Large-eddy simulations (LESs) of free-convective to near-neutral boundary layers are used to investigate the surface-layer turbulence. The article focuses on the Monin–Obukhov similarity theory (MOST) relationships that relate the structure parameters of temperature and humidity to the surface fluxes of sensible and latent heat, respectively. Moreover, the applicability of local free convection (LFC) similarity scaling is studied. The LES data suggest that the MOST function for is universal. It is shown to be within the range of the functions proposed from measurement data. It is found that follows MOST if entrainment of dry air from the free atmosphere is sufficiently small. In this case the similarity functions for and are identical. If entrainment is significant, dissimilarity between the transport of sensible heat and moisture is observed and no longer follows MOST. In the free-convection limit the LFC similarity functions should collapse to universal constants. The LES data suggest values around 2.7, which is in agreement with the value proposed in the literature. As for MOST, the LFC similarity constant for becomes nonuniversal if entrainment of dry air is significant. It is shown that LFC scaling is applicable even if shear production of turbulence is moderately high.
The measurement of area-averaged surface fluxes of sensible and latent heat at the regional scale is necessary for a better understanding of the regional (and global) energy and water cycles. It is also important in order to improve models for meteorological and hydrological processes (De Bruin et al. 1993; Li et al. 2012). Such models, in turn, play an important role in terms of parameterizations in numerical weather prediction models. Traditionally, point measurements using eddy covariance technique are used to measure surface fluxes, but they are only representative if the surface is homogeneous (Andreas 1991). Natural landscapes rarely provide such conditions, and single point measurements then cannot be considered to be representative for larger areas.
Scintillometers have been increasingly employed in the atmospheric surface layer to measure the refractive index structure parameter as a spatial average over horizontal distances of up to 10 km (e.g., Kohsiek et al. 2002; Meijninger et al. 2002, 2006; Evans et al. 2012). Such a spatial average of can, in turn, be related to the structure parameters of temperature and specific humidity (e.g., Hill 1978). Monin–Obukhov similarity theory (MOST) provides a framework that relates the estimates of and to the surface fluxes of sensible and latent heat, respectively. In this way, scintillometers offer a technique for estimating the surface fluxes at spatial scales that might be representative for an area of several square kilometers. Nevertheless, a validation of such scintillometer-based fluxes is challenging, particularly over heterogeneous terrain. A first attempt was made by using independent low-level aircraft flights along a scintillometer path during the 2009 Lindenberg Inhomogeneous Terrain–Fluxes between Atmosphere and Surface: A Long-Term Study (LITFASS-2009) experiment (Beyrich et al. 2012). Also, a first large-eddy simulation (LES)–large-aperture scintillometer comparison has been presented by Maronga et al. (2013).
To estimate the surface fluxes from measurements of and , universal MOST functions are needed that are not given by theory and must be determined experimentally (see section 2b). Several similarity functions have been proposed in the literature (Wesely 1976; Wyngaard et al. 1971b; Andreas 1988; Hill et al. 1992; Thiermann and Grassl 1992; De Bruin et al. 1993; Li et al. 2012), but there is no consensus on a precise form so far. One reason might be that there are differences in the calculation of the relevant scaling parameters—namely, the Obukhov length L (Obukhov 1971):
Here, u, υ, and w are the wind components in x, y, and z directions on a Cartesian coordinate system, respectively. The prime indicates a turbulent fluctuation. The terms θυ, , and are virtual potential temperature, friction velocity (with and being the components of the vertical surface momentum flux), and near-surface buoyancy flux, respectively; g is the gravitational acceleration and κ is the von Kármán constant with a commonly accepted value of 0.4. The overbar denotes an average (temporal or spatial). While some studies take into account the effect of moisture on the buoyancy flux and, thus, L, others rather use the Obukhov length for dry air (e.g., Wyngaard et al. 1971b; Wesely 1976; Thiermann and Grassl 1992). It is also usually assumed that the similarity functions are identical for temperature and humidity. Li et al. (2012) discussed possible reasons for dissimilarity between the turbulent transport of heat and moisture that can lead to differences in the similarity relationships of and . They found such dissimilarity in their data under weakly unstable conditions and ascribed this to nonlocal effects like nonstationarity of the flow, advection, and entrainment.
In the free-convection limit, the similarity relationships should become universal constants, but theory does not yield these constants itself. There is consensus on the constant for , but it is still an open question whether the constant for is equal to that for (see section 2c). Theoretically, local free convection (LFC) can be only considered near the surface when no mean wind is present, but in practice it is often applied also in case of weak winds (De Bruin et al. 1995; Kohsiek 1982; Kohsiek et al. 2002).
Several studies have revealed that the mean vertical profiles of and in the CBL strongly depend on the entrainment of dry warm air at the top of the mixed layer (e.g., Wyngaard and LeMone 1980; Druilhet et al. 1983; Fairall 1984, 1987, 1991). In particular, it is found that is often dominated by entrained air from the free atmosphere. Wyngaard and LeMone (1980) showed that deviations from the mixing-layer scaling laws are caused by entrainment effects that lead to a peak of the structure parameters near the capping inversion in free convection. Fairall (1984) studied the effect of wind shear on this peak and found that the wind shear enhancement of entrainment leads to an increase of the peak values. The data of Druilhet et al. (1983) showed two peaks for : one near the surface and a secondary peak in the entrainment layer. For the entrainment peak was dominant, whereas the near-surface peak was only weak. They concluded that if the entrainment characterizes the changes in humidity in the CBL, then a new mixing-layer humidity scale should be defined that incorporates the entrainment humidity flux instead of the surface flux. Based on these findings, Fairall (1987) and Fairall (1991) used LES data and the top-down (entrainment) and bottom-up (surface fluxes) approach in order to derive semiempirical profiles of and for the entire CBL. These profiles take into account the entrainment flux ratio and the boundary layer depth zi and include also an extension for the surface layer, as proposed by Wyngaard et al. (1971b). However, these semiempirical profiles all assume that the surface fluxes are dominantly determining the surface-layer part of the structure parameter profiles. Moreover, these profiles can be only used when information about the inversion layer is available. This is often challenging and hence MOST/LFC scaling is usually applied instead to relate scintillometer observations to the surface fluxes.
