# Robustness of dynamical coupling between wintertime European weather extremes and the large-scale circulation

Ane Carina Reiter, Gabriele Messori, Davide Faranda, Morten Andreas Dahl Larsen, Martin Drews

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Ane Carina Reiter, Gabriele Messori, Davide Faranda, Morten Andreas Dahl Larsen, Martin Drews. Robustness of dynamical coupling between wintertime European weather extremes and the large-scale circulation. Climate Dynamics, 2025, 63 (10), pp.399. 10.1007/s00382-025-07846-x. hal-05322102

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# Robustness of dynamical coupling between wintertime European weather extremes and the large-scale circulation

Ane Carina Reiter<fnref n="1" />,<fnref n="2" /> ID · Gabriele Messori<fnref n="3" />,<fnref n="4" />,<fnref n="5" /> ID · Davide Faranda<fnref n="6" />,<fnref n="7" /> · Morten Andreas Dahl Larsen<fnref n="2" /> ID · Martin Drews<fnref n="1" />

Received: 22 April 2025 / Accepted: 5 August 2025

© The Author(s) 2025

## Abstract

Atmospheric circulation plays a key role in driving extreme weather events. A coupling measure, termed the co-recurrence ratio (α), has been introduced to study the link between surface extremes and large-scale circulation patterns. Stemming from dynamical systems theory, the co-recurrence ratio quantifies joint recurrences of multiple atmospheric variables. Here, we evaluate the robustness of the performance of the co-recurrence ratio to the choice of geographical domains, the variables used to characterize the large-scale circulation, and the thresholds used to define extremes. We adopt a bivariate focus, diagnosing the coupling between large-scale circulation patterns and European winter extremes in temperature, wind, and precipitation. We find that the choice of domain significantly affects the interpretation of the spatial patterns and how strongly the hazard variables are coupled to the circulation variables. For some of the extremes, there are also notable differences in the coupling depending on whether the atmospheric circulation is diagnosed at the surface, by sea-level pressure, or in the mid-troposphere, by 500 hPa geopotential height. Finally, changing the threshold values used to define the extremes does not affect the robustness of our coupling results for the extreme event types that we consider. We conclude that the co-recurrence ratio provides valuable insights into the atmospheric drivers of surface extremes, but our findings also highlight the importance of choosing key parameters in the analysis with care.

**Keywords** Dynamical coupling · Extreme weather events · Dynamical systems theory · Atmospheric circulation

## 1 Introduction

Extreme weather events have profound societal impacts, posing risks to human life while causing substantial economic losses and disruptions to natural systems. In recent decades, Europe has experienced an increasing intensity and frequency of extreme events (Intergovernmental Panel on Climate Change (IPCC) 2023). Prominent examples of winter extremes in Europe include numerous storms, such as “Dennis” and “Alex” in 2020 (Davies et al. 2021; Ginesta et al. 2023) and “Eunice” in 2022 (Volonté et al. 2024a, b), as well as cold spells such as the “Beast from the East” in 2018 (Overland et al. 2020). Recent winters have also shown increasingly frequent warm spells (Tomczyk et al. 2019). In fact, Europe is the fastest warming continent in the world, and the risks are expected to become increasingly severe in the future under anthropogenic climate change (Kron et al. 2019; Bevacqua et al. 2019; Seneviratne et al. 2012).

Identifying the drivers of extremes is a crucial step in assessing future changes in their frequency and intensity. A deeper understanding of these drivers enables better

<figure><img src="image_2.png" /></figure>

<footnote>Ane Carina Reiter
acare@dtu.dk</footnote>

<footnote n="1">¹ Climate and Energy Policy Division, Technical University of Denmark, Lyngby, Denmark</footnote>

<footnote n="2">² Danish Meteorological Institute, Copenhagen, Denmark</footnote>

<footnote n="3">³ Department of Earth Sciences, Uppsala University, Uppsala, Sweden</footnote>

<footnote n="4">⁴ Swedish Centre for Impacts of Climate Extremes (climes), Uppsala University, Uppsala, Sweden</footnote>

<footnote n="5">⁵ Department of Meteorology, Stockholm University, Stockholm, Sweden</footnote>

<footnote n="6">⁶ Laboratoire des Sciences du Climat et de l'Environnement, LSCE/IPSL, CEA-CNRS-UVSQ, Université Paris-Saclay, Gif-sur-Yvette, France</footnote>

<footnote n="7">⁷ London Mathematical Laboratory, London, UK</footnote>
extreme event forecasts, more accurate risk assessments, and the development of effective climate adaptation strategies. Atmospheric circulation plays a crucial role in the occurrence of near-surface extremes in temperature, wind speed, and precipitation (Pfahl 2014; Grace et al. 2024; Jézéquel et al. 2018). In winter, the dominant mode of variability in the north Atlantic region is the North Atlantic Oscillation (NAO), dictating the westerly zonal flow of warm air from the Atlantic to Europe (Hurrell et al. 2003; Slonosky and Yiou 2002). Numerous studies have linked NAO not only to regional mean climate patterns, but also to weather extremes in winter (e.g., Kenyon and Hegerl 2008; Yiou and Nogaj 2004; Donat et al. 2010). A positive phase of the NAO, characterized by an enhanced Atlantic meridional pressure gradient, is associated with a strong jet stream that transports mild air from the Atlantic to Europe. This intensified flow also results in an increased frequency of extratropical cyclones in the North Atlantic storm track, bringing extreme winds and precipitation to western and northern Europe, while central and southern Europe experience dryer conditions. In contrast, a negative phase of the NAO is associated with cold temperatures, as a weakening of the westerly flow allows easterly winds from Siberia to transport cold air to Europe. This effect is particularly evident when a negative NAO phase coincides with blocking upstream of the continent (Tuel and Martius 2024; Sillmann et al. 2011; Kautz et al. 2022; Pfahl 2014). Summer extremes are also influenced by atmospheric circulation, such as anti-cyclonic structures. However, local factors, including thermodynamic effects from moisture limitation and dynamic processes related to small-scale convective activity, often play a more significant role than in winter (Seneviratne et al. 2006, 2010; Fischer et al. 2007; Zschenderlein et al. 2019; Da Silva and Haerter 2023).

