1. Introduction
a. Idealized modeling of the stable boundary layer
The modeling of the stable boundary layer (SBL) continues to be a challenge (Mahrt 2014) because of the presence of weak or almost collapsed turbulence and, consequently, the influence of a range of other processes [e.g., intermittent turbulence (Poulos et al. 2002), gravity waves (Mahrt 1998), baroclinicity (Mahrt 1998), surface heterogeneity (Verkaik and Holtslag 2007; Optis et al. 2014), thin and “upside down” boundary layers (Mahrt and Vickers 2002), inertial oscillations (Baas et al. 2012), and low-level jets (LLJs; van de Wiel et al. 2010)]. Research into the SBL has focused mainly on the representation of turbulence given the high sensitivity of atmospheric models to different parameterization schemes (ECMWF 2015; Beljaars and Viterbo 1999). Turbulence parameterizations are generally determined through a combination of field measurements (e.g., Beljaars and Holtslag 1991; Persson et al. 2002; Poulos et al. 2002) and modeling experiments (e.g., flux–gradient relationship analysis, 1D and 3D atmospheric models). Within the surface layer (SL), Monin–Obukhov similarity theory (MOST) is an accurate method for relating turbulent fluxes to properties of the mean flow (Monin and Obukhov 1954). Above the SL in the SBL (where MOST does not apply), single-column models (SCMs) are often used to formulate or evaluate a turbulence parameterization scheme. These models are advantageous because of their low computational requirements and the flexibility in which processes and parameterizations are included (turbulence, radiation, entrainment, land surface characteristics, etc.). The complexity of an SCM can vary from models that incorporate the complete physics of a 3D model to highly idealized representations that consider only the momentum and temperature budgets.
There is a growing body of research exploring the use of SCMs to study turbulence in the SBL (e.g., Cuxart et al. 2006; Edwards et al. 2006; Weng and Taylor 2006; Baas et al. 2010; Sterk et al. 2013; Bosveld et al. 2014b; Sorbjan 2014). The most comprehensive study has been the Global Energy and Water Cycle Experiment (GEWEX) Atmospheric Boundary Layer Study (GABLS), a series of comparisons between both operational and research-based atmospheric models focusing mainly on the representation of turbulence in the SBL (Holtslag 2014). The first phase of the experiment (GABLS1) compared 19 SCMs with large-eddy simulations using a specified surface temperature cooling rate and constant geostrophic wind representing moderately stable conditions (Cuxart et al. 2006). The second phase (GABLS2) compared the representation of the diurnal cycle for 30 different SCMs using a prescribed geostrophic wind speed and surface temperature (Svensson et al. 2011). The third phase (GABLS3) focused on the representation of the diurnal cycle for 19 different SCMs using observations over a 24-h period from the Cabauw meteorological tower in the Netherlands (Bosveld et al. 2014b). These studies demonstrated a broad range of results depending on the turbulence scheme, including large variations in the degree of turbulent mixing, surface wind speeds, temperature and turbulent fluxes, the onset of the evening and morning transitions, the evolution of the inertial oscillation, and the amplitude and altitude of the LLJ. The tendency to over- or underestimate turbulent mixing was related mainly to the tunable constants used to determine the mixing length and stability functions.
An accurate SCM simulation of the observed SBL is difficult to achieve because of the influence of 3D processes [e.g., momentum and temperature advection, baroclinic effects, internal boundary layers (IBLs)]. To facilitate comparison between SCMs, Bosveld et al. (2014a) prescribed advective tendencies as piecewise constant functions as well as a geostrophic wind vector profile based on simulations from a mesoscale model. Baas et al. (2010) demonstrated that compositing SCM results over seven LLJ events with comparable external forcings averaged out the effects of advective tendencies, facilitating comparison with similarly composited observations.
b. Wind energy context
Turbulence parameterization in the SBL is of particular importance in the field of wind power meteorology. The accurate modeling of the wind speed profile across altitudes swept out by a wind turbine blade (the “wind power altitude range” between roughly 30 and 200 m) is important for preliminary resource assessments and forecasting of the wind resource. As wind power varies with the cube of the wind speed, small errors in wind speed can lead to large errors in wind power.
