• 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 the structurally 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.

Differential Delay Equations (DDE) offer an effective modeling languageas they combine simplicity of formulation with rich behavior…

( ) ( ), 0f t f t ( ) ( ), 0f t f t

To gain some intuition, compare

( ) tf t eThe only solution is

1

( ) cos( ) sin( )kp trtk k k k

k

f t Ce e A q t B q t

, , arbitrary

- the only real root of

( ) - complex roots of

k k

x

xk k

A B C

r xe

p iq xe

The general solution is given by

In particular, oscillatory solutions do exist.

ODE DDE

i.e., exponential growth (or decay, for < 0)

Battisti & Hirst (1989)

0,0 ,)(/ TtTdtdT

SST averaged overeastern equatorial Pacific

Bjerknes’s positivefeedback

Negative feedback due to oceanic waves

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).

Battisti & Hirst (1989)

0,0 ,)(/ TtTdtdT

Suarez & Schopf (1988), Battisti & Hirst (1989)

3)(/ TTtTdtdT

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)

Battisti & Hirst (1989)

0,0 ,)(/ TtTdtdT

Suarez & Schopf (1988), Battisti & Hirst (1989)

3)(/ TTtTdtdT

Tziperman et al., (1994)

)2cos()(tanh)(tanh/ 21 ttTtTdtdT

Seasonal forcingRealistic atmosphere-ocean coupling(Munnich et al., 1991)

( ) tanh ( ) cos(2 )dh t h t b t

dt

Thermocline depth deviations from the annual mean in the

eastern Pacific

Wind-forced ocean waves (E’ward Kelvin, W’ward Rossby)

Delay due to finite wave velocity

Seasonal-cycle forcing

Strength of the atmosphere-ocean

coupling

“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)

1, 100, 0.42b

“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)

1, 4.76, 0.66b

Interdecadal variability:Spontaneous change of (1) long-term annual mean, and(2) Higher/lower positive and lower/higher negative extremes

N.B. Intrinsic, rather than forced!1, 10, 0.45b

Bursts of intraseasonal oscillationsof random amplitude

Madden-Julian oscillations, westerly-wind bursts?

1, 500, 0.0038b

( )tanh ( ) cos(2 ), 0 (1)

( ) ( ), [ ,0) (2)

dh th t b t t

dth t t t

Theorem

The IVP (1-2) has a unique solution on [0, ) for any set ( , , , ).

This solution depends continuously on initial data ( ), delay ,

and the rhs of (1) (in an appropriate norm).

b

t

CorollaryA discontinuity in solution profile indicates existence of an unstablesolution that separates attractor basins of two stable ones.

Trajectory maximum (after transient):

Smooth map

Monotonic in b

Periodic in

Trajectory maximum (after transient):

Smooth map

No longer monotonic

in b, for large No longer periodic

in for large

Trajectory maximum (after transient):

Neutral curve f (b, appears, above whichinstabilities set in.

Above this curve, the maxima are nolonger monotonic in b

or periodic in andthe map “crinkles” (i.e.,it becomes “rough”)

Trajectory maximum (after transient):

The neutral curvethat separates roughfrom smooth behavior becomes itself crinkled(rough, fractal?).

The neutral curve moves to higherseasonal forcing b and lower delays .

M. Ghil & I. Zaliapin, UCLA Working Meeting, August 21, 2007

This regionexpanded

M. Ghil & I. Zaliapin, UCLA Working Meeting, August 21, 2007

Instability point

Instability point

Delay, b

Maxima

Minima

Shape of forcing

Maxima

Minima

Time

b = 1, = 10, = 0.5

100 initial (constant) data 4 distinct solutions

Solution profile

Initi

al d

ata

(b = 1.4, = 0.57= 11)

Stable solutions (after transient)

(b = 1.0, = 0.57)

(b = 3, = 0.3)

(b = 1.6, = 1.6)

(b = 2.0, = 1.0)

(b = 1.4, = 0.57)

1. 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.

2. 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).

3. Spontaneous transitions in mean temperature, as well as in extreme annual values occur, for purely periodic, seasonal forcing.

4. 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.

5. We expect such behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.

1, 10, 0.513b

1, 10, 0.516b

Solution profile at k =1

Solution profile at k =3

Solution profile at k =3.5

Solution profile at k =4

Solution profile at k =50

Solution profile at k =100

Solution profile at k =1000

Constant history H, b=1.0,=0.5

=1.0

=3.5

=4.0

=50

Constant history H, =11, b=1.4,=0.57

H=[-1,1]

H=[0.5,0.51]

H=[0.5,0.5001]