A wide range of approaches exists to study the link between the large-scale circulation and extreme events, involving both process-based analyses and statistical methods. A widespread approach in the latter category is to relate dominant modes of variability of the climate system, often diagnosed through an empirical orthogonal function (EOF) analysis, to anomalies in the occurrence of extreme events (Yiou and Nogaj 2004). EOF analysis has also been conducted on large-scale circulation data, restricted to days with extreme events, allowing to characterize circulation variability patterns specific to these events (Gessner et al. 2023; Ionita et al. 2021). Other commonly used methods to assess the connection between the atmospheric circulation and surface extremes include canonical correlation analysis (CCA), which identifies linear relationships between two variables (Andrade et al. 2012; Friederichs and Hense 2003); nonlinear classification techniques such as self-organizing map (SOM) analysis (Ohba et al. 2015; Aragon et al. 2020; Cavazos 2000); and composites of meteorological maps.

Complementing these methods, dynamical systems theory has provided a framework, that leverages the chaotic nature of the atmosphere, to study the drivers of both individual and compound extremes. Central to this approach is the co-recurrence ratio, $\alpha$, which quantifies the instantaneous coupling between multiple variables in terms of joint recurrences of similar atmospheric configurations (Faranda et al. 2020; Messori and Faranda 2023; De Luca et al. 2020a, b). In previous studies, a close correspondence has been found between extremes in $\alpha$ and extremes in hazard variables, suggesting that $\alpha$ can provide useful information on understanding the physical drivers underlying extreme weather events. The rationale is that if a given set of extreme events is modulated by the large-scale atmospheric circulation, then variables reflecting the circulation should be coupled to hazard variables around the time when the extremes occur. For example, Faranda et al. (2020) computed $\alpha$ for large-scale temperature and sea level pressure in North America and found that cold temperature extremes over the eastern part of the continent in winter correspond to high coupling, favoring persistent cold advection from the Arctic region. Later, wet and windy extremes in western Europe have also been linked to NAO-like large-scale atmospheric circulation patterns, conditioning on extremely high coupling days (Messori and Faranda 2023). These two individual extremes have also been considered together as spatially compound extremes (De Luca et al. 2020b) and linked together through a common atmospheric driver by a trivariate implementation of $\alpha$ (Messori and Faranda 2023). The approach has also been applied successfully to reveal the drivers of compound warm-dry and cold-wet events in the Mediterranean region (De Luca et al. 2020a).

Unlike weather regimes or other analyses, which cluster the large-scale atmospheric circulation into a small number of states, the co-recurrence ratio does not rely on predefined reference states or circulation patterns. Rather, it quantifies the coupling between two or more variables at individual time steps in the data, using the spatial patterns at each time step of interest as reference states. Indeed, extreme events are often associated with unusual circulation patterns (Grotjahn et al. 2016), while the patterns obtained from clustering approaches are typically selected to be frequently visited states of circulation, representative of the overall atmospheric variability. In contrast to techniques such as EOF or CCA, which rely on linear assumptions and may miss nonlinear dependencies, the co-recurrence ratio captures nonlinear relationships between fields in a time-resolved manner.

The co-recurrence ratio approach is sensitive to the choice of key parameters, including the geographical domains, the
variables used to characterize the large-scale circulation, and the thresholds used to define extremes. However, the extent to which these choices affect the results has not been systematically investigated in the literature. The sensitivity of α to the chosen geographical domain(s) is inherent to the method, as the atmospheric configurations used to compute the co-recurrence ratio are based on all gridboxes in the domain. From a physical perspective, the size and location of the geographical domain should correspond to the scale of the phenomenon one wants to study. However, domains that are too large may incorporate irrelevant information, while overly small domains may not include remote but relevant atmospheric features. Furthermore, the selection of variables used to represent the large-scale circulation—commonly sea level pressure (SLP) or geopotential height at 500 hPa (Z500)—can have a profound impact on the results. In previous studies of the co-recurrence ratio, SLP has been adopted to analyze large-scale circulation. In fact, the dynamical systems approach is based on the theoretical assumption of stationary dynamics, and due to the thermal expansion of the lower troposphere under anthropogenic warming, Z500 exhibits a pronounced positive trend (Christidis and Stott 2015). However, the dynamics at the surface and at the mid-tropospheric levels are different and may couple differently to surface extremes. This was, for example, observed for European temperature extremes by Holmberg et al. (2023) and by Jézéquel et al. (2018). Finally, the threshold used to define extremes affects the time steps and regions identified as experiencing an extreme event, as well as the associated circulation patterns. It is unclear to what extent α can provide consistent results when applied to different definitions of extreme events.

In this paper, we aim to assess the robustness of the co-recurrence ratio to such variations in parameters, as outlined above, including geographical domains, variables, and thresholds for extremes. We consider different European domains and focus on winter extremes in temperature, wind, and precipitation. We adopt a bivariate focus, diagnosing the coupling between large-scale circulation patterns and individual hazard variables. Our goal is to better understand the implications of different methodological choices when implementing the co-recurrence ratio, providing a knowledge base to support users of this approach in applying it robustly. More broadly, a better understanding of how these parameter choices influence results is essential when using this framework to study the atmospheric drivers of surface extremes. Characterizing the relationship between local and regional-scale events and large-scale circulation is fundamental for, for example, statistical downscaling and has gained renewed relevance with the rapid emergence of machine-learning and AI-based techniques (Camps-Valls et al. 2025). Similar methodological challenges also arise in

related research initiatives such as analogue-based attribution (Faranda et al. 2024b). Assessing the robustness of the co-recurrence ratio to such choices can therefore contribute to more transferable and consistent methodologies across a range of climate science applications.

# 2 Data and methods

## 2.1 Dynamical systems framework

Here, we consider the atmosphere as a chaotic system with high-dimensional dynamics settling on a geometrical object known as a strange attractor. Adopting an “ergodic” approach from dynamical systems theory (Eckmann and Ruelle 1985), the system’s trajectories will return infinitely often to states already visited on the attractor. Recent advances, combining the study of these recurrences with extreme value statistics, enable the computation of dynamical metrics that characterize the different states of atmospheric flow (Lucarini et al. 2012, 2016; Messori et al. 2017; Faranda et al. 2017, 2024a). These properties are local in phase space, meaning that they are instantaneous in time and can be computed for any state of the system.

Extending this framework, the co-recurrence ratio, α, has been introduced as a new instantaneous measure to characterize the dynamical coupling between multiple variables, quantifying the correspondence between their recurrences (Faranda et al. 2020). Here, we consider the evolution of latitude-longitude maps of two climate variables. As an example, we take long series of daily maps of 2-metre temperature and sea-level pressure in a specific geographical domain. A large α-value for a given day indicates that when a temperature pattern closely resembles the one of that day, the sea-level pressure also exhibits patterns that resemble the one it had on that day. In other words, the two variables display co-recurrences of similar spatial patterns, which can be interpreted as a strong dynamical coupling.

