ENSO phase space dynamics in CMIP models

doi:10.21203/rs.3.rs-4727039/v1

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Research Article

ENSO phase space dynamics in CMIP models

https://doi.org/10.21203/rs.3.rs-4727039/v1

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This study analyses the El Niño-Southern Oscillation (ENSO) phase space as simulated by the Coupled Model Intercomparison Projects 5 and 6 (CMIP5 and CMIP6) models. The ENSO phase space describes the ENSO cycle between the sea surface temperature (SST) anomaly in the eastern equatorial Pacific (T) and the equatorial mean thermocline depth anomaly (h). We find that the characteristics out-of-phase cross-correlation between T and h is shifted to negative values in CMIP models, suggesting that the coupling between T and h is regionally sifted to the east compared to the observed central Pacific. If we consider the CMIP models with an eastward shifted h then the models have better agreements with the observed characteristics. While the models can capture some of the non-linear aspects with high correlations, they do largely underestimate the strength of non-linear ENSO aspects. They underestimate the likelihood of extreme El Niño and discharge states, they cannot capture the enhanced growth rates during the recharge state, the enhanced decay after the discharge state nor the reduced phase transitions after the La Niña phases. Weaker than observed wind-SST feedback and weaker h variability are likely some of the reasons why models cannot fully capture the non-linear ENSO phase space dynamics. Further, we found no indication of significant improvements from the CMIP 5 to 6 ensemble, suggesting that the two ensembles are essentially the same in terms of their ENSO dynamics. There is, however, a large spread within the model ensembles, leading to models with quite different ENSO dynamics.

El Niño-Southern Oscillation (ENSO) dynamics

recharge oscillator model

ENSO phase space

ENSO events

CMIP5 and CMIP6 models

non-linear dynamics

The El Niño-Southern Oscillation (ENSO) is a climate phenomenon that influences the global climate and weather patterns, ecosystems, and society (Tsonis et al. 2003; McPhaden et al. 2006; Latif and Keenlyside 2009; Timmermann et al. 2018; Alizadeh 2022). It is characterized by its two distinct phases which are the warm phase (El Niño) and cold phase (La Niña) in the equatorial east Pacific Ocean (Chen and Li 2021). The variations of ENSO are controlled by complex dynamics that include feedback from the atmosphere and ocean (Bellenger et al. 2014; Kim et al. 2014; Wang et al. 2017; Vijayeta and Dommenget 2018; Planton et al. 2021). The aim of this study is to contribute to a better understanding of these complex dynamics.

Key ENSO dynamics can be determined by the interaction of ocean-atmosphere processes such as sea surface temperature (SST), wind stress, and thermocline depth, known as Bjerknes feedback (Bjerknes 1969). A simplified concept for combining these processes and for understanding ENSO dynamics is described by the recharge oscillator (ReOsc) model (Jin 1997; Burgers et al. 2005). The ReOsc model describes the variability of ENSO as an interaction between the tendencies of SST anomaly in the eastern equatorial Pacific (T) and equatorial mean thermocline depth anomaly (h), with the latter representing a proxy for the subsurface equatorial heat content. It has been used extensively in previous studies to analyse ENSO dynamics (Jin 1997; Burgers et al. 2005; Frauen and Dommenget 2010; Yu et al. 2016; Vijayeta and Dommenget 2018; Wengel et al. 2018; Dommenget and Vijayeta 2019; Crespo et al. 2022; Dommenget and Al-Ansari 2022; Dommenget et al. 2023; Priya et al. 2024).

The idealised ReOsc model is an oscillation between T and h, presenting a 2-dimensional phase space, as known from physics for a harmonic oscillator or complex system analysis (Meinen and McPhaden 2000; Kessler 2002; Dommenget and Al-Ansari 2022; Dommenget et al. 2023; Priya et al. 2024). Detailed analysis of the characteristics of the ENSO phase can help to understand how the dynamics of ENSO control the amplitude, non-linear behaviour, and time evolution as functions of the ENSO phases.

The study by Dommenget et al. (2023) showed that the ENSO phase space highlights strong non-linear aspects, with enhanced extremes in the El Niño to discharge states, larger growth rates in the transition from recharge to El Niño state, strong decay after the discharge states, faster phase speeds after El Niño states and slow phase speeds after La Niña states. These non-linear aspects of the ENSO cycle have also been reported in other studies, using different measures (An and Jin 2004; An et al. 2005; Dommenget et al 2013; Sun et al. 2016; Okumura 2019; An et al. 2020).

The Coupled Model Intercomparison Projects (CMIP) models are widely used to simulate and forecast ENSO activity. They are highly capable of simulating many aspects of ENSO and do show some improvements in the consecutive generations of CMIP from phases 3 to 6 in metrics used to evaluate ENSO, such as seasonal cycle, ENSO patterns, and diversity of ENSO events (Bellenger et al. 2014; Kim et al. 2014; Planton et al. 2021). Despite their capabilities, the models do show some biases if compared against observations and they do show large variations within the model ensemble (Kim and Yu 2012; Guilyardi et al. 2012; Bellenger et al. 2014; Vijayeta and Dommenget 2018; Bayr et al. 2019; McKenna et al. 2020). The CMIP models exhibit biases in main atmospheric and oceanic feedback, which do control ENSO amplitude and stability (Guilyardi et al. 2012; Bellenger et al. 2014; Kim et al. 2014; Vijayeta and Dommenget 2018; Bayr et al. 2019; Beobide-Arsuaga et al. 2021). Bayr and Latif (2023) discussed how atmospheric feedback, specifically zonal wind feedback and heat flux feedback, are underestimated in CMIP model (CMIP5 and CMIP6) simulations of ENSO dynamics.

