# Unsupervised data-driven detection of exceptional atmospheric trajectories

Guillaume Coulaud, Davide Faranda

► To cite this version:

Guillaume Coulaud, Davide Faranda. Unsupervised data-driven detection of exceptional atmospheric trajectories. 2026. hal-05532887

HAL Id: hal-05532887

https://inria.hal.science/hal-05532887v1

Preprint submitted on 2 Mar 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Unsupervised data-driven detection of exceptional atmospheric trajectories

Guillaume Coulaud¹, a) and Davide Faranda², ³, ⁴, b)

1) University of Montpellier, Inria, CNRS, LIRMM, Montpellier,
France.

2) Laboratoire des Sciences du Climat et de l'Environnement,
UMR 8212 CEA-CNRS-UVSQ, Université Paris-Saclay & IPSL,
CEA Saclay l'Orme des Merisiers, 91191 Gif-sur-Yvette,
France

3) London Mathematical Laboratory, 8 Margravine Gardens, London W6 8RH,
UK

4) LMD/IPSL, ENS, Université PSL, École Polytechnique,
Institut Polytechnique de Paris, Sorbonne Université, CNRS,
Paris France

(Dated: 2 March 2026)
Extreme weather events in Europe are closely linked to the large-scale atmospheric circulation and often develop over several consecutive days. Most existing circulation-based approaches focus on identifying extreme weather patterns as instantaneous atmospheric states and therefore do not explicitly account for the temporal evolution of the flow. In this study, we apply a data-driven and unsupervised methodology to identify rare atmospheric trajectories from reanalysis data. The method quantifies how isolated short segments of atmospheric evolution are within the space of all observed trajectories, using daily sea-level pressure fields over Europe. We apply the approach to several decades of reanalysis data and identify the most isolated trajectories for different trajectory lengths. The detected trajectories are characterised by large-scale circulation anomalies and strong pressure gradients. A comparison with independent databases of European extreme events shows a statistically significant overlap, particularly for windstorms. Increasing the trajectory length enhances the detection of multi-day events, indicating that the method captures persistent atmospheric evolutions rather than isolated states. In addition to windstorms, the detected trajectories correspond to cold spells and blocking-like circulation patterns, as well as events that are not systematically documented in existing pan-European databases. These results indicate that analysing rare atmospheric trajectories provides complementary information to state-based approaches and offers a general framework for the detection of extreme atmospheric evolutions in reanalysis datasets.

<footnote>a) Electronic mail: guillaume.coulaud@inria.fr</footnote>

<footnote>b) Electronic mail: davide.faranda@cea.fr</footnote>
We present an unsupervised method to identify rare atmospheric trajectories
from reanalysis data by quantifying how isolated short segments of atmospheric
evolution are within the space of observed circulation patterns. Applied to daily
sea-level pressure fields over Europe, the method detects trajectories associated
with a substantial fraction of documented extreme events, particularly wind-
storms. The correspondence exceeds what would be expected under random
sampling and improves when longer trajectories are considered, highlighting
the role of temporal persistence. The method also identifies trajectories asso-
ciated with cold spells and blocking-like circulation patterns, as well as events
that are not systematically represented in existing databases. By focusing on
atmospheric evolution rather than instantaneous states, this approach provides
complementary information to traditional circulation-based analyses and offers
a general framework for detecting dynamically unusual atmospheric behaviour
in reanalysis datasets.

# I. INTRODUCTION

Extreme weather events have substantial impacts on ecosystems, infrastructure, and society, and their socio-economic costs have increased over recent decades¹. In Europe, events such as heatwaves, cold spells, windstorms, and episodes of heavy precipitation are closely linked to the large-scale atmospheric circulation²⁻⁵. Understanding the circulation conditions associated with these events, and how they may evolve under anthropogenic forcing, remains an active area of research⁶⁻¹⁰.

