ScalarFlux Similarity in the Layer Near the Surface Over Mountainous Terrain
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Abstract
The scaled standard deviations of temperature and humidity are investigated in complex terrain. The study area is a steep Alpine valley, with six measurement sites of different slope, orientation and roughness (iBox experimental site, Inn Valley, Austria). Examined here are several assumptions forming the basis of Monin–Obukhov similarity theory (MOST), including constant turbulence fluxes with height and the degree of selfcorrelation between the involved turbulence variables. Since the basic assumptions for the applicability of the MOST approach—horizontally homogeneous and flat conditions—are violated, the analysis is performed based on a local similarity hypothesis. The scaled standard deviations as a function of local stability are compared with previous studies from horizontally homogeneous and flat terrain, horizontally inhomogeneous and flat terrain, weakly inhomogeneous and flat terrain, as well as complex terrain. As a reference, similarity relations for unstable and stable conditions are evaluated using turbulence data from the weakly inhomogeneous and flat terrain of the Cabauw experimental site in the Netherlands, and assessed with the same postprocessing method as the iBox data. Significant differences from the reference curve and also among the iBox sites are noted, especially for data derived from the iBox sites with steep slopes. These differences concern the slope and the magnitude of the bestfit curves, illustrating the site dependence of any similarity theory.
Keywords
Complex terrain Horizontally inhomogeneous IBox experimental site Local similarity1 Introduction
In recent years, a growing number of studies have focused on turbulence structure in truly complex and mountainous terrain (e.g., Rotach et al. 2004; Moraes et al. 2005; Rotach and Zardi 2007; Fernando et al. 2015; Stiperski and Rotach 2016). The understanding of boundarylayer processes in complex terrain is crucial in atmospheric modelling, e.g., for numerical weather prediction, climate, air pollution and hydrosphere–cryosphere modelling (de Bruin et al. 1993; Baklanov et al. 2011). Improved boundarylayer parametrizations and turbulence closure strategies for highresolution models are needed for applications that take into account the complexity of the terrain and the flow conditions, as well as canopy flows. Although turbulent exchange processes in complex topography have been investigated over the last few decades (e.g.,Rotach and Zardi 2007; Fernando et al. 2015), relatively little work (an exception being Nadeau et al. 2013a, b) has been devoted to the systematic investigation of scaling relations in complex terrain. Therefore, in practical applications, scaling relations developed over horizontally homogeneous and flat (HHF) terrain are often employed.
Turbulent fluxes can be directly measured using the eddycovariance method, which is based on the covariance between turbulent fluctuations of scalars and the velocity vector (Aubinet et al. 2012). However, as data are not always available from routine meteorological observations, such as in operational networks, similarity theory is a useful tool, since only a few nondimensional parameters provided by basic measurements are needed to estimate the main turbulence variables (e.g.,Holtslag and Nieuwstadt 1986; Stull 1988). The early focus of similarity scaling was based mainly on ideal conditions, as well as horizontally homogeneous and flat terrain, constant fluxes with height in the surface layer, and quasistationary turbulence with very small uncertainties.
To compare turbulence characteristics between study areas in different conditions, a proper scaling method is needed. The most usual and widelyaccepted similarity scaling for the atmospheric surface layer is Monin–Obukhov similarity theory (MOST) (Monin and Obukhov 1954), in which the scaled turbulence variances of the flow depend only on the stability z / L, where z is the measurement height, and L is the Obukhov length, and is an appropriate theory over HHF terrain and under ideal conditions. Under these conditions, the MOST functions of the scaled turbulence variables are considered to be universal. We note that the MOST approach is based on the hypothesis that the surface fluxes are in equilibrium with the local fluxes, so the surface fluxes are used to define the scaling parameters at any measurement height within the surface layer. Even if originally derived for the surface layer, MOST cannot be applied to the entire surface layer, but it is only valid in its upper part, i.e., the inertial sublayer (e.g.,Rotach and Calanca 2015). Furthermore, for strongly unstable conditions, the standard deviations of the horizontal velocity components (\(\sigma _u\), \(\sigma _v\)) are influenced by large eddies extending throughout the unstable boundary layer, so that these variables do not obey surfacelayer scaling (Panofsky et al. 1977; Holtslag and Nieuwstadt 1986).
Therefore, the above criteria are not universally met, and it is questionable whether the universal functions apply in the case of inhomogeneous terrain. As an alternative, the applicability of local similarity has been investigated. The parameters of the similarity functions are assumed to be ‘localized’, i.e. specific for the area (site) under consideration. Generally, the original form of these scaling relations has not changed, except for the respective coefficients (e.g.,Andreas et al. 1998; Ramana et al. 2004; Moraes et al. 2005; Nadeau et al. 2013a).
If MOST is not applicable, a useful approach is local scaling for which turbulent quantities are a function of \(z/\varLambda \), where \(\varLambda \) is the local Obukhov length, but is a much less powerful framework than MOST. For instance, the surface fluxes cannot be retrieved from local scaling if there are no clearly verified characteristics of the fluxes (as will be demonstrated in Sect. 5). Nevertheless, if the assumptions for MOST are not fulfilled, knowledge of the surface fluxes may be of less importance, and local scaling can be considered as a first step towards a better understanding of turbulence characteristics in complex terrain.
After establishing the theoretical framework (Sect. 2) and introduction of the dataset (Sect. 3), the reference parametrizations employed are introduced in Sect. 4. Our objective is to investigate the applicability of similarity theory for normalized standard deviations of scalars in a complex Alpine valley. The crucial conditions that allow the application of Monin–Obukhov similarity theory are evaluated in Sect. 5, and local similarity theory is investigated for temperature and humidity standard deviations in Sect. 6.
2 Theoretical Background
2.1 Similarity Scaling
The majority of previous studies propose a similarity function that is only valid outside the nearneutral limit \(z/\varLambda  > 0.05\) (e.g., Moraes et al. 2005; Nadeau et al. 2013a); otherwise, \(\varPhi _{\theta }\) shows significant variability for unstable conditions in the case of nonHHF terrain. In neutral conditions over HHF terrain, the vertical heat flux tends towards zero and the temperature fluctuations become very small, so that the scaled temperature variance tends to be finite (Monin and Yaglom 1971). However, in nonideal conditions, such as in complex terrain, the temperature fluctuations remain finite (Tampieri et al. 2009), even as the heat flux goes to zero when approaching neutral conditions, thus causing the scaled temperature variance to diverge. For this reason, we treat the nearneutral (unstable) region, with the more strongly unstable ranges of \(z/\varLambda \) treated separately for \(\varPhi _{\theta }\). Recently, Tampieri et al. (2009) suggested that, in the nearneutral (unstable) region (\( 0.05\le z/\varLambda \le 0\)), the scaling relation for the temperature standard deviation as a function of \(z/\varLambda \) has an exponent \(d_{\theta } =  1\), rather than the classical \(d_{\theta } =  1/3\). On the stable side, many of the early studies suggested a constant \(\varPhi _{\theta }\) (e.g., Nieuwstadt 1984; Liu et al. 1998. However, Pahlow et al. (2001) recommended an exponent of \( 1\) for nearneutral (stable) conditions in horizontally inhomogeneous and flat terrain. As for the humidity, the variability of \(\varPhi _q\) in the nearneutral range has not been reported in the literature, because the humidity flux \(\overline{w'q'}\) does not tend to zero when approaching neutral stratification.