Further, the models exhibit biases in their representation of the SST patterns, with a westward extension compared to observed data in CMIP5 and CMIP6 models (Weller and Cai 2013; Yang and Giese 2013; Bellenger et al. 2014; Capotondi et al. 2015; Wang et al. 2015; Guilyardi et al. 2020; Jiang et al. 2021; Planton et al. 2021). The models also exhibit biases in the equatorial thermocline depth, intensity, and zonal shape due to biases in the equatorial zonal wind stress, which affects upwelling and the zonal tilt of the equatorial thermocline (Guilyardi et al. 2020).

About 80% of the variations in simulating ENSO amplitude within the CMIP model ensemble are due to variations in the linear growth rates of T and h, and in the strength of the stochastic noise forcing, with the latter also potentially being linked to non-linear processes (Wengel et al. 2018) (non-linear processes such as non-linear air-sea coupling, non-linear dynamical heating, non-linear ocean advection, non-linear relationship of precipitate-SST feedback, heat flux because of the tropical ocean instability waves, atmospheric non-linear response to the SST anomaly (wind-SST relation) (An and Jin 2004; An et al. 2005; Sun et al. 2016; Okumura 2019; An et al. 2020)). Several studies on ENSO non-linearities point out that CMIP models tend to underestimate the non-linear aspects of ENSO (An and Jin 2004; An et al. 2005; Dommenget et al. 2013; Sun et al. 2016; Okumura 2019; Planton et al. 2021).

In this study, we analyze the ENSO phase space statistics as simulated in the CMIP 5 and 6 simulations and compare them against the observed values. We examine how well the CMIP models can reproduce the observed ENSO phase space asymmetries to gain some understanding of how well models can simulate the non-linear behavior of ENSO as a function of the different phases of ENSO. We explore potential limitations in model simulations that may affect the model performance. Additionally, we are also interested in assessing the improvements from CMIP5 to CMIP6 models in representing the ENSO phase space characteristics.

This paper is structured as follows: Section 2 describes the observational datasets used, CMIP simulations, phase space statistics methods, and the ReOsc model. This is followed by the main results section, in which we present the ENSO phase space statistics for CMIP simulations and observations. Here we also analyze the linear ReOsc model fit to CMIP models and discuss some aspects of the wind, thermocline, and SST cross-relations in the context of ENSO dynamics. The study is concluded with a summary and discussion section.

2.1 Observational datasets

Observational datasets for SST and the thermocline depth estimates have been taken from the Hadley Centre global sea Ice coverage and SST (HadISST) (Rayner et al. 2003), UK Met Office Hadley Centre 'EN' series observations datasets, version 4 (EN4) (Good et al. 2013) respectively from 1980 to 2021. The reanalysis datasets for SST and thermocline depth estimates are taken from Simple Ocean Data Assimilation version 3 (SODA3) (Carton and Giese 2008) (1980–2017), Ocean Reanalysis System 5 (ORAS5) (1980–2018) (Zuo et al. 2019), and CMCC Historical Ocean Reanalysis system: CHOR AS and CHOR RL (1980–2010) (Yang et al. 2017). To observe the variability more precisely, these 5 different datasets are concatenated to create a considerable time series of SST and thermocline depth repeated for the same year. Anomalies are calculated by removing the mean seasonal cycle (calculated over the entire period) from the data. T is the SST anomaly calculated for the NINO3 region (5°S to 5°N, 150°W to 90°W) and considered for the same in this study.

Previous studies often use the depth of the 20^oC (Z₂₀) as a proxy for thermocline depth (h), however, according to Dommenget et al. (2023), the maximum gradient in the temperature profile (Z_mxg) is a better proxy of h, which is calculated by averaging Z_mxg over the equatorial Pacific region (5°S to 5°N, 130°E to 80°W), for the analysis of the ENSO dynamics within the framework of the ReOsc model than the widely used Z₂₀. The ENSO phase space for T index by using Z_mxg as a proxy of h is closer to the idealized recharge oscillator model than Z₂₀. This motivates us to use Z_mxg to estimate the equatorial Pacific Ocean thermocline depth h, following the method of Dommenget et al. (2023).

The zonal wind stress data are calculated from 10-m surface zonal wind datasets taken from the European Centre for Medium-Range Weather Forecasts (ECMWF-ERA5) reanalysis (1980–2021) (Hersbach et al. 2020) to analyse the wind dynamics of ENSO. Here, we only consider wind stress data from atmospheric reanalysis datasets, ERA5, following a similar approach to previous papers (Guilyardi et al. 2012; Bellenger et al. 2014; Vijayeta and Dommenget 2018; Guilyardi et al. 2020).

