Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-19T06:32:43.686Z Has data issue: false hasContentIssue false
  • English
  • Français

On the shape and breaking of finite amplitude internal gravity waves in a shear flow

Published online by Cambridge University Press:  12 April 2006

S. A. Thorpe
S. A. Thorpe
Affiliation:
Institute of Oceanographic Sciences, Wormley, Godalming, Surrey, England
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.

In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.

This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 85 , Issue 1 , 7 March 1978 , pp. 7 - 31
Copyright
© 1978 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1976 On the normal modes of parallel flow of inviscid stratified fluid. J. Fluid Mech. 75, 149–171.Google Scholar
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241–270.Google Scholar
Borisenko, Yu. D., Voronovich, A. G., Leonov, A. I. & Miropolskiy, YU. Z. 1976 Towards a theory of non-stationary weakly non-linear internal waves in a stratified fluid. Izv. Atoms. Ocean Phys. 12, 174–179 (English edn).Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. Roy. Soc. A 338, 101–110.Google Scholar
Eliassen, A., HøILAND, E. & Riis, E. 1953 Two-dimensional perturbation of a flow with a constant shear of a stratified fluid. Inst. Weather Climate Res., Oslo, Publ. no. 1.Google Scholar
Frankignoul, C. J. 1972 Stability of finite amplitude internal waves in a shear flow. Geophys. Fluid Dyn. 4, 91–99.Google Scholar
Garrett, C. & Munk, W. 1972 Oceanic mixing by breaking internal waves. Deep-Sea Res. 19, 823–832.Google Scholar
Grimshaw, R. 1976 The modulation and stability of an internal gravity wave. Mem. Soc. Roy. Sci. Liege, Ser. 6, 10, 299–314.Google Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–62.Google Scholar
Howard, L. N. 1963 Neutral curves and stability boundaries in stratified flow. J. Fluid Mech. 16, 333–342.Google Scholar
Kantha, L. H., Phillips, O. M. & Azad, R. S. 1977 On turbulent entrainment at a stable density interface. J. Fluid Mech. 79, 753–768.Google Scholar
Keulegan, G. H. & Carpenter, L. H. 1961 An experimental study of internal progressive oscillatory waves. Nat. Bur. Stand. Rep. no. 7319.Google Scholar
Long, R. R. 1972 The steepening of long internal waves. Tellus 24, 88–99.Google Scholar
Mcewan, A. D. & Robinson, R. M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67, 667–688.Google Scholar
Miles, J. W. 1967 Internal waves in a continuously stratified atmosphere or ocean. J. Fluid Mech. 28, 305–310.Google Scholar
Miropolskiy, Yu. Z. & Filyushkin, B. N. 1971 Temperature fluctuations in the upper ocean comparable in scale to internal gravity waves. Izv. Atmos. Ocean Phys. 7, 523–535 (English edn).Google Scholar
Orlanski, I. & Bryan, K. 1969 Formation of the thermocline step structure by large amplitude internal gravity waves. J. Geophys. Res. 74, 6975–6983.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Phillips, O. M. & Banner, M. L. 1974 Wave breaking in the presence of wind drift and swell. J. Fluid Mech. 66, 625–640.Google Scholar
Stokes, G. G. 1947 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. J. Fluid Mech. 9, 353–370.Google Scholar
Taylor, G. I. 1931 Effect of variation of density on the stability of superposed streams of fluid. Proc. Roy. Soc. A 132, 499–523.Google Scholar
Thorpe, S. A. 1968a On the shape of progressive internal waves. Phil. Trans. Roy. Soc. A 263, 563–614.Google Scholar
Thorpe, S. A. 1968b A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693–704.Google Scholar
Thorpe, S. A. 1969 Neutral eigensolutions of the stability equation for stratified shear flow. J. Fluid Mech. 36, 673–683.Google Scholar
Thorpe, S. A. 1971 Experiments on the instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299–319.Google Scholar
Thorpe, S. A. 1974 Near-resonant forcing in a shallow two-layer fluid: a model for the internal surge in Loch Ness?. J. Fluid Mech. 63, 509–627.Google Scholar
Thorpe, S. A. 1977 On the stability of internal wavetrains. In Deep-Sea Res., A Voyage of Discovery (ed. M. V. Angel), pp. 199–212. Pergamon.
Voronovich, A. G. 1976 Propagation of internal and surface gravity waves in the approximation of geometrical optics. Izv. Atmos. Ocean Phys. 12, 519–523 (English edn).Google Scholar
Yih, C. S. 1974 Progressive waves of permanent form in continuously stratified fluids. Phys. Fluids 17, 1489–1495.Google Scholar