# El Niño, the Trend, and SST Variability

El Nio, the Trend, and SST Variability or Isolating El Nio Ccile Penland and Ludmila Matrosova NOAA-CIRES/Climate Diagnostics Center Review of Linear Inverse Modeling Assume linear dynamics: dx/dt = Bx + Diagnose Green function from data: G() = exp(B) = < x(t)xT>-1 . Eigenvectors of G() are the normal modes {ui}. Most probable prediction: x(t+) = G() x(t) Optimal initial structure for growth over lead time : Right singular vector of G() (eigenvector of GTG() ) Growth factor over lead time : Eigenvalue of GTG(). SST Data used: COADS (1950-2000) SSTs in the tropical strip 30N 30S. Subjected to 3-month running mean. Projected onto 20 EOFs (eigenvectors of ) 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. This is what we call unfiltered data. This optimal initial pattern evolves into this one 6 to 9 months later. T3.4(t) Cor. = 0.65 Pat. Cor. (SST,O.S.)(t 8mo) Projection of adjoints onto O.S. and modal timescales. 1.5 decay time T = Period mo mo

mo 1 mo mo mo 0.5 Decay mode, = 31 months 0 0 5 10 15 Mode number 20 25 EOF 1 of Residual u1 of unfiltered data 10 5 0 -5 -10 -15 1950 1960 1970 1980 Date 1990 2000

2010 The pattern correlation between the longestlived mode of the unfiltered data and the leading EOF of the residual data is 0.81. Location of indices: N3.4, IND, NTA, EA, and STA. 3 2 1 0 El Nio -1 -2 -3 1950 1960 1970 3 Nio 3.4 Time Series Date 1980 1990 2000 2 El Nio + Trend 1 0 -1 -2 -3 1950 1960

1970 Date 1980 1990 2000 2 1 0 Background -1 -2 1950 1960 1970 Date 1980 1990 2000 Red: Spectrum of unfiltered Nio 3.4 SSTA Blue:1 Spectrum of residual Nio 3.4 SSTA 10 100 10 -1 10-2 10-3 10-4 1 10

100 Period (months) 1000 Spectral difference: (Spectrum of unfiltered data spectrum of residual) / Spectrum of residual. 39.9 mo 18.1 mo 66.4 mo 25 20 15.3 mo 15 10 5 0 -5 1 10 100 Period (months) 1000 Weekly SST data with its own climatology removed, then projected onto COADS EOFs. 16.5 mo 35 30 25 20 15 10 5 0 -5 0 10 5.2 mo 101 43.9 mo 102

103 Period (weeks) 104 Projection of adjoints onto O.S. and modal timescales. 1.5 decay time T = Period mo mo mo 1 mo mo mo 0.5 0 Trend mode = 31mo 0 5 10 15 Mode number 20 25 1 R = 0.36 R(Unfiltered, El Nino) = 0.36

1.5 R = 0.45 R(Unfiltered, El Nino) = 0.45 EA STA 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 1950 1960 1970 1980 Date 1990 2000 -1.5 1950 R = 0.44 1.5 1 1 0.5 0.5 0 0

-0.5 -0.5 -1 -1 1960 1970 1980 Date 1990 2000 1990 2000 R = 0.61 R(Unfiltered, El Nino) = 0.44 -1.5 1950 1970 1980 Date 1990 2000 R(Unfiltered, El Nino) = 0.61 -1.5 1950 NTA IND 1.5 1960 1960

-0.1 -0.2 -100 -50 -50 0 Lead (months) 50 100 -0.2 -100 -50 0 Lead (months) 50 100 Lagged correlation between El Nio indices and PC 1. 0 -0.5 -1 1950 1960 1970 1980 1990 R(Unfiltered, El Nino+Trend) = 0.79 1 0.5 0 -0.5 -1

-1.5 1950 1960 1970 1980 Date 1990 2000 R = 0.77 1.5 R(Unfiltered, El Nino+Trend) = 0.77 1 0.5 0 -0.5 -1 -1.5 1950 2000 R = 0.79 1.5 EA SSTA (C) 0.5 NTA SSTA (C) STA SSTA (C) IND SSTA (C) 1 RR(Unfiltered, = 0.75 El Nino +Trend) = 0.75 1960 1970 1980

