## 3 Results
### Classification of Shape of Wind Speed Gradients  
The detailed analysis of the horizontal wind speed gradients for easterly wind directions suggests that they can be classified into three main categories:  
1. The horizontal wind speed transects of \(u_{10}\) that increase with distance to the land (Figure 10a). They are hereafter referred to as \"INCS\". 2. From the coast to offshore, the horizontal wind speed transects of \(u_{10}\) that show a wind speed decrease with growing distance to land (Figure 10b). In the following these cases will be called \"DECS\". 3. Some horizontal wind speed transects of \(u_{10}\) display a delay in the wind speed increase, which begins further offshore (Figure 10c). These types are hereafter referred to as \"LINCS\" (Late INCreasing Samples). There remains only a very small percentage of the samples (\(\sim\)3%), that do not exhibit any clear trend and are thereby not classified according to the three categories defined above. As expected, the class \"INCS\" represents the highest percentage of the cases with about \(\sim\)60%. The \"DECS\" and \"LINCS\" classes represent, respectively, about 21% and 15% of the total samples (Table 2). This distribution is confirmed by the superimposed error bars for the three categories (gray vertical segments in Figure 10), which refer to the standard deviation of the estimator for the mean. The error bars are computed by dividing the sample standard deviation by the square root of the sample size. They indicate  
Figure 10: (**a–c**): Mean wind speed gradients for classes “INCS” (**a**), “DECS” (**b**) and “LINCS” (**c**) for easterly winds. Air and sea surface temperatures from DWD are, respectively, shown as dashed and solid green lines. (**d–f**): The same as (**a–c**), but for southerly winds. The blue and orange dashed lines in (**a,b,d**) refer to the fitted empirical model. the interval of confidence, which is at the lowest amplitude for the \"INCS\" samples, but becomes increasingly larger moving from \"DECS\" to \"LINCS\". The error bars around the mean wind speed gradients are bigger for southerly than for the easterly cases, because the uncertainty of the mean estimator grows due to smaller sample size. The average of \(u_{10}\) for the class \"INCS\" with easterly and southerly wind directions (Figures 10a,d and 11a) shows an increase above 2.5 m/s and 1 m/s, respectively. By fitting of the empirical model the increase \(\Delta u\) of 2.9 m/s (2.3 m/s) and an adjustment distance \(x_{95\%}\) of 90 km (38 km) for easterly (southerly) winds are found (Figure 11b,c). This indicates that the adjustment towards the equilibrium occurs over a shorter distance for southerly than easterly wind directions. Although the wind speeds far offshore are about the same magnitude (Figures 10a,d and 11a), the relative increase (Figure 11b) is also lower for easterly than southerly winds as the wind speed close to land is slightly higher along the east coast of the German Bight. The normalised relative increase (Figure 11c) for easterly and southerly winds exceeds, respectively, 40% and 30% far offshore. The parameters \(\Delta u\) and \(x_{95\%}\) for \"INCS\" cases are higher than the respective values estimated for the overall data set (Section 3.4). Interestingly, wind speed transects of \(u_{10}\) for the class \"INCS\" seem to be associated with strong instability of the boundary layer (Figure 10a,d). For the class \"DECS\", the wind speed transects of \(u_{10}\) show an absolute decrease of about 2 m/s and 1 m/s for easterly and southerly winds, respectively, (Figure 10b,e). For both wind directions, the mean of \(u_{10}\) for the class \"LINCS\" shown in Figure 10c,f seems to behave like the \"DECS\" mean in the first 40 km after which the wind speed starts increasing. The average magnitude of increase is about 2 m/s over 150 km offshore. Compared to \"INCS\", the thermal gap is reduced between the air temperature and the SST, which ultimately lowers the instability of the boundary layer for the classes \"DECS\" and \"LINCS\".