Evidence for preferred propagating terrestrial heatwave pathways due to Rossby wave activity

Domain: earth_science
Published: 2025-05-22
Editorial outcome: major_revision

Abstract
Terrestrial heatwaves are prolonged hot weather events often resulting in widespread socioeconomic impacts. Predicting heatwaves remains challenging, partly due to limited understanding of the events’ spatial evolution and underlying mechanisms. Heatwaves were mainly examined at fixed stations, with little attention given to the fact that the center of a heatwave can move a long distance. Here, we examine the spatial propagation of terrestrial heatwaves using a complex network algorithm, and find four preferred propagation pathways of terrestrial heatwaves in the northern hemisphere. Along each preferred pathway, heatwaves evolve in two ways: propagating along the pathway or being stationary. We show that the propagating heatwave pathways are consistent with the movement of Rossby wave trains, and that both are guided by enhanced Rossby wave flux activities. The detected propagation pathways are found to provide prior knowledge for occurrences of downstream heatwaves that can be used for identifying associated precursor signals. The results shed light on the mechanisms responsible for preferred propagating heatwave pathways and provide potential predictability of terrestrial heatwaves.

Introduction
Terrestrial heatwaves denote extreme weather events characterized by high temperatures persisting for consecutive days or even weeks1,2 (distinct from marine heatwaves and hereafter called “heatwaves”). These prolonged high temperatures can endanger human health3–5, strain power systems6, cause drought conditions, lead to crop failure7, and increase wildfire risk8. The severe socioeconomic impacts have sparked great interest in the mechanisms and predictability of heatwaves1,2. The regions experiencing heatwaves are typically accompanied by anomalous high-pressure weather systems1,9–12 (e.g., Supplementary Fig. S1), where the local intense heat is contributed by adiabatic warming from subsidence, and diabatic heating from solar radiation. Operational forecasting systems can reliably issue warnings of heatwaves that last more than three days, with a lead time of two to three days1. Extending the lead time of heatwave forecasts requires an improved understanding of the background atmospheric circulation patterns associated with heatwaves1,2,13,14. On the planetary scale, atmospheric high-pressure systems are dominated by persistent anomalous atmospheric circulation patterns, such as the quasi-stationary ridge or block in the atmospheric flow9,15, long-lived Rossby wave packets9,10,12, or circumglobal resonant Rossby waves16,17. However, current process-based numerical models are unable to reliably predict the locations and intensities of these relevant background patterns at the desired lead times1,18–22, which hinders both the prediction skill and our understanding of the mechanisms behind heatwaves1,22,23. Improved understanding of the relationship between heatwaves and these background patterns based on advanced data analysis techniques is hence of utmost importance.

Recent studies found that heatwaves in different regions overlap within a timeframe of approximately one week17,24–26, suggesting that specific anomalous atmospheric circulation patterns can establish teleconnection of hot weather events in distant locations. This phenomenon can amplify the overall socioeconomic impact of heatwaves5,17,24,26,27, but also there might be windows of opportunity to extend the prediction of heatwaves by further understanding the spatial evolution patterns of heatwaves and the role of anomalous atmospheric circulation1,13. Examined on the weekly timescale, heatwaves in regions separated by 4000–5000 km can occur concurrently, which has been shown to be caused by circumglobal resonant Rossby waves17,21. Here, we pay attention to the daily timescale to examine the evolution of heatwaves in space and time. Taking a heatwave over central Asia as an example (Supplementary Fig. S2), the event initially affects multiple surrounding regions with hot weather (i.e., the spatial extent or size of a heatwave28,29), and its core gradually propagates to neighboring areas in the subsequent days. It means that heatwave events at distant regions may be linked through the heatwave’s spatial extent but also by its spatial propagation. Similar phenomena can be observed in land heatwaves over Europe and North America (Supplementary Figs. S3, S4 and S5). Recent studies have confirmed that such terrestrial heatwave propagation phenomena are prevalent across global land masses26,30–32, but the mechanisms behind it remain mostly unknown. From the corresponding atmospheric circulation anomaly signals, it can be observed that as the heatwave propagates, the associated anomalous high-pressure systems aloft are also moving correspondingly (Supplementary Figs. S1–S5). This suggests that the propagation of heatwaves may be related to specific atmospheric circulation patterns, which deserves an in-depth investigation.

Here, we conduct a detailed analysis on the spatiotemporal evolution process of propagating heatwaves. We utilize a complex network algorithm31,33–35 to identify the regions with the highest frequency of propagating terrestrial heatwaves in the mid-latitudes of northern hemisphere and trace their propagation pathways. We assume that such heatwave propagation phenomena are caused by more localized Rossby wave activities, as opposed to circumglobal patterns17,21. To verify this hypothesis, we will separately analyze the propagation of heatwaves and the movement of high-pressure atmospheric systems, then examine whether these analyses yield consistent results. Additionally, understanding the linkage with atmospheric circulation patterns can enhance the predictability of extreme weather events by providing prior knowledge from constraints on larger spatiotemporal scales, but relevant investigations for heatwaves are still lacking1,19. We will therefore also assess the predictability brought about by this heatwave propagation phenomenon.

Results
Detecting heatwaves by a complex network algorithm
To capture the spatial propagation of heatwaves across distant regions, we employ a complex network algorithm to analyze the summertime heatwave events detected in the ERA5 reanalysis (Methods). In this framework, each grid cell within the surface air temperature map is represented as a network node. For identifying the network link strength for any pair of nodes A and B, we count the number of cases that a heatwave event occurs in A, and B also experiences a heatwave event, requiring that the timing of the local temperature maximum during the heatwave in nodes A and B differ by no more than 3 days. We then perform significance tests on the link strengths, retaining only those links that are unlikely to occur by chance33,35,36 (“Methods”). The probability distribution of the distances of significant links exhibit two peaks, located at 1000 km and 5100 km, respectively (see Supplementary Fig. S6). The network links with typical distances of 4000-6000 km are dominated by the teleconnection effect of circumglobal resonant Rossby waves, as reported in previous studies17,35,37,38. Here we focus on the network links with distances of 400–2000 km, which have overall higher probabilities and could be related to more localized Rossby waves, but have been rarely analyzed in past studies. For mitigating the interference of the proximity effect37 and spatial extent of heatwaves (with typical size of 300 km26,29,39) when identifying the propagation pattern of the heatwave, network links with distances shorter than 400 km are not considered during our analysis. This heatwave network is based on counting the recurrences of heatwaves in historical records using state-of-the-art data, ensuring that the subsequent network metrics capture the most frequent and recurrent patterns of heatwaves accurately.

Propagation characteristics of heatwaves
The underlying spatial propagation patterns of heatwaves can be revealed by investigating features of the constructed heatwave network. Here, we calculate a metric called directed network divergence to analyze the characteristics of the heatwave network. Network divergence is defined as the difference between the number of outgoing links and the number of incoming links at that specific node of the network, with positive values representing the source regions of propagating heatwaves and negative values representing sinks33,40 (see “Methods” for more details). Hence, nodes with positive divergence (source nodes) indicate that heatwaves at these nodes statistically tend to precede those in other connected nodes, while negative divergence indicates the opposite (sink nodes) (Fig. 1). On the same continent, source and sink nodes often appear in pairs, statistically indicating that in certain specific areas, such as Western Europe, Northern Asia, and North America, there may be a continuous occurrence of heatwaves from source to sink nodes. We focus on the heatwaves occurring at the network’s source nodes and track their spatial evolution in the following days, finding that the heatwaves indeed propagate from the source nodes to the sink nodes (Supplementary Figs. S2–S5). Inspections of all heatwaves over the source regions identified in Fig. 1 reveal that, between 1959 and 2023, such propagating heatwaves (Supplementary Note 1) have occurred 64 times at the Northern Asia source node, 76 times at the Western Europe source node, and 83 and 79 times at the North American source nodes, respectively. This encourages us to conduct further quantitative analyses as follows.

Preferred propagation pathways of heatwaves
Following upon the indication of source nodes in the heatwaves network, we use the “local search" method33,36 (“Methods”) to determine the main propagation directions outgoing from these source nodes, until the search stops at sink nodes. This establishes the most frequent and recurrent heatwave propagation pathways. As a result, four significant propagation pathways covering the mid-latitudes of the Northern Hemisphere are shown in Fig. 1: the Asian pathway, from the Urals to the coastal area of the Sea of Japan in East Asia; the Western European pathway (WE), from the eastern Atlantic to Central Europe (with the starting node chosen as Spain due to land-sea temperature differences); the North American Pathway 1 (NA1), extending from northern Canada to Quebec; and the North American Pathway 2 (NA2), from the west coast to the east coast of the United States. Along each pathway we select six representative nodes for subsequent analyses, with an approximate distance of 1000 km between adjacent nodes. Regarding the four detected pathways, we use rose diagrams to identify the dominant directions of the associated network links33,36 (Supplementary Figs. S7–S10). At each node along the propagation pathway, heatwaves can spread in various directions, but the highest probability of propagation is distributed along the direction of the heatwave pathway, which holds for all four pathways (Supplementary Fig. S11).

Consistency between heatwave propagation and RWP movement
The mechanisms underlying these four significant propagation pathways are related to the atmospheric circulation background. The composite Z500 anomalies during the heatwave propagation events along the preferred pathways (Supplementary Figs. S12–S15) suggest a strong association between the propagation of these heatwaves and mid-latitude Rossby wave packets (RWPs). RWPs, as local amplitude maxima of Rossby waves, can be approximated by the centers of anomalous high-pressure systems in the upper atmosphere10,41. To further demonstrate the association between RWPs and the revealed heatwave propagation patterns, we thus analyze the movement trajectories of high-pressure systems using the Z500 anomaly data from ERA5. The key purpose is to compare whether the high-pressure systems’ trajectories are in line with the aforesaid heatwave propagation pathways.

Taking the Asia pathway as an example, we apply the k-means clustering method to classify the movement patterns of the high-pressure system centers at the onset of heatwaves (i.e., the start node of the Asia pathway around the Ural area) into two distinct categories, and track the movement trajectories of the high-pressure system centers (Methods and Supplementary Note 2). The first category exhibits the trajectory of the moving high-pressure systems that starts at the Ural and spreads toward eastern Asia (Fig. 2), which exactly aligns with the heatwaves propagation pathway. The consistency between the heatwaves propagation pathway and the movement of high-pressure systems holds true also for the WE, NA1 and NA2 pathways (Fig. 2). In contrast, the second category remains nearly stationary near the starting node. For additional validation, we perform a similar clustering analysis on the movement of surface temperature anomaly centers and obtain two comparable categories representing moving and stationary patterns (Supplementary Fig. S16). Their trajectories also align with the high-pressure systems and the heatwave propagation pathways, further confirming the strong association between heatwaves and high-pressure systems.

The two categories of movement of high-pressure systems exhibit either propagating or standing patterns, which can further explain the spatial evolution patterns of heatwaves therein. Figure 3 shows the composite temporal evolution of surface temperature anomalies within these two categories of patterns. For the first category, the moving high-pressure system explains the heatwave propagation (Fig. 2). High-temperature anomalies develop along the pathway within 7 days, stopping at the fifth or sixth path node (Fig. 3). Hereafter, we refer to this as the “propagation pattern" of the heatwaves. For the second category, characterised by the stationary high-pressure system, the heatwaves persist near the starting node for a longer duration (more than seven days), but its influence does not extend beyond three path nodes. This persistent heatwave could cause more severe local impacts, and we refer to this as the “standing pattern" hereafter. Similar propagating and standing patterns can be identified for the cases of the WE, NA1 and NA2 pathways (Fig. 3).

By counting the number of cases within propagating and standing patterns, respectively (Table 1), we conclude that propagating cases occupy the higher proportion of heatwaves across the four pathways. Supplementary Movies 1–4 show real-time examples of the temporal evolution of 500 hPa geopotential height and surface temperature anomalies for both propagating and standing patterns. In the propagating pattern, strong correlations are observed between the movement of high-pressure systems and the heatwaves (Supplementary Fig. S17), and the movement of the high-pressure center precedes the heatwave propagation in terms of location (see also Supplementary Note 3). A more detailed analysis will be provided in the next section.

