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Evaluation of ENSO simulations in CMIP5 models: A new perspective based on percolation phase transition in complex networks

In this study, the performance of CMIP5 models in simulating the El Niño-Southern Oscillation (ENSO) is evaluated by using a new metric based on percolation theory. The surface air temperatures (SATs) over the tropical Pacific Ocean are constructed as a SAT network, and the nodes within the network are linked if they are highly connected (e.g., high correlations). It has been confirmed from reanalysis datasets that the SAT network undergoes an abrupt percolation phase transition when the influences of the sea surface temperature anomalies (SSTAs) below are strong enough. However, from simulations of the CMIP5 models, most models are found incapable of capturing the observed phase transition at a proper critical point Pc. For the 15 considered models, four even miss the phase transition, indicating that the simulated SAT network is too stable to be significantly changed by the SSTA below. Only four models can be considered cautiously with some skills in simulating the observed phase transition of the SAT network. By comparing the simulated SSTA patterns with the node vulnerabilities, which is the chance of each node being isolated during a ENSO event, we find that the improperly simulated sea-air interactions are responsible for the missing of the observed percolation phase transition. Accordingly, a careful study of the sea-air couplers, as well as the atmospheric components of the CMIP5 models is suggested. Since the percolation phase transition of the SAT network is a useful phenomenon to indicate whether the ENSO impacts can be transferred remotely, it deserves more attention for future model development.


Results
Simulated "percolation phase transition" in CMIP5 models. In this study, we analyzed the historical simulations of 15 CMIP5 models (Table 1). For each model, the surface air temperatures over the tropical Pacific with the domains 120°E to 75°W and 20°N to 20°S were constructed as a network with a resolution of 5° × 5° (Fig. 1). To study how the SAT network responds to the anomalous SST warming/cooling in the tropical central eastern Pacific, there are two quantities of interest 38 , as follows: i) The percentage of isolated nodes P. If a node is not connected to any other nodes, we consider it an isolated node. The percentage of isolated nodes P is thus defined as the fraction of isolated nodes over the total nodes 40 , which is a quantity that measures the intensity of the influences of ENSO on the upper SAT network (see Methods). ii) The giant component size S. In a network, if there are some nodes that any two of them can be connected with at least one path, we consider these nodes together as a cluster. The giant component size S is then defined as the size of the largest cluster divided by the total un-isolated nodes, which is a non-linear measure in percolation theory that can represent the network state (see Methods) [32][33][34][35] .
As reported in previous studies, close relations between the two quantities and the Niño3.4 index were found using reanalysis datasets 37,38 . In Fig. 2, we show the giant component size S (red) and the percentage of isolated nodes P (blue) for each SAT network. From reanalysis data (Fig. 2a), when the standardized monthly Niño3.4 index is larger (smaller) than 0.5 (−0.5), nodes in the upper SAT network are more likely to become isolated (P increases) and the giant component size becomes smaller (S decreases). In some cases, S falls abruptly from a high level to a low level, indicating a phase transition (Fig. 2a) 34,35,37,38,40 . From the model simulations, the anti-correlations between P and S are well reproduced for nearly all the models (Fig. 2b-p), even though the simulations from some models (CNRM-CM5, IPSL-CM5B-LR, etc.) are more violent, while from some others (INMCM4, MRI-ESM1, etc.) are more stable. However, regarding the percolation phase transitions, different SCIEnTIfIC REPORtS | (2018) 8:14912 | DOI: 10.1038/s41598-018-33340-y simulations are produced. As shown in Fig. 3, we classified all of the considered time points into two groups according to the definition of an ENSO event. That is, if the standardized three-month running mean Niño3. 4 index ≥ (≤) 0.5 (−0.5) for five consecutive overlapping three-month periods, we consider it an El Niño (La Niña) event. Time points in these events are gathered as the "ENSO" group, while other points belong to the "Normal" group. For the "Normal" group, the variations of S verus P are plotted in the left-hand panel of Fig. 3ap. In Fig. 3a (reanalysis data), one can see that the giant component sizes S of most time points are well above 0.8, implying a well connected network. There is no abrupt changes of S. This behavior was well simulated by most models (Fig. 3b-p). Only in the models GFDL-CM3, GFDL-ESM2G, HadGEM2-CC, HadGEM2-ES, and IPSL-CM5B-LR, a rightward shift of the S − P pattern is found, indicating overestimated P values. In the "ENSO" group, however, an abrupt decrease of S was observed using the reanalysis data (Fig. 3a). Although the S values are higher than 0.8 for most time points during ENSO events, as long as the fraction of isolated nodes reaches the critical points (P c = 0.48), the S drops to a very low level (approximately 0.4), indicating a percolation phase transition. This phenomenon was not reproduced by most models. As in GFDL-CM3, GFDL-ESM2G, and IPSL-CM5B-LR models, the percolation phase transition happens at either larger or smaller P values, while for HadGEM2-CC, HadGEM2-ES, INMCM4, and MRI-ESM1, no phase transition is even detected. Only in CNRM-CM5, FGOALS-s2, and NorESM1-M does a reasonable percolation phase transition seem to be reproduced.