Many studies have analysed extreme events by associating them with recurrent circulation
patterns, typically defined as spatial configurations of atmospheric variables such as sea-level
pressure or geopotential height¹¹⁻¹³. This framework has been used to investigate changes
in the frequency and persistence of circulation patterns linked to surface extremes¹⁴⁻¹⁶.
Persistence has received particular attention because long-lived circulation anomalies are
often associated with prolonged surface impacts¹⁷. However, results regarding trends in
persistence and large-scale circulation characteristics remain mixed, both in summer and
winter⁶,¹⁰,¹⁸.
A related line of work has examined changes in the occurrence of circulation patterns associated with extreme events¹²,¹⁴,¹⁹. These studies have shown that circulation changes can explain part of the observed trends in surface extremes, particularly for temperature-related events. Similar approaches have been applied to other types of extremes, including droughts and heavy precipitation²⁰,²¹. Together, these studies highlight the role of large-scale dynamics in shaping the occurrence of extreme weather events.

Most existing approaches rely on the analysis of instantaneous atmospheric states, typically at daily resolution. Circulation analogues, regime classifications, and composite analyses are generally based on individual maps of atmospheric variables. While this approach captures spatial structures, it does not explicitly account for the temporal evolution of the circulation. Many extreme events, including windstorms, cold spells, and blocking episodes, develop and persist over several days. Their impacts depend not only on the circulation pattern at a given time, but also on the sequence and duration of atmospheric states.

From a dynamical systems perspective, the atmosphere can be viewed as evolving in a high-dimensional phase space. In this framework, extreme events may correspond to trajectories that explore regions of phase space that are infrequently sampled. This perspective motivates the analysis of atmospheric evolution over finite time intervals, rather than focusing solely on individual states. While concepts related to rare trajectories have been discussed in theoretical studies of atmospheric dynamics and extremes²², their application to observational datasets has been limited.

In this study, we apply a data-driven and unsupervised methodology to identify rare atmospheric trajectories from reanalysis data. Our method extends analogue-based approaches from individual instantaneous states to short trajectories, represented as sequences of atmospheric fields over fixed time windows. For each trajectory, we quantify its degree of isolation relative to all others in the dataset using a distance-based metric. In particular, we adopt a *k*-nearest-neighbours strategy for anomaly detection in high-dimensional spaces²³⁻²⁵. Distances are computed using the Euclidean distance applied to trajectories described as tensors in time and space. Trajectories that lie far from their nearest neighbours occupy regions of low sample density, corresponding to atmospheric evolutions that are rarely observed. This enables us to define a rarity score, where low values indicate commonly occurring trajectories and high values denote rare observed events.

We apply the method to daily sea-level pressure fields over Europe and identify the most
isolated trajectories for different trajectory lengths. The detected trajectories are charac-
terised by large pressure gradients or by persistent large-scale anomalies. We compare the
identified trajectories with independent databases of documented European extreme events.
The overlap with known windstorm events increases with trajectory length, indicating that
the method captures multi-day events more effectively than state-based approaches. Addi-
tional trajectories correspond to cold spells and winter blocking episodes, while others may
reflect regional or local events not included in existing catalogues.

These results suggest that analysing atmospheric trajectories provides complementary
information to state-based methods for the identification of extreme events. The trajectory-
based framework offers a systematic way to detect episodes of unusual atmospheric evolution
without relying on predefined event thresholds or supervised classification. This approach
may be useful for the detection, classification, and further analysis of extreme weather events
in reanalysis and model datasets.

## II. DATA AND METHODS

A. Reanalysis data

We use daily sea-level pressure fields from the ERA5 reanalysis produced by the Euro-
pean Centre for Medium-Range Weather Forecasts²⁶. The analysis is restricted to a fixed
European domain and to the period covered by the reanalysis. Sea-level pressure is chosen
as it provides a direct representation of synoptic-scale circulation features relevant for mid-
latitude extreme events, including extratropical cyclones and blocking situations. All fields
are interpolated onto a common grid prior to analysis.

B. Trajectory-based rarity metric

The goal of the method is to identify atmospheric states and short evolutions that are
rarely observed in the historical record. By comparing each trajectory to all others in the
dataset, we obtain a measure of how isolated it is in the space of observed atmospheric
configurations. As depicted in Figure 1, we construct the trajectory space from the under-
lying state space using a fixed-duration temporal sliding window. A rarity score is then
assigned to each trajectory based on the average distance to its *k* nearest neighbours. This
<figure><img src="image_2.png" /><figcaption>FIG. 1. Schematic overview of the proposed methodology. In state space (left), the blue segment represents a normal trajectory observed over a few days, while the red segment corresponds to a rare trajectory. The continuous state space is partitioned using a sliding window to generate trajectory segments, which are then represented in trajectory space (right). The normal trajectory lies in a dense region, characterized by a low distance to its kth nearest neighbour, whereas the rare trajectory lies in a sparse region.</figcaption></figure>

rarity-based approach provides a direct, data-driven way to highlight unusual states.

