Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T06:39:27.747Z Has data issue: false hasContentIssue false
  • English
  • Français

On Lagrangian drift in shallow-water waves on moderate shear

Published online by Cambridge University Press:  16 July 2010

W. R. C. PHILLIPS ,
A. DAI  and
K. K. TJAN
W. R. C. PHILLIPS*
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2935, USA Department of Mathematics, Swinburne University of Technology, Hawthorn 3122, Australia
A. DAI
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2935, USA
K. K. TJAN
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2935, USA
*
†Email address for correspondence: wrphilli@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The Lagrangian drift in an O(ϵ) monochromatic wave field on a shear flow, whose characteristic velocity is O(ϵ) smaller than the phase velocity of the waves, is considered. It is found that although shear has only a minor influence on drift in deep-water waves, its influence becomes increasingly important as the depth decreases, to the point that it plays a significant role in shallow-water waves. Details of the shear flow likewise affect the drift. Because of this, two temporal cases common in coastal waters are studied, viz. stress-induced shear, as would arise were the boundary layer wind-driven, and a current-driven shear, as would arise from coastal currents. In the former, the magnitude of the drift (maximum minus minimum) in shallow-water waves is increased significantly above its counterpart, viz. the Stokes drift, in like waves in otherwise quiescent surroundings. In the latter, on the other hand, the magnitude decreases. However, while the drift at the free surface is always oriented in the direction of wave propagation in stress-driven shear, this is not always the case in current-driven shear, especially in long waves as the boundary layer grows to fill the layer. This latter finding is of particular interest vis-à-vis Langmuir circulations, which arise through an instability that requires differential drift and shear of the same sign. This means that while Langmuir circulations form near the surface and grow downwards (top down), perhaps to fill the layer, in stress-driven shear, their counterparts in current-driven flows grow from the sea floor upwards (bottom up) but can never fill the layer.

