Switzerland can be divided into several regions with distinct climates and hydrological regimes: the Jura, the Plateau, the Alps and the southern Alps (Fig. b) . These regions notably exhibit quite different seasonal cycles in extreme precipitation and discharge occurrence. In the Plateau, the heaviest precipitation occurs chiefly during summer (Fig. c) , as a result of convective instability , frequent westerly winds and Atlantic water vapor transport . In summer, however, evapotranspiration is highest and soils are less saturated than in the cold season. Consequently, extreme-discharge events are about equally likely to occur in winter, spring and summer (Fig. ). In the Jura, while the magnitude of extreme precipitation events still peaks in summer, its seasonality is less pronounced. About 20 % of extreme precipitation events indeed occur in winter and spring each (Fig. ), triggered by forced orographic ascent of moist westerlies . Extreme discharge, however, is mostly confined to winter and spring, largely driven by rain-on-snow processes .

As in the Jura, the seasonal cycle in extreme precipitation occurrence over the Alps is not strong (Fig. ) . The peak is reached in summer and fall for most catchments, when extreme precipitation occurs as a result of local convective instability , but winter and spring still concentrate 30 %–40 % of extreme events. The outlook for discharge is very different, however. Alpine catchments, especially at high elevations, are mainly driven by snowmelt and glacier melt . Thus, extreme discharge is almost exclusively confined to summer (Fig. c) . Finally, the southern Alps experience extreme precipitation mostly during summer and fall (Fig. c and d) . Such behavior results from the frequent southerly advection of moist Mediterranean air caused by upper-level troughs . These atmospheric conditions are connected to potential vorticity streamers or cut-offs centered west of the Alps, which are most frequent during fall . Extreme discharge in this region also occurs primarily during fall (50 %–60 % of events; Fig. d).

For each dataset, precipitation extremes are defined on a monthly basis as days when daily accumulated precipitation exceeds the 99th percentile of the corresponding month. For instance, January precipitation values are compared to the January 99th percentile. The percentiles are calculated using all days (both with and without precipitation). Potential trends in extreme daily precipitation percentiles are not taken into account. As the individual weather systems associated with extreme precipitation may sometimes last for several days, we remove the short-term temporal dependence in the occurrence of extreme precipitation events by applying a standard run declustering procedure with a run length of 2 d, well-suited for Switzerland . The goal of the declustering is to remove short-term dependence and to identify independent events. This procedure is for example applied prior to a peak-over-threshold statistical analysis. The declustering merges extreme events that are separated by less than 2 d into a single event. Consequently, it reduces the number of extreme events compared to the original series. Events within each season (winter: DJF; spring: MAM; summer: JJA; and fall: SON) are then analyzed together.

Extreme-discharge events for each of the 93 gauged catchments are defined as days when daily discharge exceeds its 95th or 99th percentile calculated on the whole time series (both percentiles are analyzed separately). We look at two extreme percentiles to test the sensitivity of our results to the choice of threshold and also to increase the number of detected high-discharge events. The persistence in extreme-discharge conditions is assessed by identifying periods of persistent high discharge over sub-seasonal timescales. These are defined as periods of length L containing at least N extreme-discharge days. Three sets of (L, N) values are considered: (10,5),(20,8) and (30,10). Depending on the values of L and N, no periods may be found in some catchments, in which case they are simply excluded from the corresponding analysis.

Three reasons justify the choice of seasonally varying thresholds for precipitation and fixed thresholds for discharge. First, such a choice removes the influence of the seasonality in extreme precipitation magnitude. The occurrence rate of extreme precipitation events is therefore constant across the year, and detecting clustering significance is straightforward (see below). Second, impacts of discharge extremes are usually related to their absolute rather than relative magnitude. Third, the seasonal cycles of extreme precipitation and discharge magnitudes are not in phase over much of Switzerland (Figs.  and ). The most extreme discharge does not necessarily occur after the heaviest precipitation events. Surface conditions, like soil saturation, presence of snow/ice, vegetation cover or evaporative demand, considerably shape the discharge response to heavy precipitation . As they vary substantially from one season to the next, the discharge response to the same precipitation magnitude may differ depending on the season.

