Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T21:29:57.629Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamics of vorticity waves in the coastal zone

Published online by Cambridge University Press:  26 April 2006

Victor I. Shrira  and
Vyacheslav V. Voronovich
Victor I. Shrira
Affiliation:
P. P. Shirshov Institute of Oceanology Russian Academy of Sciences, 23 Krasikov str., 117218 Moscow, Russia
Vyacheslav V. Voronovich
Affiliation:
P. P. Shirshov Institute of Oceanology Russian Academy of Sciences, 23 Krasikov str., 117218 Moscow, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 326 , 10 November 1996 , pp. 181 - 203
Copyright
© 1996 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

Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Bowen, A. J. & Holman, R. A. 1989 Shear instabilities of the mean longshore current. 1. Theory. J. Geophys. Res. 94 (C12), 18023–18030.Google Scholar
Collings, I. L. & Grimshaw, R. 1984 Stable and unstable barotropic shelf waves in a coastal current. Geophys. Astrophys. Fluid Dyn. 29, 179–220.Google Scholar
Craik A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
Defant, A. 1961 Physical Oceanography, vol. 1. Pergamon.
Dodd, N. 1994 On the destabilisation of a longshore current on a plane beach: bottom shear stress, critical conditions and onset of instability. J. Geophys. Res. 99 (C1), 811–824.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Falques, A. & Iranzo, V. 1994 Numerical simulation of vorticity waves in the nearshore. J. Geophys. Res. 99 (C1), 825–841.Google Scholar
Gorshkov, K. A. & Ostrovsky, L. A. 1981 Interactions of Solitons in nonintegrable systems. Physica D 3, 428–438.Google Scholar
Grimshaw, R. H. G. 1987 Resonant forcing of barotropic coastally trapped waves. J. Phys. Oceanogr. 17, 53–65.Google Scholar
Grimshaw, R. H. G., Pelinovsky, E. & Xian, T. 1994 Interaction of a solitary wave with an external force. Physica D 77, 405–433.Google Scholar
Joseph, R. J. 1977 Solitary waves in a finite depth fluid. J. Phys. A 10, L225–L227.Google Scholar
Kachanov, Yu. S, Ryzhov, O. S. & Smith, F. T. 1993 Formation of solitons in transitional boundary layers: theory and experiments. J. Fluid Mech. 251, 273–297.Google Scholar
LeBlond, P. H. & Mysak, L. A. 1979 Waves in the Ocean. Elsevier.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Mei, C. C. 1993 Surface waves and coastal dynamics. Ann. Rev. Fluid Mech. 25, 215–240.Google Scholar
Nayfeh, A. H. 1973 Perturbation Methods. John Wiley & Sons.
Oltman-Shay, J., Howd, P. A. & Birkemeier, W.A. 1989 Shear instabilities of the mean longshore current. 2. Field observations. J. Geophys. Res. 94 (C12), 18031–18031.Google Scholar
Olver, F. W. G. 1974 Asymptotics and Special Functions. Academic.
Pedley, T. J. & Stephanoff, K. D. 1985 Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves. J. Fluid. Mech. 160, 337–367.Google Scholar
Petviashvilli, V. I. & Pokhotelov, O. A. 1992 Solitary Waves in Plasmas and in the Atmosphere. Gordon & Breach.
Saffman, P. G. 1993 Vortex Dynamics. Cambridge University Press.
Shrira, V. I. 1989 On the subsurface waves in the oceanic upper mixed layer. Dokl. Akad. Nauk SSSR 308, 732–736; English trans.: Trans. (Dokl.) USSR Academy of Sciences, Earth Science Sections 308, 276–279.Google Scholar
Shrira, V. I., Voronovich, V. V. & Kozhelupova, N. G. 1996 On the explosive instability of vorticity waves. J. Phys. Oceanogr. (to appear).Google Scholar