# Presentation: 438960

## An Introduction to El Niño

*C**écile Penland*

- With thanks to
- 	Prashant Sardeshmukh	Ludmila Matrosova
- 	Ping Chang			Moritz Flügel
- 	Brian Ewald			Roger Temam
- 	NOAA/ESRL/Physical Sciences Division

## Outline

- 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?

## Slide 3

**Notes:**
The annual cycle of sea surface temperatures: DJF

## Slide 4

**Notes:**
The annual cycle of sea surface temperatures: MAM

## Slide 5

**Notes:**
The annual cycle of sea surface temperatures: JJA

## Slide 6

**Notes:**
The annual cycle of sea surface temperatures: SON

## Slide 7

**Notes:**
The annual cycle of sea surface temperatures: DJF

## Slide 8

**Notes:**
The annual cycle of precip rate: DJF.  Note the similarity to the SST pattern.

## Slide 9

**Notes:**
The annual cycle of precip rate: MAM

## Slide 10

**Notes:**
The annual cycle of precip rate: JJA

## Slide 11

**Notes:**
The annual cycle of precip rate: SON

## Slide 12

**Notes:**
The annual cycle of precip rate: DJF.  Note the band of precip that stretches just north of the equator in the Pacific.

## Slide 13

**Notes:**
The annual cycle of surface wind: DJF.  Note the zone of convergence and compare with the map of precip rate.

## Slide 14

**Notes:**
The annual cycle of survace winds: MAM

## Slide 15

**Notes:**
The annual cycle of survace winds: JJA

## Slide 16

**Notes:**
The annual cycle of survace winds: SON

## Slide 17

**Notes:**
The annual cycle of survace winds: DJF

## Slide 18

## Slide 19

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

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

## Dynamical Equation for Sea Surface Temperature:

*T**t ***_**__*u T*__*x *__**__*v T*__*y *__**__*w T*__*z*_* **	**	*_*F*_ (solar forcing, wind,
*    *(advection by currents) 		      evaporation, etc.)
- Let *T = T**o **T**ann **T’*(*t*).
- Similarly, decompose (*u, v, w*) and wind terms.
- Since *T**o *is so much bigger than either *T**ann* *or* *T’*, we linearize around it.
- Question: Can we treat the nonlinear terms in the equation for *T’ *as cyclostationary white noise?

**Notes:**
Note that the two-parameter linearization gives us a CONSTANT coefficient of T’.

## Linear Inverse Modeling

*Assume* linear dynamics (dropping the primes):
- 	*d****T****/dt *= **B*****T*** + ********with********(*t*)*********T ***(*t*)*** ************= ***Q**(*t*)************
- For now, we’ll assume additive noise, although that assumption is most likely false**Q**(*t*) is periodic.
- Corresponding FPE:

**Notes:**
The Fokker-Planck equation.

##       (t,) = <T(t+)TT(t +) > G() < T(t)TT(t) > GT () .   

- 	*p*(***T***,*t ************t*) is Gaussian, centered on **G**(**) ********
- where **G**(**) = exp(**B****) = <***T***(*t*+**)***T***T(*t*) >< ***T***(*t*)***T***T(*t*) >-1 .
- The covariance matrix of the prediction errors:
-       ****(*t*,**)** **=** **<***T***(*t*+**)***T***T(*t *+**) > **G**(**) < ***T***(*t*)***T***T(*t*) > **G**T** **(**) .
- Further, even if **Q** is *not* constant,

- < ***T***(*t*)***T***T(*t*) > = **B**** **< ***T***(*t*)***T***T(*t*) > + < ***T***(*t*)***T***T(*t*) > **B**T + **Q**(*t*)

## Let C  <TTT> 

- Let **C ******** **<***TT***T>
**	****CE **=**E******, where **** is the diagonal matrix of eigenvalues.
- Karhunen-Loève expansion:
- 	***T***(*t*) = **E*****c***(*t*) , where <***cc***T> = ****.
*c***  is called the “**th 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,” {*E***}.  The sum of {****} divided by the sum of *all* eigenvalues ** 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 <***TT***T>)*** ***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.

## Slide 26

## Slide 27

## The neat thing: G() ={G() } .

- Eigenvectors of **G**(**) are the ***“normal” modes ***{**u***i*}.
- Eigenvectors of **G**T(**) are the ***“adjoints” ***{**v***i*},
-          (Recall: **G**(**) = <***T***(*t*+**)***T***T(*t*) >< ***T***(*t*)***T***T(*t*) >-1)
- and **uv**T = **u**T**v **= ***1***.
- Most probable prediction: ***T***(*t*+**) = **G**(**) ***T******** ***(*t*)
- The neat thing: **G**(**) ={**G**(****) } ****.