While most previous studies in the field of MOST used experimental data, there are few studies that investigated the surface-layer similarity by means of idealized numerical experiments using turbulence-resolving LESs. Mason and Thomson (2002) showed that Smagorinsky-style subgrid-scale (SGS) closures that are commonly used in LES models fail to predict MOST relationships in the near-surface layer correctly. They found a systematic peak (“overshoot”) in the dimensionless wind shear. Khanna and Brasseur (1997) used LESs to study the effect of grid resolution and the SGS model on the MOST functions for mean fields, variances, budgets of temperature, and turbulent kinetic energy under near-neutral to moderately convective conditions. To resolve the surface layer, they nested a high-resolution mesh in the lower part of their model domain. They stated that the lowest few grid levels are always affected by the SGS model in such a way that the turbulent flow cannot be resolved. Furthermore, they found an overshoot in the normalized vertical profiles for mean shear and mean temperature gradients. With increasing grid resolution, this overshoot was moved to lower levels, but it did not vanish. Khanna and Brasseur (1997) showed that this overshoot can be ascribed to the SGS model. Recently, Brasseur and Wei (2010) focused on the mentioned overshoot in the mean gradient of the dimensionless horizontal velocity and developed criteria to design LESs that reduce this overshoot. However, they also stated that MOST scaling was not reached in the first couple of grid levels. Khanna and Brasseur (1997) also found that temperature variance satisfied LFC scaling even for conditions with considerable wind shear. Moreover, they suggested that not only z/L (where z is the height above ground) but also zi/L might be a proper scaling parameter in the surface layer. The latter was supported by field measurement data by Johansson et al. (2001), who stated that the normalized temperature variance might have a slight dependence on zi/L [see also discussion in Johansson et al. (2002)].
Peltier and Wyngaard (1995) derived and from LES data of a convective boundary layer, and they derived LFC scaling constants. They particularly found that increased entrainment decreases and increases in the lower mixed layer. This led to a higher LFC constant for humidity than for temperature. Generally, the derived constants from the LESs were smaller than the suggestions from measurement data. Cheinet and Siebesma (2009) and Cheinet and Cumin (2011) used LES to investigate the spatial variability of and , respectively. Cheinet and Siebesma (2009) found a relation of the spatial variability of to the hexagonal cellular pattern of the convective plumes in the CBL. Moreover, Cheinet and Cumin (2011) could show that the spatial distribution of in the mixed layer is dominated by air that is entrained at the top of the mixed layer, which is in agreement with previous studies (e.g., Druilhet et al. 1983; Fairall 1987; see above). Implications for surface-layer similarity were not discussed, as their numerical grid was too coarse to resolve the surface-layer turbulence. Recently, Wilson and Fedorovich (2012) used LES in order to further explore the structure parameters. Particularly, they evaluated by calculating the refractive-index structure functions. They also derived , , and (joint structure parameter of temperature and humidity) and found that, for visible radiation, temperature contributes dominantly to in the lower half of the CBL and that becomes important near the entrainment layer at top of the CBL.
To the author’s knowledge, the MOST and LFC relationships for structure parameters have not been studied by means of LES so far. First LFC predictions have been made by Peltier and Wyngaard (1995), but they could not resolve the surface layer sufficiently. In the present article, and will be determined from a set of LESs that explicitly resolve the bulk part of the surface layer. In a precursor study, Maronga et al. (2013) have shown that the structure parameters can be derived from such an LES. They evaluated different methods to obtain the structure parameters and compared these methods to semiempirical profiles after Kaimal et al. (1976) and Fairall (1987). Maronga et al. (2013) showed that all methods reproduce the proposed shape of the vertical profiles of and very well. Furthermore, it was found that the most reliable method (so-called spectral method) gave values that were comparable to those proposed by the semiempirical profiles. Moreover, they compared the LES data with in situ large-aperture scintillometer and aircraft data and found good agreement of the measurement data with the LES predictions within the surface layer. They also showed that changes in the entrainment flux of dry warm air can increase in the mixed layer, while was not modified, which is in agreement with previous numerical and experimental studies (e.g., Druilhet et al. 1983; Fairall 1987, 1991). Owing to the idealized conditions in the model, such as horizontal homogeneity and prescribed surface fluxes, LES provides a unique instrument for studying the similarity relationships for structure parameters. By using a very high grid resolution throughout the model domain the turbulence in the surface layer can be sufficiently resolved and no “nested mesh” as used by Khanna and Brasseur (1997) is required.
I will focus on the questions that are most relevant for estimating the surface fluxes from scintillometer observations: (i) Can the similarity relationships be considered universal and are they the same for and ? (ii) Is similarity between the turbulent transport of heat and moisture a general feature of the unstable surface layer? (iii) Under which conditions does LFC scaling apply? Moreover, I want to put forward the question whether neglecting the effect of moisture on buoyancy (and hence L) is appropriate for estimating the surface fluxes of sensible and latent heat.
a. Structure parameters of temperature and humidity
There is vast literature on the definition and deduction of the structure parameters of temperature and humidity from standard meteorological data (e.g., Tatarskii 1971; Wyngaard et al. 1971b; Andreas 1988). Traditionally, and are derived either directly from the structure functions or from the one-dimensional Fourier spectra. Both formulations are mathematically equivalent and require the existence of an inertial subrange. Because the spectral approach requires less computation time we will focus on this approach (see also Maronga et al. 2013). Following Wyngaard et al. (1971b), the structure parameters and are directly proportional to the spectra of temperature and humidity in the inertial subrange, respectively. They can be related to the power spectral density at a given height z by
where θ is potential temperature, q is specific humidity, k is a wavenumber in the inertial subrange, and 0.2489 = 2/3Γ(1/3) after Muschinski et al. (2004). Note that for convenience the notation with index T (actual temperature) is used.
