ENSO teleconnections with the NAE sector during December in CMIP5/CMIP6 models: impacts of the atmospheric mean state

Abstract

This study investigates how the atmospheric mean state influences the El Niño-Southern Oscillation (ENSO) teleconnections with the North Atlantic-European (NAE) region, using ERA5 and CMIP5/CMIP6 models. By isolating the contributions of heating anomalies in the Niño 3.4 and Tropical Western-Eastern Indian Ocean (TWEIO) regions, we find that in November, the Niño 3.4 teleconnection dominates, projecting onto the positive phase of the North Atlantic Oscillation (NAO). In December, the TWEIO teleconnection prevails, reinforcing the positive NAO via a zonal wavenumber-3 Rossby wave train originating from SouthEast Asia (SEA). Models that fail to simulate the December ENSO teleconnection with the NAE exhibit a weak Rossby wave source in SEA and overly strong subtropical Pacific and Atlantic jet streams, which trap Rossby waves at lower latitudes, affecting the remote atmospheric response over the NAE. This waveguide bias is likely driven by a cold bias in the northern Pacific and Atlantic, a common mean-state error in climate models.

Introduction

The El Niño-Southern Oscillation (ENSO) is the dominant mode of variability at interannual timescales and constitutes one of the most prominent sources of predictability at seasonal^1,2 to annual³ timescales. The Pacific-North-American (PNA) pattern stands out as the most recognized ENSO teleconnection with the extratropics, reaching its peak during winter^4,5. The ENSO signal in the North Atlantic and European (NAE) sector is generally less robust in terms of amplitude, spatial structure, and significance compared to the canonical PNA pattern^6,7. Nevertheless, it remains one of the major sources of predictability in this region.

Recent studies have addressed the complexity and seasonality of ENSO teleconnection with the NAE^{8,9,10,11,12,13,14,15,16,17}. Dynamical mechanisms have been proposed to explain the propagation of ENSO teleconnection with the NAE sector, including both tropospheric^7,11,15,17 and stratospheric pathways^15,16,18,19.

The ENSO-induced atmospheric circulation anomalies in the NAE feature a meridional geopotential dipole, resembling the negative phase of the North Atlantic Oscillation (NAO) in February and March, and its positive phase in November and December⁷. The transition of the ENSO-related signal in the NAE region has been linked to an interfering teleconnection triggered by heating anomalies located in the tropical Indian Ocean (IO)^{7,9,11,13,19,20}. These anomalies, linked to the Indian Ocean Dipole (IOD) mode, can initiate a wavenumber-3 Rossby wave-train, which projects onto the positive phase of the NAO in December¹¹.

General Circulation Models (GCMs) exhibit limitations in accurately simulating the November and December ENSO teleconnection with the NAE region^7,13,20,21. By analyzing historical simulations from models participating in the fifth phase of the Coupled Model Intercomparison Project (CMIP5), a previous study¹³ demonstrated that a poor December ENSO teleconnection with the NAE may be linked to a weak Indian Ocean precipitation dipole, suggesting limitations in the simulation of ENSO-IOD coupling. Further research⁷ showed that while seasonal hindcasts capture the January–February ENSO teleconnection with the NAE region reasonably well, the November–December teleconnections are often underestimated. This discrepancy may stem from model biases in simulating the interference between teleconnections originating from the Indian and Pacific Oceans.

To our knowledge, only a very limited number of studies analyze how model biases affect the simulation of ENSO and TWEIO teleconnections with the NAE region. For instance, previous research²² found that errors in the atmospheric response to ENSO in the northeastern Pacific could be partially attributed to biases in the jet stream, which can alter the propagation of Rossby waves from the tropical Pacific to the North Pacific. In contrast, it was concluded that a poor NAE response to ENSO is primarily due to biases in the tropical Rossby wave source located in the Caribbean Sea. These results are consistent with other studies emphasizing the influence of the Caribbean Sea and Gulf of Mexico Rossby wave sources on the NAE sector^8,17,23,24.

Using a bias correction method on simulated divergence and temperature tendencies, a previous work²⁵ demonstrated that model biases generally do not affect the ENSO-induced atmospheric response in the tropics and North Pacific. Using an intermediate-complexity general circulation model, it was suggested that a stronger Pacific jet stream can meridionally bend ENSO-induced Rossby waves, shifting the upper tropospheric geopotential response equatorward²⁶. Likewise, low-frequency variability in Pacific SSTs has been shown to influence the propagation of ENSO-driven Rossby waves by modifying the background flow¹⁴.

The present study serves as a continuation of previous analyses¹³, utilizing historical simulations from both the CMIP5²⁷ and CMIP6²⁸ datasets. The main objectives of this work are twofold: (1) to evaluate the ability of state-of-the-art coupled models to simulate ENSO teleconnection with the NAE during winter, and (2) to examine the influence of the models’ atmospheric mean state on the simulation of ENSO teleconnection with the NAE.

Results

ENSO teleconnection in reanalysis

Figure 1 shows the upper tropospheric (200 hPa) geopotential height and WAF anomalies regressed against the Niño 3.4 index from November to February. During November (Fig. 1a) and December (Fig. 1b), the ENSO teleconnection with the North Pacific is weaker compared to January (Fig. 1c) and February (Fig. 1d). Over the North Atlantic sector, a positive NAO-like signal emerges in November and persists into December, with its response shifted meridionally relative to the canonical NAO pattern. In November, the northern center of action, represented by an anomalous Icelandic low, is more pronounced than its southern counterpart, the anomalous Azores high. By December, the Icelandic low fades, suggesting a weaker ENSO teleconnection compared to November.

Fig. 1: ENSO teleconnection in the reference dataset during the extended boreal winter season.

ERA5 200 hPa geopotential height anomalies (Z200) and WAF regressed against the normalized Niño 3.4 precipitation index for a November, b December, c January, and d February. Shading indicates the regression of geopotential height anomalies (units m), while vectors represent the WAF (units m²/s²) with a unit vector of 0.2 m²/s². The black solid box indicates the domain (5°S–5°N, 170°W-120°W) used to compute the Niño 3.4 index. The black dashed boxes indicate the domains, namely (25°N–40°N, 25°W–5°E) and (50°N–65°N, 50°W–10°W), used to compute the Eastern Atlantic Gradient (EAG) indices listed in Tables 1 and 2. The light-gray stippling indicates the grid points in which the regression is significant at the 95% level of confidence with respect to a two-tailed t-test. The units of regression are specified per standard deviation.

