Introduction

Prolonged periods of precipitation elevate soil moisture levels. Once the soil is saturated, excess water flows directly into rivers, increasing the risk of flooding. The arrival of a storm further intensifies this risk. (The opposite is also true, as increases in extreme precipitation do not always lead to more frequent flooding. When soil moisture levels are low, the risk of flooding from heavy rainfall is reduced1,2,3.) This relationship between high antecedent soil moisture and flooding is well documented4,5,6,7,8. In this paper, we argue that the shift in the winter North Atlantic Oscillation (NAO) towards strongly positive values, see Fig. 1a, may increase this risk in north-western Europe due to increased wet weather persistence. Next to the relation between NAO states and precipitation in winter, we make our analysis broader by including the effect of the shift in the winter NAO on temperature persistence during winter. Throughout this paper, winter refers to the December–February period (DJF).

Fig. 1: NAO index patterns and precipitation anomalies in December 2023.
Fig. 1: NAO index patterns and precipitation anomalies in December 2023.
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a Kernel density estimates of the NAO index in winter for 1950–1980 in orange and 1990–2020 in red. The vertical dashed line represents the mean of the NAO index in December 2023. b NAO index in December 2023. The vertical dashed line depicts the arrival of Storm Henk. c Absolute difference in percentage of days with precipitation (0.5mm/day or more precipitation) for weather stations across Europe between December 2023 and the December months of 1950–2020. d Absolute difference in accumulated precipitation between December 2023 and the December months of 1950–2020 for weather stations across Europe. (1)–(5) denote recorded floods in the UK (1), northern France (2), eastern Germany (4) and southern Denmark (5)9,10. Record-high water levels were recorded in the Netherlands (3)11.

For illustration, in January 2024, severe flooding affected regions including the UK, eastern Germany, northern France, and southern Denmark9,10, with the Netherlands reporting record-high water levels11. In the weeks leading up to the floods, December 2023 was characterized by a strongly positive NAO phase (Fig. 1b). Figure 1c reveals highly persistent wet weather in north-western Europe during these weeks, expressed as the absolute difference in the fraction of days with precipitation exceeding 0.5mm in December 2023 compared to the average fraction of precipitation days in December months of 1950–2020. (Although 100% rainy days may correspond to a total rainfall of only 15mm, our primary interest lies in the fraction of rainy days as an indicator of persistence. Thus, this measure serves only as a lower bound.) Fig. 1d depicts the absolute difference in accumulated precipitation for the same stations between December 2023 and the average accumulated precipitation in December of 1950–2020. The dark red areas in north-western Europe indicate large accumulated precipitation values compared to historical values, leading to elevated soil moisture levels. Consequently, when Storm Henk arrived on January 2nd and 3rd, the saturated soil could not absorb additional water, resulting in widespread flooding in the regions. Thus, persistent rainfall in north-western Europe during three weeks of strongly positive NAO in December 2023 led to saturated soils, providing the precondition for flooding in early January.

This paper provides a statistical framework to formally investigate the relationship between the upward shift in the winter NAO and changes in weather persistence. The NAO is the primary mode of atmospheric variability over the North Atlantic region, with the largest impact in winter12. A simple definition of the NAO index is the difference in normalized atmospheric pressures between Lisbon (Portugal) and Stykkisholmur (Iceland)13. During winter months, positive NAO index values are associated with higher temperatures and increased precipitation in western and northern Europe, while southern Europe is more likely to experience cold and dry weather14. Since the 1950s, there has been a long-term shift towards predominantly positive phases of the NAO during winter15,16, although shorter sub-periods with opposing trends have been observed due to internal variability17,18. Projections from CMIP6 models under scenarios of substantial greenhouse gas forcing also anticipate an increasing prevalence of positive NAO states. However, the observed trajectory is now diverging from the paths generated by climate model simulations18.

Our work expands on recent findings on the influence of the NAO on precipitation and temperature patterns. Earlier research19 uses the first empirical orthogonal function (EOF1) as a proxy for the NAO index, rather than the NAO index itself. The authors report that the EOF1 is generally associated with more frequent precipitation. However, the implications of the recent shift toward extremely positive NAO index values for the persistence of precipitation remain uncertain. In our analysis, we focus specifically on the upper tail of the distribution of the NAO index. More precisely, when conditioning on the upper tail of the distribution of the NAO index, we find statistically significant deviations of the persistence of precipitation from historical patterns. Yet, when conditioning on the central regions of the distribution of the NAO index, no such deviations are found.

The authors19 also show a correlation between EOF1 and (maximum) temperature, consistent with earlier findings for the NAO index14. However, it remains unclear whether, and in what way, the recent shift in the dynamics of the winter NAO index affects different parts of the conditional temperature distribution differently. In other words, while it is well established that temperatures and the NAO index are related, it remains unclear how the temperature distribution at time t relates to the NAO index when conditioning on past temperature levels at time t−1. Hence, here we introduce a statistical method to analyze the full distribution, rather than just the mean or median, of next-day temperatures conditional on past observations. This refined perspective allows us to detect changes in the persistence of the extreme quantiles when comparing recent and earlier periods. Linking the model’s coefficients to the NAO index during winter months offers additional insights into the significance and magnitude of these changes.

The NAO index is not the only atmospheric pattern potentially linked to changes in weather persistence over Europe. To broaden and strengthen our findings, we also apply our framework to the Scandinavian Teleconnection pattern (SCAND) and the Atlantic Multidecadal Oscillation (AMO).

