# Presentation: 258300

## Extratropical Sensitivity to Tropical SST

**Prashant Sardeshmukh, Joe Barsugli, and Sang-Ik Shin**
*Climate Diagnostics Center*

## Math slide 1

**x **= tropical SST,  **y** = Extratropical Response 	 **y   **=**   G x  **+**  **O(x2**)**
- Sensitivity of individual components yi :	 	∂yi/ ∂xj 	=    **G**ij
- 						(**x** = *i* th row of **G** will maximize yi)
- Sensitivity of weighted sum of components *R*  = wTy;	 ∂R/ ∂** **xj	=  **w**T**G**    =    **g**T
- 						( **x **= **g** will maximize *R*)
**GENERAL SENSITIVITY: What patterns of x will produce the largest global r.m.s response ?**
- 	GTG   = V  VT
- 	GGT   = U  UT
- 	yTy = (Gx)TGx = xT   (GTG) x
- 	x = first eigenvector of GTG (=v1) will produce largest response y = Gv1 = 1 u1
**Thus v****1**** is optimal SST forcing pattern; u****1**** is optimal response pattern**

**QUESTION:**
**Can  G be approximated by just a few optimal forcing-response pairs in (1) ?**
**This would be an important simplification.**

**     ****What is the simplest way to think about this problem ?**

**G**  =  **U** ******V**T  =   1 **u**1 **v**1T   +   2 **u**2 **v**2T  + . . . .        (1)

## Math slide 2

- 		             **y        **=**   ****	****G x     **+**   ***O(x**2****)*****    **+ **    **********
- 		*ensemble-mean     **	**linear     **	**	**Noise **	**	**	*
*	**	**   **response   **	**	**signal                 **	**N = 32 for each patch*
- 		**We do an EOF analysis of the ensemble-mean responses.**
- 		        **C**yy 	     ≈       **G** **C**xx **G**T   +    **C**/ *N*     ≈    ****  **GG**T        ( **C**xx  ≈  ****I** )
- 		This approximation is justified if signal/noise >> 1
- 		Signal variance / Noise variance   *N*  = *N*  ( Tmax  A )2
- 		We make this ratio large by choosing relatively large *N*, Tmax and A

**We estimate G through the ensemble-mean CCM3 responses to an array **
**of localized **_**+**_**2°C SST patches (i.e. as a Fuzzy Green’s Function)**** **

**43 patches **
**(27 Indo-Pacific,  16 Atlantic)**
**16 1.5 yr runs for each patch **
**for +ve and -ve Forcing**

- => **EOF analysis of** **y** **is same as SVD analysis of** **G**  ≈  **Y**  =  [ **y**1 **y**2 **y**3 . . . **y**43 ]
- => **EOFs and PCs can be interpreted as optimal forcing-response pairs**  **v** , **u**

## Dominant Response EOFs and their statistical Significance

**Fraction of Response Variance  **
**    ****explained by each EOF**

**Signal Variance / Noise Variance**
**             ****for each EOF **

## DJF: EOFs 1 and 2 of Responses

**DJF: EOFs 1 and 2 of Responses**

_**Top and Middle Panels**_** **
**Combined EOF of  N.H. 500 mb z, **
**land surface temperature, **
**and land precipitation**

_**Bottom Panels: PC**_
**Local values show the projection **
**of the N.H. response to SST anomalies**
**at that location on the EOF.**
**The map as a whole is the**
**optimal SST anomaly pattern **
**for generating the EOF with the **
**largest amplitude**

## MAM: EOFs 1 and 2 of Responses

**MAM: EOFs 1 and 2 of Responses**

**Note**
**large precipitation response **
**over North America**

## JJA: EOFs 1 and 2 of Responses

**JJA: EOFs 1 and 2 of Responses**

**The 500 mb height response is weak in JJA,**
**but there is still a large precipitation response **

## SON:  EOFs 1 and 2 of Responses

**SON:  EOFs 1 and 2 of Responses**

**Both the 500 mb height and precipitation**
**responses are weak in SON,**
**although there is still an appreciable precipitation response**
**over the central and southeast U.S. **