Temporal clustering of precipitation extremes is quantified with Ripley's K function . Here we give a quick overview of the methodology and refer the reader to for further details. For a given window size w, Ripley's K function applied to a time series measures the average number of extreme events in a neighborhood of w days before and after a random extreme event in the series. This gives information about the tendency towards temporal clustering in the series. The larger the value of Ripley's K function for a given w, the more clustered the extreme events. The significance of temporal clustering in the series is then assessed by comparing Ripley's K values to those obtained from a Monte Carlo sample of 5000 simulated homogeneous Poisson processes with the same average event density as the observed series. In homogeneous Poisson processes, events occur independently from each other and therefore exhibit complete temporal randomness. Because we chose monthly percentiles to define extreme precipitation events, the occurrence rate of extremes is constant throughout the year. Thus we can test for clustering significance against homogeneous series. Non-homogeneous (and more complex) series would have been required if the likelihood of extreme event occurrence has been a function of time.

From this comparison we get an empirical p value for each w. As we deal with multiple hypothesis tests, we implement a false discovery rate procedure with a baseline significance level of 5 % to identify catchments where clustering is significant. Clustering significance is assessed for two intervals of w values, characteristic of sub-seasonal timescales: 15–25 and 25–35 d. Clustering is said to be significant for a given interval if it is significant for at least half of the w values in that interval.

We identify extreme precipitation cluster events over 21 d time windows with the algorithm of (see their Figs. 3 and 4). Starting from the declustered binary extreme event series, the first step is to calculate the 21 d moving sum of extreme event counts. In a second step, we select the 21 d period with the largest event count (i.e., the highest number of extreme events), if that count is larger than 2. Otherwise, no clusters are found and the algorithm stops. In the case of multiple 21 d periods with the same extreme event count, the one with the largest precipitation total is selected first. In the third step, we remove from the binary event series the extreme events that occur in the selected 21 d period. The algorithm is then run again from the first step onwards to identify the next cluster event. This procedure avoids any overlap between cluster events. The choice of the 21 d time window is well-suited to quantify clustering at sub-seasonal timescales and is generally consistent with the length of observed cluster episodes that led to major floods in Switzerland (see introduction). Results do not differ significantly for slightly shorter or longer (2–4 weeks) windows (see also ).

We then characterize clusters of precipitation extremes with two metrics related to their potential impact. The first is the average contribution of cluster periods to seasonal precipitation. This contribution increases with the frequency and total precipitation of cluster periods. The second metric is the frequency of cluster periods during extreme 21 d precipitation accumulations. It gives an idea of how often cluster periods are responsible for extreme precipitation accumulations, a frequent trigger of flood events in Switzerland .

We analyze the influence of clusters of precipitation extremes on discharge in two ways: by looking at discharge characteristics after clusters of precipitation extremes and at precipitation characteristics before persistent high-discharge periods.

First, we calculate for each catchment the probability p1 of extreme discharge for all days during and up to 5 d after 21 d clusters of precipitation extremes. We also calculate the probability p2 of extreme-discharge days for periods of the same length and same time of the year as the selected cluster periods. We can then define an odds ratio of extreme-discharge occurrence after clusters of extremes as p11-p2p21-p1 . The odds ratio compares the likelihood of extreme-discharge occurrence in the presence of a precipitation cluster to its likelihood in the absence of a precipitation cluster. The higher it is, the stronger the relationship between the occurrence of extreme discharge and precipitation clusters. From the identification of 21 d cluster periods, precipitation extremes can also be separated into “clustered” and “non-clustered” events. We then look at the likelihood of extreme-discharge occurrence after both types of events to highlight potential differences in the discharge response.

Second, for each of the persistent high-discharge periods identified as described previously, we calculate the number of precipitation extremes and the percentile of total accumulated precipitation over a window stretching from 10 d before the beginning of the persistent high-discharge period to its end. We choose to begin 10 d before because showed that moderate wet conditions occurred in the week preceding many flood events in Switzerland, hence the need to look beyond the few days preceding persistent high-discharge periods.

