Scaled Temperature Standard Deviation
The similarity functions for the temperature standard deviation are presented and discussed here. In Table 6 of the Appendix, the non-linear best-fit functions between \(\varPhi _{\theta }\) and \(z/\varLambda \) are shown for every i-Box site for unstable (\(z/\varLambda \le -0.05\)), near-neutral unstable (\(- 0.05 < z/\varLambda \le 0\)) and stable (\(z/\varLambda \ge 0\)) conditions. The number of available data points for every stability region is also listed. The derived best-fit similarity functions are only from the first level of every i-Box site. It should be noted, however, that the differences in the obtained best-fit relations between different heights where available (the CS-VFO and CS-SF8 sites) are in all aspects similar to those between sites (not shown). Tables 4 and 6 show that the number of available data points varies between the i-Box sites because of many missing or excluded data due to instrument malfunction during the operating period or low data quality. As expected, the small number of available data points increases the uncertainty in the parametrizations, especially in the near-neutral ranges, which also gives an increased possibility of self-correlation. However, these ranges are not excluded from the analysis as the fitted equations in the near-neutral region indicate the possible impact of terrain inhomogeneity.
Table 6 lists the observed range of \(z/\varLambda \) for every site, with \(|z/\varLambda |\) always much smaller than 10. In contrast, \(|z/\varLambda |\) reaches very small values in the near-neutral range (within the measurement uncertainty), with the smallest value (\(- 9.7\times 10^{-4}\)) occurring at the south-facing CS-SF8 site.
For the best-fit analysis, all the coefficients in the general formulation \(a_{\theta }\), \(b_{\theta }\), \(c_{\theta }\), \(d_{\theta }\) and \(e_{\theta }\) are fitted parameters based on Eq. 5 for \(i = \theta \),
$$\begin{aligned} \varPhi _{\theta } = \left\{ \begin{array}{ll} a_{{\theta }u}(b_{{\theta }u}-z/\varLambda )^{d_{{\theta }u}} &{}\quad \text{ for } \quad z/\varLambda \le -0.05 \\ a_{{\theta }n}(-z/\varLambda )^{d_{{\theta }n}}+e_{{\theta }n} &{}\quad \text{ for } \quad -0.05 < z/\varLambda \le 0 \\ a_{{\theta }s}z/\varLambda ^{d_{{\theta }s}}+e_{{\theta }s} &{}\quad \text{ for } \quad z/\varLambda \ge 0. \\ \end{array} \right. \end{aligned}$$
(6)
The coefficients \(c_{{\theta }u} = c_{{\theta }n} = c_{{\theta }s}\) and \(e_{{\theta }u} = b_{{\theta }n} = b_{{\theta }s}\) (see Eq. 5) are set to \(- 1\) and to zero, respectively, as these values have also been used in other studies (e.g., Tillman 1972; Tampieri et al. 2009; Pahlow et al. 2001). The subscripts u, n, s refer to unstable, near-neutral and stable ranges, respectively. The best-fit coefficients for every i-Box dataset are calculated by applying a non-linear robust fit with a bi-squared weighting, as in the case of the reference curves.
The statistical differences between individual i-Box datasets, and also between the i-Box and the reference datasets, are examined by applying the Kolmogorov–Smirnov test for every stability range separately. The purpose of this nonparametric test is to examine whether differences in the cumulative distributions of two datasets are statistically significant at the significance level of \(5\%\). For this purpose, the same stability range and the same number of data points are considered for the compared datasets. To compare two different-sized datasets, the same number of data points is obtained by randomizing the dataset with the larger number of data points using the Bootstrap method, and randomly choosing the same number of data points as in the second dataset. The above test shows statistically significant differences between all i-Box datasets and the reference for unstable, stable and near-neutral (unstable) stratification. However, between individual i-Box datasets, differences in distributions are not statistically significant in some cases, as is the case between the sites CS-NF10, CS-NF27 and CS-SF1 in the near-neutral (unstable) range, as well as between the CS-SF1 and CS-NF27 sites in the unstable range. For stable stratification, differences between all the i-Box datasets are found to be statistically significant. The few exceptions (four out of 48 pairs of datasets) suggest that the differences between the \(\varPhi _{\theta }\) distributions of the different sites are generally statistically significant.
