For our analysis, we use a 12-member ensemble of regional climate simulations for central Europe at a resolution of 50 km. The ensemble has been generated by downscaling different global climate model outputs with the regional climate model COSMO-CLM (hereafter referred to as CCLM; ). The CCLM is a non-hydrostatic climate model coupled to the soil vegetation model TERRA and is the climate version of the numerical weather model of the German weather service. Data from six different global climate models (GCMs) have been used as initial and boundary data. Two of the GCMs have used the emission scenario A1B as external forcing: CCCma3 and three realizations of ECHAM5 . The other four, ECHAM6 , CNRM-CM5 , HadGM3 and EC-EARTH have used the emission scenario RCP8.5 . Additionally the Atmospheric Forcing Shifting method was applied to the ECHAM6 data. For this method the global climate data interpolated to the 50 km grid are shifted by two grid points in all cardinal directions before being used as boundary data. This accounts for the uncertainty in positioning of synoptic systems when interpolating the GCM data to the required resolution for forcing the RCM (regional climate model) simulations. As all five ECHAM6-driven simulations obtained this way exhibit a high correlation, they are all weighed with a factor of one-fifth when calculating the mean. All other models receive a factor of 1, which leads to an effective ensemble size of eight. Additionally we use a COSMO-CLM run driven by ERA-40 boundary conditions. The ERA-40 reanalysis boundary conditions are assumed to be close to the “true” observed state. Nevertheless, they depend on, inter alia, the model, observational data and assimilation technique, and are not free from biases see e.g.,.

The simulation time periods are the recent past (1971–2000) and the near future (2021–2050). An analysis of the temperature trends of different ensemble members showed that the distribution of trend depends more strongly on the chosen global climate model than on the emission scenario. We therefore combine simulations with boundary data from GCMs with different emission scenarios to set up our ensemble.

We choose six regions, each comprising 6×6 grid points for our analysis. The regions are chosen based on the PRUDENCE regions , which could not be used because of the necessity of the same amount of grid points for each area, and due to test results that show a different behavior for these regions. We investigate 30-year periods of daily data; thus, each time series consists of ≈ 11 000 data points, yielding ≈36×11 000≈400 000 points in time for each region and ensemble member. The model domain and the six investigation areas, which are located in Spain, France, Germany, Scandinavia, Bulgaria and Russia, are shown in Fig. . These roughly match the PRUDENCE regions, which are not applicable for the analysis since equal sized areas are a requirement for comparison among regions.

For the comparison of our regional climate ensemble with observations, we use temperature and precipitation data from the gridded E-OBS data set . This data set was produced as part of the ENSEMBLES project by interpolating station data from the ECA&D station data set European Climate Assessment; to a 25 km grid. The station density is highest in Switzerland, the Netherlands and Ireland and rather low in Spain and the Balkans, which leads to an over-smoothing in these areas. This especially affects extremes and has to be taken into account when validating our ensemble against E-OBS data. Furthermore it should be noted that a comparison of E-OBS and another gridded data set, namely, Hyras (only central Europe), with respect to the dynamical behavior that we analyze in this paper, revealed differences between the two data sets . A comparison of dynamical aspects of different observational data sets yields an interesting application of the method, which, however, will not be addressed within this paper. We additionally use blended temperature and precipitation time series starting from 1900 of eight stations (all in Germany) of the ECA&D data set for a sensitivity analysis described in Sect. . The eight stations are listed in Table .

The method used in this paper consists of describing temperature and precipitation time series by a Markov chain and subsequently calculating descriptors, which characterize the dynamical (successional) behavior of the compound extreme states. The method has been used in biology to describe dynamics of succession of species in a rocky subtidal community. It has been introduced to atmospheric science by , who used it for climate classification and a comparative study of two regions. In this section, a short introduction to Markov chains is given, followed by a step by step description of the method.