## 2.2 Derivation of the co-recurrence ratio

For two climate variables, we reorganize daily time series of latitude-longitude maps into vector time series with each grid point as an entry and treat the pair of new time series as a trajectory in phase space, denoted {$x(t), y(t)$}. Since the co-recurrence ratio is instantaneous, it is computed at each time step, taking as state of interest $\zeta = \{\zeta_x, \zeta_y\}$ each time step in our data in turn.

Considering the subspace of one variable, say $x(t)$, we quantify the proximity of all other $x(t)$ states in our data to the reference state $\zeta_x$ by defining an observable, via logarithmic returns:
$$g(x(t), \zeta_x) = -\log[\text{dist}(x(t), \zeta_x)] \quad (1)$$

where dist is a distance metric. In this study, we adopt the Euclidean distance. Since the minus-log function is strictly decreasing, large values of g correspond to small distances, thereby indicating recurrences of $\zeta_x$. To formally define recurrences, we consider a high q'th quantile, $s(q, \zeta_x)$, of the time series defined by $g(x(t), \zeta_x)$, here $q = 0.98$. Recurrences of $\zeta_x$ are then identified as instances where this threshold is exceeded, that is, $g(x(t), \zeta_x) > s(q, \zeta_x)$.

We obtain recurrences of $\zeta_y$ similarly and then define the co-recurrence ratio $0 \le \alpha_{x,y}(\zeta) \le 1$ for the state $\zeta = \{\zeta_x, \zeta_y\}$ as the ratio of mutual recurrences of $\zeta_x$ and $\zeta_y$. That is,

$$\alpha_{x,y}(\zeta) = \frac{\nu[g(x(t)) > s_x(q) \cap g(y(t)) > s_y(q)]}{\nu[g(x(t)) > s_x(q)]} \quad (2)$$

where $\nu[-]$ is the number of occurrences of condition $[-]$, and where we have simplified our notation by using variable suffixes instead of $\zeta_x$ and $\zeta_y$. We note that $\nu[g(x(t)) > s_x(q)] = \nu[g(y(t)) > s_y(q)]$ since the threshold is defined as a high percentile. Thus, $\alpha$ is independent of the order of the variables, and we may not interpret $\alpha$ in terms of causality.

<figure><img src="image_3.png" /><figcaption>Fig. 1 The domains over which the co-recurrence ratio is computed</figcaption></figure>

## 2.3 Data

We use ERA5 reanalysis data from the European Centre for Medium-Range Weather Forecasts over the years 1979–2023 consistent with recent studies conducting dynamical systems analyses on atmospheric data, such as (Messori and Faranda 2023; Holmberg et al. 2023) and with a horizontal spatial resolution of 0.25°. The co-recurrence ratio, $\alpha$, is computed between a large-scale atmospheric circulation variable and a hazard variable. To describe atmospheric circulation at the surface and mid-troposphere, we adopt sea level pressure (SLP) and geopotential height at 500 hPa (Z500), respectively. For hazard variables, we use 2 m temperature (t2m), 10 m wind speed (w10m), and total precipitation (tp). The analysis is conducted on daily values. For tp, daily values are computed as the sum of hourly data. For the remaining variables, the daily values are computed as averages of hourly data.

We define anomalies relative to a calendar-day climatology, smoothed using a 31-day running mean, and consider the anomaly distributions in the winter season December-January-February (DJF). Warm extremes, as well as wind and precipitation extremes, are defined as anomalies above the 95th percentile and cold extremes as anomalies below the 5th percentile of the respective distributions at single grid points. This means that, while $\alpha$ is computed for variables over the entire domains, the extremes in the hazard variables are identified at local scales. To test the robustness of the results to the definition of extremes, we also consider the 90th (10th) and 98th (2nd) percentiles. In a similar manner, anomalies of $\alpha$ are computed relative to a calendar day climatology, and extreme coupling is defined as daily $\alpha$-values above the same upper thresholds as those for the extreme events in the hazard variables.

We investigate the relationship between weather extremes and atmospheric circulation through composite maps. We first compute composite anomalies of the circulation and hazard variables associated with extreme coupling days. Next, we compute composite $\alpha$ anomalies conditioned on days with extreme values of the hazard variables at each grid point.

## 2.4 Domains

We consider four different domains, illustrated in Fig. 1: Europe (22° W–45° E, 27° N–72° N), Western Europe (10° W–20° E, 45° N–60° N), Scandinavia (0° E–35° E, 54° N–71° N), and Denmark (7° E–16° E, 54° N–58° N). The Europe domain approximately follows the EURO-CORDEX domain but differs slightly, since simulations in the CORDEX project are performed on a rotated grid (EURO-CORDEX, http://www.euro-cortex.net/; Jacob et al. 2014).
The western European domain matches the region studied for wet-windy extremes in Messori and Faranda (2023), where the co-recurrence ratio approach was employed. Using this domain, comparable results were found to those obtained using a larger European domain. The Scandinavian domain aligns with the analogous subregion defined in the PRUDENCE project (Christensen and Christensen 2007) and re-adopted by several newer studies (Weimer et al. 2016; Hartick et al. 2021), with a slight extension southward to include all of Denmark. Finally, we define a small domain that overlaps both the western European and Scandinavian domains. While the Denmark domain is likely to be too small to capture the coupling between large-scale circulation and surface parameters, we note that domains of similar size are common in national and regional studies of surface extremes (Schleiss et al. 2020; Ferreira et al. 2014).

The co-recurrence ratio, α, does not require the variables on which it is computed to be geographically co-located. Consequently, the bivariate co-recurrence ratio which we use to determine the dynamical coupling between large-scale circulation and hazard variables can be computed on variables over two different domains. In this study, we consider different combinations of the above-described domains but limit ourselves to cases where the large-scale circulation domain matches or entirely encompasses the hazard variable domain. This approach yields nine combinations, listed in Table 1.

# 3 Results

## 3.1 Geographical domains

We first consider the robustness of the results for the different surface extremes to changes in geographical domains. Following the previous literature, we use SLP to diagnose large-scale circulation.