In Fig. 7, the five i-Box datasets are plotted together with the reference curves, with the panels (a) and (b) of Fig. 7 showing the differences in \(\varPhi _{\theta }(z/\varLambda \)) between the sites for unstable and stable stratifications, respectively. It can clearly be seen that, on average, \(\varPhi _{\theta }\) is larger than the reference for all the i-Box sites outside the near-neutral range for unstable and stable stratifications, as already noted by Rotach et al. (2017). Specifically, in the unstable region, the best-fit curve of the steepest mountain-slope site (the CS-NF27 site) exhibits the largest \(\varPhi _{\theta }\) values (Fig. 8a). However, the magnitudes of the best-fit curves for the i-Box sites do not seem to be proportional to the mountain slope. In the near-neutral (unstable) region, the curve slopes for the i-Box datasets differ from the reference. This result is better depicted in Fig. 8a where only the slope of the best-fit curve for the CS-SF8 site is smaller than the reference, while all the other curve slopes are larger than the reference, with the CS-NF10 and CS-SF1 sites having the largest curve slopes for unstable stratification. It should be noted that the CS-SF1 site is highly inhomogeneous, as it is surrounded by grass fields to the east, a house to the north, and a corn field on an escarpment to the west.
On the stable side in the near-neutral region (Fig. 8b), the CS-NF10 and CS-SF1 sites have the largest deviation from the reference, although there is no clear dependence of the slope of the curves on the terrain slope. For the region with stable stratification, the best-fit curves for the CS-VF0 and CS-NF27 sites are closest to the reference curve (Fig. 8b, Table 6). In the Appendix (Fig. 13), plots of \(\varPhi _{\theta }\)(\(z/\varLambda \)) and \(\varPhi _q\)(\(z/\varLambda \)) are shown for every site separately. On the strongly stable side, it can be seen that the best-fit i-Box and reference curves become horizontal, indicating a small dependence of \(\varPhi _{\theta }\) on \(z/\varLambda \) (Fig. 8b). This is expected, as for large values of \(z/\varLambda \), the weak turbulence inhibits any exchange with the surface and, therefore, the scaled variable \(\varPhi _{\theta }\) becomes independent of \(z/\varLambda \), corresponding to z-less scaling (Nieuwstadt 1984).
According to the best-fit equations for the i-Box sites in the near-neutral (unstable) range, the exponent \(d_{{\theta }n}\) is smallest for the valley-floor site (CS-VF0) and largest for the steep-sloped site (CS-NF10), although the coefficient does not increase proportionally to the mountain slope (Table 6). For stable stratification, the exponent \(d_{{\theta }s}\) is smallest for the site CS-SF8 and largest for the site with the steepest slope (CS-NF27). The largest differences in the curves’ slopes between the i-Box sites are detected in the near-neutral range of both unstable and stable stratifications (Fig. 8a, b, Table 6). With stronger stability (for both unstable and stable stratifications), the best-fit curves of the i-Box sites are relatively close to each other.
To examine whether the scatter around the best-fit curves is so large that the curves are not significantly different, we test whether a similarity curve from one site can be used for sites with different surface characteristics. For this purpose, the scatter of the data around the i-Box best-fit and reference curves is illustrated in Fig. 8, with shaded areas depicting the scatter of each dataset based on the median absolute deviation (MAD) of the data. The value of MAD is calculated for every stability range (unstable, near-neutral and stable) and for every site separately. These shaded areas represent the confidence intervals for each best-fit curve. It should be noted that the confidence intervals differ in width along each best-fit curve, because of the logarithmic representation. In the strongly unstable range of Fig. 8a (\(z/\varLambda < -1\)), the confidence intervals overlap, because of the large scatter of each dataset. However, for \(- 1 \le z/\varLambda < -0.05\), it can be seen that the confidence interval for the CS-NF27 site does not overlap with that for the reference site. In the near-neutral (unstable) range, the confidence intervals are not overlapping, except for site CS-NF10 with site CS-SF1. Therefore, despite the uncertainty in the datasets, the parametrizations are unique for every dataset in the near-neutral (unstable) range. For strongly stable stratification, the confidence intervals overlap, except for site CS-SF8, whose best-fit curve is much higher than all the others (Fig. 8b). However, in the near-neutral (stable) range, these intervals are separated from each other, emphasizing the uniqueness of each best-fit curve, which confirms our initial hypothesis that the universal MOST equations are not suitable for complex terrain, as the similarity functions we found are strongly site dependent.