b. Monin–Obukhov similarity theory
To derive the surface fluxes of sensible and latent heat from observations of and , MOST is used. Following MOST, the vertical turbulent fluxes over a horizontal homogeneous surface in steady-state conditions are, to a first-order approximation, constant with height within the lowest decameters of the atmosphere (e.g., Andreas 1988; Hill 1989; Foken 2006). A common definition of the surface layer is thus the lower part of the boundary layer where the fluxes vary by less than 10% of their magnitude (Stull 1988). The relevant scaling parameters are the measurement height z, L, the near-surface kinematic flux of heat (and of moisture , if humidity is considered), and the local friction velocity u*. Additionally, a temperature scale θ* and a humidity scale q* can be defined as
According to MOST, any mean turbulence quantity in the surface layer should be a universal function of z/L (or −z/L, which is commonly referred to as stability parameter) if properly scaled with θ*, q*, u*, L, and z (Andreas 1988). The structure parameters of temperature and humidity should thus satisfy
with fT and fq being universal functions that only depend on z/L. The precise form of f is not given by MOST and must be determined experimentally. In the present article only unstable conditions are considered where is positive. Several empirical functions f have been proposed for such conditions from measurement data (Wyngaard et al. 1971b; Wesely 1976; Andreas 1988; Thiermann and Grassl 1992; Hill et al. 1992; De Bruin et al. 1993; Li et al. 2012). A very common form is
with dimensionless constants cTT1, cTT2, cqq1, and cqq2 for which different values are proposed in the literature. Equations (7) and (8) are linear interpolations and the four parameters provide the blending between neutral and free-convective conditions. Li et al. (2012) pointed out that it is often assumed that f = fT = fq, but they showed that dissimilarity between the transport of heat and moisture (and hence in f) occurred in their data for weakly unstable conditions (0.01 < −z/L < 0.1). They refer to Hill (1989), who stated that if the structure parameters all follow MOST, then their similarity functions must be the same, and temperature and humidity must be perfectly correlated (correlation coefficient of ). However, often , as also shown by Beyrich et al. (2005). Hill (1989) stated that dissimilarity can be expected owing to the fact that MOST is an overidealization of the surface-layer flow dynamics.
Equations (9) and (10) reveal that it requires not only the estimate of the structure parameter but also additional measurements of the friction velocity to determine the surface fluxes. Also, L is required. In practice, u* and L are iteratively solved using wind speed data (Panofsky and Dutton 1984; De Bruin et al. 1993). Local free-convection scaling is often applied as it provides a more simple relation between structure parameters and surface fluxes.
c. Local free-convection scaling
The Obukhov length is roughly the height where mechanical and buoyant production of turbulence are equal. It is commonly expected that free convection occurs when −z/L ≥ 1 (Andreas 1991). When mechanical production is much less important than buoyant generation of turbulence (e.g., when winds are calm and u* → 0) the Obukhov length is close to zero and no longer a proper scaling parameter. Under such conditions buoyancy is the driving force and the surface layer should behave as in free convection. This is termed local free convection (e.g., Wyngaard et al. 1971a). The list of LFC scaling parameters becomes
Because of the limited scales available in LFC (number of relevant variables equals the number of dimensions), there is only one dimensionless group and the dimensionless structure parameters must follow
with AT and Aq being universal constants (Andreas 1991). There is consensus on the value of 2.7 for AT from measurements (Wyngaard et al. 1971b; Kaimal et al. 1976; Wyngaard and LeMone 1980; Kunkel et al. 1981; Andreas 1991). Wyngaard and LeMone (1980) found Aq to be around 1.5, while Andreas (1991) suggested that AT = Aq (when ), referring to the study of Hill (1989). Peltier and Wyngaard (1995) found values for the constants between 2.0 and 2.7 in their LESs. They also showed that the constant was larger for humidity than for temperature owing to entrainment effects. This contradicts the result of Wyngaard and LeMone (1980).
showing that LFC scaling can be traced back to MOST in the free-convection limit. Note that no independent measurement of the wind speed is required in Eqs. (16) and (17), but the buoyancy flux is needed [cf. Eqs. (9) and (10)]. It can be approximated without much error as (Stull 1988; Andreas 1991)
with the surface Bowen ratio and ξ = cp/Lυ (cp is the specific heat at constant pressure and Lυ is the latent heat of vaporization). It follows that
It is tempting to use Eq. (23) directly to derive from large-aperture scintillometer measurements (which give estimates of ), as neither the friction velocity nor information about humidity is required (e.g., De Bruin et al. 1995; Kohsiek 1982; Kohsiek et al. 2002). A direct consequence of this approximation is, however, that is usually systematically underestimated because when β0 > 0. This is generally the case under unstable conditions.
Note that in free convection one might also consider mixing-layer similarity throughout the CBL (e.g., Wyngaard and LeMone 1980; Fairall 1987), but then an additional estimate of zi must be provided. This usually requires additional vertical-scanning devices, which are rarely available in combination with scintillometers. Mixing-layer similarity will thus not be further studied in the present paper.
3. LES model and case description
a. LES model
The LES model PALM (revision 893) (Raasch and Schröter 2001; Riechelmann et al. 2012) was 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 as surface boundary condition locally between the surface and the first computational grid level (“local similarity model”; see also Peltier and Wyngaard 1995), including the calculation of u* from the roughness length and the local wind profiles [see also Panofsky and Dutton (1984), their chapter 6.5]. A 1.5-order flux-gradient subgrid closure scheme after Deardorff (1980) was applied, 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 were used (Williamson 1980). The virtual potential temperature is calculated from the prognostic variables θ and q as a three-dimensional quantity in the model so that the buoyancy flux profile could be directly calculated in the model. 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 to generate steady-state wind profiles as initialization for the LES.
b. Case description
The set of LESs for this study is based on a simulation of the free-convective boundary layer, which is described in detail in Maronga et al. (2013). The results from the simulation were evaluated by in situ aircraft observations of temperature and humidity at Cabauw, Netherlands. Maronga et al. (2013) derived the structure parameters of temperature and humidity from the LES data and compared them against aircraft observations of and at different heights in the boundary layer as well as with semiempirical profiles. Furthermore, they showed that , derived from the LES, was in good agreement with measurements of a large-aperture scintillometer that was operated at a height of 41 m. Hence, this simulation suits as a good reference case for the present study.