### 3.1.1 Temperature extremes

First, we consider temperature extremes. During extreme levels of $\alpha_{SLP,t2m}$, computed for SLP in Europe (Fig. 2a–d), an anticyclonic pressure anomaly is located to the south of the center of the t2m domain. The large-scale SLP pattern generally resembles a northeast-shifted positive NAO-like pattern, although the dipole structure vanishes for

Scandinavia as the anticyclonic core shifts northward with the domain. The domain's positioning relative to the anticyclone favors a strong westerly zonal flow, enhancing warm air advection from the Atlantic. This is also reflected in the t2m anomalies, which show strong positive values within the domain of interest. A similar picture emerges for SLP in Western Europe (Fig. 2e–f) and Scandinavia when considering t2m in Scandinavia (Fig. 2g), albeit with a weaker signal and a northward shift of the anticyclonic core. However, for a combination of smaller t2m and SLP domains, these configurations break down. In Fig. 2h–i, the Denmark t2m domain is located west of the negative SLP core, showing near-zero or negative t2m anomalies. This suggests that the strongest coupling element between the large-scale circulation at the surface and surface temperature in Europe during winter is the advection of warm and moist maritime air towards the continent. For very small domains, the large-scale circulation patterns become less evident, and instead other local effects come into play.

In Figs. 3 and 4, we analyze the coupling conditions associated with extreme warm and cold days locally at each grid point. For large domains, there is a spatial correspondence with the results in Fig. 2. Regions experiencing warm temperatures on extreme coupling days display positive and negative $\alpha_{SLP,t2m}$ anomalies, respectively, for warm and cold extremes. For the larger domains, Fig. 5 further confirms a linear relationship between composite grid point values of temperature and coupling anomalies: positive for warm extremes (Fig. 5a–c) and negative for cold extremes (Fig. 5e–g). Moreover, since the t2m domain generally displays positive $\alpha_{SLP,t2m}$ anomalies together with a strong sign agreement for warm extremes, we interpret this as a strong coupling of warm temperature extremes to recurrent large-scale atmospheric patterns at the surface. In contrast, cold extremes mainly demonstrate negative $\alpha_{SLP,t2m}$ anomalies with a strong sign agreement predominantly for weak coupling. We conclude that cold extremes are weakly coupled to large-scale surface-level circulation. This may be explained by the fact that upstream blocking over the North Atlantic is not fully captured in our domains, reflecting the link between Greenland blocking highs and European cold spells (see Sect. 1). Finally, for smaller domains, the link between local temperature extremes and large-scale circulation becomes more complex. For SLP and t2m in Denmark, regions with cold temperatures on extreme coupling days, including the t2m domain, display large $\alpha_{SLP,t2m}$

<table><caption>**Table 1** Domain combinations and variables used</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">t2m/w10m/tp</th></tr><tr><th>Europe</th><th>Western Europe</th><th>Scandinavia</th><th>Denmark</th></tr></thead><tbody><tr><td rowspan="4">SLP/Z500</td><td>Europe</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>Western Europe</td><td></td><td>×</td><td></td><td>×</td></tr><tr><td>Scandinavia</td><td></td><td></td><td>×</td><td>×</td></tr><tr><td>Denmark</td><td></td><td></td><td></td><td>×</td></tr></tbody></table>
<figure><img src="image_4.png" /><figcaption>**Fig. 2** Composite SLP (hPa, contours) and t2m (K, colors) anomalies for extreme coupling days (> 0.95-quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark</figcaption></figure>

(i) and for t2m (black box) in Europe (a), Western Europe (b, e), Scandinavia (c, g), and Denmark (d, f, h, i). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign

anomalies for both warm and cold extremes (Fig. 5d, h). However, the sign agreement is weak and mostly observed outside the domain of interest. Figs. S3, 4 confirm this pattern in other domain combinations. These findings reinforce the conclusion that the strongest coupling mechanism between large-scale, surface-level circulation and surface temperature is the advection of warm maritime air, which is not well captured in small domains.

### 3.1.2 Wind and precipitation extremes

We here shift the focus towards extremes of wind (Figs. 6, 7) and precipitation (Figs. 8, 9), computing $\alpha_{SLP,w10m}$ and $\alpha_{SLP,tp}$ for the different domain combinations.

The extreme coupling days computed for SLP in Europe, with respect to both w10m (Fig. 6a–d) and tp (Fig. 8a–d), show SLP configurations similar to those for t2m, resem-bling a positive NAO-like dipole, with the domain of interest centered between the positive and negative cores. We
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Composite $\alpha_{SLP,t2m}$ anomalies for extreme warm days (> 0.95-quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark (i) and for t2m (black box) in Europe (a), Western Europe (b, e), Scandinavia (c, g), and</figcaption></figure>

Denmark (d, f, h, i). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colourbar limits correspond to approximately the 95th percentile of $\alpha$ anomalies (see Figs. S1, 2 for the full distributions)

note that the magnitude of the composite SLP anomalies is greater for the coupling to wind than to temperature and precipitation. This is not surprising, as strong pressure gradients drive wind speed, making wind more directly linked to large-scale circulation than temperature or precipitation. The strongest wind anomalies are located within the w10m domains, which align with a zonal flow, favoring extratropical cyclones tracking towards the continent. While wind is directly linked to this flow, strong precipitation anomalies,

which are more local, are mainly concentrated along the Norwegian coast. This could indicate that atmospheric riv-ers, which are narrow regions of concentrated water vapor in the atmosphere, are captured by a strong coupling. Indeed, atmospheric rivers are of particular importance in moun-tainous regions such as Norway, where orographic lifting enhances extreme precipitation (Lavers and Villarini 2013; Michel et al. 2021).
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Composite $\alpha_{SLP,t2m}$ anomalies for extreme cold days (< 0.05 -quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark (i) and for t2m (black box) in Europe (a), Western Europe (b, e), Scandinavia (c, g), and Denmark (d, f, h,</figcaption></figure>

i). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colourbar limits correspond to approximately the 95th percentile of $\alpha$ anomalies (see Figs. S1, 2 for the full distributions)

For smaller SLP domains, the SLP and w10m anomalies associated with extreme $\alpha_{SLP,w10m}$ anomalies show patterns consistent with those in the SLP Europe domain (Fig. 6e-i). This indicates that, even for small domains, a strong coupling of wind in Europe to an NAO-like dipole is captured, reinforcing the direct link between wind and large-scale circulation. In contrast, extreme $\alpha_{SLP, tp}$ anomalies for smaller domains show SLP configurations that resemble those for the temperature coupling (Fig. 8c-i).