A hierarchy of models of varying complexity is used to simulate the wind profile (Fig. 1). On the left end of the spectrum are the conventional, computationally efficient, equilibrium-based (i.e., no time dependence) approaches that either lack a physical basis or are based on limited physics. These models include the power law, logarithmic wind speed profile, and the two-layer logarithmic Ekman model (Lange and Focken 2005; Emeis 2013; Optis et al. 2014). On the opposite end of the spectrum are the time-evolving 3D models that have increasingly been used in wind power meteorology over the last decade (Giebel et al. 2011). These models provide considerably more comprehensive representations of atmospheric boundary layer (ABL) physics compared to conventional equilibrium approaches, though at considerably higher computational cost. Similar to SCMs, wind profiles generated within a 3D model [such as the Weather Research and Forecasting (WRF) Model; Skamarock et al. 2008] are highly sensitive to the turbulence parameterization scheme (e.g., Shimada et al. 2011; Carvalho et al. 2012, 2014; Deppe et al. 2013; Draxl et al. 2014; Marjanovic et al. 2014). In general, turbulence schemes that incorporate nonlocal transport produce the most accurate wind profiles in unstable conditions, while local diffusion schemes perform better in stable conditions. However, the relative performance of turbulence schemes tends to vary with location.
A hierarchy of models used in simulating the wind profile. The single-column model approach (boxed in red) is the focus of this study.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
The SCM approach, which falls within the center of the spectrum in Fig. 1, is the focus of this study. SCMs occupy a potentially useful middle ground by providing more comprehensive physics than conventional extrapolation approaches while being considerably more computationally efficient than 3D models. Another advantage of an SCM approach is the ability to specify lower boundary conditions in terms of well-constrained and easily measured quantities such as wind speed, air temperature, and turbulent fluxes; by contrast, lower boundary conditions in a 3D model must be specified in terms of more poorly constrained quantities such as roughness length, surface temperature, and surface cooling. This advantage is particularly appealing within the context of wind power meteorology, as near-surface measurements of wind speeds are common in initial resource assessments. Provided a value for the geostrophic wind is specified, this approach in particular avoids the need to specify roughness lengths for momentum and temperature, which have been shown to be poorly constrained parameters to which the wind profile is highly sensitive (Verkaik and Holtslag 2007; Optis et al. 2016). Furthermore, lower boundary values of temperature or the turbulent temperature flux avoid the need to specify an SL scheme, which is generally required in atmospheric models to determine turbulent fluxes at the surface. A lower boundary above the surface also helps to mitigate the influence of horizontal heterogeneity in surface roughness and the development of IBLs (Optis et al. 2016).
To our knowledge, the application of an SCM in wind power meteorology has not been explored although it has been suggested (e.g., Rostkier-Edelstein and Hacker 2010). Furthermore, the use of an SCM with a lower boundary above the surface has not been explored in any context to our knowledge.
c. Motivation and intent of study
In Optis et al. (2014), we demonstrated the breakdown of MOST (and various MOST-based alternative models) for extrapolating wind speeds aloft in stable stratification. We now consider the extent to which an SCM approach can provide improved accuracy compared to MOST or other equilibrium approaches given its ability to incorporate a more comprehensive representation of ABL turbulence. We consider a highly idealized SCM that considers only the momentum and temperature budget equations and requires specification only of the geostrophic wind vector, the 10-m wind vector, and the 5-m turbulent temperature flux. We consider composite results over a large (10 yr) dataset in order to average out the effects of advective tendencies (as in Baas et al. 2010). We also consider a range of turbulence closure schemes identified in the GABLS3 study (Bosveld et al. 2014b; Kleczek et al. 2014). We compare the performance of the SCM to that of the two-layer model, found to be the most accurate of a range of analytic models considered in Optis et al. (2014). In section 2 we describe the data sources. The model setup including the different turbulence schemes considered is provided in section 3. In section 4 we compare the model results with observations over a range of stability classes. The influence of baroclinicity at Cabauw, methods to account for the resulting thermal wind, and the effect on the modeled wind profile are explored in section 5. A discussion is provided in section 6, and conclusions are presented in section 7.