The model was discretized in space with 1024 grid points in each horizontal direction (in the present study 2048 grid points are used). The grid resolution was 4 m in the horizontal directions (Δx = Δy = 4 m) and 832 grid points with a resolution of 2 m were used in the vertical direction (Δz = 2 m). The simulation was driven by constant kinematic surface fluxes of heat and moisture with a Bowen ratio around 0.27. A roughness length of 0.1 m was used. Neutrally stratified initial profiles of temperature and humidity with a capping inversion and the free atmosphere above were prescribed (see Fig. 1 and Table 1). For a detailed description of the initial parameters, see simulation case A in the study of Maronga et al. (2013, their section 3 and Table 1).
For the present study a set of LESs was generated based on the simulation described above. 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 (also referred to as reference simulations) that cover the relevant range of −z/L for near-neutral to free-convective conditions. First, the geostrophic wind was increased from 0 to 10 m s−1. Second, in order to cover those cases that are dominated by shear production (−z/L < 0.1), the surface fluxes were reduced to 50% and 10% of their initial values, based on the simulation with a background wind of 10 m s−1. For sensitivity studies, β0 was varied to values of 0.05 and 0.4 as well as ∞ (dry simulation). Furthermore, the lapse rate of temperature in the capping inversion γ was reduced for one case in order to study whether entrainment of warm dry air affects the surface-layer dynamics and hence the structure parameters and their similarity relationships. Because of the high computational costs of the study [one simulation took about 2 days of real time on 4096 Intel Xeon Gainestown processors (2.93 GHz) on an SGI Altix ICE 8200 Plus cluster], these sensitivity cases were simulated for two selected background wind speeds only: 0 (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
A quasi-stationary state was reached for all cases after 1 h of simulation time. The total simulation time was 2 h and instantaneous data were output every 120 s during the second hour of the simulation. As the buoyancy flux is a direct model output in PALM no approximation as introduced in Eq. (20) was needed.
The structure parameters of temperature and humidity were derived from the power spectral density of potential temperature and specific humidity by means of Eq. (2) (spectral method). First, all one-dimensional spatial spectra along the x and y directions in each horizontal plane were determined and subsequently averaged. Second, a quality check was performed to identify an inertial subrange. Third, the structure parameters were then calculated for all k in the inertial subrange and subsequently spectrally averaged. In this way, instantaneous (horizontal averaged) vertical profiles of the structure parameters could be derived. For a more detailed description of this method see Maronga et al. (2013). They particularly found that this method gave reliable estimates of the structure parameters. These instantaneous profiles were derived every 120 s over the analysis period. All figures in the following showing data points thence contain data for every 120 s of time. Because of the high computing time required to compute the profiles, the calculation was limited in the vertical direction to the lowest 150 m of the boundary layer. This was sufficient to cover the entire surface layer.
The boundary layer depth was in most cases in the order of 1 km (see section 4a and Table 1). The profiles of displayed a linear decrease with height (not shown). The top of the surface-layer zSL was first defined as the height where had decreased by 10% of its surface value. This, however, turned out to be approximately 0.1zi. For convenience and without introducing much error, the height of the surface-layer was thus defined as zSL = 0.1zi. This decision is supported by Brasseur and Wei (2010), who stated that MOST can be valid up to height levels of 0.15–0.2zi.
a. Mean profiles
Since the focus of this paper is on structure parameters in the surface layer, the mean characteristics of the boundary layers simulated shall be discussed only briefly at this point. The temperature and humidity after 1 h of simulation time all show classical boundary layer profiles with unstable stratification near the surface, a well-mixed layer, and a strong capping inversion above the mixed layer (see Fig. 1, exemplarily shown for W00). A boundary layer depth of about 1.1 km is observed (see Table 1). Note that the mixed layer does not tear down the capping inversion and hence the lapse rate of the capping inversion is a relevant parameter, while the stability of the free atmosphere is unimportant for the results. During the analysis period (1–2 h of simulation time), the mixed-layer temperature and humidity both slightly increase, whereas zi remains rather constant owing to the strong capping inversion. The entrainment flux ratio r for all reference cases is about −0.25 for sensible heat, 0.3 for moisture, and around −0.15 for the buoyancy flux. Entrainment of dry warm air is thus small compared to the input of sensible and latent heat at the surface. Therefore, one can expect that the influence of entrainment processes can be neglected in the surface layer. The increase in the geostrophic from W00 to W10 does not affect r or zi and thus indicates that shear does not increase the entrainment significantly, which is in contrast to the finding of Fairall (1984). This might be ascribed to the strong capping inversion in the present study that suppresses entrainment to a considerable degree. In cases W10_F50 and W10_F10 zi is lower (1.04 and 1.0 km, respectively) owing to the reduced surface fluxes. The sensitivity cases W00_γ07 and W08_γ07 (weak capping inversion) show a significantly increased zi of up to 1.29 km. Because of more entrainment of dry warm air, a drying out of the boundary layer is forced that is balanced by the moisture input at the surface. The mean humidity in the mixed layer for cases W00_γ07 and W08_γ07 hence is constant in time and the entrainment flux ratio is about −0.3 for temperature and 1.0 for moisture. A complete list of zi and the entrainment flux ratios for each case is given in Table 1.
b. MOST relationships for and
It is important to recognize that the LES models reality, but it gives no perfect image of nature. The observed values for the similarity relationships that are derived in the following should thus be regarded as “model truth,” but they can be different from what can be expected in nature. On the one hand, modeling errors—for example, induced by the SGS model or numerical errors—can be present. On the other hand, horizontal homogeneity is an idealization that is made for the LESs but can be rarely considered in realistic landscapes. The advantage of the LES here is the full control of all model parameters and boundary conditions. Hence, the LES allows for separating effects by varying single parameters, whereas in nature many effects superimpose each other.
