1. Introduction
Numerous studies have recognized the important role of tropical sea surface temperature (SST) variations in global climate variability, and progress in seasonal to interannual climate predictions around the globe has followed largely from this recognition (Goddard et al. 2001; Barnston et al. 2005). Anomalies associated with tropical El Niño–Southern Oscillation (ENSO) influence climate anomalies worldwide through atmospheric teleconnections (Ropelewski and Halpert 1986, 1987, 1989; Halpert and Ropelewski 1992; Klein et al. 1999; Alexander et al. 2002; Goddard and Dilley 2005). Improving ENSO predictions has therefore been an important goal of the seasonal forecasting community (see, e.g., Jin et al. 2008; Barnston et al. 2012, 2015; and references therein).
Despite significant progress, however, the ENSO prediction skill of comprehensive current climate models remains comparable to that of vastly simpler empirical models. For example, Newman and Sardeshmukh (2017) showed that the prediction skill of their simple linear inverse model (LIM) was very similar to that of the comprehensive coupled models comprising the North American Multimodel Ensemble (NMME; Kirtman et al. 2014) used operationally for seasonal predictions. Indeed, they interpreted this very similar skill, and the fact that it was also very similar to the potential skill of their LIM, to conclude that one may now be near the intrinsic predictability limit of tropical SSTs. Given its practical implications, such a conclusion obviously warrants closer scrutiny. One way to challenge it would be to provide convincing evidence that the newest versions of the coupled models consistently beat the LIM’s predictions. Another way would be to show that the potential skill estimated using a LIM is too low, possibly due to the LIM’s low-dimensionality and assumption of linearity.
On the other hand, the LIMs could also be improved. LIMs are low-dimensional empirical–dynamical models derived from the observed zero-lag and time-lag covariances of climate anomalies, and have been used extensively in many climate diagnostic and predictability studies (e.g., Penland and Sardeshmukh 1995, hereafter PS95; Penland 1996, hereafter P96; Johnson et al. 2000b; Xue et al. 2000; Kondrashov et al. 2005; Penland and Matrosova 2006; Newman 2007; Alexander et al. 2008; Hawkins and Sutton 2009; Compo and Sardeshmukh 2010; Shin et al. 2010; Newman et al. 2011a,b; Vimont 2012; Vimont et al. 2014; Capotondi and Sardeshmukh 2015; Chen et al. 2016; Huddart et al. 2016; Newman et al. 2016; Newman and Sardeshmukh 2017; and many others). In this paper, we consider one way to improve LIMs by explicitly accounting for the seasonal variation of their parameters. We call such LIMs cyclostationary LIMs (CS-LIMs).
CS-LIMs may be partly motivated by the fact that the variance of tropical SST anomalies in the east-central equatorial Pacific, where ENSO anomalies attain their largest magnitude, has a pronounced annual cycle with a minimum in boreal spring and a maximum in boreal winter, even if the anomalies themselves are defined as departures from the climatological annual cycle (Fig. 1). The winter variance maximum coincides with the timing of peak warm and cold ENSO events. This synchronization is called “ENSO phase locking” and is evident both in observations (e.g., Rasmusson and Carpenter 1982) and in forward (as opposed to inverse) models of varying complexity (Tziperman et al. 1997, 1998; Harrison and Vecchi 1999; Ham and Kug 2014). In the context of the linear inverse models considered here, this raises a basic question as to what extent the annual cycle of the SST anomaly variance arises from the annual cycle of the predictable SST dynamics versus the annual cycle of the stochastic forcing of the SSTs. P96 presented evidence that the annual cycle of the stochastic forcing may be dominant in this regard. Some subsequent studies (Vimont et al. 2003; Chang et al. 2007) also emphasized the role of extratropical atmospheric stochastic forcing, mediated through the Pacific meridional mode (Chiang and Vimont 2004), in ENSO phase-locking. However, other studies, using simple theoretical models with seasonally varying parameters as well as seasonally varying Markov models (Blumenthal 1991; Johnson et al. 2000a,b; Xue et al. 2000; Thompson and Battisti 2000, 2001; Jin et al. 2019; Liu et al. 2019), emphasized the role of seasonal variations in the deterministic system dynamics. None of these studies quantified the relative roles of the seasonal variations of the predictable SST dynamics and unpredictable stochastic forcing in ENSO phase-locking, or in determining the seasonal variation of SST predictability. The CS-LIM presented here is especially well suited for such a diagnosis.
Monthly standard deviations of Niño-3.4 (5°S–5°N; 170°–120°W) averaged SST anomalies during the period 1961–2010 derived from the HadISST dataset (Rayner et al. 2003). The thick black line shows the annual mean.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
Section 2 gives a brief overview of tropical LIM construction and discusses some additional technical aspects that need to be considered when constructing a CS-LIM. Detailed intercomparisons of stationary and cyclostationary LIMs whose anomaly state vector comprise only tropical SST anomalies (“SST-only LIMs”) are provided in section 3. Section 4 investigates the extent to which the relative roles of the annual cycles of the predictable dynamics and unpredictable stochastic forcing inferred from the SST-only CS-LIM are also reproduced in large ensembles of coupled climate model simulations, specifically the preindustrial climate simulations in the archive of phase 5 of the Coupled Model Intercomparison Project (CMIP5; Taylor et al. 2012) and the Large Ensemble simulations performed with the NCAR Community Earth System Model version 1 (CESM-LE; Kay et al. 2015). Section 5 presents an extended coupled SST–SSH CS-LIM whose anomaly state vector comprises both tropical SST and SSH anomalies, and compares its seasonally varying deterministic and probabilistic forecast skill with that of the NMME. A summary and concluding remarks follow in section 6.