Since the "Normal" group gathers the cases in which the SSTA in the tropical central eastern Pacific are moderate, and have limited influences on the upper SAT network 15,20,21,24,37,38 , the simulated S − P pattern mainly represents the models' ability to model the atmospheric processes. As shown in the left-hand panel of Fig. 3b-p, most models well reproduced the variations of S under different fractions of isolated nodes (P). On the  15,20,21,24,37,38 . Accordingly, the S − P patterns shown in the right-hand panel of Fig. 3b-p mainly indicate the models' performance in capturing the air-sea interactions, which are found to be poor even by visual inspection of Fig. 3.
It is worth noting that not all the time points within an ENSO event point to low S values. The percolation phase transition does not occur with the arising of an ENSO event. Only when the fraction of isolated nodes reaches 48% do the S values decrease from a higher level (S > 0.6, above 0.8 for most cases) to a lower level (S < 0.6, approximately 0.4 for most cases). This means that even if the SAT network is influenced by the SSTA below, as long as the impacts are not strong enough, a percolation phase transition cannot be triggered. Accordingly, we further divided the "ENSO" group into two parts: one includes the cases when the percolation phase transition is not triggered (S > 0.6), and the other is for the time points at which the percolation phase transition occurred (S < 0.6). To better evaluate the models' ability to simulate the percolation phase transition, we further presented the Taylor diagrams 11,41 for the "Normal" group, the "ENSO" group with S > 0.6, and the "ENSO" group with S < 0.6 separately. As shown in Fig. 4, results from the reanalysis dataset are considered as the "reference" (black point in Fig. 4), while results from models are marked in the diagram with other colors. The cosine of the polar angle represents the spatial correlation coefficient (SCC) between the observed (reanalysis dataset) and the modeled S − P patterns 41 . The distance from each point to the origin is the standard deviation (SD), while the distance to the "reference" point is the RMSE 41 . In general, the SCC refers to phase simulation while standard deviation refers to amplitude simulation. Thus, a model simulation that is relatively accurate would lie near the dotted arc (indicating similar variance) and close to the reference point. From Fig. 4a, it is clear that the S − P patterns simulated by more than half of the models have similar SD to that calculated from the reanalysis dataset. As for the SCC, except the GFDL-CM3, GFDL-ESM2G, HadGEM2-CC, HadGEM2-ES, and MIROC-ESM models, all of the other models have high SCCs between 0.5 and 0.8 (see Fig. 4). Accordingly, most models can reproduce the S − P pattern for the "Normal" group with small RMSE. For the S − P patterns in "ENSO" groups with S > 0.6, similar results are found as in the "Normal" groups ( Fig. 4b). Most models have reasonable simulations of the S − P pattern with similar SD and high SCC (0.6-0.8). However, regarding the "ENSO" groups with S < 0.6 ( Fig. 4c), poor simulation skills of the models are found. In