Table 1

CMIP5 and CMIP6 models with the corresponding model numbers
	CMIP5				CMIP6
Model No.	Model Name	Model No.	Model Name	Model No.	Model Name	Model No.	Model Name
1.	ACCESS1-0	21.	GISS-E2-H-CC	1.	ACCESS-CM2	21.	EC-Earth3-Veg
2.	ACCESS1-3	22.	GISS-E2-R	2.	ACCESS-ESM1-5	22.	EC-Earth3-Veg-LR
3.	bcc-csm1-1	23.	GISS-E2-R-CC	3.	BCC-ESM1	23.	FGOALS-f3-L
4.	bcc-csm1-1-m	24.	HadCM3	4.	BCC-CSM2-MR	24.	FGOALS-g3
5.	CanESM2	25.	HadGEM2-CC	5.	CanESM5	25.	FIO-ESM-2-0
6.	CCSM4	26.	HadGEM2-ES	6.	CESM2	26.	GFDL-CM4
7.	CESM1-BGC	27.	IPSL-CM5A-LR	7.	CESM2-FV2	27.	GISS-E2-1-G
8.	CESM1-CAM5	28.	IPSL-CM5A-MR	8.	CESM2-WACCM	28.	GISS-E2-1-G-CC
9.	CESM1-FASTCHEM	29.	IPSL-CM5B-LR	9.	CESM2-WACCM-FV2	29.	INM-CM4-8
10.	CESM1-WACCM	30.	MIROC5	10.	CIESM	30.	INM-CM5-0
11.	CMCC-CESM	31.	MPI-ESM-LR	11.	CMCC-CM2-SR5	31.	MIROC-ES2L
12.	CMCC-CM	32.	MPI-ESM-MR	12.	CMCC-ESM2	32.	MPI-ESM1-2-LR
13.	CMCC-CMS	33.	MPI-ESM-P	13.	CNRM-CM6-1	33.	MPI-ESM-1-2-HAM
14.	CNRM-CM5	34.	MRI-CGCM3	14.	CNRM-ESM2-1	34.	MRI-ESM2-0
15.	CNRM-CM5-2	35.	MRI-ESM1	15.	E3SM-1-0	35.	NESM3
16.	CSIRO-Mk3L-1-2	36.	NorESM1-M	16.	E3SM-1-1	36.	NorESM2-MM
17.	GFDL-CM3	37.	NorESM1-ME	17.	E3SM-1-1-ECA	37.	SAM0-UNICON
18.	GFDL-ESM2G			18.	EC-Earth3	38.	TaiESM1
19.	GFDL-ESM2M			19.	EC-Earth3-AerChem	39.	UKESM1-0-LL
20.	GISS-E2-H			20.	EC-Earth3-CC

2.1 CMIP Simulations

For the CMIP simulations, we used the historical simulations of the CMIP5 from the years 1900 to 1999 (Taylor et al. 2012) and CMIP6 for the period 1850–2014 (Eyring et al. 2016) models, utilizing every model with the variables required for this analysis. We use a single realization of each model and use all models with the required data available. See Table 1 for the 37 CMIP5 and 39 CMIP6 models taken into consideration.

2.2 ENSO Phase space

For the ENSO phase space analysis, we are following the approach by Dommenget and Al-Ansari (2022), Dommenget et al. (2023), and Priya et al. (2024). We normalise the monthly means of T and h by dividing them by their standard deviations, resulting in a non-dimensional presentation of the variables, T_n and h_n, respectively. The ENSO phase space is represented with T_n on the x-axis and h_n on the y-axis. The cartesian coordinate system has been transformed into a spherical coordinate system with phase angle, 𝜑 = 0° in h (y-direction) and 90° in T (x-direction), where the phase angle, 𝜑, rotates in a clockwise direction. The ENSO system anomaly, S, is represented by a vector with two cartesian coordinates T_n and h_n. When it is converted into the polar coordinate system, the magnitude of S remains constant for constant radius and is independent of phase angle (𝜑). The ENSO system is further described by the magnitude S and phase 𝜑.

The ENSO system's tendencies in the polar coordinate system are separated into radial and tangential components. The phase dependent growth rate is calculated by dividing the radial component of the tendencies by the magnitude of S, which is defined as the tendency to move away from the origin (positive values) and towards the origin (negative values). The phase speed is calculated by dividing the tangential component of the tendencies by the magnitude of S and is defined as the system's tendency to circle around the origin. Positive values indicate clockwise motion, while negative values indicate anti-clockwise motion; additionally, smaller positive values indicate a slower transition in the ENSO cycle. The average time required to complete one complete cycle is determined through the integration of the phase speed from 0° to 360°.

Note, that a proper analysis of the ENSO phase space statistics assumes that T and h are orthogonal to each other (Dommenget et al. 2023). Thus, they have a zero cross-correlation at time lag zero. If T and h are not orthogonal to each other (e.g., lag-zero cross-correlation is not zero), then the phase space statistics are skewed towards the diagonals between the T-axis and h-axis of the diagrams, which will result into skewed statistics of probabilities, growth rates, and phase speeds.

2.3 ReOsc Model

The linear ReOsc model is based on two tendency equations with six parameters as given below (Burgers et al. 2005):

$$\:\frac{dT\left(t\right)}{dt}={a}_{11}T\left(t\right)+\:{a}_{12}h\left(t\right)+{\xi\:}_{1}$$

$$\:\frac{dh\left(t\right)}{dt}=\:{a}_{21}T\left(t\right)+\:{a}_{22}h\left(t\right)+\:{\xi\:}_{2}$$

where a₁₁ and a₂₂ are the damping factor or growth rate of T and h respectively, a₁₂ is the coupling factor for T to h, a₂₁ is the coupling factor for h to T, ξ₁ is the noise forcing of T, and ξ₂ is the noise forcing of h. The model parameters of the ReOsc model are determined by carrying out a multivariate linear regression on the monthly mean tendencies of T and h in relation to their monthly mean values of T and h (Burgers et al. 2005; Jansen et al. 2009; Dommenget and Al-Ansari 2022; Dommenget et al. 2023; Priya et al. 2024).

In reference to the normalised presentation of the ENSO phase space (T_n and h_n), we normalise all values of the ReOsc model. The ENSO phase space resulting from the linear ReOsc model fit can be estimated by integrating the Eqs. (1–2) with random white noise forcings ξ₁ and ξ₂ over a period of 10⁴.

