## 3 Results
### Influence of the Atmospheric Stability on Wind Speed Gradients
#### 3.7.1 \"Undisturbed\" Horizontal Wind Speed Gradients  
Figure 13a,d refer to the thermally stable (solid red line) and unstable (solid black line) samples for the \"Ref\" transect for easterly and southerly winds. Similarly to the structure of the horizontal wind speed gradients in Figure 10a, stable (\"STA\") and unstable (\"UNS\") cases display a strong initial speed up with growing distance from land until an equilibrium is reached. The mean wind speeds are in fact significantly lower for the stable cases for both wind directions. This observation was already made earlier in Figure 12c,e,f, which showed that most of the distribution of the estimated parameters \(u_{land}\) and \(u_{offshore}\) associated with unstable conditions (colored blue dots) had higher magnitude. The green curves refer to thermal stability and indicate a difference between stable and unstable cases of roughly 2.5 K/m and 1.5 K/m, respectively, for easterly and southerly winds. Figure 12: Statistical distribution of estimated parameters \(x_{95\%}\), \(\Delta u\), \(R^{f}_{max}\), \(u_{land}\), \(u_{offshore}\) with respect to thermal stratification from combined wind directions. Histogram of the adjustment distance \(x_{95\%}\) (**a**). Scatter plots of wind increase \(\Delta u\) versus \(x_{95\%}\) (**b**) and versus \(u_{offshore}\) (**c**). Scatter plots of maximum normalised wind increase \(R^{f}_{max}\) versus \(x_{95\%}\) (**d**), versus \(u_{offshore}\) (**e**), and versus \(u_{land}\) (**f**). The hatched areas separate the top and low frames, which have different scaling in the vertical axis. Blue and red colors refer to unstable and stable cases, respectively. The curves of \(R_{u_{10}}\) displayed in Figure 13b,d show higher normalised increases by approximately a factor of two for the stable cases for easterly winds, whereas a factor of about 1.5 is found for southerly winds. For both wind directions, the \"UNS\" cases intensify rapidly from the coast, but stabilised wind conditions appear to be achieved over a shorter distance for southerly winds. Likely, the \"STA\" increases over an extended distance to adjust to steady wind conditions. The distance over which the wind keeps increasing is beyond \(\sim\)90 km (40 km) and \(\sim\)120 km (60 km), respectively, for \"UNS\" and \"STA\" from easterly (southerly) winds. The computed adjustment distances \(x_{95\%}\) considering easterly winds are of about 72 km and 115 km for unstable and stable thermal stratification, respectively. However, the estimated adjustment distance \(x_{95\%}\) from southerly winds is about 30 km and 50 km for \"UNS\" and \"STA\", respectively. Intuitively, this observation makes sense since higher turbulence can be expected to accelerate the adjustment processes in the boundary layer. Obviously, the adjustment distances from southerly winds are still relatively small in comparison to the values found for easterly winds. Although the wind speed increase from southerly winds seems relatively similar (not shown), the normalised wind speed increase \(R_{u_{10}}\) (Figure 13d) appears to be higher for \"STA\" than for \"UNS\". The estimated parameters are summarised in Table 3.  
Figure 13: **(a,b)**: Horizontal wind speed gradients (**a**) and normalised wind speeds (**b**) for stable (solid red) and unstable (solid black) cases from the “INCS” class for easterly winds. (**c,d**): The same as (**a,b**) for southerly wind directions. In (**a,c**) stable cases (red curves) refer to the scale of the right vertical axis (red) and unstable cases to the scale of left vertical axis (black). Green curves represent thermally stable (solid line) and unstable (dashed line) conditions and refer to the extra green scale on the right. The blue and orange dashed lines refer to the fitted empirical model. It is interesting to note that the normalised wind speeds for unstable conditions are quite similar for easterly and southerly winds with maximum values of about 30%. These are also the conditions with significantly higher wind speeds.