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Kakinuma, T., Ochi, N., Yamashita, K., & Nakayama, K. (2018). A NUMERICAL CALCULATION FOR INTERNAL WAVES OVER TOPOGRAPHY. Coastal Engineering Proceedings, 1(36), papers.64. https://doi.org/10.9753/icce.v36.papers.64


The internal waves propagating from the deep to shallow, and the shallow to deep, areas in the two-layer fluid systems, have been numerically simulated by solving the set of nonlinear equations, based on the variational principle in consideration of both the strong nonlinearity and strong dispersion of internal waves. The incident wave in the deep area, is the BO-type downward convex internal wave, which is the numerical solution obtained for the present fundamental equations. In the cases where the interface elevation is below, or equal to, the critical level in the shallow area, the disintegration of the internal waves occurs remarkably, leading to a long wave train. The lowest elevation of the interface, increases after its gradual decrease in the shallow area, where the interface is above the critical level, while the lowest elevation of the interface, increases through the internal-wave propagation in the shallow area, where the interface elevation is below, or equal to, the critical level, after its steep decrease around the boundary between the area over the upslope, and the shallow region.


Benjamin, T. B. 1967. Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29, 559-592.

Choi, W. and R. Camassa. 1999. Fully nonlinear internal waves in a two-fluid system, J. Fluid Mech., 396, 1-36.

Funakoshi, M. and M. Oikawa. 1986. Long internal waves of large amplitude in a two-layer fluid, J. Phys. Soc. Jpn, 55, 128-144.

Grimshaw, R., E. Pelinovsky, T. Talipova and A. Kurkina. 2004. Simulation of the transformation of internal solitary waves on oceanic shelves, J. Phys. Oceanogr., 34, 2774-2791.

Helfrich, K. R., W. K. Melville and J. W. Miles. 1984. On interfacial solitary waves over slowly varying topography, J. Fluid Mech., 149, 305-317.

Isobe, M. 1995. Time-dependent mild-slope equations for random waves, Coastal Eng. 1994 (ed. Edge, B. L), ASCE, 285-299.

Kakinuma, T. 2001. A set of fully nonlinear equations for surface and internal gravity waves, Proc. 5th Int. Conf. on Computer Modelling of Seas and Coastal Regions (ed. Brebbia, C. A.), Wessex Insti. Tech. Press, 225-234.

Lee, C.-Y. and R. C. Beardsley. 1974. The generation of long nonlinear internal waves in a weakly stratified shear flow, J. Geophys. Res., 79, 453-462.

Luke, J. C. 1967. A variational principle for a fluid with a free surface, J. Fluid Mech., 27, 395-397.

Nakayama, K. and T. Kakinuma. 2010. Internal waves in a two-layer system using fully nonlinear internal-wave equations, Int. J. Numer. Meth. Fluids, 62, 574-590.

Ono, H. 1975. Algebraic solitary waves in stratified fluids, J. Phys. Soc. Jpn., 39, 1082-1091.

Orr, M. H. and P. C. Mignerey. 2003. Nonlinear internal waves in the South China Sea: Observation of the conversion of depression internal waves to elevation internal waves, J. Geophys. Res., 108, C3, 9-1-9-16.

Ostrovsky, L. A. and Yu. A. Stepanyants. 1989. Do internal solitons exist in the ocean?, Rev. Geophys, 27, 293-310.

Yamashita, K. and T. Kakinuma. 2015. Properties of surface and internal solitary waves, Coastal Eng. 2014 (ed. Lynett, P. J.), ASCE, waves. 45, 15 pages.

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