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Steady periodic waves in a three-layer fluid with shear in the middle layer

Published online by Cambridge University Press:  14 December 2007

MICHAEL J. CHEN  and
LAWRENCE K. FORBES
MICHAEL J. CHEN
Affiliation:
School of Mathematics and Physics, University of Tasmania, Hobart, Australia 7001
LAWRENCE K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania, Hobart, Australia 7001
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Abstract

A three-layer intrusion flow is considered, in which all three layers are in motion, with different speeds, relative to the observer. Shear is present in the middle layer, and the lowest fluid may even move oppositely to the upper two (so giving an exchange flow). Two thin interfaces are present, above and below the moving middle layer. A linearized analysis is presented for small wave amplitudes. Nonlinear periodic solutions are then obtained using a Fourier technique, and reveal a range of nonlinear phenomena, including limiting waves, multiple solutions and resonances.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 594 , 10 January 2008 , pp. 157 - 181
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Akylas, T. R. & Grimshaw, R. H. J. 1992 Solitary internal waves with oscillatory tails. J. Fluid Mech. 242, 279–298.Google Scholar
Armi, L. & Farmer, D. M. 1986 Maximal two-layer exchange through a contraction with barotropic flow. J. Fluid Mech. 164, 27–51.CrossRefGoogle Scholar
Atkinson, K. E. 1978 An Introduction to Numerical Analysis. Wiley.Google Scholar
Candela, J. 2001 Mediterranean water and global circulation. In Ocean Circulation and Climate (ed. Siedler, G., Church, J. & Gould, J.), pp. 419–429. Academic Press.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity-capillary waves on deep water. Part II. Numerical results for finite amplitude. Stud. Appl. Maths 62, 95–111.Google Scholar
Flynn, M. R. & Sutherland, B. R. 2004 Intrusive gravity currents and internal gravity wave generation in stratified fluid. J. Fluid Mech. 514, 355–383.CrossRefGoogle Scholar
Forbes, L. K. & Hocking, G. C. 2007 An intrusion layer in stationary incompressible fluids. Part 2. Solitary waves. Eur. J. Appl. Maths 17, 577–595.Google Scholar
Forbes, L. K., Hocking, G. C. & Farrow, D. E. 2006 An intrusion layer in stationary incompressible fluids. Part 1. Periodic waves. Eur. J. Appl. Maths 17, 557–575.Google Scholar
Forbes, L. K., Chen, M. J. & Trenham, C. E. 2007 Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. J. Comput. Phys. 221, 269–287.Google Scholar
Fructus, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323–347.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Lewis, J. E., Lake, B. M. & Ko, D. R. S. 1974 On the interaction of internal waves and surface gravity waves. J. Fluid Mech. 63, 773–800.Google Scholar
Mehta, A. P., Sutherland, B. R. & Kyba, P. J. 2002 Interfacial gravity currents. Part II. Wave excitation. Phys. Fluids 14, 3558–3569.CrossRefGoogle Scholar
Meiron, D. I. & Saffman, P. G. 1983 Overhanging interfacial gravity waves of large amplitude. J. Fluid Mech. 120, 213–218.CrossRefGoogle Scholar
Michallet, H. & Dias, F. 1999 Non-linear resonance between short and long waves. In Proc. 9th Intl Offshore and Polar Engineering Conference, pp. 193–198.Google Scholar
Părău, E. & Dias, F. 2001 Interfacial periodic waves of permanent form with free-surface boundary conditions. J. Fluid Mech. 437, 325–336.Google Scholar
Phillips, O. M. 1969 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Phillips, O. M. 1974 Non-linear dispersive waves. Annu. Rev. Fluid Mech. 6, 93–108.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983 Interfacial progressive gravity waves in a two-layer shear flow. Phys. Fluids 26, 1731–1739.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
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
Sutherland, B. R., Kyba, P. J. & Flynn, M. R. 2004 Intrusive gravity currents in two-layer fluids. J. Fluid Mech. 514, 327–353.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 82, 499–523.Google Scholar
Timmermans, M.-L. E. & Pratt, L. J. 2005 Two-layer rotating exchange flow between two deep basins: Theory and application to the Strait of Gibraltar. J. Phys. Ocean Ogr. 35, 1568–1592.Google Scholar
Turner, R. E. L. & Vanden-Broeck, J.-M. 1986 The limiting configuration of interfacial gravity waves. Phys. Fluids 29, 372–375.Google Scholar
Vanden-Broeck, J.-M. & Turner, R. E. L. 1992 Long periodic internal waves. Phys. Fluids A 4, 1929–1935.Google Scholar
Williams, M. J. M., Jenkins, A. & Determann, J. 1998 Physical controls on ocean circulation beneath ice shelves revealed by numerical models. In Ocean, Ice, and Atmosphere: Interactions at the Antarctic Continental Margin (ed. Jacobs, S. S. & Weiss, R. F.), pp. 285–299. American Geophysical Union.Google Scholar
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688–700.CrossRefGoogle Scholar