• Abramowitz, M., , and I. Stegun, Eds., 1972: Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, Vol. 55, United States Government Printing Office, 1046 pp.

  • Apotsos, A., , B. Raubenheimer, , S. Elgar, , R. T. Guza, , and J. A. Smith, 2007: Effects of wave rollers and bottom stress on wave setup. J. Geophys. Res.,112, C02003, doi:10.1029/2006JC003549.

  • Ardhuin, F., , and A. D. Jenkins, 2006: On the interaction of surface waves and upper ocean turbulence. J. Phys. Oceanogr., 36, 551557, doi:10.1175/JPO2862.1.

    • Search Google Scholar
    • Export Citation
  • Bowen, A. J., , and D. L. Inman, 1969: Rip currents: 2. Laboratory and field observations. J. Geophys. Res., 74 (23), 54795490, doi:10.1029/JC074i023p05479.

    • Search Google Scholar
    • Export Citation
  • Bowen, A. J., , and D. L. Inman, 1971: Edge waves and crescentic bars. J. Geophys. Res., 76 (36), 86628671, doi:10.1029/JC076i036p08662.

    • Search Google Scholar
    • Export Citation
  • Donn, W. L., , and M. Ewing, 1956: Stokes’ edge waves in Lake Michigan. Science, 124, 12381242, doi:10.1126/science.124.3234.1238.

  • Ewing, M., , F. Press, , and W. L. Donn, 1954: An explanation of the Lake Michigan wave of 26 June 1954. Science, 120, 684686, doi:10.1126/science.120.3122.684.

    • Search Google Scholar
    • Export Citation
  • Gaster, M., 1962: A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech., 14, 222224, doi:10.1017/S0022112062001184.

    • Search Google Scholar
    • Export Citation
  • Gjevik, B., , E. Nøst, , and T. Straume, 1994: Model simulations of the tides in the Barents Sea. J. Geophys. Res., 99 (C2), 33373350, doi:10.1029/93JC02743.

    • Search Google Scholar
    • Export Citation
  • Greenspan, H. P., 1970: A note on edge waves in a stratified fluid. Stud. Appl. Math., 49, 381388.

  • Huntley, D. A., , and A. J. Bowen, 1973: Field observations of edge waves. Nature, 243, 160162, doi:10.1038/243160a0.

  • Jenkins, A. D., 1989: The use of a wave prediction model for driving a near-surface current model. Dtsch. Hydrogr. Z., 42, 133149, doi:10.1007/BF02226291.

    • Search Google Scholar
    • Export Citation
  • Johns, B., 1965: Fundamental mode edge waves over a steeply sloping shelf. J. Mar. Res., 23, 200206.

  • Kenyon, K. E., 1969: Note on Stokes’ drift velocity for edge waves. J. Geophys. Res., 74 (23), 55335535, doi:10.1029/JC074i023p05533.

    • Search Google Scholar
    • Export Citation
  • LeBlond, P. H., , and L. A. Mysak, 1978: Waves in the Ocean. Elsevier Oceanography Series, Vol. 20, Elsevier, 602 pp.

  • Llewellyn Smith, S. G., 2004: Stratified rotating edge waves. J. Fluid Mech., 498, 161170, doi:10.1017/S002211200300702X.

  • Longuet-Higgins, M. S., 1953: Mass transport in water waves. Philos. Trans. Roy. Soc. London, A245, 535581, doi:10.1098/rsta.1953.0006.

    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., 2005: On wave set-up in shoaling water with a rough seabed. J. Fluid Mech., 527, 217235, doi:10.1017/S0022112004003222.

    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., , and R. W. Stewart, 1960: Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech., 8, 565583, doi:10.1017/S0022112060000803.

    • Search Google Scholar
    • Export Citation
  • Mei, C. C., 1973: A note on the averaged momentum balance in two-dimensional water waves. J. Mar. Res., 31, 97104.

  • Mei, C. C., , C. Chian, , and F. Ye, 1998: Transport and resuspension of fine particles in a tidal boundary layer near a small peninsula. J. Phys. Oceanogr., 28, 23132331, doi:10.1175/1520-0485(1998)028<2313:TAROFP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Nøst, E., 1994: Calculating tidal current profiles from vertically integrated models near the critical latitude in the Barents Sea. J. Geophys. Res., 99 (C4), 78857901, doi:10.1029/93JC03485.

    • Search Google Scholar
    • Export Citation
  • Phillips, O. M., 1977: The Dynamics of the Upper Ocean. 2nd ed. Cambridge University Press, 336 pp.

  • Starr, V. P., 1959: Hydrodynamical analogy to E = mc2. Tellus, 11, 135138, doi:10.1111/j.2153-3490.1959.tb00013.x.

  • Stokes, G. G., 1846: Report on recent researches in hydrodynamics. Rep. 16th Brit. Assoc. Adv. Sci., 1–20.

  • Stokes, G. G., 1847: On the theory of oscillatory waves. Trans. Cambridge. Philos. Soc., 8, 441455.

  • Sverdrup, H. U., 1927: Dynamic of Tides on the North Siberian Shelf: Results from the Maud Expedition. Vol. 4. Geofysiske Publikasjoner, 75 pp.