We now turn to the analysis of Ripley's K values and their implications in terms of sub-seasonal temporal clustering. Several coherent areas exhibit K values that are significantly larger than those expected for homogeneous Poisson processes with no temporal dependence (Fig. ). In winter, significant temporal clustering of precipitation extremes is mainly found in central Switzerland, along the Alpine ridge, at the 15–25 and 25–35 d timescales (Figs. a and b). In spring, two catchments in northern Switzerland as well as a few catchments in southeastern Switzerland also exhibit significant clustering (Figs. b and d). By contrast, results for the summer season show a complete absence of temporal clustering significance at all timescales (Figs. c and e, f). Finally, in fall, significance is found at all timescales over both the western tip of Switzerland and the southern side of the Alps (Figs. d and g, h).

Similar patterns are found by comparing to the coarser-resolution precipitation datasets (ERA5, TRMM, EOBS, CPC and CMORPH), with some notable exceptions. Clustering significance over the Alps in winter is also present in the coarser-resolution data, but with a wider extent than in RhiresD (Fig. a; see also Fig. ). Temporal clustering during spring is generally less significant across Switzerland (Fig. b). Two datasets indicate significant clustering locally in northwestern Switzerland, but none do along the northern and southern borders where significance was found in RhiresD. By contrast, all datasets agree on the absence of clustering in summer (Fig. c), and on significant clustering in southern and southeastern Switzerland during the fall season (Fig. d). Temporal clustering does not appear particularly significant, however, in western Switzerland during fall. In winter, clustering significance extends over a large region stretching from the Mont Blanc massif in France to eastern Switzerland along the Alpine ridge, in good agreement with RhiresD (Figs. a and a). Similarly, southern Switzerland is part of a larger region exhibiting significant clustering, encompassing northern Lombardy and possibly extending southwards to the Mediterranean shore (Fig. d).

We now expand the statistical analysis by showing the characteristics of clustered precipitation events. In winter, extreme precipitation clusters contribute an average of ≈10 % to total winter precipitation along the Alpine ridge where clustering is statistically significant (Fig. a). Additionally, clusters occur during about 60 %–70 % of extreme 21 d precipitation accumulations (above the corresponding 99th percentile) (Fig. a). Elsewhere, clusters contribute little to both seasonal and extreme precipitation accumulations. In spring, the average contribution of clusters to seasonal precipitation is overall weak (<10 %), even for catchments where clustering in RhiresD is statistically significant (Fig. b). Yet, over western Switzerland, periods of extreme 21 d accumulations are almost always cluster periods as well (Fig. b). In summer, consistent with the absence of clustering at that time of the year, clusters do not contribute much to seasonal precipitation. Finally, in fall, cluster contribution to seasonal precipitation reaches its annual maximum of 12 %–16 % over southeastern Switzerland (particularly the southern Alps). It is also quite high (≥10 %) over western Switzerland where clustering is statistically significant as well (Fig. d). In addition, more than 80 % of extreme precipitation accumulation periods are accompanied by cluster events in the southern Alps (Fig. d). Since extreme discharge in this area is most common during fall (Fig. d), this suggests a possibly important role of extreme precipitation clusters in high-impact weather events in this region and at that time of the year.

Persistent high-discharge periods, regardless of L and N values, are systematically associated with extreme precipitation accumulations for catchments with mean elevations up to about 1500 m (Figs. a, b and a). Above 1500 m, glacial/nival runoff regimes dominate, and the link to extreme precipitation accumulations is weaker. By contrast, extreme precipitation clusters precede persistent high-discharge periods only in the southern Alps and locally over the eastern parts of the Swiss Plateau and the Jura (Fig. c and d). The dependence on catchment elevation is similar, with a much weaker intersection of cluster events and extreme discharge at high elevations, but less robust with a larger spread of values at low elevations (Fig. b).

