## 1. Introduction

Monin–Obukhov similarity theory (MOST; Obukhov 1971; Monin and Obukhov 1954) of the atmospheric surface layer has been a widely accepted pragmatic framework in the scientific community for more than 50 years (Foken 2006). It provides empirical relationships between the atmospheric stability and nondimensionalized quantities in the surface layer when using a suitable set of scaling parameters. However, as MOST does not predict the precise formulation of these relationships, they have to be determined experimentally. The MOST framework has been tested and validated in numerous field experiments, with the Kansas and Minnesota experiments in 1968 and 1973 being the most famous ones (Businger et al. 1971; Kaimal et al. 1976; Kaimal and Wyngaard 1990). Nowadays, MOST is used in numerous meteorological models, for example, to determine the turbulent fluxes at the surface as boundary condition. It is also employed by experimentalists, for example, to derive surface fluxes of sensible and latent heat from scintillometer observations at elevated heights (e.g., Kohsiek et al. 2002; Meijninger et al. 2006; Evans et al. 2012). In the free-convection limit, local-free-convection (LFC) scaling provides a different and simpler set of scaling parameters.

According to MOST, the empirical relationships for nondimensionalized quantities, once formulated, must be unique. It is also commonly assumed that these relationships are identical for thermal and moisture quantities when the correlation between (potential) temperature *θ* and (specific) humidity *q* is unity (Hill 1989). Experimental datasets, however, have revealed that such universal functions are difficult to obtain. Consequently, there coexist various “universal” functions in literature (see section 2). Some authors hypothesize that no universal functions, as demanded by MOST, exist at all, or they question the validity of MOST. Wilson (2008), for example, analyzed sonic anemometer data over a Utah salt flat and found that the neutral-limit value for the normalized standard deviation of vertical velocity deviated significantly from what is considered to be the standard value. McNaughton (2006) used a structural model of the atmospheric surface layer and found that the larger eddies of the boundary layer affect the surface layer so that a basic assumption of MOST is violated. Moreover, there is discussion on dissimilarity between the turbulent transport of heat and moisture that could affect the universality of MOST functions (Li et al. 2012a,b). Li et al. (2012a), for example, pointed out that dissimilarity between the turbulence transport of temperature and humidity occurred in their measurement data for weakly unstable conditions. Consequently, the MOST relationships for their observed structure parameters for temperature

One persisting issue in MOST is the theoretical consideration that LFC scaling predicts the slope of the MOST functions in the free-convection limit (Wyngaard et al. 1971; Businger 1973), but it has not been possible so far to find MOST functions that fulfill this requirement and follow the predicted slopes. Businger (1973) stated that this might be related to the fact that the requirement of the friction velocity

Khanna and Brasseur (1997) were the first to employ turbulence-resolving large-eddy simulations (LESs) to study the predictability of the commonly used MOST functions for mean gradients, variances, budgets of turbulent kinetic energy, and temperature variance, as well as velocity and temperature spectra under unstable conditions. They found that the temperature field in the LES model satisfied MOST, while the velocity field showed significant deviations from the proposed functions. They also reported an indirect dependency of the boundary layer depth *r* could impact the corresponding MOST relationships. In a precursor to this paper, Maronga (2014, hereafter M14) employed surface-layer-resolving LES to investigate the similarity functions for

Over the past years, direct numerical simulations have been employed to investigate MOST (e.g., van de Wiel et al. 2008; Chung and Matheou 2012). However, these studies have so far focused on flow regimes with stable static stability only, possibly owing to the fact that much larger model domains are required when simulating convective boundary layers.

As MOST is applied in almost all meteorological models, ranging from single-column models, microscale models, LES with coarse grid (where the surface layer cannot be resolved), and numerical weather predictions models, the formulation and universality of the used MOST relationships is of major importance and contributes significantly to the quality of surface-layer parameterizations. In this context we want to investigate some major open questions regarding MOST that are present for some decades now.

