The main data source for this work is simulations performed using HiGEM , a fully coupled high-resolution climate model based on the HadGEM1 configuration of the Met Office Unified Model . The horizontal resolution of the HiGEM atmospheric component is 0.83∘ latitude × 1.25∘ longitude (N144) (∼90 km in mid-latitudes) with 38 vertical levels up to 39 km. The horizontal resolution of the ocean component is 1/3∘×1/3∘ (∼30 km) and is considered to be eddy-permitting. A total of 918 years of HiGEM data are available with a 6-hourly temporal resolution. These data come from a series of four-member ensemble decadal hindcasts initialized between 1960 and 2006, and also four 59 year transient experiments initialized in 1957. Full details of the data used are described in . HiGEM has been shown to have a good representation of the North Atlantic storm tracks, and also the representation and distribution of extratropical cyclones .

For comparison, the reanalysis from the European Centre for Medium Range Weather Forecasts (ECMWF) ERA-Interim dataset is used. ERA-Interim data are available at a 6-hourly resolution, starting from January 1979, and have a T255 spectral horizontal resolution (∼80 km) with 60 vertical eta levels up to 0.1 hPa. Therefore, the resolution of ERA-Interim is comparable to that of HiGEM. In total, 36 years of ERA-Interim data are used, from 1979 to 2015. As the focus of this study is on wintertime losses resulting from extratropical cyclones our analysis will be constrained to the months of December, January, and February (DJF).

To identify extratropical cyclones in both datasets we use the tracking algorithm of applied in the same way as . This method tracks features using maxima in the 850 hPa relative vorticity field in the Northern Hemisphere. Prior to tracking the vorticity field is spectrally truncated to T42, which reduces the noise in the vorticity field. The large-scale background is also removed by removing total wavenumbers ≤5 in the spectral representation . Maxima in the vorticity field (i.e. cyclonic features) are identified every 6 h and formed into tracks. This is initially done by using a nearest neighbour approach to initialize the tracks, which are then refined by minimizing a cost function for track smoothness that is subject to adaptive constraints on the smoothness and displacement in a time step . Pressure minima are also associated with the tracks using a minimization technique within a 5∘ radial cap. In order to exclude very small-scale, noisy tracks, only tracks that travel at least 500 km and have a lifetime greater than 24 h are retained. In previous dynamical clustering studies the method of has been applied. Both cyclone tracking methods perform similarly for the tracking and clustering of North Atlantic cyclones, though the method generally tends to produce lower clustering values over the North Atlantic/European sector .

We follow the method of to calculate composite fields of RWB and the upper level jet on clustered days. RWB is calculated using the 2-D Blocking Index method from , which identifies overturning of potential temperature contours on the 2 PVU surface (dynamical tropopause; 1 PVU =1×10-6 K m2 kg-1 s-1). The upper level jet is identified as regions of high wind speed on the 250 hPa surface. These fields are composited for cyclones passing through three 700 km radii at different latitudes centred on 5∘ W; 45, 55, and 65∘ N, to focus on the impact for various locations in western Europe.

The metric developed by is used as a loss proxy for European windstorms. The SSI has been used in numerous other studies for similar purposes . It uses 10 m wind speeds in its calculation of storm severity. We follow the approach of the population-weighted SSI as used by and . The formulation of the SSI is defined in Eq.  and is constructed as follows:

Losses due to wind occur on approximately 2 % of all days ; therefore, for any losses to be produced the wind speed (Vi,j) must exceed the 98th percentile of the wind speed distribution.

SSI=∑i=1Ni∑j=1NjVi,jVi,j98-13⋅Ii,j⋅Li,j⋅popi,jVi,j=max⁡Vi,j,tifVi,j≥9ms-19ms-1ifVi,j<9ms-1,t=timeperiodIi,j=0ifVi,j<Vi,j981ifVi,j≥Vi,j98Li,j=0overseas1overlandpopi,j=Population density per grid box

The SSI is calculated at every land grid point and is used in two different forms for the main analysis in this study. Following insurance industry naming conventions, the first approach will be to calculate the maximum loss event in a year (herein referred to as the occurrence exceedance probability, OEP), and the second will calculate the total loss for an entire DJF season (herein referred to as the annual exceedance probability, AEP).

