In order to compute circulation regimes for the reanalysis ERA5 (blue pathway in Fig. ), the SLP anomaly data undergo a process of dimensionality reduction through the implementation of an empirical orthogonal function (EOF) analysis . The first ten EOFs, which correspond to the eigenvectors of the covariance matrix of the original dataset, represent a substantial proportion of the dataset's variance. Moreover, the first ten EOFs computed from the ERA5 reference dataset constitute the basis vectors for our ten-dimensional reference state space. In the literature, the choice of the dimension of the reduced reference space spanned by the leading EOFs varies from 4 to 14. Many studies used the winter season and the North-Atlantic region, where 4 EOFs already explain more than 50 % of the variance . Our own sensitivity tests for the extended summer season have shown that 4 EOFs explain about 40 % of the variance for the North-Atlantic–Eurasian region and about 34 % for the Arctic region. Moreover, we found that the spatial structure of the regimes obtained with K-Means clustering (KME) based on ERA5 SLP data remained unchanged only when retaining 6 or more EOFs (explained variance larger than 52 % and 45 %, respectively) for the North Atlantic–Eurasian region and the Arctic region. To ensure sufficient explained variance, we therefore used a 10-dimensional state space for both regions, capturing about 67 % of the variance in the North Atlantic–Eurasian region and 61 % in the Arctic region.

To obtain reference atmospheric circulation regimes, a K-Means clustering algorithm is applied in the ten-dimensional reduced state space. The PCs serve as the input data for the clustering algorithm. Each data point is assigned to its cluster centre, or centroid, which is selected to maximize inter-cluster distance while minimizing intra-cluster distance. In the case of the climate model data (red pathway in Fig. ), the SLP anomalies are not reduced in their dimensionality via an EOF analysis. As in e.g. an alternative approach was taken, whereby the anomalies were projected onto the ten-dimensional reference state space. This results in the generation of pseudo-PCs. The pseudo-PCs afterward are clustered via the K-Means clustering algorithm to obtain simulated circulation regimes for each model simulation in both time periods that differ in their spatial structure compared to the reference circulation regimes. Projecting the CMIP6 model data onto the ERA5 reference state space ensures that all calculations are performed in the same reference space, enabling consistent comparison of regime patterns. Nonetheless, the pseudo-PCs describe less variance than the models own PCs and this might introduce biases in our analysis. Following , we tested the consistency between clustering in the pseudo PC-space versus clustering in the models' own PC-space by analysing the variance ratio, i.e. the ratio between the mean inter-cluster squared distance and the mean intra-cluster variance for K-Means clustering in the pseudo PC-space versus clustering in the models' own PC-space for the common model clusters over the North-Atlantic–Eurasian region for the present-day period. We also compared these results with the variance ratio for the ERA5 reference clusters (see Appendix ). The model simulations have a lower variance ratio than the ERA5 data and are outside the ERA5 variability range. The variance ratio revealed a slight increase in the Pseudo-PC space, indicating stronger clustered data. Nonetheless, the difference to the variance ratio in the models own PC-space is not significant and the variability range is similar. In addition, we investigated whether the cluster centers obtained with both approaches represent the same patterns (Appendix ) and found very similar patterns with pattern correlations higher than 0.95. Both the analysis of the variance ratio and the evaluation of the cluster center patterns support the applicability of the projected approach for our analysis.