In this paper, we will make use of the extensive LES dataset generated by M14 in order to extend the analysis for structure parameters to the commonly used relationships for mean gradients of *θ*, *q*, and horizontal wind velocity *u*_{h}. We will also study the standard deviations of *θ*, *q*, and of the vertical velocity *w*. Finally, we will derive new MOST functions for these quantities and determine free-convection scaling constants. Our main focus will not be to provide better MOST functions than already proposed in literature. Our focus will mainly be on the following basic scientific questions: (i) Is it possible to derive universal MOST functions from LES data? (ii) Does the concept of MOST collapse in free convection? (iii) Is it possible to unify MOST relationships and LFC scaling predictions for mean gradients in the free convection limit? (iv) Can dissimilarity between the turbulent transport of heat and moisture due to entrainment alter the similarity relationships?

The LES concept provides ideal conditions for answering these questions as it allows performing simulations with well-controlled parameters, such as prescribed surface fluxes, perfect horizontal homogeneity of the surface, and stationarity of the flow. Moreover, the LES technique allows to simultaneously perform virtual measurements at different height levels within the surface layer and spatial averaging can be used instead of the commonly used time averaging (and using Taylor’s hypothesis of frozen turbulence) to improve statistics. The disadvantage is, however, that LES models generally suffer from numerical errors and effects of the small-scale turbulence parameterizations involved, particularly close to the surface. In contrast, field observations do not involve such numerical problems, but usually suffer from the lack of surface homogeneity, unsteadiness of the flow, sparse measurement density in space, and measurement uncertainty.

Despite the findings of former studies that suggested to incorporate both *z*_{i} and *r* (Steeneveld et al. 2005; Johansson et al. 2001), we will confine ourselves to the classic MOST formulation. The reasoning behind this is that, in most circumstances, the required information on the structure of the top of the boundary layer is not measured, but MOST is nonetheless used as a pragmatic solution.

The paper is organized as follows. Section 2 deals with the theoretical background of MOST and LFC scaling and the current state of the art. Section 3 gives a short overview on the LES database. The main results are presented in section 4, before we end with a summary and a short outlook in section 5.

## 2. Theory

### a. Monin–Obukhov similarity theory

*z*, the Obukhov length

*L*, the near-surface kinematic fluxes of heat

*g*is the gravitational acceleration, and

*u*and

*υ*being the horizontal wind components in

*x*and

*y*directions, respectively. Based on this set of parameters, there is an additional dimensionless group:

*z*(e.g., Andreas 1988). The gradients of mean profiles of

*θ*,

*q*, and

*C*

_{3}–

*C*

_{8}being constants, again with several different proposed values from literature (e.g., Liu et al. 1998; Andreas et al. 1998; Panofsky et al. 1977; Kaimal and Finnigan 1988; Wilson 2008).

### b. Local-free-convection scaling

The Obukhov length can be regarded as the height where mechanical and buoyant production of turbulence are roughly in balance (Andreas 1991). When buoyant production dominates the generation of turbulence (that is the case in calm winds), the mean wind collapses to zero and

*z*is the only variable with unit of length), there is only one dimensionless group and the dimensionless mean gradients should follow (Wyngaard et al. 1971)

## 3. LES model and case description

### a. LES model

The Parallelized Large-Eddy Simulation Model (PALM, revision 893) (Raasch and Schröter 2001; Maronga et al. 2015) has been used for the present study. It has been widely applied to study different flow regimes in the convective and neutral boundary layer (e.g., Raasch and Franke 2011; Letzel et al. 2008). All simulations were carried out using cyclic lateral boundary conditions. The grid was stretched in the vertical direction well above the top of the boundary layer to save computational time in the free atmosphere.

