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On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions

Published online by Cambridge University Press:  19 April 2006

T. Maxworthy
T. Maxworthy
Affiliation:
Departments of Mechanical and Aerospace Engineering, University of Southern California, Los Angeles, California 90007
Also Jet Propulsion Laboratory, Pasadena, California and the Australian National University, Canberra, Australia.
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Abstract

We show how trains of nonlinear, dispersive wavesIn some of the caaes to be described these are in fact sequences of solitary waves which are ordered by amplitude and which separate in space, as they propagate. In other cases the wave amplitude decreases as the wave propagates, but since the essential balance is between nonlinear steepening and frequency dispersion we feel justified in using the adjective ‘solitary’ to describe them though they violate the clessical description of such waves. can be produced by allowing a region of mixed fluid, with a potential energy greater than its surroundings, to collapse towards its equilibrium state. The number of waves and their amplitude depend on the properties of the mixed region and of the ambient stratification. Three different geometrical configurations have been chosen and while each gives qualitatively the same results the form taken by the generated waves and the final equilibrium shape of the mixed region depend critically on these geometrical factors. We relate the internal waves produced by this mechanism to waves produced in natural systems and show that our observations support at least one possible explanation for those found in the oceans and planetary atmospheres.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 96 , Issue 1 , 16 January 1980 , pp. 47 - 64
Copyright
© 1980 Cambridge University Press