Enhanced Rossby wave activity flux favoring heatwave propagation
Furthermore, we investigate the energy source that sustains the propagating and standing patterns by measuring the horizontal Takaya-Nakamura Rossby wave activity flux (TN flux)42. For the Asia pathway, Fig. 4 presents the stream function field and TN flux vector field at local temperature maxima during the heatwave period (Day = 0) and subsequently at Day = 2 and 4 under both propagation and standing scenarios. Similar results for the WE, NA1 and NA2 pathways are shown in Supplementary Figs. S18–S20. For the Asia pathway, at the moment of heatwave propagation, the TN flux indicates the flow of wave activity from the high-pressure system near the starting node toward the low-pressure system near the endpoint, representing the wave energy of RWPs. This wave activity flux causes the high-pressure system to move along the northwest-southeast direction, leading to the development of RWPs along the pathway. By Day = 2, the low-pressure system dominating the East Asian region significantly strengthens, further enhancing RWPs and ultimately intensifying the heatwaves downstream. The propagation of surface heatwaves is merely a manifestation of energy propagation in the overlaying atmosphere driven by RWPs. In contrast, for the case of the standing pattern, such energy propagation in the overlaying atmosphere is much weaker. The TN flux exists only at the center of the high-pressure system at Day = 0, and quickly decays thereafter.

Analyses on the TN flux for the Western European and North American regions lead to similar conclusions for attributing the heatwaves propagating patterns in those regions (Supplementary Figs. S18–S20). Propagating patterns correspond to the stronger TN flux, facilitating energy propagation along the pathway and continuously triggering the generation of surface heatwaves along the pathway. Conversely, the TN flux within the standing patterns is weak, and high-pressure systems near the starting node more easily result in locally persistent, strong heatwaves.

We have also examined additional physical processes. In the propagation pattern, we observe an anomalous increase in surface shortwave radiation, with the anomaly signal propagating along four preferred pathways (Supplementary Fig. S22). The high-pressure system can cause subsiding air that warms and compresses as it descends, resulting in higher surface temperatures. In addition, the high-pressure systems bring clear skies and reduced cloud cover, allowing more shortwave radiation to reach the surface1,2. In contrast, for the standing pattern, the shortwave radiation anomaly remains confined to the starting node.

We also note that negative anomalies in latent heat flux propagate along both the Asia and WE pathways (Supplementary Fig. S23), indicating weakened surface evaporation. This reduction in evaporation can decrease cloud formation, allowing more shortwave radiation to reach the land surface, which further promotes the development of heatwaves1,43. However, for the NA1 and NA2 pathways, weak positive anomalies in latent heat flux are observed at the starting nodes, but these anomalies do not propagate along the pathways, indicating that latent heat flux does not influence heatwave propagation in these two pathways. Furthermore, heatwaves along the WE pathway show a weak connection with sea surface temperature anomalies in the eastern North Atlantic (Supplementary Fig. S24). This connection has been noted in recent studies, where the SST anomalies and land heatwaves are linked by local RWPs44–46 (see also Supplementary Figs. S18 and S24). However, no significant SST anomaly is observed during the analysis of other preferred propagation pathways. These analyses further confirm that intensified Rossby wave activity and RWP movement are key factors in heatwave propagation.

Methods
Data
We use 2 m temperature, 500 hPa geopotential height, 300 hPa geopotential height, horizontal wind fields, surface solar radiation, surface latent heat flux and sea surface temperature from the ERA5 reanalysis dataset59 for our analysis. The spatial domain is 20°N-70°N, covering the entire longitudinal range, with a horizontal resolution of 1°×1° degree, and daily temporal resolution for the time period 1959–2023, for an extended summer season from April to September. For the time series within each spatial grid cell, we subtract its seasonal cycle and obtain the anomalies for subsequent analyses.

To define boreal summertime heatwaves, we adopt a percentile-based threshold method to identify heatwave events at each spatial grid cell1,2. For each cell, a heatwave event is defined as a period when the surface temperature anomaly exceeds the 95th percentile of the climatological distribution for that grid cell for more than three consecutive days, occurring between April and September1.

Complex network construction
We use nonlinear events synchronization (ES) measurements40 to assess the inter-regional connection of heatwave events. The basic definition of ES remains the same as in33,35,36, but we have made modifications in calculating network divergence. For a given pair of spatial grid cells i and j, we calculate the normalized number of events at i that can be uniquely associated with subsequent events at j, and vice versa: e μ i denotes the timing of the local temperature maximum during the μ-th heatwave event at grid cell i, and ev j correspondingly for grid cell j, where μ, v ∈ [1, l], l denotes the total number of events at each grid cell. The two events are counted as synchronized events if their dynamical delay d μ,v ij := e μ i - ev j does not exceed a threshold τ:

τ = min { d μ, μ-1, d μ, μ+1, dv, v-1, dv, v+1 } / 2 (1)

We set S μν ij = 1 if 0 < d μ,v ij ≤ τ and S μν ij = 0 otherwise, and calculate the normalized sum of S μν ij:

ES ij = 1/l ∑ μν S μν ij (2)

Therefore, ES ij counts the normalized number of heatwave events traveling from j to i, indicating the strength of the link that points from node j to node i in the complex network. We perform a significance test on the ES values33,35,36: From each node (18,000 in total), we generate surrogate event sequences by uniformly and randomly shuffling the blocks of events. Next, we compute the ES between all nodes with randomized event sequences and, based on the histogram of these values, we determine the 0.1-confidence level for significance test on the original ES values. The ES values that are not significant are set to 0.

Furthermore, to specifically examine synchronized heatwaves induced by localized Rossby waves, we restrict our analysis to a specific spatial range of network edge distances (i.e., 400–2000 km, see Supplementary Fig. S6), setting the ES values for node pairs outside this distance range to 0. The process results in a matrix that stores the modified ES values, serving as the adjacency matrix A of the complex network, thereby making the network both weighted and directed.

To spatially resolve the temporal sequence of extreme events, we introduce the network divergence ΔD, defined as the difference between the out-degree and in-degree for each node i:

ΔD i := Dout i - Din i := ∑ j=1 N A ji - ∑ j=1 N A ij (3)

where Dout i and Din i are the out-degree and in-degree of the node i. A positive ΔD indicates a source node in the network: the corresponding nodes’ heatwaves are statistically followed by heatwave events at other nodes. On the contrary, a negative ΔD indicates a sink node: these nodes’ heatwave events are preceded by heatwave events at other nodes.

Local searching algorithm
After selecting the source node for heatwave propagation, we employ a local search method to identify its most representative and preferred propagation pathways33,36. Since heatwaves typically do not propagate strictly in a single direction but exhibit some degree of undirected divergence. This method identifies the pathway with the highest probability of heatwave propagation. Specifically, starting from the source node as the center, we conduct a search within a radius of 800-1200 km surrounding it, considering all nodes within this range. We compare the out-degree (Dout i in Eq. 3) of each node within this circular ring around the starting node, and select the node with the highest out-degree density as the next pathway node. This process is repeated n times to determine n pathway nodes (Supplementary Fig. S31), where each path node serves as the center of the circular ring to identify the subsequent path node. In our results, we set n = 6 to ensure that all end nodes of the pathway are located in the convergence area of the heatwave.

Trajectory tracking algorithm
We employ a trajectory tracking method to determine the 10-day movement trajectory of the 500 hPa geopotential height the high-pressure center over a certain region following the occurrence of a heatwave (Supplementary Fig. S32). When a heatwave occurs in the selected region, we use the position of the maximum geopotential height within 1000 km of the heatwave center as the starting position of the high-pressure center (Day 0). Subsequently, using an Eulerian perspective, we track the movement of the high-pressure center for each subsequent day until the end of Day 10. Each heatwave event corresponds to a trajectory of the high-pressure center’s movement.

T-N wave-activity Flux
When investigating the intensity of the Rossby wave activity, we utilize the perturbation stream function at 300 hPa and the horizontal T-N flux42. The formula for the perturbation stream function ψ0 and the horizontal T-N Wave-Activity Flux W is as follows:

ψ0 = f * Z0 / g 300hpa (4)

W = 1 / (2|U|) * [ U ( (∂ψ0/∂λ)^2 - ψ0 * ∂^2ψ0/∂λ^2 ) / (a^2 * cos^2φ) + V * (∂ψ0/∂λ * ∂ψ0/∂φ - ψ0 * ∂^2ψ0/∂λ∂φ) / (a^2 * cosφ) , U * (∂ψ0/∂λ * ∂ψ0/∂φ - ψ0 * ∂^2ψ0/∂λ∂φ) / (a^2 * cosφ) + V * ( (∂ψ0/∂φ)^2 - ψ0 * ∂^2ψ0/∂φ^2 ) / a^2 ] (5)