1) MOST similarity functions from LES
The nondimensional structure parameters (horizontal average) were calculated for the reference simulations for all available time steps and height levels in the surface layer according to Eqs. (5) and (6). The results are shown as data points against −z/L in Fig. 2. It is obvious that all data points collapse to a single curve, except for the lowest values of −z/L for each simulation. Here, a significant decrease in the nondimensional structure parameters is visible. These data points relate to the lowest height levels above the surface where turbulent eddies are relatively small and cannot be sufficiently resolved. The nominal resolution of the LESs is defined by the truncation size . Smaller eddies than this size are parameterized within the SGS model, while larger eddies should be fully resolved. However, in practice the SGS model also affects larger scales, and the actual resolution is found to be typically around 6Δ ≈ 20 m. Eddies smaller than 20 m still suffer from contributions of the SGS model. This is a known feature of the LES; see, for example, Khanna and Brasseur (1997) and Brasseur and Wei (2010). This effect can in turn also artificially modify the turbulence spectra at the lowest levels for high wavenumbers as shown by Maronga et al. (2013). It is hence plausible to exclude at least the lowest levels from the analysis. Consequently, taking into account the results shown in Fig. 2, the analysis was restricted to height levels of 14 m and higher (excluding the lowest seven grid points).
Excluding these lowest levels, the decrease in the nondimensional structure parameters vanishes (not shown) and the remaining data points all follow a single function with only little scatter. This also legitimates the chosen definition of the surface-layer height (see section 3c). These data points also show no dependence on time. Moreover, it is evident from Fig. 2 that −z/L is an appropriate scaling parameter. As each simulation covers a different range of −L, it is evident from Fig. 2 that an increase (decrease) in z has the same effect on the dimensionless structure parameters as a decrease (increase) in −L (this statement will be confined later on).
Fitting functions fT and fq are determined from this (reduced) dataset for the first time from LES data and they are also shown in Fig. 2. The method of least squares using the formulation in Eqs. (7) and (8) and logarithmized values was employed to find the best fit. The data follow the proposed shape. However, it appears that the data follow a slope that is slightly steeper than the proposed −⅔ slope. This leads to an overestimation of the structure parameters by the fit in the free-convection limit (for −z/L > 10). The LES data suggest
Figure 3 shows that fT ≈ fq generally holds, which is expected when MOST is valid for both temperature and humidity (Hill 1989; Andreas 1991). The difference between the coefficients in Eqs. (24) and (25) is thus rather insignificant. Nevertheless, for weakly unstable conditions (−z/L < 0.1), it can be observed that fT and fq differ slightly. In the neutral limit (−z/L → 0), the similarity functions become
which makes a difference of 3% between fT and fq. Assuming perfect similarity (fT = fq) would give a systematic error in the flux of at most 1.5% [owing to the exponent of −½ in Eq. (16)] and can thus be neglected. Nevertheless, this difference will be discussed later in section 4b(3)(ii). Li et al. (2012), who reported distinct dissimilarity of the turbulent transport of sensible heat and moisture, found values of cTT1 = 6.7 ± 0.6 and cqq1 = 3.5 ± 0.6—differences between fT and fq that are one order of magnitude larger than what is found in the present LES fitting functions. As suggested by Li et al. (2012), their finding that the dimensionless was larger than the dimensionless in the near-neutral region might be explained by nonlocal effects, such as unsteadiness or advection, that are not considered in the LESs.
Figure 4 shows a selection of MOST similarity functions that have been previously derived from measurement data, together with the fitted similarity functions, derived from the LES data (red solid lines). The LES results follow the same shape as the proposed functions from the literature. For , many authors suggested cTT1 = 4.9 (Wyngaard et al. 1971b; Andreas 1988; De Bruin et al. 1993), while the LES data suggest a higher value of 6.7. However, the suggested functions of Thiermann and Grassl (1992), Hill et al. (1992), and Li et al. (2012) suggest even higher values of up to 8.1. The LES data thus seem to be well within the range of the functions proposed in the literature. For we find a significant difference between the LES (cqq1 = 6.3) and the study of Li et al. (2012) (cqq1 = 3.5), which might be ascribed to the fact that they observed dissimilarity between the transport of heat and moisture.
In strongly unstable conditions (−z/L > 5), all proposed functions fT, including the LES fit, converge. As was stated in section 2c, in the free-convection limit, (analogous for humidity) is the relevant parameter and usually found to be around 2.7 (e.g., Wyngaard et al. 1971b; Wyngaard and LeMone 1980; Andreas 1991). The LES data yield values of AT = 2.9 and Aq = 3.0, which is close to 2.7. We will have a detailed look at the free-convection limit later in section 4c.
2) Dissimilarity of the turbulent transport of temperature and humidity
A good measure for dissimilarity between the turbulent transport of heat and moisture is the cross correlation between temperature and humidity RTq (Li et al. 2012). Figure 5a shows RTq, calculated from the two-dimensional (horizontal) cross-correlation coefficient of the LES data fields of temperature and humidity. It is visible that RTq ≥ 0.9 for all data points. This indicates a nearly perfect correlation between temperature and humidity. For strongly unstable conditions (−z/L > 1), the data show some scatter, but the correlation does not drop below 0.9. A closer look into the data reveals that RTq generally decreases with height. In the higher levels, the influence of entrained air from the free atmosphere is more prominent, which can decrease the correlation between temperature and humidity.
For weakly unstable conditions (−z/L < 0.1), the data do not show much scatter, indicating a perfect correlation and thus similarity of the turbulent transport of sensible heat and moisture under near-neutral conditions. This is in contrast to Li et al. (2012), who observed large scatter in RTq for small −z/L, leading to dissimilarity between structure parameter relationships. They concluded that this decorrelation can be traced back to nonlocal effects like nonstationarity of the flow, advection, or entrainment effects. Since neither advection nor nonstationarity are present in the LESs, no dissimilarity has to be expected. However, we will show in section 4b(3)(ii) that entrainment effects can lead to dissimilarity, which clearly supports the conclusion of Li et al. (2012).