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References

Amen, R. & Maxworthy, T. 1980 The gravitational collapse of a mixed region into a linearly stratified fluid. J. Fluid Mech. 96, 65–80.Google Scholar
Apel, J. R., Byrne, H. M., Proni, J. E. & Charnell, R. L. 1975 Observations of oceanic and internal surface waves from the Earth Resources Technology Satellite. J. Geophys. Res. 80, 865–888.Google Scholar
Bell, T. H. & Dugan, J. P. 1974 Model for mixed region collapse in a stratified fluid. J. Engng Math. 8, 241–248.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Charba, J. 1972 Gravity current model applied to analysis of squall-line gust-front. NDAA-TM ERL NSSL-61.
Christie, D. R., Muirhead, K. J. & Hales, A. L. 1978a Density currents: a source of atmospheric solitons. Sci. Rep. R.S.E.S., Australian National University, Canberra.Google Scholar
Christie, D. R., Muirhead, K. J. & Hales, A. L. 1978b On solitary waves in the atmosphere. J. Atmos. Sci. 35, 805.Google Scholar
Chwang, A. T. & Wu, T. Y. 1976 Cylindrical solitary waves. Proc. IUTAM Symp. on Water Waves in Water of Varying Depth. Canberra, Australia.
Clarke, R. H. 1972 The Morning Glory: An atmospheric hydraulic jump. J. Appl. Met. 11, 304–311.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. & Smith, J. D. 1978 Non-linear internal waves in a fjord. In Hydrodynamics of Estuaries and Fjords (ed. J. Nihoul). Elsevier.
Gargett, A. E. 1976 Generation of internal waves in the Strait of Georgia, British Columbia. Deep Sea Res. 23, 17–32.Google Scholar
Goff, R. C. 1976 Vertical structure of thunderstorm outflow. Mon. Wea. Rev. 104, 1429–1440.Google Scholar
Halpern, D. 1971 Observations of short period internal waves in Massachusetts Bay. J. Mar. Res. 29, 116–132.Google Scholar
Hammack, J. L. & Segur, H. 1974 The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiment. J. Fluid Mech. 74, 593–610.Google Scholar
Hartman, R. J. & Lewis, H. W. 1972 Wake collapse in a stratified fluid: Linear treatment. J. Fluid Mech. 51, 613–618.Google Scholar
Hurdis, D. A. & Pao, H.-P. 1975 Experimental observation of internal solitary waves in a stratified fluid. Phys. Fluids 18, 385–386.Google Scholar
Joseph, R. I. 1977 Solitary waves in a finite depth fluid. J. Phys. (A : Math. Gen.) 10, L255–227.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersive media. Dokl. Akad. Nauk S.S.S.R. 15, 753 (Sov. Phys. Dokl. 15, 539).Google Scholar
Kao, T. W. 1976 Principal stage of wake collapse in a stratified fluid: Two-dimensional theory. Phys. Fluids 19, 1071–1074.CrossRefGoogle Scholar
Lee, C. Y. & Beardsley, R. C. 1974 The generation of long non-linear internal waves in a weakly-stratified shear flow. J. Geophys. Res. 79, 453–462.Google Scholar
Manins, P. C. 1976 Mixed-region collapse in a stratified fluid. J. Fluid Mech. 77, 177–183.Google Scholar
Maxworthy, T. 1976 Experiments on collisions between solitary waves. J. Fluid Mech. 76, 177–185.Google Scholar
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465–495.Google Scholar
Maxworthy, T. 1978 A mechanism for the generation of internal solitary waves by tidal flow over submarine topography. Ocean Modelling, no. 14, Department of Applied Mathematics & Theoretical Physics, University of Cambridge.
Maxworthy, T. 1979 A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. Submitted to J. Geophys. Res.Google Scholar
Maxworthy, T. & Browand, F. K. 1975 Experiments in rotating and stratified flows: Oceanographic application. Ann. Rev. Fluid Mech. 7, 273–305.Google Scholar
Maxworthy, T., Redekopp, L. G. & Weidman, P. D. 1978 On the production and interaction of planetary solitary waves: Applications to the Jovian atmosphere. Icarus 33, 388–409.CrossRefGoogle Scholar
Mei, C. C. 1969 Collapse of a homogeneous fluid mass in a stratified fluid. Proc. 12th Int. Cong. Appl. Mech. pp. 321–330. Springer.
Miles, J. W. 1977a Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–170.Google Scholar
Miles, J. W. 1977b Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.Google Scholar
Newell, A. C. & Redekopp, L. G. 1977 Breakdown of Zakharov-Shabat theory and soliton creation. Phy. Rev. Lett. 38, 377–380.Google Scholar
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 1082–91.Google Scholar
Padmanabhan, H., Ames, W. F., Kennedy, J. F. & Hung, T-K. 1970 A numerical investigation of wake deformation in density stratified fluids. J. Engng Math. 4, 229–241.Google Scholar
Sawyer, C. & Apel, J. R. 1976 Satellite images of ocean internal-wave signatures. N.O.A.A. S/T2401.
Schooley, A. H. & Stewart, R. W. 1963 Experiments with a self-propelled body submerged in a fluid with a vertical density gradient. J. Fluid Mech. 15, 83–96.Google Scholar
Schooley, A. H. & Hughes, B. A. 1972 An experimental and theoretical study of internal waves generated by the collapse of a two-dimensional mixed region in a density gradient. J. Fluid Mech. 51, 159–175.CrossRefGoogle Scholar
Simpson, J. E. 1972 Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53, 759–768.Google Scholar
Simpson, J. E., Mansfield, D. A. & Milford, J. R. 1977 Inland penetration of sea-breeze fronts. Quart. J. Roy. Met. Soc. 103, 47–76.Google Scholar
Smith, J. D. & Farmer, D. M. 1977 Non-linear internal waves and internal hydraulic jumps in a fjord. In Geofluiddynamic Wave Mathematics, pp. 42–53. University of Washington, Seattle.
Thorpe, S. A. 1974 Near resonant forcing in a shallow two-layer fluid: A model for the interna surge in Loch Ness. J. Fluid Mech. 63, 509–527.Google Scholar
Weidman, P. D. & Maxworthy, T. 1978 Experiments on strong interactions between solitary waves. J. Fluid Mech. 85, 417–432.Google Scholar
Weidman, P. D. & Ko, K. 1979 Weakly non-linear cylindrical gravity waves. To be submitted for publication.
Wessel, W. R. 1969 Numerical study of the collapse of a perturbation in an infinite density stratified fluid. Phys. Fluids Suppl. 12, II 171–176.Google Scholar
Whitham, G. B. 1974 Linear and Non-Linear Waves. Wiley.
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 56, 265–276.Google Scholar
Zakharov, V. E. & Shabat, A. B. 1974 A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. Funct. Anal. & Appl. 8, 226–235.Google Scholar