addition to the four models without percolation phase transition (HadGEM2-CC Fig. 3. (a) Shows the results for the "Normal" group, (b) for the "ENSO (S > 0.6)" group, while (c) the "ENSO (S < 0.6)" group. Each dot represents a model as shown in the bottom box, while the black dot stands for the results calculated from NCEP reanalysis data, which serves as the reference. The azimuthal positions represent the spatial correlation coefficients between the S − P pattern obtained from the reanalysis data and those simulated from models. The radial distance from origin to each dot stands for the standard deviation, and the distances from the black dot to the other dots are the root mean square errors (RMSE). It is worth to note that, since the percolation phase transition is missed by To quantify the models' performance in simulating the S − P patterns, we further calculated the skill scores as defined by 41 : where R is the spatial correlation coefficient, σ f the spatial standard deviation of the simulated S − P pattern divided by that of the observed S − P pattern, and R 0 the maximum correlation attainable 42 . Here, we assumed that R 0 = 1. As shown in Fig. 5 and Table 2, high skill scores are found in the "Normal" group. There are six models with skill scores of H ≥ 0.8, and nearly all the models (13 out of 15) have scores higher than 0.7. As discussed above, these high skill scores indicate reasonable abilities of the current models to simulate atmospheric processes. In other words, the percolation properties of the SAT network can be well modeled if the atmospheric field is not influenced by the SST anomalies below. As for the "ENSO" group with S > 0.6, for most models, the skill scores are close to those obtained from the "Normal" group. There are eight models with skill scores higher than 0.8, and the scores of all 15 models are higher than 0.6. Therefore, it is evident to speculate that, as long as the ENSO impacts are not strong enough to trigger a phase transition, the current models have the ability to simulate the percolation properties of the SAT network. However, regarding the "ENSO" group with S < 0.6, which represents the cases when the percolation phase transitions are triggered, much lower scores are obtained for all the models. As shown in Fig. 5 and Table 2, only four models (NorESM1-M, CNRM-CM5, FGOALS-s2 and CESM1-FASTCHEM) have scores higher than 0.7, but nine models have scores no higher than 0.5. Combined with the simulated S − P patterns shown in Fig. 3, we can only cautiously recommend four models (NorESM1-M, CNRM-CM5, FGOALS-s2 and CESM1-FASTCHEM) that have certain abilities to simulate the percolation phase transition. In other words, it is still challenging to fully capture the sea-air interactions over the Pacific. From the Figure 5. Skill scores of the 15 models on simulating the S − P patterns in "Normal" group (blue bars), "ENSO" group with S > 0.6 (red bars), and "ENSO" group with S < 0.6 (green bars). The specific values of the skill scores are listed in Table 2.  Table 2. Skill scores of the 15 models on simulating the S − P patterns in "Normal" group, "ENSO" group with S > 0.6, and "ENSO" group with S < 0.6.
