Hostname: page-component-76dd75c94c-ccc76 Total loading time: 0 Render date: 2024-04-30T08:55:50.749Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves

Published online by Cambridge University Press:  21 April 2006

M. E. Mcintyre
M. E. Mcintyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Longuet-Higgins’ exact expression for the increase in the Lagrangian-mean elevation of the free surface due to the presence of periodic, irrotational surface gravity waves is rederived from generalized Lagrangian-mean theory. The raising of the Lagrangian-mean surface as wave amplitude builds up illustrates the non-zero divergence of the Lagrangian-mean velocity field in an incompressible fluid.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 189 , April 1988 , pp. 235 - 242
Copyright
© 1988 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

Andrews, D. G. & McIntyre, M. E. 1976 Planetary waves in horizontal and vertical shear: the generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci. 33, 2031–2048.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978a An exact theory of nonlinear waves on a Lagrangianmean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978b On wave-action and its relatives. J. Fluid Mech. 89, 647–664 (Corrigendum 95, 796; also 106, 331).Google Scholar
Dunkerton, T. J. 1983 The conservation of pseudoenergy in Lagrangian time-mean flow. J. Atmos. Sci. 40, 2623–2629.Google Scholar
Dunkerton, T. J., Hsu, C.-P. F. & McIntyre, M. E. 1981 Some Eulerian and Lagrangian diagnostics for a model stratospheric warming. J. Atmos. Sci. 38, 819–843.Google Scholar
Grimshaw, R. 1979 Mean flows induced by internal gravity wave packets propagating in a shear flow. Phil. Trans. R. Soc. Lond. A 292, 391–417.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press, 738 pp.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Longuet-Higgins, M. S. 1986 Eulerian and Lagrangian aspects of surface waves. J. Fluid Mech. 173, 683–707.Google Scholar
Longuet-Higgins, M. S. 1987 Lagrangian moments and mass transport in Stokes waves. J. Fluid Mech. 179, 547–555.Google Scholar
Longuet-Higgins, M. S. 1988 Lagrangian moments and mass transport in Stokes waves. Part 2. Water of finite depth. J. Fluid Mech. 186, 321–336.Google Scholar
McIntyre, M. E. 1973 Mean motions and impulse of a guided internal gravity wave packet. J. Fluid Mech. 60, 801–811. [Note that μ should be μ−1 just above eq. (15).]Google Scholar
McIntyre, M. E. 1980 Towards a Lagrangian-mean description of stratospheric circulations and chemical transports. Phil. Trans. R. Soc. Lond. A 296, 129–148.Google Scholar
McIntyre, M. E. & Mobbs, S. D. 1988 On the ‘quasimomentum rule’ for wave-induced mean forces on obstacles immersed in a material medium. To be submitted to Proc. R. Soc. Lond. A. Presented at Lyngby IUTAM conference, summer 1984.Google Scholar
McIntyre, M. E. & Shepherd, T. G. 1987 An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech. 181, 527–565.Google Scholar
Nakamura, K. 1981 A comment on the Kelvin-Helmholtz instability. J. Met. Soc. Japan 59, 272–275.Google Scholar
Srokosz, M. A. & Longuet-Higgins, M. S. 1986 On the skewness of sea-surface elevation. J. Fluid Mech. 164, 487–498.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–473. (See §9, p. 451, or Collected Papers, p. 207.)Google Scholar
Uryu, M. 1979 Lagrangian-mean motion induced by a growing baroclinic wave. J. Met. Soc. Japan 57, 1–20.Google Scholar