Formally, let $X_t \in \mathbb{R}^{h \times w}$ denote the atmospheric state at time $t$, represented by a gridded field of sea-level pressure over a fixed region of dimensions $h \times w$. For a trajectory of duration $d$, we define the corresponding finite-time segment $\mathbf{X}_{t:t+d-1} \in \mathbb{R}^{h \times w \times d}$ such as

$$ \mathbf{X}_{t:t+d-1} = (X_t, X_{t+1}, \dots, X_{t+d-1}), $$

which represents the evolution of the atmospheric field over the time interval $[t, t+d)$.

To identify dynamically exceptional configurations, we associate each time $t$ with a rarity score $s_t$ defined as the average distance between $\mathbf{X}_{t:t+d-1}$ and its k nearest neighbours in the historical record. The set of nearest neighbours,

$$ \mathrm{NN}_t = \{i \neq t : \mathbf{X}_{i:i+d-1} \text{ is among the k closest states to } \mathbf{X}_{t:t+d-1}\}, $$
is determined using the Euclidean distance in the full field space. The rarity score is then

$$s_t = \frac{1}{k} \sum_{i \in NN_t} \| \mathbf{X}_{t:t+d-1} - \mathbf{X}_{i:i+d-1} \|_F^2,$$

with large values indicating that $\mathbf{X}_{t:t+d-1}$ lies in a sparsely sampled region of the space. Thus, $s_t$ serves as a nonparametric proxy for the local density of atmospheric states: rare or dynamically unstable configurations correspond to large distances from all observed states.

The analysis is repeated for different trajectory lengths $d$, allowing the identification of both isolated atmospheric states and multi-day atmospheric evolutions. For each value of $d$, trajectories are ranked according to their rarity score, and the most extreme trajectories are selected for further analysis.

### III. RESULTS

#### A. Structure and seasonality of rare atmospheric trajectories

For each trajectory length, we retain the 100 trajectories with the largest rarity scores. These trajectories correspond to portions of the atmospheric evolution of 5 days that are weakly represented in the historical record. The method is flexible and can be applied to different trajectory durations depending on the processes of interest. In our analyses, we also tested shorter durations of 1 and 3 days, which are suitable for capturing more transient atmospheric configurations. Here, we present the results for 5-day trajectories as an illustrative case, because this duration is particularly relevant for the detection of extreme events associated with synoptic-scale circulation patterns that develop and persist over several days. This timescale allows the method to capture the temporal structure and persistence of large-scale atmospheric configurations while remaining fully consistent with the unsupervised detection framework. No thresholds related to surface impacts or predefined event definitions are used in the detection procedure. The method is fully unsupervised and relies solely on the geometry of the atmospheric field space.

The clustering is performed on daily SLP anomaly fields derived from ERA5 reanalysis over the domain 30°N–75°N, 30°W–40°E. The anomaly fields are obtained by removing the mean seasonal cycle computed over the full analysis period (1950–present), after excluding leap days. The resulting anomaly fields are expressed in hPa.
Each anomaly field is vectorised into a single state vector of dimension $h \times w$, corresponding to the number of grid points in the spatial domain. To account for the decreasing area of grid cells towards the poles, each grid point is weighted by the cosine of latitude. This weighting ensures that the clustering reflects physically meaningful spatial structures rather than being dominated by high-latitude grid density.

The weighted anomaly vectors are then standardised by removing the mean and dividing by the standard deviation at each grid point, ensuring that all spatial locations contribute equally to the clustering regardless of their local variability.

To reduce dimensionality and improve robustness, principal component analysis (PCA) is applied to the standardised anomaly vectors. The principal components are computed using singular value decomposition, and the reduced representation retains the minimum number of components required to explain at least 90% of the total variance. For the present dataset, this corresponds to retaining the first $p$ principal components, where $p$ is determined objectively from the cumulative explained variance.