2. Linear inverse model
3. SST-only LIMs
We first consider annual-cycle effects in the simplest LIM framework, in which the state vector x comprises only SST anomalies. We call such LIMs “SST-only LIMs.” We constructed this LIM using monthly SST data from the HadISST dataset (Rayner et al. 2003) over the tropics (25°S–25°N) for the period 1961–2010. To determine the monthly anomalies on a 1° latitude × 1° longitude grid, the grand mean and the climatological annual cycle were removed at each grid point. An EOF analysis of the resulting anomalous SSTs was then performed, and the dominant 15 EOFs explaining about 80.4% of total SST anomaly variance over the domain were retained. All subsequent calculations were performed in this 15-dimensional EOF space (i.e., by defining the state vector x as a 15-component vector representing the time-varying amplitudes of the 15 EOF patterns), and then transformed to geographical space for presentation and interpretation.
a. Stability of the linear system
In the phase-averaged approach, the Floquet exponents can be determined numerically (e.g., Wang and Hale 2001) by integrating d
Figure 2 shows the sensitivity of the maximum real part of the Floquet exponents of the phase-averaged and fixed-phase CS-LIMs to the EOF truncation of the training data. Interestingly, the phase-averaged CS-LIM is unstable if more than 3 EOFs are used, whereas the fixed-phase CS-LIM remains stable using as many as 16 EOFs. We therefore decided to use the fixed-phase approach in this study, motivated by the need to retain enough SST variance in the truncated EOF space to diagnose the cyclostationary dynamics of a large fraction of tropical SST variability.
The maximum real part of Floquet exponents (μ) estimated using the phase-averaged (gray triangles) and the fixed-phase (black circles) approaches and EOF truncations of the data ranging from 2 to 20 EOFs. The filled black circles indicate cases in which one or more of the 12 monthly linear system matrices used in the fixed-phase approach are imaginary.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
b. Numerical integration
The stationary stochastic forcing covariance matrix
c. ENSO phase locking
We divided each 80 000-yr LIM run into 1600 50-yr segments and determined the 1600 sample standard deviations and power spectra of the simulated Niño-3.4 SST anomalies. The median and 5th and 95th percentiles of the standard deviations and power spectra are shown in Fig. 3. All three LIMs reproduce the main features of the observed Niño-3.4 spectrum, also shown in the figure, with a peak near 4-yr periods. In general, the LIM spectra are smoother than observed, not only because the LIMs were constructed in a truncated EOF space but also because the observed spectrum is noisier due to limited sampling. It is also evident that the CS-LIM and CSQ-LIM spectra match the observed spectrum somewhat better than does the ST-LIM spectrum. All three LIMs, however, exaggerate the power at frequencies lower than ENSO frequencies.
(top) Power spectra and (bottom) interannual standard deviations of monthly Niño-3.4 (5°S–5°N; 170°–120°W) SST anomalies derived from 80 000-yr SST-only (a) CS-LIM, (b) CSQ-LIM, and (c) ST-LIM runs. Thick black lines show the median of 1600 values derived from 1600 50-yr segments and gray shading the 90% confidence interval. The black dots show values estimated from the 50-yr HadISST (1961–2010; Rayner et al. 2003) dataset.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
The seasonally varying standard deviations of observed monthly Niño-3.4 SST anomalies are indicated by the black dots in the lower panels of Fig. 3. The standard deviations of the ST-LIM are by definition constant and cannot explain the observed annual cycle. On the other hand, introducing cyclostationarity either in the CSQ-LIM (
ENSO phase locking is further assessed in Fig. 4. This was done, as in P96, by calculating the pattern correlations of the observed leading SST EOF pattern (not shown) with the monthly LIM-generated SST anomaly patterns in each one of the 1600 50-yr segments of the simulated SST time series, and counting the number of warm (cold) events in each calendar month in which the pattern correlation was higher (lower) than 0.75 (−0.75). The CS-LIM simulates ENSO phase locking better than the other LIMs, since the warm and cold events tend to peak primarily in winter as observed. The CSQ-LIM has only weak phase locking, and the ST-LIM cannot represent phase locking at all.
(top) Number of times the pattern correlation of the simulated SST anomalies and the observed EOF-1 pattern (figure not shown) was higher than 0.75 in 50-yr segments of 80 000-yr runs of the SST-only (a) CS-LIM, (b) CSQ-LIM, and (c) ST-LIM. Thick black lines denote the median of the 1600 values obtained from the 1600 50-yr segments and gray shading the 90% confidence interval. (bottom) As at top, but for pattern correlation lower than −0.75. Note that that ordinate is reversed from that in the top panel. The black dots show values estimated from the 50-yr HadISST (1961–2010; Rayner et al. 2003) dataset.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
Interannual standard deviations of monthly Niño-3.4 (5°S–5°N, 170°–120°W) SST anomalies derived from 80 000-yr SST-only (a) CSL*-LIM, and (b) CSQ*-LIM runs. Thick black lines show the median of 1600 values derived from 1600 50-yr segments and gray shading the 90% confidence interval. The black dots show values estimated from the 50-yr HadISST (1961–2010; Rayner et al. 2003) dataset.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
This is not the only way to account for the implicit contribution of the seasonal variation of
d. The 2015–16 El Niño event
The ability of the CS-LIM to represent ENSO phase-locking results in improved hindcasts of ENSO evolution. This is clearly evident in hindcasts of the 2015–18 period, which lies outside the CS-LIM’s training period (1961–2010). Figure 6 shows 64-member ensemble hindcasts of Niño-3.4 SST anomalies during the 2015–16 El Niño event, generated using the CS-LIM and ST-LIM. Hindcasts initialized in June and December of 2015 are shown separately to assess the LIMs’ ability to capture the developing and decaying phases of the event. In both phases, the median CS-LIM hindcast captures the observed trajectory of the Niño-3.4 SST anomalies better than the median ST-LIM hindcast, which decays monotonically from the initial conditions. The CS-LIM also represents the observed evolution better beyond 12 months.