The extratropical ENSO teleconnection involves strong wave flux in the subtropical Western Pacific regions in November (Fig. 1a) and December (Fig. 1b). During December the wave flux weakens in the North Atlantic but strengthens in the western and central Pacific at mid-latitudes (Fig. 1b). The canonical PNA teleconnection pattern appears in January (Fig. 1c) and February (Fig. 1d), involving a strengthening of the Aleutian low and positive geopotential anomalies extending over the North American sector. During February, the atmospheric response in the NAE sector significantly projects onto the negative phase of the NAO (Fig. 1d).

Figure 2 illustrates the 200 hPa geopotential height and WAF anomalies regressed against the pure TWEIO and pure Niño 3.4 indices for November and December. Starting from November (Fig. 2a), the pure TWEIO regression shows a strong wave activity flux originating from SouthEast Asia (SEA: 20°N–40°N, 80°E–140°E; black box in Fig. 6a). The wave-train exhibits an anomalous trough located in the SEA, followed by an anomalous ridge over the Japanese sector. In December (Fig. 2b), the pattern exhibits a wave-train characterized by alternating geopotential height anomalies with a zonal wavelength of approximately 120°, corresponding to a zonal wavenumber of 3. The meridional scale of the wave spans the 30°N–60°N latitude band, indicating a meridional wavenumber of approximately 6. The zonal and meridional wavenumbers are consistent with the atmospheric waveguide (Fig. S4c), which exhibits total stationary wavenumbers of 5–7 over the SEA region. This wave eventually reaches the NAE sector and projects onto the positive phase of the NAO.

Fig. 2: Pure ENSO and TWEIO teleconnections in the reference dataset during boreal November and December.

ERA5 200 hPa geopotential height anomalies (Z200) and WAF regressed against the pure (i.e., independent of ENSO variability) TWEIO precipitation index for a November, b December. c, d as in a, b but for the pure (i.e., independent of TWEIO variability) Niño 3.4 regression. Shading indicates the regression of geopotential height anomalies (units m), while vectors represent the WAF (units m²/s²) with a unit vector of 0.2 m²/s². The black solid boxes in a indicate the western (10°S–10°N, 40°E–80°E) and eastern (10°S–10°N, 100°E–140°E) Indian Ocean domains used to compute the TWEIO index. The black solid box in c indicates the domain (5°S–5°N, 170°W–120°W) used to compute the Niño 3.4 index. The light-gray stippling indicates the grid points in which the regression is significant at the 95% level of confidence with respect to a two-tailed t-test. The units of regression are specified per standard deviations.

Conversely, the pure ENSO teleconnection with the NAE is stronger in November (Fig. 2c) than in December (Fig. 2d). In November, the upper tropospheric atmospheric response to the pure ENSO signal resembles the positive phase of the NAO (Fig. 2c). In December the circulation anomalies weaken, and a positive geopotential anomaly develops stretching from northwestern North America to the North Atlantic (Fig. 2d).

The monthly Niño 3.4 regressions, shown in Fig. 1, can be seen as a linear combination of the pure Niño 3.4 and TWEIO regressions, illustrated in Fig. 2. The ENSO signal in the NAE during November (Fig. 1a) seems to be dominated by the pure ENSO teleconnection (Fig. 2c). In December, the pure TWEIO signal strengthens while the pure ENSO signal weakens over the NAE, and the circulation anomalies over the NAE appear to be primarily attributable to heating anomalies in the TWEIO region.

The monthly contribution of the pure TWEIO index to the Niño 3.4 regressions can be further analyzed by calculating the regression coefficient p* (Eq. (3a)). The regression coefficient for 1950–2014 is 0.86 for November, 0.58 for December, 0.41 for January, and 0.30 for February. Additionally, our analysis of precipitation anomalies regressed against the Niño 3.4 index during the extended winter season (Fig. S1) indicates that heating anomalies in the Indian Ocean weaken during February, resulting in a weakening of the pure TWEIO teleconnection.

Although the regression coefficient in November is larger compared to later months, the partial regression analysis indicates that the pure ENSO teleconnection has a stronger influence than the pure TWEIO teleconnection over the NAE region (Fig. 2a, c). This finding is not consistent with previous research¹¹, which suggested that the pure TWEIO contribution dominates over the pure Niño 3.4 contribution in driving the NAO-like pattern during November. We believe that this discrepancy may be, at least in part, explained by the use of different reference datasets and periods.

To test the first hypothesis, we applied partial regression analysis to four datasets—ERA5, ERA5 with GPCP, NCAR-NCEP, and NCAR-NCEP with GPCP—covering the common period 1979–2024. For each dataset, we calculated the Eastern Atlantic Gradient (EAG; dashed boxes in Fig. 1a) index, defined as the difference between the average 200 hPa geopotential height anomalies in the two regions (25°N–40°N, 25°W–5°E) and (50°N–65°N, 50°W–10°W)⁷. Using partial regression analysis, we find that:

$${{EAG}}_{({Ni}{{\tilde{{n}}}}o{\it{3.4}})}={{EAG}}_{(P-{Ni}{{\tilde{{n}} }}o{\it{3.4}})}+{p}^{* }{{EAG}}_{(P-{TWEIO})}$$

(1)

where EAG_(Niño3.4), EAG_(P-Niño3.4), and EAG_(P-TWEIO) represent the regressions of the EAG index onto the Niño 3.4, pure Niño 3.4, and pure TWEIO indices, respectively. The EAG index has been shown to better capture the NAO-like pattern compared to the traditional NAO index⁷. This is because the regions used to define the EAG index are centered approximately 5 degrees south of the points canonically used for the NAO index. However, the results remain similar to those obtained using the NAO index from a previous study¹¹ (not shown). The values for terms in Eq. (1) are summarized in Tables 1 and 2 for November and December, respectively.

Table 1 Contribution of the Indian and Pacific Ocean heating anomalies to the NAO phase across different reference datasets in November

Table 2 Contribution of the Indian and Pacific Ocean heating anomalies to the NAO phase across different reference datasets in December

For November, when combining ERA5 and NCAR-NCEP with GPCP, the pure TWEIO EAG contributes approximately 27% and 26% to the Niño 3.4 EAG, respectively (Table 1). This percentage is relatively similar when using ERA5 alone (28%) but decreases drastically when using the NCAR-NCEP dataset (5%).