SCIEnTIfIC REPORtS | (2018) 8:14912 | DOI:10.1038/s41598-018-33340-y perspective of percolation, most models failed in properly simulating the responses of the SAT field to the anomalous SST warming/cooling. "Node vulnerability" and "SSTA" composites in CMIP5 models. Since the giant component size S is closely related to the node-connection pattern of a given network, to study why some models (such as NorESM1-M) reproduced the observed S − P pattern while some others (such as INMCM4, MRI-ESM1, etc.) failed, it is straightforward to investigate the node vulnerability F i in the network. As described in the "Methods" section, F i is a quantity that measures how vulnerable a node is when the network is influenced 37,38 . It is defined as the ratio of the times that a given node is isolated to the entire time period of an ENSO event. If the ratio is high, we consider the node to have high vulnerability. Based on reanalysis data (Fig. 6a), it has been found that nodes in the SAT network are highly vulnerable over the tropical central eastern Pacific. This is reasonable since ENSO events have the strongest influences on the upper SATs over this key region 1,43,44 . Consequently, node links in this region are easy broken and the nodes are more likely to be isolated 15,20,21 . From model simulations, we find that models with good skills in simulating the percolation phase transition (e.g., NorESM1-M) have similar F i patterns as that obtained from reanalysis data. However, those that cannot reproduce the desired phase transition (e.g., INMCM4, GFDL-CM3, etc.) fail in fully capturing the observed F i pattern. For instance, one can see clearly that the simulated vulnerable areas determined by GDFL-CM3 (Fig. 6g) and GFDL-ESM2G (Fig. 6h) are much larger than that obtained from reanalysis data. This overestimated node vulnerability may lead to a percolation phase transition, but at larger P values (Fig. 3g,h). For models with smaller vulnerable areas (e.g., INMCM4 and MRI-ESM1), the node links are too stable. This underestimated node vulnerability may be responsible for missing the percolation phase transition (Fig. 3o,p). For other models such as FGOALS-g2, HadGEM2-CC, HadGEM2-ES, IPSL-CM5B-LR, MPI-ESM-P, and MIROC-ESM, the simulated F i distributions are remarkably different from the observed pattern, which may contribute to the biased simulations of the percolation phase transition.
In addition to the tropical central eastern Pacific, one may also note the fairly strong node vulnerabilities F i in the far western Pacific (Fig. 6a). Since the far western Pacific is a key region for the generation of ENSO events, i.e. precursory ocean-atmosphere signals (such as the anomalous oceanic thermocline depth signal 45 and the westerly wind events in the atmosphere [46][47][48] ) that could trigger an ENSO event usually occur in the far western Pacific 49-51 , the strong F i values in this region may have physical meanings. As shown in Fig. 6c,d, some models with certain abilities to simulate the percolation phase transition also reproduced strong F i values in the far western Pacific. While for some models with poor performance in simulating the percolation phase transition, the F i patterns in the far western Pacific were not fully captured. These findings suggest that the far western Pacific may be also important for the simulations of the percolation phase transition in the SAT network.
Since the node vulnerability in the SAT network is related to the SST anomalies 37 , the simulated SSTA might give an in-depth understanding of why the percolation phase transition differs greatly among different models. Especially considering that percolation phase transition measures the models' ability to capture sea-air interactions, it is essential to further compare the simulated SSTA with the node vulnerability in the upper SAT network. As shown in Fig. 7, since the node vulnerability is related to the magnitude of SSTA, the composites of  Fig. 6, (a) shows the results from NCEP reanalysis data, while (b-p) shows the simulated results from the 15 models. The absolute SST anomalies were calculated as the composites of both the positive anomalies and negative anomalies (taking the absolute values). Nodes with stronger absolute SST anomalies are marked by dark red color. The figure is generated using Ferret (version6.9, http://ferret.pmel.noaa.gov/Ferret/documentation/release-notes/version-6-9-release-notes).
SCIEnTIfIC REPORtS | (2018) 8:14912 | DOI:10.1038/s41598-018-33340-y SSTA during ENSO events (La Niña events being represented by the absolute value) are calculated. From reanalysis data (Fig. 7a), one can see that the regions with strong absolute SSTAs are exactly the same regions with high F i (Fig. 6a). This is reasonable as the SSTA should have the strongest influences on the upper SAT of the same region 1,43,44 . In other words, if the sea-air interaction is fully captured, the F i patterns in the tropical central eastern Pacific should be similar to the SSTA patterns in Fig. 7. However, by comparing the SSTA patterns in Fig. 7 with the F i patterns in Fig. 6, we found mismatches in many models. For example, in the GFDL-CM3 and MIROC-ESM models, the vulnerable regions with high F i (Fig. 6g,n) are much larger than the regions with strong SSTA (Fig. 7g,h and n). In the FGOALS-g2, MPI-ESM-P, and INMCM4 models, the simulated vulnerable regions in the tropical central eastern Pacific (Fig. 6f,m and o) are smaller than the regions with strong SSTA (Fig. 7f,m  and o). All these models with mismatches between the SSTA patterns and the F i patterns in the tropical central eastern Pacific fail in reproducing the desired percolation phase transition. Only in the models in which the two patterns in Figs 6 and 7 are similar, such as NorESM1-M, was the percolation phase transition reasonably simulated. Accordingly, the improper simulations of percolation phase transition may rely on the mismatches of the SSTA pattern and the F i pattern in the tropical central eastern Pacific, or in other words, the imperfectly simulated sea-air interactions.