Results

Changes in temperature persistence

In the following, we refer to the quantitative impact of today’s observed temperature on tomorrow’s temperature distribution as persistence. A high persistence coefficient indicates a strong influence on tomorrow’s temperature distribution, while a low coefficient suggests a weak impact. Our quantile autoregressive model with seasonal coefficients is defined as follows:

$${Q}_{{y}_{t}}(\tau | {y}_{t-1})=\,{\mu }_{t}(\tau )+{\phi }_{t}(\tau ){y}_{t-1}.$$
(1)

Here, we model \({Q}_{{y}_{t}}(\tau | {y}_{t-1})\), the τ-quantile of the distribution of yt, which represents the daily average temperature at time t, conditional on yesterday’s average temperature yt−1. The value μt(τ) is determined by a constant α0(τ), a trend component with parameter α1(τ), and Fourier terms with parameters (αp,2, αp,3)(τ) for p ∈ {1, …, P}:

$${\mu }_{t}(\tau )=\,{\alpha }_{0}(\tau )+{\alpha }_{1}(\tau )t+\sum\limits_{p=1}^{P}\left({\alpha }_{p,2}(\tau )\sin \left(\frac{2\pi pt}{365}\right)+{\alpha }_{p,3}(\tau )\cos \left(\frac{2\pi pt}{365}\right)\right).$$

Thus, μt(τ) represents the time-varying “intercept” for quantile τ, which includes a constant, trend, and seasonality components. By allowing for a quantile-specific trend we control for the general effects of global warming in our analysis as we are primarily interested in ϕt(τ), which we refer to as the persistence coefficient. ϕt(τ) is determined by a constant β0(τ) and Fourier terms with parameters (βr,1, βr,2)(τ) for r ∈ {1, …, R}:

$${\phi }_{t}(\tau )={\beta }_{0}(\tau )+\sum\limits_{r=1}^{R}\left({\beta }_{r,1}(\tau )\sin \left(\frac{2\pi rt}{365}\right)+{\beta }_{r,2}(\tau )\cos \left(\frac{2\pi rt}{365}\right)\right).$$

By including Fourier terms, we allow the persistence coefficient ϕt(τ) to vary throughout the year. Both μt(τ) and ϕt(τ) can be estimated for any quantile τ ∈ (0, 1) of interest. We refer to the Methodology section for more information regarding the estimation procedure.

Figure 2 shows the estimated persistence coefficient curves derived from Model (1) for daily temperature data in De Bilt (The Netherlands) from 1990–2020. Panel (a) illustrates the seasonal variation in persistence coefficients throughout the year for quantiles τ ∈ 0.1, 0.2, . . . , 0.9. The lower quantiles exhibit stronger persistence during winter, while the higher quantiles show greater persistence during summer. This seasonal asymmetry is consistent with the effects of atmospheric blocking, which tends to suppress westerly winds. In Europe, this leads to colder extremes in winter and warmer extremes in summer20. These findings highlight that focusing solely on the mean or median overlooks crucial information in the extreme parts of the distribution, as averaging smoothens out variations in higher and lower quantiles. This is particularly important from an impact point of view as we are interested in the persistence of extremes, not average weather persistence.

Fig. 2: Persistence coefficients plots for De Bilt.
Fig. 2: Persistence coefficients plots for De Bilt.
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Persistence coefficients plots obtained by fitting Model (1) to data for De Bilt. Shaded areas represent 95% pointwise confidence bounds. a ϕt(τ) for τ ∈ {0.1, . . . , 0.9} fitted on the “new” data. b ϕt(0.05) fitted on “old” and “new” data. c ϕt(0.5) fitted on “old” and “new” data. d ϕt(0.95) fitted on “old” and “new” data.

Panels (b)–(d) of Fig. 2 display the persistence coefficient curves for varying quantiles τ when the statistical model is fit to the “old” (1950–1980) and the “new” data (1990–2020) in De Bilt. For τ = 0.05 in panel (b), there is a statistically significant decrease in the persistence coefficients for most days in December-February (DJF). For the median (τ = 0.5) in panel (c), there is no observable shift in the persistence coefficient, which is consistent with earlier research21. Conversely, for τ = 0.95, a statistically significant increase in the persistence coefficients is observed for early winter days. These results suggest that beyond the effects of global warming, captured by an increase in the model’s constant and trend, we observe an increased persistence of warmer quantiles and decreased persistence of colder quantiles in winter.