2.2 Terrain Influence on Turbulence: An Overview
Summarized in Tables 1 and 2 are the similarity functions for scaled temperature and humidity standard deviations as proposed in various turbulence studies for different types of terrain. The range of stability parameter \(\left z/\varLambda \right \), in which every similarity function is valid, depends on the available data range and on the definition of the nearneutral limit (usually \(\left z/\varLambda \right = 0.01  0.05\)). The different types of terrain are classified herein as follows: horizontally homogeneous and flat (HHF); horizontally inhomogeneous and flat (HIF); weakly inhomogeneous and flat (WIF) and complex terrain. The WIF terrain type is used only for the Cabauw study area as explained in Sect. 4 (Beljaars and Bosveld 1997; Bosveld 1999). Here, weakly inhomogeneous refers to terrain with very low roughness elements on the order of 0.1 m or less in height, or with taller roughness elements far from the measurement site.
The majority of the similarity functions for \(\varPhi _{\theta }\)(\(z/\varLambda \)) found in the literature have the same \(d_{\theta } = 1/3\) exponent in unstable conditions, but different coefficients \(a_{\theta }\), \(b_{\theta }\), \(c_{\theta }\) and \(e_{\theta }\). Similarly, studies over complex terrain find the same slope of the curve as the HHF studies for unstable conditions, although the values of \(\varPhi _{\theta }\) differ (see Fig. 1a). On the stable side, several studies suggest that \(\varPhi _{\theta }\) is independent of \(z/\varLambda \) (zero exponent) (e.g., Shao and Hacker 1990; Liu et al. 1998; Andreas et al. 1998), others suggest an exponent of \( 1\) (e.g., Kaimal and Finnigan 1994; Pahlow et al. 2001; Ramana et al. 2004; Nadeau et al. 2013a) or an exponent of \( 1/3\) (e.g., Quan and Hu 2009; Moraes et al. 2005). According to these functions found in the literature, no clear connection between the exponent of \(\varPhi _{\theta }\) and the terrain type is apparent.
The similarity relation between the scaled standard deviation of humidity \(\varPhi _q\) and \(z/\varLambda \) has always been a matter of controversy among different studies, especially on the stable side, probably because of the potential influence of small evaporation or condensation fluxes. As shown in Table 2, a limited number of studies address \(\varPhi _q\) for unstable conditions (e.g., Nadeau et al. 2013a), but very few for stable conditions, possibly because of the large scatter for stable stratification and the large uncertainty of humidityfluctuation measurements. Several studies suggest that humidity behaves similarly to temperature, so they use the same expression (e.g., Ramana et al. 2004). In contrast, Andreas et al. (1998) and Liu et al. (1998) suggest that \(\varPhi _q\) does not depend on \(z/\varLambda \).
List of similarity relations for the nondimensional temperature standard deviation found in the literature
A/A  References  Terrain  Unstable  Stable  Stability range  

\(a_{\theta }\)  \(b_{\theta }\)  \(c_{\theta }\)  \(d_{\theta }\)  \(e_{\theta }\)  \(a_{\theta }\)  \(b_{\theta }\)  \(c_{\theta }\)  \(d_{\theta }\)  \(e_{\theta }\)  
1  de Bruin et al. (1993)  HHF  2.9  1  − 28.4  − 1/3  0  − 100 \(\le \) \(z/\varLambda \le 0.01\)  
2  Kaimal and Finnigan (1994)  HHF  2  1  − 9.5  − 1/3  0  − 2\(\le \) \(z/\varLambda \le \) 0  
2  1  0.5  − 1  0  0 \(\le \) \(z/\varLambda \le \) 1  
3  Liu et al. (1998)  HHF  2  1  − 8  − 1/3  0  \( 40 \le z/\varLambda < 5\times 10^{3}\)  
–  –  –  –  2  0.02\(\le z/\varLambda<\) 10  
4  Nieuwstadt (1984)  HHF  –  –  –  –  3  \(z/\varLambda>\) 0  
5  Wyngaard et al. (1971)  HHF  0.95  0  \( 1\)  \( 1/3\)  0  \( 2.5< z/\varLambda < 0.05\)  
6  Andreas et al. (1998)  HIF  3.2  1  \( 28.4\)  \( 1/3\)  0  \( 4 < z/\varLambda \le 0.01\)  
–  –  –  –  3.2  0.01 \(\le z/\varLambda<\) 1  
7  Asanuma and Brutsaert (1998)  HIF  0.92  0  \(1\)  \( 1/3\)  0  \(40\) \(\le z/\varLambda < 0.3\)  
8  Cava et al. (2008)  HIF  2.3  1  \(9.5\)  \( 1/3\)  0  \( 20 \le z/\varLambda \le 0.04\)  
9  Pahlow et al. (2001)  HIF  0.05  0  1  \(1\)  3  \(4\times 10^{3} \le z/\varLambda \le \) 40  
10  Quan and Hu (2009)  HIF  1.5  0  \(1\)  \(1/3\)  0  \( 5< z/\varLambda<\) 0  
3  0  1  \( \) 1/3  0  0 \(< z/\varLambda<\) 5  
11  Ramana et al. (2004)  HIF  6.56  1  \(9.5\)  − 1/3  0  \( 2 < z/\varLambda \le 0.02\)  
6.45  1  0.25  − 1  0  0.02 \(< z/\varLambda \le \) 2  
12  Shao and Hacker (1990)  HIF  \(\sqrt{6}\)  1  \( 20\)  − 1/3  0  \( 80 < z/\varLambda \le 0.03\)  
–  –  –  –  \(\sqrt{12}\)  0.04 \(\le z/\varLambda \le \) 30  
13  Tillman (1972)  HIF  0.95  0.055  − 1  − 1/3  0  \( 60 < z/\varLambda \le 0.05\)  
14  Högstrom and Smedman (1974)  HIF  0.91  0  − 1  − 1/3  0  \( 3 < z/\varLambda \le \) 0  
15  Moraes et al. (2005)  Complex  1.7  0  − 1  − 1/3  0  \( 2 < z/\varLambda \le 0.05\)  
5.6  1  3.1  − 1/3  0  0 \(< z/\varLambda<\) 2  
16  Nadeau et al. (2013a)  Complex  2.67  1  − 16.29  − 1/3  0  \( z/\varLambda < 0.05\)  
3.22  1  0.83  − 1  0  \( z/\varLambda>\) 0.05  
17  Reference curve  WIF  0.99  0.063  − 1  − 1/3  0  \( 2.87\le z/\varLambda \le 0.05\)  
0.015  0  − 1  − 1  1.76  \( 0.05< z/\varLambda \le 7\times 10^{5}\)  
\(8.7\times 10^{4}\)  0  1  − 1.4  2.03  \(1.4\times 10^{5} \le z/\varLambda \le 5.5\) 
List of similarity relations for the nondimensional humidity standard deviation as found in the literature
A/A  References  Terrain  Unstable  Stable  Stability range  

\(a_q\)  \(b_q\)  \(c_q\)  \(d_q\)  \(e_q\)  \(a_q\)  \(b_q\)  \(c_q\)  \(d_q\)  \(e_q\)  
3  Liu et al. (1998)  HHF  2.4  1  − 8  − 1/3  0  \(40\le z/\varLambda < 0.005\)  
–  –  –  –  2.4  0.02\(\le z/\varLambda \le \) 4  
6  Andreas et al. (1998)  HIF  4.1  1  − 28.4  − 1/3  0  \(4 < z/\varLambda \le 0.01\)  
–  –  –  –  4.1  0.01 \(\le z/\varLambda < 1\)  
7  Asanuma and Brutsaert (1998)  HIF  1.42  0  − 1  − 1/3  0  \(40 \le z/\varLambda < 0.3\)  
8  Cava et al. (2008)  HIF  3.4  1  − 9.5  − 1/3  0  \(20\le z/\varLambda \le 0.04\)  
10  Quan and Hu (2009)  HIF  1.07  1  − 2.71  − 1/3  0  \(5< z/\varLambda < 0\)  
11  Ramana et al. (2004)  HIF  6.56  1  − 9.5  − 1/3  0  \(2 < z/\varLambda \le 0.02\)  
6.45  1  0.25  − 1  0  0.02 \(< z/\varLambda \le \) 2  
12  Shao and Hacker (1990)  HIF  \(\sqrt{30}\)  1  − 25  − 1/3  0  \(80 < z/\varLambda \le 0.12\)  
14  Högstrom and Smedman (1974)  HIF  1.04  0  − 1  − 1/3  0  \(5 \le z/\varLambda<\) 0  
15  Moraes et al. (2005)  Complex  1.5  0  − 1  − 1/3  0  \(2 < z/\varLambda \le 0.05\)  
16  Nadeau et al. (2013a)  Complex  3.51  1  − 21.74  − 1/3  0  \( z/\varLambda < 0.05\)  