Our main results section is divided into four subsections. We start the analysis in subsection 3.1 with investigating the cross-correlation between T and h, which is a fundamental statistic on which the ENSO phase space analysis is based on. This analysis will also focus on the regional shift in ENSO patterns, as it affects cross-correlation between T and h. In the following subsection 3.2, we analyse the ENSO phase space in CMIP simulations, which is the main focus of this study. This is followed by the analysis of the linear ReOsc model fitted to the CMIP simulations. We conclude our discussion by examining the effect of wind stress on ENSO dynamics and how well CMIP models capture this atmospheric feedback in the subsection 3.4.

3.1 Regional variations of T and h

Fig. 1a shows the observed cross-correlation between T and h. It shows the characteristic out-of-phase relation between T and h with positive cross-correlation when h leads T, zero cross-correlation at lag zero, and strong negative correlations when T leads h.

The CMIP 5 and 6 ensembles are very similar to each other, and both show significant deviations from the observed cross-correlation (Fig. 1a). All CMIP 5 and 6 simulations have negative cross-correlation at lag zero, shifting the whole ensemble of simulations into clear negative correlations at lag zero. This shift also affects the out-of-phase cross-correlations with lower positive correlation when h leads T that tend to peak at longer (10mon.) lead times than observed (7mon.). In turn, the negative cross-correlations when T leads h are stronger in the CMIP models than in the observations and tend to have short lead times (5mon.) than observed (7mon.).

The shift towards negative lag-zero cross-correlations between T and h in CMIP simulations could indicate an eastward shift in the thermocline depth variability associated with ENSO in the CMIP simulations. This would be consistent with observed regional variations in the SST cross-correlation with thermocline depth variability along the equatorial Pacific, which is known to be more positive in the eastern Pacific and more negative in the western Pacific (Burgers and Stephenson 1999; Dommenget et al. 2023).

Given that the central assumption in the ENSO phase space analysis is that the cross-correlation between T and h is zero at lag zero, we now focus on the hypothesis that regional shifts in the simulated ENSO patterns could affect ENSO dynamics (e.g., the cross-correlation between T and h) in CMIP simulations. We now analyse the equatorial patterns of the ENSO modes in the models, focussing on the SST pattern first and then on the more important regional shifts in the thermocline depth variability.

The T index in the ReOsc model is most closely related to the empirical orthogonal function (EOF) mode-1 of the tropical Pacific SST, see Fig. 2a. It is marked by a horseshoe pattern with warm SST anomalies in the eastern equatorial Pacific that transition into negative SST anomalies in the western equatorial and off-equatorial Pacific. This transition crosses zero equatorial SST anomalies at the longitude of about 159^oE (see dashed line in Fig. 2a).

In Figs. 2b-d, we show three scenarios for EOF mode-1 of SST from CMIP simulations to highlight cases closest to the observed (Fig. 2b; ACCESS-CM2), average (Fig. 2c; MRI-ESM2-0) and worst (Fig. 2d; NorESM2-MM) in relation to the observed transition line for zero equatorial SST anomalies. While the best CMIP simulation is close to the observed with some negative SST anomalies in the western equatorial Pacific, the worst model has essentially only positive SST anomalies over the whole equatorial Pacific.

In general, the EOF mode-1 of tropical Pacific SST of CMIP simulations is similar to the observed, but the transition line for zero equatorial SST anomalies is, in ensemble mean, about 20^o further west for both CMIP 5 and 6 simulations (Fig. 3a). This is consistent with the westward shift reported in previous studies (Weller and Cai 2013; Yang and Giese 2013; Bellenger et al. 2014; Capotondi et al. 2015; Wang et al. 2015; Guilyardi et al. 2020; Jiang et al. 2021; Planton et al. 2021).

The observed EOF mode-1 of thermocline depth variability is a dipole pattern (tilting mode; not shown) that has an in-phase relation with the T index in the ReOsc model and thus does not represent the h index (Meinen and McPhaden 2000). The EOF mode-2 of the equatorial Pacific is similar to a basin-wide mode, but the pattern is somewhat shifted to the eastern Pacific, see Fig. 2e. This mode is more closely correlated to h (correlation = 0.91). While the EOF mode-2 is mostly a monopole, it does have a transition from negative anomalies to positive anomalies in the far western equatorial Pacific at a longitude of about 146°E (see dashed line in Fig. 2e).

The EOF mode-2 of the CMCC-CM2-SR5 (best model) exhibits a similar spatial pattern, with near-zero values present around the same region with observation (Fig. 2f). A medium CMIP6 model, ACCESS-ESM1-5, demonstrates that the near-zero values shift towards the eastern equatorial Pacific region, and we can see a dipole-like structure with deeper thermocline depth in the western and shallower thermocline depth in the central to eastern pacific region (Fig. 2g). In addition, the CNRM-CM6-1 CMIP6 model (worst model) exhibits a clear dipole pattern in EOF mode-2, more similar to the tilting mode described in Meinen and McPhaden (2000; Fig. 2h).

In the ensemble of CMIP5 and 6 models, we can see that the EOF mode-2 patterns of thermocline depth (h) suggest an eastward shift in the equatorial Pacific region of about 20°E (Fig. 3b).

In summary, we find significant regional shifts in the SST and thermocline depth patterns associated with the ReOsc model. Interestingly, we find that the SST pattern is shifted to the west and the thermocline depth pattern is shifted to the east. At this point, we don't have a clear understanding of the reasons behind these biases, but it appears very likely that such regional shifts can affect the cross-correlation between T and h.

Following the analysis of Dommenget et al. (2023) we know that an eastward shift of the h index region leads to a more positive lag-zero cross-correlation between T and h. This could compensate for the negative lag-zero cross-correlation between T and h found in CMIP models (Fig. 1a). To fit into the ReOsc model and to follow the phase space concept, CMIP models should have an out-of-phase relationship between T and h (e.g., zero lag-zero cross-correlation).