JFM classification

Geophysical and Geological Flows: Ocean processes Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 660 , 10 October 2010 , pp. 221 - 239
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Babanin, A. V., Ganopolski, A. & Phillips, W. R. C. 2009 Wave-induced upper-ocean mixing in a climate model of intermediate complexity. Ocean Model. 29, 189–197.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Caponi, E. A., Yuen, H. C., Milinazzo, F. A. & Saffman, P. G. 1991 Water-wave instability induced by a drift layer. J. Fluid Mech. 222, 207–213.CrossRefGoogle Scholar
Christensen, K. H. & Terrile, E. 2009 Drift and deformation of oil slicks due to surface waves. J. Fluid Mech. 620, 313–332.CrossRefGoogle Scholar
Chu, V. C. & Mei, C. C. 1970 On slowly varying Stokes waves. J. Fluid Mech. 41, 873–887.CrossRefGoogle Scholar
Craik, A. D. D. 1982 a The drift velocity of water waves. J. Fluid Mech. 116, 187–205.CrossRefGoogle Scholar
Craik, A. D. D. 1982 b The generalized Lagrangian-mean equations and hydrodynamic stability. J. Fluid Mech. 125, 27–35.CrossRefGoogle Scholar
Craik, A. D. D. 1982 c Wave induced longitudinal-vortex instability in shear flows. J. Fluid Mech. 125, 37–52.CrossRefGoogle Scholar
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401–426.CrossRefGoogle Scholar
Gargett, A., Wells, J., Tejada-Martinez, A. E. & Grosch, C. E. 2004 Langmuir supercells: a mechanism for sediment resuspension and transport in shallow seas. Science 306, 1925–1928.CrossRefGoogle ScholarPubMed
Hasselmann, K. 1970 Wave-driven inertial oscillations. Geophys. Fluid Dyn. 1, 463–502.CrossRefGoogle Scholar
Hasselmann, K. 1971 Mass and momentum transfer between short gravity waves and larger scale motions. J. Fluid Mech. 50, 189–205.CrossRefGoogle Scholar
Lane, E. M., Restrepo, J. M. & McWilliams, J. 2007 Wave–current interaction: a comparison of radiation-stress and vortex-force representations. J. Phys. Oceanogr. 37, 1122–1141.CrossRefGoogle Scholar
Langmuir, I. 1938 Surface motion of water induced by wind. Science 87, 119–123.CrossRefGoogle ScholarPubMed
Larrieu, E., Hinch, E. J. & Charru, F. 2009 Lagrangian drift near a wavy boundary in a viscous oscillating flow. J. Fluid Mech. 630, 391–411.CrossRefGoogle Scholar
Lighthill, M. J. 1978 Acoustic streaming. J. Sound Vib. 61, 391–418.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. 245, 535–581.Google Scholar
Marmorino, G. O., Smith, G. B. & Lindemann, G. J. 2005 Infrared imagery of large-aspect-ratio Langmuir circulation. Cont. Shelf Res. 25, 1–6.CrossRefGoogle Scholar
McWilliams, J., Restrepo, J. M. & Lane, E. M. 2004 An asymptotic theory for the interaction of waves and currents in coastal waters. J. Fluid Mech. 511, 135–178.CrossRefGoogle Scholar
Melville, W. K., Shear, R. & Veron, F. 1998 Laboratory measurements of the generation and evolution of Langmuir circulations. J. Fluid Mech. 364, 31–58.CrossRefGoogle Scholar
Phillips, W. R. C. 1998 Finite-amplitude rotational waves in viscous shear flows. Stud. Appl. Math. 101, 23–47.CrossRefGoogle Scholar
Phillips, W. R. C. 2001 a On the pseudomomentum and generalized Stokes drift in a spectrum of rotational waves. J. Fluid Mech. 430, 209–220.CrossRefGoogle Scholar
Phillips, W. R. C. 2001 b On an instability to Langmuir circulations and the role of Prandtl and Richardson numbers. J. Fluid Mech. 442, 335–358.CrossRefGoogle Scholar
Phillips, W. R. C. 2002 Langmuir circulations beneath growing or decaying surface waves. J. Fluid Mech. 469, 317–342.CrossRefGoogle Scholar
Phillips, W. R. C. 2003 Langmuir circulation. In Wind-Over-Waves II: Forecasting and Fundamentals of Applications (ed. Sajjadi, S. & Hunt, J.), pp. 157–167. Horwood.CrossRefGoogle Scholar
Phillips, W. R. C. 2005 On the spacing of Langmuir circulation in strong shear. J. Fluid Mech. 525, 215–236.CrossRefGoogle ScholarPubMed
Phillips, W. R. C. & Shen, Q. 1996 A family of wave–mean shear interactions and their instability to longitudinal vortex form. Stud. Appl. Math. 96, 143–161.CrossRefGoogle Scholar
Phillips, W. R. C. & Wu, Z. 1994 On the instability of wave-catalysed longitudinal vortices in strong shear. J. Fluid Mech. 272, 235–254.CrossRefGoogle Scholar
Phillips, W. R. C., Wu, Z. & Lumley, J. 1996 On the formation of longitudinal vortices in turbulent boundary layers over wavy terrain. J. Fluid Mech. 326, 321–341.CrossRefGoogle Scholar
Rayleigh, L. 1883 On the circulation of air observed in Kundt's tubes and some allied acoustical problems. Phil. Trans. R. Soc. Lond. A 175, 1–21.Google Scholar
Smith, J. A. 1992 Observed growth of Langmuir circulation. J. Geophys. Res. 97, 5651–5664.CrossRefGoogle Scholar
Smith, J. A. 2006 Observed variability of ocean wave Stokes drift, and the Eulerian response to passing groups. J. Phys. Oceanogr. 36, 1381–1402.CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Ursell, F. 1950 On the theoretical form of ocean swell on a rotating earth. Mon. Not. R. Astron. Soc., Geophys. Suppl. 6, 1–8.CrossRefGoogle Scholar
Veron, F. & Melville, W. K. 2001 Experiments on the stability and transition of wind-driven water surfaces. J. Fluid Mech. 446, 25–65.CrossRefGoogle Scholar
Xu, Z. & Bowen, A. J. 1994 Wave- and wind-driven flow in water of finite depth. J. Phys. Oceanogr. 24, 1850–1866.2.0.CO;2>CrossRefGoogle Scholar