17  Reference curve  WIF  0.99  0.031  − 1  − 0.288  0  \(2.87 \le z/\varLambda \le 7\times 10^{5}\)  
–  –  –  –  2.74  \(1.4\times 10^{5} \le z/\varLambda \le 5.5 \) 
3 Data and Methods
3.1 Datasets
Main characteristics of the iBox and Cabauw measurement sites
Station name  Identification  Local slope (\(^{\circ }\))  Height of levels (m)  Vegetation  Orientation 

Kolsass  CSVF0  0  4 / 8.7 / 16.9  Mixed agricultural  Valley floor 
Terfens  CSSF8  8  6.2 / 11.3  Agricultural, carparking  Southfacing 
Eggen  CSSF1  1  6.6  Alpine meadow, corn field  Southfacing 
Weerberg  CSNF10  10  6.2  Alpine meadow  Northfacing 
Hochhauser  CSNF27  27  6.2  Alpine meadow  Northfacing 
Reference site (Cabauw)  –  0  3  Agricultural, field  Extended plain 
The list of sites used here is given in Table 3, where the names indicate the orientation and the terrain slope in degrees for every site. Three sites have one level, while the CSSF8 and CSVF0 sites have two and three levels, respectively. The turbulence measurements at the Cabauw site, which are used as a reference, are made at one measurement level (see also Sect. 4).
At the iBox sites, velocity components, air and soil temperature, humidity, and the components of the energy balance have been measured since 2013. Used here are turbulence data from CSAT3 sonic anemometers, KH20 Krypton hygrometers, and EC150 openpath gas analyzers (all from Campbell Scientific, Logan, Utah, USA) recorded at a frequency of 20 Hz, covering a period of 2.5 years (August 2013–December 2015). Necessary raw data for the current analysis include the sonic temperature and humidity fluctuations, as well as the velocity components. Lowfrequency (1min) data of air temperature, relative humidity and pressure are used mainly for the purpose of flux corrections. Detailed information about the iBox sites and the instruments can be found at https://wiki.uibk.ac.at/display/IBOX/iBox+Home. The aerodynamic roughness characteristics differ between the iBox sites and depend on the wind direction. The land use of every measurement site is shown in Table 3. For the iBox sites with only grass (CSNF27 and CSNF10), and for the sites with grass and corn (CSSF1, CSVF0, CSSF8), nonlinear functions are used to describe the height of the vegetation as a function of the day of the year (i.e. the growing period).
The CSVF0 site is located between agricultural fields with different vegetation. Using sporadic measurements of the vegetation height h through the years 2014 and 2015, the zeroplane displacement d was first calculated from the relation \(d = 0.7h\), for \(30^{\circ }\) sectors of wind direction, taking the different types of fields into consideration, which were included in every sector. By weighted averaging (dependent on the width of every field), one value of d was determined for each of the fields, for every winddirection sector, and for both stable and unstable stratifications. The footprint model of Kljun et al. (2015) was then applied three times using the value from the previous iteration as an initial value for d, until the calculated footprint became constant. With these calculated values of zeroplane displacement for every winddirection sector, a logarithmic bestfit function (following the growth rate of the plants) between d and the day of the year was determined separately for stable and unstable conditions. Finally, the calculated zeroplane displacements were subtracted from the measurement height z. The calculated bestfit functions were also used for the year 2013, for which no vegetation height for the individual fields had been assessed.
Another correction of the measurement height that may be considered relevant in the iBox environment is related to snow cover in winter. However, as the snow depth usually does not exceed the typical grass height (when data are available), corresponding to a zeroplane displacement of a few millimetres, snow cover has no discernible influence on our target variables (i.e., \(z/\varLambda \)) and is, therefore, not taken into account.
For the remaining sites, the method for determining d was simpler, because there are only two types of surface cover (grass and corn). Therefore, for these sites, the footprint model of Kljun et al. (2015) was applied to find the fields of maximum influence. Afterwards, nonlinear functions between d and the day of the year were created for stable and unstable stratifications and for the winddirection sectors that included grass or corn fields.
3.2 Methods
The 20Hz raw data from the iBox sites were processed by the software EdiRe (Version 1.4.3.1101, from R. Clement, University of Edinburgh, UK). We used a recursive filter with a time scale of 200 s to eliminate the lowfrequency motions, before calculating 30min averages of the turbulence parameters. The doublerotation method (Aubinet et al. 2012; Wilczak et al. 2001) was used to rotate the data into a slopenormal streamwise coordinate system. The procedure for ascertaining the highest quality of the data suggested by Stiperski and Rotach (2016) was followed, which includes flux corrections (Schotanus et al. 1983; Moore 1986; Webb et al. 1980), an assessment of the data uncertainty (Wyngaard 1973), a test for stationarity (Foken and Wichura 1996), skewness and kurtosis thresholds for the temperature and velocity components (Vickers and Mahrt 1997), as well as despiking. Here, we used the same threshold values for the quality criteria as Stiperski and Rotach (2016) for their ‘highquality’ dataset. Following this procedure, we only examine the highquality 30min averaged data for use in the following turbulence analysis. The test for high quality reduced the dataset to about 30\(\%\) of its original amount, which may raise the question about the generality of the results. In this respect, it should be noted that a considerable fraction of the ‘lost’ data is due to nonstationarity and statistical uncertainty, which are dataquality requirements generally applied even over ideal terrain. A useful comparison can, therefore, only be made when applying these tests also over complex terrain. Furthermore, an a posteriori assessment of the results with relaxed quality criteria (not shown) revealed that none of the results of this study changed significantly—even if the scatter using the larger but less reliable database increased. It is, therefore, argued that—despite the largely reduced amount of data through quality control—the overall characteristics of the obtained results reflect all possible flow conditions, but should nevertheless be employed with caution.