A first-order, m state (m is the number of discrete states of the Markov chain), homogeneous Markov chain is a time discrete, state discrete stochastic process, which fulfills the Markov property: Pxt|xt-1,xt-2,…,xt-n=Pxt|xt-1, meaning that the present state xt is only dependent on the preceding state xt-1. From the Markov chain, a transition probability matrix P of the order m×m can be calculated, which consists of all possible conditional probabilities Pxt|xt-1 between the m different states of the Markov chain. For a homogeneous (≡ stationary) Markov chain, the transition probability matrix is time independent. A stationary distribution π is a vector that fulfills the following equation π=Pπ. To test for homogeneity one must solve the eigenvalue problem of Eq. () to calculate the stationary distribution π. If this is identical to the empirical distribution π^j=nj∑jnj the time series is considered stationary. The entries (transition probabilities) of the transition matrix P are estimated by p^ij=nij∑inij.

In the following, the main steps of the Markov analysis are explained: a.

Partitioning and combining of univariate time series to a multivariate symbolic sequence To represent the univariate time series (here daily mean temperature anomalies and daily precipitation anomalies) by a Markov chain, each time series is partitioned into a symbolic sequence of extreme and non-extreme regimes. These univariate symbolic sequences are then combined into a multivariate symbolic sequence of m=2v different states (v number of variables). In this paper, v=2; thus, there are four possible states.

In order to get a better feeling for the descriptors and understand how they relate with each other, we will do a small thought experiment. We take a Markov chain consisting of a time series of 1000 symbols of which 10 % are extreme, the rest are normal. In this case a persistence of 0.5 would mean that in half of the 100 extreme cases, the next case is also extreme, there are 50 transitions from the extreme state to the extreme state. The maximum episode length in this case is thus 51 extreme states in a row (with all others randomly distributed). The recurrence time and entropy are inversely related to how these 50 extreme transitions are ordered. Recurrence time depends on the number of episodes (fewer episodes lead to a larger recurrence time, more episodes to a shorter recurrence time) and entropy additionally on the mean episode length. In this paper, we also look at changes in the descriptors. A change in persistence of 0.05 in the above case would mean five more extreme–extreme transitions per 1000 days, and an increase from 50/100 to 55/100 (extreme–extreme transitions/extreme–normal transitions) is surely a noticeable change. The range of actually probable values of the descriptors is smaller than the whole possible range. A persistence of 0.99 for example, would mean that there is only one extreme episode in the whole time period, all 100 extreme states occur after each other. In a climate system, this is unlikely to happen. Thus, for climate one cannot expect to observe a change of the daily persistence from, e.g., 0.5 to 0.8, because such a change would be catastrophic.

The EDI is an index for detecting drought conditions by calculating daily deviations of precipitation from a climatological mean state. It was proposed by . An important concept of the EDI is the use of effective precipitation (EP), rather than precipitation (P) itself. EP describes the depletion of water sources by a weighted summation over the 365 days preceding a given day (d): EPd=∑n=1365∑m=1nPd-mn.

By this, the memory effect of the soil is taken into account. Therefore, EP strongly correlates with soil moisture and the EDI is thus a good measure when considering droughts. Using the EP, the EDI is calculated by the following formula for a given d: EDId=EPd-EPd,rm‾σEP-EP‾d, where EPd,rm‾ is the climatological mean corresponding to a given d calculated as the 30-year average over a 5-day-running mean (rm = 5). By subtracting this climatological mean of EP from the daily value, the yearly cycle is removed from the EDI time series.

To calculated the Markov descriptors, we first calculated temperature and precipitation anomalies using the mean annual cycle of the respective time period and ensemble member/observation. We calculate the Markovian descriptors for two types of extremes:

cold and heavy precipitation (temperature anomaly (T) < 10th percentile and precipitation anomaly (P) > 75th percentile) in winter (DJF)

In order to test the spatial variability of the descriptors, we calculated them for the entire E-OBS data set for the time period 1971–2000 for the two types of extremes mentioned above.

The descriptors were calculated for each grid point, taking into account not only this grid point but also the eight neighboring grid points thus using a moving window if nine grid points. The time series for each grid point were detrended and partitioned separately before the nine partitioned time series were merged to calculate the descriptors. The reason for using this moving window of nine grid points is the fulfillment of the criteria of the Markov method (see Sect. ), which is not given for the entire area when using only single grid points. Using this moving window does not alter the general spatial pattern and smoothness of the results.