The 64-member ensemble hindcasts (thin gray lines) of the 2015–16 El Niño event using the SST-only (a) CS-LIM and (b) ST-LIM. The hindcasts were initialized in (left) January 2015 and (right) December 2015. Thick red lines show the median of the ensemble and thin red lines the 90% confidence interval. Black dots show the observed SST anomalies estimated from the 50-yr HadISST (1961–2010; Rayner et al. 2003) dataset.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
4. Annual cycles of
To further assess the importance and robustness of the annual cycles of
Coupled climate model simulations used. All data are available in the World Climate Research Programme (WCRP) CMIP5 repository (https://esgf-node.llnl.gov/projects/cmip5/). We followed the nomenclature used in the repository for the models and modeling centers; N denotes ensemble size.
Standard deviations of monthly Niño-3.4 (5°S–5°N, 170°–120°W) SST anomalies derived from the coupled climate model simulations (see Table 1) and HadISST dataset (Rayner et al. 2003). For the CESM-LE, the interensemble spread (one standard deviation) is shown in parentheses.
An important feature of LIMs is their ability to identify optimal initial perturbations with the potential to grow over finite time intervals before decaying, as they eventually must if
In Fig. 7, all CMIP5 models show some seasonal variation of the maximum singular values, although their magnitude differs substantially among the models. For time intervals τ = 7, the springtime maximum is well separated from the early fall minimum. Even for τ = 12, a clear summer minimum is evident. Similar seasonal variation is apparent in the CESM-LE ensemble. The individual ensemble members, each 50 years long, also show this behavior, although the singular values tend to be biased toward larger values than obtained from the combined ensemble equivalent to a 40 × 50 = 2000-yr simulation.
Gray curves: maximum singular values of
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
Figure 8 quantifies the relative importance of the seasonal variations of
Annual cycle of total variance tendency (tr[d
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
With regard to the observational CS-LIM, one is faced with the dilemma that while the seasonal variations of
5. Coupled SST–SSH LIMs
a. Model construction and characteristics
It is now well recognized that extending an SST-only stationary LIM by explicitly including at least some aspects of ocean dynamics in the LIM’s state vector leads to better representations and predictions of seasonal to interannual tropical SST variability (Johnson et al. 2000b; Xue et al. 2000; Hawkins and Sutton 2009; Newman et al. 2011a,b; Ault et al. 2013; Vimont et al. 2014; Capotondi and Sardeshmukh 2015; Chen et al. 2016; Newman and Sardeshmukh 2017). To assess whether this improvement is retained even in a similarly extended CS-LIM, we constructed an SST–SSH coupled CS-LIM using SST and SSH anomalies. The monthly SSTs used were derived from the HadISST dataset (Rayner et al. 2003), and the SSHs were derived from the ECMWF ocean reanalysis system ORAS4 (Balmaseda et al. 2013) over the tropics (25°S–25°N) for the 1961–2010 period. Monthly anomalies on a 1° latitude × 1° longitude grid were determined by removing the grand mean and climatological annual cycle at each grid point. EOF analyses of the SST and SSH anomalies were then performed, and 11 (5) EOFs of SST (SSH) explaining 75.8% (58.3%) of the total variance were retained. As before, all subsequent calculations were performed in EOF space, and then transformed to geographical space for presentation and interpretation.
Considering the pros and cons of the phase-averaged and fixed-phase approaches in CS-LIM construction (see section 2), we combined the two approaches by using three-month running means of the lag-1 and lag-0 covariance matrices in (5) and (6) centered on each calendar month to estimate
Retaining 11 SST and 5 SSH EOFs, the SST–SSH coupled linear system is stable (i.e., all the Floquet exponents have negative real parts). However, the stochastic forcing covariance matrices again require minor adjustments to render them positive semidefinite. Similar to the adjustments made in the SST-only LIMs, for each month j we set the negative eigenvalues of
If the linear approximation is valid, the coupled CS-LIM should be able to reproduce the observed lag-covariance statistics at much longer lags than the 1-month training lag (a.k.a. PS95’s “Tau-test”; see also Newman et al. 2011b). The lag-covariance matrices for lags from calendar month j were calculated as
(a) Observed and (b) coupled CS-LIM derived lag-covariances of SST (color shaded) and SSH (contoured) anomalies at 3-, 6-, 9-, 18-, 27-, and 36-month lags averaged over all months. Note that the observed lag covariances are derived from the full (not EOF-truncated) gridded SST (HadISST; Rayner et al. 2003) and SSH (ORAS4; Balmaseda et al. 2013) datasets.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
To assess annual cycle effects in the coupled CS-LIM, we also constructed and compared a coupled SST–SSH ST-LIM with an SST-only ST-LIM over the same domain (25°S–25°N) and using the same observations. The EOF truncations of these ST-LIMs were chosen to maximize their cross-validated ensemble-mean SST and SSH hindcast skills (presented below in section 5d) among all possible pairs of truncations, constrained by the requirements that 1)
The maximum amplification (MA) curves, depicting the maximum possible growth of SST anomalies over the domain (in the L2 SST-norm) over different time intervals, are shown for the coupled CS-LIM, coupled ST-LIM, and SST-only ST-LIM in Fig. 10. Interestingly, the shapes of the coupled CS-LIM curves (12 thin lines) are quite different from the corresponding stationary case (thick gray and black lines): they have secondary maxima and much larger amplification factors. These larger factors generally imply increased predictability (Newman et al. 2011b) and depend on the initial month of the forecast. While the curves for the ST-LIMs have a single peak at τ = 7, the coupled CS-LIM curves show primary and secondary peaks for initial anomalies that grow to maximum amplitudes in December. For example, optimal February initial conditions can lead to a peak at τ = 10 (primary) and τ = 22 (secondary), and optimal August conditions can peak at τ = 4 (secondary) and τ = 16 (primary). The largest amplification occurs with February initial conditions, at lag τ = 10.