Prior to the quality control, the turbulence data are corrected for unwanted spikes using the following despiking method as applied to the raw data by the EdiRe software; if the difference between a data point and its neighbours is larger than 10 standard deviations, the data point is replaced by the interpolated value derived from the neighbouring data points. This process is applied once for the temperature and velocity components and four times for the humidity fluctuations because the Krypton hygrometers are very sensitive to precipitation, producing many periods with spikes after and during precipitation events.
Additional quality control applied to the humidity data reduces the iBox dataset size by about 15\(\%\), including the rejection of 30min averages with a relative humidity larger than 95\(\%\), as well as data with a minimum voltage of the Krypton hygrometer lower than an instrumentspecific threshold (of about 70 mV). Periods with a voltage below this threshold are related to precipitation events and condensation on the instruments, resulting in periodic spikes not removed by the despiking algorithm, as they are systematic.
4 Weakly Inhomogeneous and Flat Terrain as Reference
Since our aim is to evaluate to what degree the scalar standard deviations in complex terrain agree with local scaling, there is a need for a reference. The reference is not derived from the HHF similarity functions, because, as shown in Fig. 1, even the HHF similarity functions exhibit some variability. Therefore, it was decided to establish new reference similarity functions based on data from a flat and only marginally horizontally inhomogeneous site, i.e., the Cabauw site in the Netherlands.
For this reason, 11 months of highfrequency data (10 Hz) are used from the database of the Cabauw Experimental Site for Atmospheric Research (CESAR), which includes turbulence measurements from a 3m mast at the Cabauw field (51.97201\(^\circ \)N, 4.924847\(^\circ \)E) recorded with a LICOR 7500 openpath gas analyzer (LICOR, Lincoln, Nebraska, USA) and a GillR3 sonic anemometer (Gill Instruments, Lymington, Hampshire, UK). The area around the tower is flat with grass fields, and the roughness length is on average 0.03 m. The surrounding terrain is characterized as flat and relatively homogeneous up to a distance of 10 km (Nieuwstadt 1984).
As we compare our results from complex terrain with the reference, and therefore need to be sure that potential differences are not due to postprocessing options or the length of the averaging interval, the raw data from Cabauw (reference data) were postprocessed with exactly the same postprocessing as the iBox data. Moreover, the highquality control process consistent with the iBox data was followed, except for the quality flags from the instruments, which reduced the reference dataset to 51\(\%\) of its initial size.
The reason for classifying the Cabauw study area as weakly inhomogeneous and not horizontally homogeneous is that, although the whole area is flat, there are some roughness elements near the measurement site (trees, houses and water canals), but at least 120 m from the measurement tower. Given the low measurement height (3 m) and the comparably large distance to the obstacles, this inhomogeneity does not likely affect the fluxes. Still, to test this assumption rigorously, we applied the footprint model of Kljun et al. (2015). Climatological footprints and several individual footprints were calculated for unstable, stable and nearneutral (slightly unstable or stable) conditions. The source area of the climatological, as well as of the individual, footprints used in the model, is the 80\(\%\) impact area. The climatological footprints are the aggregation of the footprints of all the unstable, stable and nearneutral cases, respectively, for the 11month period. This analysis shows that all the calculated footprint areas do not include the closest roughness elements. However, in some cases with stable stratification, the 80\(\%\) footprint area includes several of the nearby water canals, which is the reason we use the term weakly inhomogeneous and flat for the Cabauw study area.
In Fig. 4, \(\varPhi _{\theta }\) and \(\varPhi _q\) for the reference dataset are plotted as a function of \(z/\varLambda \) for unstable and stable stratifications, with the bestfit curves through these data points shown in Tables 1 and 2. To reduce the influence of outliers on the curve fit, the nonlinear robust bestfit method with weighting (bisquare) was used. On the unstable side, the bestfit curve was found by fitting 5 to the reference data, using \(c_{\theta } = 1\) and \(e_{\theta } = 0\), in agreement with other studies (e.g., Tillman 1972; Nadeau et al. 2013a). As the slope of \( 1\) suggested by Tampieri et al. (2009) in the nearneutral (unstable) region of the reference data fits satisfactorily to the data points, their suggested function is also used in this region, i.e. \(\varPhi _{\theta } = a_{\theta }(z/\varLambda )^{1}+e_{\theta }\).
5 ConstantFlux Hypothesis and SelfCorrelation Assessment
5.1 Variation of Turbulent Fluxes with Height
The MOST approach can be applied in the upper part (inertial sublayer) of the surface layer where the turbulent fluxes are close to being constant with height (Monin and Obukhov 1954; Nadeau et al. 2013a). According to Mahrt (1999), every study that evaluates the applicability of MOST should include an examination of the constantflux approximation, since the MOST approach is based on the assumption that the turbulence characteristics at every level depend only on the surface stability parameter. According to Nieuwstadt (1984), local scaling should be used instead of surfacelayer scaling when the heat flux and stress change significantly with height. For this reason, it is crucial to evaluate the constantflux approximation. Even in ideal conditions, the fluxes (especially of momentum) are not expected to be perfectly constant with height. Therefore, by using the term constantflux layer, we essentially refer to a ‘nearconstant’ flux layer.
As shown in Table 3, two iBox sites have more than one level: the CSVF0 and CSSF8 sites. The evaluation of the variability of fluxes with height is mainly done at the CSVF0 site, as it has three levels, and is confirmed at the CSSF8 site with two levels. For brevity, we use the term ‘temperature flux’ for the kinematic turbulent flux of sensible heat (\(\overline{w'\theta '}\)) and the term ‘humidity flux’ for the kinematic turbulent flux of latent heat (\(\overline{w'q'}\)). The variability of momentum, temperature and humidity fluxes with height is investigated. If the constantflux hypothesis is not valid on the valley floor (CSVF0), it is not expected to hold at the other sloped sites either.
Traditionally, to consider a constant flux with height requires in practice that the difference in fluxes should be \(\le 10\%\) within the surface layer (e.g., Kaimal and Businger 1970). Here, due to the complexity of the terrain, we used less strict criteria, by considering a change in magnitude of \(< 20\%\) between any two measurement levels as ‘approximately’ constant. However, small fluxes with the same sign are also considered to be constant with height. According to Klipp and Mahrt (2004), as small fluxes are often characterized by large random errors, and are affected by mesoscale trends, they are excluded from the constantflux test. Temperature and humidity fluxes are considered to be small if their absolute values are \(< 0.01\) K m s\(^{1}\) and \(10^{5}\) kg m\(^{2}\) s\(^{1}\), respectively. For momentum fluxes, this threshold is set to 0.01 m\(^2\) s\(^{2}\).
Even with these less stringent criteria, the percentage of constant fluxes is still very small. Specifically, the percentage of simultaneously constant momentum and temperature fluxes is about \(1\%\) for both stable and unstable conditions, whereas the percentage of simultaneously constant momentum and humidity fluxes is about \(0.5\%\) for both unstable and stable conditions. The percentage of constant temperature flux is about \(50\%\) for both unstable and stable conditions, and for the humidity flux, the percentages are substantially smaller, especially in unstable stratification. Concerning the momentum fluxes, only about \(1\%\) are constant, which shows that the low number of simultaneously constant momentum and temperature or humidity fluxes is mainly due to the nonconstancy of the momentum fluxes. Therefore, the hypothesis of a nearconstantflux layer does not hold, and local scaling is preferable in the case of complex terrain.
