## 4 Breakdown and Modifications of MOST under Stable Conditions
### Assessment of Similarity Functions for Predicting Wind Profiles  
Plots of the dimensionless velocity gradients versus the stability parameter in three wind direction intervals are presented in Figure 6. For the three wind direction intervals, the upslope inflows towards the mast are accelerated more at low levels than at high levels [41], and the dimensionless velocity gradient is decreased consequently. For the east and west sections in complex terrain, the fitted \(\phi_{m}\) based on field measurements is 30% lower than that of the north section, with a small slope terrain for \(0<\zeta<0.1\). Similarly, a decrease of 30% in \(\phi_{m}(=0.7+1.5\zeta)\) was also observed in a stable boundary layer over the complex terrain of the Loess Plateau, China [42]. The fitted \(\phi_{m}\) are also compared with the classical similarity function \(\phi_{m,\text{cls}}=1+\beta_{m}\zeta\) and the SHEBA profile function \(\phi_{m,\text{SHEBA}}=1+6.5\zeta(1+\zeta)^{1/3}/(1.3+\zeta)\)[43]. As shown in Figure 6, all the fitted \(\phi_{m}\) and the SHEBA profile function can limit the velocity gradient under strongly stable conditions: \(\phi_{m}\propto\zeta^{1/4}\) (fitted) and \(\phi_{m}\propto\zeta^{1/3}\) (SHEBA) for \(\zeta\rightarrow\infty\). For the north section where the inflows are dominated by a small slope, \(\phi_{m,\text{cls}}\) and \(\phi_{m,\text{SHEBA}}\) agree well with measurements for \(\zeta<0.1\) and overestimate the velocity  
Figure 5: Plots of \(\phi_{m}\) as functions of \(\zeta\) at 30 m above the ground level (AGL) for WD = \(0^{\circ}\pm 10^{\circ}\).  
gradient for \(\zeta>0.1\). Affected by complex terrain, the fitted \(\phi_{m}\) estimated for the east and west sections are much lower than \(\phi_{m,\text{cls}}\) and \(\phi_{m,\text{SHEBA}}\).  
The fitted similarity functions are evaluated for predicting wind profiles based on the Boulder Atmospheric Observatory (BAO) dataset [44]. Among the seven runs presented in [PERSON] et al. [44], only runs 3, 5, and 7 are studied, due to the criteria \(|H_{\text{S}}|\geq 10\) W\(\cdot\)m\({}^{-2}\). The wind profiles predicted by varying similarity functions are presented in Figure 7 and estimated in numerical integration by  
\[U(z)=\int_{z_{0}}^{z}\frac{u_{*}}{xz}\phi_{m}\left(\frac{z}{L}\right)\,\text{ d}z \tag{37}\]  
where the aerodynamic rounghess \(z_{0}\) is \(\sim\)0.01 m for the BAO dataset [44]. According to Figure 7, the fitted \(\phi_{m}\) of the north section performs better in predicting wind profiles for runs 3, 5, and 6, whereas \(\phi_{m,\text{cls}}\) and \(\phi_{m,\text{SHEBA}}\) overestimate the velocity gradient significantly. The wind speed is overestimated on average by 4.6 m/s, 3 m/s, and 0.5 m/s for \(\phi_{m,\text{cls}}\), \(\phi_{m,\text{SHEBA}}\) and the fitted \(\phi_{m}\) of the north section, respectively. Influenced by complex terrain, the fitted \(\phi_{m}\) of the east and west sections underpredicts the velocity of 1.2 m/s averagely. As \(\phi_{m}(0)<1\) of the east and west sections disagrees with the logarithmic law of wind profiles in flat terrain under neutral conditions [45; 46], we apply the similarity functions estimated from the dataset of the north section in the wake simulations.  
Figure 6: Plots of \(\phi_{m}\) of different models as functions of \(\zeta\): dots (measured data) and circles (bin-averaged measured data). \(n\) is the number of the measured data.