Logarithm of the maximum anomaly amplification factors (singular values of
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
The first two optimal initial structures (OS) in February and maximally evolved final structures (FS) in December (for τ = 10) in the coupled CS-LIM are shown in Fig. 11. The leading OS of SST is very similar to that in the SST-only ST-LIM (e.g., PS95) and also closely resembles the well-known meridional mode pattern over the tropical Pacific (e.g., Chiang and Vimont 2004) that evolves in 10 months into the mature phase of El Niño in December. The leading OS and FS patterns of SSH are consistent with the delayed oscillator (Schopf and Suarez 1988; Battisti and Hirst 1989) as well as the recharge oscillator (e.g., Jin 1997) mechanisms of ENSO [reviewed recently by Battisti et al. (2018)]. We do not intend here to argue for the greater relevance of one mechanism over the other, but simply to show that our coupled CS-LIM captures the essence of ENSO dynamics. Our empirically derived leading optimal initial structures of SST and SSH are also generally similar to those derived by Thompson and Battisti (2000; see their Fig. 4) from a simple dynamical linear model (Battisti 1988). The subsequent evolution to τ = 22 shows the tropical anomalies flipping sign and evolving into a La Niña–like condition. On the other hand, the second leading OS and FS of SST and SSH merely show a local amplification of the SST anomalies in the central equatorial Pacific and deepening of the thermocline, consistent with Newman et al. (2011a). The evolution from the second OS to leads to a canonical El Niño pattern at τ = 22, suggesting a possibility for the coupled CS-LIM to capture ENSO development over two years, given the “right” initial conditions (e.g., Kessler 2002).
(a) (top) Leading optimal initial patterns of SST (shaded) and SSH (contoured) anomalies in February for generating maximum rms SST-anomaly magnitude over the entire tropical domain over the next 10 months, and the evolved anomaly patterns in the coupled CS-LIM after (middle) 10 and (bottom) 22 months. The contour interval is arbitrary, but the same in all maps. Negative (zero) values of SSH are dashed (suppressed). (b) As in (a), but for the second leading optimal and evolved anomaly patterns.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
The optimal structures of the ST-LIM do not depend on season. In such a model, the system has the potential to grow into a mature ENSO pattern in any month as long as the initial SST anomaly pattern projects strongly onto the optimal structures. However, the optimal structures of the coupled CS-LIM leading to a mature ENSO pattern exist only for particular months. The pattern correlations of the OS in February for τ = 10 with the OSs in other months at τ values ranging from 1 to 22 months are shown in Fig. 12. The correlations are high mainly for winter-to-spring initial months, especially for SST, consistent with the mature phase of ENSO occurring primarily in December.
Pattern correlations of the leading initial optimal structures in February in the coupled CS-LIM shown in Fig. 11 for growth over τ = 10 months with the leading initial optimal structures in all months and for growth over all τ ranging from 1 to 22 months, for (a) SST and (b) SSH. Thick dashed lines indicate the month of December at final evolution.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
b. ENSO eigenmodes of the coupled cyclostationary system
Table 3 shows the decay times −1/Re(μ) and oscillation periods 2π/Im(μ) associated with the Floquet exponents in (9) of the eigenmodes of the coupled SST–SSH CS-LIM. While all the modes are stable, some are “temporarily unstable” in the sense described in section 3a.
Empirically derived eigenmode time scales of the coupled CS-LIM. The dominant ENSO mode is indicated by the filled square. Asterisks (*) indicate that these modes are temporarily unstable in some phases of the annual cycle.
To identify the most ENSO-relevant eigenmodes, we estimated the contribution of each mode to the Niño-3.4 SST power spectrum by filtering it out of the data and recomputing the spectrum. Figures 13a and 13b highlight the dominance of eigenmode 4/5 with a period of 38.6 months and a decay time of 19.4 months in this regard. The modal amplitude time series has a correlation of 0.86 with the Niño-3.4 SST anomaly time series, and its power spectrum closely matches the Niño-3.4 SST power spectrum, although its peak is too high. Figure 13c shows that the maximum potential for SST anomaly growth from February into a mature ENSO event 10 months later is almost entirely associated with this single mode. Consistent with this, the mode’s corresponding adjoint SST and SSH patterns (which are optimal initial patterns for exciting the mode) are highly correlated with the optimal initial SST and SSH patterns in February (shown in the top left panel of Fig. 11) at 0.93 and 0.98, respectively. This eigenmode may thus be objectively designated as the dominant ENSO mode of the system. Thompson and Battisti (2000, 2001) also reported a similarly dominant ENSO mode (with a period of 44.4 months and a decay time of 22.6 months) in their simple dynamical cyclostationary model of ENSO.