2. Data sources
Data for this analysis were taken from a range of sources. Most of the data were obtained from the Cabauw Meteorological Tower in the Netherlands, operated by the Royal Netherlands Meteorological Institute (KNMI). Measurements of meteorological variables at 10-min resolution were obtained from 1 January 2001 to 31 December 2010 (KNMI 2013). Wind speed and direction measurements are available at 10, 20, 40, 80, 140, and 200 m, and temperature measurements are available at these altitudes as well as at 2 m. Turbulent temperature flux data at 5 m at 10-min resolution were also provided. Surface pressure measurements at 10-min resolution were used to calculate the potential temperature at different heights. Turbulent momentum flux data at 10-min resolution were provided for the period July 2007–June 2008 at altitudes of 5, 60, 100, and 180 m. Two different datasets were used to estimate the geostrophic wind. The first dataset was provided by KNMI and was derived from 1-h surface pressure measurements from weather stations near Cabauw using a second-order polynomial fit. The second dataset was the 6-h-averaged wind vector data at 800 hPa taken from the interim European Centre for Medium-Range Weather Forecasts (ECMWF) global atmospheric reanalysis (ERA-Interim; available online at http://apps.ecmwf.int/datasets/data/interim_full_daily). These data were linearly interpolated horizontally to the location of Cabauw. To estimate the thermal wind, near-surface temperature measurements from 2001 to 2010 in 1-h averages were taken from nearby weather stations operated by KNMI (data are available online at http://www.knmi.nl/klimatologie/uurgegevens/). All data used in this analysis were linearly interpolated to 10-min resolution unless otherwise indicated. We consider 10-min-averaged data once every 30 min (1200, 1230, 1300 UTC, etc.) to reduce computational requirements while still obtaining a comprehensive sampling of conditions at Cabauw.
3. Model setup
a. SCM governing equations and turbulence schemes


















Turbulence closure schemes considered in this study.
Complete parameterizations of turbulence closure schemes considered in this study.
We consider a range of turbulence closure schemes based on the GABLS3 study, in which Bosveld et al. (2014b) considered 19 different SCMs and Kleczek et al. (2014) considered seven turbulence schemes within the WRF Model. Limiting the order of schemes to 1.5-order TKE closure, we identify and select for consideration in this study a total of eight different turbulence closure schemes considered in Bosveld et al. (2014b) and Kleczek et al. (2014). These schemes are summarized in Table 1 with complete parameterizations provided in Table 2.







b. The equilibrium approach
In this section, we describe the approach used to generate wind profiles that are in equilibrium (i.e., stationary in time) with observed external forcings. Of principal interest in this study is the final equilibrium wind profile, while the process used to arrive at this equilibrium profile is less relevant in the present context. We describe the process here for transparency.
To generate an equilibrium wind profile, we adopt an approach commonly used in other SCM studies of the SBL (e.g., Weng and Taylor 2003, 2006; Cuxart et al. 2006; Sorbjan 2012, 2014). Using observed external parameters at a given point in time (specifically the geostrophic wind, 10-m wind, and 5-m turbulent temperature flux), we begin from a neutral wind profile and integrate Eqs. (1a)–(1c) forward in time while keeping the external parameters constant. The goal of this approach is to reach a “quasi equilibrium” state in which the vertical wind profile and the vertical potential temperature gradient become constant in time in the lower ABL (i.e., below roughly 500 m). The potential temperature in the lower ABL does not reach equilibrium because of continued surface cooling. Previous studies have found that a period of 9 h was sufficient to reach quasi equilibrium in moderately stable conditions (Beare et al. 2006; Cuxart et al. 2006; Sorbjan 2014). We adopt the same time period in this analysis. Under very stable stratification, quasi equilibrium is generally not reached because of low turbulent mixing and the generation of inertial oscillations (Sorbjan 2014).
The initial neutral profile is solved by assuming equilibrium (i.e.,
Upper boundaries for the SCM, based on the magnitude of the geostrophic wind G.