5.2 Assessment of SelfCorrelation
Selfcorrelation occurs when the dependent and the independent variables in a functional relationship (as in Eq. 5) contain a common variable (Baas et al. 2006). For \(\varPhi _{\theta }\) and \(\varPhi _q\) as a function of \(z/\varLambda \), this common variable is \(u_{*l}\) and for \(\varPhi _{\theta }\) additionally the temperature flux \(\overline{w'\theta '}\). The characteristic velocity \(u_{*l}\) is included in both \(\varPhi _{\theta }\) (or \(\varPhi _q\)) and \(z/\varLambda \), potentially producing a spurious selfcorrelation between them. Selfcorrelation can produce erroneous confidence in scaling, so it should be examined before any attempt is made to apply local scaling (or MOST) to a dataset (Klipp and Mahrt 2004). The assessment of selfcorrelation has been investigated in the past (e.g., Hicks 1981; Andreas 2002; Klipp and Mahrt 2004; Nadeau et al. 2013a). Baas et al. (2006) found that selfcorrelation in the case of the nondimensional gradient of mean wind speed (\(\varPhi _m\)) is significant, and can lead to misleading results. Nadeau et al. (2013a) used the method of Klipp and Mahrt (2004) for the dimensionless standard deviations of wind speed, temperature and humidity to find significant selfcorrelation of the horizontal velocity components for unstable and stable conditions.
The existence of selfcorrelation in \(\varPhi _{\theta }\) and \(\varPhi _q\) as a function of stability is herein assessed for every iBox site and for every measurement level. Using the Klipp and Mahrt (2004) method, we apply a robust linear regression between \(\varPhi _{\theta }\) and \(\varPhi _q\) and \(z/\varLambda \). The variables \(u_{*l}\), \(\sigma _{\theta }\), \(\sigma _q\), \(\overline{w'\theta '}\) and \(\overline{w'q'}\) are then initially randomized 1000 times, before calculating the mean random correlation coefficient \(R_{rand}\) between \(\varPhi _{\theta }\) and \(z/\varLambda \) (\(\varPhi _q\) and \(z/\varLambda \)), and comparing with the correlation coefficient of the original data series, \(R_{data}\). If the absolute difference \(R_{data}^2R_{rand}^2 \approx 0\), this implies that the randomized dataset exhibits a similar degree of correlation as the true variables, suggesting significant selfcorrelation. In contrast, if \(R_{data}^2R_{rand}^2>> 0\), then either \(R_{data}>> R_{rand}\) (insignificant selfcorrelation) or \(R_{rand}>>R_{data}\). The second case occurs when the original dataset is not well correlated itself (as in Fig. 7d), so that the randomized dataset has a higher correlation, implying insignificant selfcorrelation again. In other words, the obtained relationship is not predominately affected by selfcorrelation if the values of \(R_{data}\) and \(R_{rand}\) differ strongly.
As no study exists that provides a specific quantitative threshold for \(R_{data}^2R_{rand}^2\) to indicate selfcorrelation, we calculate the normalized difference \(K = (R_{data}^2R_{rand}^2)/R_{data}^2\), which indicates the fraction of the variance explained by physical processes. When \(K \approx 0\), the dataset is suspected to be selfcorrelated. However, we avoid defining a certain threshold for K, because that would be arbitrary and dependent on the available datasets. In the case of \(\varPhi _{\theta }\), the degree of selfcorrelation is checked separately in the nearneutral region and in the region with stronger stability, because of the different slopes of the similarity functions in unstable and stable stratification, given that we apply linear robust regression.
Table 4 summarizes the results for \(\varPhi _{\theta }\), where \(R_{data}\) values are in all cases much larger than \(R_{rand}\) values in unstable stratification, and \(K>> 0\), indicating that selfcorrelation is not dominating the \(\varPhi _{\theta }\) relationship. On the unstable side of nearneutral stratification, K is in almost every case close to zero, indicating a considerable influence of selfcorrelation, but \(R_{data} > R_{rand}\) in the majority of cases. It is noted that the value of K is positive for all unstable cases.
On the stable side (\(z/\varLambda \ge 0.05\)), the value of K is large in magnitude for all iBox sites, but with \(R_{rand} > R_{data}\), probably because of the very small slope of the bestfit curves in this region. The horizontal bestfit curves show that there is no actual dependence of \(\varPhi _{\theta }\) on \(z/\varLambda \), so any random dataset will have a larger correlation coefficient \(R_{rand}\). The nearneutral (stable) region exhibits relatively small absolute values of K, albeit with the majority being somewhat larger than on the unstable side.
Overall, the results of Table 4 indicate that selfcorrelation is not the dominant factor influencing the \(\varPhi _{\theta }(z/\varLambda )\) relationships for nonnearneutral stratification. In the nearneutral range (especially on the unstable side), the value of K is close to zero, indicating that the obtained functional relationships are largely influenced by selfcorrelation.
Concerning \(\varPhi _q\), as mentioned before, there is no discrimination between stronger and nearneutral stabilities. From Table 5, it can be seen that K is larger for unstable than for stable stratification. On the unstable side, the values of K vary between \( 0.26\) and 0.55 among the measurement sites, suggesting weak selfcorrelation. The strongly negative values of K in the stable range are likely due to the large scatter of the data points and not to selfcorrelation. Hence, the above results indicate that, for the relationship between \(\varPhi _q\) and \(z/\varLambda \), selfcorrelation is not a dominant factor.
Although \(\varPhi _{\theta }\) was found to be seriously affected by selfcorrelation in the nearneutral regions of stable and unstable stratifications, the corresponding data are not excluded from the present analysis because it is nevertheless considered useful to study the impact of terrain complexity under these conditions. In particular, when inspecting the degree of selfcorrelation at the reference site (Tables 4, 5), it becomes clear that the degree of selfcorrelation does not seem to be primarily a question of terrain complexity. However, since the results from stability ranges potentially dominated by selfcorrelation should be considered with necessary caution, we graphically distinguish those results from those less affected (see Fig. 8).