(a) Power spectra of monthly Niño-3.4 (5°S–5°N; 170°–120°W) SST anomalies reconstructed by using the dominant ENSO eigenmode (black line; mode 4/5 in Table 3). The black dots indicate the spectra from the 50-yr HadISST (1961–2010; Rayner et al. 2003) dataset. (b) As in (a), but for the time series of Niño-3.4 SST anomalies. (c) Logarithm of the maximum anomaly amplification factors (MA) as a function of forecast lead time in the coupled CS-LIM initialized in February. The gray, solid black, and dashed lines represent the MA derived from the
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
The seasonally evolving pattern [i.e. the cyclic eigenvector p4/5 in (9)] of this dominant ENSO mode is shown in Fig. S2 of the online supplemental material. Its evolution from month j to month j + 1 can be expressed as
There is also a dominant ENSO eigenmode of the SST-only CS-LIM discussed in section 3, which is also temporarily unstable in late boreal summer and fall (figure not shown). However, it is more stable overall than the ENSO eigenmode of the SST–SSH coupled CS-LIM, with a decay time scale of 13.0 months instead of 19.4 months, and is not as dominant, with its amplitude time series correlating with the Niño-3.4 SST time series at 0.75 instead of 0.86. This suggests that the explicit inclusion of persistent SSH dynamics in the SST–SSH coupled CS-LIM, rather than cyclostationarity per se, is more important for enhancing the persistence of the ENSO eigenmode. Consistent with this, the ENSO eigenmode of the SST–SSH coupled ST-LIM (not shown) also has a long decay time scale of 20.7 months. However, this mode is also not as dominant as the ENSO mode of the SST–SSH coupled CS-LIM, with its amplitude time series correlating with the Niño-3.4 SST time series at 0.75 instead of 0.86.
The sampling uncertainties of cyclostationary LIMs derived from a limited 50-yr record are relatively greater than those of the corresponding stationary LIMs. We acknowledge that this may compromise accurate estimation of some of the SST–SSH coupled CS-LIM’s parameters (appendix A) and yield an overly dominant ENSO mode. The fact that the “contribution” of this mode to the observed Niño-3.4 SST variance in Fig. 13a exceeds the actual Niño-3.4 variance might also raise a flag in this regard. However, such an exceedance would not necessarily be problematic in itself if the time series of some other modes are negatively correlated with that of the ENSO mode in the nonnormal tropical climate system with nonorthogonal eigenmodes (PS95). Nevertheless, the fact that the CS-LIM’s total power spectrum in Fig. 14a, which accounts for all eigenmodes, also exceeds the observed power in the ENSO frequency band does suggest that the dominance of the ENSO mode in our SST–SSH coupled CS-LIM may be somewhat exaggerated.
(top) Power spectra and (bottom) interannual standard deviations of monthly Niño-3.4 (5°S–5°N; 170°–120°W) SST anomalies derived from 80 000-yr coupled CS-LIM. Thick black lines show the median of 1600 values derived from 1600 50-yr segments and gray shading the 90% confidence interval. The black dots show values estimated from the 50-yr HadISST (1961–2010; Rayner et al. 2003) dataset.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
The dominance of this single eigenmode in our coupled CS-LIM is also at odds with the combined importance of several eigenmodes highlighted in previous LIM and simple dynamical model studies of ENSO (such as PS95; Newman et al. 2011b; Xie and Jin 2018; and many others) that ignored cyclostationarity and/or explicitly coupled SST and ocean dynamics. PS95 ignored both cyclostationarity and coupling in their SST-only ST-LIM, and found that three eigenmodes were important. We constructed an additional SST-only ST-LIM* similar to theirs, and also an SST-only CS-LIM*, from the HadISST dataset for 1961–2010 (where the asterisk indicates that we used 3-month running-mean SSTs instead of monthly SSTs as in all the other LIMs in this study) and obtained nearly identical results to theirs in both cases (not shown). This strongly suggests that cyclostationarity by itself does not lead to the dominance of a single ENSO eigenmode. On the other hand, explicit coupling of the SSTs with ocean dynamics (represented by either SSH or thermocline depth) also does not guarantee the dominance of a single eigenmode, considering that the coupled SST-thermocline depth LIM of Newman et al. (2011b) and the simple coupled dynamical model of Xie and Jin (2018) both had at least two distinct ENSO-related eigenmodes. Xie and Jin (2018) indeed interpreted this as providing a dynamical basis for the “ENSO diversity” that is currently at the forefront of ENSO research (Capotondi et al. 2015; Timmermann et al. 2018). Another confounding result in this context is from Berner et al. (2020), who examined ENSO in the NCAR CESM-LE simulations of the 1920–2100 period. They found a single dominant ENSO mode even in an SST-only ST-LIM derived from the simulated SSTs, and whose dominance increased with global warming. Reconciling such disparate observational and model results with a single dominant cyclostationary ENSO mode in reality, as suggested but not conclusively established by our coupled SST–SSH CS-LIM, is an interesting issue whose detailed investigation is beyond the scope of this study, but is worth pursuing in the future.
c. Numerical integrations
Having estimated
d. Hindcast skill in the 1982–2010 period
We compared the hindcast skill of our LIMs with the skill of the nine models used in the NMME forecasting system (Table 4) over the 1982–2010 period. The NMME hindcasts were bias corrected for each model and each ensemble member by subtracting the mean difference between the hindcasts and the observations for each target month and each forecast lead time (e.g., Barnston et al. 2015). The multimodel ensemble mean (hereafter NMME-mean) hindcasts were then determined as the grand mean of all the individual ensemble-member hindcasts of all nine models.
The NMME models whose hindcast skill was assessed in this study. All models have SST hindcast datasets for the 1982–2010 period. Only four models provide SSH hindcasts for the same period. All data (except SSH from NOAA GFDL) are available on a 1° latitude × 1° longitude grid at the Earth System Grid archive (https://www.earthsystemgrid.org). The SSH data from NOAA GFDL are available at GFDL’s data portal (https://data1.gfdl.noaa.gov). We followed the nomenclature of Kirtman et al. (2014) for models and modeling centers; N denotes ensemble size, and L denotes maximum hindcast lead in months.