The projected approach (green pathway in Fig. ) allocates the pseudo-PCs directly to the reference centroids, by finding their closest reference centroids (in terms of Euclidean distance) from the ERA5 reanalysis. In other words, we assign the data from the CMIP models to the regimes from ERA5 by using the pseudo-PCs. This assignment of each day to the ERA5 reference clusters has been calculated for historical and future time periods. In order to provide a joint representation of the climate models' circulation regimes, the common simulated circulation regime framework is applied, which will be used to represent the characteristic joint regimes for the entire model ensemble. The common simulated circulation regime framework enables the possibility to compare the spatial structure between reanalysis and the entire ensemble of CMIP6 models. First the preprocessed data of each climate model are merged into a single data file along the temporal axis, either for the historical period 1985–2014 or the future period 2070–2099. Instead of performing a common EOF analysis to derive shared principal components as described, for example, in, the common climate model data are projected onto the ten-dimensional reference state space defined from the ERA5 data. This projection yields ten common pseudo-principal components (Pseudo-PCs). The common Pseudo-PCs serve as input data for the K-Means clustering algorithm that is applied. Five common simulated circulation regimes are obtained for each time period, representing the joint regimes for the entire model ensemble.

In order to investigate the sensitivity of the results to the choice of clustering method, a second method was also applied. Simulated annealing and diversified randomisation clustering SANDRA, differs from K-means clustering by introducing a randomised cluster assignment inspired by the thermally induced movement of atoms in a crystal lattice (hence the term “annealing”, from the process in metallurgy). A cooldown factor is used to gradually decrease the movements, resulting in a positioning which is numerically very efficient at finding solutions close to the global optimum. Details are found in . The application in the present study is based on the implementation in the cost733class software . Comparisons of SANDRA with other methods show very favourable results .

In contrast to the K-Means clustering algorithm, we only followed the projected approach for SANDRA, i.e. the SANDRA algorithm was trained on the ERA5 reanalysis data and then the resulting five regimes were assigned to all climate models, for both the historical and future periods. The SANDRA method was applied for both the 1.125° horizontal resolution used in KME method as well as a coarser 2.5° resolution. Unsurprisingly, the resulting regimes were almost identical (with spatial correlation > 0.998 for all 5 patterns, and maximum absolute difference below 0.5 hPa, not shown), and we decided to use the coarser resolution for computational efficiency.

In almost all clustering algorithms, the number of clusters must be predetermined, and this is a topic of extensive discussion in the literature, as referenced in the following sources: . Several methods have been applied to determine an optimal number of regimes, with most studies indicating a number of regimes between four and six for North-Atlantic winter regimes . Here, we have based the decision for a reasonable number of circulation regimes on three metrics described in detail in the Appendix : The anomaly correlation coefficient between the patterns of the clusters, following have been applied for both methods, K-Means and SAN. In addition, the elbow test and the silhouette score have been applied for K-Means. The elbow test assesses the efficiency of allocation of centroids and the silhouette score measures the quality of clustering . All three metrics support a choice of k = 5 clusters for both regions. This is also in agreement with cluster numbers used in other studies for winter North-Atlantic regimes, e.g. in .

The performance of models in reproducing the spatial structure of reference circulation regimes is evaluated using a Taylor diagram . This method incorporates two statistical measures: the spatial correlation coefficient (R) between simulated and reference regimes, and the normalized standard deviation (SD) of anomalies. These metrics are plotted on a polar diagram, where the angular axis represents R, the radial axis indicates SD. The Taylor diagram provides a compact visualization of model performance, aiding in the comparison of spatial structures.

To determine statistical significance of the changes in frequency of occurrence under the influence of rising GHG emission in the future for both regions, Welch's t-test is employed . Welch's t-test is a robust tool for comparing group means, and is particularly effective when variances are unequal. Here we use it to evaluate whether observed differences in the frequency of circulation regimes are statistically significant compared to historical records, providing insights into potential climate-related anomalies in future projections.

Five atmospheric circulation regimes are detected in the North Atlantic–Eurasian region. The reference circulation regimes for the ERA5 data 1985–2014 obtained with the K-Means clustering (KME, Fig. , upper row) and the SANDRA (SAN, Fig. ,center row) algorithm alongside with the zonal wind anomaly composites at 250 hPa for the KME regimes, lower row, are summarised here, the wind composites linked to the SAN regimes are shown in Appendix .

