Nonlinear Processes in Geophysics
www.nonlin-processes-geophys.net
1023-5809
1607-7946
15
3
2008
10.5194/npg-15-417-2008
http://www.nonlin-processes-geophys.net/15/417/2008/
http://www.nonlin-processes-geophys.net/15/417/2008/npg-15-417-2008.html
http://www.nonlin-processes-geophys.net/15/417/2008/npg-15-417-2008.pdf
417
433
2008-05-28
A delay differential model of ENSO variability: parametric instability and the distribution of extremes
M. Ghil
ghil@atmos.ucla.edu
I. Zaliapin
S. Thompson
Dépt. Terre-Atmosphère-Océan and Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure, Paris, France
Dept. of Mathematics and Statistics, University of Nevada, Reno, NV, USA
Dept. of Mathematics and Statistics, University of Radford, VA, USA
Dept. of Atmospheric and Oceanic Sciences and Institute of Geophysics and Planetary Physics, University of California Los Angeles, CA, USA
We consider a delay differential equation (DDE) model for El-Niño Southern
Oscillation (ENSO) variability.
The model combines two key mechanisms that participate in ENSO dynamics:
delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of its
physically relevant parameters.
Our results illustrate the role of these three parameters:
strength of seasonal forcing <i>b</i>, atmosphere-ocean coupling <i>κ</i>,
and propagation period <i>τ</i> of oceanic waves across the Tropical Pacific.
Two regimes of variability, stable and unstable, are separated by a sharp neutral
curve in the (<i>b</i>, <i>τ</i>) plane at constant <i>κ</i>.
The detailed structure of the neutral curve becomes very irregular and
possibly fractal, while individual trajectories within the unstable region
become highly complex and possibly chaotic, as the atmosphere-ocean coupling
<i>κ</i> increases.
In the unstable regime, spontaneous transitions occur in the mean "temperature"
(i.e., thermocline depth), period, and extreme annual values,
for purely periodic, seasonal forcing.
The model reproduces the Devil's bleachers characterizing other ENSO models,
such as nonlinear, coupled systems of partial differential equations;
some of the features of this behavior have been documented in general circulation
models, as well as in observations.
We expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.
Andronov, A. A. and Pontryagin, L. S.: Systèmes grossiers, Dokl. Akad. Nauk SSSR, 14, 5, 247–250, 1937.
Baker, C. T. H.: Retarded differential equations, J. Comp. Appl. Math., 125, 309–335, 2000.
Baker, C. T. H., Paul, C. A. H., and Willé, D. R.: A bibliography on the numerical solution of delay differential equations, Numerical Analysis Report, 269, Manchester Centre for Computational Mathematics, Manchester, England, 1995.
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.
Bhattacharya, K., Ghil, M., and Vulis, I.: Internal variability of an energy-balance model with delayed albedo effects, J. Atmos. Sci., 39, 1747–1773, 1982.
Bjerknes, J.: Atmospheric teleconnections from the equatorial Pacific, Mon. Weather Rev., 97, 163–172, 1969.
Bodnar, M.: On the differences and similarities of the first-order delay and ordinary differential equations, J. Math. Anal. Appl., 300, 172–188, 2004.
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.
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.
Cao, Y.: Uniqueness of periodic solution for differential delay equations, J. Diff. Eq., 128, 46–57, 1996.
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.
Chow, S.-N. and Walter, H. O.: Characteristic multipliers and stability of periodic solutions of $\dot x=g\left(x(t-1)\right)$, Trans. Amer. Math. Soc., 307, 127–142, 1988.
Corwin, S. P., Sarafyan, D., and Thompson, S.: DKLAG6: a code based on continuously imbedded sixth order Runge-Kutta methods for the solution of state dependent functional differential equations, Appl. Numer. Math., 24, 319–333, 1997.
Dee, D. and Ghil, M.: Boolean difference equations, I: Formulation and dynamic behavior, SIAM J. Appl. Math., 44, 111–126, 1984.
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, 16 379–16 385, 1993.
Diaz, H. F. and Markgraf, V.: El Niño: Historical and Paleoclimatic Aspects of the Southern Oscillation, Cambridge Univ. Press, New York, 1993.
Dijkstra, H. A.: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño, 2nd edition, 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.
Engelborghs, K., Luzyanina, T., and Samaey, G.: DDE-BIFTOOL v. 2.00: a Matlab package for numerical bifurcation analysis of delay differential equations, Report TW, vd330, Department of Computer Science, K.U. Leuven, Leuven, Belgium, available at http://www.cs.kuleuven.ac.be/cwis/research/twr/research/software/delay/ddebiftool.shtml, 2005.
