## 4 Breakdown and Modifications of MOST under Stable Conditions
### Experimental Data  
The field experiment used to investigate the similarity functions was carried out in the Jingbian wind farm in northwest of China [2]. In the experimental campaign, two masts, numbered M1 and M3, are installed near a wind turbine (No. 14) to capture the wake profiles (Figure 2). As the inflow from north is dominated by a small and constant terrain slope while the inflow from east and south are dominated by a significant terrain complexity, we applied measurements from the mast M1 to estimate similarity functions. On the mast M1, three Thies First Class Advanced cup anemometers were installed at 30 m, 50 m, and 70 m above the ground level (AGL) (Figure 3). Wind direction (WD) was measured at 30 m and 60 m AGL, and air temperature was measured at 30 m and 70 m AGL, respectively. Two Metek 3D sonic anemometers were installed on M1 to provide three-dimensional velocity components and sonic temperature at 30 m and 70 m AGL in a frequency of 35 Hz. A NRG logger was used to record the 10-min statistical values from the cup anemometers, wind vanes, and thermometers. The measured data are resampled to 10 min in \(\sim\)190 days. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Turbulence Model** & **Energy Equation** (14) & \(\mathbf{\mathcal{B}}\) & \(\mathbf{S_{k,\text{ABL}}}\) & \(\mathbf{C_{\text{M}}}\) & **Similarity Functions** \\ \hline AM & \(\surd\) & Equation (25) & – & \(-\)2.9 & classical \\ El-Askary & – & Equation (27) & – & 1 & classical \\ Laan & \(\surd\) & Equation (25) & Equation (32) & Equation (33) & classical \\ Proposed & \(\surd\) & Equation (25) & Equation (32) & Equation (33) & \(\phi_{m,\text{exp}},\phi_{h,\text{exp}},\phi_{h,\text{cls}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of turbulence models investigated for stable conditions, where “\(-\)” represents that the corresponding term is absence in the model, “\(\surd\)” represents that the energy equation is solved, and “classical” denotes standards for the classical similarity functions. All the anemometers are calibrated in wind tunnels to compensate flow distortion effects. The classification of cup anemometers after calibrations is class A 1.5-1.8 [32], suggesting an uncertainty within \(\pm\)2.3% for the wind speed above 3 m/s according to IEC 61400-12-1 [33]. In [34], a total uncertainty of 3.6% for the wind speed can be assessed as the root sum of squared uncertainties, due to the sensor accuracy (2.3%), the sensor calibration (2%) [35], and the mounting of anemometers on the mast (2%) [36]. For the sonic anemometers, the 10-min averaged wind speeds recorded by cup anemometers and sonic anemometers at the same heights agree well with slopes close to 1 and correlation coefficients above 0.993 (Figure 4). Figure 3: Sketch (**left**) and physical appearance (**right**) of measurement setup for M1. Figure 2: Complex terrain around M1, where Wind direction (WD) stands for the wind direction and \(D=93\) m is the diameter of wind turbine 14. Note that the field experiment was carried out in a wind farm in complex terrain. Wind turbine wakes and complex terrain could influence estimating similarity functions. As the wake structure is nonlinear and aligned with wind direction, [PERSON] et al. selected the dataset in wind direction sectors with high wind speed correlation to eliminate wake effects for the same measurement campaign [2]. Based on their work, we choose three wind direction intervals, (1) the north section: WD = \(0^{\circ}\pm 10^{\circ}\); (2) the east section: WD = \(60^{\circ}\pm 5^{\circ}\); and (3) the west section: WD = \(250^{\circ}\pm 40^{\circ}\), to study similarity functions where WD stands for the wind direction, and the intervals are kept narrow, if possible, to ensure data quality with more than 400 points of data. According to Figure 2, the inflow wind in the east and west sections are dominated by complex terrain while the inflow from the north section is dominated by a small slope terrain (slope \(\approx\) 2.4\({}^{\circ}\)).