In equations (4) and (5), U and V represent the climatological wind fields at 300 hPa, Z0 denotes the anomaly of geopotential height at 300 hPa, a stands for the radius of the Earth, λ and φ represent longitude and latitude, respectively.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript develops complex networks based approach for understanding patterns and propagation of terrestrial heat waves and attempts to relate the latter to Rossby waves. The study highlights the potential predictability of heatwaves based on the observed propagation patterns, suggesting that these pathways could inform early warning systems. This research offers insight into the mechanics of heatwave evolution, showing that Rossby wave Pathways and the Takaya-Nakamura flux play crucial roles in the propagation process. The codes provided can be reproduced without issue. While this is an interesting work, I have a few concerns which I believe must be addressed.\n1. Reproducibility is key and I would suggest that making the codes and data open access and publicly available should be a prerequisite for publication. Regarding codes, I would also request the additional codes that would allow for the reproduction of Fig. 2, Fig. 3, and Fig. 4 as part of the review process.\n2. I am not too convinced that the authors can make strong statements about causal relations between land heat wave patterns and Rossby waves based on the evidence provided. Are these chance co-occurrences or are there strong mechanistic interpretations? Could data-driven causal methods be used to develop more credibility into causation assertions? In the absence of that, should the findings be appropriately caveated? I would urge the authors to consider these questions and modify the manuscript appropriately, including in the discussion sections.\n3. I wonder the authors could make a stronger case by validating their findings on the relationship between heatwave propagation and Rossby wave pathways by examining a specific real-world situation, such as the South Asia heatwave that occurred this year.\n4. The authors could clarify the description of the ‘local Searching algorithm’ (Sec. 4.3) and ‘trajectory Tracking algorithm’, (Sec. 4.4) along with a flowchart (this could be in a supplement if needed) to facilitate easier understanding. If these algorithms are not introduced here for the first time, consider include relevant references.\n5. The authors may consider providing more information about the ‘surrogate hypothesis test’ in data (Sec. 4.1) or provide suitable references for readers’ understanding.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "Recommendation: May be suitable for publication after major revision.\nSummary:\nThe paper addresses an analysis of the preferred propagation pathways of boreal summer terrestrial heatwaves and its association with Rossby wave packets (RWPs). For this, the spatiotemporal evolution of heatwaves in a daily timescale is examined using a complex network algorithm (already used in other studies to identify propagation pathways), and verifying if it’s aligned with the movement of high-pressure systems. Thus, the manuscript represents an effort to understand the association between RWPs and heatwaves propagation. Due to the socioeconomic impacts involved in heatwaves occurrence, the article can positively contribute to potential predictability from the propagating patterns of these extreme weather events. Some suggestions are made, which may help to further improve the work.\n1. In Figure 1 is mentioned that the starting node of the Western Europe (WE) path is at the land, although the strongest positive divergence is over the ocean. However, the starting point from the land does not change the heatwave source, that is at the ocean. This could be a co-occurring terrestrial and marine heatwaves pathway? The most extreme marine heatwaves tend to occur in summer and recent investigations suggest potential interactions and common drivers between these two types of extreme events. For example, Pathmeswaran et al. (2022) found that the co-occurring terrestrial and marine heatwaves primarily occur between September and December and they are often associated with similar synoptic conditions. They also find a significant increase in the number of terrestrial heatwave days in the presence of an adjacent co-occurring marine heatwave along the coastal belt of Australia. In this sense, in Table 1 it is possible to note a smaller proportion of propagating heatwaves in WE in relation to the other pathways. Santos et al. (2024) also found co-occurring interplay between atmospheric heatwaves in southern Europe and marine heatwaves in the East Atlantic. It would be important to discuss this in the paper.\nReferences:\nPathmeswaran C, Sen Gupta A, Perkins-Kirkpatrick SE and Hart MA (2022) Exploring Potential Links Between Co-occurring Coastal Terrestrial and Marine Heatwaves in Australia. Front. Clim. 4:792730. doi: 10.3389/fclim.2022.792730\nSantos, R., Russo, A. & Gouveia, C.M. Co-occurrence of marine and atmospheric heatwaves with drought conditions and fire activity in the Mediterranean region. Sci Rep 14, 19233 (2024). https://doi.org/10.1038/s41598-024-69691-y\n2.L.142-143: “many times over the past few decades” -> How many times during the period analyzed (1959-2023)? There are any changes over time in the heatwave’s occurrence in terms of frequency? There is any propagation pathway with greater occurrence in relation to the others? It would be very interesting if the authors could briefly insert these statistics in the text.\n3.L.212-213: “high consistency” -> this is more a visual verification. The authors could insert a figure with the spatial lag correlation between the positions of the high-pressure systems and the heatwaves. This could corroborate quantitatively the statement.\n4.L.213-215: “the development of RWPs induces the heatwave propagation” - Does this mean that the heatwave propagation is systematically associated with a precursor RWP? Fragkoulidis et al. (2018) found that heatwaves of similar strength can be associated with highly distinctive daily evolutions of the upper-tropospheric circulation. Thus, mechanisms linking RWPs and temperature extremes are case-dependent with an important contribution of other physical mechanisms (e.g. radiative transfer, SST, subsidence, soil moisture, etc.). Thus, it is important to discuss this more deeply in the paper, as the authors did not mention the role of other mechanisms, just the movement of high-pressure systems.\nReference:\nG. Fragkoulidis, V. Wirth, P. Bossmann, A. H. Fink. Linking Northern Hemisphere temperature extremes to Rossby wave packets. Quarterly Royal of Meteorological Society, v. 144 (711), 2018.\n5.L260-261: Any possible explanation about these differences in RHO between the pathways? Why greater for NA2?\n6.L.354: “source node” – sink node? In Formula (3) and Figure 1, positive (negative) divergence represents the sources (sinks).\n7.L.362: “Search within a radius of 800-1200 km” – This variable circular ring is moving from the starting point or each node is considered as the starting point to find the next node in subsequent analyses? In Fig. S18, in legend, it is mentioned a circle of 1400-1800 km from the starting point, reaching the third node. However, if the distance between the nodes is approximately 1000 km, the most external circle would be more than 2000 km away from the original starting point. Please clarify.\n8.L.375: “using a Eulerian perspective” – Why not use a Lagrangian tracking of the high-pressure centers or a quasi-Lagrangian perspective?\n9.Figure 1, legend: heavewaves -> heatwaves", "reviewer_4": "Please see the attached file.\n[Editorial Note: the attachment has been appended to the very end of this file]"}, "author_response": "Please find our point-by-point responses to the comments raised by the referee in blue font below.\nReviewer 1\nThe manuscript develops complex networks based approach for understanding patterns and\npropagation of terrestrial heat waves and attempts to relate the latter to Rossby waves. The\nstudy highlights the potential predictability of heatwaves based on the observed propagation\npatterns, suggesting that these pathways could inform early warning systems. This research\noffers insight into the mechanics of heatwave evolution, showing that Rossby wave Pathways\nand the Takaya-Nakamura flux play crucial roles in the propagation process. The codes\nprovided can be reproduced without issue. While this is an interesting work, I have a few\nconcerns which I believe must be addressed.\nThank you very much for the thorough review of our paper and for providing us with very\nhelpful suggestions. We also appreciate the recognition of the interest and value of our work.\nWe have carefully taken your comments into account and have made revisions to address\neach of the points you highlighted. Please find detailed-point-by-point responses to your\nconcerns, as well as references to how we revised our manuscript, in the following.\n1. Reproducibility is key and I would suggest that making the codes and data open access and\npublicly available should be a prerequisite for publication. Regarding codes, I would also\nrequest the additional codes that would allow for the reproduction of Fig. 2, Fig. 3, and Fig. 4\nas part of the review process.\nThank you for your suggestion. We are now sharing the code and raw data used to compute\nand generate Figs. 1, 2, 3, 4 and 5 in the Code Ocean capsule. We have updated the Code\nAvailability section to indicate that our codes will be published on Github/Zenodo for public\naccess following the publication of the manuscript.\n2. I am not too convinced that the authors can make strong statements about causal relations\nbetween land heat wave patterns and Rossby waves based on the evidence provided. Are\nthese chance co-occurrences or are there strong mechanistic interpretations? Could\ndata-driven causal methods be used to develop more credibility into causation assertions? In\nthe absence of that, should the findings be appropriately caveated? I would urge the authors\nto consider these questions and modify the manuscript appropriately, including in the\ndiscussion sections.\nThank you, we agree. Existing studies have interpreted that localized Rossby Wave Packets\n(RWPs) have significant causal impact on heatwaves, playing an important role in the\nunderlying physical mechanisms (Refs. 1-4): during summertime, RWPs are accompanied\nwith local atmospheric high-pressure systems or atmospheric blocking highs, and such\nhigh-pressure systems influence heatwaves by causing subsiding air that warms and\ncompresses as it descends, resulting in higher surface temperatures. Additionally, the\nhigh-pressure systems bring clear skies and reduced cloud cover, allowing more shortwave\nradiation to reach the surface. Existing statistical studies (Refs. 4-5) have also found a close\nconnection between RWPs and heatwaves in both the Northern and Southern Hemispheres.\nIn our revised manuscript, following your suggestion, we have applied the Granger causality\ntest (Refs. 6) to investigate the causal relationship between heatwaves (HWs) and Rossby\nwave packets (RWPs) along the propagation pathway. Taking the starting node as an\nexample, we first compare the time series of the surface air temperature anomaly (SATa),\ngeopotential height anomaly at 500 hPa (z500) and the absolute amplitude of TN flux for\nRossby wave activity (WAF amplitude) during heatwaves (Fig. S21). Both local maxima of\nthe WAF amplitude and z500 precede that of the SATa, and this holds for all four preferred\npathways. Similar conclusions can be drawn from the analysis on time-lagged correlations\nbetween z500 and SATa at different path nodes (Fig. S17), where z500 precedes SATa by 1\nday during the period of propagating heatwaves. This suggests that the RWP precedes HW\nduring the period of propagating heatwaves (Refs. 7-8). We further conduct the aforesaid\nGranger causality test on the paired time series of z500 and SATa, as well as WAF amplitude\nand SATa. The test shows that the Granger causal influence from z500 to SATa is significant,\nwhile the Granger causal influence from SATa to z500 is not significant. Similarly, it is also\nfound that WAF amplitude unidirectionally Granger-causes SATa. Similar causal inference\nresults are found during our analysis on other path nodes.\nAdapted from Supplementary Fig. S21: Comparison of composite time series during propagating\nheatwaves for the geopotential height anomaly at 500 hPa (z500), the absolute amplitude of TN flux\nfor Rossby wave activity (WAF amplitude), and surface air temperature anomaly (SATa) at the\nstarting nodes of the Asia, WE, NA1, and NA2 heatwave pathways, respectively. Time = 0\ncorresponds to the timing of the local temperature maximum during the heatwave at the starting node.\nTo facilitate comparison, the time series of each variable were normalized to have a mean of 0 and a\nstandard deviation of 1.\nAdapted from Supplementary Fig. S17: Time-lagged Pearson correlation between temperature and\nz500 anomalies at individual path nodes during periods of propagating heatwaves, with results\npresented for the Asia, WE, NA1, and NA2 pathways, respectively. Positive time lag indicates that the\nz500 anomaly is lagged relative to the temperature anomaly. Only cases of propagating heatwaves are\nincluded in this analysis. The z500 anomaly is highly correlated (with Pearson correlation higher than\n0.8) with the temperature anomaly at time lag = 0, and this holds for all path nodes. Specifically, at\ntime lag = -1, where the z500 anomaly precedes the temperature anomaly by 1 day, the first three path\nnodes exhibit a maximum correlation (higher than 0.94) between the temperature and z500 anomalies,\nindicating that the z500 anomaly leads the temperature anomaly.\nIn the revised manuscript, we have included the Granger causality analyses of Rossby waves\nand heatwaves in Supplementary Note 3. Additionally, we have incorporated an analysis of\nother physical processes, such as surface shortwave radiation and surface latent heat flux\n(lines 243-265), and found that, compared to these other processes, RWPs are the key factor\ndriving heatwave propagation patterns.\n3. I wonder the authors could make a stronger case by validating their findings on the\nrelationship between heatwave propagation and Rossby wave pathways by examining a\nspecific real-world situation, such as the South Asia heatwave that occurred this year.