In December (Table 2), the ENSO teleconnection projects onto the positive phase of the NAO, albeit more weakly than in November (Table 1). The pure TWEIO EAG accounts for approximately 80% of the Niño 3.4 EAG across all datasets, except for NCAR-NCEP, which shows a lower and statistically insignificant TWEIO contribution of around 17%. As a result, in December, the pure TWEIO signal prevails over the pure ENSO signal. The discrepancies observed when using the NCAR-NCEP reanalysis compared to other datasets underscore its limitations for calculating precipitation-based indices.

To examine the second hypothesis, specifically the sensitivity to different time periods, we used a re-sampling method for the month of November. We randomly selected 45 years from the complete ERA5 dataset (covering 1940–2024) and the NCAR-NCEP dataset (covering 1948–2024) to align with the temporal coverage of the GPCP dataset (covering 1979–2024). From each of the ERA5 and NCAR-NCEP reanalysis products, we generated 10,000 sub-samples. We then calculated the frequency of instances where p*EAG_(P-TWEIO) exceeded EAG_(P-Niño3.4). This frequency constitutes the probability that the ENSO teleconnection is dominated by the pure TWEIO signal. The probability is 12% and 3% for the ERA5 and NCAR-NCEP datasets respectively, supporting a weak contribution of the pure TWEIO teleconnection with the NAE during November.

We conclude that, for the considered period and reference dataset, the influence of the pure TWEIO teleconnection on circulation anomalies in the NAE dominates over that of the pure ENSO teleconnection in December, while the opposite is true for November and February.

ENSO teleconnection in CMIP models

The ability of CMIP5 and CMIP6 models to simulate the ENSO teleconnection is assessed through Taylor diagrams²⁹ (Fig. 3) of the monthly 200 hPa geopotential height anomaly patterns regressed onto the Niño 3.4 index in the NAE sector (here defined as the domain 20°N–80°N, 60°W–0° and shown by the black box in Fig. 4a).

Fig. 3: Performance of the ENSO teleconnection with the NAE in CMIP models.

Taylor diagram based on the 200 hPa geopotential height anomalies regressed against the Niño 3.4 index over the NAE sector (20°N–80°N; 60°W–0°) for a November, b December, and c February. The black star represents the reference dataset, the black-filled square indicates the MME of CMIP6 models, and the white-filled square represents the MME of CMIP5 models. The color-filled symbols denote individual CMIP6 models, while the white-filled symbols denote individual CMIP5 models. The horizontal axis represents the standard deviation of the spatial pattern normalized over the reference standard deviation. The outer semicircle denotes the pattern correlation. Note the different sizes of the x-axis in c compared to (a), (b). CMIP6 models are represented in uppercase letters (e.g., CESM2), while CMIP5 models are shown in lowercase letters (e.g., cmcc-cm).

Fig. 4: ENSO teleconnection and ENSO-related precipitation anomalies stratified by clustered groups.

Ensemble average of 200 hPa geopotential height anomalies (Z200, units m) regressed against the normalized Niño 3.4 precipitation index for a ensemble mean of good models (GOOD) and b ensemble mean of bad models (BAD). c, d as in a, b, but for the regression of precipitation anomalies (units 10⁻⁵ kg/m²s). The black box in a indicates the NAE domain (20°N–80°N, 60°W–0°) used to compute the pattern correlation for the Taylor diagrams in Fig. 3. The light-gray stippling indicates the grid points in which the average is significant at the 95% level of confidence with respect to a one-sample t-test. The units of regression are specified per standard deviations.

The diagrams have been computed for November (Fig. 3a), December (Fig. 3b), and February (Fig. 3c). January was excluded from the analysis due to the weak and statistically insignificant ENSO signal in the NAE region during this month (Fig. 1c). Sensitivity tests demonstrate that the results are not dependent on the choice of the domain (not shown). Table 3 summarizes the key findings from the Taylor diagrams for each month.

Table 3 Summary of the Taylor diagram results stratified over CMIP5 and CMIP6 models

Overall, the models show better agreement with ERA5 in November and February than in December. Noteworthy, the results in Table 3 indicate an improvement in the simulation of ENSO teleconnection with the NAE in CMIP6 models compared to CMIP5, particularly in December. However, it’s important to acknowledge that the number of CMIP6 models is 30% higher than that of CMIP5, which may influence the MME mean performance.

By computing the spatial correlation between the reanalysis and simulated ENSO teleconnection patterns in the NAE region during December, we identified clusters of “good” and “bad” models. Good models are those that produce teleconnection patterns with a positive correlation exceeding 0.6 with the ERA5 patterns in November and December. Conversely, models with a pattern correlation below −0.35 in December are classified as bad models. The classification of clustered models, stratified across CMIP5 and CMIP6, is presented in Table 4.

Table 4 Categorization of CMIP models based on clustering

The clustering process used to select good models ensures the inclusion of those that, to some extent, can simulate the ENSO teleconnection in November and December. As part of a sensitivity test, we also selected good models based on the pattern correlation during December only, using a threshold of 0.75. In this case, the models that fall into this category are: CESM2, NorCPM1, hadgem2-cc, MPI-ESM1-2-LR, CNRM-CM6-1, IITM-ESM, ICON-ESM-LR, EC-Earth3-Veg-LR, cnrm-cm5, cnrm-cm5-2, HadGEM3-GC31-MM, and noresm1-me.). Analogous results to those presented here were obtained (not shown). We, then, analyze the fields obtained by averaging regression patterns and mean states over good models and bad models, especially for December.

Figure 4a, b shows the averaged regressions of the December 200 hPa geopotential height anomalies onto the Niño 3.4 index for the two clusters. Bad models not only fail to capture the positive NAO-like phase in the NAE region but also exhibit a weak geopotential response in the subtropical East Asian and western Pacific sectors. This deficit is also evident when taking the difference of the regressions shown in Fig. 4a, b (Fig. S2a). These findings suggest that the ENSO inter-basin teleconnection with the IO might contribute to this inter-model diversity.

Figure 4c, d shows the precipitation anomalies regressed against the Niño 3.4 index for the clustered groups. Both model groups simulate the ENSO-related precipitation patterns in the Pacific sector reasonably well when compared to the reference dataset (Fig. S1b). However, significant discrepancies are observed in the IO basin, where bad models fail at reproducing the east-west precipitation dipole associated with the ENSO inter-basin teleconnection (see also Fig. S2b for the pattern difference). It is important to note that bad models tend to reproduce a double Inter-Tropical Convergence Zone (ITCZ) bias, marked by excessively strong convergence zones in the South Pacific, Atlantic, and Indian Oceans (Fig. S3).