To define a revised h index for the CMIP models which has zero lag-zero cross-correlation between T and h, we shifted the index region for h eastward along the equator (h_shift) until we found a region that most closely follows the observed zero lag-zero cross-correlation between T and h, for all CMIP simulations. We concluded that the out-of-phase characteristics for CMIP models in the ensemble mean are very similar to observation when h_shift is considered for the east equatorial pacific region (5°S to 5°N, 190°E to 80°W) (Fig. 1b). However, individual models do have some significant variations from this overall good fit.

In addition to this, we also investigated various regions of T to understand the impact of the SST pattern shift on the ENSO phase space for CMIP models. However, we found that changing the T regions had no significant impact on the out-of-phase relationship between T and h. As a result, we have focused solely on the regional shift of h for CMIP models for the remaining analysis.

3.2 ENSO phase space analysis for CMIP models

The observed ENSO phase space is depicted in Fig. 4a. The main features show a clockwise rotation of the tendencies, with clear phase propagation through all four phases of the ENSO cycle. Unlike the phase space of an idealised linear ReOsc model, which is symmetric in all phases (Dommenget and Al-Ansari 2022), the observed variability is clearly skewed towards El Niño (positive T_n values) to discharge states (negative h_n values) (quarter Q2) and the tendencies are stronger in quarter Q2 and weaker in quarter Q4 (La Niña to recharge phases). A more detailed discussion is presented in Dommenget et al. (2023).

Examples of CMIP models are shown for the phase space estimated based on h (left column in Fig. 4) and based on h_shift (right column), see Fig. 4. We selected a best (NorESM2-MM), medium (CESM2-WACCM), and worst (BCC-ESM1) model based on the correlation values between the mean phase space estimated based on h_shift and the observed value. If we compare the left with the right column, we can notice that the phase spaces of the left column are pronounced along the diagonal of Q2 to Q4, which is a signature of a negative cross-correlation between T and h (Dommenget et al. 2023). The right column, in turn, is more equally distributed on all four quarters of the phase space and generally, closer to the observed behaviour. This highlights that the CMIP models compare better with the observed phase space when h_shift is considered.

The best model (NorESM2-MM) closely reproduces the observed ENSO phase space with high correlation in the mean phase space (0.9), clear clockwise propagation in all four phases, and variability skewed towards stronger positive T_n (El Niño), and stronger negative h_n (discharge phase). In turn, the worst model (BCC-ESM1) shows opposite asymmetries in the phase space to those observed, with a negative correlation in the mean phase space (-0.85), indicating that it has extreme recharge states instead of the observed extreme discharge states. However, it also shows clear clockwise propagation in all four phases.

The mean statistical features in the ENSO phase space are shown for all CMIP models and the observations in Fig. 5. Again, the left column shows the CMIP models based on h and the right column based on h_shift. The observed values are shown in both columns for comparison (both based on h). The CMIP models phase space based on h (left column) shows clear biases in all statistics. The mean phase space and the probability distribution of S > 2 are both more pronounced along the diagonal of Q2 to Q4, the growth rate is more pronounced along the T-axis and the phase speed along the diagonal of Q1 to Q3. All these biases directly result from the negative cross-correlation between T and h (Dommenget et al. 2023). As a result, the CMIP models phase space statistics based on h do not correlate well with the observed values.

The phase space statistics of the CMIP models based on h_shift show much less obvious biases and correlate much better with the observed values. The CMIP models generally do capture the asymmetry in the mean phase space, with the largest mean value in Q2 (El Niño and discharge state), smallest value in Q4 (La Niña and recharge state), and a correlation of the ensemble mean value in CMIP 5 and 6 of 0.88 and 0.81, respectively. However, the CMIP ensemble does not quite capture the intensity of the shift from Q4 to Q2, indicating the CMIP models are underestimating the non-linearity in the ENSO phase space.

This also holds for the probability distribution of S > 2 (extreme values), which in observations show a clear shift to Q2 and away from Q4. This is captured by the CMIP models, but again not with the right intensity. In particular, the absence of observed extremes in Q4 is not fully captured by the CMIP models. Furthermore, we see a very large spread in CMIP ensemble members, indicating widely different behaviour between the models.

The observed mean growth rate as a function of the ENSO phase shows positive values mostly around the recharge towards the El Niño state and negative values in the transition from discharge to a La Niña state (Fig. 5g). This signature is somewhat captured in the CMIP ensemble, but only with a moderate correlation. The CMIP models capture the negative growth rate in the transition from discharge to a La Niña state better than the positive values during the recharge to the El Niño state. Again, the CMIP models show a wide spread between the ensemble members. Notably, we find several models with very strong negative growth rates around the discharge and recharge phases, suggesting that these ENSO states can collapse much faster in these models than observed.

The observed phase speed is fastest, or most clear, in Q2 (after El Niño states) and slowest, or least clear in Q4 (after La Niña states; Fig. 5h). The CMIP model ensemble can partly capture this signature, but only with a moderate positive correlation. In particular, the models have problems in capturing the smaller phase speed after La Niña events. This suggests that models tend to have a faster or clearer transition from La Niña states to the recharge state. The model ensemble also has a wide range of different phase speeds. Some models transition through some ENSO phases more than twice as fast than observed, and some models have phase speeds near zero in some ENSO phases, suggesting the models have no clear ENSO cycle in these phases.