Assessment of selfcorrelation between the nondimensional temperature standard deviation and local stability for all iBox sites, for unstable and stable conditions (using robust linearregression coefficients); N is the number of data points, \(R_{data} = \sqrt{R_{data}^2}\) and \(R_{rand} = \sqrt{R_{rand}^2}\) are the linear correlation coefficients of the real data and the randomized data, respectively, and \(K = (R_{data}^2R_{rand}^2)/R_{data}^2\) is the fraction of the variance explained by physical processes
Unstable  \(z/\varLambda \le 0.05\)  \(0.05 < z/\varLambda \le 0\)  

Site  Level  N  \(R_{data}\)  \(R_{rand}\)  K  N  \(R_{data}\)  \(R_{rand}\)  K 
1st  1969  0.79  0.46  0.66  657  0.69  0.62  0.2  
CSVF0  2nd  2341  0.77  0.54  0.51  303  0.59  0.54  0.16 
3d  2385  0.7  0.58  0.31  252  0.66  0.57  0.27  
CSSF8  1st  940  0.86  0.5  0.66  390  0.89  0.66  0.44 
2nd  992  0.85  0.57  0.54  228  0.62  0.62  0.02  
CSSF1  1st  684  0.78  0.6  0.42  94  0.52  0.48  0.15 
CSNF10  1st  1001  0.78  0.59  0.42  91  0.62  0.6  0.035 
CSNF27  1st  306  0.68  0.54  0.37  58  0.5  0.49  0.05 
Reference (Cabauw)  1st  1169  0.88  0.44  0.75  1750  0.92  0.89  0.055 
Stable  \(0\le z/\varLambda < 0.05\)  \(z/\varLambda \ge 0.05\)  

Site  Level  N  \(R_{data}\)  \(R_{rand}\)  K  N  \(R_{data}\)  \(R_{rand}\)  K 
1st  824  0.67  0.59  0.21  1299  0.34  0.46  − 0.84  
CSVF0  2nd  348  0.45  0.54  \(0.48\)  1760  0.28  0.49  − 2.01 
3d  304  0.62  0.54  0.23  1811  0.3  0.51  − 2.01  
CSSF8  1st  337  0.39  0.57  \(1.11\)  207  0.36  0.49  − 0.8 
2nd  264  0.68  0.57  0.29  390  0.2  0.49  − 4.78  
CSSF1  1st  149  0.71  0.51  0.49  1119  0.26  0.5  − 2.7 
CSNF10  1st  202  0.69  0.52  0.44  2117  0.29  0.57  − 2.9 
CSNF27  1st  567  0.5  0.57  \(0.28\)  1599  0.25  0.51  − 3.28 
Reference (Cabauw)  1st  2693  0.89  0.76  0.28  2510  0.3  0.47  − 1.48 
Assessment of selfcorrelation between the nondimensional humidity standard deviation and local stability for all iBox sites, for unstable and stable conditions (robust linearregression coefficients)
Site  Level  Unstable  Stable  

N  \(R_{data}\)  \(R_{rand}\)  K  N  \(R_{data}\)  \(R_{rand}\)  K  
CSVF0  1st  2626  0.79  0.53  0.55  2123  0.6  0.84  − 0.94 
3d  2637  0.73  0.64  0.23  2115  0.69  0.76  − 0.2  
CSSF8  2nd  1220  0.75  0.67  0.21  654  0.81  0.66  0.32 
CSSF1  1st  778  0.71  0.52  0.47  1268  0.85  0.87  − 0.048 
CSNF10  1st  1092  0.73  0.82  − 0.26  2319  0.56  0.69  − 0.51 
CSNF27  1st  364  0.66  0.64  0.078  2166  0.82  0.91  − 0.24 
Reference (Cabauw)  1st  2919  0.8  0.79  0.01  5203  0.93  0.56  0.64 
6 Results
6.1 Scaled Temperature Standard Deviation
The similarity functions for the temperature standard deviation are presented and discussed here. In Table 6 of the Appendix, the nonlinear bestfit functions between \(\varPhi _{\theta }\) and \(z/\varLambda \) are shown for every iBox site for unstable (\(z/\varLambda \le 0.05\)), nearneutral 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 bestfit similarity functions are only from the first level of every iBox site. It should be noted, however, that the differences in the obtained bestfit relations between different heights where available (the CSVFO and CSSF8 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 iBox 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 nearneutral ranges, which also gives an increased possibility of selfcorrelation. However, these ranges are not excluded from the analysis as the fitted equations in the nearneutral 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 nearneutral range (within the measurement uncertainty), with the smallest value (\( 9.7\times 10^{4}\)) occurring at the southfacing CSSF8 site.
The statistical differences between individual iBox datasets, and also between the iBox 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 differentsized 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 iBox datasets and the reference for unstable, stable and nearneutral (unstable) stratification. However, between individual iBox datasets, differences in distributions are not statistically significant in some cases, as is the case between the sites CSNF10, CSNF27 and CSSF1 in the nearneutral (unstable) range, as well as between the CSSF1 and CSNF27 sites in the unstable range. For stable stratification, differences between all the iBox 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.
On the stable side in the nearneutral region (Fig. 8b), the CSNF10 and CSSF1 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 bestfit curves for the CSVF0 and CSNF27 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 bestfit iBox 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 zless scaling (Nieuwstadt 1984).
According to the bestfit equations for the iBox sites in the nearneutral (unstable) range, the exponent \(d_{{\theta }n}\) is smallest for the valleyfloor site (CSVF0) and largest for the steepsloped site (CSNF10), 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 CSSF8 and largest for the site with the steepest slope (CSNF27). The largest differences in the curves’ slopes between the iBox sites are detected in the nearneutral range of both unstable and stable stratifications (Fig. 8a, b, Table 6). With stronger stability (for both unstable and stable stratifications), the bestfit curves of the iBox sites are relatively close to each other.
To examine whether the scatter around the bestfit 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 iBox bestfit 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, nearneutral and stable) and for every site separately. These shaded areas represent the confidence intervals for each bestfit curve. It should be noted that the confidence intervals differ in width along each bestfit 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 CSNF27 site does not overlap with that for the reference site. In the nearneutral (unstable) range, the confidence intervals are not overlapping, except for site CSNF10 with site CSSF1. Therefore, despite the uncertainty in the datasets, the parametrizations are unique for every dataset in the nearneutral (unstable) range. For strongly stable stratification, the confidence intervals overlap, except for site CSSF8, whose bestfit curve is much higher than all the others (Fig. 8b). However, in the nearneutral (stable) range, these intervals are separated from each other, emphasizing the uniqueness of each bestfit 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.

Postprocessing: as iBox and reference datasets were both analyzed with exactly the same postprocessing 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 iBox dataset, and not for the reference, this excludes the postprocessing from being the reason.

The zeroplane displacement is determined in a relatively crude manner (see Sect. 3.1). However, when using extreme values for the zeroplane 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 zeroplane displacement is not the reason for the large values of \(\varPhi _{\theta }\).

Nonconstant fluxes: Nadeau et al. (2013a) used the nonconstancy 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 ‘nonconstant 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 nonconstant fluxes, for both unstable and stable stratifications at the CSVF0 site. This result confirms that the difference from the reference curves for temperature, which is observed at each iBox site, is not due to the existence of nonconstant fluxes.

Coordinate system (i.e., the frame of reference for the projection of the temperature and momentum fluxes): while it is customary to use a terrainfollowing 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 slopenormal and vertical temperature fluxes for the four iBox sites with sloping terrain (Figs. 10, 14 in the Appendix). The fluxes are converted from slopenormal 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 nearneutral (unstable) region. In the Appendix (Fig. 14), it can be seen that both sites with steeply sloping terrain (the CSNF10 and CSNF27 sites) exhibit this decrease of the curve slope in the nearneutral (unstable) range when using vertical temperature fluxes, although there are not enough data points to substantiate this statement. For the sites CSSF1 and CSSF8, this phenomenon is not observed systematically. Despite this decrease in the curves’ slopes, it is clear that the large majority of iBox data points in the unstable region do not change after axis transformation.