3) Sensitivity analysis
(i) Definition of the Obukhov length
In the definition of the Obukhov length, virtual potential temperature and the buoyancy flux are used [see Eq. (1)]. In practice, however, L is sometimes calculated using actual temperature and sometimes the kinematic surface heat flux (e.g., Wyngaard et al. 1971b; Wesely 1976; Thiermann and Grassl 1992). While it is reasonable to use actual temperature without introducing much error, neglecting the contribution of humidity to the buoyancy flux is a rather questionable approximation and requires a sufficiently dry surface and boundary layer (see also Andreas 1991). The present LES data suggest typical deviations of 10% between surface buoyancy flux and kinematic surface flux of heat (reference cases; not shown). To quantify the effect of neglecting all humidity contributions, the MOST similarity functions were additionally calculated using the Obukhov length for dry air Ldry (hereafter also referred to as dry approximation), which is defined as
Figure 3 shows the resulting fitting functions for both L and Ldry. Differences between the correct and the dry approximation in the weakly unstable region (−z/L < 0.1) are obvious. The dry approximation leads to a shift of fT and fq to lower values of −z/L and suggests fitting functions of
In the free-convection limit, this formulation yields AT,dry = 2.7 and Aq,dry = 2.8 and thus AT,dry ≈ Aq,dry. The dry approximation results are also plotted in Fig. 4, showing that they all are within the proposed range of values. Based on this result we can infer neither that one formulation is superior to the other nor that the found formulations are universal. Nevertheless, we can conclude that the definition of the Obukhov length can be at least one reason for the difference in the proposed functions from literature.
(ii) Effects of humidity
The results from the reference simulations so far suggest that fT ≈ fq. The reference cases, however, consider only conditions with specific values of the kinematic surface fluxes of heat and moisture and one special state of the capping inversion. To get an idea whether the derived functions are actually universal, those parameters were varied that can be considered to have an effect on the previous results (see Table 1). By using different surface fluxes, additional sensitivity simulations were generated with two different values for β0 of 0.05 (case W08_β05) and 0.4 (case W08_β40). Additionally, a simulation without humidity was carried out (case W08_dry, β0 = ∞). Because of limited computational resources, it was not feasible to simulate the full −z/L range (see also section 3b). Hence, a specific wind speed of 8 m s−1 was chosen. In this way, a stability range of approximately 0.1 ≤ −z/L ≤ 2 was covered. The reference cases displayed similar absolute entrainment fluxes that were around 0.2–0.3 for temperature and humidity (see Table 1). Any effect of entrainment would thus affect both similarity functions in a similar way. To simulate at least one case with a different entrainment regime, a smaller lapse rate in the capping inversion of γ = 7 K km−1 was used (case W08_γ07). Consequently, the boundary layer in case W08_γ07 was about 20% higher than for the reference cases. Table 1 shows the entrainment flux ratios for all simulated cases. On the one hand, rθ (ratio for temperature) was found to be increased for W08_dry (−0.15) and decreased for W08_β05 (−0.6) compared to the reference cases. This can be ascribed to the missing/additional contribution of moisture on the buoyancy flux [see also Eq. (20)]. In contrast, cases W08_β40 and W08_γ07 did not show a significant modification compared to the reference case W08. The entrainment flux ratio for moisture rq, on the other hand, showed only a response to the different setup for case W08_γ07 with a value of about 1.
The normalized structure parameters for these sensitivity cases in comparison with the previously derived fitting functions are shown in Figs. 6a and 6b. On the one hand, Fig. 6a shows that all data points strictly follow the fitting function proposed for fT. Differences in β0 and/or the lapse rate do not affect fT and therefore we can argue that follows MOST and fT is universal. Also, modifications due to differences in zi cannot be observed. On the other hand, Fig. 6b reveals a different behavior for . First, a change in the prescribed surface fluxes does not affect the similarity relationship and the data points follow the proposed function fq, but case W08_γ07 suggests higher values of the dimensionless . Moreover, the data points do not follow the proposed decrease for −z/L > 1 (equivalent to high values of z). Since no effect of the different lapse rate on was found we infer dissimilarity between the transport of heat and moisture that is caused by entrainment of dry air at the top of the mixed layer (cf. Fig. 6a). This is supported by RTq, which decreases rapidly with height from values of 0.9 to 0.55 for this simulation (see Fig. 5b). A consequence of this dissimilarity is that turbulent humidity fluctuations in the surface layer are affected by entrainment and no longer follows MOST. This is also in agreement with the findings of Lanotte and Mazzitelli (2013), who studied the effect of entrainment on the correlation between temperature and a passive scalar. They found that the correlation coefficient decreased, particularly in the lower boundary layer, when entrainment was significant. This result supports the assumption of Li et al. (2012) that entrainment might be a possible reason for dissimilarity between the transport of heat and moisture. The finding that fT ≈ fq in the reference simulations can be traced back to the fact that the entrainment flux ratio in the sensitivity simulations was 0.6 or less (absolute value) for both kinematic heat and moisture and was thus sufficiently small to inhibit entrainment effects on the surface-layer structure. In contrast, entrainment of dry air with an entrainment flux ratio of 1, as found in case W08_γ07, is apparently sufficient to affect the surface-layer structure. The results suggest that the preceding statement that an increase in z would have the same effect on the dimensionless structure parameters as a decrease in −L is limited to conditions where entrainment is sufficiently small. If entrainment is high, data from lower height levels are less affected by entrainment and will be determined by the surface fluxes, and MOST might be considered. As no entrainment flux ratio greater than 0.6 (absolute value, case W00_β05) was found for temperature, it is possible that there is also a critical value and that there might be conditions under which no longer follows MOST. To find such a critical value, more sensitivity studies with increased entrainment of warm air (entrainment flux ratio ≥ 1.0) would be required.