The OEP is calculated as the spatial sum of the maximum SSI within a 72 h period (i.e. the maximum SSI calculated from the 6-hourly wind speeds in a 72 h period, per grid point). The 72 h time window is consistent with that used by reinsurance companies for defining a particular event . The region used to calculate the OEP is an adapted version of the meteorological index (MI) box applied by . Our OEP region extends 10∘ W–20∘ E, 35–60∘ N and covers all of western Europe and most of central Europe. Our region differs from that of as it extends further south to 35∘ N, in order to encompass the Iberian peninsula and also all of Italy (shown by the black box in Fig. b).

The AEP is calculated in the same way as the OEP, except that instead of using the single 72 h maximum wind footprint, it sums all the individual 72 h maximum wind speed footprints in the 90-day winter period. In the calculation of the AEP all events are retained. The sensitivity to retaining all events is tested later.

There are several ways to assess the clustering of windstorms, which give different information and perspectives. Described below are the three methods which will be used in this study.

A majority of the results in this paper will be expressed in terms of return period. The return period provides a period of time in which an event of a certain magnitude is expected to occur. Return periods have been allocated in a way such that the maximum AEP year is assigned a return period of the length of the dataset divided by its rank (for the maximum event the rank is 1). Therefore the maximum AEP year from ERA-Interim has a return period of 36 years, and the highest AEP year in HiGEM has a return period of 918 years, as they both occur once in their total time period respectively. The second largest events then have return periods of 18 and 459 years for ERA-Interim and HiGEM respectively. This continues until the lowest ranked year, which has a return period of 1 year. For a majority of our analysis we rank the OEP in the order of descending AEP. This ensures we maintain a temporal connection between the OEP and AEP at all return periods and means that the largest OEP may not necessarily occur in the highest AEP year. Some analysis is performed on independently ordered AEP and OEP, which removes the connection between maximum events and the years in which they occur.

The return periods of the most extreme events are estimated using a generalized Pareto distribution (GPD) that is fitted to the AEP and OEP data above a specified threshold (“peak over threshold” method). The GPD is fit using the maximum likelihood method, following , and . Uncertainties at the 95 % level are calculated using the delta method .

HiGEM has a good representation of the large-scale tropospheric circulation and also extratropical cyclone shape, structure, and distribution . Several other studies have demonstrated that the model has skill in seasonal to decadal predictions, particularly in the North Atlantic . HiGEM provides a good representation of both the structure and amplitude of the DJF North Atlantic storm track (Fig. a, c, e). The characteristic tilt of the North Atlantic storm track is evident as cyclones in HiGEM (Fig. c) follow the SW–NE path found in ERA-Interim (Fig. a). In addition the maxima in storm numbers off the coast of Newfoundland and also over the Irminger Sea agree (anomalies within ± two cyclones per month) with ERA-Interim (Fig. e). There is an anomalous extension of the storm track in its exit region in HiGEM across Denmark and northern Germany; however the amplitude of the anomaly is small (< two cyclones per month). On the larger scale there are minimal biases present across the entire basin and the European continent. There are localized errors that are mostly below two cyclones per month when compared to 36 years of ERA-Interim reanalysis (consistent with ). The structure and amplitude of the Mediterranean storm track are also well captured. Stippling in Fig. e indicates where HiGEM and ERA-Interim are different at the 95 % level (performed using a two-tailed Student's t test). These differences are only present around the coast of Greenland and are associated with the minima in the Labrador Sea and Davis Strait, and also the maxima across the east coast of Greenland. None of the anomalies across the rest of the North Atlantic or Europe are statistically significant.

The dispersion of cyclones in the North Atlantic for ERA-Interim and HiGEM is shown in Fig. b and d respectively. The pattern of the dispersion is consistent for the two datasets, with both being characterized by a more regular behaviour in the entrance of the storm track (western North Atlantic) where storms have their main genesis region and baroclinic processes are dominant. Both datasets show overdispersive (clustered) behaviour in the exit of the storm track, e.g. the UK and Iceland. The pattern of under/overdispersion in the exit/entrance region of the storm track is comparable with the studies of . There are discrepancies in the magnitude of the dispersion (Fig. 1f); however, the large-scale pattern and sign of the dispersion are consistent between the two datasets.

