Skip to main content Accessibility help
×
Home

A regularized model for strongly nonlinear internal solitary waves

  • WOOYOUNG CHOI (a1), RICARDO BARROS (a1) and TAE-CHANG JO (a2)

Abstract

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

Copyright

Corresponding author

Email address for correspondence: wychoi@njit.edu

References

Hide All
Bogucki, D. & Garrett, C. 1993 A simple model for the shear-induced decay of an internal solitary wave. J. Phys. Oceanogr. 23, 17671776.
Bona, J. L., Chen, M. & Saut, J.-C. 2002 Boussinesq equations and other systems for small amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12, 283318.
Bona, J. L., Chen, M. & Saut, J.-C. 2004 Boussinesq equations and other systems for small amplitude long waves in nonlinear dispersive media. II. Nonlinear theory. Nonlinearity 17, 925952.
Camassa, R., Choi, W., Michallet, H., Rusas, P. & Sveen, J. K. 2006 On the realm of validity of strongly nonlinear asymptotic approximations for internal waves. J. Fluid Mech. 549, 123.
Choi, W. 2000 Modelling of strongly nonlinear internal gravity waves. In Proceedings of the Fourth International Conference on Hydrodynamics (ed. Goda, Y., Ikehata, M. & Suzuki, K.), pp. 453–458.
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 136.
Grue, J., Friis, H. A., Palm, E. & Rusas, P. E. 1997 A method for computing unsteady fully nonlinear interfacial waves. J. Fluid Mech. 351, 223353.
Grue, J., Jensen, A., Rusas, P. E., & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257278.
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.
Jo, T.-C. & Choi, W. 2002 Dynamics of strongly nonlinear solitary waves in shallow water. Stud. Appl. Math. 109, 205227.
Jo, T.-C. & Choi, W. 2008 On stabilizing the strongly nonlinear internal wave model. Stud. Appl. Math. 120, 6585.
Lamb, H. 1945 Hydrodynamics. Dover.
Michallet, H. & Barthélemy, E. 1998 Experimental study of interfacial solitary waves. J. Fluid Mech. 366, 159177.
Miyata, M. 1988 Long internal waves of large amplitude. In Proceedings of the IUTAM Symposium on Nonlinear Water Waves (ed. Horikawa, H. & Maruo, H.), Berlin: Springer–Verlag, pp. 399406.
Nguyen, H. Y. & Dias, F. 2008 A boussinesq system for two-way propagation of interfacial waves. Phys. D 237, 23652389.
Nwogu, O. 1993 Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterway, Port, Coastal, Ocean Engng 119, 618638.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

A regularized model for strongly nonlinear internal solitary waves

  • WOOYOUNG CHOI (a1), RICARDO BARROS (a1) and TAE-CHANG JO (a2)

Metrics

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.