Michael Ghil

1. A Differential Delay Model of ENSO variability: Parametric Instability and Distribution of Extremes Michael Ghil Ilya Zaliapin Skip Thompson Ecole Normale Supérieure, Paris, U of Nevada, Reno Radford U, Virginia U of California, Los Angeles Ecole Normale Supérieure Tuesday, May 20, 2008

2. Outline History & Motivation Model formulation Results Noteworthy scenarios Theoretical results Spontaneous changes in mean and extremes Concluding remarks

3. Motivation – choice of topic • Climate models -- the most sophisticated models of natural phenomena. • Still, the range of uncertainty in responses to CO2 doubling is not decreasing. • Can this be a matter of intrinsic sensitivity to model parameters and parameterizations, similar to but distinct from sensitivity to initial data? • Dynamical systems theory has, so far, interpreted model robustness in terms of structural stability; it turns out that this property is not generic. • We explore thestructurally unstable behavior of a toy model of ENSO variability, the interplay between forcing and internal variability, as well as spontaneous changes in mean and extremes.

4. Motivation – choice of "toy model" Differential Delay Equations (DDE) offer an effective modeling language as they combine simplicity of formulation with rich behavior… To gain some intuition, compare ODE DDE The general solution is given by The only solution is i.e., exponential growth (or decay, for < 0) In particular, oscillatory solutions do exist.

5. Delay models of ENSO variability Battisti & Hirst (1989) SST averaged over eastern equatorial Pacific Negative feedback due to oceanic waves Bjerknes’s positive feedback The model reproduces some of the main features of a fully nonlinear coupled atmosphere-ocean model of ENSO dynamics in the tropics (Battisti, 1988; Zebiak and Cane, 1987).

6. Delay models of ENSO variability Battisti & Hirst (1989) Suarez & Schopf (1988), Battisti & Hirst (1989) Cubic nonlinearity The models are successful in explaining the periodic nature of ENSO events. But they… • have a well defined period, which is not the case in observations • can’t explain phase locking • predict a wrong ENSO period (1.5-2 years)

7. Delay models of ENSO variability Battisti & Hirst (1989) Suarez & Schopf (1988), Battisti & Hirst (1989) Tziperman et al., (1994) Realistic atmosphere-ocean coupling (Munnich et al., 1991) Seasonal forcing

8. Model formulation Thermocline depth deviations from the annual mean in the eastern Pacific Strength of the atmosphere-ocean coupling Seasonal-cycle forcing Wind-forced ocean waves (E’ward Kelvin, W’ward Rossby) Delay due to finite wave velocity

9. Noteworthy scenarios

10. Noteworthy scenarios (1) “High-h” season with period of about 4 yr; notice the random heights of high seasons Rough equivalent of El Niño in this toy model (little upwelling near coast)

11. Noteworthy scenarios (1) “Low-h” (cold) seasons in successive years have a period of about 5 yr in this model run. Negative h corresponds to NH (boreal) winter (upwelling season, DJF, in the eastern Tropical Pacific)

12. Noteworthy scenarios (2) Interdecadal variability: Spontaneous change of long-term annual mean, and Higher/lower positive and lower/higher negative extremes N.B. Intrinsic, rather than forced!

13. Noteworthy scenarios (3) Bursts of intraseasonal oscillations of random amplitude Madden-Julian oscillations, westerly-wind bursts?

14. Existence, uniqueness, cont. dependence Theorem Corollary A discontinuity in solution profile indicates existence of an unstable solution that separates attractor basins of two stable ones.

15. Critical transitions

16. Critical transitions (1) Trajectory maximum (after transient): k = 0.5 Smooth map Monotonic in b Periodic in t

17. Critical transitions (2) Trajectory maximum (after transient): k =1 Smooth map No longer monotonic in b, for larget No longer periodic in t for larget

18. Critical transitions (3) Trajectory maximum (after transient): k =2 Neutral curve f (b, t appears, above which instabilities set in. Above this curve, the maxima are no longer monotonic in b or periodic intand the map “crinkles” (i.e., it becomes “rough”)

19. Critical transitions (4) Trajectory maximum (after transient): k =11 The neutral curve moves to higher seasonal forcing b and lower delays . The neutral curve that separates rough from smooth behavior becomes itself crinkled (rough, fractal?). M. Ghil & I. Zaliapin, UCLA Working Meeting, August 21, 2007

20. Trajectory maximum This region expanded

21. Intermediate forcing and delay M. Ghil & I. Zaliapin, UCLA Working Meeting, August 21, 2007

22. Examples of instability (1) Instability point

23. Examples of instability (2) Instability point

24. Dynamics of extrema

25. Local extrema: intermittency Maxima Minima Delay, t (k=11, b=2)

26. Local extrema: phase locking Maxima Minima Shape of forcing Time

27. Multiple (un)stable solutions

28. Multiple (un)stable solutions b = 1, k = 10, t = 0.5 100 initial (constant) data 4 distinct solutions

29. Solution profile Initial data (b = 1.4, t = 0.57, k = 11) Multiple (un)stable solutions Stable solutions (after transient)

30. (b = 1.6, t = 1.6) (b = 1.0, t = 0.57) (b = 2.0, t = 1.0) (b = 3, t = 0.3) (b = 1.4, t = 0.57) Multiple (un)stable solutions

31. Concluding remarks A simple differential-delay equation (DDE) with a single delay reproduces the realistic scenarios documented in other ENSO models, such as nonlinear PDEs and GCMs, as well as in observations. The model illustrates well the role of the distinct parameters: strength of seasonal forcing b, ocean-atmosphere coupling , and delay  (propagation period of oceanic waves across the Tropical Pacific). Spontaneous transitions in mean temperature, as well as in extreme annual values occur, for purely periodic, seasonal forcing. A sharp neutral curve in the (b–) plane separates smooth behavior of the period map from “rough” behavior; changes in this neutral curve as  changes are under study. We expect such behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.

32. Examples of instability (4)

33. Instability with respect to initial data Solution profile at k =1

34. Instability with respect to initial data Solution profile at k =3

35. Instability with respect to initial data Solution profile at k =3.5

36. Instability with respect to initial data Solution profile at k =4

37. Instability with respect to initial data Solution profile at k =50

38. Instability with respect to initial data Solution profile at k =100

39. Instability with respect to initial data Solution profile at k =1000

40. k=1.0 k=3.5 k=4.0 k=50 Constant history H, b=1.0,t =0.5

41. Instability with respect to initial data H=[-1,1] H=[0.5,0.51] H=[0.5,0.5001] Constant history H, k =11, b=1.4,t =0.57