## 2 Data and Methods
### Theoretical Background of Coastal Effects  
Typically, wind speeds increase with growing distance from the coast in situations where the wind is blowing from land to sea. The primary reason for this effect is that the land surface roughness is about two orders of magnitude higher than the roughness of the sea surface. The frictional forces are therefore reduced and the air in the boundary layer is accelerated. At the same time a transformation takes place between the boundary layers over land and the ocean, respectively. In the transition zone a so called IBL forms, which can extend to more than 100 km offshore until a balanced marine boundary layer is established. Details of this transition process depend not only on the ratio of the land and sea roughness length scales \(m=z_{land}/z_{ocean}\), but also on the stability of the boundary layer. In general, the boundary layer adjustment process is taking place over shorter length scales in unstable conditions. To illustrate these effects a simple atmospheric model as described in [PERSON] [9] can be used. The model describes the airflow over a step discontinuity of surface roughness. In the following we will denote the roughness lengths of the land and the ocean by \(z_{land}\) and \(z_{sea}\), respectively. The 2D model is based on the numerical solution of the momentum equation  
\[U\frac{\partial U}{\partial x}+W\frac{\partial U}{\partial z}=\frac{\partial \tau}{\partial z} \tag{4}\]  
and the continuity equation  
\[\frac{\partial U}{\partial x}+\frac{\partial W}{\partial z}=0 \tag{5}\]  
with the shear stress given by  
\[\tau=\kappa^{2}(z+z_{sea})^{2}(\frac{\partial U}{\partial x})^{2}\, \tag{6}\]  
where \(\kappa\) is the von [PERSON] constant. The horizontal wind speed component in \(x\) direction is denoted by \(U\) and the vertical component in \(z\) direction is represented by \(W\). Equation (6) corresponds to neutral conditions and is generalised in this study using a respective stability function \(\Phi\) according to  
\[\tau=\kappa^{2}\frac{1}{\Phi^{2}(z/L)}(z+z_{sea})^{2}(\frac{\partial U}{ \partial x})^{2} \tag{7}\]  
with Obhukov length \(L\). A similar approach was utilised in [PERSON] [8], where an additional prognostic equation for temperature was used. Following previous studies [35], the stability function was formulated for stable conditions (L\(>\)0) as  
\[\Phi(z/L)=1+5z/L \tag{8}\]  
and for unstable conditions (L\(<\)0) as  
\[\Phi(z/L)=(1-16z/L)^{-1/4}. \tag{9}\]  
More details about the numerical treatment used to solve the equations above can be found in [PERSON] [9]. Figure 3 shows a simulation of vertical profiles of normalised wind speeds (a) and normalised stresses (b) for neutral conditions (L = \(\infty\)) and a roughness ratio \(m=55\). These are extended versions of plots, which can be found in [PERSON] [9]. The black dashed line corresponds to the upstream horizontal wind speed profile over land. One can see that the highest wind speed increases near the coast are found in the lower levels and growing wind speeds at higher levels are found further offshore. The friction velocity, which is the square root of the normalised horizontal stress, shows a strong drop near the coast and then slowly recovers to higher values. As the model assumes identical momentum fluxes into the boundary layer over land and ocean from above, the friction velocity profile that eventually forms far offshore is the same as over land (dashed black lines). The order of magnitude for sea surface roughness lengths can be seen in Figure 3d, where z\({}_{sea}\) is plotted as a function of wind speed at 10 m height making the simplifying assumption that the sea state is fully developed [36]. Assuming that the roughness length z\({}_{sea}\) is 0.1 cm, the blue dashed curves in Figure 3a,b correspond to a distance of about 10 km to land and the boundary layer is obviously still far from an equilibrium at that stage. In Figure 3c, horizontal wind speed profiles of wind speeds are plotted at two normalised heights.