## 3 Analysis
This was defined as  
\[Ri=\frac{g}{T}\frac{\partial\theta/\partial z}{(\partial v/\partial z)^{2}}, \tag{3}\]where \(g\) is the acceleration of gravity, \(T\) is the mean absolute temperature, \(z\) is the height normal to the surface, and \(\partial v/\partial z\) is the mean velocity the gradient. The \(z\)-direction was assumed to be vertical. The potential temperature (\(\theta\)) is defined as  
\[\theta=T\left(\frac{p_{0}}{p}\right)^{R/c_{p}}, \tag{4}\]  
where \(p\) is the pressure, and \(p_{0}=1000\) hPa is the reference pressure, \(R=289\ \frac{\mathrm{I}}{\mathrm{Kg\,K}}\) is the specific gas constant, and \(c_{p}=1005\ \frac{\mathrm{I}}{\mathrm{Kg\,K}}\) is the specific heat capacity of air. It has to be noted that we checked for the influence of humidity and found that the influence on density as well as specific humidity [5] was below 1%. We therefore neglected all moisture corrections to Richardson's Number (Ri). In Equation (3), the quantity \(\frac{g}{2}\partial\theta/\partial z\) describes the forces introduced by heat transfer in the boundary layer. The term \((\partial v/\partial z)^{2}\) represents the momentum forces in the boundary layer. If we consider two points in the boundary layer at heights \(z_{1}\) and \(z_{2}\) where \(z_{1}<z_{2}\) and \(\Delta z=z_{2}-z_{1}\), \(Ri\) from (3) can be rewritten with the differences between the two locations as  
\[Ri=\frac{g}{T}\frac{\Delta\theta/\Delta z}{(\Delta v/\Delta z)^{2}}=\frac{g}{ T}\frac{(\theta_{2}-\theta_{1})/\Delta z}{((v_{2}-v_{1})/\Delta z)^{2}}. \tag{5}\]  
We always assumed the presence of positive velocity gradients so only the convective part of Equation (5) remained. Three typical situations were then distinguished as follows:  
\(\theta_{2}>\theta_{1}\)The surface is colder than the fluid and the gradient becomes \(\partial\theta/\partial z>0\) and therefore, \(Ri>0\). Heat is transported by conduction only, and a convection flow does not occur. In this case, the stratification is strong, and turbulence gets damped. The boundary condition is stable for \(Ri>0\). \(\theta_{2}=\theta_{1}\)The temperature gradient is zero and therefore, \(Ri=0\). There is no temperature gradient and therefore, no conduction nor convection. This condition is called neutral. \(\theta_{2}<\theta_{1}\)The surface is warmer than the fluid and the gradient becomes \(\partial\theta/\partial z<0\) and therefore, \(Ri<0\). Heat is transported by conduction and by convection from the surface to the fluid. The convection results in a vertical, upward component of the flow that interacts with the horizontal velocity component. This leads to the production of turbulence in the boundary layer and therefore, is called unstable. From our measurements we obtained the necessary values for \(p\), \(T_{air}\) at a height of 100 m, wind speed \(v_{air}\), and wind direction as well as the sensor signal of the piezoelectric microphone. By assuming the air was a perfect gas, the density \(\rho\) was calculated by the perfect gas law. For the derivatives \(\partial\theta/\partial z\) and \(\partial v/\partial z\), the temperature and wind speed at a second height needed to be known. Unfortunately, these data were not available during most of our measurements. Therefore, these gradients had to be approximated in a different way. The ground temperature (\(T_{gnd}\)) was estimated using data from the Deutscher Wetterdienst (DWD). For the onshore measurements, the ground temperature at a nearby location was available at hourly samples. The temperature of the sea for the offshore measurements was interpolated from measurement stations on the island of Helgoland and at List on the island of Sylt. The sign of \(Ri\) mostly depends on the temperature gradient, so the velocity gradient can be regarded as of lower importance. Therefore, as a rough estimate, \(v_{gnd}\approx 0\) was used to calculate the velocity gradient. This mostly affected the absolute value of \(Ri\) but not the heat transfer conditions in the atmospheric boundary layer.