Skip to main content Accessibility help

Weakly nonlinear internal waves in a two-fluid system

  • Wooyoung Choi and Roberto Camassa


We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin–Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.



Hide All
Ablowitz, M. J. & Segur, H. 1980 Long internal waves in fluids of great depth. Stud. Appl. Maths 62, 249262.
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241270.
Benjamin, T. B. 1967 Internal waves of permanent form of great depth. J. Fluid Mech. 29, 559592.
Choi, W. 1995 Nonlinear evolution equation for two-dimensional surface waves in a fluid of finite depth. J. Fluid Mech 295, 381394.
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593607.
Fletcher, C. A. J. 1990 Computational Techniques for Fluid Dynamics, Volume I: Fundamental and General Techniques. Springer
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237246.
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429464.
Johnson, R. S. 1972 Some numerical solutions of a variable-coefficient KdV equation. J. Fluid Mech. 54, 8191.
Joseph, R. I. 1977 Solitary waves in finite depth fluid. J. Phys.: A Math. Gen. 10, L225L227.
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539541.
Kakutani, T. 1971 Effect of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272275.
Kubota, T., Ko, D. R. S. & Dobbs, L. D. 1978 Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth. AIAA J. Hydrodyn. 12, 157165.
Lamb, H. 1932 Hydrodynamics. Dover
Matsuno, Y. 1992 Nonlinear evolutions of surface gravity waves of fluid of finite depth. Phys. Rev. Lett. 69, 609611.
Matsuno, Y. 1993a A unified theory of nonlinear wave propagation in two-layer fluid systems. J. Phys. Soc. Japan 62, 19021916.
Matsuno, Y. 1993b Nonlinear evolution of surface gravity waves over an uneven bottom. J. Fluid Mech. 249, 121133.
Matsuno, Y 1993c Two-dimensional evolution of surface gravity waves on a fluid of arbitrary depth. Phys. Rev. E 47, 45934596.
Matsuno, Y. 1994 Higher-order nonlinear evolution for interfacial waves in a two-layer fluid system. Phys. Rev. E 49, 20912095.
Miloh, T. 1990 On periodic and solitary wavelike solutions of the intermediate long wave equation. J. Fluid Mech. 211, 617627.
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 10821091.
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear, unsteady waves. J. Fluid Mech. 294, 7192.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley
Wu, T. Y. 1981 Long waves in ocean and coastal waters. J. Engng Mech. Div. ASCE 107, 501522.
Wu, T. Y. 1987 Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 7599.
Zhu, J., Wu, T. Y. & Yates, G. T. 1986 Generation of internal runaway solitons by moving disturbances. In Proc. 16th Symp. on Naval Hydrodynamics ed. (W. C. Webster), pp. 186197, Washington, DC: National Academy Press.
MathJax is a JavaScript display engine for mathematics. For more information see

Related content

Powered by UNSILO

Weakly nonlinear internal waves in a two-fluid system

  • Wooyoung Choi and Roberto Camassa


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.