MOST was applied locally between the surface and the first computational grid level (“local similarity model”; see also Peltier and Wyngaard 1995) to calculate the local friction velocity

A 1.5-order flux–gradient subgrid closure scheme after Deardorff (1980) was applied in the formulation of Saiki et al. (2000), which requires the solution of an additional prognostic equation for the subgrid-scale (SGS) turbulent kinetic energy. A fifth-order advection scheme of Wicker and Skamarock (2002) and a third-order Runge–Kutta time step scheme (Williamson 1980) were used to discretize the model in space and time. In case of a prescribed geostrophic wind, a one-dimensional version of the model with fully parameterized turbulence, using a mixing-length approach after Blackadar (1997) and stationary temperature and humidity profiles, was used for precursor simulations over several days to generate steady-state wind profiles as initialization for the LES. Simultaneously, inertial oscillations in the near-neutral cases were effectively damped in the precursor simulations so that a steady state could be reached early in the LES runs.

### b. Case description

The set of LES that was generated and described by M14 and is based on a reference simulation for the Cabauw area in the Netherlands from the study of Maronga et al. (2013). Details of the observed case are described in de Arellano et al. (2004). The model was discretized in space with 2048 grid points in each horizontal and 832 grid points in the vertical direction. The grid spacing was 4 m × 4 m × 2 m (Δ_{x} × Δ_{y} × Δ_{z}). The simulations were driven by constant kinematic surface fluxes of heat and moisture with a Bowen ratio ^{−1}) with a capping inversion starting at *z* = 950 m and a depth of 250 m, and the stably stratified dry free atmosphere above with a lapse rate of 7 K km^{−1} were prescribed. The lapse rates in the capping inversion are provided in Table 1. A detailed description of the setup can be found in Maronga et al. (2013, section 3 and Table 1, case A).

Overview of the LES setup (reproduced from M14), complemented by the maximum values of *γ* is lapse rate of potential temperature in the capping inversion.

Based on this setup, the geostrophic wind speed and the prescribed surface fluxes were systematically varied for each simulation. This was done in order to generate a set of simulations (herein also referred to as reference simulations) that cover a large range of ^{−1}. Second, in order to cover those cases that are dominated by shear production ^{−1}. We complemented the dataset by a run with a background wind of 10 m s^{−1} and zero surface fluxes to mimic an almost neutral boundary layer (as entrainment of warm and dry air is still possible). For sensitivity studies, *γ* was reduced for one case in order to study whether entrainment of warm and dry air affects the similarity functions. Because of the high computational costs these sensitivity cases were simulated for two selected background wind speeds only: 0 m s^{−1} (free convection) and 8 m s^{−1}. The full list of simulations with the relevant parameters is given in Table 1.

### c. Database and processing

Each simulation ran for 2 h. A quasi-steady state had reached in the surface layer for each run after at most 1 h of simulation time. Horizontally averaged profiles (spatial average over 2048^{2} grid points) of temperature, humidity, and horizontal wind were thus output every 120 s during the second hour of the simulation. In addition, the (horizontal) variance profiles of temperature, humidity, and vertical wind were stored. The mean gradients of *θ*, *q*, and *L*, the mean vertical profile of *θ* and *q* as a three-dimensional quantity in PALM, the buoyancy flux was a direct model output.

The boundary layer depth (see Table 1) was defined as the height where the gradient of the horizontally averaged potential temperature had a maximum. Moreover, this height had to fulfill the requirement that the gradient was at least 0.002 K m^{−1} and greater than that of the four grid levels above. The value of *z*_{i} was in most cases in the order of 1 km. The top of the surface layer *z*_{i}. M14 showed that this choice did not introduce any error.

Nondimensional mean gradients and standard deviations according to MOST and LFC were calculated in a postprocessing for each available time step and each height level within the surface layer. M14 explained that the surface layer could not be resolved sufficiently within the lowest seven grid points owing to the LES numerics [in agreement with the former LES of Khanna and Brasseur (1997) and Brasseur and Wei (2010)], but he showed that the nondimensionalized LES data for structure parameters collapsed to single curves when excluding data from these lowest grid levels (see his Fig. 2). In the present study we thus decided to remove the lowest seven grid points directly from our analysis.