The Scandinavian Blocking regime (SCAN), shown in Fig. a, indicates a positive pressure anomaly centered above Scandinavia and a low pressure anomaly, centered over the North Atlantic Ocean westwards of the British Isles (refer to KME, Fig. a, upper row). The regime pattern obtained by the SAN algorithm displays a slightly weaker positive pressure anomaly over Scandinavia and a slightly stronger low pressure anomaly over the North Atlantic (Fig. a, center row). The SCAN is associated with a southward shifted Atlantic jet.

analysed circulation regimes over the same region for the summer season June to August (JJA) and found similar MSLP anomaly patterns for each regime. and , on the other hand, analysed summer circulation regimes for JJA over a smaller North Atlantic region (20–80° N, 90° W–30° E), using SLP and Z500 data, respectively. In both studies, four circulation regimes were detected. These four regimes comprise the negative and positive phases of the NAO (called NAO- and Blocking, BL), an Atlantic low (AL) and an Atlantic ridge (AR) regime, which bear many similarities with the NAO-, NAO+, ATL and SCAN regimes detected in our study.

The agreement between the characteristic regime patterns of both algorithms, KME and SAN, is evaluated by means of a Taylor plot in Fig. , where the KME patterns serve as reference. The correlation coefficient for all clusters are above 0.85 and the spatial standard deviation of the patterns obtained with the SAN method are close to the reference values. Two patterns, ATL and NAO-, exhibit even very high spatial correlation values above 0.95. The spatial similarity between the clusters calculated from KME and SAN algorithm is validated by these findings.

In Sect.  two different methods to calculate circulation regimes for the model simulations were presented: the projected approach and the simulation circulation regime approach. The simulated common circulation regimes for the extended boreal summer season for the future period, revealing the joint regimes for the whole ensemble of climate models, exhibit small differences in their spatial structure compared to the reference circulation regimes computed from reanalysis ERA5 data. The evaluation by means of a Taylor plot is presented in Fig.  for the North Atlantic–Eurasian region. The calculated common simulated circulation regimes in the future period exhibit high spatial correlation coefficients with the ERA5 reference circulation regimes. The correlation coefficients reach values above 0.85 for KME regimes and above 0.7 for SAN patterns and the standard deviations of the common simulated circulation regimes are found to be between 0.8 and 1.4 compared to the normalized reference standard deviation of 1.

These results support the hypothesis of and introduced in Section 1, namely that a rather weak external forcing acting on the dynamical system atmosphere does not alter the spatial structure of the regime patterns. In our case, even under the high emission scenario SSP5-8.5 the spatial structure remains largely unchanged in the future time period.

This allows us to apply the projected approach for the calculation of the frequency of regime occurrence for the climate models in the historical and future period. Additionally, the projected approach is able to calculate the frequency of occurrence of the circulation regimes of the CMIP6 models more accurately than the simulated circulation regime approach. This is visualised in Fig.  for both regions, the absolute difference between the simulated respective projected and reference circulation regimes frequencies averaged over all regimes for each model in the historical period, i.e. the frequency bias as defined in is shown. The mean frequency bias is smaller for the projected circulation regimes compared to the simulated circulation regimes. The interquartile range of the projected circulation regimes' frequency bias is lower compared to the interquartile range of the simulated circulation regimes' frequency bias. suggested that a lower frequency bias results in a higher confidence to project changes in the frequency of occurrence under climate change. Due to this, the projected approach is preferable in the analysis of changes in the frequency of occurrence for both regions.