c. Two-layer model setup
The two-layer model [described in detail in Optis et al. (2014)] consists of a MOST-based logarithmic wind speed profile applied within the SL and the Ekman equations applied above. Required observational data include the 10-m wind speed, the bulk Richardson number between 10 m and the surface (with the assumption that 2-m temperatures are representative of surface values), and the magnitude of the geostrophic wind. The height of the SL is computed internally based on the nondimensional parameter
4. Results

Stability classes considered in this analysis, based on
In Fig. 2, we compare modeled and observed probability density functions (PDFs) of Δθ between 200 and 10 m (i.e.,
PDFs of modeled and observed
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
This bias toward low modeled stratifications can be related to the existence of two physically meaningful equilibrium solutions for the SBL for a fixed value of
Joint PDFs of the observed 5-m turbulent temperature flux to both the observed and modeled (UKMO scheme) near-surface stratifications for the different stability classes.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
Van de Wiel et al. (2007) demonstrated that models generally tend toward the computationally stable weak stratification equilibrium and away from the computationally unstable strong stratification equilibrium. If the value of
Frequency of model breakdown by stability class for the different turbulence schemes.
The mean vertical profiles of modeled and observed wind speed are shown in Fig. 4 for the different stability classes. Observed wind speeds tend to decrease with increasing stratification and demonstrate an LLJ below 200 m on average in extremely stable conditions. The different turbulence closure schemes result in a broad range of mean profiles across all stability classes. The modeled profiles all overestimate the wind speed in weakly stable conditions (Fig. 4a), and are more evenly distributed around the mean observed profile in the other stability classes. By comparing Figs. 2 and 4, it is evident that the tendency to underestimate stability (i.e., overestimate mixing) is associated with low wind speed shear below 200 m, as expected. Conversely, models that best represent the stratification demonstrate the highest wind speed shear below 200 m. The two-layer model shows strong agreement with the mean observed profiles for weakly to moderately stable conditions. This result was demonstrated in Optis et al. (2016) and is not surprising given that the model parameters were tuned to the Cabauw data. In very to extremely stable conditions, the two-layer model overestimates the wind speeds.
Mean vertical profiles of modeled and observed wind speeds for the different stability classes. Note that the MYJ and RAC profiles overlap with the QNSE and ECMWF profiles, respectively, and are therefore difficult to distinguish.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
It is interesting to note that all modeled profiles in Fig. 4 overestimate wind speeds in the lower stability classes despite the tendency to underestimate stratification. We highlight this tendency in Fig. 5, where mean modeled and observed wind profiles are shown for weakly to moderately stable conditions but using higher-resolution stability classes. We show results only for the UKMO model given that it best represented the stratification in Fig. 2. The tendency to overestimate wind speeds is highest at the lowest stratifications. With increasing stratification, the mean modeled profiles show stronger agreement with the observed profiles. The tendency to overestimate wind speeds and wind shear in weakly stable conditions is caused by the local IBL at Cabauw (Beljaars 1982; Verkaik and Holtslag 2007; Optis et al. 2016). The immediate surroundings at Cabauw (within 200 m) have relatively low roughness, while farther from the tower (within 1–2 km) roughness increases significantly as a result of the presence of small towns and belts of trees. Turbulence at 10 m (and therefore the wind speed) in weakly stable conditions is representative of the low local roughness. At higher stratification, the height of the IBL is reduced and the 10-m wind speeds become more representative of regional roughness.
