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Resonant flow of a stratified fluid over topography

Published online by Cambridge University Press:  21 April 2006

R. H. J. Grimshaw  and
N. Smyth
R. H. J. Grimshaw
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia Current address: School of Mathematics, University of N.S.W., P.O. Box 1, Kensington, N.S.W. 2033, Australia.
N. Smyth
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia Current address: School of Mathematics, University of N.S.W., P.O. Box 1, Kensington, N.S.W. 2033, Australia.
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Abstract

The flow of a stratified fluid over topography is considered in the long-wavelength weakly nonlinear limit for the case when the flow is near resonance; that is, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. It is shown that the amplitude of this mode is governed by a forced Korteweg-de Vries equation. This equation is discussed both analytically and numerically for a variety of different cases, covering subcritical and supercritical flow, resonant or non-resonant, and for localized forcing that has either the same, or opposite, polarity to the solitary waves that would exist in the absence of forcing. In many cases a significant upstream disturbance is generated which consists of a train of solitary waves. The usefulness of internal hydraulic theory in interpreting the results is also demonstrated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 169 , August 1986 , pp. 429 - 464
Copyright
© 1986 Cambridge University Press

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References

Akylas, T. R. 1984 On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466.Google Scholar
Baines, P. G. 1977 Upstream influence and Long's model in stratified flows. J. Fluid Mech. 82, 147–159.Google Scholar
Baines, P. G. 1979 Observation of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus 31, 351–371.Google Scholar
Baines, P. G. 1984 A unified description of two-layer flow over topography. J. Fluid Mech. 146, 127–167.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Benjamin, T. B. 1970 Upstream influence. J. Fluid Mech. 40, 49–79.Google Scholar
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579–587.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.Google Scholar
Farmer, D. M. & Denton, R. A. 1985 Hydraulic control of flow over the sill in Observatory Inlet. J. Geophys. Res. 90, 9051–9068.Google Scholar
Farmer, D. M. & Smith, J. D. 1980 Tidal interaction of stratified flow with a sill in Knight Inlet. Deep-Sea Res. 27A, 239–254.Google Scholar
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A 289, 373–404.Google Scholar
Gear, J. & Grimshaw, R. 1983 A second order theory for solitary waves in shallow fluids. Phys. Fluids 26, 14–29.Google Scholar
Grimshaw, R. 1983 Solitary waves in density stratified fluids. In Nonlinear Deformation Waves (ed. V. Nigul & J. Engelbrecht), pp. 431–447. Springer.
Gurevich, A. V. & Pitaevskii, L. P. 1974 Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38, 291–297.Google Scholar
Hammack, J. L. & Segur, H. 1978 Modelling criteria for long water waves. J. Fluid Mech. 84, 359–373.Google Scholar
Huang, D.-B., Sibul, G. J., Webster, W. C., Wehausen, J. V., Wu, D.-M. & Wu, T. Y. 1982 Ships moving in the transcritical range. In Proc. Conf. on Behaviour of Ships in Restricted Waters, Varna, vol. 2, pp. 26–1-26-10.
Keady, G. 1971 Upstream influence in a two-fluid system. J. Fluid Mech. 49, 373–384.Google Scholar
Lee, S.-J. 1985 Generation of long water waves by moving disturbances. Ph.D. thesis. California Institute of Technology.
Mcintyre, M. E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow. J. Fluid Mech. 52, 209–242.Google Scholar
Malanotte-Rizzoli, P. 1984 Boundary-forced nonlinear planetary radiation. J. Phys. Oceanogr. 14, 1032–1046.Google Scholar
Miles, J. W. 1978 On the evolution of a solitary wave for very weak nonlinearity. J. Fluid Mech. 87, 773–783.Google Scholar
Miles, J. W. 1979 On internal solitary waves. Tellus 31, 456–562.Google Scholar
Patoine, A. & Warn; T. 1982 The interaction of long, quasi-stationary baroclinic waves with topography. J. Atmos. Sci. 39, 1018–1025.Google Scholar
Smyth, N. F. 1986 Modulation theory solution for resonant flow over topography. Proc. R. Soc. Lond. (to appear).
Warn, T. & Brasnett, B. 1983 The amplification and capture of atmospheric solitons by topography: theory of the onset of regional blocking. J. Atmos. Sci. 40, 28–38.Google Scholar
Whitham, G. B. 1965 Non-linear dispersive waves. Proc. R. Soc. Lond. A 283, 238–261.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Wu, D.-M. & Wu, T. Y. 1982 Three-dimensional nonlinear long waves due to moving surface pressure. In Proc. 14th Symp. on Naval Hydrodynamics, Ann Arbor.