Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 17 2 2010 10.5194/npg-17-123-2010 http://www.nonlin-processes-geophys.net/17/123/2010/ http://www.nonlin-processes-geophys.net/17/123/2010/npg-17-123-2010.html http://www.nonlin-processes-geophys.net/17/123/2010/npg-17-123-2010.pdf 123 135 2010-03-22 A delay differential model of ENSO variability – Part 2: Phase locking, multiple solutions and dynamics of extrema I. Zaliapin zal@unr.edu M. Ghil Department of Mathematics and Statistics, University of Nevada, Reno, Nevada, USA Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), Ecole Normale Supérieure, Paris, France Department of Atmospheric & Oceanic Sciences and Institute of Geophysics & Planetary Physics, University of California, Los Angeles, California, USA We consider a highly idealized model for El Niño/Southern Oscillation (ENSO) variability, as introduced in an earlier paper. The model is governed by a delay differential equation for sea-surface temperature <i>T</i> in the Tropical Pacific, and it combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform a theoretical and numerical study of the model in the three-dimensional space of its physically relevant parameters: propagation period τ of oceanic waves across the Tropical Pacific, atmosphere-ocean coupling κ, and strength of seasonal forcing <i>b</i>. Phase locking of model solutions to the periodic forcing is prevalent: the local maxima and minima of the solutions tend to occur at the same position within the seasonal cycle. Such phase locking is a key feature of the observed El Niño (warm) and La Niña (cold) events. The phasing of the extrema within the seasonal cycle depends sensitively on model parameters when forcing is weak. We also study co-existence of multiple solutions for fixed model parameters and describe the basins of attraction of the stable solutions in a one-dimensional space of constant initial model histories. Battisti, D. S.: The dynamics and thermodynamics of a warming event in a coupled tropical atmosphere/ocean model, J. Atmos. Sci., 45, 2889–2919, 1988. Battisti, D. S. and Hirst, A. C.: Interannual variability in the tropical atmosphere-ocean system: Influence of the basic state and ocean geometry, J. Atmos. Sci., 46, 1687–1712, 1989. Bjerknes, J.: Atmospheric teleconnections from the equatorial Pacific, Mon. Weather Rev., 97, 163–172, 1969. Boulanger, J. P., Menkes, C., and Lengaigne, M.: Role of high- and low-frequency winds and wave reflection in the onset, growth and termination of the 1997–1998 El Nino, Clim. Dynam., 22(2–3), 267–280, 2004. Burgers, G. and Stephenson, D. B.: The "normality" of El Niño, Geophys. Res. Lett., 26, 1027–1030, 1999. Cane, M.: The evolution of El Niño, past and future, Earth Planet. Sc. Lett., 230(3–4), 227–240, 2005. Cane, M., Munnich, M., and Zebiak, S. E.: A study of self-excited oscillations of the tropical ocean-atmosphere system. Part I: Linear analysis, J. Atmos. Sci., 47(13), 1562–1577, 1990. Chang, P., Wang, B., Li, T., and Ji, L.: Interactions between the seasonal cycle and the Southern Oscillation: Frequency entrainment and chaos in intermediate coupled ocean-atmosphere model, Geophys. Res. Lett., 21, 2817–2820, 1994. Chang, P., Ji, L., Wang, B., and Li, T.: Interactions between the seasonal cycle and El Niño – Southern Oscillation in an intermediate coupled ocean-atmosphere model, J. Atmos. Sci., 52, 2353–2372, 1995. Diaz, H. F. and Markgraf, V. (Eds.): El Niño: Historical and Paleoclimatic Aspects of the Southern Oscillation, Cambridge Univ. Press, New York, 1993. Delcroix, T., Eldin, G., McPhaden, M., and Morlière, A.: Effects of westerly wind bursts upon the western equatorial Pacific Ocean, February–April 1991, J. Geophys. Res., 98(C9), 16379–16385, 1993. Dijkstra, H. A.: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño, 2nd edn., Springer-Verlag, 2005. Dijkstra, H. A. and Ghil, M.: Low-frequency variability of the ocean circulation: a dynamical systems approach, Rev. Geophys., 43, RG3002, doi:10.1029/2002RG000122, 2005. Feigenbaum, M. J.: Quantitative universality for a class of non-linear transformations, J. Stat. Phys., 19, 25–52, 1978. Gebbie, G., Eisenman, I., Wittenberg, A., and Tziperman, E.: Modulation of westerly wind bursts by sea surface temperature: A semistochastic feedback for ENSO, J. Atmos. Sci., 64, 3281–3295, 2007. Ghil, M. and Jiang, N.: Recent forecast skill for the El Niño/Southern Oscillation, Geophys. Res. Lett., 25, 171–174, 1998. Ghil+02 Ghil, M., Allen, M. R., Dettinger, M. D., Ide, K., Kondrashov, D., Mann, M. E., Robertson, A. W., Saunders, A., Tian, Y., Varadi, F., and Yiou, P: Advanced spectral methods for climatic time series, Rev. Geophys., 40(1), 1003, doi:10.1029/2000RG000092, 2002. Ghil, M. and Robertson, A. W.: Solving problems with GCMs: General circulation models and their role in the climate modelling hierarchy, in: General Circulation Model Development: Past, Present and Future, edited by: Randall, D., Academic Press, San Diego, 285–325, 2000. Ghil, M., Zaliapin, I., and Coluzzi, B.: Boolean delay equations: A simple way of looking at complex systems, Physica D, 237, 2967–2986, doi: 10.1016/j.physd.2008.07.006, 2008a. Ghil, M., Zaliapin, I., and Thompson, S.: A delay differential model of ENSO variability: parametric instability and the distribution of extremes, Nonlin. Processes Geophys., 15, 417–433, 2008b. Ghil, M., Chekroun, M. D., and Simonnet, E.: Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237, 2111–2126, 2008c. Glantz, M. H., Katz, R. W., and Nicholls, N. (Eds.): Teleconnections Linking Worldwide Climate Anomalies, Cambridge Univ. Press, New York, 545~pp., 1991. Grebogi, C., Ott, E., and Yorke, J A.: Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Science, 238(4827), 632–638, 1987. Hale, J K.: Theory of Functional Differential Equations, Springer-Verlag, New-York, 1977. Hale, J K. and Verduyn Lunel, S.: Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. Harrison, D. E. and Giese, B.: Remote westerly wind forcing of the eastern equatorial Pacific; some model results, Geophys. Res. Lett., 15, 804–807, 1988. Hoerling, M. P., Kumar, A., and Zhong, M.: El Niño, La Niña, and the nonlinearity of their teleconnections, J. Climate, 10, 1769–1786, 1997. Hurrell, J. W. and K. E. Trenberth: Global sea surface temperature analyses: Multiple problems and their implications for climate analysis, modelling, and reanalysis, B. Am. Meteorol. Soc., 80, 2661–2678, 1999. Jiang, N., Neelin, J. D., and Ghil, M.: Quasi-quadrennial and quasi-biennial variability in the equatorial Pacific, Clim. Dynam., 12, 101–112, 1995. Jin, F.-F., Neelin, J. D., and Ghil, M.: El Niño on the Devil's Staircase: Annual subharmonic steps to chaos, Science, 264, 70–72, 1994. Jin, F.-F., Neelin, J. D., and Ghil, M.: El Niño/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity, Physica D, 98, 442–465, 1996. Kadanoff, L P.: Roads to chaos, Phys. Today, 12, 46–53, 1983. \bibitem[Kondrashov et al.(2005)] KKGR05 Kondrashov, D., Kravtsov, S., Robertson, A W., and Ghil, M.: A hierarchy of data-based ENSO models, J. Climate, 18, 4425–4444, 2005. Latif, M., Barnett, T. P., Flügel, M., Graham, N. E., Xu, J.-S., and Zebiak, S. E.: A review of ENSO prediction studies, Clim. Dynam., 9, 167–179, 1994. Lengaigne, M., Guilyardi, E., Boulanger, J. P., et al.: Triggering of El Nino by westerly wind events in a coupled general circulation model, Clim. Dynam., 23(6), 601–620, 2004. Madden, R. A. and Julian, P. R.: Description of a 40–50 day oscillation in the zonal wind in the tropical Pacific, J. Atmos. Sci., 28, 702–708, 1971. Madden, R. A. and Julian, P. R.: Description of global-scale circulation cells in the tropics with a 40–50 day period, J. Atmos. Sci., 29, 1109-,1123, 1972. Madden, R. A. and Julian, P. R.: Observations of the 40–50-day tropical oscillation – A review, Mon. Weather Rev., 122(5), 814–37, 1994. McPhaden, M J., Busalacchi, A J ., Cheney, R., Donguy, J R., Gage, K S., Halpern, D., Ji, M., Julian, P., Meyers, G., Mitchum, G T., Niiler, P P., Picaut, J., Reynolds, R W., Smith, N., and Takeuchi, K.: The Tropical Ocean-Global Atmosphere observing system: A decade of progress, J. Geophys. Res., 103(C7), 14169–14240, 1998. Munnich, M., Cane, M., and Zebiak, S. E.: A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: Nonlinear cases, J. Atmos. Sci., 48(10), 1238–1248, 1991. Neelin, J. D., Latif, M., and Jin, F.-F.: Dynamics of coupled ocean-atmosphere models: the tropical problem, Annu. Rev. Fluid Mech., 26, 617–659, 1994. Neelin, J. D., Battisti, D. S., Hirst, A. C., Jin, F.-F., Wakata, Y., Yamagata, T., and Zebiak, S.: ENSO Theory, J. Geophys. Res., 103(C7), 14261–14290, 1998. Nussbaum, R. D.: Functional Differential Equations, available at: http://citeseer.ist.psu.edu/437755.html, 1998. Philander, S. G. H.: El Niño, La Niña, and the Southern Oscillation, Academic Press, San Diego, 1990. Reynolds, R. W. and Smith, T. M.: Improved global sea surface temperature analyses using optimum interpolation, J. Climate, 7, 929–948, 1994. Sardeshmukh, P. D., Compo, G. P., and Penland, C.: Changes of probability associated with El Niño, J. Climate, 13, 4268–4286, 2000. Saunders, A. and Ghil, M.: A Boolean delay equation model of ENSO variability, Physica D, 160, 54–78, 2001. Saynisch, J., Kurths, J., and Maraun, D.: A conceptual ENSO model under realistic noise forcing, Nonlin. Processes Geophys., 13, 275–285, 2006. Shampine, L. F. and Thompson, S.: A friendly Fortran 90 DDE solver, Appl. Numer. Math., 56(2–3), 503–516, 2006. Suarez, M. J. and Schopf, P. S.: A delayed action oscillator for ENSO, J. Atmos. Sci, 45, 3283–3287, 1988. Trenberth, K. E.: The definition of El Niño, B. Am. Meteorol. Soc., 78, 2771–2777, 1997. Tziperman, E., Stone, L., Cane, M., and Jarosh, H.: El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science, 264, 72–74, 1994. Tziperman, E., Cane, M. A., and Zebiak, S. E.: Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos, J. Atmos. Sci., 50, 293–306, 1995. Verbickas, S.: Westerly wind bursts in the tropical Pacific, Weather, 53, 282–284, 1998.