For the coupled CS-LIM (ST-LIM), the cross-validated hindcast skills of SST and SSH were determined by subsampling the data: by successively removing one 5-yr segment (10% of the observations) at a time, re-estimating
Local anomaly correlation (AC) skill and root-mean-square error (RMSE)-based skill score (e.g., Barnston et al. 2015) were used to assess the deterministic skill of the ensemble-mean hindcasts, and the relative operating characteristic (ROC; e.g., Mason 1982) curve was used to assess the probabilistic skill (see appendix B for details of the prediction skill measures used). For the deterministic skill, unless stated otherwise, we show the AC skill of the NMME-mean and the coupled CS-LIM. We also show the impacts of the “coupling effect” (coupled ST-LIM skill minus SST-only ST-LIM skill), and the “annual cycle effect” (coupled CS-LIM skill minus coupled ST-LIM skill) on the AC skill of the coupled CS-LIM. Results obtained using the RMSE-based skill score (shown in the supplemental material) are largely consistent with those obtained using the AC skill, and lead to the same conclusions.
1) Deterministic skill
The AC skills at months 6 and 12 of the coupled CS-LIM and NMME-mean are compared in Fig. 15 for SST and in Fig. 16 for SSH (see Figs. S3 and S4 of the supplemental material for the RMSE-based skill scores). For SST, the NMME-mean skill is somewhat higher than the coupled CS-LIM skill (Figs. 15a,b), except in the western Pacific warm pool region. This is because of the well-known erroneous tendency of virtually all climate models to extend ENSO too far west compared to observations, as also highlighted for the NMME models by Newman and Sardeshmukh (2017). The coupled CS-LIM does not have this deficiency.
(a) Local SST anomaly correlation skill of (left) month 6 and (right) month 12 NMME-mean hindcasts in the 1982–2010 period. (b) As in (a), but of the coupled CS-LIM hindcasts. (c) Impact on month 6 and month 12 coupled ST-LIM skill from the SST–SSH coupling (= coupled ST-LIM minus SST-only ST-LIM skill), and (d) impact on month-6 and month-12 coupled CS-LIM skill from including annual cycle effects (= coupled CS-LIM minus coupled ST-LIM skill). The sum of (c) + (d) quantifies the total impact on skill of including both SST–SSH coupling and annual cycle effects in the coupled CS-LIM. Regions in (a) and (b) of insignificant skill at the 95% confidence level are dotted. See Fig. S6 of the supplemental material for the local anomaly correlation skill of the SST-only and coupled ST-LIMs.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
(a),(b) As in Figs. 15a and 15b, but for the SSH anomaly hindcasts. (c) As in Fig. 15d, but for the impact of including annual cycle effects in the coupled CS-LIM on the SSH hindcast skill (= coupled CS-LIM minus coupled ST-LIM skill). See Fig. S7 of the supplemental material for the local anomaly correlation skill of the coupled ST-LIM.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
Figures 15c and 15d show how the skill of the coupled CS-LIM is improved by incorporating both SST–SSH coupling (Fig. 15c) and cyclostationarity (Fig. 15d). The impact of coupling is especially marked over the Indian and Atlantic Oceans, leading to an increase in AC skill of as much as 0.4 at month 12. Over the central equatorial Pacific (Niño-3.4), coupling leads to about a 0.1 increase in skill. Introducing the annual cycle in
For SSH, the NMME-mean skill is slightly higher than the CS-LIM skill over the Indo-Pacific domain (Figs. 16a,b) except in the Atlantic where it is negative whereas the coupled CS-LIM skill is relatively high (>0.6). The SSH skill of the coupled CS-LIM is also improved by introducing cyclostationarity (Fig. 16c), especially over the western Pacific at month 12. Also notable is the skill improvement along the region of the Pacific North Equatorial Countercurrent (NECC), located between approximately 2° and 7°N, at both 6- and 12-month lead times. As reviewed by Battisti et al. (2018), SSH anomalies along the NECC impact ENSO evolution, and are associated with westward propagating oceanic Rossby waves that reflect off the western boundaries as eastward propagating Kelvin waves into the central equatorial Pacific (e.g., Wyrtki 1973). The improved representation of dynamics in the NECC region in the coupled CS-LIM likely also contributes to its improved SST hindcast skill in Fig. 15.
Figure 17 provides a more detailed comparison of the seasonally dependent NMME-mean and coupled CS-LIM AC skill, as well as the seasonally dependent impacts of SST–SSH coupling and cyclostationarity on the CS-LIM skill (see Fig. S5 of the supplemental material for the RMSE-based skill scores). The results presented are for the Niño-3.4 index (N34; e.g., Trenberth 1997), defined as the area averaged SST anomaly over the central and eastern equatorial Pacific (120°–170°W, 5°S–5°N). The seasonal variation of the coupled CS-LIM skill is generally similar to that of the NMME-mean skill. The NMME-mean skill is somewhat higher, especially in the summer verification months. The SST–SSH coupling (Fig. 17c) improves the CS-LIM skill in summer and fall (June–October) at longer than 3-month leads. This improvement, especially in summer, may be related to oceanic memory in the thermocline (equivalent to the SSH near the equator) of information from the previous winter. On the other hand, the beneficial impact of including the annual cycle in the CS-LIM (Fig. 17d) is greatest in months in which the N34 index typically attains its peak values. The impact on the N34 skill is thus largest in late winter (December–March) at longer than 6-month leads.