In what follows, \({\bar{\Delta }}_{i,\phi }(\tau )=\frac{1}{\#\{t\in \,{{\mbox{DJF}}}\,\}}{\sum }_{t\in {\mbox{DJF}}}\left({\phi }_{t}^{i,new}-{\phi }_{t}^{i,old}\right)(\tau )\) represents the average difference in persistence coefficients for weather station i in winter (see Equation (4)). Figure 3a–c presents \({\bar{\Delta }}_{i,\phi }(\tau )\) for weather stations across Europe. A positive \({\bar{\Delta }}_{i,\phi }(\tau )\) means that the persistence coefficient has increased in winter for quantile τ while a negative \({\bar{\Delta }}_{i,\phi }(\tau )\) implies that the persistence coefficient of quantile τ has decreased in winter. Panel (a) shows a considerable reduction in the average persistence coefficient for τ = 0.05 at weather stations across western Europe during winter. In contrast, for τ = 0.5 in panel (b), there appears to be no shift in \({\bar{\Delta }}_{i,\phi }(\tau )\). However, for τ = 0.95 in panel (c), there is a pronounced increase in persistence across western Europe, suggesting a nuanced pattern of persistence shifts depending on the quantile considered. Supplementary Fig. 1 shows the same figure where we removed the stations where the difference between the persistence curves in winter is insignificant using the uniform confidence bands for the difference between \({\phi }_{t}^{i,new}\) and \({\phi }_{t}^{i,old}\) as discussed in the Methodology section. Here we use uniform confidence bands to assess whether the entire difference curve during winter is significantly different from zero, rather than testing significance at individual time points. We also perform a leave-one-year-out analysis to ensure that the persistence shift is not driven by a single year in the “new” data. Supplementary Fig. 2 demonstrates that removing individual years does not significantly alter the distribution of \({\bar{\Delta }}_{\phi }(0.95)\).

Fig. 3: Shifts in persistence across Europe in winter.
Fig. 3: Shifts in persistence across Europe in winter.
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a–c \({\bar{\Delta }}_{i,\phi }(\tau )\) for weather stations across Europe with (a) τ = 0.05 (b) τ = 0.5 (c) τ = 0.95. d–i \({\bar{\Delta }}_{i,{\psi }_{s}}(\tau )\) for weather stations across Europe with (d) τ = 0.05 and s = NAO − (e) τ = 0.5 and s = NAO − , (f) τ = 0.95 and s = NAO − , (g) τ = 0.05 and s = NAO + (h) τ = 0.5 and s = NAO +, (i) τ = 0.95 and s = NAO + . The blank regions in Poland and Belarus result from the unavailability of data in the ECA dataset (see Data section).

So far, we have shown decreased persistence in low quantiles and increased persistence in high quantiles for daily temperatures in western Europe during winter. Combined with the global warming trend, this suggests that warm and cold spells in winter are becoming more and less likely, respectively. We now proceed to investigate the impact of the winter NAO index shift on temperature persistence. Supplementary Table 1 illustrates how the winter NAO index is linked with the likelihood of weather regimes22 during 1950-2020, with higher winter NAO index values increasing the probability of both NAO + and Scandinavian Trough (ScTr) regimes. This highlights the central role of the NAO index in modulating North Atlantic atmospheric variability12. Therefore, we focus on the NAO index in our analysis, but, for completeness, we also consider SCAND and AMO, which influence the northwestern European climate on seasonal to decadal scales23,24,25.

First, we perform two-sample Kolmogorov–Smirnov tests for the equality of the “old” and “new” distribution in the winter of the indices. The tests reveal a significant difference for the daily winter NAO index distributions (p-value = 0.0) and a significant difference in the monthly winter AMO index distributions (p-value = 0.029). In contrast, no significant difference is observed in the monthly SCAND index distributions between the two periods (p-value = 0.332). Note that a significant shift in the AMO index is expected as it is characterized by variability on decadal timescales. As shown in Supplementary Fig. 3, the winter NAO index distribution gradually increases over time, suggesting a potential long-term shift rather than natural variability, consistent with earlier findings18. This highlights the importance of analyzing changes in temperature and precipitation patterns conditionally on the NAO index to better understand the climatic implications of such a trend.

A shift in the persistence of weather patterns may stem from changes in the persistence of the underlying atmospheric modes26,27. To test whether winter NAO persistence has changed, we split the winter NAO index into quintiles for each period (see Supplementary Table 2) and apply a test for equality of first-order Markov transition matrices28 (the matrices indicating the switching probabilities Pi,j between NAO quintiles, from quintile i to j with i, j ∈ {1, 2, 3, 4, 5}). The resulting p-value of 0.517 indicates no significant change. See also Supplementary Fig. 4 for a statistical analysis of transition probabilities Pi,i for i ∈ {1, 2, 3, 4, 5} for the “old” and the “new” period. Thus, the increased persistence associated with the NAO is not due to the NAO itself becoming more persistent, but to more persistent warm and wet weather when the NAO index reaches strongly positive values.

For illustration, Supplementary Fig. 5 panel (d) shows visually that when we condition on winter NAO index values above the 95th percentile for the “old” and “new” data separately, the temperature range of the conditional distribution for the “new” data in De Bilt is not only shifted to the right but is also narrower than that of the “old” data. However, we also demonstrate that when the NAO index quantiles are fixed to their historical values, the current climate system exhibits behavior similar to the past. Panel (b) illustrates this stability, as conditioning on the 92.5 to 97.5 percentile of the “old” distribution (rather than conditioning separately by period) results in conditional distributions that are comparable in location and shape.