The key difference is that, even for the smallest domains (Fig. 8h-i), the Denmark tp domain remains north of the anticyclonic core. We further note that for smaller domains, the positive tp anomalies remain concentrated in Norway.

Figs. 7 and 9 show composite $\alpha_{SLP,w10m}$ anomalies and $\alpha_{SLP, tp}$ anomalies associated with extreme wind and precipitation, respectively. The spatial anomaly patterns mostly match the w10m and tp anomalies in Figs. 6 and 8 for all domain combinations. This is consistent with the
<figure><img src="image_7.png" /><figcaption>**Fig. 5** Scatterplots of composite t2m (K) anomalies for extreme coupling days (> 0.95-quantile) vs. composite $\alpha_{SLP,t2m}$ anomalies for extreme warm days (> 0.95-quantile) (a–d) and extreme cold days (< 0.05-quantile) (e–h). The co-recurrence ratio is computed for SLP and t2m in Europe (a, e), Western Europe (b, f), Scandinavia (c, g), and Denmark (d, h). Orange points represent grid points within the</figcaption></figure>

positive linear relationships seen in Fig. 10 (in Figs. S5, 6, we show all scatterplots, confirming this relationship for the remaining domain combinations). Since the w10m domain predominantly displays strong coupling within the wind domain and most grid points display large sign agreement (Fig. 10a–d), we conclude that the wind extremes are strongly coupled to large-scale SLP configurations for all combinations of domains. For extreme precipitation, the sign agreement is generally strong in both the composited tp anomalies and $\alpha_{SLP,tp}$ anomalies for SLP and tp in the Europe domain (10e). However, for smaller domains the match in strong sign agreement in the pair of composited anomalies is limited to grid points outside the tp domain (10f–h). This can mainly be explained by a limited sign agreement for the positive composite tp anomalies on the mountainous Norwegian coast in Fig. 8. These findings point to a strong coupling of precipitation extremes to large-scale surface-level circulation patterns, although for smaller domains, the most extreme coupling days only capture extreme precipitation events to a limited degree.

co-recurrence domain, and blue points represent grid points outside this domain. The black lines bound high density regions in the scatterplots, determined by the Kernel Density Estimate (KDE), where at least 2/3 of the composited anomalies, for both t2m and $\alpha_{SLP,t2m}$, have the same sign. The dotted red line is a linear regression line of all grid points

## 3.2 Large-scale circulation variable

We next examine how the coupling results between surface hazard variables and the large-scale circulation are affected by the choice of circulation variable. We specifically compare SLP to Z500, as both have been used in the literature to diagnose large-scale circulation.

### 3.2.1 Temperature extremes

We compute $\alpha_{t2m,Z500}$ for the four domain combinations in which the t2m and Z500 domains are the same (Fig. 11).

Panels a, d, g, j show composite anomalies of t2m and Z500 associated with extreme $\alpha_{t2m,Z500}$ anomalies. The resulting atmospheric configurations distinctly differ from those in Fig. 2, with a cyclonic structure dominating much of Europe. The core of this structure is roughly aligned with the t2m domain but slightly shifted eastward for coupling in the Europe, Scandinavia, and Denmark domains. This displacement, along with strong negative temperature anomalies predominantly to the northwest of the low, suggests that the configurations are associated with the advection of cold continental air masses from the north-east. A strong coupling of cold temperatures to a mid-tropospheric low, in the
<figure><img src="image_8.png" /><figcaption>(b) Europe-Western Europe</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(c) Europe-Scandinavia</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(d) Europe-Denmark</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(e) Western Europe-Western Europe</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(f) Western Europe-Denmark</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(g) Scandinavia-Scandinavia</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(h) Scandinavia-Denmark</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(i) Denmark-Denmark</figcaption></figure>

<figure><img src="image_16.png" /></figure>

**Fig. 6** Composite SLP (hPa, contours) and w10m (m s⁻¹, colors) anomalies for extreme coupling days (> 0.95-quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark (i) and for w10m (black box) in Europe (a), Western Europe (b),

e), Scandinavia (c, g), and Denmark (d, f, h, i). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign

absence of a corresponding surface-level signal, could be explained by the presence of an upper-level trough or cut-off low, typically associated with cold air moving equatorward (Schemm et al. 2020).

Computing composite $\alpha_{Z500,t2m}$ anomalies associated with local warm extremes (panels b, e, h, k) shows a complex link to Z500 coupling extremes (see also scatterplots in Fig. S8). For Europe, the pattern of $\alpha_{Z500,t2m}$ anomalies resembles the pattern of $\alpha_{SLP,t2m}$ anomalies, although with

a weaker signal (cf. Fig. 4a). This points to a strong coupling for warm extremes, although not captured by the most extreme coupling days. For the three smaller domains, we find a spatial match with negative $\alpha_{Z500,t2m}$ anomalies in the regions displaying negative t2m anomalies in panels d, g, j, respectively. This points to a weaker coupling to large-scale circulation in the mid-troposphere on warm days.

Finally, panels c, f, i, l show composite $\alpha_{Z500,t2m}$ anomalies associated with local cold extremes. A strong spatial
<figure><img src="image_17.png" /><figcaption>**Fig. 7** Composite $\alpha_{SLP,w10m}$ anomalies for extreme windy days (> 0.95-quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark (i) and for w10m (black box) in Europe (a), Western Europe (b, e), Scandinavia (c, g), and Denmark</figcaption></figure>

(d, f, h, i). Stippling indicates regions where at least 2/3 of the compos-
ited anomalies share the same sign. To ensure consistency, the colorbar
limits correspond to approximately the 95th percentile of α anomalies
(see Figs. S1, 2 for the full distributions)

match is observed, with positive $\alpha_{Z500,t2m}$ anomalies in
regions with negative t2m anomalies in panels a, d, g, j and
strong sign agreement in both composite maps. This sug-
gests that cold extremes are strongly coupled to mid-tropo-
spheric large-scale atmospheric configurations.

3.2.2 Wind and precipitation extremes

Here, we consider the link between local wind and pre-
cipitation extremes and their coupling to mid-tropospheric
large-scale circulation (Fig. 12).