Clustering is then performed in the reduced PCA space using the $k$-means algorithm with squared Euclidean distance as the similarity metric. The algorithm is initialised using random seeds and repeated 50 times with different initialisations. The solution with the smallest total within-cluster variance is retained. The maximum number of iterations is set to 1000 to ensure convergence.

The optimal number of clusters $c$ is determined using the silhouette coefficient$^{27}$, which measures both cluster cohesion and separation. For each candidate value $c = 2, \dots, 8$, clustering is repeated independently and the mean silhouette score is computed across the 100 trajectories. The mean silhouette score reaches a clear maximum at $c = 3$, with a value of 0.401, compared to 0.373 for $c = 2$ and values below 0.37 for all larger values of $c$. This indicates that three clusters provide the most robust partition of the rare trajectories.

The distribution of individual silhouette values for $c = 3$ is shown in Fig. 2. Most trajectories exhibit positive silhouette values, typically between 0.3 and 0.7, indicating good separation between clusters. Only a small number of trajectories have silhouette values close to zero, corresponding to borderline cases between neighbouring regimes. The absence of strongly negative silhouette values confirms that the clustering identifies well-separated and physically coherent circulation regimes.

Using $c = 3$, the 100 rare trajectories are partitioned into clusters of sizes:
$$n_1 = 47, \quad n_2 = 35, \quad n_3 = 18.$$

The composite SLP anomaly fields for each cluster are shown in Fig. 3. These composites are computed as the arithmetic mean of the anomaly fields belonging to each cluster.

Cluster 1 (Fig. 3a) is characterised by negative SLP anomalies over northern Europe and positive anomalies over southern Europe, corresponding to an enhanced meridional pressure gradient across western and central Europe.

Cluster 2 (Fig. 3b) exhibits a broad negative anomaly centred over the European continent, corresponding to a large-scale low-pressure regime affecting much of the domain.

Cluster 3 (Fig. 3c) displays the opposite polarity of Cluster 1, with positive anomalies over northern Europe and lower pressure towards the south-west of the domain.

These results demonstrate that the rare trajectories correspond to a small number of dynamically coherent circulation regimes rather than a continuum of unrelated anomaly patterns.

The seasonal distribution of the detected trajectories is shown in Fig. 4. All clusters exhibit a pronounced winter maximum, with the majority of trajectories occurring between December and February. This reflects the stronger dynamical variability of the North Atlantic–European circulation during winter, when baroclinic instability and large-scale circulation anomalies are most pronounced.

Cluster 1 and Cluster 3 occur primarily in mid-winter, whereas Cluster 2 shows a particularly strong concentration in early winter. This indicates that the rarity metric captures multiple classes of winter circulation anomalies with distinct dynamical characteristics.

It is to be noted that the preprocessing used for clustering and the definition of the rarity score serve different purposes. Clustering relies on problem-dependent preprocessing designed to highlight meaningful spatial structures. In contrast, the rarity score is intentionally defined in a generic manner regardless of the structures present in the data. The rarity scoring method is independent of the choice of preprocessing and can be applied to both raw and preprocessed data, depending on the context and objective.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Silhouette values of the 100 rare trajectories for the optimal clustering with $c = 3$. Each horizontal bar represents one trajectory, and its length indicates the silhouette value, which measures the degree of similarity to its assigned cluster relative to other clusters. Positive values indicate well-separated trajectories, while values near zero correspond to trajectories located near cluster boundaries.</figcaption></figure>

## B. Verification against documented extreme weather events

To evaluate whether the detected rare trajectories correspond to documented extreme weather events, we perform an independent verification using the catalogue of historical European extreme events compiled in Table S1. This verification is not used as input to the rarity detection algorithm, but serves as an independent validation of its physical relevance.

For each of the 100 detected trajectories, we identify documented extreme events occurring on the same date. A trajectory is considered to correspond to a documented event if its date falls within the reported duration of a known extreme event. Because many atmospheric extreme events persist for multiple consecutive days, a single event may correspond to several detected trajectories.

