## 3 Results
### Classification of Shape of Wind Speed Gradients
The detailed inspection of the horizontal wind speed gradients for \"LINCS\" cases indicates that these are mostly satellite measurements taken in the morning (around 5 UTC). In the following, the rest of the analysis is mainly based on \"INCS\" cases since they are the most expected and frequent situations. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Shape Types** & **Samples (Percentage)** & \(<\)\(\Delta\)**\(u_{10}\) \(>\) (m/s)** \\ \hline \multicolumn{3}{c}{_Easterly wind_} \\ \hline Increasing (“INCS”) & 60 \% & \(\sim\)3 \\ Decreasing (“DECS”) & 22 \% & \(\sim\)\(-\)2 \\ Late increasing (“LINCS”) & 15 \% & \(\sim\)2 \\ \hline \multicolumn{3}{c}{_Southerly wind_} \\ \hline Increasing (“INCS”) & 62 \% & \(\sim\)2 \\ Decreasing (“DECS”) & 19 \% & \(\sim\)\(-\)1 \\ Late increasing (“LINCS”) & 15 \% & \(\sim\)2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Three different groups of the shape of the horizontal wind gradients found in four-year period for easterly and southerly wind directions. Distribution of the Parameters \(x_{95\%}\), \(R^{f}_{max}\), \(u_{land}\) and \(u_{offshore}\), and Atmospheric Stability Dependence  
The parameters \(x_{95\%}\), \(R^{f}_{max}\), \(\Delta u\), \(u_{land}\), and \(u_{offshore}\) are estimated from SAR derived \(u_{10}(x)\) transects with the empirical model Equation (10) to get an impression about their statistical distribution. The computation of these parameters are performed on the samples from the class \"INCS\" from combined easterly and southerly wind directions. Figure 12 shows respective histograms and scatter plots of these parameters. The histogram of the adjustment distance \(x_{95\%}\) in Figure 12 indicates that in most of the cases the adjustment distance is below 100 km, but still distances of 200 km and above can occur. As can be seen in Figure 12b, the wind speed increase \(\Delta u\) grows with the adjustment distance \(x_{95\%}\). The average \(\Delta u\) and \(x_{95\%}\) of about 2.6 m/s and \(\sim\)67 km are, respectively, estimated. The distribution in Figure 12b-f also illustrates the stability dependence of the parameters \(x_{95\%}\), \(u_{land}\), \(u_{offshore}\), \(\Delta u\) and \(R^{f}_{max}\) (Equation (16)). Overall, the blue and red dots refer to the parameters derived from \(u_{10}\) associated with unstable and stable thermal stratification, respectively. Figure 12c shows that the wind speed increase grows with the equilibrium wind speed offshore. According to the theoretical model [PERSON] [9] one would in fact expect a linear relationship, but as already explained in Section 2.2, this model is based on a number of simplifying assumptions. The growth of the wind speed increase with the wind speed offshore is seemingly higher for unstable than stable stratification. The behaviour of the parameter \(R^{f}_{max}\) as a function of \(x_{95}\), \(u_{offshore}\) and \(u_{land}\) is illustrated in Figure 12d-f. For better visualisation the scale for \(R^{f}_{max}\) is not continuous, but is split into two parts separated by the hatched area. The parameter \(R^{f}_{max}\) noticeably grows with increasing adjustment distance and drops for higher values of \(u_{land}\) or \(u_{offshore}\). It is interesting once again to remember that according to the theoretical model in Section 2.2 the normalised wind increase \(R^{f}_{max}\) should stay constant with regard to changes of either \(u_{land}\) or \(u_{offshore}\), if a constant surface roughness ratio is assumed. Figure 12f indicates that the stable cases are usually linked to lower land wind speeds and that higher normalised increases can be observed in these conditions. Figure 12d-f also show that the longest adjustment distances are found for stable conditions. As explained before, this is to be expected, because unstable conditions accelerate the adjustment process. In the following we have a closer look at the stability dependence of the horizontal wind speed gradients.