  • Ursell, F., 1951: Trapping modes in the theory of surface waves. Math. Proc. Cambridge Philos. Soc., 47, 347358, doi:10.1017/S0305004100026700.

    • Search Google Scholar
    • Export Citation
  • Ursell, F., 1952: Edge waves on a sloping beach. Proc. Roy. Soc. London, A214, 7997, doi:10.1098/rspa.1952.0152.

  • Weber, J. E. H., 2012: A note on trapped Gerstner waves. J. Geophys. Res., 117 (C3), 3048, doi:10.1029/2011JC007776.

  • Weber, J. E. H., , and A. Melsom, 1993: Transient ocean currents induced by wind and growing waves. J. Phys. Oceanogr., 23, 193206, doi:10.1175/1520-0485(1993)023<0193:TOCIBW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Weber, J. E. H., , and P. Ghaffari, 2009: Mass transport in the Stokes edge wave. J. Mar. Res., 67, 213224, doi:10.1357/002224009789051182.

    • Search Google Scholar
    • Export Citation
  • Weber, J. E. H., , and E. Støylen, 2011: Mean drift velocity in the Stokes interfacial edge wave. J. Geophys. Res.,116, C04002, doi:10.1029/2010JC006619.

  • Weber, J. E. H., , and M. Drivdal, 2012: Radiation stress and mean drift in continental shelf waves. Cont. Shelf Res., 35, 108116, doi:10.1016/j.csr.2012.01.001.

    • Search Google Scholar
    • Export Citation
  • Weber, J. E. H., , G. Broström, , and Ø. Saetra, 2006: Eulerian versus Lagrangian approaches to the wave-induced transport in the upper ocean. J. Phys. Oceanogr., 36, 21062117, doi:10.1175/JPO2951.1.

    • Search Google Scholar
    • Export Citation
  • Weber, J. E. H., , K. H. Christensen, , and C. Denamiel, 2009: Wave-induced setup of the mean surface over a sloping beach. Cont. Shelf Res., 29, 14481453, doi:10.1016/j.csr.2009.03.010.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 69 69 5
PDF Downloads 25 25 4

Mass Transport in the Stokes Edge Wave for Constant Arbitrary Bottom Slope in a Rotating Ocean

View More View Less
  • 1 Department of Geosciences, University of Oslo, Oslo, Norway
© Get Permissions Rent on DeepDyve
Restricted access

Abstract

The Lagrangian mass transport in the Stokes surface edge wave is obtained from the vertically integrated equations of momentum and mass in a viscous rotating ocean, correct to the second order in wave steepness. The analysis is valid for bottom slope angles β in the interval 0 < βπ/2. Vertically averaged drift currents are obtained by dividing the fluxes by the local depth. The Lagrangian mean current is composed of a Stokes drift (inherent in the waves) plus a mean Eulerian drift current. The latter arises as a balance between the radiation stresses, the Coriolis force, and bottom friction. Analytical solutions for the mean Eulerian current are obtained in the form of exponential integrals. The relative importance of the Stokes drift to the Eulerian current in their contribution to the Lagrangian drift velocity is investigated in detail. For the given wavelength, the Eulerian current dominates for medium and large values of β, while for moderate and small β, the Stokes drift yields the main contribution to the Lagrangian drift. Because most natural beaches are characterized by moderate or small slopes, one may only calculate the Stokes drift in order to assess the mean drift of pollution and suspended material in the Stokes edge wave. The main future application of the results for large β appears to be for comparison with laboratory experiments in rotating tanks.

Corresponding author address: Peygham Ghaffari, Department of Geosciences, University of Oslo, P.O. Box 1022, Blindern, 0315 Oslo, Norway. E-mail: peygham.ghaffari@geo.uio.no

Abstract

The Lagrangian mass transport in the Stokes surface edge wave is obtained from the vertically integrated equations of momentum and mass in a viscous rotating ocean, correct to the second order in wave steepness. The analysis is valid for bottom slope angles β in the interval 0 < βπ/2. Vertically averaged drift currents are obtained by dividing the fluxes by the local depth. The Lagrangian mean current is composed of a Stokes drift (inherent in the waves) plus a mean Eulerian drift current. The latter arises as a balance between the radiation stresses, the Coriolis force, and bottom friction. Analytical solutions for the mean Eulerian current are obtained in the form of exponential integrals. The relative importance of the Stokes drift to the Eulerian current in their contribution to the Lagrangian drift velocity is investigated in detail. For the given wavelength, the Eulerian current dominates for medium and large values of β, while for moderate and small β, the Stokes drift yields the main contribution to the Lagrangian drift. Because most natural beaches are characterized by moderate or small slopes, one may only calculate the Stokes drift in order to assess the mean drift of pollution and suspended material in the Stokes edge wave. The main future application of the results for large β appears to be for comparison with laboratory experiments in rotating tanks.

Corresponding author address: Peygham Ghaffari, Department of Geosciences, University of Oslo, P.O. Box 1022, Blindern, 0315 Oslo, Norway. E-mail: peygham.ghaffari@geo.uio.no
Save