A delay differential model of ENSO variability – Part 2: Phase locking, multiple solutions and dynamics of extrema

We consider a highly idealized model for El Ni˜ no/Southern Oscillation (ENSO) variability, as introduced in an earlier paper. The model is governed by a delay differ- ential equation for sea-surface temperature T 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 of oceanic waves across the Tropical Pacific, atmosphere-ocean coupling , and strength of seasonal forcing b. Phase locking of model solutions to the periodic forcing is prevalent: the local max- ima and minima of the solutions tend to occur at the same po- sition within the seasonal cycle. Such phase locking is a key feature of the observed El Ni ˜ no (warm) and La Ni˜ na (cold) events. The phasing of the extrema within the seasonal cy- cle 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.

[1]  Michael Ghil,et al.  El Ni�o on the Devil's Staircase: Annual Subharmonic Steps to Chaos , 1994, Science.

[2]  Michael Ghil,et al.  Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy , 2000 .

[3]  Bin Wang,et al.  Interactions between the seasonal cycle and the Southern Oscillation - Frequency entrainment and chaos in a coupled ocean-atmosphere model , 1994 .

[4]  J. D. Neelin,et al.  Dynamics of Coupled Ocean-Atmosphere Models: The Tropical Problem , 1994 .

[5]  E. Tziperman,et al.  Modulation of Westerly Wind Bursts by Sea Surface Temperature: A Semistochastic Feedback for ENSO , 2007 .

[6]  Bin Wang,et al.  Interactions between the Seasonal Cycle and El Niño-Southern Oscillation in an Intermediate Coupled Ocean-Atmosphere Model , 1995 .

[7]  Michael Ghil,et al.  El Nin˜o/Southern Oscillation and the annual cycle: subharmonic frequency-locking and aperiodicity , 1996 .

[8]  J. Bjerknes.  ATMOSPHERIC TELECONNECTIONS FROM THE EQUATORIAL PACIFIC1 , 1969 .

[9]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[10]  P. R. Julian,et al.  Detection of a 40–50 Day Oscillation in the Zonal Wind in the Tropical Pacific , 1971 .

[11]  James W. Hurrell,et al.  Global Sea Surface Temperature Analyses: Multiple Problems and Their Implications for Climate Analysis, Modeling, and Reanalysis , 1999 .

[12]  Matthieu Lengaigne,et al.  Triggering of El Niño by westerly wind events in a coupled general circulation model , 2004 .

[13]  D. Battisti,et al.  Dynamics and Thermodynamics of a Warming Event in a Coupled Tropical Atmosphere–Ocean Model , 1988 .

[14]  Kevin E. Trenberth,et al.  The Definition of El Niño. , 1997 .

[15]  Michael Ghil,et al.  ADVANCED SPECTRAL METHODS FOR CLIMATIC TIME SERIES , 2002 .

[16]  Thomas M. Smith,et al.  Improved Global Sea Surface Temperature Analyses Using Optimum Interpolation , 1994 .

[17]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[18]  Mojib Latif,et al.  A review of ENSO prediction studies , 1994 .

[19]  Michael Ghil,et al.  A delay differential model of ENSO variability: parametric instability and the distribution of extremes , 2007, 0712.1312.

[20]  Henk A. Dijkstra,et al.  Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño, , 2000 .

[21]  J. Hale.  Functional Differential Equations , 1971 .

[22]  Prashant D. Sardeshmukh,et al.  Changes of Probability Associated with El Niño , 2000 .

[23]  M. Mcphaden,et al.  Effects of westerly wind bursts upon the western equatorial Pacific Ocean, February–April 1991 , 1993 .

[24]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[25]  Lawrence F. Shampine,et al.  A friendly Fortran DDE solver , 2006 .

[26]  Anthony C. Hirst,et al.  Interannual variability in a tropical atmosphere−ocean model: influence of the basic state, ocean geometry and nonlinearity , 1989 .

[27]  J. Yorke,et al.  Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics , 1987, Science.

[28]  Westerly wind bursts in the tropical Pacific , 1998 .

[29]  M. Ghil,et al.  Boolean delay equations: A simple way of looking at complex systems , 2006, nlin/0612047.

[30]  Eli Tziperman,et al.  Irregularity and Locking to the Seasonal Cycle in an ENSO Prediction Model as Explained by the Quasi-Periodicity Route to Chaos , 1995 .

[31]  Mark A. Cane,et al.  A study of self-excited oscillations of the tropical ocean-atmosphere system , 1990 .

[32]  Neville Nicholls,et al.  Teleconnections Linking Worldwide Climate Anomalies , 2009 .

[33]  J. Kurths,et al.  A Conceptual Enso Model under Realistic Noise Forcing , 2022 .

[34]  M. Lengaigne,et al.  Role of high- and low-frequency winds and wave reflection in the onset, growth and termination of the 1997–1998 El Niño , 2004 .

[35]  P. R. Julian,et al.  Description of Global-Scale Circulation Cells in the Tropics with a 40–50 Day Period , 1972 .

[36]  Michael Ghil,et al.  Recent forecast skill for the El Niño/Southern Oscillation , 1998 .

[37]  Mark A. Cane,et al.  The evolution of El Nino, past and future , 2005 .

[38]  David B. Stephenson,et al.  The “normality” of El Niño , 1999 .

[39]  P. R. Julian,et al.  Observations of the 40-50-day tropical oscillation - a review , 1994 .

[40]  Min Zhong,et al.  El Niño, La Niña, and the Nonlinearity of Their Teleconnections , 1997 .

[41]  Steve Horstmeyer,et al.  El Niño, La Niña, and the Southern Oscillation , 2011 .

[42]  Michael Ghil,et al.  A boolean delay equation model of ENSO variability , 2001 .

[43]  Michael Ghil,et al.  A Hierarchy of Data-Based ENSO Models , 2005 .

[44]  Eli Tziperman,et al.  El Ni�o Chaos: Overlapping of Resonances Between the Seasonal Cycle and the Pacific Ocean-Atmosphere Oscillator , 1994, Science.

[45]  Antonio J. Busalacchi,et al.  The Tropical Ocean‐Global Atmosphere observing system: A decade of progress , 1998 .

[46]  Max J. Suarez,et al.  A Delayed Action Oscillator for ENSO , 1988 .

[47]  Michael Ghil,et al.  Low‐frequency variability of the large‐scale ocean circulation: A dynamical systems approach , 2005 .

[48]  D. E. Harrison,et al.  Remote westerly wind forcing of the eastern equatorial Pacific; Some model results , 1988 .

[49]  Michael Ghil,et al.  Quasi-quadrennial and quasi-biennial variability in the equatorial Pacific , 1995 .

[50]  Leo P. Kadanoff,et al.  Roads to chaos , 1983 .

[51]  S. Deo.  Functional Differential Equations , 1968 .

[52]  Michael Ghil,et al.  Climate dynamics and fluid mechanics: Natural variability and related uncertainties , 2008, 1006.2864.