Panels a, d, g, j display composites of Z500 and w10m associated with extreme $\alpha_{Z500,w10m}$ anomalies, and panels b, f, j, n conversely display composite $\alpha_{Z500,w10m}$ anomalies associated with wind extremes. In all four domain
<figure><img src="image_18.png" /><figcaption>**Fig. 8** Composite SLP (hPa, contours) and tp (mm day⁻¹, colors) anomalies for extreme coupling days (> 0.95-quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark (i).</figcaption></figure>

mark (i) and for tp (black box) in Europe (a), Western Europe (b, e), Scandinavia (c, g), and Denmark (d, f, h, i). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign

combinations, the Z500 and w10m anomaly patterns as well as the α<sub>Z500,w10m</sub> anomaly patterns closely match the corresponding patterns for coupling to large-scale circulation at the surface (cf. Figs. 6, 7). This suggests a strong coupling of surface wind extremes to a recurrent large-scale NAO-like dipole structure, both at surface level and at mid-tropospheric level.

A similar correspondence is found between large-scale Z500 circulation patterns for extreme α<sub>Z500, tp</sub> anomalies (panels c, g, k, o) and the SLP anomaly dipole structure for

extreme coupling at surface-level (cf. Fig. 8). However, a key difference is that the positive core does not shift northward for the Western Europe and Scandinavia domains, leading to a more dominant negative core and a better preservation of the dipole structure. Furthermore, the composite tp anomalies display distinct patterns compared to those observed for SLP coupling, with positive values distributed mainly within the tp domain, rather than over the Norwegian coastal mountains. This is particularly pronounced for the Western Europe and Denmark domains, which do not
<figure><img src="image_19.png" /><figcaption>**Fig. 9** Composite $\alpha_{SLP, tp}$ anomalies for extreme precipitation days (> 0.95-quantile), computed for SLP in Europe (a–d), Western Europe (e, f), Scandinavia (g, h), and Denmark (i) and for tp (black box) in Europe (a), Western Europe (b, e), Scandinavia (c, g), and</figcaption></figure>

Denmark (d, f, h, i). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colorbar limits correspond to approximately the 95th percentile of $\alpha$ anomalies (see Figs. S1, 2 for the full distributions)

include Norway. Finally, regions that display strong positive tp anomalies associated with extreme coupling also display strong $\alpha_{Z500, tp}$ anomalies associated with extreme local precipitation (panels d, h, l, p; see also scatterplots in Fig. S9). In contrast to the previous surface-level coupling results, there is a strong sign agreement both for positive tp anomalies and positive $\alpha_{Z500, tp}$ anomalies within the tp domains. This suggests a strong coupling of precipitation extremes to large-scale circulation in the mid-troposphere.

### 3.3 Threshold for extremes

In the above results, extremes have been defined as anomalies above the 95th percentile of the respective anomaly distributions. Here, we test the robustness of our findings regarding the coupling of the surface extremes to large-scale circulation at surface-level to varying thresholds used to define the extremes.
<figure><img src="image_20.png" /><figcaption>**Fig. 10** Scatterplots of composite w10m (m s⁻¹) (a–d) and tp (mm day⁻¹) (e–h) anomalies for extreme coupling days (> 0.95-quantile) vs. composite αSLP,w10m respectively αSLP,tp anomalies for extreme windy and extreme precipitation days (> 0.95-quantile). The co-recurrence ratio is computed for SLP, w10m, and tp in Europe (a, e), Western Europe (b, f), Scandinavia (c, g), and Denmark (d, h). Orange points represent grid points within the co-recurrence domain,</figcaption></figure>

In Fig. 13, we consider the 98th percentile (2nd for cold extremes) as a new threshold for extreme temperature. For extreme coupling days, the composite SLP and t2m anomalies qualitatively match those using the 95th percentile threshold, although with a stronger signal (cf. panels a, d, g, j with Fig. 2). Likewise, the composite αSLP,t2m anomalies associated with warm and cold temperatures exhibit a similar yet more pronounced pattern compared to the lower threshold (cf. panels b, e, h, k and panels c, f, i, l with Figs. 3 and 4). In Fig. S10, the same analysis is conducted for the 90th (10th) percentile threshold, and again consistent anomalies are observed although weaker than for the higher thresholds. We interpret this as evidence supporting the robustness of our findings that warm extremes in Europe are strongly coupled to a positive NAO-like dipole, while cold extremes show a weaker coupling to large-scale circulation at the surface.

Also for wind and precipitation extremes, we find that the co-recurrence ratio yields consistent results when varying the threshold. As for temperature, we consider the 98th percentile as a new threshold for extremes (Fig. 14). Composite anomalies in SLP, w10m, tp associated with extreme coupling and composite anomalies in αSLP,w10m and αSLP,tp

and blue points represent grid points outside this domain. The black lines bound regions of high density, determined by the Kernel Density Estimate (KDE), where at least 2/3 of the composited anomalies, for both w10m and αSLP,w10m respectively tp and αSLP,tp, have the same sign. The dotted red line is a linear regression line of all grid points

associated with extreme wind and precipitation match qualitatively the patterns for 95th percentile, albeit with a stronger signal (cf. panels a, e, i, m with Fig. 6; panels b, f, j, n with Fig. 7; panels c, g, k, o with Fig. 8; panels d, h, l, p with Fig. 9). In keeping with the result for temperature, Fig. S11 shows weaker signals for the 90th percentile. These results support the robustness of our findings, indicating that the link between extreme wind and precipitation events and large-scale circulation at surface level remains consistent across varying thresholds.

# 4 Discussion and conclusion

The co-recurrence ratio (α), an instantaneous measure of coupling between multiple atmospheric variables, has been used to link the large-scale atmospheric circulation to extreme weather events (Faranda et al. 2020; Messori and Faranda 2023). In this paper, we evaluated the performance of the co-recurrence ratio in diagnosing atmospheric drivers under varying methodological choices, which includes different geographical domains, large-scale circulation variables, and thresholds for extremes. We specifically focused
<figure><img src="image_21.png" /><figcaption>**Fig. 11** Composite Z500 (m² s⁻², contours) and t2m (K, colors) anomalies for extreme coupling days (> 0.95-quantile) (**a**, **d**, **g**, **j**) and composite α<sub>Z500,t2m</sub> anomalies for extreme warm days (> 0.95-quantile) (**b**, **e**, **h**, **k**) and extreme cold days (< 0.05-quantile) (**c**, **f**, **i**, **j**). The co-recurrence ratio is computed for Z500 and t2m in Europe (**a**–**c**),</figcaption></figure>

