## 2 Data and Methods
### Downwind Horizontal Wind Speed Gradient Estimations  
In this study, surface wind fields derived from SAR data (Section 2.1) with a particular focus on offshore wind directions, i.e., winds blowing from land towards the sea, are analysed. For the land boundaries of the German Bight, offshore winds can be found for a wide wind direction sector of more than 90\({}^{\circ}\) from northeast to southwest (see Figures 1 and 4). Atmospheric wakes downstream of wind farms in the German Bight (see Figures 1 and 2b) are superimposed on coastal effects especially for southwesterly wind directions. Cluster N-4 (Figure 1) is relatively isolated from neighbouring wind farms and the surrounding area was therefore chosen to analyse coastal effects with and without wake interaction. The investigation of the horizontal wind gradient focuses on southerly and easterly wind directions. The geometric structures of the southern and eastern coastlines of the German Bight are different. For the eastern coastline the barrier islands are distributed over a larger area and the coastline therefore has a more fuzzy appearance. As we suspected that these differences have an impact on the boundary layer transition process, both wind directions were analysed separately (Figure 2b). More precisely, we consider easterly and southerly winds with wind directions ranging, respectively, from 60\({}^{\circ}\) to 120\({}^{\circ}\) and from 160\({}^{\circ}\) to 200\({}^{\circ}\) (see Table 1) to increase the probability to collect SAR wind field acquisitions with respect to offshore winds for more robust statistics. To closely follow the evolution of the offshore winds with growing distance from the shore, two transects are defined parallel to the wind direction coming from the land: (1) One transect (named \"Ref\") is considered in the area that is free of wind farms in order \begin{table}
\begin{tabular}{c c c} \hline \hline
**Offshore Winds** & **Wind Direction Range (\({}^{\circ}\))** & **Samples** \\ \hline Easterly & 60 \(\leq\,w_{d}\,\leq\) 120 & 70 \\ Southerly & 160 \(\leq\,w_{d}\,\leq\) 200 & 47 \\ All wind directions & & 447 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Offshore wind directions, the corresponding interval and samples considered for the study. Figure 4: Wind rose computed from DWD data for the wind speed at 10 m height around the cluster N-4 location. to minimise their effects on the horizontal wind speed profiles (Figure 2b); (2) A second transect is placed in such a way that it crosses the wind farms (e.g., cluster N-4 for easterly and southerly wind) to analyse potential interactions of coastal effects with the atmospheric wakes. This transect is hereafter named \"\(N\)4\" as it crosses the cluster N-4. The width of the transects are chosen to be 13 km, which is wide enough to represent the large-scale variability. Furthermore, the width of the transects is large enough to include wake effects, as the length scale of most wind farms in the German Bight is of the order of 10 km. It is well known that SAR data in very shallow water can be affected by oceanic effects, which are not related to surface wind speeds [37]. As the water depths in the German Bight close to land is in fact small for the most part (see Figures 1 and 2b), the start point of the transects on the land side is defined as the 10 m water depth isobath. The bathymetry effects on the surface roughness and the retrieved wind speed as reported in Horn Rev by [PERSON] and [PERSON] [38] are negligible around the cluster N-4 and do not impact the results. The elongated transects are subdivided into 2 km bins, which leads to boxes of 2 km \(\times\) 13 km each. The wind speed is therefore spatially averaged over each 26 km\({}^{2}\). This finally leads to one-dimensional horizontal SAR wind speed transect of \(u_{10}\) that depends on the distance from the shore.