Discussion and Conclusions
In this study, we evaluated the ability of 15 CMIP5 models to simulate ENSO from a new perspective, namely as a climate network. By constructing the upper SAT as a network and studying its percolation properties under different SST conditions, we found that most models can simulate the percolation properties of the SAT network successfully at weak SST anomalies, but fail to correctly simulate the percolation phase transition when the SST anomalies were strong enough to trigger a state change of the SAT network. For the 15 models, four models missed the observed percolation phase transition, seven models reproduced the percolation phase transition at incorrect critical point P c . Only four models (NorESM1-M, CNRM-CM5, FGOALS-s2, and CESM1-FASTCHEM) seemed to be able to capture the desired percolation phase transition, but their skill scores calculated from Taylor diagrams were relatively low. As the percolation property essentially measures the fragmentation and functionality of a network, its phase transition indicates a state conversion 40 . Therefore, whether the models simulated the percolation phase transition of the SAT network successfully is related to the models' ability to fully capture the influences of SSTA on the upper SAT field, or, in other words, the sea-air interaction. In this work, we find most models fail to fully capture the sea-air interactions. By comparing Figs 6 with 7, it is found that for most models the regions with strong absolute SSTA are different from the regions with high node vulnerability in the SAT network. Only a few models, such as NorESM1-M and FGOALS-s2, produced similar spatial patterns of the node vulnerability ( Fig. 6) and the SSTA (Fig. 7). This inconsistency implies the improper simulations of the percolation phase transition may rely on the mismatches of the SSTA pattern and the F i pattern in the tropical central eastern Pacific, which suggests a new direction for further improvement of dynamical models.
It is worth noting that when studying the simulated percolation properties, we used the simulated Niño3.4 index to classify both the "Normal" and "ENSO" groups. Therefore, the low skills of most models in simulating the percolation phase transition are not related to the models' ability to simulate the SST anomalies. More likely, it is the inappropriate atmospheric component, or the coupler of the CMIP5 models that leads to the poor simulations. As discussed in 37 , it has been proposed that the differences in performance between the FGOALS-s2 and FGOALS-g2 models in simulating the percolation phase transition mainly arise from the different atmospheric components. In this study, the simulations of these two models show good agreement with the previous study, although the data length is slightly different between the two works 37 . However, for the other two pairs of models that are the different versions of the same model, GFDL-CM3/GFDL-ESM2G and HadGEM2-CC/HadGEM2-ES, significantly different simulation performance is not shown. In fact, both versions of the HadGEM2 model (HadGEM2-CC and HadGEM2-ES) miss the percolation phase transition (Fig. 3), while both versions of the GFDL model (GFDL-CM3 and GFDL-ESM2G) fail in simulating the transition at the correct P c value. Therefore, more studies are required to improve these models' ability to simulate sea-air interactions.
Finally, we emphasize that the percolation phase transition is not only a metric for the evaluation of ENSO simulations. It may also serve as an indicator for the ENSO impacts. In previous studies 37, 38 , it has been conjectured that the transition of the percolation phase indicates that the upper SAT field has been fully changed by the SSTA, and the influences of ENSO may be further transported to remote regions through an atmospheric bridge 52,53 . If there is no percolation phase transition, however, the influences of ENSO may be mainly trapped in nearby regions. In this study, we can see that even in the "ENSO" group there are also stages in which the phase state stays stable with S > 0.6. This indicates that despite a ENSO event arises, the SAT network may still stay stable until the percentage of isolated nodes exceeds the threshold P c . Accordingly, compared to the Niño3.4 index, checking the percolation properties of the SAT network maybe a better way to judge the ENSO impacts. Although this discussion is beyond the scope of the present work, it deserves more detailed study in the future.

