (NW Mediterranean) [J]. Marine Geology, 2001, 178(1–4): 81–93.

    Article  Google Scholar 

  12. [12]

    Quaresma L. S., Vitorino J., Oliveira A. Evidence of sediment resuspension by nonlinear internal waves on the western Portuguese mid-shelf [J]. Marine Geology, 2007, 246(2): 123–143.

    Article  Google Scholar 

  13. [13]

    Boegman L., Ivey G. N. Flow separation and resuspension beneath shoaling nonlinear internal waves [J]. Journal of Geophysical Research Oceans, 2009, 114(C2): 309–321.

    Google Scholar 

  14. [14]

    Aghsaee P., Boegman L., Diamessis P. J. Boundary-layer-separation-driven vortex shedding beneath internal solitary waves of depression [J]. Journal of Fluid Mechanics, 2012, 690: 321–344.

    Article  Google Scholar 

  15. [15]

    Zhu H., Wang L. L., Tang H. W. Large-eddy simulation of suspended sediment transport in turbulent channel flow [J]. Journal of Hydrodynamics, 2013, 25(1): 48–55.

    Article  Google Scholar 

  16. [16]

    Grimshaw R., Pelinovsky E., Talipova T. et al. Internal solitary waves: Propagation, deformation and disintegration [J]. Nonlinear Processes in Geophysics, 2010, 17(6): 633–649.

    Article  Google Scholar 

  17. [17]

    Zhang H. S., Jia H. Q., Gu J. B. Numerical simulation of the internal wave propagation in continuously density-stratified ocean [J]. Journal of Hydrodynamics, 2014, 26(5): 770–779.

    Article  Google Scholar 

  18. [18]

    Thorpe S. A. Models of energy loss from internal waves breaking in the ocean [J]. Journal of Fluid Mechanics, 2018, 836: 72–116.

    MathSciNet  Article  Google Scholar 

  19. [19]

    Sakai T., Redekopp L. G. A parametric study of the generation and degeneration of wind-forced long internal waves in narrow lakes [J]. Journal of Fluid Mechanics, 2010, 645: 315–344.

    Article  Google Scholar 

  20. [20]

    Fang X. H., Jiang M. S., Du T. Dispersion relation of internal waves in the western equatorial Pacific Ocean [J]. Acta Oceanologica Sinica, 2000, (4): 37–45.

    Google Scholar 

  21. [21]

    Ostrovsky L. A. Evolution equations for strongly nonlinear internal waves [J]. Physics of Fluids, 2003, 15(15): 2934–2948.

    MathSciNet  Article  Google Scholar 

  22. [22]

    Zhu H., Wang L. L., Tang H. W. Large-eddy simulation of the generation and propagation of internal solitary waves [J]. Science China Physics Mechanics and Astronomy, 2014, 57(6): 1128–1136.

    Article  Google Scholar 

  23. [23]

    Thompson D. A., Karunarathna H., Reeve D. Comparison between wave generation methods for numerical simulation of bimodal seas [J]. Water Science and Engineering, 2016, 9(1): 3–13.

    Article  Google Scholar 

  24. [24]

    Aghsaee P., Boegman L., Lamb K. G. Breaking of shoaling internal solitary waves [J]. Journal of Fluid Mechanics, 2010, 659: 289–317.

    MathSciNet  Article  Google Scholar 

  25. [25]

    Masunaga E., Homma H., Yamazaki H. Mixing and sediment resuspension associated with internal bores in a shallow bay [J]. Continental Shelf Research, 2015, 110(8): 807–807.

    Google Scholar 

  26. [26]

    Wang W., Huai W. X., Gao M. Numerical investigation of flow through vegetated multi-stage compound channel [J]. Journal of Hydrodynamics, 2014, 26(3): 467–473.

    Article  Google Scholar 

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This work was supported by the Special Fund of State Key Laboratory of China (Grant No. 20185044412), the 111 Project (Grant No. B17015).

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Corresponding author

Correspondence to Ling-ling Wang.

Additional information

Project supported by the National Key Research and Development Program of China (2016YFC0401703, 2017YFC0405605) and the National Natural Science Foundation of China (Grant Nos. 51879086, 51609068, 51709126)

Biography: Jin Xu (1992-), Male, Ph. D.

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Xu, J., Wang, L., Tang, H. et al. Scalar transport by propagation of an internal solitary wave over a slope-shelf. J Hydrodyn 31, 317–325 (2019).

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Key words

  • Internal solitary waves
  • slope-shelf
  • scalar transport
  • wave inversion
  • dimensionless time scale