Erminia Capodicasa, Pietro Scandura, Foti Enrico


A numerical model aimed at computing the mean velocity generated by a sea wave propagating over a sloping bottom, offshore the breaker line, is presented. The model is based on the assumption that the fluid domain can be partitioned into two boundary layers and a core region where at a first order of approximation the flow can be regarded as irrotational. The irrotational flow is computed by using a theory based on the assumption of small amplitude waves which allows both fully absorbed waves and partially reflected waves at the coastline to be considered. The distribution of the mean velocity is controlled by the ratio between the thickness of the boundary layer and the wave amplitude. When this ratio is small, the mean velocities are rather constant along the depth and a second boundary layer develops close to the bottom. In the case of fully reflected waves such boundary layer separates and the mean vorticity can be convected far from the bottom.


sea waves; boundary layers; steady currents


Bagnold, R.A. 1947. Sand movements by waves: some small scale experiments with sand of very low density, J. Inst. Civil Engrs, 27, 447-469.

Blondeaux, P., Brocchini, M. and G. Vittori. 2002. Sea waves and mass transport on a sloping beach, Proc. R. Soc. London A, 458, 2053-2082.

Dore, B.D. 1977. On mass transport velocity due to progressive waves, Q. J. Mech. Appl. Math. 30(2), 157-173.

Hunt, J.N., and B. Johns. 1963. Currents induced by tides and gravity waves, Tellus, 15, 343-351.

Hwung, J.N., and C. Lin. 1990. The mass transport of waves propagating over a sloping bottom, Proc. 22 nd

Int. Conf. Coastal Engineering, ASCE, pp. 544-556, Reston.

Iskandarani, M., and P.L.F. Liu. 1991. Mass transport in three dimensional water waves, J. Fluid Mech., 231, 417-437.

Liu, P.L.F. 1977. Mass transport in the free surface boundary layers, Coastal Engng, 1, 207-219.

Longuet-Higgins, M.S. 1953. Mass transport in water waves, Phil. Trans. R. Soc. Landon A, 245, 535-581.

Mei, C.C., Stiassnie M., and D.K-P. Yue. 2005. Theory and applications of ocean surface waves. Part 2: Nonlinear Aspects, Advanced Series on Ocean Engineering 23. Singapore, World Scientific.

Roache, P.J. 1972. Computational fluid dynamics, Albuquerque, NM: Hermosa, 1972.

Sani, R.L., and P.M. Gresho. 1994. Résumé and remarks on the open boundary condition minisymposium, Int. J. Num. Meth. In Fluids, 18, 983-1008.

Stive, M.J.F., and H.G. Wind. 1986. Cross-shore mean flow in the surf zone, Coastal Engng, 10, 325-340.

Stoker, J.J. 1947. Surface waves in water of variable depth, Quart. Appl. Math., 5, 1-54.

Stokes, G.G. 1847. On the theory of oscillatory waves, Trans. Camb. Phil. Soc., 8, 441-455.

Stuart, J.T. 1966. Double boundary layer in an oscillating viscous fluid, J. Fluid. Mech., 24, 673-687.

Svendsen, I.A. 1984. Mass flux and undertow in a surf zone, Coastal Engng., 8, 247-356

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