Out of the 100 detected trajectories, 79 correspond to confirmed documented extreme
<figure><img src="image_4.png" /><figcaption>FIG. 3. Composite SLP anomalies for the three clusters identified among the 100 rarest trajectories ($d = 5$). (a) Cluster 1 ($n = 47$). (b) Cluster 2 ($n = 35$). (c) Cluster 3 ($n = 18$).</figcaption></figure>

atmospheric conditions, while the remaining 21 correspond to circulation states requiring further archival confirmation. Among the confirmed events, 40 trajectories are associated with windstorms or intense extratropical cyclones, corresponding to 51% of confirmed cases and 40% of all detected trajectories. This fraction is substantially higher than expected from random sampling of winter days, indicating that the rarity metric preferentially identifies dynamically extreme circulation states.

The correspondence between clusters and documented windstorms is highly structured. Cluster 1 contains 26 windstorm trajectories out of 35 confirmed events, corresponding to a fraction of 74%. Cluster 2 contains 13 windstorm trajectories out of 29 confirmed events (45%), while Cluster 3 contains only 1 windstorm trajectory out of 15 confirmed events (7%). Expressed relative to the total cluster sizes, windstorms represent 55% of Cluster 1, 37% of Cluster 2, and only 6% of Cluster 3.

These results demonstrate a strong association between windstorms and Cluster 1 circulation patterns. As shown in Fig. 3a, Cluster 1 is characterised by a pronounced meridional pressure gradient across Europe, with negative pressure anomalies over northern Europe and positive anomalies over southern Europe. This configuration is dynamically consistent with enhanced zonal flow and strong baroclinic activity, which are known to favour the development and propagation of intense extratropical cyclones.

Cluster 2 exhibits a weaker but still substantial association with windstorms. The composite anomaly of this cluster (Fig. 3b) corresponds to a large-scale low-pressure regime centred over Europe, which may reflect mature cyclone stages or broader low-pressure envi-
<figure><img src="image_5.png" /><figcaption>FIG. 4. Seasonal distribution of rare trajectories by cluster. Histograms show the number of trajectories as a function of calendar month.</figcaption></figure>

ronments associated with storm activity.

In contrast, Cluster 3 shows almost no association with windstorms. Its composite anomaly pattern (Fig. 3c), characterised by positive pressure anomalies over northern Europe, is more consistent with blocking or anticyclonic regimes. These circulation states are dynamically rare but are not typically associated with intense windstorms but they are rather associated with cold spells.

The strong concentration of windstorm events in Cluster 1 confirms that the trajectory-based rarity metric identifies physically meaningful circulation regimes associated with major European windstorms. At the same time, the existence of additional clusters dominated by non-storm circulation anomalies demonstrates that the rarity metric captures a broader class of dynamically unusual atmospheric trajectories, including blocking regimes and persistent
large-scale pressure anomalies.

These results provide independent validation that the rarity metric isolates dynamically extreme and physically relevant atmospheric states, rather than statistical artefacts of the trajectory representation.

## IV. DISCUSSION

The results show that a rarity-based analysis of atmospheric trajectories isolates dynamically coherent and physically meaningful circulation regimes. Rather than forming a homogeneous set, the 100 rarest trajectories partition robustly into three distinct clusters, as objectively determined by silhouette analysis. The optimal partition ($c = 3$) exhibits the highest mean silhouette score (0.401), with most trajectories showing positive silhouette values, confirming that the identified clusters correspond to well-separated regions of atmospheric state space.

The cluster composites reveal that the rare trajectories correspond to a small number of large-scale circulation regimes characterised by persistent and spatially coherent sea-level pressure anomalies. These regimes include configurations with strong meridional pressure gradients across Europe, large-scale low-pressure anomalies centred over the continent, and anticyclonic or blocking-like anomalies over northern Europe. These circulation patterns are dynamically distinct and represent qualitatively different atmospheric evolutions, rather than small perturbations around a single extreme state.

The correspondence with documented extreme weather events provides an independent validation of the dynamical relevance of the rarity metric. Out of the 100 rare trajectories, 79 coincide with independently documented extreme atmospheric conditions. In particular, 40 trajectories correspond to documented windstorms or intense extratropical cyclones. The association between windstorms and specific circulation regimes is highly structured: windstorms account for 55% of Cluster 1 trajectories, compared to 37% in Cluster 2 and only 6% in Cluster 3. This strong concentration demonstrates that the rarity metric identifies circulation regimes dynamically favourable to cyclone development and propagation.