Niño-3.4 SST index anomaly correlation skill in each verification month of the (a) NMME-mean and (b) coupled CS-LIM hindcasts, (c) the impact on skill of SST–SSH coupling (= coupled ST-LIM minus SST-only ST-LIM skill), and (d) the impact on skill of including annual cycle effects in the coupled CS-LIM (= coupled CS-LIM minus coupled ST-LIM skill). See Fig. S8 of the supplemental material for the local anomaly correlation skills of the SST-only and coupled ST-LIMs.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
2) Probabilistic skill
We also intercompared the cross-validated probabilistic skill of the 114-member ensemble SST hindcasts made using the coupled CS-LIM, coupled ST-LIM, and the NMME. The skill was assessed in terms of relative operating characteristic (ROC) curves for three N34 categories (below normal, neutral, and above normal), shown in Fig. 18 for the three prediction systems. At month 6, the ROC curves for the coupled CS-LIM are very similar to those of the NMME in the neutral and above-normal categories, while the NMME is clearly better than the coupled CS-LIM in the below-normal category. This is partly due to the inability of the CS-LIM to capture asymmetric aspects of ENSO (e.g., Okumura and Deser 2010). Recently, Martinez-Villalobos et al. (2019) attributed the asymmetry of El Niño (above normal) and La Niña (below normal) events to the correlated additive and multiplicative (CAM) noise forcing (Sardeshmukh and Penland 2015; Sardeshmukh et al. 2015) of ENSO events, which is ignored in the LIMs developed here.
Probabilistic skill measures of the NMME-mean (blue), coupled CS-LIM (red), and coupled ST-LIM skill (black) hindcasts of the N34 SST index in the 1982–2010 period. (top) ROC curves for the (a) below-normal category N34 index hindcasts of the NMME-mean at 5.5-month lead and the coupled CS-LIM and coupled ST-LIM (black) at 6-month leads, and (bottom) the ROC scores as a function of forecast lead time. (b),(c) As in (a), but for the neutral and above-normal N34 index categories, respectively.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
The beneficial impact of accounting for annual cycle effects in the CS-LIM is clearly reflected in the higher ROC scores of the CS-LIM than the ST-LIM in the below-normal and above-normal categories at all leads longer than 3 months. This impact increases with forecast lead time. Interestingly, the CS-LIM ROC scores for these categories are slightly higher than even the NMME scores at leads longer than 10 months. In the neutral category, all three prediction systems show more or less similar ROC scores at all leads.
e. The 2015–16 El Niño event
The generally similar deterministic and probabilistic skill of the coupled CS-LIM and the NMME is evident even in forecasts of strong El Niño events. Figure 19 compares ensemble forecasts of the strong 2015–16 event using the two forecast systems, initialized separately in July 2015 and January 2016 during the event’s developing and decaying phases. We stress again that this event was not included in the CS-LIM’s training period. The NMME simulations were initialized on (or near) the first day of July 2015 and January 2016, whereas the LIM hindcasts were initialized with the monthly mean observations centered on the previous months of June 2015 and December 2015.
(left) Ensemble hindcasts (thin gray lines) of the 2015–16 El Niño event using (a) the NMME (maximum of 114 members) and (b) the coupled CS-LIM (114 members), all initialized in July 2015. (c) Thick blue line shows the NMME median hindcast, light blue shading shows the ensemble spread (90% confidence interval). Thin blue lines are median hindcasts of the 9 individual NMME models (see Table 4). Thick red solid line shows the coupled CS-LIM ensemble median hindcast and the thick red dashed lines the ensemble spread (90% confidence interval). Black dots represent observed SST anomalies (HadISST; Rayner et al. 2003). (right) As at left, but for hindcasts initialized in January 2016.
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
In both the developing and decaying phases, the median ensemble forecasts of the CS-LIM and NMME are reasonably close to the observed trajectory of the N34 SST anomalies. The NMME spread tends to be larger than the CS-LIM spread, implying greater forecast uncertainty. The CS-LIM forecasts underpredict the observed warm anomalies during the developing phase and overpredict the cold anomalies during the decaying phase of the event. We should note, however, that the CS-LIM was constructed in a truncated EOF space of 11 SST EOFs explaining 75.8% of the SST variance over the entire domain, whereas the observed N34 SST evolution shown is in the untruncated space.
6. Summary and concluding remarks
In this study we developed cyclostationary linear inverse models (CS-LIMs) of monthly tropical SST and SSH anomalies from their observed zero-lag and 1-month lag covariance statistics over the 1961–2010 period. We constructed two versions of these LIMs, using only SST data (“SST-only” LIMs) and using both SST and SSH data (“coupled SST–SSH” LIMs). One of our major goals was to assess the impacts of the seasonal variations of the background state (encapsulated in the model’s deterministic feedback matrix
The spectra of Niño-3.4 SST anomalies in long integrations of the SST-only CS-LIM, ST-LIM, and CSQ-LIM reproduce the main features of the observed Niño-3.4 spectrum with a peak near 4-yr periods. The CS-LIM represents ENSO phase-locking better than the CSQ-LIM, which suggests that the seasonal variation of
SST-only CS-LIMs constructed from CMIP5 model and CESM-LE output suggest the importance of the seasonal variations of
We compared the deterministic hindcast skill of the extended SST–SSH coupled CS-LIM with that of the comprehensive NMME prediction system over the 1982–2010 period at forecast leads of a month to a year. The CS-LIM is clearly improved by including both SST–SSH coupling and cyclostationarity, and is more skillful especially in the Pacific NECC region, which is of importance to ENSO. With these improvements, the CS-LIM skill becomes similar to the NMME skill and may indeed be used to benchmark the NMME performance.
We also compared the probabilistic hindcast skill of the coupled CS-LIM with that of the NMME, in terms of ROC scores of N34 anomalies in the above-normal, neutral, and below-normal categories. The scores for the coupled CS-LIM are generally comparable to the NMME scores, and higher than the scores for coupled ST-LIM at all leads beyond 3 months in the above-normal and below-normal categories. This highlights the importance of accounting for annual cycle effects in predicting extreme ENSO events. Still, the CS-LIM scores in the below-normal category are lower than the NMME scores between 3 and 9 months, perhaps due to the inability of the version of the CS-LIM developed here to represent asymmetric aspects of ENSO.
