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Geostrophic adjustment in a channel: nonlinear and dispersive effects

Published online by Cambridge University Press:  26 April 2006

G. G. Tomasson  and
W. K. Melville
G. G. Tomasson
Affiliation:
R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
W. K. Melville
Affiliation:
R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Scripps Institution of Oceanography, University of California, San Diego, La Jolla CA 92093-0213, USA.
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Abstract

We consider the general problem of geostrophic adjustment in a channel in the weakly nonlinear and dispersive (non-hydrostatic) limit. Governing equations of Boussinesq-type are derived, based on the assumption of weak nonlinear, dispersive and rotational effects, both for surface waves on a homogeneous fluid and internal waves in a two-layer system. Numerical solutions of the Boussinesq equations are presented, giving examples of the geostrophic adjustment in a channel for two different kinds of initial disturbances, both with non-zero perturbation potential vorticity. The timescales of rotational separation (that is, the separation of the Kelvin and Poincaré waves due to their dispersive properties) and that of nonlinear evolution are considered, with particular concern for the resonant interactions of nonlinear Kelvin waves and linear Poincaré waves described by Melville, Tomasson & Renouard (1989). A parameter measuring the ratio of the two timescales is used to predict when the free and forced Poincaré waves may be separated in the solution. It also distinguishes the cases in which the linear solutions are valid for the rotational separation from those requiring the full Boussinesq equations. Finally, solutions for the evolution of nonlinear internal waves in a sea strait are presented, and the effects of friction on the wavefront curvature of the nonlinear Kelvin waves are briefly considered.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 23 - 57
Copyright
© 1992 Cambridge University Press

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References

Armi, L. & Farmer D. M. 1988 The flow of Mediterranean water through the Strait of Gibraltar. Prog. Oceanogr. 21, 1–105.Google Scholar
Farmer D. M. 1978 Observations of long nonlinear internal waves in a lake. J. Phys. Oceanogr. 8, 63–73.Google Scholar
Farmer, D. M. & Armi L. 1988 The flow of Atlantic water through the Strait of Gibraltar. Prog. Oceanogr. 21, 1–105.Google Scholar
Farmer, D. M. & Smith J. D. 1988 Nonlinear internal waves in a fjord. In Hydrodynamics of Estuaries and Fjords. Elsevier.
Gargett A. E. 1976 Generation of internal waves in the Strait of Georgia, British Columbia. Deep-Sea Res. 23, 17–32.Google Scholar
Gill A. E. 1976 Adjustment under gravity in a rotating channel. J. Fluid Mech. 77, 603–621.Google Scholar
Gill A. E. 1982 Atmosphere–Ocean Dynamics. Academic.
Grimshaw R. 1985 Evolution equations for weakly nonlinear, long internal waves in a rotating fluid. Stud. Appl. Maths 73, 1–33.Google Scholar
Grimshaw, R. & Melville W. K. 1989 On the derivation of the modified Kadomtsev-Petviashvili equation. Stud. Appl. Maths 80, 183–202.Google Scholar
Hammack, J. L. & Segur H. 1978 Modelling criteria for long water waves. J. Fluid Mech. 84, 359–373.Google Scholar
Hermann A. J., Rhines, P. B. & Johnson E. R. 1989 Nonlinear Rossby adjustment in a channel: beyond Kelvin waves. J. Fluid Mech. 205, 469–502.Google Scholar
Hunkins, K. & Fliegel M. 1973 Internal undular surges in Seneca Lake: A natural occurrence of solitons. J. Geophys. Res. 78, 539–548.Google Scholar
Katsis, C. & Akylas T. R. 1987 Solitary internal waves in a rotating channel: A numerical study. Phys. Fluids 30, 297–301.Google Scholar
Lacombe, H. & Richez C. 1982 The regime of the Straits of Gibraltar. In Hydrodynamics of Semi-Enclosed Seas (ed. J. C. J. Nihoul), pp. 13–73. Elsevier.
La Violette, P. E. & Arnone, R. A. 1988 A tide-generated internal waveform in the western approaches to the Strait of Gibraltar. J. Geophys. Res. 93, 15653–15667.Google Scholar
Martinsen, E. A. & Weber J. E. 1981 Frictional influence on internal Kelvin waves. Tellus 33, 402–410.Google Scholar
Maxworthy T. 1979 A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. J. Geophys. Res. 84, 338–346.Google Scholar
Maxworthy T. 1983 Experiments on solitary internal Kelvin waves. J. Fluid Mech. 129, 365–383.Google Scholar
Melville W. K., Tomasson, G. G. & Renouard D. P. 1989 On the stability of Kelvin waves. J. Fluid Mech. 206, 1–23.Google Scholar
Pedersen G. 1988 On the numerical solution of the Boussinesq equations. Res. Rep. University of Oslo, Department of Mathematics.Google Scholar
Pedersen, G. & Rygg O. B. 1987 Numerical solution of the three dimensional Boussinesq equations for dispersive surface waves. Res. Rep. 1. University of Oslo, Institute of Mathematics.Google Scholar
Pierini S. 1989 A model for the Alboran Sea internal solitary waves. J. Phys. Oceanogr. 19, 755–772.Google Scholar
Renouard D. P., Chabert D'Hieres, G. & Zhang, X. 1987 An experimental study of strongly nonlinear waves in a rotating system. J. Fluid Mech. 177, 381–394.Google Scholar
Renouard D. P., Tomasson, G. G. & Melville W. K. 1992 An experimental and numerical study of nonlinear internal waves.
Rygg O. B. 1988 Nonlinear refraction-diffraction of surface waves in intermediate and shallow water, Coastal Engng. 12, 191–211.Google Scholar
Thorpe S. A., Hall, A. & Crofts I. 1972 The internal surge in Loch Ness. Nature 237, 96–98.Google Scholar
Tomasson G. G. 1991 Nonlinear waves in a channel: Three-dimensional and rotational effects. Doctoral thesis, Department of Civil Engineering, Massachusetts Institute of Technology.
Tomasson, G. G. & Melville W. K. 1990 Nonlinear and dispersive effects in Kelvin waves Phys. Fluids A 2, 189–193.Google Scholar
Wesson, J. C. & Gregg M. C. 1988 Turbulent dissipation in the Strait of Gibraltar and associated mixing. In Small-scale Turbulence and Mixing in the Ocean: Proc. 19th Intl. LieGge Coll. on Ocean Hydrodyn (ed. J. C. J. Nihoul & B. M. Jamart), pp. 201–212. Elsevier.
Ziegenbein J. 1969 Short internal waves in the Strait of Gibraltar. Deep-Sea Res. 16, 479–487.Google Scholar