For both type of extremes, the descriptors show smooth spatial patterns (see Fig. ); nevertheless, variations between different regions can be identified.

The persistence for the winter extremes (left side of Fig. ) is lower than for summer extremes; specifically, in northern and central Europe compound cold and wet events are most likely events of a short duration and rather rare (with recurrence times of up to 400 days). Along the Mediterranean coast and southeastern Europe, the values are higher and probabilities of residing in a compound extreme state of over 50 % are observed. The recurrence time for these events is also comparatively low (around 100 days). Interpreting the results, one has to keep in mind that we are always referring to relative compound extremes. The entropy is around 0.9 for most of the area with small regions showing lower entropies down to 0.5. These high values can be explained by the low persistence – as compound winter extremes are grouped in very short episodes (low persistence), they are very hard to predict. The highest persistence for summer events (right side of Fig. ) are observed in Scandinavia and the eastern part of the E-OBS domain and lowest in central Europe and the northern coast of Spain. The persistence is above 50 % for the whole domain, which means that the probability of the system residing in a compound extreme state is high and these events are grouped in episodes of long duration. The recurrence time lies between 40 and 100 days and is as such also lower than that for compound winter events. The lowest values are observed in the Balkan region. The entropy lies between 0.4 and 0.65, which means that the extreme events are not so easy to predict, especially for parts of central Europe where the entropy is the highest. However, according to our definitions, summer extremes can better be predicted than winter extremes.

For the main analysis in this paper we apply the method to six regions, which we chose in rough agreement with the PRUDENCE regions. The crucial point for being able to compare the descriptors of different regions is that each region contains the same amount of grid/data points. Since the descriptors do not vary strongly within the PRUDENCE regions, we chose regions consisting of 6×6 grid points from within these widely used regions. The regions that will be analyzed in the further sections of this paper are shown in Fig. .

Note that the results shown in Fig.  can only be qualitatively compared to those of the regions considered later or the station data in the next section as the number of grid points (or stations) contributing to the analysis differs.

To assess the temporal variability of the descriptors, we calculated the descriptors for 30-year moving windows of observational station data from the ECA&D station data set . Since we are interested in the daily values of temperature and precipitation, only stations were chosen with a continuous daily record (with an allowance of 50 missing values at most). Using these criteria there are eight stations with temperature and precipitation time series from 1900 to 2015 of which all are in Germany (see Table ). One station has 15 missing values for temperature. These days were excluded from the analysis, considering the 30-year time windows consisting of 10 950 days, this amounts to roughly 0.1 % of the values and does not alter the value of the descriptors. Of these eight stations, five are in the vicinity of Hamburg and have the same values for the first 17–22 years. The records of two stations in Hamburg are identical throughout the whole time period; therefore, only one of them is included in the analysis, which leaves a total of seven stations.

The descriptors were again calculated for both types of compound events. Linear temperature trends were removed separately for each of the 30-year time windows and in order to fulfill the criteria of the Markov method (stationarity and non-zero entries of the transition probability matrix; see Sect. ), the partitioned data of these seven stations were combined to one time series to calculate the descriptors.

The results are shown in Fig.  for both winter (black) and summer (gray) extremes. Especially for the persistence and recurrence time, a clear shift is visible between 1930 and 1950. This time range is not preindustrial, but the crucial point is that the observed shift coincides with an observed shift in the global increase in CO2 around 1950 see, e.g., Fig. SPM.1d. From this finding we observe two main points: 1.

The descriptors (especially persistence and recurrence time) seem to be sensitive to changes of the CO2 increase. That means a stronger increase of CO2 (e.g., from 1950 on) yields a lower level of persistence and to a higher level of recurrence time. Although CO2 is still increasing after 1950, the recurrence time, e.g., remains constant. Hence, the recurrence time seems not to be dependent on the absolute CO2 concentration, but on the increase of latter.

2.

