Irregularity and decadal variation in ENSO: a simplified model based on Principal Oscillation Patterns

Abstract

A new method of estimating the decay time, mean period and forcing statistics of El Niño-Southern Oscillation (ENSO) has been found. It uses a two-dimensional stochastically forced damped linear oscillator model with the model parameters estimated from a Principal Oscillation Pattern (POP) analysis and associated observed power spectra. It makes use of extended observational time series of 150 years of sea surface temperature (SST) and sea level pressure (SLP) as well as climate model output. This approach is motivated by clear physical relationships that SST and SLP POP patterns have to the ENSO cycle, as well as to each other, indicating that they represent actual physical modes of the climate system. Moreover, the leading POP mode accounts for 20–50 % of the variance on interannual time scales. The POP real part is highly correlated with several Niño indices near zero lag while the imaginary part exhibits a 6–9 month lead time and thus is a precursor. The observed POP power spectra show markedly different behavior for the peak and precursor, the former having more power at ENSO frequencies and the latter dominating at low frequencies. The results realistically suggest a period of oscillation of 4–6 years and a decay time of 8 months, which corresponds to the practical ENSO prediction limit. A fundamental finding of this approach is that the difference between the observed peak and precursor spectra at low frequencies can be related to the forcing statistics using the simple model, as well as to the difference between patterns of decadal and interannual variability in the Pacific.

Appendices

Appendix

1 Additional data sets

In addition to the data sets presented in this study the analyses described above were applied to the following data sets as well. The extensively documented Hadley Centre global SLP data version 2 (HadSLP) is available at 5° resolution from January 1850 to December 2004 (Allan and Ansell 2006). NOAA reconstructed sea surface temperature data version 3b (ERSST) available on a 2° grid from January 1854 to February 2011 (Smith et al. 2008). For comparison the Hadley Centre global 5° SST anomaly data version 3 (HadSST) (Kennedy et al. 2011a, b), available from January 1850 to December 2007, and the NOAA extended reconstructed 2° SLP (ERSLP) data set (Smith and Reynolds 2004) available from January 1854 to December 1997 were also used.

Results obtained using these data sets are consistent with those from HadISST and 20CrSLP. The POP patterns for the first pair are almost identical with only small differences. The time series for the peak phase are highly correlated between the data sets and with Niño3. Correlations for the precursors are lower and show higher variability (Fig. 4). The POP power spectra for different data sets and variables have similar characteristics (Fig. 5) and the fitted parameter values agree well (Table 1).

2 Theoretical spectrum for 2 modes

The theoretical spectrum for a 2 mode model is computed to analyze how the inclusion of more than one mode in the theoretical model influences the power spectra for the first mode. Analogous to the damped harmonic oscillator with stochastic forcing described in Sect. 4 we can compute the spectrum for a 2 mode oscillation:

$$\begin{aligned} d\left[ \begin{array}{l} x_1\\ y_1\\ x_2\\ y_2 \end{array}\right] = {\mathbf {A}}\left[ \begin{array}{l} x_1\\ y_1\\ x_2\\ y_2 \end{array}\right] dt + {\mathbf {F}}dW, \end{aligned}$$

(9)

where

$$\begin{aligned} {\mathbf {A}}= \left[ \begin{array}{llll} \varepsilon _1&{}- \eta _1&{}0&{}0\\ \eta _2&{} \varepsilon _2&{}0&{}0\\ 0&{}0&{} \varepsilon _2&{}- \eta _2\\ 0&{}0&{} \eta _2&{} \varepsilon _2 \end{array}\right] \end{aligned}$$

(10)

with $\mathbf {F}$ the (4D) forcing matrix and $W$ the (vector) Wiener stochastic process.

The stochastic forcing in Eq. (9) has an associated covariance matrix $\mathbf {R}$. This contains as parameters the variance of the forcing on the four phases as well as the correlation between them. It can be written as

$$\begin{aligned} {\mathbf {R}}= {\mathbf {F}}{\mathbf {F}}^T = c _1 \left[\begin{array}{llll} 1&r _{12}\sqrt{ \alpha _2}&r _{13}\sqrt{ \alpha _3}&r _{14}\sqrt{ \alpha _4}\\ r _{12}\sqrt{ \alpha _2}&\alpha _2&r _{23}\sqrt{ \alpha _2 \alpha _3}&r _{24}\sqrt{ \alpha _2 \alpha _4}\\ r _{13}\sqrt{ \alpha _3}&r _{23}\sqrt{ \alpha _2 \alpha _3}&\alpha _3&r _{34}\sqrt{ \alpha _3 \alpha _4}\\ r _{14}\sqrt{ \alpha _4}&r _{24}\sqrt{ \alpha _2 \alpha _4}&r _{34}\sqrt{ \alpha _3 \alpha _4}&\alpha _4 \end{array}\right] \end{aligned}$$

where $\alpha _i = c _i/ c _1$ is the ratio of the variance of phase $i$ and phase 1.