The phase speed analysis suggests that CMIP models are oscillating a bit faster than observed. Figure 6 shows the observed and CMIP simulated power spectrum of the T index. We can first notice that the ensemble mean power spectrum of CMIP 5 and 6 are in very good agreement with the observed, with no clear shift in the power to higher frequencies. However, both CMIP ensembles have a small tendency to peak at slightly shorter periods (~ 3yrs period) than observed (~ 4yrs period). Individual members of the ensemble can, however, behave quite differently, with some models having clear peaks at different periods and other models having less obvious peaks with a more continuous power spectrum.

3.3 Fitted linear ReOsc model for CMIP

The ENSO phase space is based on the ReOsc model. It therefore does help to analyse the ReOsc model fitted to the data of the CMIP models to understand what is causing the characteristics of the ENSO phase space. Here we focus on the linear model, thus not considering non-linearities, and without considering any seasonality in the fitted parameters. Neglecting non-linearities is a strong limitation, as will be shown below, and is only reflecting that we yet do not know what non-linear model can describe the observed asymmetries in the ENSO phase space (Dommenget and Al-Ansari 2022; Dommenget et al. 2023).

Figure 7 shows the fitted ReOsc model parameters for all CMIP models and observations for models based on h (left column) and based on h_shift (right column). The observed estimate is the same in both columns. Here all parameters are normalized (as in T_n and h_n) to allow a better discussion of the dynamical implication for the ENSO phase space.

We can first-of-all note that there is quite some spread between models, with many models having parameters being far away from the observed values in both estimates. Further, we find that the two estimates of the ReOsc models are quite different from each other. The models based on h are strongly biased in the growth rates and the noise strength in h relative to the observed values but are much less biased in the coupling parameters. In turn, the models based on h_shift are more strongly biased in the coupling parameters.

The results for CMIP5 models are also somewhat different from Vijayeta and Dommenget (2018; hereafter VD18), which used Z₂₀ to estimate h instead of Z_mxg. This does alter the ReOsc model parameter fits. However, the CMIP5 models tend to overestimate the damping (negative a₂₂) of h in both estimates present here and in VD18.

We now focus on the ReOsc model fits based on h_shift as they better represent the observed ENSO phase space. Here we are primarily interested in what dynamical elements of the ReOsc model are causing the simulated ENSO phase space characteristics. Any deviation of the ENSO phase space from a perfect cycle (a perfectly symmetric phase space), can be linked to asymmetries (deviations for the dashed lines in Fig. 7) in the ReOsc models parameter pairs: growth rates (a₁₁, a₂₂), coupling (a₁₂, a₂₁) or noise forcings (ξ₁, ξ₂). In a non-linear model, they can also result from non-linearities in any of these aspects.

Individual models do show strong deviations from the dashed lines in the growth rates and coupling parameters, but less so in the noise forcings. In the ensemble mean the largest asymmetries are found in the coupling parameters, where nearly all models are below the dashed line, indicating a stronger coupling of T to h (a₁₂) than the coupling of h to T (a₂₁), which is quite different from the observed symmetric coupling. The coupling parameters are most important for the period of ENSO (Wyrtki 1985). The Wyrtki-index, which measures the peak period of ENSO as a function of a₁₂ and a₂₁, is for most models slightly shifted to shorter periods (Fig. 7e), consistent with somewhat larger phase speed values seen in the phase space diagrams (Fig. 5h).

Integrating the linear ReOsc model fitted to each CMIP model allows us to compute the phase space statistics based on the resulting T_n and h_n values, see left column in Fig. 8. The phase spaces resulting from the ReOsc model are in general similar to those of the CMIP models, but, due to the linear approach, the phase spaces are by construction symmetric (e.g., all structures are mirroring at the origin) and therefore lack all asymmetric variations. This is particularly important when compared against the observed phase space, because most of the interesting observed phase space characteristics are asymmetric. None of the observed asymmetries can be captured by the analysis of the linear ReOsc model, indicating that such structures must result from the non-linear processes. Thus, a first and important outcome of the linear ReOsc model fit is that the CMIP model mismatch to the observed phase space characteristics are likely to result from non-linear processes.

Despite the limitation of the linear ReOsc model fit, we can still gain some understanding about the CMIP model ensemble spread and how different linear aspects of the ReOsc model affect the ENSO phase space statistics. The mean S, the probability of extreme S > 2 and the phase speed are all only varying along the diagonals of the phase space but have no variation along the T-axis and h-axis of the diagrams (left column in Fig. 8). The growth rates are only varying along T-axis and h-axis, but not along the diagonals. However, the actual data of the CMIP ensembles does show more complex variations in all statistical aspects (right column Fig. 5). This in turn suggests that variations not captured by the linear model parameters are a result of non-linear processes.

We can further test the sensitivity of the ENSO phase space to the individual ReOsc model parameter variations in the CMIP ensembles, by varying only a subset of the parameters, while holding the other parameters at the ensemble mean values following the approach of VD18. This approach can allow us to determine which ReOsc model dynamics are causing asymmetries or variations in the ENSO phase space. The following three scenarios are presented: asymmetries in the growth rates (second column of Fig. 8), coupling (third column of Fig. 8), and forcing strength (right column of Fig. 8).

The spread in the mean S values in the phase space of the linear ReOsc model fits (first row Fig. 8) is much smaller in each of the three sensitivity experiments, suggesting that the spread in mean S is resulting from a more complex combination of several parameters, but does not result from any of the three asymmetries tested. The ensemble mean S value is mostly affected by asymmetries in the coupling and forcings, which both have opposing effects. Here the asymmetries in linear coupling do have some similarities with the observed, but they lack the non-linear elements. Asymmetries in the growth rates have little impact on the mean S.

The phase space variations of the extreme values (S > 2) are affected by all their parameter asymmetries, but most strongly by the asymmetries in the coupling and forcings (second row Fig. 8). The phase space variations of the growth rates are clearly linked to the asymmetries in the growth rates, but asymmetries in the coupling and forcings do also have a significant effect (third row Fig. 8).

