Editorial Type:
Article
Article Type:
Research Article

Explosive Instability of Vorticity Waves

V. I. Shrira P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia

Search for other papers by V. I. Shrira in
Current site
Google Scholar
PubMed Close
,
V. V. Voronovich P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia

Search for other papers by V. V. Voronovich in
Current site
Google Scholar
PubMed Close
, and
N. G. Kozhelupova P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia

Search for other papers by N. G. Kozhelupova in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Apr 1997
DOI:
https://doi.org/10.1175/1520-0485(1997)027<0542:EIOVW>2.0.CO;2
Page(s):
542–554
Restricted access

Abstract

The weakly nonlinear dynamics of “vorticity waves” (VW), specific wavelike motions occurring nearshore in the presence of an alongshore shear current is examined. By means of a standard asymptotic technique starting with the shallow-water equations, the authors derive the equations governing field evolution due to resonant interactions for the arbitrary current and bottom profiles and show that the VW interactions occur in the lowest order. Among them there are always explosive interactions; that is, the resonant triplets where all interacting waves growing synchronously tend to infinity in a finite time. The explosive instability is studied as a potential mechanism for VW generation, their main implications being the following: 1) The range of explosively excited scales appears to be much wider than the domain of linear instability, with the low-frequency cutoff absent; 2) the explosive instability occurs even when all linear perturbations are damped due to bottom friction, provided the initial amplitudes of disturbances exceed a certain threshold; and 3) the weakly nonlinear evolution most likely results in the emergence of strongly nonlinear motions. The dependence of the explosive processes on the background parameters is analyzed for the simplest model of alongshore current and topography.

Corresponding author address: Dr. V. I. Shrira, P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Krasikova 23, Moscow 117218, Russia.E-mail: shriraglas.apc.org

Abstract

The weakly nonlinear dynamics of “vorticity waves” (VW), specific wavelike motions occurring nearshore in the presence of an alongshore shear current is examined. By means of a standard asymptotic technique starting with the shallow-water equations, the authors derive the equations governing field evolution due to resonant interactions for the arbitrary current and bottom profiles and show that the VW interactions occur in the lowest order. Among them there are always explosive interactions; that is, the resonant triplets where all interacting waves growing synchronously tend to infinity in a finite time. The explosive instability is studied as a potential mechanism for VW generation, their main implications being the following: 1) The range of explosively excited scales appears to be much wider than the domain of linear instability, with the low-frequency cutoff absent; 2) the explosive instability occurs even when all linear perturbations are damped due to bottom friction, provided the initial amplitudes of disturbances exceed a certain threshold; and 3) the weakly nonlinear evolution most likely results in the emergence of strongly nonlinear motions. The dependence of the explosive processes on the background parameters is analyzed for the simplest model of alongshore current and topography.

Corresponding author address: Dr. V. I. Shrira, P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Krasikova 23, Moscow 117218, Russia.E-mail: shriraglas.apc.org

Save
Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Allen, J. S., P. A. Newberger, and R. A. Holman, 1996: Nonlinear shear instabilities of alongshore currents on plane beaches. J. Fluid Mech.,310, 181–213.

  • Benjamin, T. B., 1963: The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech.,16, 436–450.

  • Bowen, A. J., and R. A. Holman, 1989: Shear instabilities of the mean longshore current. 1.Theory. J. Geophys. Res.,94(C12), 18 023–18 030.

  • Cairns, R. A., 1979: The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech.,92, 1–14.

  • Craik, A. D. D., 1985: Wave Interactions and Fluid Flows. Cambridge University Press, 325 pp.

  • Deigaard, R., E. D. Christensen, J. S. Damgaard, and J. Fredsoe, 1994: Numerical simulation of finite amplitude shear waves and sediment transport. Proc. 24th Int. Coastal Eng. Conf., Kobe, Japan, Amer. Soc. Civil Eng., in press.

  • Dodd, N., 1994: On the destabilization of a longshore current on a plane beach: Bottom shear stress, critical conditions and onset of instability. J. Geophys. Res.,99 (C1), 811–824.

  • ——, and E. B. Thornton, 1990: Growth and energetics of shear waves in the nearshore. J. Geophys. Res.,95 (C9), 16 075–16 083.

  • ——, and ——, 1992: Longshore current growth to finite amplitude. Proc. 23d Int. Coastal Eng. Conf., Venice, Italy, Amer. Soc. Civil Eng., 2655–2668.

  • ——, J. Oltman-Shay, and E. B. Thornton, 1992: Shear instabilities in the longshore current: A comparison of observation and theory. J. Phys. Oceanogr.,22, 62–82.

  • Falques, A., and V. Iranzo, 1994: Numerical simulation of vorticity waves in the nearshore. J. Geophys. Res.,99 (C1), 825–841.

  • Ignatov, A. M., 1984: Wave interactions in active conservative medium. Zh. Eksp. Teor. Fiz.,87, 1652–1659.

  • ——, 1989: What is the energy of a wave in an unstable medium? Comm. Plasma Phys.,12, 205–213.

  • Kaup, D. J., A. Reiman, and A. Bers, 1979: Space-time evolution of nonlinear three-wave interactions. I. Interactions in a homogeneous media. Rev. Mod. Phys.,51, 275–310.

  • Landahl, M. T., 1962: On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech.,13, 609–632.

  • Mei, C. C., and P. L.-F. Liu, 1993: Surface waves and coastal dynamics. Annu. Rev. Fluid Mech.,25, 215–240.

  • Oltman-Shay, J., P. A. Howd, and W. A. Birkemeier, 1989: Shear instabilities of the mean longshore current. 2. Field observations. J. Geophys. Res.,94 (C12), 18 031–18 042.

  • Ostrovsky, L. A., S. A. Rybak, and L. Sh. Tsimring, 1986: Negative energy waves in hydrodynamics. Uspekhi Fizicheskikh Nauk,150, 417–437.

  • Putrevu, U., and I. A. Svendsen, 1992: Shear instability of longshore currents: A numerical study. J. Geophys. Res.,97 (C5), 7283–7303.

  • Reniers, A. J. H. M., J. A. Battjes, A. Falques, and D. A. Huntley, 1994: Shear-wave laboratory experiment. Proc. Int. Symp. Waves—Physical and Numerical Modelling, Amer. Soc. Civil Eng., Vancouver, BC, Canada, 356–365.

  • Romanova, N., and V. I. Shrira, 1988: Explosive generation of surface waves by wind. Izv. Acad. Sci. USSR, Atmos. Oceanic Phys.,24, 528–535.

  • Shemer, L., N. Dodd, and E. B. Thornton, 1991: Slow-time modulation of finite-depth water waves: Relation to longshore current oscillations. J. Geophys. Res.,96 (C4), 7105–7113.

  • Thornton, E. B., and R. T. Guza, 1986: Surf zone longshore currents and random waves: Field data and models. J. Phys. Oceanogr.,16, 1165–1178.

  • Weiland, J., and H. Wilhelmsson, 1977: Coherent Non-Linear Interaction of Waves in Plasmas. Pergamon Press, 187 pp.

  • Zakharov, V. E., and S. V. Manakov, 1975: The theory of resonant interactions of wave packets in nonlinear media. Zh. Eksp. Teor. Fiz.,69, 1654–1673.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 434 190 58
PDF Downloads 225 28 1