Thus, we can conclude that the natural variability of the descriptors can be approximated by the variability observed before and after the shift. This natural variability is smaller than the shift of the mean.

To assess the estimation error of the descriptors we used the Multivariate Iterated Amplitude Adjusted Fourier Transform (MIAAFT) algorithm as described by and . With this algorithm, the data are shuffled and thus the original distribution is preserved. In addition, the auto- and cross-correlation of the temperature and precipitation time series are approximately preserved. We constructed 100 MIAAFT surrogates for the temperature and precipitation anomalies (or the EDI time series for summer events, respectively) for the E-OBS data set for the reference period (1971–2000). We then estimated the standard deviation of the descriptors calculated from these surrogate time series. It is important to note that this standard deviation, under the framework of such a bootstrap test, already represents the standard error of the mean, which corresponds to the normal standard deviation divided by N. The errors for both types of extremes and the six regions are listed in Table . The errors do not vary much between the different extremes and regions, the error of the persistence is on the order of 0.01 or lower, for the recurrence time between 1 and 2.6 and the error of the entropy on the order of 0.005. Adopting these errors to the values of the E-OBS descriptors for the reference period (shown in Figs.  and in Sect. ), the error of the persistence is about 2–10 %, for the recurrence time about 2 % and for the entropy about 1–2 % (cf. Table ).

This estimation error is much smaller than the ensemble uncertainty and can approximately be neglected. This shows that the estimation of the descriptors is robust. Further, we will consider the E-OBS data approximately as truth and we will use the ensemble uncertainty as the error for our main analysis.

In order to get an idea about the order of magnitude of the change signal, the observational E-OBS data set was split into two equal parts of 30 years, 1951–1980 and 1981–2010. The descriptors were calculated for both time periods and a change signal derived.

For cold and wet extremes (see Fig. ) all regions except France show a decrease in persistence, regions 5 and 6 (Russia and Balkan) show the strongest absolute decrease (≈ 0.15) and Germany the highest relative decrease of -72 % (relative changes are shown above the respective bars). The recurrence time does not change much for all regions except region 5 (Russia) where it decreases by 150 days. In this region, compound cold and wet extremes occurred more frequently but were of shorter duration in 1981–2010. The entropy shows a decrease of more than 5 % in Spain and Germany where the system becomes more regular. In Spain an increase of entropy is observed and the compound extremes are harder to predict in 1981–2010 with respect to 1951–1980. The change signal for all descriptors and seasons (except for the entropy of France and Russia) are greater than the estimated error by FT-surrogates (see Table ); thus, these changes are robust.

Changes for hot and dry extremes in summer (see Fig. ) are below 10 % for most regions. Nevertheless for most regions these changes are still greater than the estimated errors by FT surrogates (see Table ). In Scandinavia, both persistence and recurrence time show a decrease, the extreme episodes are of shorter duration but occur more often. In Spain and Germany, both descriptors show an increase – especially in recurrence time; thus, episodes of compound extremes occur less frequently. An increase in recurrence time can also be seen in Russia. The entropy increases in regions 3–6 (Germany, Scandinavia, Russia and Balkan), in these regions the system becomes less regular with respect to compound hot and dry events and harder to predict, whereas in Spain the Entropy shows a decrease – these compound events are easier to predict.