The spatial structure of Cluster 1 is consistent with this interpretation. Its composite anomaly pattern exhibits a pronounced meridional pressure gradient over Europe, a configuration associated with enhanced zonal flow and baroclinic instability. These conditions
are known to favour the development and intensification of extratropical cyclones. Cluster 2, characterised by broad negative pressure anomalies over Europe, likely corresponds to mature cyclone stages or extended low-pressure regimes. In contrast, Cluster 3 is dominated by anticyclonic anomalies over northern Europe and shows little association with windstorms, indicating that the rarity metric also identifies dynamically unusual blocking or high-pressure regimes.

The seasonal distribution of the rare trajectories further supports their dynamical interpretation. All clusters exhibit a strong winter maximum, consistent with the enhanced variability and baroclinic activity of the North Atlantic–European circulation during this season. This seasonality reflects the underlying dynamical processes governing atmospheric variability rather than any explicit seasonal constraint in the detection algorithm.

An aspect of the detected trajectories is that a fraction of them do not correspond to events documented in existing European extreme event databases. Many of these cases exhibit circulation anomalies resembling blocking regimes or regional pressure anomalies. This may reflect limitations of existing catalogues, which often focus on specific impact thresholds or event types. Alternatively, these trajectories may correspond to dynamically rare atmospheric evolutions that do not necessarily produce extreme surface impacts. The rarity metric therefore identifies dynamically unusual circulation states independently of predefined impact criteria.

A strength of the methodology is that it does not rely on predefined circulation regimes, thresholds, or supervised classification. The detection is entirely unsupervised and based solely on the geometry of atmospheric trajectories in phase space. As a result, the identified rare trajectories emerge from the intrinsic structure of atmospheric variability. The clustering analysis further demonstrates that these rare trajectories are organised into a small number of dynamically coherent regimes.

Several limitations should nevertheless be noted. First, the analysis is based exclusively on sea-level pressure, which captures large-scale dynamical structure but does not directly represent surface impacts such as precipitation or wind speed. Second, the rarity metric identifies trajectories that are uncommon in the historical record but does not directly identify their physical causes. Finally, the results are based on a specific spatial domain and trajectory length, although the methodology itself is general and can be applied to other regions, variables, and temporal scales.
V. CONCLUSION

In this study, we introduced and applied a trajectory-based rarity metric to identify dynamically unusual atmospheric evolutions in daily sea-level pressure fields over Europe. By extending analogue-based approaches from instantaneous states to finite-time trajectories, the method captures the temporal structure of atmospheric evolution and identifies circulation regimes that are weakly represented in the historical record.

The rare trajectories partition into three dynamically coherent clusters, corresponding to distinct large-scale circulation regimes. The clustering structure is robust, as demonstrated by silhouette analysis, and reveals that atmospheric rarity is organised around a small number of well-defined dynamical configurations rather than isolated anomalous states.

A quantitative comparison with documented extreme events shows that a large fraction of rare trajectories correspond to known extreme atmospheric conditions, particularly windstorms. Windstorm trajectories are strongly concentrated in one cluster characterised by enhanced meridional pressure gradients and dynamically favourable conditions for cyclone development. This demonstrates that the rarity metric identifies circulation regimes associated with major extreme events while also capturing other classes of dynamically unusual atmospheric states, including blocking regimes.

Because the methodology is fully unsupervised and relies only on the intrinsic geometry of atmospheric trajectories, it provides a general framework for identifying dynamically rare atmospheric evolutions. The approach does not require predefined circulation regimes, impact thresholds, or supervised training, making it broadly applicable to different variables, spatial domains, and datasets.

Future work could extend this framework to additional atmospheric variables, longer trajectories, and climate model simulations. This would enable systematic investigation of how the frequency, structure, and persistence of rare atmospheric trajectories evolve under changing climate conditions, and how these changes relate to the occurrence of extreme weather events.
## ACKNOWLEDGMENTS

We acknowledge funding by Région Occitanie, from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 101003469 (XAIDA), the ANR project TASE PowDev (Strategic Development of the Power Grids of the Future, ANR-22-PETA-0016), the ANR project Templex (ANR-23-CE56-0002), and the ANR project TRACCS, Transformer la modélisation du climat pour les services climatiques (ANR-22-EXTR-0005, Extending).

