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An Introduction to El Ni ño

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An Introduction to El Niño

With thanks to

Prashant SardeshmukhLudmila Matrosova

Ping ChangMoritz Flügel

Brian EwaldRoger Temam

NOAA/ESRL/Physical Sciences Division

Cécile Penland

- Some phenomenology: What is El Niño
- The Annual Cycle
- Deviations from it, especially El Niño

- An empirical-dynamical model
- Justification and uses

- A look at the predictions?

Size of the annual cycle (oC). Typical size of El Niño ~ 1-3 oC

Dynamical Equation for Sea Surface Temperature:

Tt =-u Tx -v Ty -w Tz+ F (solar forcing, wind,

(advection by currents) evaporation, etc.)

Let T = To + Tann + T’(t).

Similarly, decompose (u, v, w) and wind terms.

Since To is so much bigger than either TannorT’, we linearize around it.

Question: Can we treat the nonlinear terms in the equation for T’ as cyclostationary white noise?

Linear Inverse Modeling

Assume linear dynamics (dropping the primes):

dT/dt = BT + x, with< x (t+t) xT (t)>= Q(t)d(t)

For now, we’ll assume additive noise, although that assumption is most likely false. Q(t) is periodic.

Corresponding FPE:

Further simplification: let Q be constant. From the FPE.

p(T,t +t|To,t) is Gaussian, centered on G(t) To

where G(t) = exp(Bt) = <T(t+t)TT(t)>< T(t)TT(t)>-1 .

The covariance matrix of the prediction errors:

S (t,t)=<T(t+t)TT(t +t)> - G(t) < T(t)TT(t)> GT(t) .

Further, even if Q is not constant,

< T(t)TT(t)> = B< T(t)TT(t)> + < T(t)TT(t)> BT + Q(t)

Brief digression: Empirical Orthogonal Functions (EOFs).

Let C <TTT>

CE =EL, where L is the diagonal matrix of eigenvalues.

Karhunen-Loève expansion:

T(t) = Ec(t) , where <ccT> = L.

cais called the “ath Principal Component of T.”

If the data are noisy, the higher-ordered eigenvectors are essentially sampling errors, and the data can be “cleaned up” by projecting onto a leading subset of eigenvectors, or “EOFs,” {Ea }. The sum of {la} divided by the sum of all eigenvalues l is called the “fraction of explained variance.”

SST Data used:

- COADS (1950-2000) SSTs in 30E-70W, 30N – 30S, or in the global tropical strip, consolidated onto a 4x10-degree grid.
- Subjected to 3-month running mean.
- Projected onto 20 EOFs (eigenvectors of <TTT>)explaining about two-thirds of the variance.
- T, then, represents the vector of SST anomalies, each component representing a location, or else it represents the vector of Principal Components.

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

Eigenvectors of GT(t) are the “adjoints” {vi},

(Recall: G(t) = <T(t+t)TT(t)>< T(t)TT(t)>-1)

and uvT = uTv = 1.

Most probable prediction: T(t+t) = G(t) To(t)

The neat thing: G(t) ={G(to) } t/ to .

If LIM’s assumptions are valid, the prediction error e = T(t+t) - G(t) Todoes 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. What is plotted: Tr(e) vs lead.

The annual cycle:

dT/dt = BT + x< x (t+t) xT (t)>= Q(t)d(t)

Given stationary B use the fluctuation-dissipation relation to diagnose Q(t):

< T(t)TT(t)> = B< T(t)TT(t)> + < T(t)TT(t)> BT + Q(t).

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

Phase locking?

Note: SST difference map between El Niño years and neutral years turns out to be the leading EOF in the data.

EOF 1

# |pattern correlation| > 0.5

(~600 months of COADS data)

A model generated with the stationary Band the stochastic forcing with cyclic statistics Q(t) does reproduce the correct phase-locking of El Niño/ La Niña with the annual cycle.

(Note: Generate model using Kloeden and Platen (1992) )

# |pattern correlation| > 0.5

(~24 000 months of model data)

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?

- Again, we run the numerical model.
- 80 realizations, each having 500 samples
- It is run in EOF space, with 20 degrees of freedom
- Three different versions of the model:

- a) Original model: dT/dt = BT + x
- b) Multiplicative noise: dT/dt=(B*+ A x*)T+x**
- with Aestimated from the one timestep error
- c) Multiplicative noise: dT/dt=(B*+ A x*)T+x**
with A= B*.

With Stratonovich linear multiplicative noise:

G(t) = <T(t+t)TT(t)>< T(t)TT(t)>-1 = exp {(B* +1/2A2) t}

and we have a revised FDR:

< T(t)TT(t)> = A< T(t)TT(t)> AT + Q**(t) +…

…+(B* +1/2A2)< T(t)TT(t)> + < T(t)TT(t)> (B* +1/2A2)T

We examine the sample skew and trend of the first PC.

Additive Noise Model

dT/dt = BT + x

Multiplicative Noise Model 1

dT/dt=(B*+Ax*)T+x**

Multiplicative Noise Model 2

dT/dt=B’(I+Ix’)T+x’’

Optimal initial structure for growth over lead time t:

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

Growth factor over lead time t:

Eigenvalue g of GTG(t).

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

a)

b)

Temp. Anom. in Niño 3.4 region (6oN-6oS, 170oW-120oW): dT3.4

This time series virtually identical with PC 1.

c)

Does it? Judge for yourself! 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.

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!

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

- Stochastic forcing:
S (t,t)=<T(t+t)TT(t +t)> - G(t) < T(t)TT(t)> GT(t)

- Uncertain initial conditions:
<dT(t+t)dTT(t +t)>i.c. = G(t) <dT(t)dTT(t)> GT(t)

- Sampling errors when estimating G(t) :
<dT(t+t)dTT(t +t)>Samp= <dG(t) T(t)TT(t)dGT(t) >

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.
- The observed skew in El Niño –La Niña events IS NOT significant compared with realistic null hypotheses.
- The trend IS significant compared with realistic null hypotheses

- An optimal initial structure for growth precedes a mature El Niño event by 6 to 9 months.
- Predictions are better in cold events than warm events.
- Initial condition errors project onto the optimal structure so that nonnormal dynamics amplify them in the forecast.
- A lot of the “noise” might be predicted with higher temporal resolution.
- In progress: weekly forecasts!

Be patient.