To include the NAO index in our analysis we extend Model (1) as follows:

$$ {Q}_{{y}_{t}}(\tau \vert {y}_{t-1},{{\rm{NAO}}}_{t-1}) \\ =\left\{\begin{array}{ll}{{\nu }_{t,+}({\tau})+{\gamma}_{+}(\tau){\mathbb{1}}_{\{t-1 = w\}}{{\rm{NAO}}}_{t-1}+{\psi }_{t,+}({\tau}){y}_{t-1}} & {{{\rm{if}}}\,{{\rm{NAO}}}_{t-1}\ge 0,}\\ {{\nu}_{t,-}(\tau)+{\gamma}_{-}({\tau}){\mathbb{1}}_{\{t-1 = w\}}{{\rm{NAO}}}_{t-1}+{\psi}_{t,-}({\tau}){y}_{t-1}} & {{{\rm{if}}}\,{{\rm{NAO}}}_{t-1} < 0.}\end{array}\right.$$
(2)

In (2), we split \({Q}_{{y}_{t}}(\tau | {y}_{t-1},{{\mbox{NAO}}}_{t-1})\) based on past NAO index values NAOt−1. For clarity, we rename the coefficients of the model where we condition on the NAO index. The value νt,s(τ) denotes the quantile intercept, instead of μt(τ), for a given τ and state s ∈ {NAO − , NAO + }, where s = NAO − if NAOt−1 < 0, and s = NAO + otherwise. Similarly, ψt,s(τ) represents the persistence coefficient, instead of ϕt(τ). The parameter γs(τ) reflects that higher NAO index values are associated with warmer temperatures and lower values with colder temperatures. For the monthly AMO and SCAND indices, we apply the same modeling approach, assigning the same index value to each day within a given month.

Supplementary Fig. 6 illustrates that when we apply Model (2) to the daily De Bilt data and thus condition on the NAO index values, the decrease in the persistence coefficient for τ = 0.05 is statistically significant for most winter days during NAO − phases but not during NAO + phases. Furthermore, the difference between the “old” and “new” curves has widened compared to the results shown in Fig. 2b. The persistence coefficients of the median remain statistically insignificant. For τ = 0.95 we observe a more pronounced difference between the “old” and “new” curves in winter compared to Fig. 2d. Additionally, the difference between the persistence coefficient curves is statistically significant during winter days in NAO + phases, while it remains insignificant during NAO − phases. Note that these differences in persistence coefficients are also not simply due to positive NAO index values being generally warmer, as this effect is captured by the model’s constant, trend and the lagged NAO variable.

We conduct a rolling window analysis for ψt,s(τ) on January 1st in De Bilt to verify whether the shifts in persistence coefficients evolve gradually. See Supplementary Fig. 7 for the rolling window analysis for all combinations of τ ∈ {0.05, 0.5, 0.95} and s ∈ {NAO − , NAO + }. Supplementary Fig. 7a shows that the persistence coefficient gradually decreases for τ = 0.05 and that there seems to be a negative relation between ψt,−(0.05) and the 80th percentile of the NAO index over time. Supplementary Fig. 7f shows that the persistence coefficient gradually increases for τ = 0.95 over time. Moreover, it provides evidence that there is a positive relation between the shift in ψt,+(0.95) and the 80th percentile of the NAO index over time.

In Fig. 3d–i, we present \({\bar{\Delta }}_{i,{\psi }_{s}}(\tau )=\frac{1}{\#\{t\in \,{\mbox{DJF}}\,\}}{\sum }_{t\in {\mbox{DJF}}}\left({\psi }_{s,t}^{i,new}-{\psi }_{s,t}^{i,old}\right)(\tau )\) (see Equation (5)) for weather stations i across Europe and state s ∈ {NAO + , NAO − }. Supplementary Fig. 1d–i again shows the same plots where we remove the stations where the difference between the persistence curves in winter is insignificant according to the uniform confidence bands. Figure 3d shows that for τ = 0.05, western Europe appears darker green when conditioned on NAO—compared to the lighter green seen when conditioned on NAO + in panel (g). Panels (e) for NAO—and (h) for NAO + indicate that there is no significant shift in persistence coefficients when conditioning on NAO index values for τ = 0.5. For τ = 0.95, there is a dark red area over western Europe in panel (i) when conditioned on NAO + , in contrast to the lighter red seen in panel (f). These results underscore a shift in persistence within the climate system for some of its characteristics: higher NAO + index values are associated with increased persistence coefficients for τ = 0.95, while decreased persistence coefficients are observed for τ = 0.05 when conditioned on NAO − .

A similar pattern is seen for SCAND in Supplementary Fig. 8. For SCAND − and τ = 0.95 (panel f), persistence decreases over the Nordics but increases over central and western Europe, aligning with the increasing probability of ScTr regimes at high NAO index values. The reverse holds for τ = 0.05 (panel d). For SCAND + (panels (g)–(i)), no notable differences appear compared to the unconditional results. Similarly, the AMO (Supplementary Fig. 9) shows no important deviations, suggesting it does not influence the persistence shift.

Changes in precipitation persistence

Next, we address the potential changes in winter precipitation persistence, which raises antecedent soil moisture levels and, consequently, elevates flood risk4,5,6,7,8. We begin by presenting the results for De Bilt, followed by the findings for weather stations across Europe. In Supplementary Fig. 10, the probabilities of precipitation are plotted conditional on the quintiles of the NAO index for the “old” and “new” datasets separately. Although the probability of precipitation increases with the NAO index, there is no statistically significant increase in the probabilities when conditioning on the lower and central regions of the NAO index distribution. However, for the highest quintile, we observe a statistically significant increase in the probability of precipitation by approximately 20%. This suggests that when the climate system exhibits NAO index values above the 80% percentile, De Bilt is much more likely to experience a sequence of days with precipitation in a row compared to the “old” period.