MOST fitting functions are derived using the classical nonlinear least squares regression method. The data were not weighted; that is, regions with more data points led to an increased weighting for this region during the regression. Braam et al. (2014) studied the effect of the regression method, scaling, and weighting of the data on MOST fitting functions for structure parameters. From their results, they concluded that orthogonal regression would be more appropriate than classical least squares regression because uncertainties in the independent variables would be considered. They also suggested performing the regression on logarithmized data. However, their results are based on datasets from field observations that traditionally display large scatter, so that these specific details of the regression analysis indeed can have a strong impact. When scatter in the data is small, these impacts will be negligible. In section 4 we will see that the LES data in most cases indeed displays significantly less scatter than observational data so that we feel confident that our regression method does not imply uncertainties that we could overcome with an adjusted regression method as proposed by Braam et al. (2014).

## 4. Results

### a. General features of the simulated boundary layers

After having reached a steady state (≈1-h simulation time), all cases displayed classical convective boundary layer profiles with a shallow, highly unstable region close to the surface, a well-mixed layer that extended up to a capping inversion, and the stably stratified and dry free atmosphere above (see Fig. 1 of M14). The profiles of *z*_{i} was about 1.1 km (see Table 1). In the course of the simulations, the surface heating and turbulent mixing were not strong enough to tear down the capping inversion. The lapse rate within the capping inversion was thus a relevant parameter that steered the amount of entrainment of dry and warm air into the boundary layer, whereas the lapse rate in the free atmosphere was of minor importance. In the reference cases, *z _{i}* (1.04 and 1.0 km, respectively) because of a reduced surface forcing so that thermals were weaker and not able to penetrate the capping inversion as rigorously as in the reference cases. Sensitivity cases W00_

*γ*07 and W08_

*γ*07 (weak capping inversion) are characterized by a significantly increased

*z*

_{i}of up to 1.29 km. The entrainment flux ratio here is about −0.3 for temperature and 1.0 for moisture, showing that the entrainment flux (causing a drying of the boundary layer) became the same magnitude as the surface forcing (causing a moistening of the boundary layer), whereas no increase in the entrainment of sensible heat was present. M14 showed that this had a distinct effect on the similarity function for

*z*

_{i}values and the entrainment flux ratios, see Table 1.

In the following, we will derive MOST and LFC functions based on LES data. First, however, we should recognize that the LES technique aims at modeling reality, but it hardly gives a perfect image of nature (e.g., owing to the turbulence parameterization or numerical errors). The similarity functions to be derived in the following should thus be regarded as the “model truth” but might be somewhat different from what we must expect in the field. The obvious advantage of LES is that we can control the boundary conditions for the simulations very precisely. Key issues for field experiments, like surface heterogeneity and unsteadiness of the flow, are not present. Also we do not need to deal with problems of measurement uncertainty and we can replace the commonly used time average in field experiments by the more straightforward horizontal average to derive mean quantities.

### b. MOST relationships for mean gradients

The nondimensional mean (horizontally averaged) gradients of *θ*, *q*, and

#### 1) Mean temperature and humidity gradients

Figures 1a and 1b show that the data points collapse to a single curve with remarkably little scatter in the data. As *L* remains rather constant during each simulation, the variations in *z* has basically the same effect on the dimensionless gradients as a variation in *L*. This is in agreement with the findings for structure parameters in the precursor of this study (see M14).

*ϕ*functions than the commonly used −1/2 in the BD form. Otherwise, the suggested shape of the fitting functions is very well met. Furthermore, it is noteworthy that even the data of case W00 (true free convection) behaves according to MOST. While this might be somehow surprising, it clearly supports the argument of Businger (1973) and Schumann (1988) that the large eddies in a buoyancy-driven boundary layer create a local wind shear that, when averaged, yields a friction velocity larger than zero. In our simulation data we observe

As LFC scaling would suggest the slope of the fitting functions to be −1/3 instead of −1/2, Figs. 1a and 1b were complemented by a line indicating the −1/3 slope. It is visible that the LES data do not back up this behavior for most of the