Date 1990 2000 R = 0.62 R(Unfiltered, El Nino + Trend) = 0.62 1.5 1 0.5 0 -0.5 -1 -1.5 1950 1960 1970 1980 1990 2000 Indices. Black: Unfiltered data. Green: El Nio signal + Trend. This optimal initial condition evolves into this one 6 to 9 months later. T3.4 (t) Cor. = 0.65 Pat. Cor. (SST,O.S.)(t-8mo) MA Curve Eigenvalue of GTG() and expected error. 4 3.5 Black: Unfiltered 3 2.5

2 Red: El Nio 1.5 1 Green: El Nio + Trend 0.5 0 0 5 10 15 Lead (months) 20 10 15 Lead (months) 20 25 Blue: El Nio + Parabolic Trend 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0 5 25 Lagged correlation C(): O.S., Nio 3.4 Nio3.4 (Observed Nio3.4 (Expected Error Variance)

Error Variance) Nio 3.4 (AR1 Error Variance) 1.5 1 0.5 0 0 5 10 15 20 Lead (months) 25 0 5 10 15 20 Lead (months) 25 1.5 1 0.5 0 1.5 1 0.5 0 0 5 10 15 20 Lead (months) 25

Error variance normalized to climatology IND (AR1 Error Variance) IND (Expected Error Variance) 1 1 0.5 0.5 0 0 5 10 15 20 Lead (months) 25 0 1.5 1.5 1 1 0.5 0.5 0 0 5 10 15 20

Lead (months) 25 0 1.5 1 1 0.5 0.5 0 0 5 10 15 20 Lead (months) 25 0 5 10 15 20 Lead (months) 25 0 5 10 15 20 Lead (months) 25 0

5 10 15 20 Lead (months) 25 Error variance normalized to climatology NTA (Observed Error Variance) 1.5 0 NTA (Expected Error Variance) IND (Observed Error Variance) 1.5 NTA (AR1 Error Variance) 1.5 EA (AR1 Error Variance) EA (Expected Error Variance) 1 1 0.5 0.5 0 0 5 10

15 Lead (months) 20 25 0 1.5 1.5 1 1 0.5 0.5 0 0 5 10 15 20 Lead (months) 25 0 1.5 1 1 0.5 0.5 0 0

5 10 15 20 Lead (months) 25 0 5 0 0 10 15 20 25 5 10 15 20 Lead (months) 25 5 10 15 20 Lead (months) 25 Lead (months) Error variance normalized to climatology STA (Observed Error Variance) 1.5

0 STA (Expected Error Variance) EA (Observed Error Variance) 1.5 STA (AR1 Error Variance) 1.5 R = 0.36 R = 0.36 R=0.36 R=0.36 1 1 0.5 0.5 0 0 -0.5 -1 1950 -0.5 1960 1970 1980 Date 1990 2000

-1 1950 R = 0.30 1 0.5 0.5 0 0 -0.5 -0.5 1970 1980 Date 1990 2000 1990 2000 R=0.48 1 1960 1970 1980 Date R = 0.48 R=0.30 -1 1950 1960 1990 2000

-1 1950 1960 1970 1980 Date Black: Unfiltered data. Blue: Background (No Nio, no Trend) R = -0.37; Trend = u1 1 BLUE: NTA No Nio, No Trend 0.5 0 -0.5 -1 1950 1960 1970 1980 Date 1990 2000 R = -0.49; Trend = PC1 of residual 1 RED: STA No Nio, No Trend 0.5 0 -0.5 -1 1950

1960 1970 1980 Date 1990 2000 Conclusions Two different ways of identifying the trend lead to qualitatively similar results. The pattern-based filter can be applied to data of any temporal resolution. The El Nio signals in the tropical Indian and North tropical Atlantic are highly correlated (R = 0.84). El Nio signals in EA and STA precede that in Nio 3.4 by about 8 months. This wont help the predictions, though. Conclusions (cont.) El Nio plus the trend appear to dominate SSTA variability in IND, EA and STA. The trend seems to cause overestimation of nonmodal growth of El Nio. Isolating the signals with this filter seems to be more valuable for diagnosis than prediction, except in IND. The tropical Atlantic dipole is significant in the background SSTA field.

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