Figure 4 shows the results of a rolling window analysis of the probability of precipitation conditional on the highest quintile of the NAO index in winter over time in De Bilt. It can be seen that the shift in the probability of precipitation is not sudden but evolves gradually. When comparing the 95% confidence bounds for the probabilities of precipitation before 1990 and after 2015 it becomes evident that the conditional probability of precipitation has increased significantly over time, which is consistent with Supplementary Fig. 10 Q5. For comparison, we plot the 80th percentile of the NAO index in winter over time in red, which shows a similar trend as the probability of precipitation in the highest quintile of the NAO index.

Fig. 4: Rolling window analysis of precipitation probabilities and the NAO index in winter.
Fig. 4: Rolling window analysis of precipitation probabilities and the NAO index in winter.
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Rolling window analysis with a window length of 30 years and ending years 1980–2020 of the probability of precipitation conditional on the highest quintile of the NAO index in winter for De Bilt, with 95% confidence intervals. 80th percentile of the winter NAO index of the respective window depicted in red.

Figure 5a shows

$$ {\Delta }_{i,{Q}_{5}}^{NAO}= P({y}_{t}^{i,new}= 1| t\in DJF,{Q}_{NA{O}_{t-1}^{new}}\\ = 5)-P({y}_{t}^{i,old}=1| t\in DJF,{Q}_{NA{O}_{t-1}^{old}}=5),$$

see Equation (6), for weather stations across Europe. Consistent with the results shown in Supplementary Fig. 10, our analysis reveals that north-western Europe experiences increased precipitation persistence in the upper tail of the NAO index in winter. We refer to Supplementary Fig. 11 for the version of Fig. 5a where the insignificant stations are removed. Figure 5b presents the fraction of winter days with more than 0.5 mm of precipitation, considering only those days with NAO index values above the highest quintile of the NAO index of the “new” dataset in winter. The probability of precipitation is observed to approach values close to 100% as one moves towards north-western Europe. The pattern observed in Fig. 5b closely resembles the precipitation pattern observed in Fig. 1c for December 2023, prior to the floods in north-western Europe.

Fig. 5: Precipitation probabilities associated with the highest NAO quintile in winter.
Fig. 5: Precipitation probabilities associated with the highest NAO quintile in winter.
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a \({\Delta }_{i,{Q}_{5}}\) for weather stations across Europe. b Fraction of winter days with more than 0.5 mm of precipitation, considering only those days with NAO index values in the highest quintile of the NAO index of the “new” dataset in winter.

To verify whether this pattern is not driven by one year with extremely high accumulated precipitation during winter, we perform a leave-one-year-out analysis, implying that we performed the same analysis but with one year of the “new” dataset excluded. We plot the values for \({\Delta }_{i,{Q}_{5}}^{NAO}\) throughout Europe in boxplots in Supplementary Fig. 12. The leave-one-year-out analysis provides evidence that leaving one year out does not impact the probability of precipitation in the highest quintile of the NAO index.

A similar analysis is performed for the SCAND and AMO index. Given the link between positive NAO values and the Scandinavian Trough regime, we conditioned on the lowest SCAND quintile. The resulting patterns, denoted by \({\Delta }_{i,{Q}_{1}}^{\,{\mbox{SCAND}}\,}\) and shown in Supplementary Fig. 13, are consistent with the findings obtained when conditioning on the highest NAO quintile. For the AMO, we analyzed both \({\Delta }_{i,{Q}_{1}}^{\,{\mbox{AMO}}\,}\) and \({\Delta }_{i,{Q}_{5}}^{\,{\mbox{AMO}}\,}\), presented in Supplementary Fig. 14a, b, respectively. Panel (a) shows no evident shift in precipitation. In contrast, panel (b) indicates that the influence of the AMO on the probability of precipitation has changed. However, the spatial pattern differs from those observed for the NAO and SCAND indices.

Discussion

As temperatures increase, the atmosphere’s water-holding capacity grows which can drive more frequent rainfall extremes29,30,31,32. In addition, heavy winter precipitation events over northern Europe are expected to be modulated by a trend towards positive NAO index values by the end of the 21st century33, which enhances flood risk34. Here we have shown that wet (and warm) weather becomes more persistent in the “new” upper tail of the NAO distribution and that this trend in persistence can already be detected in observations.

We verify that the observed increase in persistence is not due to the NAO itself becoming more persistent. Specifically, we test whether the transition matrices differ between the “old” and “new” periods. We find no evidence of a significant difference between the transition matrices of the “old” and “new” periods. Instead, when the NAO index reaches extremely positive values, a very strong and zonally-oriented jet stream develops. This strong westerly flow brings moist air masses, with mild temperatures, from the Atlantic towards western Europe. Also, the jet stream steers North Atlantic storm systems directly towards Europe. Hence, strongly positive NAO states lead to persistent warm and rainy conditions.

The observed winter NAO trend goes outside historical climate simulations, making it difficult to attribute it to anthropogenic forcing with any level of confidence18. However, there is limited evidence for a connection with the potential weakening of the Atlantic Meridional Overturning Circulation (AMOC). The potential slowdown of the AMOC is supported by evidence such as the “cold blob”, and researchers suggest that it has been occurring since the 1950s16. Under strong greenhouse gas forcing and a weakened AMOC, climate models replicate observed oceanic and atmospheric changes34. NAO-induced changes in the AMOC may have significantly contributed to the rapid shifts observed in Arctic sea ice, extratropical temperature variations and Atlantic tropical storm activity in recent decades15. Conversely, there is also evidence indicating a link between the shift towards positive NAO index values and the weakening of the AMOC16. The authors find a negative correlation between AMOC and NAO indices, suggesting that the AMOC can at least partly drive NAO changes via changes in North Atlantic SSTs. Whether this slowdown is predominantly anthropogenic or driven by natural variability remains uncertain. Irrespective of whether the “cold blob” is due to AMOC weakening or not, it increases the north-south temperature gradient, which in turn favors a positive NAO phase35,36,37.