According to Oldroyd et al. (2016a), the change from slopenormal 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 slopenormal to vertical coordinates according to the approach of Oldroyd et al. (2016a), a change of sign is detected in about 100 cases at the CSNF10 site, and for a few at the other sites. More importantly, on the stable side, the move from slopenormal 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 stronglysloped sites (the CSNF10 and CSNF27 sites). This leads to the separation of the datasets into two main clusters for the sites CSNF10 and CSNF27, likely because of two different flow types (i.e., katabatic vs. dynamicallymodified flows under stable conditions). It does not, however, alter the fact that \(\varPhi _{\theta }\) is systematically larger than predicted from the reference curve.
6.2 Scaled Humidity Standard Deviation
Figure 8c shows that the bestfit curve of the CSNF27 site is always higher in magnitude than the reference on the unstable side, whereas the other iBox bestfit 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 bestfit function for stable stratification is a constant, which is also suggested by the scatter of the present data. In Fig. 8d, the CSNF27 and CSSF1 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 bestfit for stable stratification in the case of metrescale heterogeneous terrain, whereas Liu et al. (1998) found \(\varPhi _q = 2.4\) in the case of HHF terrain. For the iBox sites, bestfit 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 bestfit curve of the CSVF0 site is closer to the reference curve than those of the other iBox 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 CSSF1 and CSSF8, as well as between sites CSNF27 and CSSF8 (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 bestfit curves in Fig. 8c, d represent the MAD values of the datasets and, therefore, the confidence intervals of the bestfit curves. In Fig. 8c, it is noted that almost all the confidence intervals overlap for strong instability. In contrast, the intervals for sites CSVF0 and CSNF10 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 CSVF0 site is almost the same as the reference site, as their bestfit curves are very similar, and both exhibit little scatter in the data.
6.3 Comparison Between Temperature and Humidity Similarity Functions
Many studies have suggested using the bestfit 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 bestfit similarity curves of temperature and humidity for unstable and stable stratifications (\(z/\varLambda  \ge 0.05\)). The bootstrapping method, combined with the Student’s ttest, 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 nearneutral 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 bestfit curves for \(\varPhi _{\theta }\) and \(\varPhi _q\) are very similar for the sites with a small terrain slope (the reference, CSVF0, CSSF8 and CSSF1 sites). In contrast, the slopes of the two curves differ noticeably for the CSNF10 site, as well as in magnitude for the CSNF27 site. In the nearneutral (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 CSVF0 site, which gives almost identical curves, and the CSSF8 site where the \(\varPhi _q\) curve is lower in magnitude. The largest differences in the curves’ magnitudes are noted for the CSSF1 and CSNF27 sites. For the nearneutral (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 ttest, the bestfit 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 bestfit 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.
7 Summary and Conclusions
Our objective was to examine the applicability of local scaling to fluxvariance relationships of temperature and humidity in the complex terrain of an Alpine valley. For this we used the iBox dataset, consisting of data from five different sites of different slope angle, orientation and roughness, located in one of the major valleys in the Alps. As a reference for horizontally (weakly) inhomogeneous and flat terrain, an 11month dataset from the Cabauw site in the Netherlands was used.
For the scaled standard deviation of temperature \(\varPhi _{\theta }\), the reference formulation is based on Eq. 5 for strong stability (\(z/\varLambda  \ge 0.05\)) and the suggestion of Tampieri et al. (2009) for the nearneutral range to use a curve of slope \( 1\). Data from the Cabauw study area agree with the ‘classical’ formulations from the literature for horizontally homogeneous and flat terrain in magnitude and slope in the range of their respective applicability, which is also true for the nondimensional standard deviation of humidity (\(\varPhi _q\)).
The classical Monin–Obukhov equations for \(\varPhi _{\theta }\) are not applicable to the reference dataset, which do not account for the slope of the referencedata curve in the nearneutral (unstable) region. For the data outside the nearneutral range (\(z/\varLambda  \ge 0.05\)), the reference bestfit curves are found to be similar to results from previous studies for HHF terrain, and, therefore, were used as a reference here. Although the same quality control and corrections were applied to both the reference dataset and to the iBox data, the reference data fitted perfectly to the Tillman (1972) curve on the unstable side, but the iBox data points lie mostly above this curve (Fig. 7).
The iBox sites are characterized by turbulent fluxes that usually vary with height, considering the strong requirement that all turbulent fluxes have to be simultaneously constant with height. Specifically, the value of \(\overline{w'\theta '}\) is only found to be approximately constant with height about 50\(\%\) of the time, while momentum fluxes are usually height dependent for the study period. Although only two sites were examined for the constantflux hypothesis (one at the valley floor and one with a weak slope), it can be safely assumed that, if the constantflux hypothesis fails at the valley floor, which approaches HHF terrain to some degree, it will likely also fail for the mountain slopes. Since constant turbulent fluxes with height are an assumption of Monin–Obukhov similarity, the basic research question was the applicability of local scaling in highly complex terrain.
The selfcorrelation test for \(\varPhi _{\theta }\) and \(\varPhi _q\) as a function of \(z/\varLambda \) following Klipp and Mahrt (2004) shows that data are not selfcorrelated outside the nearneutral range. The nearneutral data for both stable and unstable stratification, however, showed selfcorrelation (Tables 4, 5). Although the nearneutral regions probably ‘suffer’ from selfcorrelation, data from these regions were not excluded from the analysis in order to investigate their dependence on the inhomogeneity of the study area. It should be mentioned though that the results from the nearneutral regions should be treated with caution, because either the currently available methods for assessing selfcorrelation (e.g., Klipp and Mahrt 2004) need improvement, or a completely new method is needed to represent data with nonlinear relationships more adequately. Also, conditions for a threshold for \(R_{data}^2R_{rand}^2\) should be defined to allow a more quantitative determination of selfcorrelation.
The analysis of the \(\varPhi _{\theta }\) similarity functions to iBox data indicates that the bestfit similarity functions for every iBox site are different, and these differences are statistically significant (Fig. 8, Table 6). While similarity curves differ in terms of both slope and magnitude of the curves, some similarities—such as a large slope of the curve in the nearneutral region—are found between the sites with the steepest terrain slopes (the CSNF10 and CSNF27 sites). Furthermore, the bestfit curve of the valleyfloor site (the CSVF0 site) is found to be more similar to the reference curve than the other sites for stable stratification, but still (in a statistical sense) significantly different (Fig. 8b, Table 6).
Similarity functions for \(\varPhi _{\theta }\) were found to be significantly larger in magnitude than the reference at all sites, and for stabilities outside the nearneutral (unstable) range. Four possible reasons for this were investigated: the postprocessing method, the determination of the zeroplane displacement, the nonconstancy of the fluxes with height, and the slopefollowing coordinate system. As shown in the results, significantly larger \(\varPhi _{\theta }\) values than the reference are caused neither by the postprocessing method, nor by the nonconstancy of the fluxes. It is further demonstrated that the large values are not due to the crude method used to determine the zeroplane displacement in \(z/\varLambda \), or the chosen coordinate system. Having rejected these potential reasons, the most probable reason for this difference is the complexity of the terrain. It should be noted that all iBox sites exhibit differences in the magnitude of the curve, even the valleyfloor site, and so cannot be the result of the local terrain slope of the sites, but possibly the general terrain inhomogeneity of the study area.