The TRWSs in reanalysis and CMIP models

Results from ERA5 (top row in Fig. S4), good models (middle row in Fig. S4), and bad models (bottom row in Fig. S4) reveal three main waveguides extending from subtropical Africa, the western Pacific, and the subtropical Atlantic: the Sub-Saharan Africa (SSA: 10°N–30°N, 10°E–40°E; black box in Fig. 6b) waveguide, the subtropical South Asian waveguide, and the subtropical Atlantic waveguide. The lower tropospheric convergence (divergence) associated with anomalous convection (subsidence) in the tropics is balanced in the upper troposphere by divergent (convergent) flow. This mechanism interacts with the atmospheric waveguide and constitutes an essential trigger for the initiation of quasi-stationary planetary Rossby wave-trains^30,31.

Figure 5 shows the 200 hPa TRWS and irrotational wind anomalies regressed against the Niño 3.4 index for the reference dataset (Fig. 5a) and the clustered groups (Fig. 5b, c). In ERA5, the Niño 3.4 heating anomalies induce upper tropospheric divergence, resulting in a negative horseshoe-like TRWS pattern (Fig. 5a). In the Indian Ocean, upper tropospheric divergence (convergence) over the western (eastern) basin interacts with the African (Asian) waveguide. This mechanism, in turn, generates a negative (positive) TRWS located in the SSA (SEA) region. Two secondary Rossby wave sources are also identified north of the Gulf of Mexico and east of the Caribbean Sea, primarily driven by the anomalous rearrangement of the Walker and Hadley circulations in response to Niño 3.4 heating anomalies. Hence, five key regions involving significant TRWSs have been identified: the SSA (10°N–30°N, 10°E–40°E; black box in Fig. 6b), the SEA (20°N–40°N, 80°E–140°E; black box in Figs. 6a and Fig. 9a), the Tropical Pacific (TP: 0°–30°N, 180°E–120°W; black box in Fig. 9c), the Caribbean Sea (CS: 5°N–25°N, 70°W–20°W; black box in Fig. 9e), and the Gulf of Mexico (GM: 25°N–45°N, 110°W–60°W; black box in Fig. 9g).

Fig. 5: ENSO-related Tropical Rossby Wave Source (TRWS) in ERA5 and clustered groups.

December 200 hPa TRWS and irrotational wind regressed against the normalized Niño 3.4 precipitation index for a ERA5, b ensemble mean of good models (GOOD), and c ensemble mean of bad models (BAD). Shading indicates the regression of TRWS anomalies (units 10⁻¹¹ s⁻²), while vectors represent the irrational wind (units m/s) with a unit vector of 1 m/s. Only TRWS that is significant at the 95% level of confidence with respect to two-tailed (ERA5) and one-sample (MME) t-tests is shown. Winds are displayed at the 90% level of confidence with respect to two-tailed (ERA5) and one-sample (MME) t-tests. The units of regression are specified per standard deviations.

Fig. 6: Upper tropospheric atmospheric response to Rossby wave sources in the SouthEast Asia (SEA) and Sub-Saharan Africa (SSA) regions.

ERA5 December 200 hPa geopotential height anomalies (Z200) and WAF regressed against the pure a SEA (20°N–40°N, 80°E–140°E), and b SSA (10°N–30°N, 10°E–40°E) TRWS indices. Shading indicates the regression of geopotential height anomalies (units m), while vectors represent the WAF (units m²/s²) with a unit vector of 0.2 m²/s². The black box in a indicates the domain used to compute the SEA index. The black box in b indicates the domain used to compute the SSA index. The light-gray stippling indicates the grid points in which the regression is significant at the 95% level of confidence with respect to two-tailed t-test. The units of regression are specified per standard deviations.

Many studies have investigated the role of TRWSs in triggering Rossby waves over the tropical and subtropical Pacific, the Caribbean Sea, and the Gulf of Mexico^{8,16,17,22,23,24}. However, much less is known about the TRWSs in the SEA and SSA regions. To fill this gap, we first defined two indices by averaging the 200 hPa TRWS time series over the SEA and SSA regions, indicated by the black boxes in Fig. 6a, b, respectively. Subsequently, we regressed the 200 hPa geopotential height anomalies against the pure TRWS indices (i.e., independent of the Niño 3.4 index), as shown in Fig. 6. Sensitivity tests indicate that the results shown in Fig. 6 are not affected by the size of the boxes used to compute the indices (not shown).

The SEA TRWS explains a zonal wavenumber-3 Rossby wave-train, consistent with the regression based on the pure TWEIO index (Fig. 2b). The regression patterns show a strong wave flux originating from the SEA, where an anomalous trough appears. The wave-train reaches the NAE sector, where geopotential anomalies resemble the positive phase of the NAO. Therefore, we argue that in December, the SEA TRWS plays a key role in modulating the atmospheric response to ENSO in the NAE sector.

The regression of 200 hPa geopotential anomalies and WAF onto the pure SSA TRWS index (Fig. 6b) shows a wave-train similar to that in Figs. 2b and 6a. However, a key difference emerges in the North Atlantic, where strong WAF originates from the Caribbean Sea, arches northeastward, and then propagates back to the SSA. This analysis suggests that the TRWS in the SSA sector is a response to a Rossby wave originating from the Caribbean Sea.

A comparison of the upper tropospheric TRWS simulated by good models (Fig. 5b) and bad models (Fig. 5c) shows that the latter significantly underestimate TRWS in the SSA and SEA regions. The difference in the SEA is primarily attributed to a weaker upper tropospheric anomalous divergent wind field (Fig. S5b) rather than discrepancies in the climatological absolute vorticity gradient (Fig. S5a). In contrast, no substantial differences in the TRWS field are observed across the tropical and subtropical Pacific, the Caribbean Sea, and the Gulf of Mexico.

As a further step, we have calculated for each model and the reanalysis the ENSO-related December 200 hPa TRWS averaged over the five key regions previously identified: the SSA, the SEA, the TP, the GM, and the CS. The values of the averaged TRWS in each of the five regions are plotted in Fig. 7 against the spatial correlation coefficients of the ENSO-related 200 hPa geopotential height anomalies in the NAE sector (see Fig. 3b).

