Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-07T18:42:57.091Z Has data issue: false hasContentIssue false
  • English
  • Français

Wavenumber transport: scattering of small-scale internal waves by large-scale wavepackets

Published online by Cambridge University Press:  26 April 2006

David L. Bruhwiler  and
Tasso J. Kaper
David L. Bruhwiler
Affiliation:
Grumman Research and Development Center, 4 Independence Way, Princeton, NJ 08540, USA
Tasso J. Kaper
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work, we treat the problem of small-scale, small-amplitude, internal waves interacting nonlinearly with a vigorous, large-scale, undulating shear. The amplitude of the background shear can be arbitrarily large, with a general profile, but our analysis requires that the amplitude vary on a length scale longer than the wavelength of the undulations. In order to illustrate the method, we consider the ray-theoretic model due to Broutman & Young (1986) of high-frequency oceanic internal waves that trap and detrap in a near-inertial wavepacket as a prototype problem. The near-inertial wavepacket tends to transport the high-frequency test waves from larger to smaller wavenumber, and hence to higher frequency. We identify the essential physical mechanisms of this wavenumber transport, and we quantify it. We also show that, for an initial ensemble of test waves with frequencies between the inertial and buoyancy frequencies and in which the number of test waves per frequency interval is proportional to the inverse square of the frequency, a single nonlinear wave–wave interaction fundamentally alters this initial distribution. After the interaction, the slope on a log-log plot is nearly flat, whereas initially it was -2. Our analysis captures this change in slope. The main techniques employed are classical adiabatic invariance theory and adiabatic separatrix crossing theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 289 , 25 April 1995 , pp. 379 - 405
Copyright
© 1995 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

Broutman, D. & Grimshaw, R. 1988 The energetics of the interaction of short, small-amplitude internal waves and inertial waves. J. Fluid Mech. 196, 93–106.Google Scholar
Broutman, D. & Young, W. R. 1986 On the interaction of small-scale oceanic internal waves with near-inertial waves. J. Fluid Mech. 166, 341–358 (referred to herein as B & Y).Google Scholar
Bruhwiler, D. L. 1990 Scattering and diffusion of particles in slowly-varying, large-amplitude waves. PhD thesis, University of Colorado, Boulder.
Bruhwiler, D. L. & Cary, J. R. 1989 Diffusion of particles in a slowly-modulated wave. Physica D 40, 265–282.Google Scholar
Bruhwiler, D. L. & Cary, J. R. 1992 Particle dynamics in a large-amplitude wave packet. Phys. Rev. Lett. 68, 255–258.Google Scholar
Bruhwiler, D. L. & Cary, J. R. 1994 Dynamics of particles trapping and detrapping in coherent wave packets. Phys. Rev. E 50, 3949–3961.Google Scholar
Byrd, P. F. & Friedman, M. D. 1954 Handbook of Elliptic Integrals for Engineers and Physicists, pp. 9–10. Springer.
Cary, J. R., Escande, D. F. & Tennyson, J. L. 1986 Adiabatic invariant change due to separatrix crossing. Phys. Rev. A 34, 4256–4275.Google Scholar
Cary, J. R. & Skodje, R. 1989 Phase change between separatrix crossings. Physica D 36, 287–316.Google Scholar
Drummond, W. E. & Pines, D. 1964 Nonlinear plasma oscillations. Ann. Phys. 28, 478–499.Google Scholar
Elskens, Y. & Escande, D. F. 1991 Slowly pulsating separatrices sweep homoclinic tangles where islands must be small: an extension of classical adiabatic theory. Nonlinearity 4, 615–667.Google Scholar
Fuchs, V., Krapchev, V., Ram, A. & Bers, A. 1985 Diffusion of electrons by coherent wavepackets. Physica D 14, 141–160.Google Scholar
Graham, K. N. & Fejer, J. A. 1976 One-dimensional numerical study of charged particle trajectories in turbulent electrostatic wave fields. Phys. Fluids 19, 1054–1058.Google Scholar
Henrard, J. 1982 Capture into resonance: an extension of the use of adiabatic invariants. Celest. Mech. 27, 3–22.Google Scholar
Henrard, J. 1993 The adiabatic invariant in classical mechanics. Dynamics Reported 2, New Series, 117–236.Google Scholar
Henyey, F. S. & Pomphrey, N. 1983 Eikonal description of internal wave interactions: a non-diffusive picture of “induced diffusion.” Dyn. Atm. Oceans 7, 189–219.Google Scholar
Kaper, T. J. & Kovačič, G. 1994 A geometric criterion for adiabatic chaos. J. Math. Phys. 35, 1202–1219.Google Scholar
Kaper, T. J. & Wiggins, S. 1991 Lobe area in adiabatic Hamiltonian systems. Physica D 51, 205–212.Google Scholar
Kaper, T. J. & Wiggins, S. 1993 An analytical study of transport in Stokes flows exhibiting large-scale chaos in the eccentric journal bearing. J. Fluid Mech. 253, 211–243.Google Scholar
Kruskal, M. 1962 Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. J. Math. Phys. 3, 806–828.Google Scholar
McComas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82, 1397–1422.Google Scholar
Marmorino, G. O., Rosenblum, L. J. & Trump, C. L. 1987 Fine-scale temperature variability: the influence of near-inertial waves. J. Geophys. Res. 92 (C12), 13049–13062.Google Scholar
Meiburg, E. & Newton, P. K. 1991 Particle dynamics and mixing in a viscously decaying shear layer. J. Fluid Mech. 227, 211–244.Google Scholar
Meiss, J. D. & Watson, K. M. 1982 Internal wave interactions in the induced-diffusion approximation. J. Fluid Mech. 117, 315–341.Google Scholar
Menyuk, C. R. 1985 Particle motion in the field of a modulated wave. Phys. Rev. A 31, 3282–3290.Google Scholar
Mora, P. 1992 Particle acceleration in a relativistic wave in the adiabatic regime. Phys. Fluids B 4, 1630–1634.Google Scholar
Munk, W. H. 1981 Internal waves and small-scale processes. In Evolution of Physical Oceanography (ed. B.A. Warren & C. Wunsch) p. 623. MIT Press.
Neishtadt, A. I. 1975 Passage through a separatrix in a resonance problem with a slowly-varying parameter. Prikl. Matem. Mekhan. 39, 621–632.Google Scholar
Neishtadt, A. I. 1986 Change in adiabatic invariant at a separatrix. Sov. J. Plasma Phys. 12, 568–573; Erratum: Sov. J. Plasma Phys. 13, 441.Google Scholar
Percival, I. & Richards, D. 1982 Introduction to Dynamics, Chapter 6, Cambridge University Press.
Pinkel, R. 1983 Doppler sonar observations of internal waves: wavefield structure. J. Phys. Oceanogr. 13, 804–815.Google Scholar
Robinson, C. 1983 Sustained resonance for a nonlinear system with slowly-varying coefficients. SIAM J. Math. Anal. 14, 847–860.Google Scholar
Yoder, C. F. 1979 Diagrammatic theory of transition in pendulum like systems. Celest. Mech. 19, 3–29.Google Scholar