Figures 6c and 6d show the data from the sensitivity simulations with dry approximation of L. A significant gap is present between the data from case W08_β05 and the proposed fitting functions, whereas the proposed slope remains. Case W08_dry suggests higher dimensionless than the reference simulations (see Fig. 6c). This already proves that fT,dry cannot be considered universal. For case W08_β40 an effect is hardly visible, possibly owing to the fact that the Bowen ratio was similar than in the reference simulation (0.27 and 0.4 for W08 and W08_β40, respectively). We must conclude that the dry approximation is not valid and no universal functions can be derived using Ldry.
c. Local free-convection scaling
The LFC similarity constants AT and Aq have been calculated from case W00 using the formulation in Eqs. (14) and (15). The lowest seven grid points have been excluded from the analysis according to the results discussed in section 4b(1). Averaging of the remaining data points yields AT ≈ 2.7 and Aq ≈ 2.8 (specification is given below). Figure 7 shows and in the surface layer derived from the LES data, including the lowest grid points. The data are complemented by the LFC predictions using AT and Aq as given above [see Eqs. (14) and (15)]. It is visible that the lowest seven grid points deviate significantly from the LFC prediction, whereas there is very good agreement between LES and LFC prediction at height levels above 14 m. There the data show only minor scatter due to variation in time and height. On closer inspection (hardly visible in Fig. 7), however, it can be observed that AT and Aq both show a slight height dependency up to height levels around 60 m with a peak value at a height of 40 m. Khanna and Brasseur (1997) and Brasseur and Wei (2010) found similar peaks for the dimensionless wind shear and temperature gradient and ascribed these to the SGS model. However, one would expect that this overshoot moves to higher (lower) levels if coarser (finer) grid resolutions are used in the LES (Sullivan et al. 1994; Brasseur and Wei 2010). Test simulations with grid resolutions between 5 and 20 m did not show such an effect for the present dataset. We can thus conclude that this peak is not an effect of the SGS model. Other possible explanations are local shear-induced turbulence generation that is not considered in LFC scaling (Businger 1973) and the fact that similarity theory itself is an overidealization of the surface-layer dynamics (Hill 1989). However, as it is visible from Fig. 7, this height dependency is not significant and we observe scatter in the data with deviations of not more than 7% from the mean values of AT and Aq. The variation in time and height leads to AT = 2.7 ± 0.2 (95% confidence interval), which is in remarkable agreement with observations in the lower part of the boundary layer (Wyngaard et al. 1971b; Kaimal et al. 1976; Wyngaard and LeMone 1980; Kunkel et al. 1981; Andreas 1991). The value is somewhat lower than 2.9, which was derived from the MOST fitting function earlier in section 4b(1). This can be traced back to an overestimation of fT for large −z/L by the fitting (see Fig. 2b) and also applies to Aq, where the LES data suggest Aq = 2.8 ± 0.2 for LFC scaling and the MOST fitting function gave 3.0. Peltier and Wyngaard (1995) found values between 2.0 and 2.7 in their LES data, suggesting a tendency to lower values than in the present LESs. Note that the present LES data suggest that the value of AT is roughly equal to that of Aq anyway, with a tendency of Aq to be higher than AT. This can be ascribed to effects of entrainment that were already discussed in section 4b(3)(ii). This will also be discussed in section 4c(1).
As was stated in section 2c, LFC is often applied for convenience, even under conditions where shear significantly contributes to turbulence production. Moreover, the effect of humidity is often neglected as independent measurements are not available. To estimate the error of this approximation, the surface fluxes were calculated for all reference cases [Eqs. (21) and (22)] and additionally for the formulation for dry air [Eq. (23)] using AT = 2.7 and Aq = 2.8 (we will show later that no universal constants exist for the dry approximation). Figures 8a and 8b show that for case W00 (free convection, −1/L ≈ 0.4 m−1) the uncertainty in the kinematic surface fluxes of heat and moisture due to variation of AT in height and time (see Fig. 7) is about 5% (95% confidence interval). The mean value, however, is exact, because the values for AT and Aq have been derived from case W00. With decreasing −1/L (0.02 ≤ −1/L ≤ 0.2 m−1) it can be observed that the surface fluxes are overestimated by up to 5%. The scatter in the derived fluxes (due to variation in time and height) is slightly increasing to 8%–10%. For −1/L < 0.017 m−1, shear production starts to dominate the surface-layer flow and the flux estimates decrease while the scatter of the estimates increases. For case W00_F10, the estimate of the flux is 40% too small with scatter of about 20% around this value. From Fig. 8c it becomes evident that the dry approximation leads to a systematic underestimation of the kinematic surface flux of heat. This was already predicted in section 2c because h ≥ 1 under unstable conditions. For case W00, this leads to an underestimation of the flux of about 10%. On the one hand, the shape of the curve in Fig. 8c coincides with that shown in Fig. 8a because the Bowen ratio was equal in all reference simulations. The scatter, on the other hand, is slightly reduced, owing to the fact that fluctuations in humidity do not lead to additional fluctuations in θυ. The underestimation of depends mainly on the Bowen ratio [see Eqs. (21) and (23)] and is thus obviously sensitive to changes in the surface fluxes.
In summary, the LES results suggest that LFC scaling can be applied even under moderate wind conditions (here for wind speeds up to 8 m s−1 with −1/L ≥ 0.017 m−1) without accepting a systematic error of more than 5% and scatter (due to variation in time and height) of more than 10% around this value. This estimate of the error is of course based on virtual measurements at different heights that are already representing the horizontal average. In practice, the error in the flux will increase if sufficient averaging of turbulent motions (e.g., by spatial or temporal averaging) is not feasible. Moreover, it is shown that the kinematic surface flux of heat is underestimated if humidity is significant in the boundary layer, but neglected in the buoyancy term [see Eq. (23)]. The magnitude of this underestimation is mainly given by the surface Bowen ratio. By an independent estimate of β0 (e.g., De Bruin et al. 1999) and using the approximation h by Andreas (1991), one might overcome the underestimation of .