Figure 2a shows the derived fitting functions

Overview of fitting functions (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Overview of fitting functions (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Overview of fitting functions (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Overview of MOST functions proposed in literature. The formulations have been partly modified for use with

#### 2) Mean horizontal wind gradient

*L*remains constant during each run as already noted above. At low levels, the wind gradients show smaller values than expected by MOST, whereas higher levels display a clear overestimation. For highly unstable cases (W04 and W02) we observe additional scatter that is not related to a height dependency. This finding is in agreement with the former LES study of Khanna and Brasseur (1997), who also found significant departures of the velocity field from MOST, whereas temperature behaved much better. The dependence of the data on height is possibly related to deficiencies of the LES model in capturing the near-surface wind profile accurately. Such behavior was previously reported in literature (e.g., Andren et al. 1994; Sullivan et al. 1994) and is commonly ascribed to effects of the subgrid-scale model that plays an important role close to the surface. To minimize these effects, we cut off the lowest seven grid levels as already mentioned. Figures 1a,b reveal that the same effect is visible for the scalars, even though it is much less pronounced. Another reason could be the fact that MOST itself is a known oversimplification of the surface layer structure. In particular, the assumption of a constant flux layer is known to be a rough approximation and might lead to height-dependent nondimensionalized gradients. Despite this height dependency and the implied scatter in the data, Fig. 1c shows that the data do nevertheless group along a single line. The scatter involved compares to that of random errors reported in observations (e.g., Johansson et al. 2001) so that we are confident that the LES data are well suited to derive MOST relationships, even though the uncertainty in the functions will be larger than for the scalar gradients. Note also that no data points are visible for case W00 owing to the fact that the mean wind gradient here collapses to zero and is no longer a meaningful quantity. Consequently, it is difficult to derive suitable MOST fitting functions from these data. The best fits read

Because of the significant scatter in Fig. 1c, it is difficult to judge which of the above functions performs better. As for the gradients of the scalars, we have complemented Fig. 1c by a line with a slope of 1/3, which would be the local-free-convection prediction (Businger 1973). Amazingly, the slope of the nondimensional wind gradients approaches the 1/3 slope for

This interesting result might explain two key issues of past research on MOST: first, the fact that the traditional MOST fitting functions do not back up LFC scaling when looking at the free-convection limit, and second, the larger scatter observed in the mean wind gradient compared to the scatter in mean scalar gradients, at least near free-convective conditions.

Despite the large scatter, Fig. 2b shows that the

In summary, we must acknowledge that the BD fitting functions from the LES data are in much better agreement with those proposed in literature than the new fitting functions. However, they do not capture the steeper slope in the LES data for

#### 3) Spurious correlations

In a comment on the paper of Johansson et al. (2001), Andreas and Hicks (2002) pointed out that spurious self-correlation cannot be avoided when using MOST scaling, because *z*/*L*–*ϕ*_{m} pair normal to the fitting function, whereas a *z*/*L*–*ϕ*_{h} pair will move along the fitting function under unstable conditions. They conclude that the larger scatter observed in

To check whether our newly derived fitting functions suffer from self-correlation, we followed the approach of Hartogensis and De Bruin (2005) and calculated the mean gradients indirectly using the new MOST fitting functions and directly from the raw LES data. As ^{−3} s^{−1}. This is related to the scatter we have seen for large values of

Mean gradients of (a) temperature, (b) humidity, and (c) the horizontal wind velocity derived from the new MOST fitting functions against directly calculated values from the LES data. The relative error (RE) of the MOST results in relation to the directly measured data is listed in each frame.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Mean gradients of (a) temperature, (b) humidity, and (c) the horizontal wind velocity derived from the new MOST fitting functions against directly calculated values from the LES data. The relative error (RE) of the MOST results in relation to the directly measured data is listed in each frame.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Mean gradients of (a) temperature, (b) humidity, and (c) the horizontal wind velocity derived from the new MOST fitting functions against directly calculated values from the LES data. The relative error (RE) of the MOST results in relation to the directly measured data is listed in each frame.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