For a general phase vector $v = \left[ A,B,C,D\right]$ the spectrum is:

$$\begin{aligned} S _v = v S (\omega )v^* = \frac{ c _1}{8\pi }\left[ S _v^{1}(\omega ) + S _v^2(\omega ) + S _v^{12}(\omega ) \right] \end{aligned}$$

(11)

with the first 2 terms equal to the spectra for each mode

$$\begin{aligned}&S _v^1(\omega ) = (1+ \alpha _2)(A^2+B^2) \left[ S ^1_{+}(\omega )+ S ^1_{-}(\omega )\right. \nonumber \\&\quad \left. +\, 2f_{12}\cos \left( \gamma _{\mathrm {AB}}+\arccos ( \varepsilon _1 S ^1_{-} (\omega ))-\arccos ( \varepsilon _1 S ^1_{+}(\omega ))\right) \sqrt{ S ^1_{+}(\omega ) S ^1_{-}(\omega )} \right] \end{aligned}$$

(12)

and similarly for the second mode spectrum $S _v^2(\omega )$. The third term in Eq. (11) is the spectral contribution due to the correlation between the mode forcings. This can be written as

$$\begin{aligned} S _v^{12}(\omega )&= 2\sqrt{(A^2+B^2)(C^2+D^2)} \left[ f_{+} \left( X^{12}_{+} + X^{12}_{-}\right) \right. \nonumber \\&\quad \left. +\, f_{-} \left( Y^{12}_{+} +Y^{12}_{-} \right) \right] \end{aligned}$$

(13)

with

$$\begin{aligned} X^{12}_{+}&= \cos (\phi _{+}^{12})\sqrt{ S ^1_{-}(\omega ) S ^2_{-}(\omega )}\end{aligned}$$

(14)

$$\begin{aligned} X^{12}_{-}&= \cos (\phi _{-}^{12})\sqrt{ S ^1_{+}(\omega ) S ^2_{+}(\omega )}\end{aligned}$$

(15)

$$\begin{aligned} Y^{12}_{+}&= \cos (\psi _{+}^{12})\sqrt{ S ^1_{-}(\omega ) S ^2_{+}(\omega )} \end{aligned}$$

(16)

$$\begin{aligned} Y^{12}_{-}&= \cos (\psi _{-}^{12})\sqrt{ S ^1_{+}(\omega ) S ^2_{-}(\omega )}\end{aligned}$$

(17)

$$\begin{aligned} f_{\pm }&= \sqrt{\left( r _{13}\sqrt{ \alpha _3}\pm r _{24}\sqrt{ \alpha _2 \alpha _4}\right) ^2 +\left( r _{23}\sqrt{ \alpha _2 \alpha _3}\mp r _{14}\sqrt{ \alpha _4}\right) ^2} \end{aligned}$$

(18)

and

$$\begin{aligned} \phi _{+}^{12}&= \arctan \left( \frac{BC-AD}{AC+BD}\right) \nonumber \\&\quad + \arccos \left( \frac{ r _{13}\sqrt{ \alpha _3}+ r _{24}\sqrt{ \alpha _2 \alpha _4}}{f_{+}}\right) - \arccos ( \varepsilon _1 S ^1_{-}(\omega )) + \arccos ( \varepsilon _2 S ^2_{-}(\omega ))\nonumber \\ \phi _{-}^{12}&= -\arctan \left( \frac{BC-AD}{AC+BD}\right) \nonumber \\&\quad + \arccos \left( \frac{ r _{13}\sqrt{ \alpha _3}+ r _{24}\sqrt{ \alpha _2 \alpha _4}}{f_{+}}\right) - \arccos ( \varepsilon _1 S ^1_{+}(\omega )) + \arccos ( \varepsilon _2 S ^2_{+}(\omega ))\nonumber \\ \psi _{+}^{12}&= \arctan \left( \frac{BC+AD}{AC-BD}\right) \nonumber \\&\quad + \arccos \left( \frac{ r _{13}\sqrt{ \alpha _3}- r _{24}\sqrt{ \alpha _2 \alpha _4}}{f_{-}}\right) - \arccos ( \varepsilon _1 S ^1_{-}(\omega )) + \arccos ( \varepsilon _2 S ^2_{+}(\omega ))\nonumber \\ \psi _{-}^{12}&= -\arctan \left( \frac{BC+AD}{AC-BD}\right) \nonumber \\&\quad + \arccos \left( \frac{ r _{13}\sqrt{ \alpha _3}- r _{24}\sqrt{ \alpha _2 \alpha _4}}{f_{-}}\right) - \arccos ( \varepsilon _1 S ^1_{+}(\omega )) + \arccos ( \varepsilon _2 S ^2_{-}(\omega )). \end{aligned}$$