Fig. 7: Relationship between the simulation of ENSO teleconnection with the NAE and the TRWSs averaged over five key regions in CMIP models.

December multi-model relation of the spatial correlation of the ENSO-related 200 hPa geopotential height anomalies over the NAE region (see Fig. 3b) and TRWS averaged over a the SSA (10°N–30°N, 10°E–40°E), b the SEA (20°N–40°N, 80°E–140°E), c the TP (0°–30°N, 180°E–120°W), d the CS (0°–30°N, 80°W–20°W), and e the GM (25°N–45°N, 120°W–60°W). The black star represents the reference dataset, the black-filled square indicates the MME of CMIP6 models, and the white-filled square represents the MME of CMIP5 models. The color-filled symbols denote individual CMIP6 models, while the white-filled symbols denote individual CMIP5 models. The gray dashed line indicates the best linear fit. The Pearson correlation is shown in the top-left corner, with values shown in bold if they are significant at the 95% level of confidence. CMIP6 models are represented in uppercase letters (e.g., CESM2), while CMIP5 models are shown in lowercase letters (e.g., cmcc-cm).

In the models, the relationship of the averaged TRWS over the SSA (Fig. 7a) exhibits a significant correlation, suggesting that models failing to simulate the positive NAO phase during an El Niño event in December also underestimate the SSA TRWS. This finding is consistent with the hypothesis that the atmospheric anomalies linked to the positive NAO phase may propagate to the tropics over the African subcontinent, thereby generating the negative SSA TRWS.

Also, for the SEA case (Fig. 7b), models with a weak TRWS tend to simulate a poor December ENSO teleconnection with the NAE. The simulated relations for the other three key regions, namely the TP (Fig. 7c), the CS (Fig. 7d), and the GM (Fig. 7e), were found to be insignificant at the 95% level of confidence, confirming that a missed positive NAO response cannot be attributed to differences of TRWSs in these regions. However, the following question arises from the previous analysis: Why do bad models fail to reproduce the SSA TRWS, despite their ability to simulate the Caribbean Sea TRWS? This point will be addressed in the following section.

The role of the atmospheric waveguide

Previous studies have highlighted the influence of the jet stream on the propagation of atmospheric Rossby wave^14,26. Here, we examine whether differences in the simulated climatological jet stream can explain variations in the December ENSO teleconnection with the NAE across different models.

Figure 8 shows the differences in the December mean states of upper tropospheric zonal wind and SST between the MME means of bad and good models. Compared to good models, bad models exhibit subtropical Pacific and Atlantic jet streams that are 20–30% stronger and shifted equatorward (Fig. 8a). These stronger waveguides are likely linked to a cold SST bias extending across the northern Pacific and Atlantic basins (Fig. 8b). The cold bias, in fact, tends to enhance (reduce) the meridional SST gradient at low (high) latitudes, thereby strengthening (weakening) upper tropospheric zonal winds through thermal wind balance. Additionally, the stronger Pacific (Atlantic) waveguide, coupled with a colder North Pacific (Atlantic) surface mean state, is also evident in the ensemble mean of bad models relative to the reference (Fig. S6). The weak and statistically not significant differences in tropical Pacific SST mean states (Fig. 8b) suggest that substantial variations in the seasonal phase and amplitude of simulated ENSO events are unlikely.

Fig. 8: Jet stream and surface ocean mean states stratified by clustered groups.

Difference between the ensemble averages of bad and good models (BAD–GOOD) of the December climatology, denoted by the overbar, for a 200 hPa zonal wind (units m/s) and b sea surface temperature (units °C). The contour lines in a represent the 200 hPa zonal wind climatology for good models (units m/s), while the contour lines in b represent the SST climatology for good models (units °C). The light-gray stippling indicates the grid points in which the difference is significant at the 95% level of confidence with respect to one-sample t-test.

At the surface, the Polar high, Aleutian and Icelandic lows are stronger in bad models (Fig. S7). These patterns are in agreement with the difference in the lower tropospheric wind field, which exhibits a cyclonic (anti-cyclonic) pattern over the North Pacific and Atlantic (Arctic). The pressure systems detected in Fig. S7 show a quasi-barotropic structure (not shown) and resemble the leading mode of variability of the extratropical circulation, namely the Arctic Oscillation (AO)^32,33.

To assess the impact of the stronger jet streams on the propagation of planetary waves during December, we implement the Rossby wave ray tracing algorithm in two configurations:

• 1. using the ERA5 200 hPa zonal wind December climatology, hereafter referred to as $\bar{{u}}$_ERA5.

• 2. using $\bar{u}$_ERA5 superposed to the difference between bad and good models of the 200 hPa zonal wind climatology, hereafter referred to as $\bar{u}$_{BAD - GOOD}.

The rays are initiated from the TRWSs located in four key regions previously identified: the SEA, the TP, the CS, and the GM. The SSA TRWS has been excluded because we interpret it as a result of atmospheric anomalies propagating from the Euro-Atlantic sector to tropical Africa. The starting points for the Rossby wave ray algorithm are chosen as the grid point exhibiting maxima/minima of TRWS. The ray position and group velocity are calculated and then stepped forward in time for 10 days for zonal wavenumbers spanning from 1 to 5. The results are shown in Fig. 9 for the $\bar{u}$_ERA5 (left panels) and $\bar{u}$_ERA5 + $\bar{\,u}$_{BAD - GOOD} (right panels) configurations. When interpreting the results presented in Fig. 9, it is important to keep in mind that the wave-tracking algorithm relies on the linear theory for stationary barotropic Rossby waves. This theoretical basis limits the conclusions we can draw, and thus the results should be interpreted with some caution⁶.

Fig. 9: The influence of the atmospheric mean state on Rossby wave ray paths.

December ERA5 200 hPa TRWS regressed against the normalized Niño 3.4 precipitation index (shading with 10⁻¹¹ s⁻²) and Rossby rays (colored scatters) for different zonal wavenumber k, starting from the gridpoint of maximum TRWS in the SEA (20°N–40°N, 80°E–140°E) for a ERA5 ($\bar{{\rm{u}}}$_ERA5) and b ERA5 superposed to the difference of zonal wind climatology between bad and good models ($\bar{{\rm{u}}}$_ERA5 + $\bar{{\rm{u}}}$_{BAD - GOOD}). c–h as in a, b but for the TP (0°–30°N, 180°E–120°W), the CS (5°–25°N, 70°W–20°W), and the GM (25°N–45°N, 110°W–60°W) sectors, respectively. The boxes in the left panels highlight the key areas where the rays are initiated and where TRWS-based indices are computed.