Conclusion

Our novel statistical modeling procedure quantifies changes in the persistence coefficients across the full temperature distribution, conditional on past observations, rather than focusing solely on the mean or median. The models employed allow for distinct trends and seasonal components for each quantile considered. This way, we show an upward and downward shift in the persistence coefficients of the high and low quantiles in western Europe during winter, respectively, even after accounting for a long-term upward trend in temperatures. These shifts become even more pronounced when we consider a model that conditions on NAO states. Furthermore, we observe a statistically significant increase in the persistence of precipitation in winter in north-western Europe when the climate system is in the upper tail of the distribution of the NAO index, favoring more persistent wet weather. The results conditioned on the SCAND index align with those based on the NAO index, reflecting the association between high NAO index values and the ScTr regime. In contrast, no relationship is found between the AMO and shifts in weather persistence.

Regarding the climate impacts in western and northern Europe, our findings indicate that warm and wet conditions persist longer during winters characterized by extreme positive NAO values. This enhanced persistence is distinct from the broader effects of global warming and exceeds the well-established link between high NAO phases and milder, wetter winters. As the distribution of the NAO shifts upwards, previously observed low temperatures no longer occur, and the probability of precipitation in locations such as De Bilt increases by approximately 20% relative to the highest quintile of the “old” period (from 0.556 to 0.652). Figure 6 illustrates this change: Panel (a) shows a two-day increase in the expected number of precipitation days over a three-week period, while panel (b) reveals a threefold increase in the probability of experiencing more than 14 wet days (from 0.1054 to 0.3634). These findings indicate that increasingly positive NAO conditions substantially raise the likelihood of prolonged wet periods, potentially leading to elevated soil moisture and, consequently, heightened flood risk and runoff volumes4,5,6,7,8,38, as exemplified by the events of January 2024.

Fig. 6: Density functions for winter precipitation probabilities in De Bilt.
Fig. 6: Density functions for winter precipitation probabilities in De Bilt.
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a Probability density function of a binomial distributed variable with n = 21 and p = 0.6519 (in red) and p = 0.5556 (in orange). b Cumulative density function of (a) with horizontal dashed lines denoting the cumulative probability of less than 14 days with precipitation exceeding 0.5mm.

Methods

Conditional quantile autoregression for daily temperature data with seasonal coefficients

We now discuss the model used to model daily temperatures in this paper. When modeling temperatures, we need to consider seasonality as we expect higher temperatures in summer than in winter. Additionally, because heat waves are more likely in summer and cold spells are more common in winter20, the persistence parameter may also show signs of seasonality. In our model, not only the constant but also the persistence coefficient depends on time index t. 

Our modeling procedure relates to ref. 39, who utilize the model proposed by Giraitis and Marotta40 to estimate properties of unevenly spaced time series and apply it to a daily Central England Temperature (CET) series starting in 1772. Their comparison of model outcomes between the periods 1780–1820 and 1980–2020 reveals a statistically significant increase in the persistence coefficient of the mean during winter. However, the study does not provide an inferential analysis for comparing more recent years and different weather stations. Others21 analyze global and recent data and find no shifts in the persistence coefficients of the mean after adjusting for trends and noise.

Including more recent years and different weather stations, as well as using quantile instead of mean autoregression, allows us to substantially extend these results from the literature. Quantile autoregression41 is the time series version of quantile regression42. Theory for nonparametric quantile estimations for dynamic smooth coefficient models is presented in ref. 43, where the quantile regression coefficients are allowed to vary over a smoothing parameter, which can be one of the regressors, time or an exogenous variable. In their model, however, the authors do not account for the nonstationarity of the data if the coefficients vary over time. Their results are extended to nonparametric inference for time-varying coefficient quantile regression, where the nonstationarity is taken into account44. The emphasis of the mentioned studies is on estimating the coefficient functions rather than the quantile regression surface \({Q}_{{y}_{t}}(\tau | {y}_{t-1})\), aligning with our setting. In our analysis, the main interest lies in ϕt(τ), the persistence coefficient for quantile τ on day t. For fixed τ, our model without incorporating the NAO index is defined as follows:

$${Q}_{{y}_{t}}(\tau | {y}_{t-1}) = \,{\mu }_{t}(\tau )+{\phi }_{t}(\tau ){y}_{t-1}\\ = \,{\alpha }_{0}(\tau )+{\alpha }_{1}(\tau )t+{\sum }_{p = 1}^{P}\left({\alpha }_{p,2}(\tau )\sin \left(\frac{2\pi pt}{365}\right)+{\alpha }_{p,3}(\tau )\cos \left(\frac{2\pi pt}{365}\right)\right)\\ \;\;\;\;\;+ \left({\beta }_{0}(\tau )+{\sum }_{r = 1}^{R}\left({\beta }_{r,1}(\tau )\sin \left(\frac{2\pi rt}{365}\right)+{\beta }_{r,2}(\tau )\cos \left(\frac{2\pi rt}{365}\right)\right)\right){y}_{t-1}.$$
(3)