We now turn to analyse the projected future changes under the SSP5-8.5 scenario. The changes in the frequency of occurrence for circulation regimes computed by both methods, KME and SAN, are shown in Fig. . The change in the frequency of occurrence for a pattern is the difference between the percentage of days assigned to that particular pattern in the future (2070–2099) and in the historical time period (1985–2014) in the extended boreal summer season from May to October. This is calculated for each of the twenty CMIP6 models, with the spread of the models visualised in the box plot. Both methods show consistent results for all five clusters, only the SAN method indicate non-significant results for the ATL pattern. Regarding the SCAN pattern, both methods detected a decrease in the frequency of occurrence in the future for most of the models. Here models linked to BK+/AA- storylines exhibit a stronger decrease in their frequency of occurrence in the future. In contrast, the BK-/AA+ linked models exhibit only small changes, even a positive change in the frequency for KME method, BK+/AA+ associated models are in close proximity to the model mean. For the ATL pattern, both methods show a decrease in its frequency of occurence under strong global warming, only significant for the KME algorithm. Models connected with the BK-/AA+ storylines show rather stronger decreases in their frequency of occurrence under global warming. For both methods, i.e. SAN and KME, the NAO+ pattern occurs significantly more frequent in the future period compared to the historical period under the GHG forcing projected by the SSP5-8.5 scenario. The NAO+ pattern is linked to a tilted, polewards shifted Atlantic jet structure, thus this results suggests the jet will be shifted northwards under strong global warming in the season from May to October, which is concurrent with the changes from June to August found in and the annual mean in . The NAO- pattern displays a significant decrease in occurrence frequency in the future, as detected with both algorithms. As the NAO- pattern is associated with a southward shifted Atlantic jet, this decrease in frequency of occurrence underlines the suggestion made analysing the NAO+ pattern, i.e. the Atlantic jet will be shifted northwards. In , who analysed a more confined region in the summer season utilising CMIP3 climate models, an increase in NAO+ and decrease in NAO- was also found, which is in line with the presented findings. For the DIPOLE pattern, both methods show a significant increase in their frequency of occurrence, that is stronger when applying the SAN method. Likewise to the NAO+ pattern, the DIPOLE regime is linked to the northward shifted jet structure and shows a positive change in its frequency of occurrence under global warming.

The changes in frequency of occurrence may arise from two different sources: change in how often events occur or for how long they persist. The latter is analysed in the following for the North Atlantic–Eurasian region, shown in Fig. . The analysis reveals that changes in persistence largely mirror those in the frequency of occurrence. For the KME method, the significant shifts identified in occurrence frequency are likewise evident in persistence, with the exception of the SCAN and NAO- regimes, whose persistence change is not statistically significant yet follows the same directional trend, suggesting that the future changes are due to the regimes becoming less persistent as well as occurring less often in the future. For the SAN algorithm, the significant alterations in frequency of occurrence for SCAN and DIPOLE are also reflected in persistence. By contrast NAO+ and NAO- do not exhibit a corresponding significant change, meaning the change in frequency of occurrence for the NAO patterns may be due to both: a change in persistence of the regimes as well as the events occurring more/less often in general. For both algorithms, as well as for both metrics, i.e. frequency of occurrence and persistence, the BK-/AA+ storyline for the ATL regime display a more robust negative trend. All other regimes do not show any dependence of the changes in occurrence frequency on the storyline. One reason for that might be the small sample size of only one or two models representing each storyline.

To summarise, both methods, the KME and SAN calculate typical circulation regimes for the North Atlantic–Eurasian region, which are similar in their spatial structure. The simulated changes in the spatial structure due to the influence of climate change in the atmosphere are small for the whole ensemble of climate models, but the frequency of occurrence is altered as suggested by , see also the Introduction (Sect. ). Both methods detect a significant increase in the occurrence of the NAO+ and DIPOLE pattern in the future under strong GHG forcing, the change of the SAN method is higher in its median and interquartile range. The same applies for the future change of the NAO- regime, both methods simulate a significant decrease in their frequency of occurrence. Most of the results are in line with the findings regarding the persistence. Overall we found a high correspondence between future changes in occurrence and persistence. The analyses of the related zonal wind composites revealed that regimes associated with a northward shifted jet stream are simulated to occur more often and to be more persistent, while patterns linked to a southward shifted jet stream occur less often under strong global warming.

For the Arctic region, five atmospheric circulation regimes are determined based on the arguments in Sect. . The reference patterns calculated by both methods, KME and SAN, for the ERA5 data are shown in Fig. , upper two rows. Additionally, the zonal wind anomaly composites for KME regimes at 250 hPa are shown in the lower row. The zonal wind anomaly composites respective to the SAN regimes are shown in Appendix .