Data and Methods
Data. In this study, historical simulations of 15 CMIP5 models were analyzed. Information on the 15 models is listed in Table 1. For each model, the simulated daily SAT and monthly SST data were downloaded from the Earth System Grid Federation (ESGF) via https://esgf-data.dkrz.de/search/cmip5-dkrz/. All of the data end in 2005 and cover at least 100 years. Using SAT data, the SAT network was constructed over the domain 120°E to 75°W and 20°N to 20 o S (Fig. 1) psd/data/climateindices/ 54 . The reanalysis SAT data were used to construct a SAT network. Since it covers only 69 years (1948-2016) with a spatial resolution of 2.5° × 2.5°, to maintain consistency among different datasets, the SAT networks were constructed using data from 1950 to 2005 with the spatial resolution set as 5° × 5°. As for the standardized three-month running mean Niño3.4 indices, we defined an ENSO event when the indices ≥(≤) +0.5(−0.5) for five consecutive month periods. Monthly SST anomalies (with annual cycle removed) were also used in this study (see Fig. 7). The data were downloaded from NOAA via https://www.esrl.noaa.gov/psd/ data/gridded/data.noaa.ersst.v4.html.

Methods
Surface air temperature Network. In this study, we employed the nonlinear synchronization measure to construct a SAT network. As shown in Fig. 1, each node is marked with numbers from 1 to 306 as node indexes according to the sequence from west to east and from north to south. For the SAT in each node, we first calculated the anomalies by subtracting the long-term mean annual cycle T k (d), where k represents the node index (1-306) and d is the calendar date. For every 30th day t in the considered time span between January 1950 and August 2005, we then computed the time-delayed cross-correlations for each pair of nodes i and j over 365 days before t, with time lags τ between −200 days and 200 days. The coefficient is denoted as . For each time point t, the link strength between nodes i and j can thus be defined as 15,21 : , , , A pair of nodes is considered connected if their link strength is above a threshold Q. In view of systematic biases between different models, it is necessary to standardize link strength W i j t , so that the biases could be eliminated. Here, we determined a threshold Q = 3.1 for the confidence level of 99% (see Supplementary Materials, Fig. S1). Using the Heaviside function, we represented this definition as: where i is the node index from 1 to 306.
Percentage of isolated nodes. In the SAT network, the isolated nodes can be considered as the result of influences from ENSO events. Accordingly, for each time point t, we defined the intensity of influences as 37,38,55 : where P t denotes the fraction of the isolated nodes at time point t (see Fig. 2). Node Vulnerability. To quantify the chance of a node to be isolated under influences, we defined another quantity as 37,38 : i t T i t where L(T) is the length of a given time period (or the number of the total time points), and F i is the fraction of the time points when node i is isolated (see Fig. 6).
Giant component size. Giant component size is used to indicate the phase state of a network and measure its fragmentation and functionality. In a network, if there are some nodes such that any two of them can be connected with at least one path, we consider these nodes together as a cluster. By definition, there is no isolated node in a cluster. By counting the number of the nodes, cluster size is quantified. For the cluster with the largest size (highest number of nodes), we define it as the largest cluster. The giant component size S can then be defined as 37 where N LC t is the number of nodes in the largest cluster, and S t represents the giant component size at time point t (see Fig. 2). According to S, the phase state of a given network could be theoretically classified as stable (S = 1), unstable (S → 0), or metastable (0 < S < 1) [32][33][34][35] . However, in practice, it is difficult to find a full stable network with S = 1, and an unstable network with S → 0. Therefore, the concepts of stable/unstable are normally discussed relatively. As in our study, the phase state of a SAT network is considered relatively stable when S is around 0.8 (Fig. 3a, left-hand panel), while it is considered metastable or relatively unstable when S is around 0.4 (Fig. 3a,  right-hand panel).