\nThank you for your suggestion. We added a case for the propagating heatwave occurring in\nSouthern and Eastern Asia in the year of 2022 (Supplementary Fig. S1). On June 10th, 2022,\na heatwave was detected in the Urals and Central Asia, with high temperatures gradually\nspreading to Southern and Eastern Asia over the following 10 days. During the development,\nthe location of the positive center of 500 hPa geopotential height anomalies keeps\noverlapping with that of the high temperature center. The spatial domain of our study covers\nthe mid-latitudes of the Northern Hemisphere, ranging from 20°N to 70°N, and does not\nextend to the subtropical region of South Asia like India and Pakistan. Additionally, recent\nstudies (Refs. 1, 9 and 10) suggest that the spatial spread of heatwaves in the subtropical\nregion of South Asia is related to the late-monsoon effect and the strong local northerly wind\nanomalies. In the revised manuscript, we have added the case of the propagating heatwave\noccurring at the south and middle of East Asia in the year of 2022 in Supplementary Fig. S1.\nAdapted from Supplementary Fig. S1: On June 10, 2022, the distribution of surface temperature\nand 500 hPa geopotential height anomalies (contours) across Asia. Days 0 to 10 represent the\nevolution of surface temperature anomalies from the onset of the heatwave to 10 days thereafter,\nillustrating a complete process of heatwave propagation from the Urals to the south and middle of\nEast Asia.\n4. The authors could clarify the description of the ‘local Searching algorithm’ (Sec. 4.3) and\n‘trajectory Tracking algorithm’, (Sec. 4.4) along with a flowchart (this could be in a\nsupplement if needed) to facilitate easier understanding. If these algorithms are not\nintroduced here for the first time, consider include relevant references.\nThank you for your suggestion. In supplementary Figs. S31 and S32 of the revised\nmanuscript, we have implemented the flowcharts for ‘Local Searching algorithm’ and\n‘Trajectory Tracking algorithm’, respectively. The method of ‘Local Searching algorithm’\nwas initially developed by Boers et al. 2014 (Ref. 11) and Li et al., 2024 (Ref. 12), and we\nhave added the corresponding references in Methods.\n5. The authors may consider providing more information about the ‘surrogate hypothesis test’\nin data (Sec. 4.1) or provide suitable references for readers’ understanding.\nThank you for your suggestion. In the revised manuscript, we have added a more detailed\nexplanation for the surrogate hypothesis test in Supplementary Note 4, including the precise\ndefinition of 𝑅 and the proper revision on Supplementary Fig. S26, as below:\n𝐻𝑂\n“Supplementary Note 4: Calculation and surrogate hypothesis test for 𝑅\n𝐻𝑂\nWe define an index 𝑅 to assess the impact of each preferred propagation pathway on\n𝐻𝑂\nheatwave occurrences downstream. Taking the Asia pathway as an example, 𝑅 represents\n𝐻𝑂\nthe probability of heatwave occurrences at the k-th path node within 7 days following a\nheatwave at the starting node, where k ∈ [1,6] represents the order of the six representative\nnodes along the pathway (Fig. 1).\n𝑘 𝑁\n𝑅 = 1|𝑘 (5)\n𝐻𝑂 𝑁\n𝑘\nwhere 𝑁 is the total number of heatwave events at the k-th path node, and 𝑁 denotes the\n𝑘 1|𝑘\nnumber of cases that a heatwave occurs at the k-th path node within 7 days following a\n𝑘\nheatwave at the starting node. Therefore, for each path node, 𝑅 is used to estimate the\n𝐻𝑂\nimpact of the preferred propagation pathway on heatwave occurrence at that node (Fig. 5a).\nTo verify the significance of the preferred propagation pathway's impact on heatwave\noccurrences at its downstream nodes, we use surrogates to compute 𝑅 and test the null\n𝐻𝑂\nhypothesis. Specifically, when a heatwave occurs at the starting node of the Asia pathway, we\nestimate the probability of a heatwave occurring at different nodes within the following 7\ndays (i.e., 𝑅 ), where each of these nodes is at the same distance from the starting node.\n𝐻𝑂\nThus, these nodes are distributed in a circle centered on the starting node (Supplementary\nFig. S26): one of these nodes lies along the Asia pathway, with the probability of heatwave\noccurrence denoted as 𝑅 (Path); while the remaining nodes, which lie outside the Asia\n𝐻𝑂\npathway, serve as surrogates, with their probability of heatwave occurrence denoted as 𝑅\n𝐻𝑂\n(Non-path). Hence, the null hypothesis states that 𝑅 (Path) will not differ significantly from\n𝐻𝑂\n𝑅 (Non-path).\n𝐻𝑂\nWe calculate 𝑅 (Non-path) values for the surrogates, and obtain the mean, 99th percentile\n𝐻𝑂\nand 1st percentile of them. The results show that 𝑅 (Path) is beyond the 99% confidence\n𝐻𝑂\ninterval for the surrogates (as in Fig. 5), confirming the significant impact of preferred\npropagation pathway on the heatwave occurrence at its downstream nodes. Similar\nsignificance tests using surrogates were also performed for the West Europe, North America\n1, and North America 2 pathways (not shown here), where the impacts of these pathways\nwere found to be significant as well. ”\nAdapted from Supplementary Fig. S26. Illustration of surrogate generation for the 𝑅 significance\n𝐻𝑂\ntest. Taking the 3rd path node of the Asia pathway as an example, we consider all grid cells at a\nsimilar distance from the starting node, comparable to the distance between the starting node and the\n3rd path node (around 1600 km). These grid cells, represented by the red dots within black rings with\nradii of 1400 km and 1800 km, are not part of the Asia pathway but serve as surrogates for the 3rd\npath node in subsequent significance tests. The brown arrow indicates that the 𝑅 calculation is\n𝐻𝑂\nperformed on the 3rd path node of the Asia pathway, and the result is denoted as 𝑅 (Path). The gray\n𝐻𝑂\narrow, instead, indicates that the 𝑅 calculation is performed on the surrogates, denoted as 𝑅\n𝐻𝑂 𝐻𝑂\n(Non-path). See Supplementary Note 4 for the details of calculating 𝑅 .\n𝐻𝑂\nReviewer #1 (Remarks on code availability):\nWe have successfully reproduced all the provided codes, and they work flawlessly. We kindly\nrequest the authors to include three additional codes that would allow for the reproduction of\nFig. 2, Fig. 3, and Fig. 4. Furthermore, the codes should be made open and publicly\naccessible with appropriate links.\nThank you. We are now sharing all code and raw data used to compute and generate Figs. 1,\n2, 3, 4 and 5 in the Code Ocean capsule. We have updated the Code Availability section to\nindicate that our codes will be published on Github/Zenodo for public access following the\npublication of the manuscript.\nReference\nRef. 1: Domeisen, D.I., Eltahir, E.A., Fischer, E.M., Knutti, R., Perkins-Kirkpatrick, S.E., Schar, C.,\nSeneviratne, S.I., Weisheimer, A., Wernli, H.: Prediction and projection of heatwaves. Nature Reviews\nEarth & Environment 4(1), 36–50410 (2023)\nRef. 2: Barriopedro, D., Garcia-Herrera, R., Ordonez, C., Miralles, D., Salcedo-Sanz, S.: Heat waves:\nPhysical understanding and scientific challenges. Reviews of Geophysics 61(2), 2022–000780 (2023)\nRef. 3: Ma, Q., Franzke, C.L.: The role of transient eddies and diabatic heating in the maintenance of\neuropean heat waves: a nonlinear quasi-stationary wave perspective. Climate Dynamics 56(9-10),\n3003–3004 (2021)\nRef. 4: Risbey, J.S., O’Kane, T.J., Monselesan, D.P., Franzke, C.L., Horenko, I.: On the dynamics of\naustral heat waves. Journal of Geophysical Research: Atmospheres 123(1), 38–57 (2018)\nRef. 5: Fragkoulidis, G., Wirth, V., Bossmann, P., Fink, A.: Linking northern hemisphere temperature\nextremes to Rossby wave packets. Quarterly Journal of the Royal Meteorological Society 144(711),\n553–566 (2018)\nRef. 6: Granger, C.W.: Investigating causal relations by econometric models and cross-spectral\nmethods. Econometrica: journal of the Econometric Society, 424–438 (1969)\nRef. 7: McGraw, M.C., Barnes, E.A.: Memory matters: A case for granger causality in\nclimate variability studies. Journal of climate 31(8), 3289–3300 (2018)\nRef. 8: Huang, Y., Franzke, C.L., Yuan, N., Fu, Z.: Systematic identification of causal relations in\nhigh-dimensional chaotic systems: application to stratosphere-troposphere coupling. Climate\nDynamics 55, 2469–2481 (2020)\nRef. 9: Röthlisberger, M., Sprenger, M., Flaounas, E., Beyerle, U., Wernli, H.: The substructure of\nextremely hot summers in the Northern Hemisphere. Weather and Climate Dynamics, 1(1), 45-62\n(2020)\nRef. 10: Dar, J.A., Apurv, T.: Spatiotemporal characteristics and physical drivers of heatwaves in\nindia. Geophysical Research Letters 51(15), 2024–109785 (2024)\nRef. 11: Boers, N., Bookhagen, B., Barbosa, H.M., Marwan, N., Kurths, J., Marengo, J.A.: Prediction\nof extreme floods in the eastern central andes based on a complex networks approach. Nature\nCommunications 5, 5199 (2014)\nRef. 12: Li, K., Huang, Y., Liu, K., Wang, M., Cai, F., Zhang, J., Boers, N.: Key propagation pathways\nof extreme precipitation events revealed by climate networks. npj Climate and Atmospheric Science\n7(1), 165 (2024)\nReviewer 2\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This\nis part of the Nature Communications initiative to facilitate training in peer review and to\nprovide appropriate recognition for Early Career Researchers who co-review manuscripts.\nThank you very much for the very thoughtful co-review of our manuscript and for providing\nus with very helpful suggestions. We have carefully revised our manuscript according to your\ncomments. Please find our detailed-point-by-point responses to your comments and\nsuggestions, as well as references to how we revised our manuscript accordingly.\nReviewer 3\nThe paper addresses an analysis of the preferred propagation pathways of boreal summer\nterrestrial heatwaves and its association with Rossby wave packets (RWPs). For this, the\nspatiotemporal evolution of heatwaves in a daily timescale is examined using a complex\nnetwork algorithm (already used in other studies to identify propagation pathways), and\nverifying if it’s aligned with the movement of high-pressure systems. Thus, the manuscript\nrepresents an effort to understand the association between RWPs and heatwaves propagation.\nDue to the socioeconomic impacts involved in heatwaves occurrence, the article can\npositively contribute to potential predictability from the propagating patterns of these extreme\nweather events. Some suggestions are made, which may help to further improve the work.\nThank you very much for the accurate summary of our study and the recognition of the interest\nand value of our work. Your suggestions have been very valuable in improving our work. Please\nfind point-by-point responses to your comments, as well as references to how we revised our\nmanuscript, in the following.\n1. In Figure 1 is mentioned that the starting node of the Western Europe (WE) path is at the\nland, although the strongest positive divergence is over the ocean. However, the starting point\nfrom the land does not change the heatwave source, that is at the ocean. This could be a\nco-occurring terrestrial and marine heatwaves pathway? The most extreme marine heatwaves\ntend to occur in summer and recent investigations suggest potential interactions and common\ndrivers between these two types of extreme events. For example, Pathmeswaran et al. (2022)\nfound that the co-occurring terrestrial and marine heatwaves primarily occur between\nSeptember and December and they are often associated with similar synoptic conditions.\nThey also find a significant increase in the number of terrestrial heatwave days in the\npresence of an adjacent co-occurring marine heatwave along the coastal belt of Australia.\nIn this sense, in Table 1 it is possible to note a smaller proportion of propagating heatwaves in\nWE in relation to the other pathways. Santos et al. (2024) also found co-occurring interplay\nbetween atmospheric heatwaves in southern Europe and marine heatwaves in the East\nAtlantic. It would be important to discuss this in the paper.\nReferences:\nPathmeswaran C, Sen Gupta A, Perkins-Kirkpatrick SE and Hart MA (2022) Exploring\nPotential Links Between Co-occurring Coastal Terrestrial and Marine Heatwaves in\nAustralia. Front. Clim. 4:792730. doi: 10.3389/fclim.2022.792730\nSantos, R., Russo, A. & Gouveia, C.M. Co-occurrence of marine and atmospheric heatwaves\nwith drought conditions and fire activity in the Mediterranean region. Sci Rep 14, 19233\n(2024). https://doi.org/10.1038/s41598-024-69691-y\nThank you for your suggestion. We presented the composited sea surface temperature (SST)\nanomalies corresponding to heatwaves along the Western Europe pathway and found the\nco-existing SST anomalies in the North Atlantic Ocean and heatwaves in Western Europe\n(Supplementary Fig. S24), although the composited SST anomalies are relatively weak\n(around 0.5°C). In the propagating pattern, the SST anomalies also show a slight eastward\nmovement from Day = -4 to Day = 6 (Supplementary Fig. S24a). Additionally, the patterns of\nSST anomalies differ between the standing and propagating heatwave patterns. A recent\nstudy (Ref. 1) also found that European heatwaves could be linked to North Atlantic SST\nanomalies, with high-pressure systems and local Rossby waves playing key roles in bridging\nthis connection. Our focus is on the propagating pathway of the heatwaves, and the SST\nanomalies observed here need further investigation in future work.\nAdapted from Supplementary Fig. S24: Composite sea surface temperature anomaly during (a) the\npropagating heatwaves and (b) the standing heatwaves on the Western Europe pathway. Only values\nsignificant at the 5% significance level are displayed.\n.\nIn lines 258-265 of the revised manuscript, we have properly included the discussion about\nthe North Atlantic SST anomalies, as below:\n“Furthermore, heatwaves along the WE pathway show a weak connection with sea surface\ntemperature anomalies in the eastern North Atlantic (Supplementary Fig. S24). This\nconnection has been noted in recent studies, where SST anomalies and land heatwaves are\nlinked by local RWPs [42–44] (see also Supplementary Figs. S18 and S24). However, no\nsignificant SST anomaly is observed during the analysis of other preferred propagation\npathways. These analyses further confirm that intensified Rossby wave activity and RWP\nmovement are key factors in heatwave propagation.”\n2. L.142-143: “many times over the past few decades” -> How many times during the period\nanalyzed (1959-2023)? There are any changes over time in the heatwave’s occurrence in\nterms of frequency? There is any propagation pathway with greater occurrence in relation to\nthe others? It would be very interesting if the authors could briefly insert these statistics in the\ntext.