Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-15T04:58:03.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple equilibria and stable oscillations in thermosolutal convection at small aspect ratio

Published online by Cambridge University Press:  26 April 2006

Charles Quon  and
Michael Ghil
Charles Quon
Affiliation:
Climate Dynamics Center, Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1565, USA Department of Fisheries and Oceans, Bedford Institute of Oceanography, Dartmouth, Nova Scotia, Canada
Michael Ghil
Affiliation:
Climate Dynamics Center, Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1565, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

For thermosolutal convection in an enclosure of arbitrary vertical aspect ratio, mixed boundary conditions – with salt flux and temperature prescribed on a horizontal boundary – can lead to symmetry breaking via pitchfork bifurcation. In the present paper we consider an enclosure of very small height-to-length aspect ratio δ, as encountered in the world's oceans. In this case, if the ratios of the vertical to horizontal components of viscosity, [Pscr ], and of diffusivity, [Lscr ], are of order unity, advective transport cannot set in even at very high Rayleigh numbers. The ratios [Pscr ] and [Lscr ] must be substantially less than unity in order for convection to dominate the heat and solute transport.

We have investigated numerically the effects of monotonic and non-monotonic temperature and salinity boundary conditions in a two-dimensional domain at constant δ = 0.01 and constant [Pscr ] = [Lscr ] = 0.01. This ratio of eddy-mixing coefficients reflects the different scales of motion – vertical and horizontal – in the ocean, rather than a physically realizable laboratory fluid. It is found that when the salt-flux strength, γ, is sufficiently large, the system undergoes a second bifurcation for both types of boundary conditions. It is a Hopf bifurcation, leading from the asymmetric steady states produced by the first one to oscillatory solutions. These periodic solutions are stable and very robust. An approximate Hopf bifurcation diagram has been produced. We conclude that non-monotonic salt-flux conditions are neither necessary nor sufficient to induce the oscillations, while the strength of the salt flux is crucial.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 291 , 25 May 1995 , pp. 33 - 56
Copyright
© 1995 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

Allen, M. R., Read, P. L. & Smith, L. A. 1992a Temperature time series? Nature 355, 686.Google Scholar
Allen, M. R., Read, P. L. & Smith, L. A. 1992b Temperature oscillations. Nature 359, 679.Google Scholar
Bryan, F. 1986 High-latitude salinity effects and interhemispheric thermohaline circulation. Nature 323, 301–304.Google Scholar
Bryan, K. & Cox, M. D. 1968 A nonlinear model of an ocean driven by wind and differential heating. Parts I and II. J. Atmos. Sci. 25, 945–978.Google Scholar
Cessi, P. & Young, W. R. 1992 Multiple equilibria in two-dimensional thermohaline circulation. J. Fluid Mech. 241, 291–309.Google Scholar
Chen, F. & Ghil, M. 1995 Interdecadal variability of the thermohaline circulation and high-latitude surface fluxes. J. Phys. Oceanogr. (in the press).Google Scholar
Ghil M., Benzi R. & Parisi G. (eds.) 1985 Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland.
Ghil, M. & Childress, S. 1987 Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics Springer.
Ghil, M. & Vautard, R. 1991 Interdecadal oscillations and the warming trend in global temperature time series. Nature 350, 324–327.Google Scholar
Gregg, M. C. 1987 Diapycnal mixing in a thermocline: A review. J. Geophys. Res. 92, 5249–5286.Google Scholar
Kagan, B. A. & Maslova, N. B. 1990 Multiple equilibria of the thermohaline circulation in a ventilated ocean model. Ocean Modelling, issue 90 (unpublished manuscript).
Levitus, S. 1982 Climatological Atlas of the World Ocean. NOAA Professional Paper 13. US Department of Commerce, National Oceanic and Atmospheric Administration.
Mikolajewicz, U. & Maier-Reimer, E. 1990 Internal secular variability in an ocean general circulation model. Climate Dyn. 4, 145–156.Google Scholar
Quon, C. & Ghil, M. 1992 Multiple equilibria in thermosolutal convection due to salt-flux boundary conditions. J. Fluid Mech. 245 449–483 (referred to herein as QG).Google Scholar
Quon, C. & Sandstrom, H. 1990 A numerical algorithm to study internal solitary waves. J. Comput. Phys. 86, 168–186.Google Scholar
Sandstrom, H. & Quon, C. 1994 On time-dependent, two-layer flow over topography. II. Evolution and propagation of solitary waves. Fluid Dyn. Res. 13, 197–215.Google Scholar
Stommel, H. 1961 Thermohaline convection with two stable regimes of flow. Tellus 13, 224–228.Google Scholar
Thual, O. & McWilliams, J. C. 1992 The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box model. Geophys. Astrophys. Fluid Dyn. 64, 67–95.Google Scholar
Walin, G. 1985 The thermohaline circulation and the control of ice ages. Paleogeogr., Paleoclimatol., Paleoecol. 50, 323–332.Google Scholar
Weaver, A. J., Sarachik, E. S. & Marotzke, J. 1991 Freshwater flux forcing of decadal and interdecadal oceanic variability. Nature 353, 836–838.Google Scholar
Weaver, A. J., Marotzke, J., Cummins, P. F. & Sarachik, E. S. 1993 Stability and variability of the thermohaline circulation. J. Phys. Oceanogr. 23, 39–60.Google Scholar