Enright, W. H. and Hayashi, H.: A delay differential equation solver based on a continuous Runge-Kutta method with defect control, Numer. Alg., 16, 349–364, 1997.
Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, USA, 2002.
Falbo, C. E.: Analytical and numerical solutions to the delay differential equation $\dot y=\alpha y(t-\delta)$, Proc. Joint Mtg. California sections of the MAA, San Luis Obispo, CA, USA, available at: http://www.sonoma.edu/math/faculty/falbo/pag1dde.html, 21 October 1995.
Fuglestvedt, J. S., Berntsen, T. K., Godal, O., Sausen, R., Shine, K. P., and Skodvin, T.: Metrics of climate change: Assessing radiative forcing and emission indices, Climatic Change, 58, 267–331, 2003.
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., 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 Childress, S.: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Clim. Dynam., Springer, Verlag, 1987.
Ghil, M. and Mullhaupt, A. P.: Boolean delay equations. II: Periodic and aperiodic solutions, J. Stat. Phys., 41, 125–173, 1985.
Ghil, M. and Robertson, A. W.: Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy, Randall, D.(Ed.) General Circulation Model Development: Past, Present and Future, Academic Press, San Diego, 285–325, 2000.
Glantz, M. H., Katz, R. W., and Nicholls, N.: Teleconnections Linking Worldwide Climate Anomalies, Cambridge Univ. Press, New York, 545 pp., 1991.
Hale, J.: Theory of Functional Differential Equations, Springer-Verlag, New-York, 1977.
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.
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.
Katok, A. and Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 802 pp., 1995.
Kolmanovskii, V. B. and Nosov, V. R.: Stability of Functional Differential Equations, Academic Press, 1986.
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, 814–837, 1994.
\textscMatlab 7.4: The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA 01760, 2007.
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.
Murphy J. M., Sexton, D. M. H., Barnett, D. N., Jones, G. S., Webb, M. J., and Collins, M.: Quantification of modelling uncertainties in a large ensemble of climate change simulations, Nature, 430, 7001, 768–772, 2004.
Neelin, J. D., Latif, M., and Jin, F.-F.: Dynamics of coupled ocean-atmosphere models: the tropical problem, Ann. 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), 14 261–14 290, 1998.
Nussbaum, R. D.: Uniqueness and nonuniqueness for periodic solutions of $\dot x(t)=-g\left(x(t-1)\right)$, J. Diff. Eq., 34, 25–54, 1979.
Nussbaum, R. D.: Functional Differential Equations, available at http://citeseer.ist.psu.edu/437755.html, 1998.
Paul, C. A. H.: A user-guide to ARCHI, Numerical Analysis Report 283, Mathematics Department, University of Manchester, UK, 1995.
Philander, S. G. H.: El Niño, La Niña, and the Southern Oscillation, Academic Press, San Diego, 1990.
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. Proc. Geophys., 13, 3, 275–285, 2006.
Shampine, L. F.: Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, NY, 1994.
Shampine, L. F.: Solving ODEs and DDEs with residual control, Appl. Numer. Math., 52, 113–127, 2005.
Shampine, L. F. and Thompson, S.: Solving DDEs in \textscMatlab, Appl. Numer. Math., 37, 441–458, 2001.
Shampine, L. F. and Thompson, S.: A friendly Fortran 90 DDE solver, Appl. Num. Math., 56, 2–3, 503–516, 2006.
Stainforth D. A., Aina, T., Christensen, C., et al.: Uncertainty in predictions of the climate response to rising levels of greenhouse gases, Nature, 433, 7024, 403–406, 2005.
Suarez, M. J. and Schopf, P. S.: A delayed action oscillator for ENSO, J. Atmos. Sci, 45, 3283–3287, 1988.
Taylor, K. E.: Summarizing multiple aspects of model performance in a single diagram, J. Geophys. Res., 106, 7183–7192, 2001.
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.
Yanai, M. and Li, C.: Interannual variability of the Asian summer monsoon and its relationship with ENSO, Eurasian snow cover, and heating, Proc. Int. Conf. on Monsoon Variability and Prediction, Trieste, Italy, World Meteorological Organization, 27–34, 9–13 May 1994.
Zaliapin, I., Keilis-Borok, V., and Ghil, M.: A Boolean delay equation model of colliding cascades. Part I: Multiple seismic regimes, J. Stat. Phys., 111, 815–837, 2003.
Zebiak, S. and Cane, M. A.: A model El-Niño Southern Oscillation, Mon. Weather Rev., 115, 2262–2278, 1987.