**Notes:**
Use the Cayley-Hamilton Theorem to raise the Green function matrix to a power.

## Below, different colors correspond to different lags used to identify the parameters. What is plotted: Tr() vs lead.

- Below, different colors correspond to different lags used to identify the parameters. What is plotted: *Tr*() *vs* lead.

**Notes:**
Passing the tau-test not verifies only linearity, but also verifies that B is essentially a constant.

## 	dT/dt = BT + (t)T (t) = Q(t)

*	**d****T****/dt *= **B*****T*** + ******(*t*)*********T ***(*t*)*** ************= ***Q**(*t*)************
- Given stationary **B** use the fluctuation-dissipation relation to diagnose **Q**(*t*):
-      < ***T***(*t*)***T***T(*t*) > = **B**** **< ***T***(*t*)***T***T(*t*) > + < ***T***(*t*)***T***T(*t*) > **B**T + **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.

- 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 B and 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: *d****T****/dt *= **B*****T*** + ******
******b) Multiplicative noise: *d****T****/dt*=(**B***** **+ **A ******** *********)***T***+******
- 			with **A**** **estimated from the one timestep error
- 		c) Multiplicative noise: *d****T****/dt*=(**B***** **+ **A ******** *********)***T***+******
- 			with **A**** **= **B***.

## With Stratonovich linear multiplicative noise:

**G**(**) = <***T***(*t*+**)***T***T(*t*) >< ***T***(*t*)***T***T(*t*) >-1 = exp (**B* **+1/2** ****A**2) **
- 	and we have a revised FDR:
-       < ***T***(*t*)***T***T(*t*) > = **A**** **< ***T***(*t*)***T***T(*t*) > **A**T + **Q****(*t*) +...
-  ...+** **(**B* **+1/2** ****A**2)< ***T***(*t*)***T***T(*t*) > + < ***T***(*t*)***T***T(*t*) > (**B* **+1/2**A**2)T
- We examine the sample skew and trend of the first PC.

## Additive Noise Model 

*Additive Noise Model*
*d****T****/dt *= **B*****T*** + ******

*Multiplicative Noise Model 1*
*d****T****/dt*=(**B***+**A*********)***T***+******

*Multiplicative Noise Model 2*
*d****T****/dt*=**B**’(**I**+**I********’)***T***+*********’’***

**Notes:**
The vertical line indicates the sample skew and sample trend as estimated from data.  Note that
The skew is NOT significant against a realistic null hypothesis.

## Slide 38

## Optimal initial structure for growth over lead time : 

-  	Right singular vector of **G**(**) (eigenvector of **G**T**G**(**))
- Growth factor over lead time **:
- 	Eigenvalue of **G**T**G**(**).

## Temp. Anom. in Niño 3.4 region (6oN-6oS, 170oW-120oW): T3.4

- Temp. Anom. in Niño 3.4 region (6oN-6oS, 170oW-120oW): ***T**3.4*
- This time series virtually identical with *PC 1*.

- 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)*

## 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.

*c)*

## Scatter plot of the El Niño index and pattern correlations shown in (c).

- 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.

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

## Slide 44

## Several sources of expected error and uncertainty:

-  Stochastic forcing:
****(*t*,**)** **=** **<***T***(*t*+**)***T***T(*t *+**) > **G**(**) < ***T***(*t*)***T***T(*t*) > **G**T** **(**)
- Uncertain initial conditions:
- 	   <*****T***(*t*+**)*****T***T(*t *+**) >*i.c.* = **G**(**) <*****T***(*t*)*****T***T(*t*) > **G**T** **(**)
- Sampling errors when estimating **G**(**) :
- 	    <*****T***(*t*+**)*****T***T(*t *+**) >*Samp**= <*****G**(**) ***T***(*t*)***T***T(*t*) ****G**T** **(**) >

## Slide 46

**Notes:**
Top: Niño 3.4 SSTA (heavy solid line), 6-month prediction (light solid line), 9 month prediction (dashed line)

Below: contributions to uncertainty from stochastic forcing (solid line), sampling errors of the Green function (dashed line),
and uncertain initial conditions(dotted line).

## Slide 47

## Slide 48

**Notes:**
It would be nice to do weekly forecasts....

## Slide 49

## 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

## Conclusions (cont.)

- 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.

**Notes:**
Truth comes out eventually.  That’s why we’re scientists. :-)