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A parametric study of the generation and degeneration of wind-forced long internal waves in narrow lakes

Published online by Cambridge University Press:  04 February 2010

TAKAHIRO SAKAI  and
L. G. REDEKOPP
TAKAHIRO SAKAI*
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1191, USA
L. G. REDEKOPP
Affiliation:
Department of Aerospace & Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1191, USA
*
†Email address for correspondence: tsakai@usc.edu
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Abstract

The generation and energy downscaling of wind-forced long internal waves in strongly stratified small-to-medium sized narrow lakes are studied. A two-layer nonlinear model with forcing and damping is employed. Even though the wave field is fundamentally bidirectional in nature, a domain folding technique is employed to simulate the leading-order internal wave field in terms of a weakly nonlinear weakly dispersive model equation of Korteweg–deVries type. Parametric effects of wind-forcing and environmental conditions, including variable topography and variable basin width, are examined. Energy downscaling from basin-scale waves to shorter scales are quantified by means of a time evolution of the wave energy spectra. It is demonstrated that an internal wave resonance is possible when repetitive wind-forcing events arise with a frequency near the linear seiche frequency. An attempt is made to apply the model to describe the shoaling of long waves on sloping endwall boundaries. Modelling of the energy loss and energy reflection during a shoaling event is calibrated against laboratory experiments.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 645 , 25 February 2010 , pp. 315 - 344
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Baines, P. G. 1995 Topographic Effects in Strafified Flows. Cambridge University Press.Google Scholar
Boegman, L., Ivey, G. N. & Imberger, J. 2004 An internal solitary wave parameterization for hydrodynamic lake models. In Proceedings of Fifteenth Australasian Fluid Mechanics Conference. The University of Sydney, Sydney, Australia.Google Scholar
Boegman, L., Ivey, G. N. & Imberger, J. 2005 a The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 1620–1637.CrossRefGoogle Scholar
Boegman, L., Ivey, G. N. & Imberger, J. 2005 b The energetics of large-scale internal wave degeneration in lakes. J. Fluid Mech. 531, 159–180.CrossRefGoogle Scholar
Bourgault, D. & Kelly, D. 2007 On the reflectance of uniform slopes for normally incident interfacial solitary waves. J. Phys. Oceanogr. 37, 1156–1162.CrossRefGoogle Scholar
Drazin, P. G. & Johnson, R. S. 1989 Solitons: An Introduction, pp. 90–95. Cambridge University Press.Google Scholar
Farmer, D. M. 1978 Observations of long nonlinear internal waves in lake. J. Phys. Oceanogr. 8, 63–73.Google Scholar
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonliear wave phenomena. Phil. Trans. R. Soc. Lond. A 289, 373–404.Google Scholar
Gardner, C. S. 1971 The Korteweg–deVries equation and generalizations. IV. The Korteweg–deVries equation as a Hamiltonian system. J. Math. Phys. 12, 1548–1551.CrossRefGoogle Scholar
Grimshaw, R. 2002 Environmental Stratified Flow. Kluwer Academic.Google Scholar
Grimshaw, R., Pelinovsky, E. & Talipova, T. 2007 Modelling internal solitary waves in the coastal ocean. Surv. Geophys. 28, 273–298.CrossRefGoogle Scholar
Heaps, N. S. 1984 Vertical structure of current in homogeneous and stratified waters. In Hydrodynamics of Lakes (ed. Hutter, K.), pp. 153–207. Springer Verlag Wien.Google Scholar
Heaps, N. S. & Ramsbottom, A. E. 1966 Wind effects on the water in a narrow two-layered lake. Phil. Trans. R. Soc. Lond. A 259, 391–430.Google Scholar