As mentioned before, standard deviations of the non-dimensional temperature for the i-Box sites have larger magnitudes than those from the reference site, which is especially true for the unstable range. Figure 1 shows that this is also the case for all the HIF and complex-terrain cases from the literature review for stable stratification, but only for a limited number of cases in the unstable range. Comparing the \(\varPhi _{\theta }\) best-fit equations of Cabauw and the i-Box sites (Table 6), it can be seen that, for unstable stratification, the coefficients \(a_{{\theta }u}\), which determine the magnitudes of the curves, are larger for the i-Box sites than for the reference site. Similarly, in the stable region, the coefficients \(e_{{\theta }s}\), which determine the curves’ magnitudes—because \(a_{{\theta }s}\) is very small for all sites—are much larger for all i-Box sites than for the reference site. In Fig. 8, it is shown that the differences between the i-Box and the reference best-fit curves are substantial. The question arises as to what the possible reasons for this enhanced temperature variability may be compared with ‘ideal’ sites, i.e., larger \(\varPhi _{\theta }\) values at sites over complex terrain.
We have identified four possible reasons for this:
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Post-processing: as i-Box and reference datasets were both analyzed with exactly the same post-processing method by the EdiRe software, with \(\varPhi _{\theta }\) only found to be much larger than in the HHF and HIF curves (e.g., Kaimal and Finnigan 1994; Tillman 1972) for the i-Box dataset, and not for the reference, this excludes the post-processing from being the reason.
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The zero-plane displacement is determined in a relatively crude manner (see Sect. 3.1). However, when using extreme values for the zero-plane displacement (e.g., \(d = 0\), or double the estimated values), no significant differences from the original values shown in Fig. 8 and Table 6 are observed (not shown), which suggests that the estimation of the zero-plane displacement is not the reason for the large values of \(\varPhi _{\theta }\).
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Non-constant fluxes: Nadeau et al. (2013a) used the non-constancy of the turbulent fluxes as one of the arguments as to why the MOST approach may not be applicable in complex topography (but rather local scaling). While we certainly agree with this argument, we investigated whether periods of ‘non-constant fluxes’ exhibit particularly large \(\varPhi _{\theta }\) values. Figure 9 shows that the data points with constant fluxes are not closer to the reference curves than data for non-constant fluxes, for both unstable and stable stratifications at the CS-VF0 site. This result confirms that the difference from the reference curves for temperature, which is observed at each i-Box site, is not due to the existence of non-constant fluxes.
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Coordinate system (i.e., the frame of reference for the projection of the temperature and momentum fluxes): while it is customary to use a terrain-following coordinate system over sloped surfaces, so that the temperature and momentum fluxes are normal to the surface, this introduces some difficulty because even if the local (perturbation) isentropes are parallel to the slope, the dominant direction of heat fluxes at some distance away from the surface is vertical (Stiperski and Rotach 2016; Oldroyd et al. 2016a, b; Lobocki 2017). We have, therefore, tested the hypothesis that the vertical (rather than the normal) heat fluxes constitute the appropriate scaling variable by comparing \(\varPhi _{\theta }\) calculated with slope-normal and vertical temperature fluxes for the four i-Box sites with sloping terrain (Figs. 10, 14 in the Appendix). The fluxes are converted from slope-normal to vertical coordinates, following the method of Oldroyd et al. (2016a). Figure 10 shows that this coordinate transformation does not affect the magnitude of \(\varPhi _{\theta }\) on the unstable side, but does affect the curve’s slope in the near-neutral (unstable) region. In the Appendix (Fig. 14), it can be seen that both sites with steeply sloping terrain (the CS-NF10 and CS-NF27 sites) exhibit this decrease of the curve slope in the near-neutral (unstable) range when using vertical temperature fluxes, although there are not enough data points to substantiate this statement. For the sites CS-SF1 and CS-SF8, this phenomenon is not observed systematically. Despite this decrease in the curves’ slopes, it is clear that the large majority of i-Box data points in the unstable region do not change after axis transformation.
According to Oldroyd et al. (2016a), the change from slope-normal to vertical coordinates may lead to a significant change in \(z/\varLambda \) under stable conditions, and even a change in sign (while the thermal stratification remains stable). Indeed, when moving from slope-normal to vertical coordinates according to the approach of Oldroyd et al. (2016a), a change of sign is detected in about 100 cases at the CS-NF10 site, and for a few at the other sites. More importantly, on the stable side, the move from slope-normal to vertical coordinates changes the distribution of \(\varPhi _{\theta }\) data points (not shown) in the sense that the scatter of the datasets increases, especially at the strongly-sloped sites (the CS-NF10 and CS-NF27 sites). This leads to the separation of the datasets into two main clusters for the sites CS-NF10 and CS-NF27, likely because of two different flow types (i.e., katabatic vs. dynamically-modified flows under stable conditions). It does not, however, alter the fact that \(\varPhi _{\theta }\) is systematically larger than predicted from the reference curve.
