Michael Ghil

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A toy model of ENSO variability Structural
instability and spontaneous transitions in
extremes
Michael Ghil
Ilya Zaliapin
and
Ecole Normale Supérieure, Paris,
Univ. of Nevada, Reno, NV, USA
and Univ. of California, Los Angeles
zal_at_unr.edu
ghil_at_lmd.ens.fr
European Geosciences Union, General Assembly,
1520 April 2007, Vienna
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Outline
Motivation
Model formulation
Results
Theoretical results
Phase locking and Devil's staircase
Spontaneous changes in mean and extremes
Concluding remarks
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Motivation choice of topic

• Climate models are among the most detailed and
  sophisticated models
• of natural phenomena in existence.
• Still, the range of uncertainty in responses to
  either CO2 doubling or to various emission
  scenarios is not decreasing from one IPCC report
  to the next.
• Is this merely a stubborn engineering problem in
  tuning model parameters or is it a matter of
  intrinsic sensitivity to such parameters and to
  model parameterizations, similar to but distinct
  from sensitivity to initial data?
• Dynamical systems theory has, so far, interpreted
  model robustness mostly in terms of structural
  stability.
• 5. It turns out that this property is not
  generic (see also Session NP5.01 "Robust
  estimates of climate change and the
  generalization of structural stability" by
• M. Ghil, M. Chekroun and E. Simonnet, Lecture
  Room 22, Thursday at 1030 a.m.).
• 6. We explore the structurally unstable
  behavior of a very simple, but interesting model
  of ENSO variability, the interplay between
  forcing and internal variability, as well as
  spontaneous changes in mean and extremes.

M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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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 ? lt 0)
In particular, oscillatory solutions do exist.
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Model formulation
Thermocline depth deviations from the annual
mean in the eastern Pacific
Wind-forced ocean waves (Eward Kelvin, Wward
Rossby)
Delay due to finite wave velocity
Seasonal-cycle forcing
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Model parameters
Wind-forced ocean waves (Kelvin, Rossby)
Strength of the atmosphere-ocean coupling
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Model parameters (cont'd)
The seasonal-cycle forcing has the period P0
P0 (2??)1 1 yr,
and we consider the following parameter ranges
The initial data for our DDE are given by the
constant history (warm event)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Model general results
With no seasonal forcing we have
For large delays, the solution is
asymptotically periodic, with period 4t
For small delays, the solution is
asymptotically zero, as it is for no delay (ODE
case)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Model general results (cont'd)
accordingly, for
For large delays, there are nonlinear
interactions between periodic solutions with
periods 4t and 1
For small delays, the solution is
asymptotically periodic with period 1, as for
the no- delay (ODE) case
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Examples
Period 4t
No period
Simple period 1
Complex period 1
Rough period 1
Period 1
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Noteworthy scenarios (1)
Low-h (cold) seasons in successive years have
a period of about 5 yr in this model run. N.B.
Negative h corresponds to NH (boreal) winter
(upwelling season, DJF, in the
eastern Tropical Pacific)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Noteworthy scenarios (2)
High-h season with period of about 4 yr notice
the random heights of high seasons N.B. Rough
equivalent of El Niño in this toy
model (little upwelling near coast)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Noteworthy scenarios (3)
Bursts of intraseasonal oscillations () of
random amplitude () Madden-Julian
oscillations, westerly-wind bursts?
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Noteworthy scenarios (4)

• Interdecadal variability
• Spontaneous change of
• long-term annual mean, and
• Higher/lower positive and
• lower/higher negative extremes
• N.B. Intrinsic, rather than forced!

M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Critical transitions
Our toy model produces several types of temporal
behavior.
Is transition from one type to another smooth or
sudden?
We answer this question by studying
the period map
trajectory statistics (max, tail, etc.)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Devil's staircase phase locking
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Devil's staircase small forcing and short delays
Period dependence on delay for b 0.03, k ? 100
The period is always given by
and it is always close to an integer.
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Devil's bleachers for small forcing and small
delay
Regime diagram for the period index
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Critical transitions (1)
Trajectory maximum (after transient) k 0.5
Smooth map
Monotonic in b
Periodic in t
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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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 in t??and the map
crinkles (i.e., it becomes rough)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Examples of instability (1)
Instability point
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Examples of instability (2)
Instability point
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Concluding remarks

• A simple differential-delay equation (DDE) with a
  single delay reproduces the
• Devils staircase scenario documented in other
  ENSO models, such as
• nonlinear PDEs and GCMs, as well as in
  observations.
• 2. The model illustrates well the role of the
  distinct parameters, such as strength of seasonal
  forcing b vs. nonlinearity ? (ocean-atmosphere
  coupling) and delay ? (propagation period of
  oceanic waves across the Tropical Pacific).
• Spontaneous transitions in mean temperature
  (i.e., thermocline depth), as well
• as in extreme annual values occur, for purely
  periodic, seasonal forcing.
• 4. The model generates intraseasonal
  (Madden-Julian?) oscillations of various periods
  and amplitudes (westerly wind bursts?), as well
  as interdecadal variability.
• 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.
• The various types of instabilities across this
  neutral curve are being explored.
• We expect such behavior in much more detailed and
  realistic models, where it is harder to describe
  its causes as completely.

M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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References

• Ghil, M., and A. W. Robertson, 2000 Solving
  problems with GCMs General circulation models
  and their role in the climate modeling hierarchy.
  General Circulation Model Development Past,
  Present and Future, D. Randall (Ed.), Academic
  Press, San Diego, pp. 285325.
• Hale, J. K., 1977 Theory of Functional
  Differential Equations, Springer-Verlag, New
  York, 365 pp.
• Jin, F.-f., J. D. Neelin and M. Ghil, 1994 El
  Niño on the Devil's Staircase Annual subharmonic
  steps to chaos, Science, 264, 7072.
• Saunders, A., and M. Ghil, 2001 A Boolean delay
  equation model of ENSO variability, Physica D,
  160, 5478.
• Tziperman, E., L. Stone, M. Cane and H. Jarosh,
  1994 El Niño chaos Overlapping of resonances
  between the seasonal cycle and the Pacific
  ocean-atmosphere oscillator. Science, 264,
  7274.
• Munnich, M., M. Cane, and S. Zebiak, 1991 A
  study of self-excited oscillations of the
  tropical ocean atmosphere system 2. Nonlinear
  cases , J. Atmos. Sci., 48, 12381248.

M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Reserve slides
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Critical transitions (2)
Trajectory maximum (after transient) k 1
Smooth map
No longer monotonic in b, for large t
No longer periodic in t?? for large t
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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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, EGU Assembly, April 16,
2007
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Examples of instability (3)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007
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Examples of instability (4)
M. Ghil I. Zaliapin, EGU Assembly, April 16,
2007