## 4 Discussion
The wind speed over land \(u_{land}\) and offshore \(u_{offshore}\) associated with thermally stable stratification are related to lower wind speed amplitudes. Higher values of \(R_{max}^{f}\) appear to be associated with higher \(x_{95\%}\), but are correlated with lower \(u_{land}\) and \(u_{offshore}\). However, the relationship between \(R_{max}^{f}\) and the wind speeds is in contradiction with the theoretical model presented in Section 2.2. The fact that higher values of \(R_{max}^{f}\) are observed for lower wind speeds could be an indication that the missing sea-state dependence of the ocean surface roughness is a relevant deficit of the model. Overall, higher values of the normalised wind speed increase are also found in stable conditions. The atmospheric stability has been proven to influence the structure of the horizontal wind speed gradients. Regardless of stability impacts, the horizontal winds commonly follow the usual two-phase process regarding the evolution of the wind field. The first phase originates in response to the wind crossing a step of two different surface roughnesses and characterised by an increase of wind speed [8; 9; 60]. The second phase is then determined by the stratification of the IBL above the sea. The adjustment distance \(x_{95\%}\) of about 75 km and 30 km for, respectively, easterly and southerly winds are estimated for the horizontal wind speed gradients associated with unstable conditions. For thermally stable stratification the adjustment distance \(x_{95\%}\) of about 115 km (50 km) for easterly (southerly) winds are found. Indeed, the adjustment distance is rather shorter in thermally unstable than stable conditions. This result is consistent with previous studies [7; 19; 62]. In unstable stratifications, the short distance is explained by the fact that strong mixing and turbulence force the winds to quickly adjust to an equilibrium and stable state. On the contrary, the induced strong wind shear in the stable stratification slows wind mixing and consequently extends the adjustment distance towards an equilibrium state. The parameter \(R_{u_{10}}\) is higher in general for stable than unstable conditions for both wind directions, but the average thermal stability is relatively lower for southerly winds. This is quite counter-intuitive, because one would expect smaller relative increases for higher stability as the airflow is more decoupled from surface roughness changes in this case. This could be another hint that nonlinear sea surface roughness effects related to sea-state development may be a factor to be considered in more details. In stable conditions and near the coast the wind speed appear to be lower for easterly winds than southerly ones. This difference indicates that the evolution of the boundary layer may be significantly different for easterly and southerly wind directions in stable conditions and this should be investigated in more details in future studies using additional information on vertical profiles. The thermal stability varies with different seasons. Strong stability occurs in spring, while strong instability are found in autumn [10; 13; 51; 63]. As the adjustment distance is affected by the atmospheric stability, it is also impacted by the seasonal variability of the atmospheric stratification. For instance, in early winter or autumn with unstable thermal stratification, the adjustment to equilibrium is accelerated. This leads to shorter adjustment distances in autumn than in spring, where stable conditions are expected. Horizontal wind speed gradients are obviously influenced by wakes generated behind the wind farm. The impact of the wakes are detectable in the first 20-30 km right behind the cluster N-4, characterised by a lower wind amplitude in comparison to the incoming wind speed upstream the wind farm. As expected, the wind speed gradients are weakly impacted by wakes during unstable stratification, since wakes are short or recovered in these conditions because of the turbulence and mixing [13; 17; 53; 64; 65].