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Nonlinear dynamics in Langmuir circulations and in thermosolutal convection

Published online by Cambridge University Press:  21 April 2006

Sidney Leibovich ,
Sanjiva K. Lele  and
Irene M. Moroz
Sidney Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-7501, USA
Sanjiva K. Lele
Affiliation:
NASA Ames Research Center, 202A-1, Moffett Field, CA 94035, USA
Irene M. Moroz
Affiliation:
School of Mathematics and Physics, University of East Anglia, Norwich Norfolk NR4 7TJ, UK
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Abstract

Two-dimensional motions generated by Langmuir circulation instability of stratified layers of water of finite depth are studied under a simplifying assumption making it mathematically analogous to double-diffusive thermosolutal convection with constant solute concentration and constant heat flux at the boundaries. The nature of possible motions is mapped over a significant region in (S, R) parameter space, where S and R are parameters measuring, respectively, the stabilizing and destabilizing agencies in the problem. In the Langmuir circulation problem R measures the effects of wind and surface wave action, and S measures the stabilizing effect of buoyancy: in the thermosolutal problem, R measures the destabilizing effects of heating, while S measures the stabilizing effect of solute concentration. Effects of lateral boundary or symmetry conditions are found to be crucial in determining the qualitative behaviour. Complex temporal behaviour, including intermittently chaotic flows, are found under suitably constrained (no flux) lateral conditions but are unstable and not realized when these constraints are relaxed and replaced by periodic lateral conditions. Multiple steady states also arise, with those found under constrained lateral conditions losing stability either to travelling waves, or to other steady states when the lateral boundary conditions are relaxed. In some regions of the parameter space, multiple stable nonlinear motions have been found under periodic boundary conditions. The multiple stable states may either be coexisting travelling waves and steady states (different from those found under the constrained lateral conditions). The existence of robust travelling waves may explain some field observations of laterally drifting windrows associated with Langmuir circulations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 198 , January 1989 , pp. 471 - 511
Copyright
1989 Cambridge University Press