\nThank you for your suggestion. Between 1959 and 2023, the propagating heatwaves have\noccurred 64 times at the Northern Asia source node, 76 times at the Western Europe source\nnode, and 83 and 79 times at the North American source nodes, respectively. We found\nsignificantly increasing trends of the heatwave occurrence frequency in the Northern\nHemisphere and also the regions near preferred pathways (Supplementary Fig. S25c). We\nalso inspected the frequency of heatwave occurrences in the mid-latitudes of the Northern\nHemisphere, and the proportion of the cases of propagating heatwaves. The frequency of\nheatwave occurrence along these four pathways does not exceed the overall distribution\nacross northern-hemisphere mid-latitudes (Supplementary Fig. S25a), but the proportion of\npropagating cases is notably higher near these pathways (Supplementary Fig. S25b).\nIn the revised manuscript, we have revised the descriptions in lines 142-146, and\nincorporated the analyses on frequency of heatwave occurrences in lines 275-282, and\nSupplementary Fig. 25, as follows:\n“Inspections of all heatwaves over the source regions identified in Fig. 1 reveal that, between\n1959 and 2023, such propagating heatwaves (Supplementary Note 1) have occurred 64 times\nat the Northern Asia source node, 76 times at the Western Europe source node, and 83 and 79\ntimes at the North American source nodes, respectively. This encourages us to conduct\nfurther quantitative analyses as follows” (lines 142-146),\nand “Additionally, we note that while the frequency of heatwave occurrence along the four\npreferred pathways does not exceed the overall distribution across northern-hemisphere\nmid-latitudes (Supplementary Fig. S25a), the proportion of propagating cases is notably\nhigher near these pathways (Supplementary Fig. S25b), and heatwave occurrences in these\nregions have shown increasing trends in recent decades (Supplementary Fig. S25c) [24, 47].\nThis suggests that the four propagation pathways are located in regions with more active\nheatwave propagation and a rising frequency of heatwave occurrences.” (lines 275-282).\nAdapted from Supplementary Fig. S25: Statistics of heatwave occurrences (1959-2023) in the\nNorthern Hemisphere from 20°N to 70°N, and comparison with the preferred propagation pathways\nof heatwaves revealed in our study (yellow lines). (a) Frequency of heatwave occurrence; (b)\nproportion of the cases of propagating heatwaves (Supplementary Note 1); and (c) trend of heatwave\noccurrence frequency.\n3. L.212-213: “high consistency” -> this is more a visual verification. The authors could\ninsert a figure with the spatial lag correlation between the positions of the high-pressure\nsystems and the heatwaves. This could corroborate quantitatively the statement.\nThank you for your suggestion. To visualize the consistency between the movement of\nheatwaves and high-pressure systems along the preferred pathway, we examine the\ntime-lagged Pearson correlation between temperature and Z500 anomalies at individual path\nnodes during propagating heatwave periods, as shown in Fig. S17. Taking the Asia pathway\nas an example, the Z500 anomaly is highly correlated (Pearson correlation > 0.8) with the\ntemperature anomaly at time lag = 0, and this holds for all path nodes. Specifically, at time\nlag = -1, where the Z500 anomaly precedes the temperature anomaly by 1 day, the first three\npath nodes exhibit a maximum correlation (Pearson correlation > 0.94) between the\ntemperature and Z500 anomalies, indicating that the Z500 anomaly leads the temperature\nanomaly. Similar conclusions can be drawn from examination on the WE, NA1 and NA2\nheatwave pathways. In the revised manuscript, we have included the time lagged Pearson\ncorrelations at individual path nodes into Supplementary Fig. S17.\nAdapted from Supplementary Fig. S17: Time-lagged Pearson correlation between temperature and\nZ500 anomalies at individual path nodes during periods of propagating heatwaves, with results\npresented for the Asia, WE, NA1, and NA2 heatwave pathways, respectively. Positive time lag\nindicates that the Z500 anomaly is lagged relative to the temperature anomaly. Only cases of\npropagating heatwaves are included in this analysis. The Z500 anomaly is highly correlated (with\nPearson correlation higher than 0.8) with the temperature anomaly at time lag = 0, and this holds for\nall path nodes. Specifically, at time lag = -1, where the Z500 anomaly precedes the temperature\nanomaly by 1 day, the first three path nodes exhibit a maximum correlation (higher than 0.94)\nbetween the temperature and Z500 anomalies, indicating that the Z500 anomaly leads the temperature\nanomaly.\n4. L.213-215: “the development of RWPs induces the heatwave propagation” - Does this\nmean that the heatwave propagation is systematically associated with a precursor RWP?\nFragkoulidis et al. (2018) found that heatwaves of similar strength can be associated with\nhighly distinctive daily evolutions of the upper-tropospheric circulation. Thus, mechanisms\nlinking RWPs and temperature extremes are case-dependent with an important contribution\nof other physical mechanisms (e.g. radiative transfer, SST, subsidence, soil moisture, etc.).\nThus, it is important to discuss this more deeply in the paper, as the authors did not mention\nthe role of other mechanisms, just the movement of high-pressure systems.\nReference:\nG. Fragkoulidis, V. Wirth, P. Bossmann, A. H. Fink. Linking Northern Hemisphere\ntemperature extremes to Rossby wave packets. Quarterly Royal of Meteorological Society, v.\n144 (711), 2018.\nThank you for your helpful suggestion. In our manuscript, we examine the Takaya-Nakamura\nRossby wave activity flux in the upper troposphere (at the 300 hPa pressure level), which\ndistinguishes between the propagation and standing patterns of heatwaves. Additionally, we\nconducted analyses on additional physical processes, as follows:\n(1) In the propagation pattern, we observe an anomalous increase in surface shortwave\nradiation, with the anomaly signal propagating along four preferred pathways\n(Supplementary Fig. S22). The high-pressure systems can cause subsiding air that warms and\ncompresses as it descends, resulting in higher surface temperatures. Additionally, the\nhigh-pressure systems bring clear skies and reduced cloud cover, allowing more shortwave\nradiation to reach the surface. In contrast, for the standing pattern, the shortwave radiation\nanomaly remains confined to the starting node.\nAdapted from Supplementary Fig. S22: Spatiotemporal evolutions of surface solar radiation\nanomaly during propagating and standing heatwaves. Composite surface solar radiation anomaly as a\nfunction of time and the spatial nodes on the detected propagation pathways. Horizontal axis denotes\nthe six representative nodes on the four detected propagation pathways (see Fig. 1). Time = 0\ncorresponds to the timing of the local temperature maximum during the heatwave at the starting node.\n(2) We also note that negative anomalies in latent heat flux propagate along both the Asia and\nWE pathways (Supplementary Fig. S23), indicating weakened surface evaporation. This\nreduction in evaporation can decrease cloud formation, allowing more shortwave radiation to\nreach the land surface, which further promotes heatwave development. However, for the NA1\nand NA2 pathways, weak positive anomalies in latent heat flux are observed at the starting\nnodes, but these anomalies do not propagate along the pathways, indicating that latent heat\nflux does not influence heatwave propagation in these two pathways.\nAdapted from Supplementary Fig. S23: Spatiotemporal evolutions of surface latent heat flux\nanomalies during propagating and standing heatwaves. Composite surface latent heat flux anomaly as\na function of time and the spatial nodes on the four detected propagation pathways. The horizontal\naxis denotes the six representative nodes on the detected propagation pathways (see Fig. 1). Time = 0\ncorresponds to the timing of the local temperature maximum during the heatwave at the starting node.\n(3) Furthermore, heatwaves along the WE pathway show a weak connection with sea surface\ntemperature anomalies in the eastern North Atlantic (Supplementary Fig. S24). This\nconnection has been noted in recent studies (Refs. 5-6), where SST anomalies and land\nheatwaves are linked by local RWPs (see also Supplementary Figs. S18 and S24). However,\nno significant SST anomaly is observed during the analysis of other preferred propagation\npathways.\nThese analyses further confirm that intensified Rossby wave activity and RWP movement are\nkey factors in heatwave propagation of the four preferred pathways.\nAdapted from Supplementary Fig. S24: Composite sea surface temperature anomaly during (a) the\npropagating heatwaves and (b) the standing heatwaves on the Western Europe pathway. Only values\nsignificant at the 5% significance level are displayed.\nIn the revised manuscript, we have incorporated the analyses on the additional physical\nprocesses in lines 243-265, as follows:\n“We have also examined additional physical processes. In the propagation pattern, we\nobserve an anomalous increase in surface shortwave radiation, with the anomaly signal\npropagating along four preferred pathways (Supplementary Fig. S22). The high-pressure\nsystem can cause subsiding air that warms and compresses as it descends, resulting in higher\nsurface temperatures. In addition, the high-pressure systems bring clear skies and reduced\ncloud cover, allowing more shortwave radiation to reach the surface [1, 2]. In contrast, for\nthe standing pattern, the shortwave radiation anomaly remains confined to the starting node.\nWe also note that negative anomalies in latent heat flux propagate along both the Asia and\nWE pathways (Supplementary Fig. S23), indicating weakened surface evaporation. This\nreduction in evaporation can decrease cloud formation, allowing more shortwave radiation\nto reach the land surface, which further promotes the development of heatwaves [1, 41].\nHowever, for the NA1 and NA2 pathways, weak positive anomalies in latent heat flux are\nobserved at the starting nodes, but these anomalies do not propagate along the pathways,\nindicating that latent heat flux does not influence heatwave propagation in these two\npathways. Furthermore, heatwaves along the WE pathway show a weak connection with sea\nsurface temperature anomalies in the eastern North Atlantic (Supplementary Fig. S24). This\nconnection has been noted in recent studies, where SST anomalies and land heatwaves are\nlinked by local RWPs [42–44] (see also Supplementary Figs. S18 and S24). However, no\nsignificant SST anomaly is observed during the analysis of other preferred propagation\npathways. These analyses further confirm that intensified Rossby wave activity and RWP\nmovement are key factors in heatwave propagation.”\n5. L260-261: Any possible explanation about these differences in RHO between the\npathways? Why greater for NA2?\n𝑅 represents the probability of heatwave occurrences at a path node within seven days\n𝐻𝑂\nfollowing the onset of a heatwave event at the starting node. The 𝑅 for the second path\n𝐻𝑂\nnode in the NA2 pathway (around 80%) is a bit higher than in other pathways (around 70%).\nHowever, from the third to the sixth path nodes in the NA2 pathway, the 𝑅 decays more\n𝐻𝑂\nrapidly, falling below the values observed in the other pathways. This may be due to the\nslower propagation of heatwaves in the NA2 pathway (Fig. 3), allowing the heatwaves to\npersist longer around the second path node compared to the other pathways. This prolonged\npresence can facilitate the increase of the 𝑅 values at the second path node. Additionally,\n𝐻𝑂\nthe 𝑅 values shown in Fig. 5a include all heatwave cases, which also account for\n𝐻𝑂\nheatwaves from the standing pattern category. In this category, heatwaves at the second path\nnode of the NA2 pathway can persist for a longer duration than those in other pathways\n(bottom right of Fig. 3). As a result, more heatwave events at the second path node of the\nNA2 pathway are included in the 𝑅 than in other pathways. Heatwave occurrences can also\n𝐻𝑂\nbe influenced by the local land surface conditions in different continents. The comparison\nacross different continents is complex and warrants further investigation in future studies.\nAdapted from Fig. 3: Spatiotemporal evolutions of propagating and standing heatwaves. Composite\nsurface temperature anomalies as a function of time and the spatial nodes on the detected propagation\npathways. The horizontal axis denotes the six representative nodes on the detected propagation\npathways (see Fig. 1). Time = 0 corresponds to the timing of the local temperature maximum during\nthe heatwave at the starting node.\n6. L.354: “source node” – sink node? In Formula (3) and Figure 1, positive (negative)\ndivergence represents the sources (sinks).\nSorry, this was a typo, and we have modified it in line 381 of the revised manuscript. Thank\nyou for the reminder.\n7. L.362: “Search within a radius of 800-1200 km” – This variable circular ring is moving\nfrom the starting point or each node is considered as the starting point to find the next node in\nsubsequent analyses? In Fig. S18, in legend, it is mentioned a circle of 1400-1800 km from\nthe starting point, reaching the third node. However, if the distance between the nodes is\napproximately 1000 km, the most external circle would be more than 2000 km away from the\noriginal starting point. Please clarify.\nThank you for your suggestion. In the method for identifying the preferred propagation\npathway, each path node is treated as the center of the circular ring, then to identify the next\nnode in subsequent analyses. This method was initially developed by Boers et al. 2014 (Ref.\n2) and Li et al., 2024 (Ref. 3), and we have properly added the corresponding references in\nMethods.\nFor clarity, in the revised manuscript, we have modified the method introduction in lines\n394-398 and added a flowchart in Supplementary Fig. S31, illustrating this method for\nidentifying the preferred propagation pathway, as shown below:\n“This process is repeated n times to determine n pathway nodes (Supplementary Fig. S31),\nwhere each path node serves as the center of the circular ring to identify the subsequent path\nnode. In our results, we set n = 6 to ensure that all end nodes of the pathway are located in\nthe convergence area of the heatwave.”