As Fig. 9a shows, Rossby wave-trains of zonal wavenumber-3 (green scatters) follow a track that originated in the SEA region, crosses the Pacific, and eventually reaches the NAE sector. This is consistent with the results from the partial regression analysis of 200 hPa geopotential height anomalies (Figs. 2b and 6a). By superposing a stronger subtropical Pacific jet stream onto the reference basic flow, the trajectory of Rossby waves is affected, particularly those of zonal wavenumber-3 (Fig. 9b).

In ERA5, Rossby wave trains with zonal wavenumbers-2 (yellow scatters) and 3 (green scatters) originating from the TP successfully propagate to the NAE sector (Fig. 9c). However, as shown in Fig. 9d, the stronger Atlantic and Pacific jet streams constrain wavenumber-3 Rossby waves to lower latitudes, potentially altering the Euro-Atlantic atmospheric response to ENSO. Similarly, zonal wavenumber-1 to 3 Rossby waves originating from the CS (Fig. 9e) and GM (Fig. 9g) regions can freely propagate toward the northern Atlantic and European sectors in ERA5. Interestingly, waves with zonal wavenumbers-4 and 5 forced from the CS sector, as well as waves with zonal wavenumbers-2 and 3 from the GM sector, eventually reach the Sub-Saharan region. This finding is consistent with the hypothesis that the SSA TRWS is not a direct ENSO-driven source but rather a response to atmospheric anomalies propagating from the Euro-Atlantic region back to the tropics.

In bad models, the stronger Atlantic waveguide appears to alter the ray paths of wavenumber-3 waves originating from the CS (Fig. 9f) and GM (Fig. 9h) regions, constraining them to lower latitudes. This finding may explain why bad models fail to produce a strong upper-tropospheric atmospheric response (Fig. 4b) and TRWS (Fig. 5c) in the SSA region, despite their ability to simulate the TRWSs over the CS and GM sectors. Furthermore, a comparison of the permitted total wavenumbers in the atmospheric waveguide of bad models (Fig. S4i) with the reference (Fig. S4c) and good models (Fig. S4f) reveals the presence of a “forbidden area” (i.e., a region of imaginary total stationary wavenumber) over the northeastern sector of North America in bad models. The location of the forbidden region supports the hypothesis that Rossby waves originating from the CS and GM are effectively trapped at lower latitudes, preventing their poleward propagation.

These findings indicate that bad models, which significantly underestimate the SEA TRWS linked to ENSO inter-basin teleconnections, are also characterized by stronger Pacific and Atlantic waveguides. The associated intensification of the subtropical jet streams leads to an equatorward shift of the refraction latitudes of zonal wavenumber-3 Rossby waves originating from the tropics, potentially altering the remote atmospheric response in the Euro-Atlantic region.

Discussion

This study examines how the atmospheric mean states of climate models influence the ENSO teleconnection with the North Atlantic and European sector in winter.

The results obtained from ERA5 show that the ENSO teleconnection with the NAE region projects onto the positive phase of the NAO in November and December. Using partial regression analysis and a subsampling technique across multiple reanalysis datasets, we demonstrated that in November, the NAO-like pattern is primarily driven by the pure ENSO teleconnection. This finding contrasts with the results of previous research, which suggested that the November positive NAO is primarily driven by the pure TWEIO signal^7,11.

Tropical rainfall variability in reanalyses is influenced by observational constraints, which improved significantly with the advent of satellite data in the late 1970s. Before this period, reanalysis-derived precipitation over tropical oceans is increasingly uncertain due to sparse observational coverage. Additionally, intraseasonal variability in the Indian Ocean, particularly the Madden-Julian Oscillation (MJO), introduces further discrepancies among datasets on monthly timescales, potentially affecting our teleconnection indices in the pre-1979 decades. However, we have shown that the dominance of the pure ENSO teleconnection over the pure TWEIO signal remains robust even in the post-1979 decades.

Furthermore, decadal to multi-decadal variability, such as that associated with the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO), can modulate the contribution of the pure TWEIO teleconnection to the NAE. To explore this, we computed the November contribution of the pure TWEIO teleconnection to the ENSO teleconnection (see the rightmost column in Table 1) using a 35-year sliding window applied to ERA5 over the 1940–2024 period. The results, presented in Fig. S8, reveal a strong non-stationary signal, even in the post-1979 decades, with a strong decrease in the pure TWEIO contribution after 1985.

Another possible dynamical explanation for the dominance of the pure ENSO signal in November is that the IOD SST anomalies, which peak in October, exhibit a 2-months delayed connection with the atmospheric circulation over the NAE via a precipitation dipole over the Indian Ocean³⁴. Consequently, the November TWEIO precipitation index may not be fully influenced by the October IOD SSTs, thereby reducing the impact of the pure TWEIO signal on the NAE region.

By December circulation anomalies appear to be predominantly influenced by the pure TWEIO teleconnection. The TWEIO heating anomalies are associated with the initiation of a zonal wavenumber-3 Rossby wave-train originating from the SEA region and extending to the NAE sector. In the North Atlantic, this wave-train spatially projects onto the positive phase of the NAO, further supporting the role of the TWEIO in influencing the atmospheric response to ENSO in the NAE sector^{7,9,11,13,19,20,21,35}.

Through a clustering analysis, we identified models that can reasonably simulate ENSO teleconnections with the NAE region (“good models”), distinguishing them from those that cannot (“bad models”). Results indicate that bad models tend to underestimate the Indian Ocean heating dipole, often concurrent with ENSO episodes, leading to a weak tropical Rossby wave source in the SEA sector. In turn this wave source is directly linked to the initiation of a zonal wavenumber-3 Rossby wave-train that impacts the NAE sector with a positive NAO-like fingerprint; thus, its weakness results in feeble teleconnections.

Previous studies have highlighted the existence of a secondary tropical Rossby wave source situated over the Caribbean Sea, associated with the Atlantic waveguide^{8,17,22,23,24}. According to our results, no significant differences are detected in the simulated TRWSs over the Gulf of Mexico and Caribbean Sea between bad and good models.

