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Optimal two-layer approximation for continuous density stratification

Published online by Cambridge University Press:  05 January 2011

R. CAMASSA and
R. TIRON
R. CAMASSA
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
R. TIRON*
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
*
†Email address for correspondence: tiron@kaist.ac.kr
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Abstract

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Two-layer fluids of finite depth under gravity are the simplest configuration capable of supporting internal wave motion. The extent to which such systems can be used to provide quantitative information on smoothly stratified fluids, in configurations relevant for geophysical applications, is analysed and a model of practical interest derived. The model is based on long-wave asymptotic expansions and on first-principle criteria for an optimal choice of effective two-layer parameters for the incompressible, smoothly stratified Euler equations. The accuracy of the model is extensively tested, via fully resolved numerical computations, on the class of travelling wave solutions supported by smooth stratification systems. It is found that, despite the severe restrictions posed by the discrete two-layer density assumption, solitary wave solutions corresponding to experimentally realizable parametric values can be accurately predicted, in both wave and fluid parcel markers, such as phase speed and density fields, respectively. Thanks to this analysis, explicit closed-form solutions are available for all relevant physical quantities. The agreement between the simple, optimized two-layer model and the parent smooth-stratification Euler system persists up to extreme cases, such as that of internal fronts, and even up to thicknesses of the pycnocline comparable to that of the effective layers.

JFM classification

Geophysical and Geological Flows: Internal wavesGeophysical and Geological Flows: Shallow water flowsGeophysical and Geological Flows: Stratified flows

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011, pp. 32 - 54
Copyright
Copyright © Cambridge University Press 2011

References

REFERENCES

Camassa, R., Choi, W., Michallet, H., Rusås, P.-O. & Sveen, J. K. 2006 On the realm of validity of strongly nonlinear asymptotic approximations for internal waves. J. Fluid Mech. 549, 1–23.CrossRefGoogle Scholar
Camassa, R., Rusås, P.-O. Saxena, A. & Tiron, R. 2010 Fully nonlinear periodic internal waves in a two-fluid system of finite depth. J. Fluid Mech. 652, 259–298.Google Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
Fructus, D., Carr, M., Grue, J., Jensen, A. & Davies, P. 2009 Shear-induced breaking of large internal solitary waves. J. Fluid Mech. 620, 1–29.Google Scholar
Fructus, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323–347.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.Google Scholar
James, G. 2001 Internal traveling waves in the limit of a discontinuously stratified fluid. Arch. Rat. Mech. Anal. 160, 41–90.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lamb, K. G. & Wan, B. 1998 Conjugate flows and flat solitary waves for a continuously stratified fluid. Phys. Fluids 10, 2061–2079.Google Scholar
Love, A. E. H. 1891 Wave-motion in heterogeneous heavy fluids. Proc. Lond. Math. Soc. 22, 307–316.Google Scholar
Makarenko, N. I. & Maltseva, J. L. 2008 An analytical model of large amplitude internal solitary waves. In Extreme Ocean Waves (ed. Pelinovsky, E. & Kharif, C.), pp. 179–189. Springer.CrossRefGoogle Scholar
Miyata, M. 1985 An internal solitary wave of large amplitude. La Mer 23, 4348.Google Scholar
Miyata, M. 1988 Long internal waves of large amplitude. In Nonlinear Water Waves, IUTAM Symp., Tokyo 1987 (ed. Horikawa, K. & Maruo, H.), pp. 399–406. Springer.Google Scholar
Rusås, P.-O. 2001 On nonlinear internal waves in two- and three-layer fluids. DSc thesis, Department of Mathematics, University of Oslo.Google Scholar
Rusås, P.-O. & Grue, J. 2002 Solitary waves and conjugate flows in a three-layer fluid. Eur. J. Mech. B Fluids 21, 185–206.CrossRefGoogle Scholar
Sakai, T. & Redekopp, L. G. 2007 Models for strongly-nonlinear evolution of long internal waves in a two-layer stratification. Nonlinear Process. Geophys. 14, 31–47.Google Scholar
Tiron, R. 2009 Strongly nonlinear internal waves in near two-layer stratifications: generation, propagation and self-induced shear instabilities. PhD thesis, Mathematics Department, University of North Carolina at Chapel Hill.Google Scholar
Tung, K. K., Chan, T. F. & Kubota, T. 1984 Large amplitude internal waves of permanent form. Stud. Appl. Math. 66, 187–199.Google Scholar
Turkington, B., Eydeland, A. & Wang, S. 1991 A computational method for solitary internal waves in a continuously stratified fluid. Stud. Appl. Math. 85, 93–127.CrossRefGoogle Scholar
Yih, C.-S. 1969 Stratified flows. Annu. Rev. Fluid Mech. 1, 73–110.CrossRefGoogle Scholar