Greenland-Siberia Dipole(Fig. a) is characterized by a negative pressure anomaly above Iceland and Greenland facing a positive pressure anomaly above Siberia resulting in a strong westerly flow across the Arctic region. The zonal wind composites at 250 hPa show a strengthening of the Atlantic jet exit.

From visual inspection of Fig. , it is clear that the two methods agree quite well on the first four patterns, while Cluster 5 is more different. We note that the low pressure in the Bering–Svalbard Dipole cluster is located over Alaska in KME, while it is more towards Kamchatka in SAN. This is compensated in SAN by grouping together days with low pressure over Alaska in Cluster 5 (Of the days that are assigned to the Bering–Svalbard KME pattern only 48.8 % is assigned to the Bering–Svalbard SAN pattern, while 35.1 % is assigned to Cluster 5 SAN underlining the spatial similarity between the two patterns).

Similar to the North Atlantic–Eurasian region, the spatial patterns obtained by both methods are compared by a Taylor plot in Fig. , where the patterns calculated by the KME algorithm serve as reference. The regimes show correlation coefficients between 0.78 and 0.95, thus the spatial structure of every pattern is similar for both methods, KME and SAN, except for Cluster 5. The standard deviations of the magnitudes of the SLP anomalies are close to the reference for all five patterns, slightly under the normalized value of one for Cluster5, AO+ and AO-. To conclude, both methods, KME and SAN, calculate four clusters with similar spatial structure for the Arctic region in the extended boreal summer season May to October in the historical time period 1985–2014 from ERA5 reanalysis data, but the 5th cluster shows substantial differences.

In Fig.  the Taylor diagram comparing the circulation regime structure changes under the strong SSP5-8.5 scenario compared to the reference historical ERA5 regimes for KME and SAN algorithm is shown. The standard deviations differ from the reference (normalized to 1), more pronounced for the SAN algorithm, as the standard deviation for the Bering–Svalbard Dipole is 0.57 and for Cluster 5 above 1.5, indicating a change in the magnitudes of the pressure anomalies. However, the correlation coefficient remains above 0.7 for all patterns across both algorithms. Alongside with the arguments in Sect.  and Appendix , the projected approach is applied to analyse changes in regime occurrence frequencies between the historical and future periods for the Arctic region. These results are shown in Fig.  for both methods, KME and SAN. The future frequency of occurrence of the Greenland–Siberia Dipole shows no consistent change in both methods, as the SAN regime is simulated to occur more often and the KME regime less often in the ensemble median. For the regimes calculated by the KME algorithm, the AO+ regime occurs less frequently in the future which is in agreement with the results obtained by the SAN algorithm. The AO- regime occurs significantly more often in the future time period for both methods. Cluster 5 shows a significant positive change in its frequency of occurrence for the KME algorithm and a strong negative trend regarding the SAN method. This finding underlines the assumption that the SAN Cluster 5 is compensating the shifted low pressure anomaly above Kamchatka in the Bering–Svalbard Dipole present in the KME method. A decrease in the frequency of low pressure events in this area therefore shows up as a negative frequency change in Bering–Svalbard dipole for KME but Cluster 5 in SAN. Additionally, the KME Bering–Svalbard Dipole and SAN Cluster 5 are both linked to a stengthening in the Pacific jet. Similar to the North Atlantic–Eurasian region, the interquartile range of the frequency changes for the SAN regimes is greater compared to the KME regimes, especially regarding the Bering–Svalbard Dipole and AO-.