To conclude, none of the four reasons discussed can be responsible for the large values detected in the plots of (especially) the unstable \(\varPhi _{\theta }\)(\(z/\varLambda \)) values (Fig. 7). Therefore, we conclude that the inhomogeneity of the study area and the complexity of the terrain are likely the reasons for this difference.
Scaled Humidity Standard Deviation
In Table 6 (in the Appendix), the best-fit coefficients for the five i-Box sites are shown, together with the available number of data points for every region. The unstable side is not divided into unstable and near-neutral (unstable) regions, because there is no near-neutral variability, as in the case of \(\varPhi _{\theta }\). For stable stratification, only one coefficient is fitted to the data. Due to the large scatter in the stable range (Fig. 7d), the best-fit curve appears to be a horizontal line, indicating no significant dependence of \(\varPhi _q\) on \(z/\varLambda \). As for \(\varPhi _{\theta }\), the coefficients \(a_q\), \(b_q\) and \(d_q\) in the similarity functions were fitted to the data for unstable stratification, so that differences in the slope and the magnitude of the curves can be shown. To compare the best-fit equations of \(\varPhi _{\theta }\) and \(\varPhi _q\) (Sect. 6.3), we consider the same general formulation for \(\varPhi _q(z/\varLambda )\) as for \(\varPhi _{\theta }(z/\varLambda )\) in the unstable region,
$$\begin{aligned} \varPhi _q = \left\{ \begin{array}{ll} a_{qu}(b_{qu}-z/\varLambda )^{d_{qu}} &{} \quad \text{ for } \quad z/\varLambda \le 0 \\ e_{qs} &{} \quad \text{ for } \quad z/\varLambda \ge 0 \\ \end{array} \right\} , \end{aligned}$$
(7)
where the subscripts u and s denote unstable and stable conditions, respectively. In Fig. 7c, d, \(\varPhi _q\) as a function of the stability is shown, and for a better distinction between the sites, Fig. 13 in the Appendix depicts each i-Box dataset separately. On the unstable side, the sites with steep-terrain slopes (the CS-NF10 and CS-NF27 sites) present high scatter, whereas the CS-VF0 dataset has the smallest scatter around the reference curve (Fig. 7c). In contrast to temperature, the \(\varPhi _q\) curves do not generally exhibit a larger magnitude than the reference curve, illustrating why the best-fit coefficients \(a_{qu}\) of the i-Box curves are similar to the coefficient for the reference best-fit (cf. Table 6). In the near-neutral region of the unstable range, data show no increasing scatter for all the i-Box sites (Fig. 7c).
Figure 8c shows that the best-fit curve of the CS-NF27 site is always higher in magnitude than the reference on the unstable side, whereas the other i-Box best-fit curves are higher in the strongly unstable range, but smaller in magnitude than the curves reported in the literature for \(z/\varLambda > -0.1\). In accordance with the literature review (e.g., Andreas et al. 1998; Liu et al. 1998), the chosen formulation for the best-fit function for stable stratification is a constant, which is also suggested by the scatter of the present data. In Fig. 8d, the CS-NF27 and CS-SF1 site curves are the highest in magnitude (\(\varPhi _q = 5.28\) and \(\varPhi _q = 5.25\), respectively), whereas for the other sites, the value of \(\varPhi _q\) decreases, following the decrease of the terrain slope. Andreas et al. (1998) found \(\varPhi _q = 4.1\) as the best-fit for stable stratification in the case of metre-scale heterogeneous terrain, whereas Liu et al. (1998) found \(\varPhi _q = 2.4\) in the case of HHF terrain. For the i-Box sites, best-fit curves of \(\varPhi _q\) are found to vary between 2.58 and 5.28 (Table 6). It should be noted that, in all cases, the best-fit curve of the CS-VF0 site is closer to the reference curve than those of the other i-Box sites. All of the above suggests that the magnitude of \(\varPhi _q\) for stable stratification is affected by the terrain slope or the heterogeneity of the terrain, without, however, any direct relationship between them.