For most catchments, particularly those below 1500 m elevation, the discharge response after an extreme precipitation event differs between single extreme events and events that are part of clusters (Fig. ). In the 5 d following an extreme precipitation event, the fraction of days exceeding either the 95th or the 99th percentile of daily discharge is higher when that event belongs to a cluster. The difference is particularly large over northern Switzerland and in the southern Alps where, for instance, the 99th percentile of daily discharge values is exceeded on average 20 %–30 % of the time (1–1.5 d) in the 5 d following an extreme precipitation event during a cluster but only 10 %–15 % of the time if the extreme occurred outside of a cluster. Given that daily discharge exceeds its 99th percentile on average only ≈3.5 d a year, this implies an important effect of clustered extremes. The occurrence of a cluster of precipitation extremes greatly increases the likelihood of high-discharge events, particularly at low elevations, as evidenced by high odds ratio values (Fig. ). Daily discharge is more than 7 (10) times more likely to exceed its 95th (99th) percentile during and after a cluster in the Jura, for instance. The distribution of daily discharge percentiles after extreme precipitation events confirms these results: while the probability of exceeding high-discharge thresholds on the day following an extreme event is not very different between clustered and non-clustered extremes, that probability decreases much faster in the days following non-clustered extremes (Fig. ).

Our definition of extreme precipitation events based on the exceedance of monthly percentiles of daily precipitation removes the influence of seasonality in extreme precipitation magnitude (Fig. ). By applying Ripley's K function to the resulting time series, we can thus focus on short-term temporal dependence in extreme precipitation occurrence driven by sub-seasonal dynamics or intra-annual variability (e.g., climate modes). Clustering significance is found mainly over the Alps during winter and in southern and southeastern Switzerland during fall (Figs.  and ). The spatial coherence and robustness of the results over these two regions across timescales and datasets (Fig. ) suggests that specific physical processes are responsible for the clustering.

Both in winter and in fall, clustering significance is chiefly concentrated at timescales below 30 d, which seems to preclude any dominant role of seasonal sea-surface temperature anomalies .

First, we discuss the spatial pattern during winter, when significant clustering occurs in the central Alps. During winter, extreme precipitation events in northern Switzerland often occur in connection with extreme integrated water vapor transport (e.g., linked to atmospheric rivers) that interacts with the orography . The concentration of clustering significance over the Alps during winter possibly results from a spatial anchoring of precipitation at high elevations, through orographic lifting and convergence of the moist air masses, while precipitation in the lower-lying area is spatially more dependent on the presence of cold-air pools upstream of the mountains . The spatial extent of these cold pools may vary from event to event, along with the location of the strongest precipitation.

Next, we discuss the spatial pattern during fall when clustering occurs south of the Alps. During fall, several major clusters of precipitation extremes south of the Alps were related to recurrent Rossby wave breaking over western and southwestern Europe . The wave breaking leads to enhanced low-level moisture transport from the Mediterranean towards the Alps . discuss why wave breaking was recurrent for four case studies. Recurrent cyclogenesis and extratropical transition of tropical cyclones upstream over the western Atlantic seem to play a role, but not necessarily a systematic one. Persistence in blocking conditions in the northwestern Atlantic also contributed to upper-level wave amplification and the subsequent occurrence of extreme-precipitation clusters in southern Switzerland .

The apparent lack of agreement among precipitation datasets regarding the significance of the clustering over western Switzerland in fall (Figs. d and d) is less straightforward to interpret. Despite the precautions taken to control the false discovery rate, it is still possible that significance in this region is detected by pure chance. Yet, it is not altogether obvious that the gridded satellite or reanalysis datasets are completely reliable either. If thunderstorms are responsible for many extreme precipitation events, their representation in reanalysis products can be questioned. Their small spatial scale may also cause them to be missed in satellite products. While coarse-resolution gridded datasets seem to agree on the timing and magnitude of precipitation extremes in western Switzerland during fall , a more detailed comparison with RhiresD data and an analysis of the type of events responsible for these extremes would be required to conclude.

Persistent high-discharge periods at elevations lower than 1500 m a.s.l. are almost systematically associated with extreme precipitation accumulations in the preceding days (Fig. a and b). This is consistent with the mainly pluvial regimes at lower elevations, as well as with the results of . Such accumulations are not always the consequence of clusters of extremes, however. Outside southern Switzerland during fall, western Switzerland during spring and, to a lesser extent, Alpine catchments during winter, the overlap of cluster events and extreme precipitation accumulations is generally smaller than 50 %, and often smaller than 25 % (Fig. ). Results in summer and spring can be understood by the scarceness of cluster events in those seasons. In winter, despite being frequent, clusters over the Alps are less systematically associated with extreme accumulations, which are often the result of a single heavy-precipitation event.