In a second step we calculate the change signal between 1971–2000 and 2021–2050 for all members of the CCLM-ensemble. An additional information of interest for the interpretation of the results is the change in the number of compound extreme days. The number of univariate extreme days are kept constant when partitioning the data (see Sect. ) but the combination can change. The climate change signal is calculated separately for each ensemble member and then the mean climate change signal (bar in the following plots) as well as the interquartile range (marked by the whiskers) of the individual change signals are calculated and pictured. The number of compound cold and wet extreme days increases in all regions except region 5 (Russia) between the two time periods 1971–2000 and 2021–2050 and the number of compound extreme days differs between the regions. Regions 1 and 6 (Spain and Bulgaria) show the highest number of compound extreme events (see Fig. ). The ensemble mean values of the descriptors for cold and wet extremes in winter are shown in Fig. , whiskers give the interquartile range. The significance of the change signal was calculated using the nonparametric Mann–Whitney–Wilcoxon test, which tests for a difference in location of the values of the ensemble for the two different time periods. The p values are shown below the bars in the respective figures. About one-third of these p values are smaller than 0.5, thus significant at the 5 % significance level; e.g., region 5 (Russia) shows a significant change signal for the persistence and some changes are significant at the 10 or 20 % significance level (p value ≤ 0.1 or ≤ 0.2). Nevertheless, we follow , who question hypotheses testing on future climate ensembles and instead propose to better use “a simple descriptive approach for characterizing the information in an ensemble of scenarios”. Being conscious about the difficulties, which may arise during hypotheses testing, we look at the ensemble spread in the form of the interquartile range to assess the robustness of the results, and consult the significance test to support our findings. In many cases, the majority of ensemble members show a change signal in the same direction and the change signal is of a similar order of magnitude as the observed past changes in the preceding section (Figs. , ). In addition, a comparison to the results of the error estimation using FT-surrogate time series (Table ) yields that the changes are higher than the estimated error. Therefore, we conclude that future changes of the succession of cold and wet extremes in winter in some regions in Europe can be expected. These changes are, for the significant cases, larger than 50 % for the persistence, larger than 20 % for the recurrence time and larger than 5 % for the entropy. Regarding the findings from our sensitivity analysis (Sect. ) such changes are larger than the natural variability of the descriptors, which hence can be ruled out as the cause. Further, the sensitivity study has shown that such changes in the past occurred concurrently with a strong increase in CO2 emissions. As explained in Sect. , the only difference between the model runs for the periods 1971–2001 and 2021–2050 is the CO2 forcing; thus, the most probable reason for these changes in the future is the increase in CO2 emissions.

Figure  reveals three regions that seem to be particularly susceptible to changes of the dynamics/succession, namely, regions 2 (France), 3 (Germany) and 5 (Russia). The persistence changes for all regions and cold and wet episodes are likely to be of longer duration in the future. In regions 2 and 3 (France and Germany) the recurrence time decreases. The consequences of these changes are that these regions will probably experience more and longer cold and wet events in winter. Furthermore, these are less predictable (increase of entropy). The situation is different for region 5 (Russia), here the duration of cold and wet periods probably increases as well, but the number of events stays constant. Thus, the system resides for longer times in the non-extreme states (increase in recurrence time).

The change in the number of compound hot and dry extreme days is depicted in Fig. . Here, the number of compound extreme days varies with the region (although the number of univariate extremes are kept the same). Region 1 (Spain) shows a relatively low number of compound hot and dry days (note: all extremes in this paper are relative), regions 5 and 6 (Russia and Bulgaria) have a high number and also the highest decrease between the two time periods. Except for region 3 (Germany), which shows a slight increase, the number of compound extremes decreases in all regions. However, the change is generally small, < 10 %. Thus, the observed changes of the descriptors can mostly be attributed to the change in the dynamics and not to a change in the numbers of events, except maybe for regions 4 and 5 (Russia and Bulgaria). The change signal of the descriptors is pictured in Fig. . Two regions are most probably susceptible to changes in the dynamics of the hot and dry state, namely, regions 1 (Spain) and 6 (Bulgaria). Region 1 shows a small increase in persistence and a quite strong increase in recurrence time (on the order of 20 %) of the hot and dry state, the entropy does not change. The hot and dry periods get longer but less frequent. Regarding again the sensitivity study (Sect. ) it can be seen that a change of 20 % of the recurrence time in summer (JJA) is at least twice as large as the variability of the recurrence time (about 10 %) from 1900 to 2015 and constitutes a fairly large jump. The situation for region 6 is similar to that of region 1, with an increase in persistence and recurrence time and only a very small change in entropy. In addition, region 3 (Germany) shows an increase in persistence and a decrease in entropy. This means the episodes will be longer and more regular, whereas in region 5 (Russia) the persistence slightly decreases and the recurrence time increases. This implies changes towards shorter and less frequent events.