The application of the Kolmogorov–Smirnov test shows that almost all differences in dataset distributions are statistically significant. For unstable stratification, the differences in data distribution are not statistically different between the sites CS-SF1 and CS-SF8, as well as between sites CS-NF27 and CS-SF8 (note that the statistically similar site pairs are not the same as those for temperature). These few exceptions again suggest that, overall, the differences between the \(\varPhi _q\) distributions of the different sites are statistically significant.
The shaded areas around the best-fit curves in Fig. 8c, d represent the MAD values of the datasets and, therefore, the confidence intervals of the best-fit curves. In Fig. 8c, it is noted that almost all the confidence intervals overlap for strong instability. In contrast, the intervals for sites CS-VF0 and CS-NF10 start to diverge from the rest for \(z/\varLambda > -0.1\). For stable stratification in Fig. 8d, the high scatter of all datasets causes all the confidence intervals to overlap. It should be noted that the confidence interval for the CS-VF0 site is almost the same as the reference site, as their best-fit curves are very similar, and both exhibit little scatter in the data.
Comparison Between Temperature and Humidity Similarity Functions
Many studies have suggested using the best-fit function of \(\varPhi _{\theta }\) for \(\varPhi _q\), because the characteristics of humidity and temperature fluctuations are considered to be similar (e.g., Ramana et al. 2004). Here we investigate how useful this is by comparing the best-fit similarity curves of temperature and humidity for unstable and stable stratifications (\(|z/\varLambda | \ge 0.05\)). The bootstrapping method, combined with the Student’s t-test, is followed for all the cases to determine whether the differences between the two similarity functions are statistically significant. It should be noted that the temperature and humidity curves are not compared in the near-neutral regions, since the slope of the curves and the form of the functions are different there anyway.
The differences between \(\varPhi _{\theta }\) and \(\varPhi _q\) for unstable and stable stratification are shown in Fig. 11. For unstable stratification, the best-fit curves for \(\varPhi _{\theta }\) and \(\varPhi _q\) are very similar for the sites with a small terrain slope (the reference, CS-VF0, CS-SF8 and CS-SF1 sites). In contrast, the slopes of the two curves differ noticeably for the CS-NF10 site, as well as in magnitude for the CS-NF27 site. In the near-neutral (unstable) range, the differences in slopes of the curves are large as expected (see Fig. 8a, c, e, g, i, k). For stable stratification, the \(\varPhi _q\) curves are much higher in magnitude than \(\varPhi _{\theta }\) for most of the sites (see Fig. 11b, d, f, h, j, l), exceptions being the CS-VF0 site, which gives almost identical curves, and the CS-SF8 site where the \(\varPhi _q\) curve is lower in magnitude. The largest differences in the curves’ magnitudes are noted for the CS-SF1 and CS-NF27 sites. For the near-neutral (stable) regions, the slopes for \(\varPhi _q\) are zero, whereas for \(\varPhi _{\theta }\), they are larger than zero; therefore, the curves in this region are not compared.
By applying the bootstrapping method with the Student’s t-test, the best-fit coefficients \(a_{{\theta }u}\), \(b_{{\theta }u}\), \(d_{{\theta }u}\) of \(\varPhi _{\theta }\) are compared with the coefficients \(a_{qu}\), \(b_{qu}\), \(d_{qu}\) of \(\varPhi _{qu}\), respectively, for unstable stratification (\(z/\varLambda \le -0.05\)). For stable stratification (\(z/\varLambda \ge 0.05\)), the coefficients \(e_{{\theta }s}\) are compared with the coefficients \(e_{qs}\) for all the sites. The above analysis shows that most of the best-fit coefficients differ statistically significantly, the exception being the difference between \(e_{{\theta }s}\) and \(e_{qs}\) values for the reference site, indicating that, overall, the curves \(\varPhi _{\theta }\)(\(z/\varLambda \)) and \(\varPhi _q\)(\(z/\varLambda \)) are different for both stable and unstable stratifications when considering sites in complex terrain.
As a second step, the confidence intervals of each best-fit curve were calculated (not shown), and represent the areas between the 10th and the 90th quantile of the set of all possible best-fit curves derived from randomizing the datasets 1000 times. Even with these large quantiles, the confidence intervals of the best-fit curves do not overlap, which supports the conclusion that differences in the best-fit curves of \(\varPhi _{\theta }\) and \(\varPhi _q\) as a function of \(z/\varLambda \) are statistically significant. As a result, the use of the \(\varPhi _{\theta }\) similarity functions for \(\varPhi _q\) is not recommended, especially in complex terrain.