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Statically unstable layers produced by overturning internal gravity waves

Published online by Cambridge University Press:  26 April 2006

S. A. Thorpe
S. A. Thorpe
Affiliation:
Department of Oceanography, The University, Southampton SO9 5NH, UK
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Abstract

Internal waves in a uniformly stratified fluid of sufficiently large amplitude develop tilted layers in which the fluid is statically unstable. To investigate the evolution and subsequent development of this structure, experiments are made in which a horizontal rectangular tube containing a fluid of uniform density gradient is gently rocked at a selected frequency about a horizontal axis normal to the tube length. Large-amplitude standing internal gravity waves of the first mode are generated, and these steepen and overturn, the isopycnal surfaces folding to produce a vertically thin and horizontally extensive layer in which the fluid is statically unstable. In experiments with relatively small forcing, the layer persists for some 6 buoyancy periods, with no detected evidence of secondary instability, and static stability is re-established as the periodic flow reverses. The layer however breaks down, with consequent diapycnal mixing, when greater forcing is applied.

The scale and growth rates of instability in the overturning internal gravity waves are estimated using the theory developed in a companion paper by Thorpe (1994a). For the parameters of the laboratory experiments with relatively small forcing, the growth rates are small, consistent with the absence of signs of secondary instability. Larger growth rates and disturbance amplification factors of about 70 are predicted for the conditions in the experiment in which mixing was observed to occur. The experimental observations are consistent with an instability having a longitudinal structure.

We conclude that the form and development of breaking in internal gravity waves will vary according to the circumstances in which waves break, but depend on the Prandtl number of the fluid and, in particular, on the Rayleigh and Reynolds numbers of regions of static instability which develop as the waves overturn.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 260 , 10 February 1994 , pp. 333 - 350
Copyright
© 1994 Cambridge University Press

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References

Batchelor, G. K. & Nitsche, J. M. 1991 Instability of stationary unbounded stratified fluid. J. Fluid Mech. 227, 357–391.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Blumen, W. 1988 Wave breaking in a compressible atmosphere. Geophys. Astrophys. Fluid Dyn, 43, 311–332Google Scholar
Broutman, D. 1984 The focussing of short internal waves by an inertial wave. Geophys. Astrophys. Fluid Dyn. 30, 199–225.Google Scholar
Broutman, D. 1986 On internal wave caustics. J. Phys. Oceanogr. 16, 1625–1635.Google Scholar
Castro, I. P., Snyder, W. H. & Marsh, G. L. 1983 Stratified flow over three-dimensional ridges. J. Fluid Mech. 153, 261–282.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.Google Scholar
Drazin, P. G. 1977 On the instability of an internal gravity wave. Proc. R. Soc. Lond. A 356, 411–432.Google Scholar
Ivey, G. N. & Nokes, R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479–500.Google Scholar
Klostermeyer, J. 1982 On parametric instabilities of finite-amplitude internal gravity waves. J. Fluid Mech. 119, 367–377.Google Scholar
Koop, C. G. & Mcgee, B. 1986 Measurements of internal gravity waves in a continuously stratified shear flow. J. Fluid Mech. 172, 453–480.Google Scholar
Lin, C.-L., Ferziger, J. H., Koseff, J. R. & Monismith, S. G. 1993 Simulation and stability of two dimensional internal gravity waves in a stratified shear flow. Dyn. Atmos. Oceans (to appear).Google Scholar
Mcewan, A. D. 1971 Degeneration of resonantly excited standing internal gravity waves. J. Fluid Mech. 50, 431–448.Google Scholar
Mcewan, A. D. 1973 Interactions between internal gravity waves and their traumatic effect on a continuous stratification. Boundary-Layer Met. 5, 159–175.Google Scholar
Mcewan, A. D. 1983a The kinematics of stratified mixing through internal wavebreaking. J. Fluid Mech. 128, 47–57.Google Scholar
Mcewan, A. D. 1983b Internal mixing in stratified fluids. J. Fluid Mech. 128, 59–81.Google Scholar
Mcewan, A. D. & Robinson, R. M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67, 667–687.Google Scholar
Mcintyre, M. E. & Palmer, T. N. 1984 The ‘surf zone’ in the stratosphere. J. Atmos. Terr. Phys. 46, 825–849.Google Scholar
Mcintyre, M. E. & Palmer, T. N. 1985 A note on the general concept of wave breaking for Rossby and gravity waves. Pure Appl. Geophys. 123, 964–975.Google Scholar
Mied, R. P. 1976 The occurrence of parametric instability in finite-amplitude internal gravity waves. J. Fluid Mech. 78, 763–784.Google Scholar
Miranda, P. M. A. & James, I. N. 1992 Non-linear three-dimensional effects on gravity-wave drag: Splitting flow and breaking waves. Q. J. R. Met. Soc. 118, 1057–1082.Google Scholar
Rottman, J. & Smith, R. B. 1989 A laboratory model of severe downslope winds. Tellus 41A, 401–415.Google Scholar
Taylor, J. 1992 The energetics of breaking events in a resonantly forced internal wave field. J. Fluid Mech. 239, 309–340.Google Scholar
Thorpe, S. A. 1968a On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32, 489–528.Google Scholar
Thorpe, S. A. 1968a On the shape of progressive internal waves. Phil. Trans. R. Soc. Lond. A 263, 563–614.Google Scholar
Thorpe, S. A. 1978a On the shape and breaking of finite-amplitude internal gravity waves in a shear flow. J. Fluid Mech. 85, 7–31.Google Scholar
Thorpe, S. A. 1978b On internal gravity waves in an accelerating shear flow. J. Fluid Mech. 88, 623–639.Google Scholar
Thorpe, S. A. 1981 An experimental study of critical layers. J. Fluid Mech. 103, 321–344.Google Scholar
Thorpe, S. A. 1984 A laboratory study of stratified accelerating flow over a rough boundary. J. Fluid Mech. 138, 185–196.Google Scholar
Thorpe, S. A. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279–302.Google Scholar
Thorpe, S. A. 1988 A note on breaking waves. Proc. R. Soc. Lond. A 419, 323–335.Google Scholar
Thorpe, S. A. 1989 The distortion of short internal waves by a longwave, with application to ocean boundary mixing. J. Fluid Mech. 208, 395–415.Google Scholar
Thorpe, S. A. 1994a The stability of statically unstable layers. J. Fluid Mech. 260, 315–331.Google Scholar
Thorpe, S. A. 1994b Observations of parametric instability and breaking waves in an oscillating tilted tube. J. Fluid Mech. 261, 33–45.Google Scholar
Winters, K. B. & D'asaro, E. A. 1989 Two-dimensional instability of finite amplitude internal gravity wave packets near a critical layer. J. Geophys. Res. 94, 12709–12719.Google Scholar
Winters, K. B. & Riley, J. J. 1992 Instability of internal waves near a critical level. Dyn. Atmos. Oceans 16, 249–278.Google Scholar
Woods, J. D. 1968 Wave-induced shear instability in the seasonal thermocline. J. Fluid Mech. 32 791–800.Google Scholar
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131–144.Google Scholar