## EOF1 and Sensitivity of North American Precipitation 

- DJF
- 44%

- MAM
- 44%

- JJA
- 49%

- SON
- 32%

**Note! Warm SST = ****RED****    and  Wet Precip = ****BLUE**

## DJF: EOF 1 of Tropical Precipitation  Responses (left) and Noise  (right)

**DJF: EOF 1 of Tropical Precipitation  Responses (left) and Noise  (right)**

**Why is EOF 1 *****so***** dominant ? **
**Why does it explain more than 57% of the responses in DJF ? **
**What is responsible for organizing the responses to *****such an extent***** **
**           ****even for effectively random SST forcing **( **C**xx  ≈  ****I** ) **?**
**One possible explanation: Tropical Precipitation response is highly organized**
**	****	****	****            ****(It tends to be relatively stronger over the warm pool)**

## Math slide 3

-         SST  **x** ---> Tropical Precipitation Response **p**  ---> Extratropical Response **y**
- 		                                    **x             p = G****1****x + e****1**
-     EOF1 of p responses explains 20%. Some of the dominance of EOF 1 of **y** is because of this.
- 		      **p              y = G****2****p + e****2****    =  G****2****G****1****x + (G****2****e****1**** + e****2****)  = Gx + e**
-              This explains the other 57%  20 % = 37% of the dominance of EOF1 of **y**.
** ****Both the relative insensitivity of the Rossby Wave Source associated with the precip response**
**	****	****	****	****	****AND **
**the organization of the response to that  source by the stationary waves are responsible for this.**

**But the dominance of EOF1 of the DJF extratropical responses (57%)**
**is only **_*** ***__***partly***_*** *****explained by the dominance of EOF1 of the tropical **
**precipitation responses (20 %)**

**This explains why:**
**1.****	****EOF 1 of response is more dominant than is the EOF1 of noise**
**2.****	****EOF 1 of response always explains more than 20% of responses**
**3.****	****EOF 1 of responses is most dominant in DJF  (57%)**

## Sensitivity of Local Tropical Precipitation Response	(as a function of background SST)

**Sensitivity of Local Tropical Precipitation Response****	****(as a function of background SST)**

**For +2° C anomalies                For - 2° C anomalies**

**What about Nonlinearity ? How Important is it ?**

##        EOF 1  of DJF Responses to warm and cold SST anomalies

**       ****EOF 1  of DJF Responses to warm and cold SST anomalies**

**The patterns of the optimal DJF **
**SST forcing and response fields **
**are slightly different for warm and cold SST forcing.**
**There is also a slightly weaker **
**response to cold forcing.**

## Dominant SST sensitivity patterns are VERY DIFFERENT  from            the dominant pattern of observed SST variability 

**Dominant SST sensitivity patterns are VERY DIFFERENT  from ****           the dominant pattern of *****observed***** SST variability**** **

**DJF   57%**

**MAM   30%**

**JJA   40%**

**SON   41%**

**EOF 1 of Observed SST   32%**

**There is much greater sensitivity to **
**SST changes in the Central and **
**West Pacific and Indian oceans.**
**The sensitivity is generally *****opposite***** **
**to SST changes in the Indian and **
**West  Pacific oceans .  **

## Summary

**Extratropical sensitivity to tropical SST changes can be characterized by just **
**	****2 or 3 optimal forcing-response pairs in each season.**
**The dominance of these pairs results partly from the relatively large sensitivity of the local precipitation response in regions of warmer background SST such as the warm pool, and partly from a “modal” dynamical extratropical response to tropical diabatic forcing  (Ting and Sardeshmukh 1993). **
**The optimal SST forcing patterns are VERY DIFFERENT from the ENSO pattern. The only area of significant overlap is Nino-4.**
**The extratropical climate generally shows opposite sensitivities to  SST changes in the western and eastern halves of the warm pool.**
**	**
**5.****	****These results have important implications for understanding and modeling global climate changes from monthly to millenial time scales.**
**	**

## An Array of localized SST patches

**An Array of localized SST patches**

## The SST seasonal cycle 

**The SST seasonal cycle**** **

## DJF: Local Variance Explained by Response EOFs 1 and 2