As in Fig. 4, but using higher-resolution Ri stability classes and only the UKMO turbulence closure scheme.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
We explore the effect of the IBL on the SCM wind profile in Fig. 6, which shows observed and modeled mean momentum flux profiles and mean wind speeds for weakly stable conditions using data collected between 1 July 2007 and 30 June 2008 (for which turbulent flux profile observations are available). We use the UKMO turbulence scheme and consider three different lower boundary conditions: specified winds at 10 m, specified surface roughness of
Influence of the local IBL at Cabauw for weakly stable conditions and considering different SCM lower boundary heights for the period 1 Jul 2007–30 Jun 2008. Shown are the (a) mean modeled and observed momentum flux profiles and (b) mean modeled and observed wind speed profiles.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
Box plots of the relative error between modeled and observed winds at different altitudes and stability classes are shown in Fig. 7. In general, the spread of the error increases with stratification. Within individual stability classes, there is little variation in spread between the different SCM turbulence closure schemes. Models that use a
Box plots of the relative error between modeled and observed winds [i.e.,
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
5. Accounting for baroclinicity in the geostrophic wind profile
As demonstrated in the previous section, the effect of the local IBL results in a strong tendency for the SCM (in which horizontal homogeneity was assumed) to overestimate wind speeds in weakly stable conditions. However, the local IBL may not be the only factor producing this bias. It is possible that the vertical structure of the geostrophic wind may be important. We assumed in the previous section that the geostrophic wind vector (calculated from surface pressure measurements) was constant with altitude. In general, this is not the case, particularly at near-coastal sites where the land–sea temperature gradient results in baroclinic conditions and a nonzero thermal wind. Given the high sensitivity of the wind speed profile throughout the ABL to small changes in the geostrophic wind (Baas et al. 2010; Bosveld et al. 2014a), an accurate representation of the geostrophic wind profile is important. In this section, we explore two different approaches to determining the geostrophic wind profile.
a. Horizontal temperature gradient approach
Cabauw is approximately 50 km from the North Sea (Fig. 8) and is subject to mesoscale temperature gradients due to the land–sea temperature contrast (Tijm et al. 1999; Bosveld et al. 2014a). We demonstrate this temperature gradient and the resulting thermal wind in Fig. 9 for the different stability classes. Distributions of the differences in 2-m temperatures measured at Cabauw and at Hoek van Holland (located about 50 km west of Cabauw and along the coastline) are shown in Fig. 9a. The temperature difference is generally negative as a result of a relatively warmer sea temperature in stable conditions. Furthermore, the difference is larger for higher stability classes (often more than 6 K in extremely stable conditions), which can be attributed to colder land temperatures in higher stability classes.
A map of weather stations operated by KNMI. Cabauw is circled in red, and the remaining weather stations considered in section 5a are circled in blue. (Courtesy of KNMI.)
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
Characteristics of the thermal wind between 200 m and the surface by stability class, based on 1.5-m temperature measurements from 11 KNMI weather stations (Fig. 8). Shown are PDFs of (a) ΔT between Cabauw and Hoek van Holland, (b) the direction of the thermal wind at Cabauw, and (c) the magnitude of the thermal wind at Cabauw.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1







Using Eq. (12), and assuming the horizontal temperature gradient is constant with height, we calculate the thermal wind between the surface and 200 m. The spatial scale of the observational network is on the threshold between the mesoscale and synoptic scale; therefore, we expect this thermal wind approximation to be reasonable. Distributions of the thermal wind direction are shown in Fig. 9b for the different stability classes. The thermal wind is predominately from the north-northeast for all stability classes, indicative of a temperature gradient toward the west-northwest and consistent with the expectation that
b. Synoptic interpolation approach
Temperature measurements from nearby weather stations may in general not be available for estimation of the thermal wind. An alternative measure of the thermal wind can be made by comparing the angle between the geostrophic wind vector aloft and that at the surface. For this analysis, we consider the 800-hPa (roughly 2000 m) wind vector from the ERA-Interim model as an estimate of the geostrophic wind vector at 2000 m. The thermal wind between the surface and 2000 m is then calculated as the vector difference between the 2000-m and surface geostrophic winds. We consider only cases where both wind vectors have magnitudes greater than 5 m s−1 to exclude the high variability in the thermal wind direction during low wind speed events. Distributions of the resulting thermal wind direction are shown in Fig. 10a for the different stability classes. Relative to the surface–200-m thermal wind direction PDFs estimated in the previous section, the distributions in Fig. 10a demonstrate a broader range of values and in particular a larger representation of westerly thermal winds. This broader range is expected given that the temperature gradient is not generally uniform with altitude between the surface and 2000 m. The westerly thermal winds may be attributed to the planetary-scale north–south temperature gradient, which is expected to have some influence well above the surface. Distributions of the magnitude of the thermal wind are shown in Fig. 10b. Differences in the magnitudes for different stability classes are much smaller than are found for the surface–200-m estimates. Furthermore, the magnitudes are considerably higher than those found for the surface–200-m estimates, which is expected given the larger (by a factor of 10) altitude range.