It was shown in that clustering events occurring at different latitudes of western Europe are associated with a specific set of dynamical conditions (as discussed in Sect. 1). The clustering periods identified are characterized by a strong and extended upper level jet that was associated with anomalous RWB on one or both flanks, which acts to drive the jet further north or further south and then anchor it in position. These persistent conditions allow the cyclones to track in similar directions and leads to clustering over different regions of western Europe. This analysis has been repeated for all 918 years of HiGEM data in order to assess how well HiGEM dynamically represents these events (Fig. ). The same analysis of ERA-Interim is shown in the Appendix (Fig. ). Figure a shows that for cyclones clustering at 65∘ N, the jet is extended toward the northern UK with speeds in excess of 40 ms-1. The jet is associated with large amounts of RWB on its southern flank. For clustered days at 55∘ N (Fig. b), the extended jet is more zonal than for events at 65∘ N. The strong jet has anomalous RWB on the northern and southern flanks; however the RWB on the southern flank is of a smaller magnitude than that seen in ERA-Interim (Fig. b). The clustered events at 45∘ N show a very zonal upper level jet with a dominance of RWB on the northern flank.

All of the composites of clustered days (Fig. a–c) show a marked departure from the climatology (Fig. d). The jet is stronger and more zonally extended than the climatology for all three cases, and each features anomalous amounts of RWB on one or both flanks of the upper level jet, all of which are comparable to ERA-Interim. The climatological state of the jet and RWB in HiGEM (Fig. d) is comparable with ERA-Interim (Fig. d), with a slightly reduced amount of RWB across central Europe. Hence, clustering events in HiGEM appear dynamically consistent with those in ERA-Interim, albeit with a lower frequency of anticyclonic RWB on the southern flank of the jet.

HiGEM has been found to have a good representation of North Atlantic extratropical cyclones and cyclone clustering, as well as the large-scale circulation driving this behaviour. This demonstrates the suitability of using HiGEM to investigate clustered windstorm-related losses.

The SSI is widely used for quantifying losses related to windstorms. We now compare the SSI for both HiGEM and ERA-Interim. The characteristic of the SSI is that it is calculated above a set threshold, the 98th percentile of the local distribution of 10 m wind speed (Vi,j98). The structure of Vi,j98 for ERA-Interim, HiGEM, and the difference between the two datasets is shown in Fig. . Both datasets show a similar large-scale structure, with maxima over the North Atlantic Ocean and minima over the high orography of the Alps and Pyrenees. However, HiGEM values are systematically lower than ERA-Interim across almost all of Europe by 1–3 ms-1 (Fig. c), whereas across the North Atlantic Ocean the bias is smaller and slightly positive. This systematic difference suggests there may be differences in the boundary layer scheme of HiGEM compared to ERA-Interim as this bias is not present for wind speed at 850 hPa or higher (not shown). Similar differences have also been found in the biases of 10 m and 925 hPa wind speeds in four reanalysis datasets .

To address the lower European wind speeds, a simple bias correction is applied to the 10 m wind speeds in HiGEM. This is done by correcting the 10 m wind speeds by the spatially averaged offset in the Vi,j98 field between ERA-Interim (Vi,j98ERA-I‾) and HiGEM (Vi,j98HiGEM‾) for all land grid points within our area of interest (black box in Fig. b). As a result all HiGEM wind speeds are uniformly increased by 18.75 % over land. The resulting bias-corrected HiGEM 10 m wind speeds will be called HiGEM_bc herein and its formulation is shown in Eq. (). The corrected Vi,j98 wind field (difference relative to ERA-Interim) is shown in Fig. d and shows much reduced differences across our core European region. In some regions (northern Germany, Benelux, northern and northwestern France) the differences have changed sign and are now positive, and there are also some regions that still have negative anomalies, resulting in an overall neutral anomaly compared to ERA-Interim. HiGEM_bc=HiGEM⋅Vi,j98ERA-I‾Vi,j98HiGEM‾

Spatial maps of the DJF average SSI are shown in Fig.  and are consistent between ERA-Interim and HiGEM across large parts of northern and northwestern Europe. Both datasets show a peak in SSI across northwestern Europe from London across to northern and northwestern Germany, as would be expected from the population weighting. Other densely populated regions are also identifiable. There are further regions of noticeable SSI across Germany and extending east toward Russia. There are also peaks in SSI across the Iberian peninsula and Italy.

