El Niño and how to get rid of it

Cécile Penland

With thanks to

Prashant Sardeshmukh Ludmila Matrosova

Ping Chang Moritz Flügel

Brian Ewald Roger Temam

The Climate Diagnostics Center OGP

Review of Linear Inverse Modeling

Assume linear dynamics:

dx/dt = Bx +

Diagnose Green function from data:

G() = exp(B) = <x(t+)xT(t) >< x(t)xT(t) >-1 .

Eigenvectors of G() are the “normal” modes {ui}.

Most probable prediction: x’(t+) = G() x(t)

The neat thing: G() ={G() }

SST Data used:

• COADS (1950-2000) SSTs in 30E-70W, 30N – 30S consolidated onto a 4x10-degree grid.

• Subjected to 3-month running mean.• Projected onto 20 EOFs (eigenvectors of <xxT>)

containing 66% of the variance.• x, then, represents the vector of SST anomalies,

each component representing a location, or else it represents the vector of Principal Components.

El Niño can be described this way. 

If LIM is successful, prediction error does not depend on the lag at which the covariance matrices are evaluated. This is true for El Niño; it is not true for the chaotic Lorenz system. Below, different colors correspond to different lags used to identify the parameters.

The annual cycle:

dx/dt = Bx + (t)T (t) = Q(t)

Given stationary B use (time-dependent) conservation of variance to diagnose Q(t).

Result: The annual cycle of Q(t) looks nothing like the phase locking of either El Niño or the optimal structure to the annual cycle.

But

A model generated with the stationary B and the stochastic forcing with cyclic statistics Q(t) does reproduce the correct phase-locking in both.

What do models say?

• Hybrid coupled model (Chang 1994)• Dynamical core of GFDL Ocean model,

considerably simplified• Statistical atmosphere based on EOFs of

observed wind stress• Interactive annual cycle• Strength of coupling determines dynamical

regime (Note: an artificial parameter )

Optimal initial structure for growth over lead time :

Right singular vector of G() (eigenvector of GTG())

Growth factor over lead time :

Eigenvalue of GTG().

(a)

The transient growth possible in a multidimensional linear system occurs when an El Niño develops. LIM predicts that an optimal pattern (a) precedes a mature El Niño pattern (b) by about 8 months.

(b)

Does it? Judge for yourself! (c and d)

c)d)

(c) (d)

In (c), the red line is the time series of pattern correlations between pattern (a) and the sea surface temperature pattern 8 months earlier. The blue line is a time series index of how strong pattern (b) is at the date shown; the blue line is an index of El Niño when it is positive and of La Niña when it is negative. In (d) we see a scatter plot of the El Niño index and pattern correlations shown in (c).Pattern (a) really does precede El Niño! Pattern (a) with the signs reversed precedes La Niña!

Decay mode, = 31 months

0

0.5

1

1.5

0 5 10 15 20 25Mode number

momo

momo

momo

decay timeT = Period

Projection of adjoints onto O.S. and modal timescales.

Location of indices: N3.4, IND, NTA, EA, and STA.

-3

-2

-1

0

1

2

3

1960 1970 1980 1990 2000

Nino3.4

months

-3

-2

-1

0

1

2

3

1960 1970 1980 1990 2000

Filtered Nino3.4

months

10-4

10-3

10-2

10-1

100

101

1 10 100 1000

Period (months)

-5

0

5

10

15

20

25

1 10 100 1000

Period (months)

66.4 mo39.9 mo

18.1 mo

15.3 mo

-505

101520253035

100 101 102 103 104

43.9 mo

16.5 mo

5.2 mo

Period (weeks)

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino) = 0.36

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino) = 0.44

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino) = 0.45

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000

R(Unfiltered, El Nino) = 0.61

EA

ST

AIN

D

NT

A

R = 0.36 R = 0.45

R = 0.44 R = 0.61

Indices. Black: Unfiltered data. Red: El Niño signal.

-0.8

-0.6

-0.4

-0.2

0

0.2

-100 -50 0 50 100Lead (months)

8 months

STA leads PC1 leads

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-100 -50 0 50 100Lead (months)

IND leads PC1 leads

-0.6

-0.4

-0.2

0

0.2

0.4

-100 -50 0 50 100Lead (months)

EA leads PC1 leads

9 months

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-100 -50 0 50 100Lead (months)

NTA leads PC1 leads

Lagged correlation between El Niño indices and PC 1.

STA leads PC1 leads PC1 leads

PC1 leads PC1 leads

EA leads

IND leads NTA leads

Decay mode, = 31 months

0

0.5

1

1.5

0 5 10 15 20 25Mode number

momo

momo

momo

decay timeT = Period

Projection of adjoints onto O.S. and modal timescales.

EOF 1 of Residual

-15

-10

-5

0

5

10

1950 1960 1970 1980 1990 2000 2010Date

u1 of un-filtered data

The pattern correlation between the longest-lived mode of the unfiltered data and the leading EOF of the residual data is 0.81.

-1

-0.5

0

0.5

1

1950 1960 1970 1980 1990 2000

R(Unfiltered, El Nino +Trend) = 0.75

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino+Trend) = 0.79

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000Date

R(Unfiltered, El Nino+Trend) = 0.77

-1.5

-1

-0.5

0

0.5

1

1.5

1950 1960 1970 1980 1990 2000

R(Unfiltered, El Nino + Trend) = 0.62

EA

SS

TA

(C

)

ST

A S

ST

A (

C)

IND

SS

TA

(C

)

NT

A S

ST

A (

C)

R = 0.75 R = 0.77

R = 0.79 R = 0.62

Indices. Black: Unfiltered data. Green: El Niño signal + Trend.

All this results from SST dynamics being essentially linear.

But linear dynamics implies symmetry between El Niño and La Niña events.

SST anomalies appear to be positively skewed.

Is the skew significant?

Additive Noise Model

dx/dt = Bx +

Multiplicative Noise Model 1

dx/dt=(B*+A*)x +

Multiplicative Noise Model 2

dx/dt=B’(I+I’)x +

Conclusions• El Niño appears to be a damped system forced by

stochastic noise.• There is evidence that the phase locking of El

Niño to the annual cycle is due to the annually-varying statistics of the stochastic noise.

• An optimal initial structure for growth precedes a mature El Niño event by 6 to 9 months.

• The nonnormal dynamics are dominated by 3 nonorthogonal modes.

• The essential linearity of the system allows isolation of the El Niño signal.

• El Niño indices in the equatorial Indian Ocean and the North Tropical Atlantic Ocean are very similar.

• El Niño and the (parabolic) trend dominate the SST variability in the Indian Ocean as well as in the Equatorial and S. Tropical Atlantic Ocean.

• The observed skew in El Niño –La Niña events IS NOT significant compared with realistic null hypothses.

• The trend IS significant compared with realistic null hypotheses.