#### 4) Turbulent Prandtl number

_{t}= 0.74–1.0 (Kays 1994; Foken 2006). It is also expected that

Turbulent Prandtl number

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Turbulent Prandtl number

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Turbulent Prandtl number

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

### c. MOST relationships for standard deviations

The nondimensional standard deviations of *θ*, *q*, and *w* were calculated analogous to the mean gradients. Please note that no data for

#### 1) Standard deviation of temperature and humidity

Normalized standard deviations of (a) temperature, (b) humidity, and (c) the vertical velocity against stability parameter *ϕ*, while the black dashed lines show the best fit with −1/3 slope.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Normalized standard deviations of (a) temperature, (b) humidity, and (c) the vertical velocity against stability parameter *ϕ*, while the black dashed lines show the best fit with −1/3 slope.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Normalized standard deviations of (a) temperature, (b) humidity, and (c) the vertical velocity against stability parameter *ϕ*, while the black dashed lines show the best fit with −1/3 slope.

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

The comparison of the LES fittings with literature indicates that the fitting function with a −1/3 power law is within the range of the proposed functions (Figs. 6a,b). In particular, good agreement is found for

Overview of fitting functions (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Overview of fitting functions (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Overview of fitting functions (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

#### 2) Standard deviation of vertical velocity

### d. Sensitivity analysis

In this section we aim at analyzing whether the results obtained so far (i.e., the new MOST functions based on LES data) are indeed universal, or whether effects of humidity, entrainment, or possibly of the surface Bowen ratio modify these functions. The latter would support the findings of some recent studies (McNaughton 2006; Wilson 2008; Li et al. 2012a) and the precursor of this paper (see M14).

Figure 7 shows the dimensionless data from case W08 as well as the data from the sensitivity runs as described in section 3b, complemented by the new LES-based fitting functions derived from the reference cases. For the mean gradients of temperature and humidity, as well as

Nondimensional mean gradients and standard deviations from sensitivity simulations and MOST fitting function (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Nondimensional mean gradients and standard deviations from sensitivity simulations and MOST fitting function (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Nondimensional mean gradients and standard deviations from sensitivity simulations and MOST fitting function (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

It is seen that *γ*07 the spatial correlation coefficient between temperature and humidity rapidly decreased with height. As

### e. LFC scaling

The LFC constants *A* were calculated for the free-convection case W00 for all available time steps and height levels within the surface layer according to Eqs. (18)–(23). Figure 8 shows these data against height.

Vertical profiles of (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Vertical profiles of (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

Vertical profiles of (a)

Citation: Journal of the Atmospheric Sciences 74, 4; 10.1175/JAS-D-16-0186.1

#### 1) LFC constants for mean gradients

From Figs. 8a and 8b a distinct height dependency in the nondimensional mean gradients is visible, with increasing values from the lowest levels to more or less constant values at

#### 2) LFC constants for standard deviations

As for the mean gradients, the normalized standard deviations also show a clear height dependence (see Figs. 8c–e). Unlike the mean gradients, however, we observe high values close to the surface and a decrease with height. We see also some scatter due to variability in time, which is apparently higher for

Overview of similarity constants for LFC scaling and their double standard deviation (95% confidence interval), and the estimated values from the derived MOST fitting functions.

Table 3 shows all LFC constants for the references and sensitivity cases, together with the estimates for standard deviations from MOST fitting functions, calculated by means of Eqs. (24)–(26). The MOST estimate of

#### 3) Sensitivity analysis

In analogy to the analysis in section 4d, the LFC constants were derived from the sensitivity cases (i.e., W00_dry, W00_*β*05, W00_*β*40, and W00_*γ*07). The results are shown in Table 3. It is striking that only *β*40 as well as a clearly higher value of 1.4 in case W00_*γ*07. Also note that the scatter has significantly increased in the latter case. This supports the findings obtained from the sensitivity analysis for the MOST relationships that

## 5. Summary and conclusions

The present LES study focused on the derivation of MOST similarity relationships for mean gradients and standard deviations in the unstable atmospheric surface layer. A comprehensive dataset of surface-layer-resolving LES cases, covering convective to near-neutral boundary layers, that was previously used by M14 to derive similarity relationships for structure parameters was further investigated for this purpose.