In contrast to \(\varPhi _{\theta }\), similarity functions for the scaled standard deviation of humidity \(\varPhi _q\) as a function of stability for unstable conditions show no scatter of the data points in the nearneutral region in agreement with the literature review. However, the magnitude of the bestfit curves are affected by the mountain slope of each iBox site (Fig. 8, Table 6). The magnitude of the bestfit curve of the iBox site with the maximum mountain slope (the CSNF27 site) is significantly higher than the curves for the other sites in the unstable range, and exhibits much larger scatter in the stable regimes. However, this difference, which is influenced by the mountain slope, is not evident in the slope of the curves in contrast to \(\varPhi _{\theta }\). It should be noted that, although there is a remarkable difference in the magnitude of \(\varPhi _q\) between the iBox sites with a large mountain slope, and those with zero or a small slope, the similarity between the curves for the site with the largest local slope and the one with a minor slope suggests that this difference is not directly related to the terrain slope (or other site characteristics).
As a final step, the comparison between the bestfit curves of \(\varPhi _{\theta }\) and \(\varPhi _q\) was conducted for all sites for non nearneutral stratification (Fig. 11), illustrating that differences in the two types of curves are generally statistically significant for all cases, except for the reference curves, with differences in curve magnitude more profound for stable stratification. Additionally, as the differences are affected by the mountain slope of the site, it is not recommended to use \(\varPhi _{\theta }\) to describe the humidity fluctuations as a function of \(z/\varLambda \), especially in nonhomogeneous terrain.
List of bestfit similarity relations for the nondimensional temperature \(\varPhi _{\theta }\) and humidity \(\varPhi _q\) standard deviations, for every iBox site, for unstable, nearneutral (unstable) and stable stratifications
\(\varPhi _{\theta }\)  

iBox site  N  \(a_{\theta }\)  \(b_{\theta }\)  \(c_{\theta }\)  \(d_{\theta }\)  \(e_{\theta }\)  Stability  Stability range 
CSVF0  1969  1.29  0.081  − 1  − 0.33  0  Unstable  \(3.3\le z/\varLambda \le 0.05\) 
657  0.011  0  − 1  − 1.1  2.24  Nearneutral (unstable)  \(0.05< z/\varLambda \le 0.001 \)  
2123  \(2.8\times 10^{5}\)  0  1  − 2.1  2.32  Stable  \(1.7\times 10^{3}\le z/\varLambda \le \) 1.53  
CSSF8  940  1.32  0.17  − 1  − 0.32  0  Unstable  \( 7.59\le z/\varLambda \le 0.05 \) 
390  \(3.6\times 10^{3}\)  0  − 1  − 1.2  2.025  Nearneutral (unstable)  \(0.05< z/\varLambda \le 9.7\times 10^{4} \)  
544  0.018  0  1  − 0.9  3.39  Stable  \(6.1\times 10^{4} \le z/\varLambda \le \) 4.38  
CSSF1  684  1.14  0.069  − 1  − 0.4  0  Unstable  \(2.71\le z/\varLambda \le 0.05\) 
94  \(3\times 10^{3}\)  0  − 1  − 1.5  2.35  Nearneutral (unstable)  \(0.05< z/\varLambda \le 3.7\times 10^{3}\)  
1268  \(1.05\times 10^{4}\)  0  1  − 2.07  2.67  Stable  \(2.5\times 10^{3}\le z/\varLambda \le \) 1.84  
CSNF10  1001  1.13  0.081  − 1  − 0.37  0  Unstable  \( 5.28\le z/\varLambda \le 0.05 \) 
91  \(1.2\times 10^{3}\)  0  − 1  − 1.7  2.19  Nearneutral (unstable)  \(0.05< z/\varLambda \le 3\times 10^{3}\)  
2319  \(6.8\times 10^{5}\)  0  1  − 1.96  2.54  Stable  \( 2.2\times 10^{3} \le z/\varLambda \le \) 4.98  
CSNF27  306  1.38  0.2  − 1  − 0.5  0  Unstable  \(1.82\le z/\varLambda \le 0.05 \) 
58  \(4.5\times 10^{3}\)  0  − 1  − 1.3  2.57  Nearneutral (unstable)  \(0.05< z/\varLambda \le 4.6\times 10^{3} \)  
2166  \(3.37\times 10^{6} \)  0  1  − 2.6  2.46  Stable  \(3.2\times 10^{3} \le z/\varLambda \le \) 1.39  
Reference (Cabauw)  1169  0.99  \(6.3\times 10^{2}\)  − 1  v1/3  0  Unstable  \(2.87\le z/\varLambda \le 0.05 \) 
1750  \(1.5\times 10^{2}\)  0  − 1  − 1  1.76  Nearneutral (unstable)  \(0.05< z/\varLambda \le 7\times 10^{5} \)  
5203  \(8.7\times 10^{4} \)  0  1  − 1.4  2.03  Stable  \(1.4\times 10^{5} \le z/\varLambda \le \) 5.5 
\(\varPhi _q\)  

iBox site  N  \(a_q\)  \(b_q\)  \(c_q\)  \(d_q\)  \(e_q\)  Stability  Stability range 
CSVF0  2626  1.37  0.43  − 1  − 0.51  0  Unstable  \(3.3< z/\varLambda \le 0.001\) 
2123  –  –  –  –  2.58  Stable  \(1.7\times 10^{3} \le z/\varLambda \le \) 1.53  
CSSF8  1220  1.1  0.03  − 1  − 0.25  0  Unstable  \(7.59 < z/\varLambda \le 9.7\times 10^{4}\) 
654  –  –  –  –  3.12  Stable  \( 6.1\times 10^{4} \le z/\varLambda \le \) 4.38  
CSSF1  778  1.11  0.08  − 1  − 0.33  0  Unstable  \(2.71 < z/\varLambda \le 3.7\times 10^{3}\) 
1268  –  –  –  –  5.25  Stable  \(2.5\times 10^{3}\le z/\varLambda \le \) 1.84  
CSNF10  1092  1.07  0.26  − 1  − 0.36  0  Unstable  \(5.28 < z/\varLambda \le 3\times 10^{3} \) 
2319  –  –  –  –  3.83  Stable  \(2.2\times 10^{3} \le z/\varLambda \le \) 4.98  
CSNF27  364  1.18  0.04  − 1  − 0.27  0  Unstable  \(1.82 < z/\varLambda \le 4.6\times 10^{3}\) 
2166  –  –  –  –  5.28  Stable  \(3.2\times 10^{3} \le z/\varLambda \le \) 1.39  
Reference (Cabauw)  2919  0.99  \(3.1\times 10^{2}\)  − 1  − 0.288  0  Unstable  \(2.87 < z/\varLambda \le 7\times 10^{5}\) 
5203  –  –  –  –  2.74  Stable  \(1.4\times 10^{5} \le z/\varLambda \le \) 5.5 
Notes
Acknowledgements
Open access funding provided by University of Innsbruck and Medical University of Innsbruck. The authors are grateful to all the collaborators at the Department of Atmospheric and Cryospheric Sciences (ACINN) who worked for many years to make the reliable and continuous database of the iBox study area available. The authors would also like to thank the managers of the database of the Cabauw Experimental Site for Atmospheric Research (CESAR) for their contribution and helpful support. Financial support for this work from the Austrian Research Fund (FWF) through grant \(\#P 26290N26\) is thankfully acknowledged. We are thankful to the anonymous reviewers for their thoughtful and important comments, which helped to greatly improve the paper. Many thanks go to Georg Mayr (ACINN) for his valuable support in the statistical analysis.
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