Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-24T16:03:56.994Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric focusing of internal waves

Published online by Cambridge University Press:  26 April 2006

Leo R. M. Maas  and
Frans-Peter A. Lam
Leo R. M. Maas
Affiliation:
Netherlands Institute for Sea Research, PO Box 59, 1790 AB Texel, The Netherlands
Frans-Peter A. Lam
Affiliation:
Netherlands Institute for Sea Research, PO Box 59, 1790 AB Texel, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

The spatial structure of the streamfunction field of free, linear internal waves in a two-dimensional basin is governed by the canonical, second-order, hyperbolic equation on a closed domain. Its solution can be determined explicitly for some simple shapes of the basin. It consists of an algorithm by which ‘webs’ of uniquely related characteristics can be constructed and the prescription of one (independent) value of a field variable, related to the streamfunction, on each of these webs. The geometric construction of the webs can be viewed as an alternative version of a billiard game in which the angle of reflection equals that of incidence with respect to the vertical (rather than to the normal). Typically, internal waves are observed to be globally attracted (‘focused’) to a limiting set of characteristics. This attracting set can be classified by the number of reflections it has with the surface (its period in the terminology of dynamical systems). This period of the attractor is a fractal function of the normalized period of the internal waves: large regions of smooth, low-period attractors are seeded with regions with high-period attractors. Occasionally, all internal wave rays fold exactly back upon themselves, a ‘resonance’: focusing is absent and a smooth pattern, familiar from the cellular pattern in a rectangular domain, is obtained. These correspond to the well-known seiching modes of a basin. An analytic set of seiching modes has also been found for a semi-elliptic basin. A necessary condition for seiching to occur is formulated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 300 , 10 October 1995 , pp. 1 - 41
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

Abdullaev, S. S. 1993 Chaos and Dynamics of Rays in Waveguide Media Gordon and Breach.
Azetsu-Scott, K., Johnson, B. D. & Petrie, B. 1995 An intermittent, intermediate nepheloid layer in Emerald Basin, Scotian Shelf. Cont. Shelf Res. 15, 281–293.Google Scholar
Baines, P. G. 1971a The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid Mech. 46, 273–291.Google Scholar
Baines, P. G. 1971b The reflexion of internal/inertial waves from bumpy surfaces. Part 2. Split reflexion and diffraction. J. Fluid Mech. 49, 113–131.Google Scholar
Baines, P. G. 1973 The generation of internal tides by flat-bump topography. Deep-Sea Res. 20, 179–205.Google Scholar
Baines, P. G. & Fang, X.-H. 1985 Internal tide generation at a continental shelf/slope junction: a comparison between theory and a laboratory experiment. Dyn. Atmos. Oceans 9, 297–314.Google Scholar
Berry, M. V. 1981 Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ‘billiard’. Eur. J. Phys. 2, 91–102.Google Scholar
Bunimovich, L. A. 1980 Concerning certain properties of internal ocean waves with horizontally varying Väisälä-Brunt frequency. Izv. Atmos. Ocean Phys. 16, 354–359.Google Scholar
Cacchione, D. & Wunsch, C. 1974 Experimental study of internal waves over a slope. J. Fluid Mech. 66, 223–239.Google Scholar
Cushman-Roisin, B. 1993 Natural resonance of internal tides. IAPSO Proc., PS-10 18, 321.Google Scholar
Cushman-Roisin, B. & Svendsen, H. 1983 Internal gravity waves in sill fjords: vertical modes, ray theory and comparison with observations. In Coastal Oceanography, (ed. H. G. Gade, A. Edwards & H. Svendsen), pp. 373–396, Plenum.
Cushman-Roisin, B., Tverberg, V. & Pavia, E. G. 1989 Resonance of internal waves in fjords: a finite-difference model. J. Mar. Res. 47, 547–567.Google Scholar
Defant, A. 1941 Physical Oceanography, Vol. II. Pergamon Press.
Dewitt, L. M., Levine, M. D., Paulson, C. A. & Burt, W. V. 1986 Semidiurnal internal tide in JASIN: observations and simulation. J. Geophys. Res. 91, 2581–2592.Google Scholar
Garrett, C. 1991 Marginal mixing theories. Atmos.-Oceans 29, 313–339.Google Scholar
Görtler, H. 1943 Über eine Schwingungserscheinung in Flüssigkeiten mit stabiler Dichteschichtung. Z. Angew. Math. Mech. 23, 65–71.Google Scholar
Groen, P. 1948 Two fundamental theorems on gravity waves in inhomogeneous incompressible fluids. Physica 14, 294–300.Google Scholar
Ivey, G. N. & Nokes, R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479–500.Google Scholar
Krauss, W. 1973 Interne Wellen Gebrüder Bornträger.
Larsen, L. H. 1969 Internal waves incident upon a knife edge barrier. Deep-Sea Res. 16, 411–419.Google Scholar
Lighthill, J. 1978 Waves in Fluids Cambridge University Press.
Longuet-Higgins, M. S. 1969 On the reflexion of wave characteristics from rough surfaces. J. Fluid Mech. 37, 231–250.Google Scholar
Magaard, L. 1962 Zur Berechnung interner Wellen in Meeresräumen mit nicht-ebenen Böden bei einer speziellen Dichteverteilung. Kieler Meeresforsch. 18, 161–183.Google Scholar
Magaard, L. 1968 Ein Beitrag zur Theorie der internen Wellen als Störungen geostrophischer Strömungen. Deutsche Hydrogr. Z. 21, 241–278.Google Scholar
Manton, M. J. & Mysak, L. A. 1971 Construction of internal wave solutions via a certain functional equation. J. Math. Anal. Appl. 35, 237–248.Google Scholar
Mowbray, D. E. & Rarity, B. S. H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16.Google Scholar
Munk, W. H. 1966 Abyssal recipes. Deep Sea-Res. 13, 207–230.Google Scholar
Münnich, M. 1993 On the influence of bottom topography on the vertical structure of internal seiches. PhD thesis, ETH Zürich.
Münnich, M. 1994 The influence of bottom topography on internal seiches in continuously stratified media. In Preprints of the Fourth Intl Symp. on Stratified Flows, Vol. 2 (ed. E. Hopfinger, B. Voisin and G. Chavand). LEGI, Institut de Mécanique de Grenoble.
Pingree, R. D. & New, N. L. 1991 Abyssal penetration and bottom reflection of internal tidal energy in the Bay of Biscay. J. Phys. Oceanogr. 21, 28–39.Google Scholar
Robinson, R. M. 1969 The effects of a vertical barrier on internal waves. Deep-Sea Res. 16, 421–429.Google Scholar
Sandstrom, H. 1969 Effect of topography on propagation of waves in stratified fluids. Deep-Sea Res. 16, 405–410.Google Scholar
Sandstrom, H. 1976 On topographic generation and coupling of internal waves. Geophys. Astrophys. Fluid Dyn. 7, 231–270.Google Scholar
Schuster, H. G. 1984 Deterministic Chaos. Physik Verlag.
Thorpe, S. A. 1968 On the shape of progressive internal waves. Phil. Trans. R. Soc. Lond. A 263, 563–614.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids Cambridge University Press.
Wunsch, C. 1968 On the propagation of internal waves up a slope. Deep-Sea Res. 15, 251–258.Google Scholar
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131–141.Google Scholar