## 3 Results
### Joint Probabilities of Hydroclimatic Conditions and Wind Speeds  
Based on the discussion in Section 1, it is clear that the closing velocity between hydrometeors and wind turbine blade and the kinetic energy transfer to the blades are maximized under the joint occurrence of high RR with a large fraction of high diameter droplets (Figure 5) and/or hail and high wind speeds when the blades are likely to be rotating at their maximum speed. Accordingly, any blade lifetime estimate is going to be critically contingent on the joint distributions of RR (and DSD and phase) and WS. Examples of the joint distributions of RR and hub-height wind speed are given in Figures 10-13 for four illustrative sites; the US SGP, coastal Canada, coastal Norway and the onshore site in Denmark. In construction of these illustrative joint-probabilities, wind speeds are assumed to be constant over each 1 min period within each t-minute periods over which the wind speed data are collected. Laboratory studies and numerical models have repeatedly shown that impact responses on the wind turbine blade coating system increase with \"increasing droplet size and increasing droplet impact angle with maximum impulses, stresses and damages developed for normal impingement (90\({}^{\circ}\))\" [15]. A numerical modeling study found that the number of impacts required for onset of erosion damage increases by a factor over seven when impact velocities are decreased from 140 ms\({}^{-1}\) to 80 ms\({}^{-1}\) under RR \(>\) 25 mmhr\({}^{-1}\)[15]. Hence, in much of the following discussion we focus on the relative frequency of joint occurrence of high RR and high wind speeds. Figure 10: Heatmaps of the joint probabilities (in %, colorbar and also numeric values shown in each cell) of RR (rain only) and wind speeds at the US SGP site. The joint probabilities are computed for periods with precipitation and thus the sum of all cells is 99.8% since 0.2% of wind speeds are above 25 ms\({}^{-1}\). Data for periods with hail (WC = 88 or 89) are processed separately and are shown in the lower panel. The values for hail sum to 98.8% because a few hail events are associated with wind speeds at 90 m in excess of 25 ms\({}^{-1}\). At the US SGP site >2.4% of 1 min precipitation events, or 0.1% of all 1 min periods, are associated with wind speeds at which typical wind turbines will be rotating at their maximum speed (or close there-to) (WS = 10-25 ms\({}^{-1}\)) and RR > 5 mmhr\({}^{-1}\) (Figure 10). Conversely at the coastal Canada site (WEICan) only 1.2% of 1 min precipitation events, which is 0.09% of all 1 min periods, are associated with these conditions (Figure 11). At the site in Bergen (noting the WS are from the NORA3 hindcast), 2.3% of all 1 min periods with precipitation and 0.47% of all 1 min periods in an average year are associated with WS between 10 and 25 ms\({}^{-1}\) and RR > 5 mmhr\({}^{-1}\) (Figure 12). At the site in Denmark 0.46% of all 1 min periods with precipitation and 0.04% of all 1 min periods in an average year are associated with WS between 10 and 25 ms\({}^{-1}\) and RR > 5 mmhr\({}^{-1}\) (Figure 13). Figure 11: Heatmaps of the joint probabilities (in %, colorbar and also numeric values shown in each cell) of RR and wind speeds at coastal Canada. The joint probabilities are computed for periods with precipitation and thus the sum of all cells is 99.8% since 0.2% of wind speeds are above 25 ms\({}^{-1}\). Focusing on the even more extreme portion of the joint distributions; RR \(>\) 10 mmhr\({}^{-1}\) and WS of 12.5-25 ms\({}^{-1}\), these occur on average on \(\sim\)200 min per year at the US SGP site, on \(\sim\)60 min in an average year at coastal Canada and 30 min in Denmark. Thus, all other things being equal material stresses in blades on wind turbines operating in the US SGP will be higher than at these other two locations. The implied enhanced material stress at the US SGP site is amplified by the high frequency of co-occurrence of WS of 12.5-25 ms\({}^{-1}\) and hail (\(\sim\)100 min in an average year).