The analysis of the MOST relationships showed that the dimensionless gradients of potential temperature and specific humidity strictly follow universal functions that only depend on stability (i.e.,

By comparing the estimates of the mean gradients from the newly derived fitting functions with the “true” gradients derived from the LES data, we could exclude possible errors in the fitting functions due to spurious self-correlation. The relative error of the predicted mean gradients due to scatter in the data was found to be around 6% for temperature and humidity and 10% for wind.

For the dimensionless standard deviations of potential temperature, specific humidity, and vertical velocity, we found little scatter in the data but slightly steeper slopes of the fitting functions for temperature and humidity than given by the proposed −1/3 power law in literature. The vertical velocity, however, behaves as expected from theory and follows a 1/3 power law. The fitting functions derived from the LES data are within the range of the previously proposed functions in literature.

In the free-convection limit, LFC scaling is more appropriate and the similarity relationships should reduce to constants. For the mean gradients, these constants are

To study the universality of the derived MOST–LFC relationships, data from a sensitivity study were analyzed, where the surface Bowen ratio and the lapse rate in the capping inversion were varied. The analysis revealed that almost all similarity relationships appear to be universal and both mean gradients and standard deviations do follow MOST. It is generally found that the similarity relationships for temperature quantities are roughly equal to those for humidity quantities. The analysis for

In summary, our results confirm that the MOST and LFC scaling are suitable frameworks for practical modeling and measurement applications and that the LES technique is a useful addition to achieve a better understanding of the uncertainties induced by the assumptions of MOST that are often violated in nature. Moreover, we were able to show that the LFC predicted slopes for the MOST fitting functions of mean gradients do not compete with the slopes observed directly in measurement data and that this long-standing issue was most likely due to lacking data for large values of

Despite the obvious violation of one assumption of MOST, the horizontal homogeneity of the flow, atmospheric models generally apply MOST even over very heterogeneous surface (e.g., Patton et al. 2005; Huang et al. 2009; Mironov and Sullivan 2016). Maronga et al. (2014) demonstrated that the use of MOST–LFC for structure parameters over heterogeneous terrain gave reliable estimates of the surface sensible heat flux. In a follow-up study we will more thoroughly investigate the MOST framework over heterogeneous terrain and whether effects of surface heterogeneity should be incorporated in the similarity relationships.

Also, the use of direct numerical simulations is a promising technique to gain a better understanding of the applicability of MOST in the atmospheric surface layer. In such simulations, the near-surface region can be better resolved and the technique does not suffer from limitations implied by the use of a subgrid-scale model when performing LES, particularly near the surface of the model.

## Acknowledgments

First of all, we would like to emphasize the quality of the very helpful comments of the anonymous reviewers that identified deficiencies in our initial manuscript and who provided excellent ideas for further analysis of our data that entered the revised version. Moreover, we thank Siegfried Raasch, Dieter Etling, and Christoph Knigge (all at Leibniz Universität Hannover, Germany) for reading a draft of the manuscript and/or for various discussions on the topic. Most part of the analyses in this paper were done during a research visit of the first author at the Geophysical Institute at the University of Bergen in May 2016, for which financial support was provided by Leibniz Universität Hannover and the University of Bergen, which is gratefully acknowledged. The simulations conducted for this study were supported by the German Research Foundation (DFG) under Grants RA 617/20-1 and RA 617/20-3. All simulations were performed on the former SGI Altix ICE at The North-German Supercomputing Alliance (HLRN), Hannover/Berlin. NCL (NCAR 2013) has been used for data analysis and visualization.

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