\nAdapted from Supplementary Fig. S31: Flowchart diagram illustrating the local searching\nalgorithm for identifying preferred propagation pathway of heatwaves in the complex network.\nFig. S18 in the original manuscript was intended to explain the surrogate generation and\nsignificance test of 𝑅 , rather than the method for identifying the preferred propagation\n𝐻𝑂\npathway. Fig. S18 used the third path node as an example, illustrating how to perform the\nsignificance test for the impact of the source node on heatwave occurrence at the third path\nnode. The distance between the source node and the third path node is approximately 1600\nkm. Therefore, the black rings with radii of 1400 km and 1800 km are used to identify\nsurrogate nodes at similar distances from the source node to the third path node.\nIn the revised manuscript, Fig. S18 in the previous manuscript has been moved to Fig. S26.\nFor clarification, we have modified the figure caption and added Supplementary Note 4 to\nexplain this figure in more detail, as below:\n“Supplementary Note 4: Calculation and surrogate hypothesis test for 𝑅\n𝐻𝑂\nWe define an index 𝑅 to assess the impact of each preferred propagation pathway on\n𝐻𝑂\nheatwave occurrences downstream. Taking the Asia pathway as an example, 𝑅 represents\n𝐻𝑂\nthe probability of heatwave occurrence at the k-th path node within 7 days following a\nheatwave at the starting node, where k ∈ [1,6] represents the order of the six representative\nnodes along the pathway (Fig. 1).\n𝑘 𝑁\n𝑅 = 1|𝑘 (5)\n𝐻𝑂 𝑁\n𝑘\n𝑁 is the total number of heatwave events at the k-th path node, and 𝑁 denotes the number\n𝑘 1|𝑘\nof cases that a heatwave occurs at the k-th path node within 7 days following a heatwave at\n𝑘\nthe starting node. Therefore, for each path node, 𝑅 is used to estimate the impact of the\n𝐻𝑂\npreferred propagation pathway on heatwave occurrence at that node (Fig. 5a).\nTo verify the significance of the preferred propagation pathway's impact on heatwave\noccurrences at its downstream nodes, we use surrogates to compute 𝑅 and test the null\n𝐻𝑂\nhypothesis. Specifically, when a heatwave occurs at the starting node of the Asia pathway, we\nestimate the probability of a heatwave occurring at different nodes within the following 7\ndays (i.e., 𝑅 ), where each of these nodes is at the same distance from the starting node.\n𝐻𝑂\nThus, these nodes are distributed in a circle centered on the starting node (Supplementary\nFig. S26): one of these nodes lies along the Asia pathway, with the probability of heatwave\noccurrence denoted as 𝑅 (Path); while the remaining nodes, which lie outside the Asia\n𝐻𝑂\npathway, serve as surrogates, with their probability of heatwave occurrence denoted as 𝑅\n𝐻𝑂\n(Non-path). Hence the null hypothesis states that 𝑅 (Path) will not differ significantly from\n𝐻𝑂\n𝑅 (Non-path).\n𝐻𝑂\nWe calculate 𝑅 (Non-path) values for the surrogates, and obtain the mean, 99th percentile\n𝐻𝑂\nand 1st percentile of them. Result shows 𝑅 (Path) is beyond the 99% confidence interval for\n𝐻𝑂\nthe surrogates (as in Fig. 5), confirming the significant impact of preferred propagation\npathway on the heatwave occurrence at its downstream nodes. Similar significance tests\nusing surrogates were also performed for the West Europe, North America 1, and North\nAmerica 2 pathways (not shown here), where the impacts of these pathways were found to be\nsignificant as well. ”\nAdapted from Supplementary Fig. S26. Illustration of surrogate generation for the 𝑅 significance\n𝐻𝑂\ntest. Taking the 3rd path node of the Asia pathway as an example, we consider all grid cells at a\nsimilar distance from the starting node, comparable to the distance between the starting node and the\n3rd path node (around 1600 km). These grid cells, represented by the red dots within black rings with\nradii of 1400 km and 1800 km, are not part of the Asia pathway but serve as surrogates for the 3rd\npath node in subsequent significance tests. Therown arrow indicates that the 𝑅 calculation is\n𝐻𝑂\nperformed on the 3rd path node of the Asia pathway, and the result is denoted as 𝑅 (Path). Whereas\n𝐻𝑂\ngray arrow indicates that the 𝑅 calculation is performed on the surrogates, denoted as 𝑅\n𝐻𝑂 𝐻𝑂\n(Non-path). See Supplementary Note 4 for the details of calculating 𝑅 .\n𝐻𝑂\n8. L.375: “using a Eulerian perspective” – Why not use a Lagrangian tracking of the\nhigh-pressure centers or a quasi-Lagrangian perspective?\nThank you. The Lagrangian perspective could be an alternative approach, but it typically\ndemands high-resolution data and could be computationally intensive (e.g., Ref. 4). On the\nother hand, here our complex network algorithm is based on the Eulerian perspective. We\nbelieve that using the Eulerian perspective here better maintains the consistency and\nsimplicity of the methodology. A comparison of our results to an approach based on\nLagrangian tracking would be a very interesting subject of future work.\n9. Figure 1, legend: heavewaves -> heatwaves\nThank you. We have modified it in the revised manuscript.\nReference\nRef. 1: Duchez, A., Frajka-Williams, E., Josey, S.A., Evans, D.G., Grist, J.P., Marsh, R., McCarthy,\nG.D., Sinha, B., Berry, D.I., Hirschi, J.J.: Drivers of exceptionally cold north atlantic ocean\ntemperatures and their link to the 2015 european heatwave. Environmental Research Letters 11(7),\n074004 (2016)\nRef. 2: Boers, N., Bookhagen, B., Barbosa, H.M., Marwan, N., Kurths, J., Marengo, J.A.: Prediction\nof extreme floods in the eastern central andes based on a complex networks approach. Nature\nCommunications 5, 5199 (2014)\nRef. 3: Li, K., Huang, Y., Liu, K., Wang, M., Cai, F., Zhang, J., Boers, N.: Key propagation pathways\nof extreme precipitation events revealed by climate networks. npj Climate and Atmospheric Science\n7(1), 165 (2024)\nRef. 4: Papritz L, Röthlisberger M. A novel temperature anomaly source diagnostic: method and\napplication to the 2021 heatwave in the Pacific Northwest. Geophysical Research Letters. 50, 23:\ne2023GL105641 (2023)\nRef. 5: Santos, R., Russo, A., Gouveia, C.M.: Co-occurrence of marine and atmospheric heatwaves\nwith drought conditions and fire activity in the mediterranean region. Scientific Reports 14(1),\n19233 (2024)\nRef. 6: Duchez, A., Frajka-Williams, E., Josey, S.A., Evans, D.G., Grist, J.P., Marsh, R., McCarthy,\nG.D., Sinha, B., Berry, D.I., Hirschi, J.J.: Drivers of exceptionally cold north atlantic ocean\ntemperatures and their link to the 2015 european heatwave. Environmental Research Letters 11(7),\n074004 (2016)\nReviewer 4\nThis work investigates the mechanisms underlying the spatial propagation of the terrestrial\nheatwaves and find out four preferred propagation pathways of terrestrial heatwaves in the\nnorthern hemisphere. And each preferred pathways are consistent with the movement of\nRossby wave trains. The detected propagation pathways are found to provide prior\nknowledge for occurrences of downstream heatwaves and thus can be used for identifying\nassociated precursor signals. I find the results interesting and innovative, as they provide a\ncompelling explanation for the occurrence of simultaneous extremely high temperatures\nacross the globe within a short period. This insight sheds light on the underlying mechanisms\nthat drive these widespread heatwave events. However, the method and expressions are a\nlittle hard to follow. Please see the detailed comments below.\nThank you very much for taking the time to review our paper and for offering valuable\nsuggestions. We greatly appreciate the recognition of the interest and value of our work. In\nresponse, we have carefully revised the manuscript and made improvements to the Methods\nsection. Below, we provide a detailed point-by-point response to your comments and\nsuggestions, along with references to the corresponding revisions in the manuscript.\nOverall, the method parts are very complex and confusing,\nl. Definition of Heatwaves: The authors state in the manuscript, “A heatwave with the spatial\nextent (around 300 km [26]) could be detected by multiple surrounding regions, and this is\ntaken into account when analyzing heatwaves by the used complex network algorithm.”\nHowever, it remains unclear how many hot regions a heatwave comprises or what the\nminimum spatial extent should be for an event to qualify as a heatwave. Could the authors\nclarify these points to ensure a consistent and precise definition of heatwaves in their\nanalysis?\nThank you for this comment. We apologize, this could have been much clearer. In our study\nand methodology, we identify a heatwave event using the traditional definition (Ref. 1): “at\neach spatial grid cell, if its surface temperature anomaly exceeds the 95th percentile of the\nclimatological distribution for that grid cell for more than three consecutive days and occurs\nbetween April and September, it is considered a heatwave event”. The spatial size of\nheatwave was not used for identifying heatwave events at each spatial grid cell in our study.\nWe note that heatwave events at different spatial grid cells may be linked through the\nheatwave’s spatial extent but also by its spatial propagation, but our study merely focuses on\nthe heatwave’s spatial propagation. Therefore, in the subsequent complex network algorithm,\nit is important to note that only the average or maximum spatial extents are considered, as\nthey help mitigate the interference of the heatwave’s spatial extent when identifying the\nheatwave’s propagation pattern. Earlier studies (Ref. 2 and Ref. 3) suggested that the\nminimum, maximum and average spatial extent of a heatwave are 151000 𝑘𝑚 (219 km in\nradius), 400000 𝑘𝑚 (356 km in radius) and 300000 𝑘𝑚 (309 km in radius), respectively,\nwith the typical size being approximately 300 km in radius. Therefore, we set a cut-off\ndistance of 400 km, analyzing only the complex network’s links with distances greater than\n400 km. The minimum spatial extent is not involved during the analysis. We have revised the\nsentences in lines 347-351, 78-80 and 119-122 of the manuscript, as follows:\n“Definition of boreal summertime heatwaves: Here we adopt the percentile-based threshold\nmethod to identify heatwave events at each spatial grid cell [1, 2]. For each spatial grid cell,\na heatwave event is defined as a period when the surface temperature anomaly exceeds the\n95th percentile of the climatological distribution for that grid cell for more than three\nconsecutive days, occurring between April and September [1]” (lines 347-351) ,\n“It means that heatwave events at distant regions may be linked through the heatwave’s\nspatial extent but also by its spatial propagation” (lines 78-80),\nand “For mitigating the interference of the proximity effect [32] and spatial extent of\nheatwaves (with typical size of 300 km [26, 27, 34]) when identifying the heatwave’s\npropagation pattern, network links with distances shorter than 400 km are not considered\nduring our analysis” (lines 119-122).\n2. The calculation of ES measurements: I am confused about the definition of 𝝁,𝝊, e and d.\nThey are not clearly explained, which make the method hard to be understood. More specific,\nwhat exactly do the event time series ei/ej represent? The average temperature or maximum\ntemperature or the start date/end date for a heatwave event?\n𝝁\nThank you for raising this point. For a given pair of spatial grid cells i and j, 𝑒 denotes the\n𝑖\n𝑣\ntiming of the local temperature maximum during the μ-th heatwave event in grid cell i, and 𝑒\n𝑗\ncorrespondingly for grid cell j, where μ, v ∈ [1, l], l denotes the total number of heatwave\n𝝁, 𝑣 𝝁 𝑣\nevents at each grid cell. 𝑑 is equal to 𝑒 - 𝑒 , and denotes the time delay between these two\n𝑖, 𝑗 𝑖 𝑗\nevents. We have improved the descriptions in lines of 355-361 of the revised manuscript:\n“For a given pair of spatial grid cells i and j, we calculate the normalized number of events\n𝝁\nat i that can be uniquely associated with subsequent events at j, and vice versa: 𝑒 denotes\n𝑖\nthe timing of the local temperature maximum during the μ-th heatwave event at grid cell i,\n𝑣\nand 𝑒 correspondingly for grid cell j, where μ, v ∈ [1, l], l denotes the total number of\n𝑗\nevents at each grid cell. The two events are counted as synchronized events if their dynamical\n𝝁, 𝑣 𝝁 𝑣\ndelay 𝑑 := 𝑒 - 𝑒 does not exceed a threshold τ ”\n𝑖, 𝑗 𝑖 𝑗\n3. Does the matrix A consist of modified ES values ?\nYes, the modified ES values are stored in matrix A. We have improved the sentences in lines\n374-376 of the revised manuscript:\n“This process results in a matrix that stores the modified ES values, serving as the adjacency\nmatrix A of the complex network, thereby making the network both weighted and directed.”\n4. Local Searching algorithm: how is the out-degree obtained?\nThank you. The out-degree is calculated from the network’s adjacent matrix A, as in Eq.3. In\nthe revised manuscript, we have supplemented the descriptions in lines 380 and 392-394:\n𝑜𝑢𝑡 𝑖𝑛\n“where 𝐷 and 𝐷 are the out-degree and in-degree of the node i ” (line 380),\n𝑖 𝑖\n𝑜𝑢𝑡\nand “We compare the out-degree (𝐷 in Eq.3) of each node within this circular ring around\n𝑖\nthe starting node”(line 392-394).\n5. Also, I am confused about the expression of heatwaves in “region A or B” and “node A\nand B”.\nThank you. The terms \"regions A and B\" are intended to have the same meaning as \"nodes A\nand B\". We have modified this expression as \"nodes A and B\" throughout the revised\nmanuscript.\nThe result descriptions:\n6. As stated in Line 142 “Inspections of all heatwaves over the source regions identified from\nFig. 1 suggests that such propagating heatwaves have occurred many times over the past few\ndecades.” Could you be more specific, or provide a global map for the frequency of heatwave\noccurrences.\nThank you. Between 1959 and 2023, propagating heatwaves have occurred 64 times at the\nNorthern Asia source node, 76 times at the Western Europe source node, and 83 and 79 times\nat the North American source nodes, respectively. Supplementary Fig. S25 illustrates the\nfrequency of heatwave occurrences in the mid-latitudes of the Northern Hemisphere, and the\nproportion of the cases of propagating heatwaves. The frequency of heatwave occurrence\nalong these four pathways does not exceed the overall distribution across\nnorthern-hemisphere mid-latitudes (Supplementary Fig. S25a), but the proportion of\npropagating cases is notably higher near these pathways (Supplementary Fig. S25b).