Figure  shows the AEP from ERA-Interim and HiGEM as a function of return period, up to a maximum return period of 36 years. Also shown in Fig.  are 10 000 bootstrap samples of the 918 years of HiGEM_bc AEP data in 36-year samples, and the associated 95 % confidence intervals. The AEP from ERA-Interim is within the confidence intervals of the HiGEM_bc samples at all return periods, with ERA-Interim being at the upper end of the spread at the lowest return periods of 1–2 years, and in the middle of the spread for the remaining return periods. Figure  suggests that HiGEM_bc can capture the variation of AEP as a function of return period that is found in ERA-Interim.

In Fig.  we have ordered the AEP and OEP independently by return period in order to assess how the SSI from ERA-Interim and HiGEM directly compares in magnitude at varying return periods. Also shown in Fig.  are the GPD fits of the ERA-Interim AEP and OEP, and associated confidence intervals (non-filled). The GPD provides a good estimation of the data above a 5-year return period; however due to the small amount of data used to fit the distribution, uncertainties start to become very large above a return period of 20 years. By a return period of 50 years they have diverged greatly.

Also shown in Fig.  are the 95 % confidence intervals for the GPD fits (shaded regions) of the HiGEM_bc AEP and OEP (ordered independently). These are shown above a return period of 10 years. The confidence intervals for HiGEM_bc are much narrower than ERA-Interim, which is to be expected as there is considerably more data being used in the GPD fit (consistent with ). For return periods for which both datasets have data (<36 years) the GPD fit of the HiGEM_bc AEP and OEP are within the confidence intervals of the ERA-Interim AEP and OEP. This is also the case for return periods greater than 36 years for the AEP and the OEP, although the confidence intervals for the GPD fit for ERA-Interim become extremely large for return periods greater than 50 years. HiGEM_bc is therefore consistent with the accumulated seasonal losses and the individual events found in ERA-Interim. This suggests HiGEM is a useful climate model for investigating AEP and OEP for large return periods.

Figure  shows the AEP and OEP for HiGEM_bc. Both the AEP and OEP curves of HiGEM_bc extend beyond the respective maxima from ERA-Interim, suggesting that more severe windstorm seasons and also single cyclone events may be possible than those seen in the ERA-Interim period. For example, the 918-year return period season in HiGEM is approximately twice the magnitude of the 1 in 36-year season in ERA-Interim. There is more noise in the ERA-Interim curves compared to their HiGEM_bc counterparts due to the smaller number of years. The OEP and AEP of HiGEM_bc are sorted by AEP magnitude in Fig. . Consequently, there is substantially more spread in the OEP values than seen in Fig. . However, a general increase in OEP with return period is still found, with low AEP years generally having a lower OEP and high AEP years having a higher OEP, but there are some specific deviations from this (for example, note the four very high OEP years between return periods 40 and 100).

To test the sensitivity of Figs.  and to the default definition of AEP as the sum of all events, we repeated the analysis, but only retaining on average the top three events of each year. This equates to an order of magnitude reduction in the number of events (see Fig. ). The magnitudes of the reduced event AEPs are marginally lower than the original AEPs, as would be expected with the filtering of events, but the main features of the curves in Fig.  are very similar.

previously showed how the number of cyclones within a winter is strongly related to their intensities. Similarly, earlier results (Fig. e) showed that cyclones in HiGEM tend to occur in groups. To quantify the contribution of windstorm clustering to the large AEP values we compare the HiGEM AEPs to randomized series of loss events. The ratio between the AEP and the AEP_random may be particularly important for the insurance industry as it characterizes the importance of the clustering of cyclones in seasonal losses.

We have performed a non-replacement randomization of the 72 h periods that make up the HiGEM_bc AEP with 10 000 samples, and this randomization ensures that each random sample contains the exact same data as the original 918 years. This randomization allows us to assess how the intensity of losses and the associated number of cyclones acts to influence the AEP in HiGEM_bc, compared to a time series in which windstorms are occurring randomly. The mean of these random samples is shown by the black line in Fig. , with the grey shading indicating the 95 % confidence interval of these samples. Below a return period of ∼3 years the AEP_random is greater than that from the HiGEM_bc AEP, and above the 3-year return period the AEP_random is consistently less than the AEP. Therefore, low (high) return period loss years tend to have a lower (higher) AEP than random.