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study are openly available. ERA5 is the latest climate reanalysis produced by ECMWF as part of the implementation of the EU-funded Copernicus Climate Change Service (C3S), providing hourly data on atmospheric, land-surface, and sea-state parameters, together with estimates of uncertainty, from 1950 to the present day. ERA5 data are available through the C3S Climate Data Store on regular latitude-longitude grids at a 0.25° × 0.25° resolution (https://cds.climate.copernicus.eu/#!/home). The Python code to compute the rarity score is available on the following GitHub repository (https://github.com/GuillaumeCld/Trajectory-kNN).

## REFERENCES

¹G. Forzieri, A. Bianchi, F. B. e Silva, M. A. M. Herrera, A. Leblois, C. Lavalle, J. C. Aerts, and L. Feyen, “Escalating impacts of climate extremes on critical infrastructures in europe,” Global environmental change **48**, 97–107 (2018).

²P. M. Sousa, D. Barriopedro, R. García-Herrera, C. Ordóñez, P. M. Soares, and R. M. Trigo, “Distinct influences of large-scale circulation and regional feedbacks in two exceptional 2019 european heatwaves,” Communications Earth & Environment **1**, 1–13 (2020).

³J. Lehmann and D. Coumou, “The influence of mid-latitude storm tracks on hot, cold, dry and wet extremes,” Scientific reports **5**, 1–9 (2015).

⁴S. Zanardo, L. Nicotina, A. G. Hilberts, and S. P. Jewson, “Modulation of economic losses from european floods by the north atlantic oscillation,” Geophysical Research Letters **46**, 2563–2572 (2019).
<footnote n="5">⁵J. Hanley and R. Caballero, “The role of large-scale atmospheric flow and rossby wave breaking in the evolution of extreme windstorms over europe,” Geophysical Research Letters **39** (2012).</footnote>

<footnote n="6">⁶J. Cattiaux, Y. Peings, D. Saint-Martin, N. Trou-Kechout, and S. J. Vavrus, “Sinuosity of midlatitude atmospheric flow in a warming world,” Geophysical Research Letters **43**, 8259–8268 (2016).</footnote>

<footnote n="7">⁷R. M. Horton, J. S. Mankin, C. Lesk, E. Coffel, and C. Raymond, “A review of recent advances in research on extreme heat events,” Current Climate Change Reports **2**, 242–259 (2016).</footnote>

<footnote n="8">⁸M. E. Mann, S. Rahmstorf, K. Kornhuber, B. A. Steinman, S. K. Miller, S. Petri, and D. Coumou, “Projected changes in persistent extreme summer weather events: The role of quasi-resonant amplification,” Science advances **4**, eaat3272 (2018).</footnote>

<footnote n="9">⁹J. Cohen, X. Zhang, J. Francis, T. Jung, R. Kwok, J. Overland, T. Ballinger, U. Bhatt, H. Chen, D. Coumou, et al., “Divergent consensuses on arctic amplification influence on midlatitude severe winter weather,” Nature Climate Change **10**, 20–29 (2020).</footnote>

<footnote n="10">¹⁰J. Riboldi, F. Lott, F. d'Andrea, and G. Rivière, “On the linkage between rossby wave phase speed, atmospheric blocking, and arctic amplification,” Geophysical Research Letters **47**, e2020GL087796 (2020).</footnote>

<footnote n="11">¹¹S. Corti, F. Molteni, and T. Palmer, “Signature of recent climate change in frequencies of natural atmospheric circulation regimes,” Nature **398**, 799–802 (1999).</footnote>

<footnote n="12">¹²D. E. Horton, N. C. Johnson, D. Singh, D. L. Swain, B. Rajaratnam, and N. S. Diffenbaugh, “Contribution of changes in atmospheric circulation patterns to extreme temperature trends,” Nature **522**, 465–469 (2015).</footnote>

<footnote n="13">¹³A. Jézéquel, P. Yiou, S. Radanovics, and R. Vautard, “Analysis of the exceptionally warm december 2015 in france using flow analogues,” Bulletin of the American Meteorological Society (2017).</footnote>

