Soft loss of stability in an ocean circulation box model with turbulent fluxes

  • A. A. Davydov
  • N. B. Melnikov


For a 2D system of ordinary differential equations that gives a qualitative description of the thermohaline circulation in the ocean, we prove the existence of a limit cycle for a large class of transfer functions. We show that this cycle arises in the system as a result of the soft loss of stability of a steady state when a step transfer function is smoothed by functions from the above-mentioned class.


Transfer Function STEKLOV Institute Hopf Bifurcation Turbulent Flux Phase Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. V. Alekseev and A. M. Gusev, “Free Convection in Geophysical Processes,” Usp. Fiz. Nauk 141(2), 311–342 (1983) [Sov. Phys. Usp. 26 (10), 906–922 (1983)].Google Scholar
  2. 2.
    F. W. Taylor, Elementary Climate Physics (Oxford Univ. Press, Oxford, 2005).Google Scholar
  3. 3.
    S. Rahmstorf, M. Crucifix, A. Ganopolski, H. Goosse, I. Kamenkovich, R. Knutti, G. Lohmann, R. Marsh, L. A. Mysak, Z. Wang, and A. J. Weaver, “Thermohaline Circulation Hysteresis: A Model Intercomparison,” Geophys. Res. Lett. 32, L23605 (2005), doi:10.1029/2005GL023655.Google Scholar
  4. 4.
    K. Keller and D. McInerney, “The Dynamics of Learning about a Climate Threshold,” Clim. Dyn. (2008) (in press).Google Scholar
  5. 5.
    S. Titz, T. Kuhlbrodt, and U. Feudel, “Homoclinic Bifurcation in an Ocean Circulation Box Model,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 12(4), 869–875 (2002).CrossRefGoogle Scholar
  6. 6.
    H. Stommel, “Thermohaline Convection with Two Stable Regimes of Flow,” Tellus 13, 224–230 (1961).CrossRefGoogle Scholar
  7. 7.
    P. A. Cessi, “Simple Box Model of Stochastically Forced Thermohaline Flow,” J. Phys. Oceanogr. 24(9), 1911–1920 (1994).CrossRefGoogle Scholar
  8. 8.
    J. Marotzke, “Abrupt Climate Change and Thermohaline Circulation: Mechanisms and Predictability,” Proc. Natl. Acad. Sci. USA 97(4), 1347–1350 (2000).CrossRefGoogle Scholar
  9. 9.
    P. A. Welander, “A Simple Heat-Salt Oscillator,” Dyn. Atmos. Oceans 6(4), 233–242 (1982).CrossRefGoogle Scholar
  10. 10.
    V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed. (Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 1999; Springer, New York, 1988).Google Scholar
  11. 11.
    A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985; Kluwer, Dordrecht, 1988).Google Scholar
  12. 12.
    A. A. Davydov and N. B. Melnikov, “Andronov-Hopf Bifurcation in Simple Double Diffusion Models,” Usp. Mat. Nauk 62(2), 175–176 (2007) [Russ. Math. Surv. 62 (2), 382–384 (2007)].MathSciNetGoogle Scholar
  13. 13.
    A. A. Davydov and N. B. Melnikov, “Existence of Self-sustained Oscillations in an Ocean Circulation Box Model with Turbulent Fluxes,” IIASA Interim Rep. IR-06-049 (Laxenburg, 2006).Google Scholar
  14. 14.
    P. Hartman, Ordinary Differential Equations (Wiley, New York, 1964; Mir, Moscow, 1970).zbMATHGoogle Scholar
  15. 15.
    V. I. Arnold, Ordinary Differential Equations (Nauka, Moscow, 1984; MIT Press, Cambridge, MA, 1980).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  1. 1.Vladimir State UniversityVladimirRussia
  2. 2.International Institute for Applied Systems AnalysisLaxenburgAustria
  3. 3.Central Economics and Mathematics InstituteRussian Academy of SciencesMoscowRussia
  4. 4.Moscow State UniversityLeninskie gory, MoscowRussia

Personalised recommendations