Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-19T18:06:02.153Z Has data issue: false hasContentIssue false
  • English
  • Français

The generation of Langmuir circulations by an instability mechanism

Published online by Cambridge University Press:  12 April 2006

A. D. D. Craik
A. D. D. Craik
Affiliation:
Department of Applied Mathematics, University of St Andrews, Fife KY16 9SS, Scotland
Get access Rights & Permissions [Opens in a new window]

Abstract

Equations governing the current system in the upper layers of oceans and lakes were derived by Craik & Leibovich (1976). These incorporate the dominant effects of both wind and waves. Solutions comprising the mean wind-driven current and a system of ‘Langmuir’ cells aligned parallel to the wind were found for cases in which the wave field consisted of just a pair of plane waves. However, it was not clear that such cellular motions would persist for the more realistic case of a continuous wave spectrum.

The present paper shows that, in the latter case, infinitesimal spanwise periodic perturbations will grow on account of an instability mechanism. Mathematically, the instability is closely similar to the onset of thermal convection in horizontal fluid layers. Physically, the mechanism is governed by kinematical processes involving the mean (Eulerian) wind-driven current and the (Lagrangian) Stokes drift associated with the waves. The relationship of this mechanism to instability models of Garrett and Gammelsrød is clarified.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 81 , Issue 2 , 24 June 1977 , pp. 209 - 223
Copyright
© 1977 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Chen, C. F. & Kirchner, R. P. 1971 Stability of time-dependent rotational Couette flow. Part 2. Stability analysis. J. Fluid Mech. 48, 365–384.Google Scholar
Craik, A. D. D. 1970 A wave-interaction model for the generation of windrows. J. Fluid Mech. 41, 801–821.Google Scholar
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401–426.Google Scholar
Currie, I. G. 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337–347.Google Scholar
Elder, J. W. 1968 The unstable thermal interface. J. Fluid Mech. 32, 69–96.Google Scholar
Faller, A. J. 1964 The angle of windrows in the ocean. Tellus 16, 363–370.Google Scholar
Faller, A. J. 1966 Sources of energy for the ocean circulation and a theory of the mixed layer. Proc. 5th U.S. Nat. Cong. Appl. Mech. pp. 651–669.
Faller, A. J. 1971 Oceanic turbulence and the Langmuir circulations. Ann. Rev. Ecol. System. 2, 201–235.Google Scholar
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 1249–1257.Google Scholar
Foster, T. D. 1968 Effect of boundary conditions on the onset of convection. Phys. Fluids 11, 1257–1262.Google Scholar
Gammelsrød, T. 1975 Instability of Couette flow in a rotating fluid and origin of Langmuir circulations. J. Geophys. Res. 80, 5069–5075.Google Scholar
Garrett, C. J. R. 1976 Generation of Langmuir circulations by surface waves-a feedback mechanism. J. Mar. Res. 34, 117–130.Google Scholar
Gresho, P. M. & Sani, R. L. 1971 The stability of a fluid layer subjected to a step change in temperature: transient vs. frozen time analyses. Int. J. Heat Mass Transfer 14, 207–221.Google Scholar
Kenyon, K. E. 1969 Stokes drift for random gravity waves. J. Geophys. Res. 74, 6991–6994.Google Scholar
Kraus, E. B. 1967 Organised convection in the ocean surface layer resulting from slicks and wave radiation stress. Phys. Fluids Suppl. 10, S 294.Google Scholar
Leibovich, S. 1977 On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. Theory and the averaged current. J. Fluid Mech. 79, 715–743.CrossRefGoogle Scholar
Leibovich, S. & Radhakrishnan, K. 1977 On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 2. Structure of the Langmuir vortices. J. Fluid Mech. 80, 481–507.CrossRefGoogle Scholar
Leibovich, S. & Ulrich, D. 1972 A note on the growth of small-scale Langmuir circulations. J. Geophys. Res. 77, 1683–1688.Google Scholar
Lick, W. 1965 The instability of a fluid layer with time-dependent heating. J. Fluid Mech. 21, 565–567.Google Scholar
Liu, D. C. S. & Chen, C. F. 1973 Numerical experiments on time-dependent rotational Couette Flow. J. Fluid Mech. 59, 77–96.Google Scholar
Pierson, W. J. & Moskowitz, L. 1964 A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. J. Geophys. Res. 69, 5181–5190.Google Scholar
Pollard, R. T. 1976 Observations and theories of Langmuir circulations and their role in nearsurface mixing. Deep-Sea Res., Sir George Deacon Anniversary Suppl.
Robinson, J. L. 1976 Theoretical analysis of convective instability of a growing horizontal thermal boundary layer. Phys. Fluids 19, 778–791.Google Scholar
Stewart, R. W. 1967 Mechanics of the air-sea interface. Phys. Fluids Suppl. 10, 547–555.Google Scholar