Characteristics of the thermal wind by stability class, calculated as the vector difference between the 800-hPa wind vector and the surface geostrophic wind vector. Shown are PDFs of (a) the direction and (b) the magnitude of the thermal wind.
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
c. Applying the baroclinic correction to the wind speed profiles
Having demonstrated two reasonable and broadly consistent approximations of the thermal wind, we now examine the influence of the thermal wind on the wind speed profile up to 200 m at Cabauw.
We focus on the
Different representations of the geostrophic wind vector profiles considered in this analysis.
Mean modeled and observed wind profiles are shown in Fig. 11 for different wind direction sectors (based on the observed 200-m wind direction). The influence of the thermal wind on the modeled wind profile is strongly dependent on wind direction. For the mesoscale temperature gradient approach, the influence is largest in the southwest (SW) sector and smallest in the northeast (NE) sector. These results indicate a surface–200-m thermal wind from the northeast on average and are consistent with results found in Fig. 9 for moderately stable conditions. For the synoptic interpolation approach, the influence is largest in the NE and southeast (SE) sectors and negligible in the northwest (NW) and SW sectors. These results indicate a surface–2000-m thermal wind from the west on average, broadly consistent with the results found in Fig. 10a. We note that the Syn log approach produces much larger corrections to the wind profile for the NE and SE directions relative to the other approaches, generally producing unrealistic-looking profiles.
Mean modeled and observed wind speed profiles for the NW, NE, SW, and SE wind direction sectors. Different models account for different representations of the geostrophic wind profile (Table 6). The UKMO turbulence scheme is used, and the
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
Box plots of the relative error between modeled and observed winds at different altitudes and stability classes are shown in Fig. 12. In general, there is little variation in the spread between different models apart from the Syn log model, which shows substantial spread in the NE and SE sectors. The Syn linear approach tends to show slightly less spread than the other models, while the G 1000 approach tends to show slightly more spread.
As in Fig. 7, but for the NW, NE, SW, and SE wind direction sectors. Different models account for different representations of the geostrophic wind profile (Table 6). The UKMO turbulence scheme is used, and the
Citation: Journal of Applied Meteorology and Climatology 55, 4; 10.1175/JAMC-D-15-0075.1
6. Discussion
To our knowledge, this is the first study to carry out an observationally based assessment of SCM wind and temperature profiles using an equilibrium approach. In previous studies, equilibrium approaches have been employed for intermodel comparisons (Weng and Taylor 2003; Cuxart et al. 2006) or for exploring the general characteristics of the ABL (Weng and Taylor 2006; Sorbjan 2014) without comparison to atmospheric observations. Furthermore, to our knowledge this is the first SCM study to use an observational dataset sufficiently large (10 years) to obtain a comprehensive sampling of atmospheric conditions. Previous observation-based SCM studies have focused only on one or several case studies (Baas et al. 2010; Bosveld et al. 2014b).
Results from this study clearly demonstrate the limitations of an equilibrium-based SCM in modeling the SBL under stable stratification. Specifically, the use of near-surface
The assumption of horizontal homogeneity also contributed to the bias between the SCM results and observations. In particular, the local IBL at Cabauw resulted in a strong tendency for the SCMs to overestimate wind speeds in weakly stable conditions. In contrast, the two-layer model was accurate in this stability class. This result can be attributed to the degree to which the two-layer model was tuned to the Cabauw data. As described in Optis et al. (2014), the two-layer model uses a MOST-based stability function within the surface layer that was derived based on Cabauw data (Beljaars and Holtslag 1991). Furthermore, in cases where the diagnosed surface-layer height was less than 10 m (i.e., very to extremely stable conditions), the model reduced to an Ekman model and a parameterization of the diffusivity coefficient was selected that best matched the mean wind profile at Cabauw. Finally, surface stability was determined from the Richardson number calculated between 10 m and the surface, based on the assumption that 2-m temperatures were representative of surface values. This assumption tended to underestimate near-surface stability and often modeled neutral stratification in weakly to moderately stable conditions. This unintentional bias toward neutral conditions resulted in a wind profile that matched the observed IBL-influenced wind profile at Cabauw. In very to extremely stable conditions, the breakdown of the two-layer model was evident and the equilibrium SCM was more accurate.