In analogy to the analysis in section 4b(3), AT and Aq were derived from the sensitivity cases W00_dry, W00_β05, W00_β40, and W00_γ07. Additionally, the dry approximations AT,dry and Aq,dry (no humidity considered in the calculation of wLF) were calculated. The results are shown in Table 2 together with the estimates from MOST fitting functions. It is apparent that AT is constant for the sensitivity cases with a value around 2.7 and can thus be considered universal. The MOST fitting function gives a slightly higher value of 2.9, which can be ascribed to the overestimation of the dimensionless by the fitting function fT as already discussed. Also, Aq can be only considered universal with a value around 2.8 if entrainment is sufficiently small. In this case the assumption that AT = Aq by Andreas (1991) is valid. When entrainment becomes significant (case W00_γ07), Aq is higher (here 3.2 ± 0.5) and Aq is thus no longer universal. This is in agreement with Peltier and Wyngaard (1995), who found a larger value for Aq than for AT owing to entrainment effects in their LES data. The fact that the LFC prediction becomes nonuniversal for humidity can explain why no consensus was reached on the exact value of Aq so far.
Despite the fact that both MOST and LFC scaling suggest AT,dry ≈ Aq,dry (except case W00_γ07), the dry approximation reveals varying values between 1.5 ± 0.1 and 2.8 ± 0.5. These results clarify that neglecting the effect of humidity to buoyancy is not a valid approach for either MOST or for LFC scaling, and no universal similarity functions can be obtained.
The present LES study focused on the derivation of similarity relationships for structure parameters in the unstable atmospheric surface layer. Because of a grid resolution of 2–4 m and today’s computing capacities, it was possible to actually resolve the surface layer with an LES model for a whole set of LESs, covering convective to near-neutral boundary layers. This was necessary to cover all relevant stability ranges that are commonly expressed in terms of the parameter −z/L. The LES results showed that the flow within the lowest grid levels of the LESs should be generally interpreted carefully because effects of the SGS model are present and the flow is not well resolved.
The analysis of the MOST relationships showed that the dimensionless structure parameters of temperature and humidity strictly follow functions (fT and fq) that only depend on −z/L, as proposed by theory and previous experimental data. Only a little scatter in the LES data was found so that MOST fitting functions, linking the structure parameters to the kinematic surface fluxes of heat and moisture, were derived for the first time from LES data. While there is a lack of explicit fitting functions for in the literature, the fitting function for was found to be well within the range of the previously suggested similarity functions from measurement data. It could be shown that increasing (decreasing) the measurement height z had the same effect as decreasing (increasing) L. On the one hand, it thus appears to be a logical approach to place measurement instruments (e.g., scintillometers) as high as possible in order to approach the free-convection limit. On the other hand, this would increase the size of the footprint of the measurement.
The LES results strongly suggest that follows MOST and that fT is a universal function. However, for it is found that MOST relationships are only valid if entrainment at the top of the mixed layer is sufficiently small (here if the absolute value of the entrainment flux ratio is less than 0.6). In this case it holds that fT ≈ fq. For an entrainment flux ratio of 1, dry air that is entrained into the mixed layer can also affect the surface-layer structure. Consequently, no longer follows MOST and fq cannot be considered to be a universal function. Furthermore, it could be shown that higher levels are more affected by entrainment than low levels, which is in agreement with previous experimental and LES results (e.g., Fairall 1987, 1991; Cheinet and Cumin 2011). This was also visible in the correlation coefficient between temperature and humidity, which decreased with height, suggesting that dissimilarity between the turbulent transport of heat and moisture can be induced by entrainment. This finding might explain the dissimilarity observed by Li et al. (2012). The preceding conclusion to perform measurements as high as possible must thus be limited to conditions where entrainment effects can be neglected. However, a more extensive sensitivity study would be required to define critical values of the entrainment flux ratio.
In the free-convection limit, where LFC scaling is applicable, the dimensionless structure parameters reduce to constants AT and Aq. The common value of 2.7 for AT was reproduced by the LES model, with scatter of 5% in the data. If entrainment is sufficiently small, Aq is found to be around 2.8, so that the assumption AT = Aq is roughly valid, even though Aq appears to be consistently larger than AT, which is in agreement with the data of Li et al. (2012) for convective conditions. This difference between AT and Aq is, however, within the 95% confidence bounds (see Table 2). Otherwise, no universal value for Aq can be derived. The scatter was mostly related to a height dependence of AT with an overshoot at a height of 40 m. Possible reasons for this overshoot might be the fact that similarity theory is an overidealization of the surface-layer dynamics (Hill 1989) and local shear production by the turbulence near the surface (Businger 1973). An error analysis showed that LFC scaling can be applied even for moderate wind conditions (with −1/L ≥ 0.017 m−1) without introducing a systematic error of more than 5% and scatter of more than 10% (95% confidence interval).
Neglecting the contribution of humidity to the buoyancy flux (and hence to L and wLF) was found to lead to dimensionless structure parameters that can no longer be expressed in terms of universal functions or constants. Using the formulation for dry air is thus a rather questionable approach and limited to sufficiently dry conditions (see also Andreas 1988, 1991). It could be shown that the kinematic surface flux of heat is systematically underestimated when humidity is significant. At least an estimate of the surface Bowen ratio is required to account for the effect of humidity on buoyancy when using LFC scaling.
In a follow-up study the MOST and LFC relationships for structure parameters will be derived in the CBL over the heterogeneous LITFASS terrain in the southeast of Berlin, Germany, in order to investigate the effect of surface heterogeneity on structure parameters and their similarity relationships.
The author would like to thank Frank Beyrich (German Weather Service, DWD) and Siegfried Raasch (Leibniz Universität Hannover, Germany) for reading a draft of this article. Sincere thanks are given to the Meteorology and Air Quality Group at Wageningen University (Netherlands) for their valuable comments and the discussion during a visit to Wageningen in 2012. Particularly, the author would like to thank Jordi Vilà as well as Arnold Moene and Miranda Braam. The latter two also provided additional valuable comments on the manuscript, which are gratefully acknowledged. Finally, the authors would like to thank the three anonymous reviewers for their comments and suggestions that helped to improve the manuscript. This study was supported by the German Research Foundation (DFG) under Grants RA 617/20-1 and RA 617/20-3. All simulations were performed on the SGI Altix ICE at The North-German Supercomputing Alliance (HLRN), Hannover/Berlin. The NCAR Command Language (version 5.2.1) (http://dx.doi.org/10.5065/D6WD3XH5NCL) has been used for data analysis and visualization.