Helfrich, K. R. 1992 Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech. 243, 133–154.CrossRefGoogle Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.Google Scholar
Horn, D. A., Imberger, J. & Ivey, G. N. 2001 The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181–207.CrossRefGoogle Scholar
Horn, D. A., Imberger, J., Ivey, G. N. & Redekopp, L. G. 2002 A weakly nonlinear model of long internal waves in closed basins. J. Fluid Mech. 467, 269–267.Google Scholar
Hunkins, K. & Fliegel, M. 1973 Internal undular surges in Seneca lake: a natural occurrence of solitons. J. Geophys. Res. 78, 539–548.CrossRefGoogle Scholar
Imberger, J. 1994 Transport processes in lakes: a review. In Limnology Now: A Paradigm of Planetary Problems (ed. Margalef, R.), pp. 99–193. Elsevier.Google Scholar
Imberger, J. 1998 Coastal and Esturine Studies. American Geophysical Union.Google Scholar
Imberger, J. & Patterson, J. C. 1990 Physical limnology. Adv. Appl. Mech. 27, 303–475.Google Scholar
Keulegan, G. H. 1948 Gradual damping of solitary waves. J. Res. Natl Bur. Stand. 40, 487–498.Google Scholar
Koop, C. G. & Butler, G. 1981 An investigation of internal solitary waves. J. Fluid Mech. 112, 225–251.Google Scholar
Lamb, K. G. & Nguyen, V. T. 2009 Calculating energy flux in internal solitary waves with an application to reflectance. J. Phys. Oceanogr. 39, 559–580.CrossRefGoogle Scholar
Michallet, H. & Ivey, G. N. 1999 Experiments on mixing due to internal solitary waves. J. Geophys. Res. 104 (C6), 13467–13477.CrossRefGoogle Scholar
Miles, J. W. 1976 Damping of weakly nonlinear shallow-water waves. J. Fluid Mech. 76, 251–257.Google Scholar
Monismith, S. 1986 An experimental study of the upwelling response of stratified reservoirs to surface shear stress. J. Fluid Mech. 171, 407–439.CrossRefGoogle Scholar
Mortimer, C. H. 1952 Water movements in lakes during summer stratification; evidence from the distribution of termperature in Windermere. Phil. Trans. Roy. Soc. Lond. B 236, 355–404.Google Scholar
Ostrovsky, L. A. & Grue, J 2003 Evolution equations for strongly nonlinear internal waves. Phys. Fluids 15, 2934–2948.Google Scholar
Ostrovsky, L. A. & Stepanyants, Y. A. 2005 Internal solitons in laboratory experiments: Comparison with theoretical models. Chaos 15, 037111.CrossRefGoogle ScholarPubMed
Ostrovsky, I., Yacobi, Y. Z., Walline, P. & Kalikhman, I. 1996 Seiche-induced mixing: its impact on lake productivity. Limnol. Oceanogr. 41 (2), 323–332.Google Scholar
Saggio, A. & Imberger, J. 1998 Internal wave weather in a strafified lake. Limnol. Oceanogr. 43, 1780–1795.CrossRefGoogle Scholar
Sakai, T. & Redekopp, L. G. 2007 Models for strongly-nonlinear evolution of long internal waves in a two-layer stratification. Nonlinear Process. Geophys. 14, 31–47.CrossRefGoogle Scholar
Sakai, T. & Redekopp, L. G. 2009 A weakly nonlinear model for multi-modal evolution of wind-generated long internal waves in a closed basin. Nonlinear Process. Geophys. 16, 487–502.CrossRefGoogle Scholar
Stevens, C. & Imberger, J. 1996 The initial response of a stratified lake to surface shear stress. J. Fluid Mech. 312, 39–66.Google Scholar
Teng, M. H. & Wu, T. Y. 1992 Nonlinear water waves in channels of arbitrary shape. J. Fluid Mech. 242, 211–233.CrossRefGoogle 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, 509–527.Google Scholar
Thorpe, S. A. & Hall, A. 1972 The internal surge in Loch Ness. Nature 23, 96–98.Google Scholar
Thorpe, S. A., Keen, J. M., Jiang, R. & Lemmin, U. 1996 High-frequency internal waves in Lake Geneva. Phil. Trans. R. Soc. Lond. A 354, 237–257.Google Scholar
Wiegand, R. C. & Carmack, E. 1986 The climatology of internal waves in deep temperate lake. J. Geophys. Res. 91, 3951–3958.CrossRefGoogle Scholar
Wüest, A. & Lorke, A. 2003 Small-scale hydrodynamics in lakes. Annu. Rev. Fluid Mech. 35, 373–412.Google Scholar