Collectively, our results indicate that a CS-LIM that includes the annual cycles of the background state and stochastic forcing of tropical SST and SSH anomalies is better at representing the seasonal modulation of ENSO-related SST anomalies and their phase locking to the annual cycle than a traditional LIM that ignores such annual cycles. The explicit inclusion of these cycles in the SST–SSH coupled CS-LIM improves the forecast skill of both SST and SSH anomalies through improved SST–SSH coupling. The higher CS-LIM skill results from improving the representation of both ENSO phase-locking and Pacific NECC variations. These improvements result not only from explicitly accounting for the annual cycle of the background state, but also that of the stochastic forcing.
Our results suggest that the coupled CS-LIM may be a useful new tool to benchmark NMME forecasts of tropical SST and SSH anomalies at seasonal-to-interannual scales. They also suggest that the CS-LIM could be a useful component of a global two-tiered seasonal-to-interannual prediction system in which skillful CS-LIM forecasts of tropical SST and SSH would provide evolving boundary conditions to a global model. This is a topic of current research.
Acknowledgments
This work was supported by the Physical Sciences Laboratory of NOAA. SS and MN were also supported by NSF (Grant AGS-1624831) and DOE (Grant 0000238382). We wish to thank two reviewers, Dr. David Battisti and an anonymous reviewer, for their constructive reviews that greatly improved the manuscript. We also wish to thank Dr. Alex Voronovich for his internal review.
APPENDIX A
Sufficiency of Data Record Length to Build a Reliable CS-LIM
We define a “reliable” CS-LIM for this purpose as one whose seasonal variation of maximum monthly singular values, associated with optimal anomaly growth from initial anomalies in each month over the next 7 months (see section 3), is statistically significant. We addressed this issue using an 80 100-yr run of the SST-only CS-LIM (see section 3), dividing the last 80 000 years into 1600 50-yr segments. We constructed a CS-LIM from each segment and determined the maximum singular values of
(a) Estimated uncertainties (spreads) of the maximum singular values of
Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0291.1
Figure A1a compares the spreads of the maximum monthly singular values derived from the 50-, 100-, 200-, and 400-yr CS-LIM run segments with the values derived from the single 50-yr observational record. For the 50-yr SST-only CS-LIM segments, the spread is large. For longer segments the spread is tighter, and all the observed singular values tend to be located within the spread. We assess the seasonal variation of the observed singular values as significant if the value for any month lies outside the spread for at least one other month. This condition is clearly met for longer than 100-yr segments. It is nearly met even for 50-yr segments in that the observed value for May lies outside the spread for all other months, and the observed values for all other months lie outside the spread for May. The largest increase in the ratio of the signal (i.e., the seasonal variation of the singular values) to noise (i.e., their spread) occurs when the segment length increases from 50 to 100 years. Figure A1a thus suggests that a 100-yr data record may be sufficient to build a reliable SST-only CS-LIM of the tropical climate system, although even a 50-yr record may be adequate for some purposes.
Figure A1b further supports this conclusion. For a 50-yr data record, the uncertainty of the correlation coefficients (i.e., the width of their PDF) is large. For longer data records, the uncertainty is smaller. The estimated probabilities of obtaining a correlation with the “truth” (defined here as the 12-component “vector” of observed singular values) of larger than 0.8 (0.9) are 57.9% (19.6%), 91.9% (62.4%, 99.8% (95.8%), and 100% (100%) for 50-, 100-, 200-, and 400-yr records, respectively. Thus, Fig. A1b also suggests that a 100-yr data record may be sufficient to build a reliable SST-only CS-LIM of the tropical climate system, although even a 50-yr record may be adequate for some purposes.
We repeated this significance analysis using an 80 100-yr run of the SST–SSH coupled CS-LIM (see section 5), determining the maximum singular values associated with optimal ENSO amplification over τ = 10 months (see Fig. 10). The results are shown in Figs A1c and A1d, in an identical format to Figs A1a and A1b, respectively. The maximum singular values in Fig A1c for winter are well separated from those for other seasons for longer than 100-yr records, and reasonably well separated even for 50-yr records. The estimated probabilities (Fig. A1d) of obtaining a correlation with “truth” of larger than 0.8 (0.9) are 84.9% (55.4%), 99.5% (93.6%), 100% (100%), and 100% (100%) for 50-, 100-, 200-, and 400-yr records, respectively. Note that the probabilities are generally larger than in Fig A1b, likely due to the utilization of the extra information contained in SSH as well as the “hybrid” 3-month phase smoothing approach used to construct the SST–SSH coupled CS-LIM. Collectively, Figs. A1c and A1d suggest that a 100-yr data record may also be sufficient to build a reliable coupled CS-LIM of the tropical climate system, although even a 50-yr record may be adequate for some purposes.
APPENDIX B
Prediction Skill Measures
We defined the deterministic prediction skill as the skill of the ensemble-mean forecasts and measured it by the local anomaly correlation (AC) and RMSE-based skill scores. While AC measures the extent to which the prediction is temporally in phase with the observed anomaly, the RMSE-based skill score emphasizes the amplitude of the predicted anomaly in relation to that of the observed anomaly (e.g., Barnston et al. 2015). A positive (negative) RMSE-based skill score indicates that a prediction model performs better (worse) than a model which merely predicts observed climatological values. We estimated the statistical significance of our estimated AC using the two-tailed Student’s t test, testing a null hypothesis of zero skill. We converted our AC values to Fisher’s z statistic prior to applying the t test and used the effective degrees of freedom estimated from observations during 1982–2010.
We measured the probabilistic prediction skill by the ROC curve (e.g., Mason 1982). A ROC curve indicates the degree of correct probabilistic discrimination (hit rate vs false alarm rate) in a set of forecasts, where discrimination is defined as the ability to distinguish one categorical outcome from another, even if the forecast probabilities have biases or calibration problems. The area underneath the ROC curve is the ROC score. ROC scores above 0.5 reflect positive discrimination skill.
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