In this paper, we set P = R = 2, but the conclusions remain consistent when other values of P and R are used. We leave a data-driven approach for determining P and R open for further research. In the second part of our analysis of the daily temperature data, we incorporate the S index in winter for S ∈ {NAO, AMO, SCAND}. For clarity, we now define the constant and persistence coefficient to be νt,s(τ) and ψt,s(τ) respectively with s ∈ { + , − } (instead of μt(τ) and ϕt(τ)). The model is then defined as follows for fixed τ:

$$ {Q}_{{y}_{t}}({\tau} \vert {y}_{t-1},{S}_{t-1})\\ =\left\{\begin{array}{rcl}{\nu}_{t,+}({\tau})+{\gamma}_{+}({\tau}){\mathbb{1}}_{\{t-1 = w\}}{S}_{t-1}+{\psi}_{t,+}({\tau}){y}_{t-1} & {{\rm{if}}}\,{S}_{t-1}\ge 0,\\ {\nu}_{t,-}({\tau})+{\gamma}_{-}({\tau}){\mathbb{1}}_{\{t-1 = w\}}{S}_{t-1}+{\psi}_{t,-}({\tau}){y}_{t-1} & {{\rm{if}}}\,{S}_{t-1} < 0.\end{array}\right.$$
(4)

St represents the value of the S index on date t. \({{\mathbb{1}}}_{\{t-1 = w\}}\) is a dummy variable that equals 1 if t − 1 is a winter day and otherwise 0. Note that α0, . . . , βR,2 are functions \([0,1]\to {\mathbb{R}}\) that we want to estimate at a fixed point τ. For each τ, the parameter vector θ(τ) can be estimated by minimizing

$${\hat{\theta }}_{n}(\tau )={\arg\min}_{\theta \in {{\mathbb{R}}}^{p+1}}{\sum}_{t=1}^{n}{\rho }_{\tau }({y}_{t}-{x}_{t}^{{\prime} }\theta ).$$

Here, \({\rho }_{\tau }(u)=u\left(\tau -{{\mathbb{1}}}_{\{u < 0\}}\right)\) is the usual check function n denotes the length of the time series and \({x}^{{\prime} }\) represents the transpose of a vector x. Given the estimated parameter vector \({\hat{\theta }}_{n}(\tau )\) we obtain the estimated model coefficients \({\hat{\mu }}_{t,n}(\tau )\) and \({\hat{\phi }}_{t,n}(\tau )\) (or \({\hat{\nu }}_{t,\cdot ,n}(\tau )\) and \({\hat{\psi }}_{t,\cdot ,n}(\tau )\) when we condition on S index values). We are particularly interested in ϕt(τ) (and ψt,⋅(τ)), so we provide pointwise confidence intervals around the estimated curves by employing the delta method. Using the covariance matrix \({\hat{\Sigma }}_{n}\) obtained after parameter estimation, we can construct 100 × (1 − α)% pointwise confidence intervals by calculating:

$$C{I}_{{\hat{\phi }}_{t,n}(\tau )}(\alpha )=\,{\hat{\phi }}_{t,n}(\tau )\pm \frac{{\Phi }^{-1}(\alpha /2)}{\sqrt{n}}\sqrt{\frac{\partial {\hat{\phi }}_{t,n}(\tau )}{\partial \beta (\tau )}{\hat{\Sigma }}_{\beta ,n}{\frac{\partial {\hat{\phi }}_{t,n}(\tau )}{\partial \beta (\tau )}}^{{\prime} }},$$

which can be easily obtained analytically or numerically. Here, Σβ ⊂ Σ denotes the subset of the full covariance matrix for the parameters determining ϕt(⋅), Φ−1(⋅) denotes the inverse of the normal cumulative density function. Note that \(C{I}_{{\hat{\psi }}_{t,\cdot ,n}(\tau )}(\alpha )\) can be obtained in a similar manner. These confidence bounds are used in Fig. 2 and Supplementary Fig. 6. This pointwise interval has the interpretation that, for any fixed t, the probability that the interval contains the true value ϕt(τ) is approximately 1 − α. However, it does not guarantee simultaneous coverage for all t ∈ [a, b] with a < b. In contrast, uniform confidence bands provide coverage over the entire domain with a specified confidence level. Therefore, to test whether the persistence curves ϕt(τ) are different for the “old” and the “new” winter, we use uniform confidence bands. The uniform confidence bands are constructed as follows and are used in Supplementary Fig. 1.

Consider a time interval [a, b] with a < b, in our case DJF. For any t ∈ [a, b] let a random variable X(t) ~ Y(t). Then

$${\sup }_{t\in [a,b]}| X(t)| \sim {\sup }_{t\in [a,b]}| Y(t)| .$$

Now, let Z ~ N(0, 1) and let \(f:[a,b]\to {\mathbb{R}}\) be a function with f(t) ≠ 0 for at least one t ∈ [a, b]. Then X(t) ≔ f(t)Z is distributed as Y(t) ≔ N(0, f(t)2) for any t ∈ [a, b]. Hence, for any ε > 0: 

$${\mathbb{P}}\left({\sup}_{t\in [a,b]}| Y(t)| \le \varepsilon \right) = {\mathbb{P}}\left({\sup}_{t\in [a,b]} {|} X(t){|} \le \varepsilon \right) \\ = {\mathbb{P}}\left({|} Z{|} \le \frac{\varepsilon}{{\sup}_{t\in [a,b]}{|} f(t){|}}\right) \\ = \Phi \left(\frac{\varepsilon}{\mathop{\sup}_{t\in [a,b]}{|} f(t){|}}\right) - \Phi \left(-\frac{\varepsilon}{\mathop{\sup}_{t\in [a,b]}{|} f(t){|}}\right).$$
(5)