Western Europe (**d**–**f**), Scandinavia (**g**–**i**), and Denmark (**j**–**l**). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colorbar limits correspond to approximately the 95th percentile of α anomalies (see Figs. S1, 2 for the full distributions)
<figure><img src="image_22.png" /><figcaption>**Fig. 12** Composite Z500 (m² s⁻², contours) and w10m (m s⁻¹, colors), respectively, tp (mm day⁻¹) anomalies for extreme coupling days (> 0.95-quantile) (**a**, **e**, **i**, **m**, respectively **c**, **g**, **k**, **o**) and composite α<sub>Z500,w10m</sub> anomalies for extreme windy days (> 0.95-quantile) (**b**, **f**, **j**, **n**) and α<sub>Z500,tp</sub> anomalies for extreme precipitation days (> 0.95-quantile) (**d**, **h**, **l**, **p**). The co-recurrence ratio is computed for Z500,</figcaption></figure>

w10m, and tp in Europe (**a**–**d**), Western Europe (**e**–**h**), Scandinavia (**i**–**l**), and Denmark (**m**–**o**). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colorbar limits correspond to approximately the 95th percentile of α anomalies (see Figs. S1, 2 for the full distributions)

on European winter extremes in temperature, wind, and precipitation, diagnosing the coupling of these extremes to large-scale circulation patterns.

We considered different combinations of the four domains Europe, Western Europe, Scandinavia, and Denmark, in which the large-scale circulation domain either matches or entirely encompasses the hazard variable domain. For

large domains, our results show a strong coupling of warm extremes and wind and precipitation extremes to large-scale, surface-level circulation patterns, as represented by SLP, resembling a positive NAO-like dipole structure. This aligns with the widely established link between a positive NAO, associated with strong zonal advection of mild air from the North Atlantic, and warm, windy, and wet winter
<figure><img src="image_23.png" /><figcaption>**Fig. 13** Composite SLP (hPa, contours) and t2m (K, colors) anomalies for extreme coupling days (> 0.98-quantile) (**a**, **d**, **g**, **j**) and composite $\alpha_{SLP,t2m}$ anomalies for extreme warm days (> 0.98-quantile) (**b**, **e**, **h**, **k**) and extreme cold days (< 0.02-quantile) (**c**, **f**, **i**, **j**). The recurrence ratio is computed for SLP and t2m in Europe (**a**–**c**), Western</figcaption></figure>

Europe (**d**–**f**), Scandinavia (**g**–**i**), and Denmark (**j**–**l**). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colourbar limits correspond to approximately the 95th percentile of $\alpha$ anomalies (see Figs. S1, 2 for the full distributions)
<figure><img src="image_24.png" /><figcaption>**Fig. 14** Composite SLP (hPa, contours) and w10m (m s⁻¹, colors, respectively, tp (mm day⁻¹) anomalies for extreme coupling days (> 0.98-quantile) (a, e, i, m, respectively c, g, k, o) and composite α<sub>SLP,w10m</sub> anomalies for extreme windy days (> 0.98-quantile) (b, f, j, n) and α<sub>SLP,tp</sub> anomalies for extreme precipitation days (> 0.98-quantile) (d, h, l, p). The co-recurrence ratio is computed for SLP,</figcaption></figure>

w10m, and tp in Europe (a–d), Western Europe (e–h), Scandinavia (i–l), and Denmark (m–o). Stippling indicates regions where at least 2/3 of the composited anomalies share the same sign. To ensure consistency, the colorbar limits correspond to approximately the 95th percentile of α anomalies (see Figs. S1, 2 for the full distributions)

days in Europe. Cold extremes instead occur under weak coupling conditions, and correspond to some extent to a large-scale negative NAO pattern. We hypothesize that this finding can be explained by the role of upstream blocking over the North Atlantic, not included in our considered domains. When considering smaller domains, the strong coupling is preserved for wind extremes, reflecting a direct

link between wind and strong pressure gradients. In contrast, α fails to capture the large-scale circulation picture for temperature extremes. Although extreme precipitation days show stronger coupling to the large-scale circulation, these events are not fully represented on the days with the strongest coupling.
A key finding of this study is therefore that the choice of geographical domain can significantly influence the degree of coupling of local weather extremes to the large-scale circulation. Jézéquel et al. (2018) also explored the importance of domain size, examining the relationship between European summer heatwaves and synoptic circulation characteristics. While the domains they considered were larger than those in this study, they found that smaller domains were more effective at capturing the influence of synoptic circulation characteristics on summertime heatwaves. In contrast, our results for wintertime extremes suggest that, at least for temperature and precipitation extremes, larger domains yield stronger coupling. This indicates that the optimal domain size is not universal but instead depends on the season, the type of extreme being studied, and the associated physical drivers. Following this reasoning, it is possible that for cold extremes, a stronger coupling might emerge if an even larger circulation domain, encompassing the North Atlantic, was considered. In essence, domain choice considerations must be carefully incorporated into studies using the co-recurrence ratio.

In addition to SLP, we also computed the coupling between the hazard variables and Z500. The aim was to analyze how the coupling of surface extremes to large-scale circulation patterns varies across different atmospheric levels. No large differences were found for the coupling of wind and precipitation extremes. However, for precipitation, mid-level coupling tends to capture extremes within the domains of interest rather than orographic precipitation over the mountainous Norwegian coast. While warm extremes in Europe are strongly coupled to surface-level atmospheric configurations, cold extremes are strongly coupled to mid-level atmospheric configurations, characterized by a dominant cyclonic structure and the advection of cold Arctic wind from the north-east. These findings highlight the differences in the atmospheric dynamics that drive surface extremes at different levels. While the surface variable may better reflect the different phases of the NAO, the mid-tropospheric variable reflects features such as upper-level troughs. These features, associated with cold winter extremes, are more common in southern Europe and the Mediterranean and less frequent downstream of Greenland (Schemm et al. 2020; Muñoz et al. 2020; Nieto et al. 2005), the region on which this study is primarily focused. However, when these features do occur downstream of Greenland, they tend to be more transient (Schemm et al. 2020). Our findings suggest that even in regions where these large-scale circulation patterns are unusual, they can still result in a strong coupling to surface temperature extremes.

Finally, by changing the threshold used to define the extremes, we found qualitatively similar results for all types

of events. This underscores the robustness of $\alpha$ to the exact definition of what is “extreme”.

Our findings contribute to the growing body of research on dynamical systems theory to investigate how the large-scale circulation at different atmospheric levels links to surface extremes. For example, Jézéquel et al. (2018) found that Z500 is better suited to study European summer heat waves due to the blurring of the SLP pattern signal caused by heat lows. Hochman et al. (2022) demonstrated different temporal evolutions of the dynamical characteristics of cold spells in the Eastern Mediterranean computed for SLP and Z500. Holmberg et al. (2023) found a link between summertime heatwaves and mid-tropospheric atmospheric persistence and, in turn, a link between wintertime warm spells and atmospheric persistence at the surface. We add a new dimension to this body of work, by elucidating how local extremes across Europe couple to the large-scale circulation for different choices of geographical domains, circulation variables, and extreme event definitions.