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References

Arakawa, A.: 1966 Computational design for long-term numerical integration of the equations of motion: two-dimensional incompressible flow. Part. 1. J. Comput. Phys. 1, 119–143.Google Scholar
Baines, P. G. & Gill, A. E., 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37, 289–306.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.: 1982 The drift velocity of water waves. J. Fluid Mech. 116, 187–205.Google Scholar
Craik, A. D. D. & Leibovich, S.1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401–426.Google Scholar
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A., 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
da Costa, L. N., Knobloch, E. & Weiss, N. O., 1983 Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25–43.Google Scholar
Dangelmayr, G., Armbruster, D. & Neveling, M., 1985 A codimension three bifurcation for the laser with saturable absorber. Z. Phys. B 59, 365–370.Google Scholar
Glendinning, P. & Sparrow, C., 1984 Local and global behaviour near homoclinic orbits. J. Statist. Phys. 35, 645–696.Google Scholar
Golubitsky, M. & Stewart, I. N., 1985 Hopf bifurcation in the presence of symmetry. Arch. Rat. Mech. Anal. 87, 107–165.Google Scholar
Guckenheimer, J. & Holmes, P. J., 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Guckenheimer, J. & Knobloch, E., 1983 Nonlinear convection in a rotating layer; Amplitude expansions and centre manifolds. Geophys. Astrophys. Fluid Dyn. 23, 247–272.Google Scholar
Huppert, H. E. & Moore, D. R., 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–852.Google Scholar
Iooss, G. & Joseph, D. D., 1980 Elementary Stability and Bifurcation Theory. Springer.
Knobloch, E.: 1986 On the degenerate Hopf bifurcation with circular symmetry. In Multiparameter Bifurcation Theory (ed. M. Golubitsky & J. Guckenheimer). Contemporary Mathematics, vol. 56, pp. 193–201. American Mathematical Society.
Knobloch, E., Deane, A. E., Toomre, J. & Moore, D. R., 1986a Doubly diffusive waves. In Multiparameter Bifurcation Theory (ed. M. Golubitsky & J. Guckenheimer). Contemporary Mathematics, vol. 56, pp. 203–216. American Mathematical Society.
Knobloch, E., Moore, D. R., Toomre, J. & Weiss, N. O., 1986b Transitions to chaos in two-dimensional double-diffusive convection. J. Fluid Mech. 166, 409–448.Google Scholar
Lamb, H.: 1932 Hydrodynamics, 6th edn, p. 372. Dover.
Langmuir, I.1938 Surface motion of water induced by wind. Science 87, 119–123.Google Scholar
Leibovich, S.: 1977a Convective instability of a stably stratified water in the ocean. J. Fluid Mech. 82, 561–585.Google Scholar
Leibovich, S.: 1977b 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.: 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.
Lele, S. K.: 1985 Some problems in hydrodynamic stability arising in geophysical fluid dynamics. Ph.D. dissertation, Cornell University.
Lilly, D. K.: 1965 On the computational stability of numerical solutions of time-dependent nonlinear geophysical fluid dynamics problems. Mon. Weather Rev. 93, 11–26.Google Scholar
Marcus, P. S.: 1981 Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241–255.Google Scholar
Meyer, G. E.: 1969 A field study of Langmuir circulations. In Proc. 12th Conf. Great Lakes Res. Intl Assoc. Great Lakes Res., pp. 652–663.Google Scholar
Moore, D. R., Toomre, J., Knobloch, E. & Weiss, N. O., 1983 Period doubling and chaos in partial differential equations for thermosolutal convection. Nature 303, 663–667.Google Scholar
Moroz, I. M.: 1985 Heteroclinic orbits in a model for Langmuir circulations. Phys. Lett. 110A, 109–112.Google Scholar
Moroz, I. M.: 1986 Codimension-three bifurcations arising in Langmuir circulations. In Multiparameter Bifurcation Theory (ed. M. Golubitsky & J. Guckenheimer). Contemporary Mathematics, vol. 56, pp. 223–235. American Mathematical Society.
Moroz, I. M. & Holmes, P. J., 1985 Double Hopf bifurcation and quasi-periodic flow in a model for baroclinic instability. J. Atmos. Sci. 41, 3147–3159.Google Scholar
Moroz, I. M. & Leibovich, S., 1985 Competing instabilities in a nonlinear model for Langmuir circulations. Phys. Fluids 28, 2050–2061.Google Scholar
Moroz, I. M. & Leibovich, S., 1987 Codimension-two Hopf bifurcations in a model for Langmuir circulations. Proc. Intl Conf. Fluid Mech., pp. 157–162. Peking University Press.
Nield, D. A.: 1967 The thermohaline Rayleigh-Jeffreys problem. J. Fluid Mech. 29, 545–558.Google Scholar
Orszag, S. A.: 1971 Numerical simulation of incompressible flows within simple boundaries: accuracy. J. Fluid Mech. 49, 75–112.Google Scholar
Orszag, S. A. & Patera, A., 1983 Secondary instability of wall bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Piascek, S. A. & Toomre, J., 1980 Nonlinear evolution and structure of salt fingers. In Marine Turbulence (ed. J. Nihoul), pp. 193–219. Elsevier.
Pomeau, Y. & Manneville, P., 1980 Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197.Google Scholar
Roache, P. J.: 1976 Computational Fluid Dynamics. 2nd edn. Hermosa.
Schecter, S.: 1976 Bifurcations with symmetry. In The Hopf Bifurcation and its Applications (ed. J. Marsden & M. McCracken), pp. 224–249. Springer.
Stern, M. E.: 1960 The ‘salt fountain’ and thermohaline convection. Tellus 12, 172–175.Google Scholar
Swartztrauber, P. & Sweet, R., 1975 Efficient Fortran subprograms for the solution of elliptic partial differential equations. NCAR Tech. Note IA-109, Nat. Center for Atmos. Res., Boulder.Google Scholar
Thorpe, S. A.: 1985 Small-scale processes in the upper ocean boundary layer. Nature 318, 512–522.Google Scholar
Thorpe, S. A. & Hall, A. J., 1982 Observations of the thermal structure of Langmuir circulations. J. Fluid Mech. 114, 237–250.Google Scholar
Thorpe, S. A. & Hall, A. J., 1983 The characteristics of breaking waves, bubble clouds and near-surface currents observed using side-scan sonar. Continental Shelf Res. 1, 353–384.Google Scholar
Van Straaten, L. M. J. U.1950 Periodic patterns of rippled and smooth areas on water surfaces induced by wind action. KoninklŸke Nederlandse Akademie der Wetenschappen B B53, 1217–1227.Google Scholar
Veronis, G.: 1965 On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 1–17.Google Scholar
Williams, G. P.: 1969 Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow. J. Fluid Mech. 37, 727–750.Google Scholar
Weller, R. A., Deran, 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. (submitted).Google Scholar