The mean state analysis of clustered groups reveals that bad models tend to simulate a stronger Pacific and Atlantic waveguides. These patterns reflect the jet stream bias commonly observed in many climate models^{26,36,37,38,39,40}. The stronger subtropical Pacific and Atlantic jet streams are related via thermal wind balance to a colder northern Pacific and Atlantic Oceans, respectively, which mirror a common systematic error in older-generation models^40,41,42,43, usually characterized by a poor resolution in the ocean.

Using a ray tracing algorithm based on linear theory, we have shown that, in December, a stronger subtropical Pacific and Atlantic jet streams are associated with a southward displacement of the refraction latitudes of zonal wavenumber-3 Rossby waves. As a result, Rossby waves originating from the tropics are meridionally bent and may, therefore, fail to reach the NAE sector. This could explain why bad models fail to capture the ENSO-induced remote atmospheric response over the Euro-Atlantic and Sub-Saharan sectors, despite their ability to simulate the TRWSs over the Gulf of Mexico and Caribbean Sea.

In conclusion, in the analyzed CMIP models, two key sources of errors concerning the simulation of the December ENSO teleconnection with the North Atlantic-European sector have been identified: (1) an underestimated SEA TRWS, which might be the consequence of a poorly simulated ENSO inter-basin connection with the Indian Ocean; (2) stronger Pacific and Atlantic waveguides, in turn related via thermal wind balance to overly cold Pacific and Atlantic Oceans, respectively.

This study clearly demonstrates the strong impact that models’ systematic errors have on the simulation of teleconnections. Addressing these biases would significantly improve the models’ ability to simulate the atmospheric extratropical teleconnection patterns and thus the skill of climate predictions in mid-latitude.

Data and methods

Models

The historical simulations from 48 and 37 models participating in phases 5 and 6 of CMIP, respectively, are used. To distinguish between CMIP6 and CMIP5 models, the names of CMIP6 models are written in uppercase letters, while CMIP5 model names are written in lowercase letters (e.g., CESM2 and cmcc-cm). The lists of CMIP6 and CMIP5 models can be found in Tables 5 and 6, respectively.

Table 5 CMIP6 models

Table 6 CMIP5 models

The model data include the monthly upper tropospheric geopotential height (200 hPa), upper tropospheric zonal and meridional winds (200 hPa), total precipitation, sea surface temperature, Sea Level Pressure (SLP), and lower tropospheric zonal and meridional winds (925 hPa). Following the methodology of a previous study¹³, only one ensemble member (i.e., ‘r1i1p1f1’ and ‘r1i1p1’, when available) is included to ensure equal weightage. Other members were included for those models not having the first realization of the ensemble. The last 64 years of model simulations were analyzed, with CMIP6 data assessed for the period 1950–2014 and CMIP5 data for the period 1940–2004. Similar results are obtained when analyzing CMIP6 data for the period 1940–2004 (not shown).

Reanalysis

The reference dataset used in this study is the ERA5 reanalysis product⁴⁴, which includes monthly data of upper tropospheric geopotential height (200 hPa), upper tropospheric zonal and meridional winds (200 hPa), and total precipitation. The dataset covering the period from 1950 to 2014 is analyzed, and similar results are obtained when using the dataset spanning from 1940 to 2004 (not shown). To assess the sensitivity of the results to the reference dataset, additional data from the NCEP/NCAR Reanalysis 1 product⁴⁵ and the Global Precipitation Climatology Project (GPCP) Version 2 dataset⁴⁶ are used. Both model and reference datasets are bilinearly interpolated to a common grid with a horizontal resolution of 2.5° × 2.5°.

Linear and partial regression analysis

Given that teleconnections primarily originate from heating anomalies produced by intense convective activities in tropical regions, monthly precipitation anomalies are used to calculate the indices instead of SST anomalies^7,11,47. The normalized Niño 3.4 precipitation index, hereafter referred to as Niño 3.4 index, is obtained by averaging the standardized monthly precipitation anomalies over the Niño 3.4 region (5°S–5°N, 170°W–120°W; black box in Fig. 1a). To assess the link between heating anomalies located in the Indian Ocean and ENSO teleconnections, the normalized Tropical Western-Eastern Indian Ocean (TWEIO; black boxes in Fig. 2a) precipitation index is used. Following previous studies^11,47, the TWEIO index is defined as the difference of standardized precipitation anomalies averaged over the western (10°S–10°N, 40°E–80°E) and eastern (10°S–10°N, 100°E–140°E) Indian Ocean.

Linear regression analysis is applied to monthly data, with linearly detrended anomaly fields regressed onto normalized indices. The resulting regression patterns are consistent with those obtained from composite analysis (not shown). The partial regression method is implemented to separate the contributions of different processes from an anomaly field. The TWEIO and Niño 3.4 indices can be decomposed as follows:

$${\alpha }_{{TWEIO}}(t)={\beta }_{{TWEIO}}(t)+{p}^{*}{\alpha }_{{Ni}{\tilde{\rm{n} }}o3.4}(t)$$

(2a)

$${\alpha }_{{Ni} {\tilde{\rm{n} }}o3.4}(t)={\beta }_{{Ni} {\tilde{\rm{n} }}o3.4}(t)+{s}^{* }{\alpha }_{{TWEIO}}(t)$$

(2b)

where α_TWEIO and α_Niño3.4 denote the TWEIO and Niño 3.4 indices, respectively, β_TWEIO represents the pure TWEIO index (i.e., independent of the Niño 3.4 index), and β_Niño3.4 represents the pure Niño 3.4 index (i.e., independent of the TWEIO index). The regression coefficients p* and s* are computed as follows:

$${{\rm{p}}}^{* }=\frac{{\sum }_{t=0}^{N}[{\alpha }_{{TWEIO}}(t){\alpha }_{{Ni}{\tilde{\rm{n} }}o3.4}(t)]}{{\sum }_{t=0}^{N}{\alpha }_{{Ni} {\tilde{\rm{n} }}o3.4}^{2}(t)}$$

(3a)

$${{\rm{s}}}^{* }=\frac{{\sum }_{t=0}^{N}[{\alpha }_{{TWEIO}}(t){\alpha }_{{Ni} {\tilde{\rm{n} }}o3.4}(t)]}{{\sum }_{t=0}^{N}{\alpha }_{{TWEIO}}^{2}(t)}$$

(3b)

The Niño 3.4 regression patterns can be expressed as a linear combination of the regressions against the pure Niño 3.4 and TWEIO indices, as follows:

$$R(x)={R}_{0}(x)+{p}^{* }{P}_{0}(x)$$

(4)

where R(x) is the Niño 3.4 regression, R₀(x) is the pure Niño 3.4 regression, P₀(x) is the pure TWEIO regression, and p* is the regression coefficient shown in Eq. (3a). From now on, regressions against the pure TWEIO and pure Niño 3.4 indices will also be referred to as pure TWEIO and ENSO teleconnections, respectively.