\nAdapted from Supplementary Fig. S25: Statistics of heatwave occurrences (1959-2023) in the\nNorthern Hemisphere from 20°N to 70°N, and comparison with the preferred propagation pathways\nof heatwaves revealed in our study (yellow lines). (a) Frequency of heatwave occurrence, and (b)\nproportion of the cases of propagating heatwaves (Supplementary Note 1).\nIn the revised manuscript, we have revised the descriptions in lines 142-146, and\nincorporated an analysis of the frequency of heatwave occurrences in lines 275-282, and\nSupplementary Fig. S25:\n“Inspections of all heatwaves over the source regions identified in Fig. 1 reveal that, between\n1959 and 2023, such propagating heatwaves (Supplementary Note 1) have occurred 64 times\nat the Northern Asia source node, 76 times at the Western Europe source node, and 83 and 79\ntimes at the North American source nodes, respectively. This encourages us to conduct\nfurther quantitative analyses as follows” (lines 142-146).,\nand “Additionally, we note that while the frequency of heatwave occurrence along the four\npreferred pathways does not exceed the overall distribution across northern-hemisphere\nmid-latitudes (Supplementary Fig. S25a), the proportion of propagating cases is notably\nhigher near these pathways (Supplementary Fig. S25b), and heatwave occurrences in these\nregions have shown increasing trends in recent decades (Supplementary Fig. S25c) [24, 47].\nThis suggests that the four propagation pathways are located in regions with more active\nheatwave propagation and a rising frequency of heatwave occurrences” (lines 275-282).\n7. Figure 1 is interesting. It gives the four typical heatwave propagation pathways across the\nwhole globe. The pathway over Asia indicates that heatwaves over the eastern coast regions\nof China generally originate from Ural. I am curious about where the heatwaves in southern\nChina originate. Do they originate from the same source as the eastern part of China?\nAdditionally, it would be helpful to clarify how much these four typical propagation\npathways contribute to the overall heatwave pathways worldwide.\nThank you for your suggestion. Some of the heatwaves in the southern and middle part of\nChina can be related to the Ural: Taking the case of June 2022 as an example (Supplementary\nFig. S1), for a heatwave in the Ural, the subsequent spatial movement of high temperature\nanomalies involves a broad spatial range over eastern and southern China. This suggests that\nheatwaves can spread in various directions around the Asia pathway. But most of the\ndirections (61.5%) are toward the southeast and align with the extension of the Asia pathway,\nas shown by the percentages in the rose diagram of movement directions for heatwaves at the\nAsia pathway (Supplementary Fig. S11). As designed by the algorithm, the spatial extension\nof the preferred Asia pathway is along the direction with the highest probability of movement\ndirection, illustrating the predominant direction and representative pathway for most\nheatwave movements. For the preferred pathways in Western Europe and North America we\ncreated similar rose diagrams to present the percentages of movement directions of heatwaves\nin their respective regions, as shown in Supplementary Fig. S11.\nAdapted from Supplementary Fig. S1 On June 10, 2022, the distribution of surface temperature and\n500 hPa geopotential height anomalies (contours) across Asia. Days 0 to 10 represent the evolution of\nsurface temperature anomalies from the onset of the heatwave to 10 days thereafter, illustrating a\ncomplete process of heatwave propagation from the Ural to the south and middle of East Asia.\nAdapted from Supplementary Fig. S11 Rose diagrams of movement directions for the heatwaves at\nstarting nodes of the four preferred pathways (Asia, WE, NA1 and NA2 pathways), respectively. The\ndots connected with yellow lines denote the spatial routes of the four preferred pathways.\nIn the revised manuscript, we added Fig. S1 to present the case of the heatwave propagating\nto the southern and middle part of China. We also added Supplementary Fig. S11 to illustrate\nthe percentages of movement directions of heatwaves, compared with the extension\ndirections of preferred pathways in Asia, Western Europe and North America, respectively.\n8. How to obtain the centers of the 500-hPa pressure system and their uncertainty range in\nFigure 2?\nTaking the Asia pathway as an example, our procedure is:\n(1) For each heatwave event, we first identify the geographic location with the maximum of\nthe 500-hPa anomaly around the starting node (on Day = 0, i.e., the central moment (the day\nof local temperature maximum) of the heatwave event), designating this as the center of the\nhigh-pressure system on Day = 0.\n(2) For the subsequent days, we identify the centers of the high-pressure system from Day = 1\nto Day = 10, allowing us to track the movement of the high-pressure system throughout the\nheatwave event.\n(3) This process yields the ensemble movement trajectories for all heatwave events, and we\napply k-means clustering to categorize them into propagating and standing patterns.\n(4) For each day's set of trajectory points, we calculate the average geographic location as\nthe center. The uncertainty for each point is determined as twice the standard deviation of its\nlatitude and longitude, as shown in Fig. 2.\nIn the revised manuscript, we have supplemented this method description into Supplementary\nNote 2:\n“Supplementary Note 2: Identifying trajectory and uncertainty of the high-pressure system\nmovement\nFigure 2 presents the trajectory and uncertainty of the high-pressure system. Taking the Asia\npathway as an example, our procedure is as follows:\n(1) For each heatwave event, we first identify the geographic location with the maximum of\nthe 500-hPa anomaly around the starting node (on Day = 0, i.e., the central moment (the day\nof local temperature maximum) of the heatwave event), designating this as the center of the\nhigh-pressure system on Day = 0.\n(2) For the subsequent days, we identify the centers of the high-pressure system from Day = 1\nto Day = 10, allowing us to track the movement of the high-pressure system throughout the\nheatwave event.\n(3) This process yields the ensemble movement trajectories for all heatwave events, and we\napply k-means clustering to categorize them into propagating and standing patterns.\n(4) For each day's set of trajectory points, we calculate the average geographic location as\nthe center. The uncertainty for each point is determined as twice the standard deviation of its\nlatitude and longitude, as shown in Fig. 2.”\n9. Line 186-187 The stationary heatwaves are firstly mentioned here but without a detail\ndescription. Please clarify how to distinguish the propagating and stationary heatwaves?\nThank you for this suggestion. The stationary heatwaves were identified through clustering\nanalysis on the movement trajectories of the high-pressure system’s center:\n(1) When we performed unsupervised k-means clustering on the movement trajectories of the\nhigh-pressure system’s center, we obtained two categories: one exhibits long-traveling\nbehavior, and another exhibits nearly stationary behavior (as shown in Fig. 2). (2) To\ndouble-check, we applied the same clustering analysis but to the surface air temperature’s\nhighest value center, and obtained long-traveling and stationary categories as well (as shown\nin Supplementary Fig. S16). (3) Next, we composited the surface air temperature anomalies\ncorresponding to these two categories of the high-pressure system’s movement, and found\nthat the composition of the “stationary-behavior” category shows the pattern of stationary\nheatwaves (as shown in Fig. 3), and the composition of the “long-traveling-behavior”\ncategory shows the pattern of propagating heatwaves that is in line with the analysis result by\nusing complex network.\nIn the revised manuscript, we have improved the descriptions in lines 179-193:\n“Taking the Asia pathway as an example, we apply the k-means clustering method to classify\nthe movement patterns of the high-pressure system centers at the onset of heatwaves (i.e., the\nstart node of the Asia pathway around the Ural area) into two distinct categories, and track\nthe movement trajectories of the high-pressure system centers (Methods and Supplementary\nNote 2). The first category exhibits the trajectory of the moving high-pressure systems that\nstarts at the Ural and spreads towards eastern Asia (Fig. 2), which exactly aligns with the\nheatwaves propagation pathway. The consistency between the heatwaves propagation\npathway and the movement of high-pressure systems holds true also for the WE, NA1 and\nNA2 pathways (Fig. 2). In contrast, the second category remains nearly stationary near the\nstarting node. For additional validation, we perform a similar clustering analysis on the\nmovement of surface temperature anomaly centers and obtain two comparable categories\nrepresenting moving and stationary patterns (Supplementary Fig. S16). Their movement\ntrajectories also align with the high-pressure systems and the heatwave propagation\npathways, further confirming the strong association between heatwaves and high-pressure\nsystems.”\nl0. How is RHO calculated? Is it a distance or a ratio? So confused.\nWe apologize, this could have been much clearer. Indeed, 𝑅 is a ratio, not a distance, and it\n𝐻𝑂\nquantifies the impact of each preferred propagation pathway on downstream heatwave\noccurrences. We adopted this quantifier based on the work of Boers et al. (2014) and Li et al.\n(2024) (Refs. 4 and 5). In the revised manuscript, we have added a more detailed explanation,\nincluding the precise definition and mathematical formulation of 𝑅 in Supplementary Note\n𝐻𝑂\n4:\n“Supplementary Note 4: Calculation and significant test for 𝑅\n𝐻𝑂\nWe define an index 𝑅 to assess the impact of each preferred propagation pathway on\n𝐻𝑂\nheatwave occurrences downstream. Taking the Asia pathway as an example, 𝑅 represents\n𝐻𝑂\nthe probability of heatwave occurrence at the k-th path node within 7 days following a\nheatwave at the starting node, where k ∈ [1,6] represents the order of the six representative\nnodes along the pathway (Fig. 1).\n𝑘 𝑁\n𝑅 = 1|𝑘 (5)\n𝐻𝑂 𝑁\n𝑘\nwhere 𝑁 is the total number of heatwave events at the k-th path node, and 𝑁 denotes the\n𝑘 1|𝑘\nnumber of cases that a heatwave occurs at the k-th path node within 7 days following a\n𝑘\nheatwave at the starting node. Therefore, for each path node, 𝑅 is used to estimate the\n𝐻𝑂\nimpact of the preferred propagation pathway on heatwave occurrence at that node (Fig. 5a).”\nl1. How can we recognize the duration of blocking system from Fig. S19?\nWe presented the spatiotemporal distribution of blocking indices for the standing heatwave\npatterns in the four pathways, respectively (Supplementary Figs. S27-S30). Taking the\nstanding heatwave patterns in the Asia pathway as an example (Supplementary Fig. S27), the\nnegative blocking index values around Ural denote the pattern of a blocking system, and this\npattern can persist from Day = -4 to Day = 2, hence the duration time of this blocking system\nis approximately 6 days. For the WE, NA1 and NA2 pathways, a similar analysis is\nconducted.\nIn the revised manuscript, we have added the spatiotemporal distribution of the blocking\nindex for the standing heatwave patterns in the four pathways, respectively, in Supplementary\nFigs. S27-S30.\nAdapted from Supplementary Fig. S27: Spatiotemporal distribution of GHGN blocking index\n(Supplementary Note 4) for the standing heatwave patterns in the Asia pathway. Day=0, 2, and 4\nrespectively represent the central moment (the date of local maxima) of heatwave occurrence at the\nstarting node and the subsequent 2 and 4 days. Day= -4 and -2 denote the 2 and 4 days before the\ncentral moment of the heatwave. A more negative GHGN value indicates a stronger blocking.\nReviewer #4 (Remarks on code availability):\nI have briefly reviewed the code, and its format seems to be correct and functional.\nThank you. We are now sharing all code and raw data used to compute and generate Figs. 1,\n2, 3, 4 and 5 in the Code Ocean capsule. We have updated the Code Availability section to\nindicate that our codes will be published on Github/Zenodo for public access following the\npublication of the manuscript.\nReference\nRef. 1: Domeisen D.I., Eltahir E.A., Fischer E.M., Knutti R., Perkins-Kirkpatrick S.E., Schar C.,\nSeneviratne S.I., Weisheimer A., Wernli H.: Prediction and projection of heatwaves. Nature Reviews\nEarth & Environment 4(1), 36–50414 (2023).\nRef. 2: Keellings D., Bunting E., Engström J.: Spatiotemporal changes in the size and shape of heat\nwaves over North America. Climatic Change 147, 165–178 (2018).\nRef. 3: Luo M., Wu S., Lau G.N.-C., Pei T., Liu Z., Wang X., Ning G., Chan T.O., Yang Y., Zhang W.:\nAnthropogenic forcing has increased the risk of longer-traveling and slower-moving large contiguous\nheatwaves. Science Advances 10(13), 1598 (2024).\nRef. 4: Boers, N., Bookhagen, B., Barbosa, H.M., Marwan, N., Kurths, J., Marengo, J.A.: Prediction\nof extreme floods in the eastern central andes based on a complex networks approach. Nature\nCommunications 5, 5199 (2014)\nRef. 5: Li, K., Huang, Y., Liu, K., Wang, M., Cai, F., Zhang, J., Boers, N.: Key propagation pathways\nof extreme precipitation events revealed by climate networks. npj Climate and Atmospheric Science\n7(1), 165 (2024)"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I would like to commend the authors on meticulously addressing all my comments. I have read the full response and find their results convincing. The revised version effectively addresses three of my key comments: (a) a causality test, (b) details on their surrogate hypothesis test, and (c) validation of the findings with a specific real-world case. I would recommend acceptance.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "Comments, Round 2:\nI really appreciated the effort made by the authors to improve the document. All my comments on the original version of the paper have been satisfactorily addressed and in my opinion the manuscript is ready for publication.\nSome minor issues to be corrected:\n1.Legend Fig S32 – “Central moment od the heatwave denote the the” – “Central moment of the heatwave denote the”\n2.Supplementary Note 4 - P.40 – “where each of these nodes is 5at the same distance” – remove 5", "reviewer_4": "Reviews on “Evidence for preferred propagating terrestrial heatwave pathways due to Rossby wave activity”\nThe authors' extensive analyses demonstrate the robustness and reproducibility of the findings, offering valuable insights into the spatial propagation mechanisms of heatwaves. Additionally, the authors have provided a well-considered response to my questions. However, I still have two remaining concerns.