The ratio of AEP to AEP_random is shown in Figure . Values >1 (<1) indicate a higher (lower) AEP in the actual HiGEM_bc years compared to the AEP_random years. Above a return period of 3 years the realistic, dynamically consistent, representation of the grouping of events in HiGEM_bc tends to generate more losses and a greater AEP than the random grouping of events. The median contribution above the 3-year return period is generally in the range of 1.1 to 1.2 times the random AEP. At a return period of 200 years the 95 % confidence intervals range from 1 to 1.3. For low return periods (<3 years) the occurrence of events in HiGEM_bc leads to a lower AEP than in the random realizations, with HiGEM_bc AEP values in the range of 0.6 to 1. This suggests that during low loss winters, there are a smaller number of windstorms and weather loss events occurring in HiGEM_bc than would be expected from considering a randomized series of events.

The difference can be interpreted physically by considering two recent DJF periods in the UK. Firstly, the winter of 2009/2010 was characterized by a strongly negative North Atlantic Oscillation and an absence of extratropical cyclones influencing the UK for this period . Secondly, the winter of 2013/2014 was almost the complete opposite and was associated with the continuous presence of deep cyclones, occurring in groups, for almost the entire DJF period . The nature of these two seasons would result in 2009/2010 having a very low AEP, and 2013/2014 having a very high AEP. Randomizing these two seasons, the result would be two synthetic seasons with AEP values between these two extremes. Hence the clustering of cyclones in 2013/2014 results in a higher AEP than expected from a random distribution and 2009/2010 having a lower AEP than expected.

In Fig.  it appears that the realistic, dynamically consistent, representation of clustering in HiGEM_bc causes higher losses above a 3-year return period, with losses being 10 %–20 % higher than a random distribution at all return periods. Despite the nearly constant value of AEP / AEP_random above a return period of 3 years, the absolute difference between the two values is increasing with return period (the AEP is more than 4 times larger at a 918-year return period compared to a 3-year return period). Hence the physically consistent representation of clustering in HiGEM_bc is causing larger increases to the AEP with increasing return period.

A different view of clustering can be gained by examining how a single event can affect the accumulated seasonal losses through the ratio of the OEP to AEP. A high value implies that the single largest event is causing most of the losses in a season, and a lower value implies a contribution to the overall seasonal losses from many cyclones in that particular season. The results from ERA-Interim and HiGEM are compared in Fig. a. This shows 10 000 random samples of 36 years of the HiGEM_bc OEP / AEP with associated confidence intervals as well as the ERA-Interim OEP / AEP. The values are sorted into return periods by order of descending AEP. There is considerable spread in the ERA-Interim OEP / AEP values, with a minimum of ∼0.2 at a return period of 1–2 years, and a maximum of ∼0.8 at a return period of 10 years. There is no clear systematic increase or decrease in the value of OEP / AEP in the ERA-Interim data, which is consistent with the median values and confidence intervals from the HiGEM_bc samples. The 95 % confidence intervals range from 0.15 to 0.65 at a return period of 1 year, and from 0.2 to 0.8 at a return period of 36 years. This suggests that there is a wide range of OEP / AEP values that may characterize a high or a low AEP season.

Figure b shows the OEP / AEP values for HiGEM_bc. As in Fig. a there is considerable spread variation in the value of OEP / AEP at all return periods, with no clear systematic increase or decrease with return period. All return periods have a majority of the data with values of ∼0.25–0.5, with the extremes ranging from 0.15 to 0.9. This indicates that in terms of the contribution of a single event to the overall seasonal loss, there is no direct relationship with return period. At any return period it appears that 25 %–50 % of losses will come from the largest event.

As with the ratio of AEP / AEP_random in Fig. , the ratio of OEP / AEP in Fig. b has a relatively constant value at all return periods and this suggests a constant relationship between the OEP and AEP. However, as the return period is increasing, the OEP and AEP are also increasing, so despite the AEP / AEP_random ratio being constant at a return period of 5 and 200 years, the absolute difference between OEP and AEP at the two return periods would be very different. The higher return periods have a higher absolute difference between the AEP and OEP; hence the additional losses that are not the OEP are increasing with return period. Hence the relative difference between the OEP and AEP is consistent with return period, but the absolute difference is continuing to increase with return period. This absolute increase is likely a result of more severe events in the high return period AEP years.