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Oceanological and Hydrobiological Studies

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Volume 44, Issue 4


A note on the vertical distribution of momentum transport in water waves

Jerzy Kołodko / Gabriela Gic-Grusza
  • Corresponding author
  • Institute of Oceanography, Faculty of Oceanography and Geography, University of Gdańsk, Al. M. Piłsudskiego 46, 81–378 Gdynia, Poland
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Published Online: 2015-12-09 | DOI: https://doi.org/10.1515/ohs-2015-0053


In this paper, the classical problem of horizontal waveinduced momentum transport is analyzed once again. A new analytical approach has been employed to reveal the vertical variation of this transport in the Eulerian description.

In mathematical terms, this variation is shown to have (after “smoothing out” the surface corrugation) the character of a generalized function (distribution) and is described by a classical function in the water depths and by an additional Dirac-delta-function component on the averaged free surface.

In terms of physics, the considered variation consists of two entities: (i) a continuous distribution of the mean momentum transport flux density (tensorial radiation pressure) over the entire water column, and (ii) an additional momentum transport flux concentrated on the mean free surface level (tensorial radiation surface pressure). Simple analytical formulae describing this variation have been derived.

This allowed a conventional expression to be derived, describing the depth-integrated excess of horizontal momentum flux due to the presence of waves (the so-called “radiation stress”), confirming to some extent the correctness of the whole analysis carried out.

The results obtained may be important to the ocean dynamics, especially in view of their possible application in the field of hydrodynamics of wave-dominated coastal zones.

Keywords: radiation stress; coastal zone; wave-induced currents; momentum transport; three-dimensional modeling


  • Cieślikiewicz W. & Gudmestad T. (1993). Stochastic characteristics of orbital velocities of random water waves. J. Fluid Mech. 255: 275-299.Google Scholar

  • Cieślikiewicz W. & Gudmestad O. T. (1994). Random water wave kinematics. Archives of Hydro-Engineering and Environmental Mechanics, 41 (1-2), Part 1. Theory: 3-35; Part 2. Experiment: 37-85.Google Scholar

  • Deigaard R. (1993). A note on the three-dimensional shear stress distribution in a surf zone, Coastal Engng. 20: 157-171.Google Scholar

  • De Vriend H. J. & Stive M. J. F. (1987). Quasi-3D modelling of nearshore currents. Coastal Engng. 11: 565-601.Google Scholar

  • Dingemans M. A. (1997). Water wave propagation over uneven bottoms. Part I - Linear wave propagation. Singapore: World Scientific, 471 pp.Google Scholar

  • Grusza G. (2007). Three-dimensional modelling of wave-generated currents in coastal zone. Ph.D. Diss. (in Polish), Inst. of Oceanogr., University of Gdańsk, Gdynia, Poland, 97 pp.Google Scholar

  • Herman A. (2006). Three-dimensional structure of waveinduced momentum flux in irrotational waves in combined shoaling-refraction conditions. Coastal Engng. 53: 545−555.Google Scholar

  • Krauss W. (1973). Dynamics of the homogeneous and the quasihomogeneous ocean. Methods and Results of Theoretical Oceanography, vol. 1. Berlin: Gebr. Borntraeger.Google Scholar

  • Lin P. & Zhang D. (2004). The Depth-Dependent Radiation Stresses and Their Effect on Coastal Currents. The 6 th International Conference on Hydrodynamics, 23-27, November 2004 , West Leederville , Western Australia.Google Scholar

  • Longuet-Higgins M. S. & Stewart R. W. (1960). Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech. 8: 565-583.Google Scholar

  • Longuet-Higgins M. S. & Stewart R. W. (1962). Radiation stress and mass transport in gravity waves with application to „surf beats”. J. Fluid Mech. 13: 481-504.Google Scholar

  • Longuet-Higgins M. S. & Stewart R. W. (1964). Radiation stresses in water waves; a physical discussion with applications Deep Sea Res. 11: 529-562.Google Scholar

  • Lundgren H. (1962). The concept of the wave thrust. Coastal Engng. Lab., Tech. Univ. Denmark, Basic Res. - Progr. Rep. 3: 1-5.Google Scholar

  • Lundgren H. (1963). Wave thrust and energy level. Proc. 10th Congr. Int. Assoc. Hydr. Res., London, Paper 1.20, 1: 147-151.Google Scholar

  • Mei C. C. (1989). The Applied Dynamics of Ocean Surface Waves. Singapore: World Scientific, 740 pp.Google Scholar

  • Mellor G. L. (2003). The three-dimensional current and surface wave equations. J. Phys. Oceanogr., 33: 1978-1989.Web of ScienceGoogle Scholar

  • Mellor G. L. (2011). Wave radiation stress. Ocean Dyn., 61 (5), pp. 563-568. DOI: 10.1007/s10236-010-0359-2.CrossrefGoogle Scholar

  • Mellor G. L. (2013). Waves, circulation and vertical dependence. Ocean Dynamics, 63, 447-457.Google Scholar

  • Nielsen P. (1992). Coastal bottom boundary layers and sediment transport. Advanced Series in Ocean Engineering, vol. 4, World Scientific Publishing Co., Singapore.Google Scholar

  • Nobuoka H., Mimura N. & Kato H. (1998). Three-dimensional nearshore currents model based on vertical distribution of radiation stress. Proceedings of the International Conference on Coastal Engineering, ASCE, 1: 829-842.Google Scholar

  • Phillips O. M. (1977). The dynamics of the upper ocean. Cambridge University Press, Cambridge e.a., 336 pp.Google Scholar

  • Stive M. J. F. & Wind H. G. (1986). Cross-shore mean flow in the surf zone. Coastal Engng. 10: 325-340.Google Scholar

  • Webb B. M. & Slinn D. N. (2004). Vertical distribution of radiation stress for non-linear shoaling waves. AGU Fall Meeting.Google Scholar

  • Xia H., Xia Z. & Zhu L. (2004). Vertical variation in radiation stress and wave-induced current. Coastal Engng. 51: 309-321. Google Scholar

About the article

Received: 2014-05-16

Accepted: 2015-06-18

Published Online: 2015-12-09

Published in Print: 2015-12-01

Citation Information: Oceanological and Hydrobiological Studies, Volume 44, Issue 4, Pages 563–568, ISSN (Online) 1897-3191, ISSN (Print) 1730-413X, DOI: https://doi.org/10.1515/ohs-2015-0053.

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