As demonstrated in this and in previous studies, the modeled wind profiles are highly sensitive to the choice of the turbulence closure scheme. Schemes with constant or no asymptotic mixing length limits resulted in the largest underestimates of stratification, while those schemes with asymptotic mixing length limits that scaled with the boundary layer height (i.e.,
This analysis also demonstrated the influence of baroclinicity on the wind profile at a near-coastal site. Although the effects of baroclinicity at Cabauw during unstable conditions have been well demonstrated (Tijm et al. 1999; Bosveld et al. 2014a), the effects in stable stratification have to our knowledge not been explored. We demonstrated that the land–sea temperature difference in stable stratification is often large and we considered several representations of the geostrophic wind profile. Contrary to unstable conditions, where accounting for the thermal wind has been shown to have substantial influence on the wind profile below 200 m (Bosveld et al. 2014a), the influence in stable stratification was shown here to be modest.
In general, an equilibrium-based SCM approach for modeling the wind profile is fundamentally limited. A natural question is whether an SCM that makes use of time-evolving observations will result in more accuracy across all stability classes. Such an approach has the added benefit of less computational cost compared to the equilibrium approach (which evolved for 9 h for each fixed point in time) and less likelihood of model breakdown since an equilibrium state is not required. Furthermore, such an approach would allow for a distinction between errors arising from the equilibrium assumption and that of horizontal homogeneity. By considering the time-evolving problem, we can determine the overall utility of the single-column approach. The performance of a time-evolving SCM relative to the equilibrium SCM as well as to a time-evolving 3D mesoscale model will be the subject of a subsequent study.
7. Conclusions
In this study, we used an idealized equilibrium SCM to extrapolate 10-m winds within the altitude range most relevant to wind power. We explored the sensitivity of the wind profile to different turbulence closure schemes and to different estimates of the geostrophic wind vector profile accounting for baroclinic conditions. We compared model results with 10 yr of 10-min-averaged observations at the 213-m Cabauw tower in the Netherlands. Results from this study demonstrated several limitations to the equilibrium approach. First, the existence of two physically meaningful equilibrium solutions for a given value of the surface turbulent temperature flux (used as a lower boundary in the SCM) generally resulted in either a tendency to underestimate stratification or the breakdown of the model as a result of runaway cooling and collapsed turbulence. Second, the equilibrium approach was by design unable to accurately account for time-evolving phenomena such as the inertial oscillation and low-level jet. We further demonstrated in this study no clear association between the accuracy of the wind profile and the order of turbulence closure. Rather, the accuracy of the diffusivity coefficient (calculated using appropriate mixing length and stability function formulations) varied across all orders of turbulence closure and had predominant influence on wind profile accuracy. Baroclinic influences due to the land–sea temperature gradient were shown to have only modest influence on the wind speed profile below 200 m for moderately stable conditions. The IBL at Cabauw resulted in a strong tendency for the SCM to overestimate wind speeds in weakly to moderately stable conditions. In very stable conditions (where the IBL influence was low), SCM accuracy was improved. Despite these limitations, the equilibrium SCM was found to outperform a highly tuned two-layer logarithmic Ekman model. Results from this study indicate the need to assess the role of time dependence relative to the other limitations of the equilibrium SCM approach.
Acknowledgments
We thank Fred Bosveld of KNMI for providing the turbulence and geostrophic wind data and for the many comments and useful dialog pertaining to this research. We also acknowledge access to the CESAR database, which provided the remaining observational data at Cabauw used in this analysis.
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