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On Lagrangian drift in shallow-water waves on moderate shear

  • W. R. C. PHILLIPS (a1) (a2), A. DAI (a1) and K. K. TJAN (a1)


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.


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Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609646.
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, 189197.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
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, 207213.
Christensen, K. H. & Terrile, E. 2009 Drift and deformation of oil slicks due to surface waves. J. Fluid Mech. 620, 313332.
Chu, V. C. & Mei, C. C. 1970 On slowly varying Stokes waves. J. Fluid Mech. 41, 873887.
Craik, A. D. D. 1982 a The drift velocity of water waves. J. Fluid Mech. 116, 187205.
Craik, A. D. D. 1982 b The generalized Lagrangian-mean equations and hydrodynamic stability. J. Fluid Mech. 125, 2735.
Craik, A. D. D. 1982 c Wave induced longitudinal-vortex instability in shear flows. J. Fluid Mech. 125, 3752.
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401426.
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, 19251928.
Hasselmann, K. 1970 Wave-driven inertial oscillations. Geophys. Fluid Dyn. 1, 463502.
Hasselmann, K. 1971 Mass and momentum transfer between short gravity waves and larger scale motions. J. Fluid Mech. 50, 189205.
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, 11221141.
Langmuir, I. 1938 Surface motion of water induced by wind. Science 87, 119123.
Larrieu, E., Hinch, E. J. & Charru, F. 2009 Lagrangian drift near a wavy boundary in a viscous oscillating flow. J. Fluid Mech. 630, 391411.
Lighthill, M. J. 1978 Acoustic streaming. J. Sound Vib. 61, 391418.
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. 245, 535581.
Marmorino, G. O., Smith, G. B. & Lindemann, G. J. 2005 Infrared imagery of large-aspect-ratio Langmuir circulation. Cont. Shelf Res. 25, 16.
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, 135178.
Melville, W. K., Shear, R. & Veron, F. 1998 Laboratory measurements of the generation and evolution of Langmuir circulations. J. Fluid Mech. 364, 3158.
Phillips, W. R. C. 1998 Finite-amplitude rotational waves in viscous shear flows. Stud. Appl. Math. 101, 2347.
Phillips, W. R. C. 2001 a On the pseudomomentum and generalized Stokes drift in a spectrum of rotational waves. J. Fluid Mech. 430, 209220.
Phillips, W. R. C. 2001 b On an instability to Langmuir circulations and the role of Prandtl and Richardson numbers. J. Fluid Mech. 442, 335358.
Phillips, W. R. C. 2002 Langmuir circulations beneath growing or decaying surface waves. J. Fluid Mech. 469, 317342.
Phillips, W. R. C. 2003 Langmuir circulation. In Wind-Over-Waves II: Forecasting and Fundamentals of Applications (ed. Sajjadi, S. & Hunt, J.), pp. 157167. Horwood.
Phillips, W. R. C. 2005 On the spacing of Langmuir circulation in strong shear. J. Fluid Mech. 525, 215236.
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, 143161.
Phillips, W. R. C. & Wu, Z. 1994 On the instability of wave-catalysed longitudinal vortices in strong shear. J. Fluid Mech. 272, 235254.
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, 321341.
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, 121.
Smith, J. A. 1992 Observed growth of Langmuir circulation. J. Geophys. Res. 97, 56515664.
Smith, J. A. 2006 Observed variability of ocean wave Stokes drift, and the Eulerian response to passing groups. J. Phys. Oceanogr. 36, 13811402.
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.
Ursell, F. 1950 On the theoretical form of ocean swell on a rotating earth. Mon. Not. R. Astron. Soc., Geophys. Suppl. 6, 18.
Veron, F. & Melville, W. K. 2001 Experiments on the stability and transition of wind-driven water surfaces. J. Fluid Mech. 446, 2565.
Xu, Z. & Bowen, A. J. 1994 Wave- and wind-driven flow in water of finite depth. J. Phys. Oceanogr. 24, 18501866.
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On Lagrangian drift in shallow-water waves on moderate shear

  • W. R. C. PHILLIPS (a1) (a2), A. DAI (a1) and K. K. TJAN (a1)


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