The significance testing for ERA5 and CMIP models is assessed by implementing a two-tailed t-test, while for the Multi-Model Ensemble (MME) mean a Student’s one-sample t-test is implemented.

Tropical Rossby wave source analysis

The Tropical Rossby Wave Source (TRWS) formula is used to investigate the anomalous source of upper tropospheric vorticity, which is responsible for the initiation of planetary Rossby waves^8,16. The TRWS is calculated as the advection of climatological absolute vorticity by anomalous irrotational wind^48,49:

$${TRWS}=-{\upsilon }_{x}^{{\prime} }\cdot \nabla (\bar{\zeta }+f).$$

(5)

In Eq. (5), ${\upsilon }_{x}^{{\prime} }$ represents the anomalous irrotational wind, $\bar{\zeta }$ and $f$ are the relative and planetary climatological vorticities, respectively. The irrotational wind is calculated by first integrating the velocity potential χ from the divergence field, and then deriving it to obtain the irrotational wind, similarly to previous research¹⁶. The advection of the climatological absolute vorticity by anomalous irrotational flow is the dominant source of upper tropospheric vorticity in the tropics⁴⁹. We recognize that the term representing the vorticity tendency by vortex stretching or shrinking, the Extratropical Rossby Wave Source (ERWS)⁴⁹, is excluded from our analysis. However, analogous conclusions hold when considering the ERWS (not shown).

Atmospheric waveguide analysis

The meridional gradient of climatological upper tropospheric absolute vorticity and zonal wind serve as a proxy for the atmospheric waveguide³⁰. The total stationary wavenumber, indicating the permitted wavenumbers of stationary Rossby waves along a given waveguide, is calculated using the linear theory for barotropic Rossby waves as follows:

$$K=\sqrt{(\beta /\bar{u})},$$

(6)

where $\beta$ is the meridional gradient of absolute vorticity and $\bar{u}$ is the climatological upper tropospheric zonal wind. Regions of negative $K$², which indicate imaginary wavenumbers, are regarded as “forbidden regions” for the meridional propagation of stationary Rossby waves and therefore serve as effective refraction zones.

Wave activity analysis

The propagation of stationary Rossby wave packets is assessed using the Wave Activity Flux (WAF) formula⁵⁰. This formula diagnoses Rossby wave propagation independently of its phase and parallel to its local group velocity under the geostrophic approximation. It is a widely used tool in extratropical teleconnection studies^11,13,51. The zonal and meridional WAF components are:

$${{WAF}}_{x}=\frac{{pcos}\phi }{2\bar{U}}\left(\frac{\bar{u}}{{a}^{2}{co}{s}^{2}\phi }\left[{({\partial }_{x}\psi ^{\prime} )}^{2}-\psi ^{\prime} {{\partial }^{2}}_{x}\psi ^{\prime} \right]+\frac{\bar{v}}{{a}^{2}\cos \phi }({\partial }_{x}\psi ^{\prime} {\partial }_{y}\psi ^{\prime} -\psi ^{\prime} {{\partial }^{2}}_{{xy}}\psi ^{\prime} )\right)$$

(7a)

$${{WAF}}_{y}=\frac{{pcos}\phi }{2\bar{U}}\left(\frac{\bar{u}}{{a}^{2}\cos \phi }\left[{\partial }_{x}\psi ^{\prime} {\partial }_{y}\psi ^{\prime} -\psi ^{\prime} {{\partial }^{2}}_{{xy}}\psi ^{\prime} \right]+\frac{\bar{v}}{{a}^{2}}\left[{({\partial }_{y}\psi ^{\prime} )}^{2}-\psi ^{\prime} {{\partial }^{2}}_{y}\psi ^{\prime} \right]\right)$$

(7b)

where p is normalized pressure, $\bar{u}$ and $\bar{v}$ are the climatological zonal and meridional winds, $\bar{U}$ is the absolute value of the vector $\vec{\bar{U}}=(\bar{u},\,\bar{v})$, a is the Earth radius, and ${\partial }_{x}$ and ${\partial }_{y}$ are the partial derivatives along the zonal and meridional directions, respectively. The term $\psi$’ stands for the anomalous geostrophic streamfunction, defined as the anomalous geopotential $\varphi$’ divided by the Coriolis parameter f. Both TRWS and WAF are calculated at the 200 hPa level in the upper troposphere, where Rossby wave sources typically reach their maximum intensity⁶.

Rossby wave ray tracing algorithm

An algorithm based on linear theory and the assumption of stationarity^30,52 is applied to investigate the pathway of Rossby wave rays^6,8. The zonal and meridional group velocity of stationary barotropic Rossby waves are:

$${c}_{{gx}}=\frac{2[{\bar{u}]}^{2}{k}^{2}}{(\beta -{{\partial }^{2}}_{y}[\bar{u}])}$$

(8a)

$${c}_{{gy}}=\frac{2{[\bar{u}]}^{2}k{\left(\frac{\beta -{{\partial }^{2}}_{y}[\bar{u}]}{[{\bar{u}}]}-{k}^{2}\right)}^{1/2}}{(\beta -{{\partial }^{2}}_{y}[\bar{u}])}$$

(8b)

where $[\bar{u}]$ is the zonal average of the climatological zonal wind, ${\partial }_{y}$ is the partial derivative along the meridional direction, k is the zonal wavenumber, and β is the meridional gradient of the Coriolis parameter. The zonal group velocity, c_gx, is consistently positive and it increases at a faster rate than c_gy, indicating that barotropic Rossby waves propagate more predominantly in the zonal direction^30,52. We consider the barotropic case because these waves propagate more easily, thereby influencing the remote response⁶. From a starting point, typically identified within a region exhibiting a local TRWS maximum, the group velocities and position are determined. Subsequently, they are projected forward in time by 1 h over a span of 10 days^6,8. The grid-point group velocities and positions are calculated using nearest-neighbor interpolation of the fields described in Eqs. (8a) and (8b), following the approach of previous research^6,8.