The following statements can be made when comparing the results for Arctic and North Atlantic–Eurasian region: Considering the spatial patterns shown in Fig. , the circulation regime characterized by a negative pressure anomaly above the Arctic center; AO- is simulated to occur more frequently in the future period for both investigated algorithms. This change is consistent with the increase in the frequency of occurrence of the NAO+ regime found in the North Atlantic–Eurasian region in the future time period, since both patterns, AO- and NAO+ exhibit spatial similarities, i.e. a negative pressure pressure anomaly above Greenland and a broad positive pressure anomaly above Northern Europe (see Figs. c and d). Both patterns are linked to a northward shifted Atlantic jet, and projected to occur more often in the future. The circulation regimes associated with positive pressure anomalies above the Arctic; Bering–Svalbard Dipole and AO+ are expected to occur less frequently in the future. Again, AO+ and NAO- (from the analysis over the North Atlantic–Eurasian region) are both characterized by a positive pressure anomaly above Greenland and a negative pressure anomaly extending from the Ural to the North Atlantic. The AO+ regime is connected with a southward shifted jet structure above the Atlantic, likewise to the NAO- regime, supporting the findings in the North Atlantic–Eurasian region.

Most of the found changes in frequency of occurrence can be also observed by analysing the change in persistence for the Arctic region (refer to Fig. ), although the results are not as corresponding as for the North Atlantic–Eurasian region. The indicated change in persistence for the Greenland-Siberia Dipole for both methods is positive, which supports the trend for the frequency of occurrence for this pattern regarding the SAN method, although the change is not significant. For the Bering–Svalbard Dipole there are non-significant future changes in persistence that are inconsistent with the changes indicated in the frequency of occurrence, refer to Fig. , meaning the change in frequency of occurrence for the Bering–Svalbard Dipole may be due to the event occurring less often in the future rather than a decreasing persistence. Only for the KME method the changes in frequency of occurrence of the AO+ pattern are consistent with the future persistence changes, where the change in persistence is almost zero and non-significant. This finding indicates that the occurrence of shorter AO+ events may be more prevalent in the SAN method. The AO- shows more robust results, indicating a significant positive change in persistence for both methods. Here, the models linked to the BK+/AA+ storylines exhibit stronger signals compared to the opposing BK-/AA- storylines' models. Cluster 5 again shows a higher persistence that supports the findings from the frequency of occurrence for KME. In contrast for the SAN method, Cluster 5 is simulated to occur significantly less frequently in the future and the persistence even shows a slight longer persistence, indicating events to appear less often. The found results suggest that the correspondence between persistence and frequency of occurrence metrics is less robust in the Arctic compared to North Atlantic–Eurasian region.

As shown in Fig. 4, the CMIP6 multi-model mean (MMM) shows a poleward shifted Atlantic jet, but the strength of this poleward shift depends on the storyline, with a weaker poleward shift in storylines BK+/AA- and BK-/AA- and slightly stronger poleward shift in storylines BK+/AA+ and BK-/AA+ (compared to the MMM). The projected increase in the occurrence of NAO+, DIPOLE, and AO-, which are associated with a poleward shifted jet, supports this result. However, the specific changes in the storylines could not be attributed to shifts in occurrence frequency, likely due to the small sample size, as each storyline is based on only one or two CMIP6 models. To summarize, the calculated circulation regimes of both methods, KME and SAN, exhibit small changes under the influence of global warming in their spatial structure when analysing the Arctic region in the extended boreal summer season from May to October but the frequency of occurrence is changing significantly for most of the patterns. The simulated changes in the frequency of occurrence utilising the projected approach show a significant increase for the regime AO- for both methods. The changes in the frequency of occurrence are in line with the changes in persistence for this pattern, meaning the observed increasing the frequency of occurrence is due to longer lasting events. The AO+ pattern is projected to decrease significantly in frequency of occurrence under the influence of a strong GHG scenario in the future for both algorithms. In the Arctic region, the changes in frequency and persistence showed less agreement between the two methods and across both metrics than in the North Atlantic–Eurasian region. In contrast to the North Atlantic–Eurasian domain, where we detected a correspondence between occurrence and persistence changes for all regimes, in the Arctic region this correspondence is only found for the AO- regime.