\n1. The article concludes that the propagation paths of Rossby waves influence the trajectories of heatwaves, which may not be particularly novel. However, this finding is crucial for heatwave forecasting, given that propagation-type heatwaves account for more than 50% of cases along the four identified paths, albeit with regional variations—for instance, only around 50% for the Western European (WE) path. These regional differences could have implications for the predictability of heatwaves based on Rossby wave propagation. A key question is whether this proportion remains stable over time and what physical mechanisms govern whether a heatwave remains stationary or follows a propagation pattern in different regions. Understanding the temporal stability of propagation-type heatwaves along these paths and the underlying physical processes is essential for improving heatwave predictability.\n2. The article examines the relationship between Rossby wave propagation and heatwave forecasting. Since temperature-based forecasting (traditional numerical weather forecasting) already provides a certain lead time (7-10 days) for predicting heatwaves, another key question is, compared to temperature-based methods, what are the advantages of using Rossby wave propagation for improving heatwave prediction?\nTypo errors:\n1. Abstract “fix stations” should be “fixed stations”\n2. Line 54 three days two days in advance"}, "author_response": "RESPONSE TO REVIEWER COMMENTS\nPlease find our point-by-point responses to the comments raised by the referee in blue font\nbelow.\nReviewer 1\nI would like to commend the authors on meticulously addressing all my comments. I have\nread the full response and find their results convincing. The revised version effectively\naddresses three of my key comments: (a) a causality test, (b) details on their surrogate\nhypothesis test, and (c) validation of the findings with a specific real-world case. I would\nrecommend acceptance.\nThank you for your valuable feedback and recognition. We are pleased to hear that you are\nsatisfied with the revised version and appreciate your recommendation for acceptance.\nReviewer #1 (Remarks on code availability):\nThe authors have addressed all our prior comments.\nThank you.\nReviewer 2\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This\nis part of the Nature Communications initiative to facilitate training in peer review and to\nprovide appropriate recognition for Early Career Researchers who co-review manuscripts.\nThank you for the thoughtful co-review of our manuscript and for providing us with helpful\nsuggestions. Please find our detailed-point-by-point responses to your comments and\nsuggestions, as well as references to how we revised our manuscript accordingly.\nReviewer 3\nComments, Round 2:\nI really appreciated the effort made by the authors to improve the document. All my\ncomments on the original version of the paper have been satisfactorily addressed and in my\nopinion the manuscript is ready for publication.\nThank you for your valuable feedback and recognition. We are pleased to hear that you are\nsatisfied with the revised version and appreciate your recommendation for acceptance.\nSome minor issues to be corrected:\n1.Legend Fig S32 – “Central moment od the heatwave denote the the” – “Central moment of\nthe heatwave denote the”\nThanks. We have corrected this typo in the revised manuscript.\n2.Supplementary Note 4 - P.40 – “where each of these nodes is 5at the same distance” –\nremove 5\nThanks. We have corrected this typo.\nReviewer 4\nReviews on “Evidence for preferred propagating terrestrial heatwave pathways due to Rossby\nwave activity”\nThe authors' extensive analyses demonstrate the robustness and reproducibility of the\nfindings, offering valuable insights into the spatial propagation mechanisms of heatwaves.\nAdditionally, the authors have provided a well-considered response to my questions.\nHowever, I still have two remaining concerns.\nThank you for your valuable feedback and recognition. We have carefully considered your\ncomments and suggestions. Please find below our detailed point-by-point responses, along\nwith references to the corresponding revisions made in the manuscript.\n1. The article concludes that the propagation paths of Rossby waves influence the trajectories\nof heatwaves, which may not be particularly novel. However, this finding is crucial for\nheatwave forecasting, given that propagation-type heatwaves account for more than 50% of\ncases along the four identified paths, albeit with regional variations—for instance, only\naround 50% for the Western European (WE) path. These regional differences could have\nimplications for the predictability of heatwaves based on Rossby wave propagation. A key\nquestion is whether this proportion remains stable over time and what physical mechanisms\ngovern whether a heatwave remains stationary or follows a propagation pattern in different\nregions. Understanding the temporal stability of propagation-type heatwaves along these\npaths and the underlying physical processes is essential for improving heatwave\npredictability.\nThank you for this suggestion. (1) We analyzed the temporal trend of propagating heatwave\nproportions at each grid cell in the Northern Hemisphere, from 20°N to 70°N, covering the\nperiod from 1959 to 2023. Most of the trends were statistically insignificant, indicating that\nthe proportion of propagating heatwaves has remained stable over time. This temporal\nstability of the proportion holds for most grid cells and across the four preferred pathways\n(Supplementary Fig. S25d).\n(2) The identified heatwave propagation pathways can help to improve the heatwaves’\npredictability downstream of the pathway. We agree that understanding the onset of\npropagation patterns is crucial for further enhancing heatwave predictability over the\npathway’s start point and also the entire pathway. Our findings show that the propagating and\nstationary patterns of heatwaves are closely linked to the movement of high-pressure systems\nassociated with Rossby wave packets (Figs. 2 and 4). Additionally, we observed that land soil\nmoisture exerts the feedback effect on the propagation of heatwaves (Supplementary Fig.\n23). To thoroughly understand and predict the onset and development of the propagation\nand standing patterns, a comprehensive numerical-simulation experiment would be needed.\nHowever, current physical-processes-based climate models are unable to correctly predict the\nlocation, strength and movement direction of the Rossby wave trains associated with\nheatwaves at desired lead times (see Refs. 1-4). On the other hand, as previous studies\nsuggested (e.g., Refs. 1, 4, 5 and 6), topography, surface friction and soil moisture play a\ncritical role in determining the ability of numerical climate models to characterize the\nstability, frequency, and transition phases of Rossby wave packets, thereby influencing their\npredictability (e.g., Refs. 3, 4 and 6). The performance to reproduce the link between Rossby\nwave packets and topography in state-of-the-art numerical weather prediction models and\nclimate models also remains an open question (e.g., Refs. 1, 4 and 5). Therefore, it is\nchallenging to anticipate the onset of the propagating and standing patterns earlier. A\ncomprehensive research addressing this question would be a very interesting subject of future\nwork.\nIn the revised manuscript, we have added a figure illustrating the trend analysis of the\nproportion of the propagating heatwaves, as shown in Supplementary Fig. 25d. We also have\nincorporated the relevant discussion in lines 60-63, and lines 309-316:\nLines: 60-63: “However, current process-based numerical models are unable to reliably\npredict the locations and strengths of these relevant background patterns at the desired lead\ntimes [1, 18–22], which hinders both the prediction skill and our understanding of the\nmechanisms behind heatwaves[1, 22, 23].”\nLines 309-316: “The proportion of propagating heatwaves across the four pathways has\nremained stable over recent decades (Supplementary Fig. 25d), suggesting that the\npredictability explained by recurrent propagation patterns is consistent over time. To further\nenhance the ability to predict heatwaves in practical forecasting, in the future, a deeper\nunderstanding of RWPs is needed for anticipating the onset of propagating and standing\npatterns earlier, and integrating the link between propagation patterns and RWPs into the\nmachine-learning or numerical forecast models [22, 23, 50].”\nAdapted from Supplementary Figure 25d: Statistics of heatwave occurrences (1959-2023) in the\nNorthern Hemisphere from 20°N to 70°N, and comparison with the preferred propagation pathways\nof heatwaves revealed in our study (yellow lines). (a) Frequency of heatwave occurrences; (b)\nproportion of the cases of propagating heatwaves (Supplementary Note 1); (c) trend of the frequency\nof heatwave occurrences; and (d) trend of the proportion of the propagating heatwaves. Cross marks\nindicate trend values that are significant at the 5% significance level.\n2. The article examines the relationship between Rossby wave propagation and heatwave\nforecasting. Since temperature-based forecasting (traditional numerical weather forecasting)\nalready provides a certain lead time (7-10 days) for predicting heatwaves, another key\nquestion is, compared to temperature-based methods, what are the advantages of using\nRossby wave propagation for improving heatwave prediction?\nThank you for raising this important point. As recent studies have suggested, numerical\nweather forecasting systems can reliably issue the warning of heatwave, with a lead time of 2\nto 3 days (Ref. 1); however, as the forecast lead time is further extended, the accuracy of\nheatwave predictions significantly decreases (Refs. 1, 3, and 7). This decline is primarily due\nto the inadequate representation of the evolution of Rossby waves and the relationship\nbetween Rossby wave patterns and heatwaves in numerical weather forecasting and climate\nmodels (Refs. 1, 3, 4, and 7). Specifically, it was found that the current numerical forecast\nmodels are unable to predict the location, strength, and movement direction of local Rossby\nwave packets associated with the heatwaves at the desired lead times (Refs. 1, 2 and 3). Our\nstudy reveals the association between propagating heatwaves pathways and the recurrent\nRossby waves patterns based on real-world cases. Integrating or reproducing this linkage in\nthe state-of-the-art numerical models or the machine-learning models, would be crucial for\nfurther enhancing practical predictability of heatwaves in the follow-up work.\nIn the revised manuscript, we have incorporated the relevant discussions in lines 53-63 and\nlines 312-316:\nLines 53-63: “Operational forecasting systems can reliably issue warnings of heatwaves that\nlast more than three days, with a lead time of two to three days [1]. Extending the lead time\nof heatwave forecasts requires an improved understanding of the background atmospheric\ncirculation patterns associated with heatwaves [1, 2, 13, 14]. On the planetary scale,\natmospheric high-pressure systems are dominated by persistent anomalous atmospheric\ncirculation patterns, such as the quasi-stationary ridge or block in the atmospheric flow [9,\n15], long-lived Rossby wave packets [9, 10, 12], or circumglobal resonant Rossby waves [16,\n17]. However, current process-based numerical models are unable to reliably predict the\nlocations and strengths of these relevant background patterns at the desired lead times [1,\n18–22], which hinders both the prediction skill and our understanding of the mechanisms\nbehind heatwaves[1, 22, 23]. ”\nLines 312-316: “To further enhance the ability to predict heatwaves in practical forecasting,\nin the future, a deeper understanding of RWPs is needed for anticipating the onset of\npropagating and standing patterns earlier, and integrating the link between propagation\npatterns and RWPs into the machine-learning or numerical forecast models [22, 23, 50].”\nTypo errors:\n1. Abstract “fix stations” should be “fixed stations”\nThanks. We have modified this typo in the revised manuscript.\n2. Line 54 three days two days in advance\nThanks. We have improved this sentence in the revised manuscript, as below:\n“Operational forecasting systems can reliably issue warnings of heatwaves that last for more\nthan three days, with a lead time of two to three days [1]”\nReference\nRef. 1: Domeisen, D.I., Eltahir, E.A., Fischer, E.M., Knutti, R., Perkins-Kirkpatrick, S.E.,\nSchar, C., Seneviratne, S.I., Weisheimer, A., Wernli, H.: Prediction and projection of\nheatwaves. Nature Reviews Earth & Environment 4(1), 36–50 (2023).\nRef. 2: Kornhuber, K., Lesk, C., Schleussner, C.F., Jagermeyr, J., Pfleiderer, P., Horton, R.M.:\nRisks of synchronized low yields are underestimated in climate and crop model projections.\nNature Communications 14(1), 3528 (2023).\nRef. 3: Lin, H., Mo, R., Vitart, F.: The 2021 western north american heatwave and its\nsubseasonal predictions. Geophysical Research Letters 49(6), 2021–097036 (2022).\nRef. 4: Lembo, V., Bordoni, S., Bevacqua, E., Domeisen, D. I., Franzke, C. L., Galfi, V. M.,\net al.: Dynamics, statistics, and predictability of Rossby waves, heat waves, and spatially\ncompounding extreme events. Bulletin of the American Meteorological Society, 105(12),\nE2283-E2293 (2024).\nRef. 5: Tibaldi, S., Molteni, F.: Atmospheric blocking in observation and models. In Oxford\nResearch Encyclopedia of Climate Science (2018).\nRef. 6: Schubert, S. and Lucarini, V.: Dynamical analysis of blocking events: spatial and\ntemporal fluctuations of covariant Lyapunov vectors. Quarterly Journal of the Royal\nMeteorological Society, 142(698), pp.2143-2158 (2016).\nRef. 7: Chattopadhyay, A., Nabizadeh, E., Hassanzadeh, P.: Analog forecasting of\nextreme‐causing weather patterns using deep learning. Journal of Advances in Modeling\nEarth Systems, 12(2), e2019MS001958 (2020)."}, {"round": 3, "reviewer_comments": {"reviewer_4": "Thank you to the authors for carefully addressing my previous comments. I believe the paper is now suitable for publication."}, "author_response": ""}]

Decision letter