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Fully nonlinear periodic internal waves in a two-fluid system of finite depth

Published online by Cambridge University Press:  19 May 2010

R. CAMASSA ,
P.-O. RUSÅS ,
A. SAXENA  and
R. TIRON
R. CAMASSA*
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
P.-O. RUSÅS
Affiliation:
Faculty of Computer Sciences, Østfold University College, N-1757 Halden, Norway
A. SAXENA
Affiliation:
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
R. TIRON
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
*
†Email address for correspondence: [email protected]
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Abstract

Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 259 - 298
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Amick, C. J. & Turner, R. E. L. 1986 A global theory of internal solitary waves in two-fluid systems. Trans. Amer. Math. Soc. 298, 431–460.CrossRefGoogle Scholar
Benjamin, T. B. 1995 Verification of the Benjamin–Lighthill conjecture about steady water waves. J. Fluid Mech. 295, 337–356.CrossRefGoogle Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On Cnoidal waves and bores. R. Soc. Lond. Proc. A 224, 448–484.Google Scholar
Bona, J. L., Lannes, D. & Saut, J.-C. 2008 Asymptotic models for internal waves. J. Math. Pures Appl. 89, 536–566.CrossRefGoogle Scholar
Bridges, T. J. & Donaldson, N. M. 2007 Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves. Phys. Fluids 19, 072111.CrossRefGoogle Scholar
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
Choi, W. & Camassa, R. 1996 Weakly nonlinear internal waves in a two-fluid system. J. Fluid Mech. 313, 83–103.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
Craig, W., Guyenne, P. & Kalisch, H. 2005 Hamiltonian long-wave expansions for free surfaces and interfaces. Commun. Pure Appl. Math. 58, 1587–1641.CrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 2003 On internal fronts. J. Fluid Mech. 479, 145–156.CrossRefGoogle Scholar
Drociuk, R. J. 2004 On the closed form solution for the geodesics in SdS space. ArXiv General Relativity and Quantum Cosmology e-prints, arXiv:gr-qc/0402093.Google Scholar
Duda, T. F., Lynch, J. F., Irish, J. D., Beardsley, R. C., Ramp, S. R., Chiu, C.-S., Tang, T. Y. & Yang, Y. J. 2004 Internal tide and nonlinear internal wave behaviour at the continental slope in the northern South China Sea. IEEE J. Ocean Eng. 29, 1105–1131.CrossRefGoogle Scholar
Funakoshi, M. & Oikawa, M. 1986 Long internal waves of large amplitude in a two-layer fluid. J. Phys. Soc. Japan 55, 128–144.CrossRefGoogle Scholar
Gavrilov, N. V. 1994 Internal solitary waves and smooth bores which are stationary in a laboratory coordinate system. J. Appl. Mech. Tech. Phys. 35, 29–33.CrossRefGoogle Scholar
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.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.CrossRefGoogle Scholar
Helfrich, K. & Melville, K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
Holyer, J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.CrossRefGoogle Scholar
Klopman, G. 1990 A note on integral properties of periodic gravity waves in the case of a non-zero mean Eulerian velocity. J. Fluid Mech. 211, 609–615.CrossRefGoogle Scholar
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves in a two-fluid system. J. Fluid Mech. 112, 225–251.CrossRefGoogle Scholar
James, G. 2001 Internal travelling waves in the limit of a discontinuously stratified fluid. Arch. Ration. Mech. Anal. 160, 41–90.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. R. Soc. Lond. Proc. A 342, 157–174.Google Scholar
Makarenko, N. I. & Maltseva, Zh. L. 2007 Phase velocity spectrum of internal waves in a weakly-stratified two-layer fluid. Fluid Dyn. 2, 278–294.Google Scholar
Makarenko, N. I., Maltseva, Zh. L. & Kazakov, A. Yu. 2009 Conjugate flows and amplitude bounds for internal solitary waves. Nonlinear Process. Geophys. 16, 169–178.CrossRefGoogle Scholar
McIntyre, M. E. 1981 On the ‘wave momentum’ myth. J. Fluid Mech. 106, 331–347.CrossRefGoogle Scholar
Meiron, D. I. & Saffman, P. G. 1983 Overhanging interfacial gravity waves of large amplitude. J. Fluid Mech. 129, 213–218.CrossRefGoogle Scholar
Mielke, A. 1995 Homoclinic and heteroclinic solutions in two-phase flow. In Proceedings of the IUTAM/ISIMM Symposium on Structure and Dynamics of Nonlinear Waves in Fluids (ed. Mielke, A. & Kirchgässner, K.), pp. 353–362. World Scientific.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. (ed. Horikawa, K & Maruo, H), pp. 399–406. Springer.CrossRefGoogle Scholar
Miyata, M. 2000 A note on broad narrow solitary waves. PRC Report 00-01, SOEST 00-05, 1–26.Google Scholar
Monismith, S. G, Cowen, E. A., Nepf, H. M., Magnaudet, J. & Thais, L. 2007 Laboratory observations of mean flows under surface gravity waves. J. Fluid Mech. 573, 133–147.CrossRefGoogle Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1988 Finite-amplitude solitary waves at the interface between two homogeneous fluids. Phys. Fluids 31, 3550–3559.CrossRefGoogle 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
Saffman, P. G. & Yuen, H. C. 1982 Finite amplitude interfacial waves in the presence of a current. J. Fluid Mech. 123, 459–476.CrossRefGoogle Scholar
Su, C. H. & Gardner, C. S. 1969 Korteweg–de Vries equation and generalization III: derivation of the Korteweg–de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539.CrossRefGoogle Scholar
Thorpe, S. A. 1968 On the shape of progressive internal waves. Phil. Trans. R. Soc. Lond. A. 263, 563–614.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.Google Scholar
Troy, D. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136.CrossRefGoogle 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
Turner, R. E. L. & Vanden-Broeck, J.-M. 1985 The limiting configuration of interfacial gravity waves. Phys. Fluids 29, 372–375.CrossRefGoogle Scholar
Williams, J. M. 1981 Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A. 263, 139–188.Google Scholar
Yih, C.-S. 1959 Gravity waves in a stratified fluid. J. Fluid Mech. 8, 481–508.CrossRefGoogle Scholar
Yih, C.-S. 1997 The role of drift mass in the kinetic energy and momentum of periodic waves and sound waves. J. Fluid Mech. 331, 429–438.CrossRefGoogle Scholar