3 Forcing for stochastic model

To generate time series using the theoretical model and the estimated model parameters, it is necessary to get an expression for the stochastic forcing matrix. The covariance matrix $\mathbf {R}$ of the stochastic forcing is directly given using the estimated parameters $c , r$ and $\alpha$. The forcing $\mathbf {F}$ can be obtained from the covariance matrix by computing its eigendecomposition:

$$\begin{aligned} \mathbf {R}= \mathbf {L}\mathbf {D}\mathbf {L}^{-1} \end{aligned}$$

As $\mathbf {R}$ is a real, positive definite, symmetric matrix $\mathbf {L}$ is hermitian and satisfies $\mathbf {L}^{-1}=\mathbf {L}^T$ and all the eigenvalues in $\mathbf {D}$ are positive. With $\mathbf {F}=\mathbf {L}\sqrt{\mathbf {D}}$ we define a forcing with the correct covariance matrix since:

$$\begin{aligned} \mathbf {F}dB_t \left( \mathbf {F}dB_t\right) ^T = \mathbf {F}dB_t \left( dB_t\right) ^T \mathbf {F}^T= \mathbf {F}\mathbf {F}^T = \mathbf {L}\mathbf {D}\mathbf {L}^{-1} = \mathbf {R}, \end{aligned}$$

where $W = \left[ W^1,W^2\right] ^T$ and the $W^i, i=\left[ 1,2\right]$ are uncorrelated Wiener processes.

4 Peak and precursor phase recovery

We denote the modeled Niño3 time series by $z _r$ and the modeled precursor phase by $z _i$. The phase angles for these time series are $\phi$ and $\psi = \phi +\frac{\pi }{2}$ respectively. These phases are related to the theoretical model time series $x$ and $y$ in Eq. 1 through linear combinations with coefficients $\cos \phi$ and $\sin \phi$ for the mature phase and similarly with phase angle $\psi$ for the precursor. This means that we can write $z _r$ and the modeled $z ^{\mathrm {slp}}_i$ or precursor phase time series (denoted by $z _i$) as the following linear combination of $x$ and $y$:

$$\begin{aligned} \left[\begin{array}{l} z _r\\ z _i\end{array}\right] = \left[\begin{array}{l} \cos \left( \phi \right) y+\sin \left( \phi \right) x\\\cos \left( \psi \right) y+\sin \left( \psi \right) x \end{array}\right] = \left[\begin{array}{ll} \sin \left( \phi \right)&\cos \left( \phi \right) \\\sin \left( \psi \right)&\cos \left( \psi \right) \end{array} \right] \left[\begin{array}{l} x\\y \end{array}\right]. \end{aligned}$$

Defining

$$\begin{aligned} {\mathbf {Q}}= \left[\begin{array}{ll} \sin \left( \phi \right)&\cos \left( \phi \right) \\\sin \left( \psi \right)&\cos \left( \psi \right) \end{array}\right] \end{aligned}$$

this is the linear transformation from the $\left( x,y\right)$ basis to the $\left( z _r, z _i\right)$ basis.

The same linear transformation also transforms the forcing matrix $\mathbf {F}$. The forcing covariance matrix in $\left( z _r, z _i\right)$ coordinates is given by ${\tilde{\mathbf {R}}} = {\mathbf {Q}}{\mathbf {R}}{\mathbf {Q}}^T$ where the diagonal elements of $\tilde{\mathbf {R}}$ now give the forcing variances onto $z _r$ and $z _i$ and the off-diagonals represent the correlation between the forcings onto $z _r$ and $z _i$. In the case of Niño3 and 20CrSLP results the phase angles are $\phi =-0.18$ and $\psi =1.39$ and the transformation is

$$\begin{aligned} {\mathbf {Q}}= \left[\begin{array}{ll} -0.18&0.98 \\ 0.98&0.18 \end{array}\right]. \end{aligned}$$

The covariance matrix in $\left( z _r, z _i\right)$ coordinates is given by

$$\begin{aligned} {\tilde{\mathbf {R}}} = 1.5 \left[\begin{array}{ll} 1&-0.31 \\ -0.31&0.27 \end{array}\right], \end{aligned}$$

where now the $\left( 1,1\right)$ entry gives the variance of the forcing onto the peak phase $z _r$, and the $\left( 2,2\right)$ entry gives the ratio of the forcing variance of the precursor $z _i$ to the peak. For all considered data sets and both variables this ratio is always less than 1, indicating that the forcing variance on the precursor is smaller than on the peak phase.