<footnote n="14">¹⁴A. Jézéquel, J. Cattiaux, P. Naveau, S. Radanovics, A. Ribes, R. Vautard, M. Vrac, and P. Yiou, “Trends of atmospheric circulation during singular hot days in europe,” Environmental Research Letters **13**, 054007 (2018).</footnote>

<footnote n="15">¹⁵K. Kornhuber, S. Osprey, D. Coumou, S. Petri, V. Petoukhov, S. Rahmstorf, and L. Gray, “Extreme weather events in early summer 2018 connected by a recurrent hemispheric wave-7 pattern,” Environmental Research Letters **14**, 054002 (2019).</footnote>
<footnote n="16">¹⁶C. D. Rogers, K. Kornhuber, S. E. Perkins-Kirkpatrick, P. C. Loikith, and D. Singh, "Sixfold increase in historical northern hemisphere concurrent large heatwaves driven by warming and changing atmospheric circulations," Journal of Climate 35, 1063-1078 (2022).</footnote>

<footnote n="17">¹⁷M. Röthlisberger, L. Frossard, L. F. Bosart, D. Keyser, and O. Martius, "Recurrent synoptic-scale rossby wave patterns and their effect on the persistence of cold and hot spells," Journal of Climate 32, 3207-3226 (2019).</footnote>

<footnote n="18">¹⁸P. Hassanzadeh, Z. Kuang, and B. F. Farrell, "Responses of midlatitude blocks and wave amplitude to changes in the meridional temperature gradient in an idealized dry gcm," Geophysical Research Letters 41, 5223-5232 (2014).</footnote>

<footnote n="19">¹⁹M. E. Mann, S. Rahmstorf, K. Kornhuber, B. A. Steinman, S. K. Miller, and D. Coumou, "Influence of anthropogenic climate change on planetary wave resonance and extreme weather events," Scientific reports 7, 1-12 (2017).</footnote>

<footnote n="20">²⁰S. M. Vicente-Serrano and J. I. López-Moreno, "The influence of atmospheric circulation at different spatial scales on winter drought variability through a semi-arid climatic gradient in northeast spain," International Journal of Climatology: A Journal of the Royal Meteorological Society 26, 1427-1453 (2006).</footnote>

<footnote n="21">²¹M. Barlow, W. J. Gutowski, J. R. Gyakum, R. W. Katz, Y.-K. Lim, R. S. Schumacher, M. F. Wehner, L. Agel, M. Bosilovich, A. Collow, et al., "North american extreme precipitation events and related large-scale meteorological patterns: a review of statistical methods, dynamics, modeling, and trends," Climate Dynamics 53, 6835-6875 (2019).</footnote>

<footnote n="22">²²D. Faranda, G. Messori, and P. Yiou, "Dynamical proxies of north atlantic predictability and extremes," Scientific reports 7, 1-10 (2017).</footnote>

<footnote n="23">²³M. Amer and M. Goldstein, "Nearest-neighbor and clustering based anomaly detection algorithms for rapidminer," in Proc. of the 3rd RapidMiner Community Meeting and Conference (RCOMM 2012) (2012) pp. 1-12.</footnote>

<footnote n="24">²⁴F. Angiulli and C. Pizzuti, "Fast outlier detection in high dimensional spaces," in European conference on principles of data mining and knowledge discovery (Springer, 2002) pp. 15-27.</footnote>

<footnote n="25">²⁵S. Ramaswamy, R. Rastogi, and K. Shim, "Efficient algorithms for mining outliers from large data sets," in Proceedings of the 2000 ACM SIGMOD international conference on Management of data (2000) pp. 427-438.</footnote>
<footnote n="26"><sup>26</sup>H. Hersbach, C. Peubey, A. Simmons, P. Berrisford, P. Poli, and D. Dee, “Era-20cm: A twentieth-century atmospheric model ensemble,” Quarterly Journal of the Royal Meteorological Society **141**, 2350–2375 (2015).</footnote>

<footnote n="27"><sup>27</sup>P. J. Rousseeuw, “Silhouettes: a graphical aid to the interpretation and validation of cluster analysis,” Journal of computational and applied mathematics **20**, 53–65 (1987).</footnote>