The sensitivities of the phase speed to the linear ReOsc parameters show complex behaviour (last row Fig. 8). When we only consider the asymmetries in the growth rates or forcing strength, we get much larger phase speeds than observed in the CMIP ensemble. This suggests that the asymmetry in the coupling parameters has a strong impact on the phase speed. We can further notice that none of the three asymmetries reflect the overall variations in the phase speed, suggesting that more complex combinations of different ReOsc model parameters affect the phase speed.

3.4 Wind dynamics for CMIP models

The surface wind response to SST anomalies is a key element of the Bjerknes feedback and is one of the main processes that control ENSO growth rate and period. While this relation does not directly relate to the ENSO phase space or the ReOsc model, it is implicitly related to both as it affects all parameters of the ReOsc model, in particular the growth rate of T.

Figure 9a shows the cross-correlation between the NINO4 region (5°S to 5°N, 160°E to150°W) zonal wind stress (τ_x) and T for the observations and the CMIP models. The observed cross-correlation shows a strong and mostly in-phase relation between τ_x and T, consistent with a strong response of τ_x to variations in T. The relation is, however, strongest when τ_x leads T by about one month, indicating also that variations in τ_x are forcing variations in T.

Overall, we see a close agreement between CMIP 5 and 6, and between the models and the observations. Despite the good agreement, there are some deviations in the CMIP models from observations. The CMIP models have a slightly low cross-correlation at lag zero, indicating a weaker response of τ_x to T. The maximum cross-correlation in the CMIP ensembles is when τ_x leads T by about two months, indicating a stronger forcing of τ_x on variations in T than observed. The spread in these relations is quite large in the CMIP ensembles, indicating quite diverse behaviour in the individual models.

Figure 9b shows the regression value of τ_x per T, which is often referred to as the atmospheric Bjerknes feedback (Bjerknes 1969). The observed atmospheric Bjerknes feedback is approximately 12.5*10^− 3 Nm^− 2/^oC, whereas the CMIP ensemble averages are 7.4 (CMIP5) and 7.8*10^− 3 (CMIP6) Nm^− 2/^oC. Thus, CMIP models underestimate this important feedback by approximately 40%, with none of the CMIP models reaching the observed value and no significant improvement from CMIP 5 to 6. There is a substantial spread within the CMIP model ensemble that is also of similar magnitude in both CMIP 5 and 6. These results are largely consistent with previous studies (Guilyardi et al. 2012; Bellenger et al. 2014; Vijayeta and Dommenget 2018; Guilyardi et al. 2020).

To further examine the variability in T, τ_x and h we looked at the standard deviation of these variables (Fig. 10). The standard deviation of T in the CMIP model averages closely resembles observations, with the CMIP5 model average being smaller than the CMIP6 value (Fig. 10a). However, there are significant differences between individual CMIP models, with some models having significantly smaller and others larger standard deviations than the ensemble model or observed.

The observed τ_x has a significantly larger standard deviation than both CMIP5 and CMIP6 model averages (Fig. 10b). This would be consistent with the weaker T forcing on τ_x. Some CMIP models show standard deviations that are closer to the observation, while others have noticeably smaller standard deviations. The results suggest that the weaker τ_x variations may be a cause or a reflection of the weaker τ_x relation to T in the models.

The CMIP6 ensemble mean standard deviation for h is very close to the observed, whereas the CMIP5 ensemble mean is a bit smaller, and some show large deviations for the ensemble mean (Fig. 10c). However, more relevant for the ENSO phase space discussion is the variability of h_shift (Fig. 10d). This is significantly weaker than the observed h for both CMIP ensembles. Given a similar cross-correlation between h_shift and T in the CMIP model as in the observations (Fig. 1b) and the smaller standard deviation of h_shift in the CMIP as shown above, we would expect larger values of a₁₂ in CMIP models than observed (see Fig. 7e), since the regression values are divided the smaller standard deviations of h_shift. Thus, per h_shift anomaly in the CMIP models we get a larger T anomaly than in observations. It should be noted here that this does not need necessarily imply a larger sensitivity of T to h_shift, because correlation does not allow to determine causality here.

In this study, we examined how well the CMIP5 and CMIP6 models can simulate the observed ENSO phase space characteristics. In the context of the ReOsc model, the ENSO phase space presents ENSO as a cycle for which we can analyze important statistics, such as extreme value probabilities, growth rates, and phase speeds, as a function of the different phases of ENSO. This allows us to consider linear and non-linear aspects of the amplitude, growth rates, and phase transition speeds at the same time.

A key starting point for the analysis of the ENSO phase space is the cross-correlation between T and h, which is assumed and observed to be out-of-phase (orthogonal) to allow for a proper evaluation of the phase space statistics, and it is an assumed relation for an idealized linear ReOsc model. We found that this relationship is clearly not out-of-phase for the CMIP models but is shifted to negative lag-zero cross-correlations. This shift in the cross-relation between T and h is linked to a regional shift in the ENSO patterns in CMIP models, with thermocline depth variability being shifted to the eastern equatorial Pacific.

If the eastward shift in the thermocline depth variability (h_shift) is considered in estimating the ENSO phase space, then we find a much better agreement between models and observations. This suggests that CMIP models do simulate the ENSO cycle with some regional displacements in the ocean-atmosphere interactions.