A limitation of this study is the assumption of stationarity in the dynamical systems approach. Subscribing to the chaotic hypothesis by Gallavotti and Cohen (1995), we assume that the atmosphere, being chaotic and with many degrees of freedom, exhibits a behavior that allows it to be treated as autonomous and ergodic. In reality, the climate system is time-dependent, both in terms of natural periodic seasons and in terms of external forcing, including solar radiation, volcanic activity, and anthropogenic climate change. While theoretical results on nonstationary extreme value theory for dynamical systems remain limited, recent advances offer promising foundations for employing the recurrence framework for non-stationary dynamics. In particular, with the application to climate dynamics in mind, so-called sequential systems, with parameters changing deterministically in time, have been studied (Galatolo and Faranda 2025; Freitas et al. 2017). These results have been applied to highly idealized systems, and the extension to more complex real-world examples represents an opportunity for future research. Moreover, several studies have demonstrated that dynamical metrics are capable of detecting gradual changes in the long-term behavior of geophysical systems, supporting the application of this method to weakly nonstationary cases, where the system does not drift too rapidly away from the original attractor (Faranda et al. 2019; Rodrigues et al. 2018). To assess the potential influence of non-stationarities on the present analysis, we show in Fig. S12 scatterplots of $\alpha$ computed on raw vs. detrended data in Western Europe. We find a strong correlation between the two sets of data for all variables. This suggests that the results are robust with respect to slow trends, consistent with previous findings on other dynamical metrics (Faranda et al. 2024a).
Another caveat of our work is the effect of autocorrelation on the identified recurrences in a given atmospheric variable. Indeed, the persistence of the variables that we use can introduce biases, as multiple days from the same events might be selected as recurrences to compute the coupling. One way to address this limitation is to impose a minimum separation between recurrences. However, this would reduce the sample size and, therefore, a larger data set than the one we have available would be required. While winter-time warm spells have been associated with enhanced persistence in surface-level atmospheric configurations, these anomalies are generally weak and often lack statistical significance at regional scales (Holmberg et al. 2023). Moreover, precipitation and wind extremes are typically linked to shorter-lived, synoptic-scale weather systems, and their persistence is expected to be lower than that of temperature extremes. Therefore, persistence is unlikely to be the dominant factor driving the identified coupling patterns.

An additional caveat pertains specifically to precipitation extremes. Unlike temperature or wind, precipitation events, particularly during summer, are often highly localized and influenced by small-scale drivers such as convective activity, which may weaken their link to large-scale circulation. However, this study focuses on winter extremes, for which strong evidence links heavy precipitation events over Europe to the large-scale atmospheric circulation, such as the positive phase of the NAO (Pfahl 2014; Yiou and Nogaj 2004; Casanueva et al. 2014). Furthermore, previous studies have successfully applied the co-recurrence ratio framework to winter precipitation extremes in Europe, demonstrating its relevance in this context (Messori and Faranda 2023; De Luca et al. 2020b).

In conclusion, adopting a dynamical systems approach for studying extreme weather events can provide valuable insights into the underlying atmospheric drivers. This approach complements more conventional methods by coupling local extremes to large-scale atmospheric dynamics instantaneously in time. Our findings highlight the importance of carefully choosing key parameters, especially the geographical domains and the large-scale circulation variable, as these choices significantly influence the diagnosed coupling dynamics. This is a general result, which extends beyond the current study, and which we believe could have implications for a wide range of current and future investigations.

In this light, we propose some general guidelines that may help inform parameter choices. For wind extremes, strong coupling can be captured using relatively small domains for both the circulation and hazard variables, suggesting that a limited spatial extent is sufficient to characterize the large-scale circulation patterns. In contrast, for temperature and precipitation extremes, a larger circulation

domain is necessary to capture a strong coupling, while the hazard variable domain can be smaller. Furthermore, while our study identifies differences in coupling strength between surface-level and mid-tropospheric circulation variables for European winter extremes, such vertical dependencies may vary regionally. For instance, (Faranda et al. 2017) found a strong coupling of cold temperature extremes to surface-level circulation patterns in North America. This underscores the importance of considering multiple atmospheric levels when applying the co-recurrence ratio framework in different geographic contexts, rather than assuming that a fixed vertical level will be optimal.

Although this study focuses on European winter extremes, the methodological framework introduced is broadly transferable as evidenced by the wide range of applications in previous studies. Large-scale circulation patterns that drive extremes may be climatologically rare in some regions but can still result in strong coupling and severe local impacts when they occur. Our findings therefore support the use of co-recurrence analysis as a flexible tool for diagnosing atmospheric drivers of extremes across regions, provided that local climatological characteristics are carefully considered. While this study applies the method in a bivariate setting, its extension to more variables offers promising avenues for analyzing compound extremes and their atmospheric drivers. Ultimately, advancing our understanding of these drivers is essential for improving forecasts and risk management in a changing climate.

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1007/s00382-025-07846-x.

**Author contributions** A.C. Reiter, G. Messori, and M. Drews contributed to the study conception and design. A.C. Reiter carried out the analysis and drafted the first version of the manuscript. A.C. Reiter, G. Messori, M. Drews, M.A.D. Larsen, and D. Faranda commented on previous versions of the draft and read and approved the final manuscript.

**Funding** Open access funding provided by Technical University of Denmark. This work was supported by a Ph.D. grant partly funded by the Danish Klimaatlas (Climate Atlas) under the Danish Meteorological Institute (DMI) and partly by the EU Horizon project Artificial Intelligence and Machine Learning for Enhanced Representation of Processes and Extremes in Earth System Models (AI4PEX, Grant agreement ID: 101137682). The work was further supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant no. 948309, CENÆ project) and by the Swedish Research Council Vetenskapsrådet (grant no. 2022-06599).

**Data availability** The ERA5 reanalysis data used in this research can be downloaded from https://cds.climate.copernicus.eu/datasets/reanalysis-era5-single-levels?tab=overview and https://cds.climate.copernicus.eu/datasets/reanalysis-era5-pressure-levels?tab=overview (last access: 14 April 2025).
## Declarations

**Conflict of Interest** The authors have no financial or non-financial interest to declare.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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