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Large-scale Langmuir circulation and double-diffusive convection: evolution equations and flow transitions

Published online by Cambridge University Press:  26 April 2006

Stephen M. Cox  and
Sidney Leibovich
Stephen M. Cox
Affiliation:
Department of Applied Mathematics, The University of Adelaide, Adelaide 5005, Australia
Sidney Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA
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Abstract

Two-dimensional Langmuir circulation in a layer of stably stratified water and the mathematically analogous problem of double-diffusive convection are studied with mixed boundary conditions. When the Biot numbers that occur in the mechanical boundary conditions are small and the destabilizing factors are large enough, the system will be unstable to perturbations of large horizontal length. The instability may be either direct or oscillatory depending on the control parameters. Evolution equations are derived here to account for both cases and for the transition between them. These evolution equations are not limited to small disturbances of the nonconvective basic velocity and temperature fields, provided the spatial variations in the horizontal remain small. The direct bifurcation may be supercritical or subcritical, while in the case of oscillatory motions, stable finite-amplitude travelling waves emerge. At the transition, travelling waves, standing waves, and modulated travelling waves all are stable in sub-regimes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 276 , 10 October 1994 , pp. 189 - 210
Copyright
© 1994 Cambridge University Press

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References

Cessi, P. & Young, W. R. 1992 Multiple equilibria in two-dimensional thermohaline circulation. J. Fluid Mech. 241, 291–309.Google Scholar
Chapman, C. J., Childress, S. & Proctor, M. R. E. 1980 Long wavelength thermal convection between non-conducting boundaries. Earth & Planet. Sci. Lett. 51, 362–369.Google Scholar
Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh-Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101, 759–782.Google Scholar
Cox, S. M. 1994 Thermosolutal convection between poorly conducting plates. To appear in J. Engng Maths.Google Scholar
Cox, S. M. & Leibovich, S. 1993 Langmuir circulations in a surface layer bounded by a strong thermocline. J. Phys. Oceanogr. 23 1330–1345 (referred to herein as CLI).Google Scholar
Cox, S. M. & Leibovich, S. 1994 Nonlinear scale selection in Langmuir circulation and in thermohaline convection. To be submitted (referred to herein as CLII).
Cox, S. M., Leibovich, S., Moroz, I. M. & Tandon, A. 1992a Nonlinear dynamics in Langmuir circulations with O(2) symmetry. J. Fluid Mech. 241, 669–704.Google Scholar
Cox, S. M., Leibovich, S., Moroz, I. M. & Tandon, A. 1992b Hopf bifurcations in Langmuir circulations. Physica 59D, 226–254.Google Scholar
Craik, A. D. D. 1977 The generation of Langmuir circulations by an instability mechanism. J. Fluid Mech. 81, 209–223.Google Scholar
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401–426.Google Scholar
Dangelmayr, G. & Knobloch, E. 1987 The Takens–Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A 322, 243–279.Google Scholar
Depassier, M. C. & Spiegel, E. A. 1982 Convection with heat flux prescribed on the boundaries of the system. I. The effect of temperature dependence of material properties. Geophys. Astrophys. Fluid Dyn. 21, 167–188.Google Scholar
Gertsberg, V. L. & Sivashinsky, G. I. 1981 Large cells in nonlinear Rayleigh-Bénard convection. Prog. Theor. Phys. 66, 1219–1229.Google Scholar
Hefer, D. & Pismen, L. M. 1987 Long-scale thermodiffusional convection. Phys. Fluids 30, 2648–2654.Google Scholar
Huang, N. E. 1971 Derivation of Stokes drift for a deep–water random gravity wave field. Deep-Sea Res. 18, 255–259.Google Scholar
Kantha, L. H. & Phillips, O. M. 1976 On turbulent entrainment at a stable density interface. J. Fluid Mech. 79, 753–768.Google Scholar
Knobloch, E. 1989 Nonlinear binary fluid convection at positive separation ratios. In Cooperative Dynamics in Complex Physical Systems (ed. H. Takayama). Springer Series in Synergetics, vol. 43.
Langmuir, I. 1938 Surface motion of water induced by wind. Science 87, 119–123.Google Scholar
Leibovich, S. 1977a On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. Theory and averaged current. J. Fluid Mech. 79, 715–743.Google Scholar
Leibovich, S. 1977b Convective instability of stably stratified water in the ocean. J. Fluid Mech. 82, 561–581.Google Scholar
Leibovich, S. 1983 The form and dynamics of Langmuir circulations. Ann. Rev. Fluid Mech. 15, 391–427.Google Scholar
Leibovich, S. 1985 Dynamics of Langmuir circulations in a stratified ocean. In The Ocean Surface (ed. Y. Toba & H. Mitsuyasu), pp. 457–464. Reidel.
Leibovich, S., Lele, S. & Moroz, I. M. 1989 Nonlinear dynamics in Langmuir circulations and in thermosolutal convection. J. Fluid Mech. 198, 471–511.Google Scholar
Nield, D. A. 1967 The thermohaline Rayleigh–Jeffreys problem. J. Fluid Mech. 29, 545–558.Google Scholar
Pismen, L. M. 1988 Selection of long-scale oscillatory convective patterns. Phys. Rev. 38, 2564–2572.Google Scholar
Roberts, A. J. 1985 An analysis of near-marginal, mildly penetrative convection with heat flux prescribed on the boundaries. J. Fluid Mech. 158, 71–93.Google Scholar
Sivashinsky, G. I. 1982 Large cells in nonlinear Marangoni convection. Physica 4D, 227–235.Google Scholar
Sivashinsky, G. I. 1983 On cellular instability in the solidification of a dilute binary alloy. Physica 8D, 243–248.Google Scholar
Smith, J., Pinkel, R. & Weller, R. A. 1987 Velocity structure in the mixed layer during MILDEX. J. Phys. Oceanogr. 17, 425–439.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1964 Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. J. Fluid Mech. 18, 513–528.CrossRefGoogle Scholar
Weller, R. A., Dean, J. P., Marra, J., Price, J. F., Francis, E. A. & Boardman, D. C. 1985 Three-dimensional flow in the upper ocean. Science 227, 1552–1556.Google Scholar
Weller, R. A. & Price, J. F. 1988 Langmuir circulation within the oceanic mixed layer. Deep-Sea Res. 35, 711–747.Google Scholar