The main characteristics of the observed phase space are mostly well simulated by the CMIP ensembles. The key feature of the observed higher likelihoods of extreme El Niño to discharge states and reduced likelihoods of extreme La Niña to recharge states are captured by the CMIP ensembles mean with a high correlation. However, the models strongly underestimate the magnitude of non-linearity in this aspect. This suggests that although models, in their ensemble mean, capture the non-linearity of phases of ENSO, the non-linear processes responsible for these characteristics are not strong enough (weaker non-linear processes in CMIP models than observation as suggested by An et al. 2005; Sun et al. 2016) or are too linear.

The observed non-linearity in growth rates of the ENSO phase space are less well captured by the CMIP ensembles. While they do capture the enhanced growth rates during the El Niño states, they lack increased growth rates during the recharge state, and they lack strong decay (negative growth rates) of ENSO amplitudes in the transition from discharge to La Niña states.

The observed variations in the ENSO phase speed are also captured by the CMIP ensembles, but only with low correlations. The models can capture the phase speed in the transition from El Niño to the discharge state, but the transition is too fast (or too clear) from the La Niña to the recharge state, where the observed transition is much slower (or less clear). Combined with the lack of variations in the growth rates this suggests that the ENSO cycle in CMIP models is too regular, with too little differences between the El Niño and La Niña phases.

The ENSO phase space is strongly linked to the idea of the ReOsc model. It therefore does make sense to explain the observed and simulated characteristics of the phase space by the ReOsc model dynamics. We therefore fitted the linear ReOsc model parameters to the observed and simulated T and h data but encountered a number of problems. First, the regional shift in the ENSO pattern in CMIP models does strongly affect the ReOsc model estimates. Second, the most interesting aspects of the observed ENSO phase space must result from non-linear dynamics, where it is unclear what kind of dynamics these are (Dommenget et al. 2023).

We further analyzed the relation between SST (T), wind stress (τ_x) and thermocline depth (h) variability. Like previous studies, we find that the linear relation of τ_x and T is much weaker than observed. This weaker relation is likely to contribute to the weaker non-linear ENSO phase space characteristics, since previous studies have indicated that the wind-SST relation is a key element of the non-linear behavior of ENSO. The variability of h in the eastward shifted region of the Pacific, which is most important for CMIP models, is also weaker than observed. This, combined with the weaker wind-SST relation indicates that the ENSO mode in the CMIP model is more wind-driven and less influenced by the thermocline variability. A previous study by Zhao and Sun 2022 found that the CMIP models have a shallower thermocline depth in the eastern equatorial Pacific than observed, which could explain the weaker response of the above-mentioned processes. However, further analysis into the SST, wind stress and thermocline depth interactions are needed to understand how they impact the ENSO phase space. It is worth noting that Planton et al. 2021 made no mention of thermocline depth biases in their ENSO metrics package, which is quite surprising, given that thermocline depth and their biases have a significant impact on the ENSO phase space characteristics and its dynamics as we found in our study.

While the ensemble mean of the CMIP models does show good agreement with the observations in many aspects, there is a widespread within the model ensembles. Some models show opposite non-linear ENSO phase spaces from those observed, marking severe biases in the model behavior. Other models clearly outperform the ensemble mean, indicating that much more realistic ENSO simulations are possible within the CMIP model framework.

In the comparison of the CMIP5 versus the CMIP6 ensembles, we find no substantial differences in the ENSO phase space dynamics, indicating no substantial improvements from CMIP5 to CMIP6. In particular, we find no differences in the skill of the model in simulating the significant non-linear ENSO behavior. This is a bit disappointing and worrying, as it indicates that no progress has been made from CMIP5 to CMIP6. This finding is largely consistent with Planton et al. 2021.

More future work is required to better understand the limitations of the CMIP models in simulating the non-linear ENSO phase space characteristics. A better understanding of how the ReOsc model can incorporate the non-linear aspects to simulate the observed ENSO phase space would be important to better evaluate CMIP models. The regional shift in the simulation of ENSO dynamics are also an important bias. We still don't know why these biases exist and what are the reason for these discrepancies, necessitating further analysis and a deeper understanding of this question, as well as opportunities for future studies to address these concerns.

Competing Interests

None

Funding

This research is funded by the Australian Research Council (ARC), discovery project “Improving projections of regional sea level rise and their credibility” (DP200102329) and the Centre of Excellence for Climate Extremes (CLEX) Grant Number: CE170100023.

Authors Contributions

Priyamvada Priya formulated the concept and designed this study. All authors contributed to the material preparation, collecting data, and analysis of this study. Priyamvada Priya wrote the first draft of the manuscript, and all authors commented and provided feedback on that. All authors read and approved the final version of the manuscript.

Acknowledgements

The authors would like to thank Monash University Melbourne, Australia, for providing all the resources and instruments required to conduct this research. This research is funded by the Australian Research Council (ARC) Centre of Excellence for Climate Extremes (CLEX), CLEX Grant Number: CE170100023, and the ARC discovery project “Improving projections of regional sea level rise and their credibility” (DP200102329). The authors would also like to thank Australia's computing facility National Computational Infrastructure (NCI) and CLEX for providing us with a wide range of facilities and computational tools for this study.

Data Availability

The data sources for this study are cited in the text.

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ENSO phase space dynamics in CMIP models

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Abstract

Figures

1. Introduction

2. Data and Methods

2.1 Observational datasets

2.1 CMIP Simulations

2.2 ENSO Phase space

2.3 ReOsc Model

3. CMIP simulations of ENSO dynamics

3.1 Regional variations of T and h

3.2 ENSO phase space analysis for CMIP models

3.3 Fitted linear ReOsc model for CMIP

3.4 Wind dynamics for CMIP models

4. Summary and discussions

Declarations

Competing Interests

Funding

Authors Contributions

Acknowledgements

Data Availability

References

Status:

Version 1