Recall that ϕt(τ) is a function of β(τ). Consequently, if \({\phi }_{t}^{new}(\tau )\) and \({\phi }_{t}^{old}(\tau )\) are equal for t ∈ 1, . . . , T* we must have βnew(τ) = βold(τ) for T* not too small. Under the assumption of \({\phi }_{t}^{new}(\tau )={\phi }_{t}^{old}(\tau )\) we thus have asymptotically

$$\sqrt{n}({\hat{\beta }}^{new}(\tau )-{\hat{\beta }}^{old}(\tau )) \sim N\left(0,{\Sigma }_{{\beta }^{new}}+{\Sigma }_{{\beta }^{old}}\right).$$

Because a linear combination of jointly normally distributed random variables is also normally distributed, we also have asymptotically

$$\sqrt{n}\left({\hat{\phi }}_{t}^{new}-{\hat{\phi }}_{t}^{old}\right) \sim N\left(0,\frac{\partial {\hat{\phi }}_{t,n}(\tau )}{\partial \beta (\tau )}\left({\hat{\Sigma }}_{{\beta }^{new},n}+{\hat{\Sigma }}_{{\beta }^{old},n}\right){\frac{\partial {\hat{\phi }}_{t,n}(\tau )}{\partial \beta (\tau )}}^{{\prime} }\right).$$

Denoting this variance by \(\hat{f}(t)\), a uniform confidence band for the difference in \({\phi }_{t}^{new}(\tau )\) and \({\phi }_{t}^{old}(\tau )\) can be constructed according to Equation (5).

To verify whether the model fits the data well, we verified whether the ‘hits’, which are defined as

$${{\mbox{hit}}}_{t}^{\tau > \ 0.5} = \left\{\begin{array}{ll}0& {{\rm{if}}} \,{y}_{t} < {Q}_{{y}{t}}(\tau | {y}_{t-1})\\ 1& {{\rm{if}}} {y}_{t} > {Q}_{{y}{t}}(\tau | {y}_{t-1})\end{array}\right.\\ {{\mbox{and}}}\,{{\mbox{hit}}}_{t}^{\tau \le 0.5}= \left\{\begin{array}{ll}0& {{\rm{if}}} \,{y}_{t} > {Q}_{{y}{t}}(\tau | {y}_{t-1})\quad \\ 1 & {{\rm{if}}} \,{y}_{t} < {Q}_{{y}{t}}(\tau | {y}_{t-1}),\hfill \end{array}\right.$$

for τ > 0.5 and τ ≤ 0.5, respectively, are uniformly distributed over (I) months throughout the sample and (II) days within the year. With (I) we test whether the hits are evenly distributed over the 30 years whereas (II) tests whether the hits are not clustered in a, for instance, particular month or season within the years. For De Bilt, we cannot reject uniformity of the hits for all quantiles considered and for any common level of significance. In the testing procedure we account for multiple testing using the Bonferroni–Holm method. A small-scale Monte Carlo simulation study is performed to verify the validity of the asymptotic confidence intervals of θ(τ) for τ ∈ {0.05, 0.5, 0.95} and is presented in Supplementary Table 3. Here we used the parameter estimates obtained by fitting Model (1) to the “new” data from De Bilt. 5000 Monte Carlo iterations are performed. The results suggest that the asymptotic confidence intervals are appropriate, as the empirical coverage rates closely align with their nominal values.

Coefficient comparisons

To visualize the shift in the (persistence) coefficients for the temperature and precipitation models, we use the following equations to quantify the differences. For the temperature model without conditioning we use:

$${\bar{\Delta }}_{i,\phi }(\tau )=\frac{1}{\#\{t\in \,{\mbox{DJF}}\,\}}\mathop{\sum }_{t\in {\mbox{DJF}}}\left({\phi }_{t}^{i,new}-{\phi }_{t}^{i,old}\right)(\tau ).$$
(6)

Here we take the mean of the difference in persistence coefficients \({\phi }_{t}^{i,new}(\tau )\) and \({\phi }_{t}^{i,old}(\tau )\) during the winter months over time ("new” minus “old”). For the model with conditioning on index values s ∈ {S − , S + } for S ∈ {NAO, AMO, SCAND} we use:

$${\bar{\Delta }}_{i,{\psi }_{s}}(\tau )=\frac{1}{\#\{t\in \,{\mbox{DJF}}\,\}}{\sum }_{t\in {\mbox{DJF}}}\left({\psi }_{s,t}^{i,new}-{\psi }_{s,t}^{i,old}\right)(\tau ).$$
(7)

For the differences in the precipitation probabilities conditional on the upper (i.e. 5th) quintile of the S index we use:

$$ {\Delta }_{i,{Q}_{j}}^{S}=P({y}_{t}^{i,new} =1| t\in DJF,{Q}_{{S}_{t-1}^{new}}=j)\\ -P({y}_{t}^{i,old}=1| t\in DJF,{Q}_{{S}_{t-1}^{old}}=j).$$
(8)

Here, we calculate the difference in the probability of precipitation in winter conditional on the previous index St−1, belonging to the jth quintile.