Still, from the perspective of surface impacts, clusters remain relevant, regardless of their overall frequency, if they increase flood hazard. The discharge response to both clustered and non-clustered extreme precipitation events typically peaks 1 d after the event (Fig. ), consistent with the findings of . However, our results show that clusters of precipitation extremes increase the likelihood of occurrence and the duration of high-discharge events, particularly at low elevations (Figs.  and ). This influence is noticeably larger than for non-clustered precipitation extremes (Fig. ). On average, daily accumulated precipitation during clustered and non-clustered extremes is similar. Instantaneous precipitation rates might be different, but it is not possible to verify it given the daily resolution of the precipitation data. However, the first extreme in a cluster event likely increases soil moisture, which enhances the discharge response to the subsequent precipitation extremes . The role of antecedent soil moisture in flood generation and volume is well-documented for Switzerland and Alpine catchments e.g.,. This may explain why extreme-discharge probability decreases more slowly after clustered precipitation extremes compared to non-clustered events (Fig. ).

This difference is quite high in the southern Alps (e.g., Fig. c and d), possibly due to the fact that floods in this area generally occur in the fall (Fig. d; ) when clusters bring substantial amounts of precipitation (Fig. d). There, frequent clusters leading to extreme precipitation accumulations are likely to be an important precursor of major flood events, as confirmed by observations of several damaging clustering periods . This region of Switzerland also experiences the largest precipitation extremes . Additionally, it is characterized by poor infiltration rates, steep slopes and weak soils . Infiltration excess (connected to Hortonian-type storm runoff generation) may therefore be more rapidly reached than in the rest of the country. Coupled with saturation excesses following the first extreme event in a cluster, it might explain why the region stands out in most of our analyses.

By contrast, in the Alps during winter, though clustering is statistically significant, its impact on extreme discharge is quite limited. This results most likely from the fact that discharge in Alpine catchments is lowest in winter, when much of the precipitation falls as snow and the magnitude of precipitation extremes is generally lower.

Finally, the case of western Switzerland during spring is interesting. Though rare, clusters are responsible for almost all extreme precipitation accumulations (Fig. b). Over this region, floods are somewhat less frequent in spring than in winter (Fig. a and b), despite similar extreme precipitation likelihood (Fig. a and b). This may result from fewer rain-on-snow events, a major flood process for the region , but also drier soils coupled to high infiltration rates . Yet, spring floods can still be quite devastating, since precipitation generally falls as rain instead of snow, and limited vegetation cover makes erosion more likely. Consequently, cluster events that affect western Switzerland during spring should be the focus of further research.

Our approach to quantify links between clustered extremes and flood response has a number of limitations. First, we based our analysis on the exceedance of given precipitation and discharge percentiles, not taking into account either flood volumes or precipitation intensities. This is naturally quite restrictive and in particular fails to capture potential non-linearities in relationships above the selected percentiles. In addition, spatial variability in extreme discharge or average total cluster precipitation were not analyzed.

Our results only focus on the hazard component of flood risk. We do not take into account exposure and vulnerability, which may differ substantially between catchments due to variability in population density, infrastructure, flood management capacities, etc. Similarly, we did not take into account the influence of catchment regulation in this work. While the analyzed catchments are generally not heavily regulated ones, human influence may still be felt, especially when it results in the smoothing over time of extreme-discharge conditions, which would impact our analysis of persistent flood events.

Finally, our definition of floods and flooding persistence was also somewhat simplistic. First, while from the perspective of impacts it makes sense to define floods based on annual discharge percentiles, in snow-driven or glaciated catchments, this choice may discard potential high-discharge conditions occurring outside summer. Second, our definition of flooding persistence based on a minimal number of flood days in a given time window lumps together single, long floods and recurrent short ones, two kinds of events with potentially different impacts, and which may have to be managed differently.