1. Introduction
During the sunset transition the incoming shortwave radiation from the sun gradually lessens. As a result, the convective boundary layer (CBL) weakens and a temperature inversion starts to form near the surface, which is the onset of the stable boundary layer (SBL). Typically, the temperature inversion grows within a few hours and then remains relatively constant until sunrise. It is the early growth that we are interested in for this study. In particular, we use both observations and numerical models to investigate what parameters determine how fast the SBL evolves during this period.
During the late afternoon, turbulence in the boundary layer is typically decaying (Stull 2000). Much research effort has been aimed at understanding this period on the full range from fundamental studies (van Heerwaarden and Mellado 2016), to practical models (Nadeau et al. 2011), and observations (Blay-Carreras et al. 2014; Jensen et al. 2016). The same is true for the established SBL (Armenio and Sarkar 2002; Sorbjan 2010; Ansorge and Mellado 2016; Mahrt 2014; van Hooijdonk et al. 2017).
These previous studies of the SBL have mostly relied on Monin–Obukhov similarity theory (MOST; Monin 1970), which, once the SBL is established, provides a reasonable description for a broad range of conditions (Sorbjan 2010; Grachev et al. 2013). The decaying turbulence itself, before the onset of the SBL, may also exhibit self-similar behavior, at least for strongly convective situations with weak synoptic forcing (e.g., Nadeau et al. 2011; van Heerwaarden and Mellado 2016). Thus, our understanding of both the (quasi) steady CBL and the (quasi) steady SBL has progressed significantly, despite their own challenges (Holtslag et al. 2013; Lothon et al. 2014).
This understanding and resulting parameterizations may be of limited use during the sunset transition itself (e.g., Sun et al. 2003), since the onset of the SBL typically occurs a few hours before the net radiation becomes negative (van der Linden et al. 2017). Consequently, the decay of convective turbulence and the SBL coexist and interact during the sunset transition (Nadeau et al. 2011; Blay-Carreras et al. 2014). This period is therefore very challenging for numerical weather prediction models (Lothon et al. 2014). Moreover, large-eddy simulation is complicated owing to the changing resolution and domain requirements during the transition (Basu et al. 2008) and the importance of other processes, such as radiative transfer (Edwards 2009). Therefore, it is very important to gain observationally based insight into the period just after the onset of the SBL. As an initial step, we investigate the evolution of the temperature inversion for a broad range of conditions.
Most studies that have focused on the near-surface temperature (inversion) in the SBL have related the instantaneous temperature inversion to other flow variables (André and Mahrt 1982; Sorbjan 2006; Vignon et al. 2017), on detailed investigation of individual cases (Sun et al. 2003), or on mechanistic understanding of the “steady state” (van de Wiel et al. 2012; van Hooijdonk et al. 2015; van de Wiel et al. 2017). However, the temporal behavior starting at the onset of the SBL has received relatively little attention.
A few studies have explicitly aimed at the temporal evolution of the temperature during this period. Earlier work often assumed a square root dependence of the inversion strength with time (e.g., Stull 1983). Whiteman et al. (2004) studied the onset of the SBL within sinkholes, as measured by the evolution of the temperature inversion. Since the wind was always weak inside the sinkholes, the evolution could be modeled considering radiative processes only, and hence the sky-view factor was identified as a key parameter for the temporal evolution inside the sinkholes. A follow-up study by De Wekker and Whiteman (2006) extended to plains and basins, using an exponentially decaying function to fit the time series of the absolute temperature to determine a “typical” time scale during clear and calm nights.
Another study by Pattantyús-Ábrahám and Jánosi (2004) also studied the absolute temperature evolution during 2 years at two distinct sites. They specifically limited their analysis to cases where an exponential fit provided excellent agreement with the time series, which again mostly corresponded to clear and calm nights with wind speed less than 2 m s−1. Despite the large number of (predominantly moisture- and radiation-related) quantities they investigated, no clear relation of the time scale was found with any of these. Edwards (2009) used an idealized single-column model (SCM) to study the early evolution of the SBL at weak winds. His main focus was the importance of radiative processes on the near-surface heat budget, while the time scales of the boundary layer evolution did not receive attention. Later, Edwards (2011) imposed the tendency of the surface temperature in a dimensionless model to study the evolution of the sensible heat flux. Nieuwstadt and Tennekes (1981) used a similar imposed tendency to model the evolution of the boundary layer height.
We employ a similar approach to De Wekker and Whiteman (2006) in the sense that a typical time scale is determined for each night. In our case this time scale is based on the growth of the temperature inversion rather than the decay of the absolute temperature. Therefore, we use a logistic growth function, instead of an exponential function, to fit the time series of the temperature inversion (see section 2). Realizing that our approach is heuristic, the typical time scale is considered to be representative of the rate at which the SBL evolves.
The main analysis focuses on a large number of observations from the Cabauw Experimental Site for Atmospheric Research (CESAR) observatory (the Netherlands). These are supported by observations from the Karlsruhe station (Germany) and the Dome C observatory (Antarctica). The Cabauw and Dome C sites are essentially flat, and the Karlsruhe site is located in a small forest clearing in a wide valley. To aid the interpretation, we use three types of numerical simulations: a bulk model for the SBL, which is based on van de Wiel et al. (2017); a single-column model for the SBL (A. M. Holdsworth and A. H. Monahan 2017, unpublished manuscript); and the Regional Atmospheric Climate Model (RACMO)-SBL, the single-column version of an operational-level regional climate model of the Royal Dutch Meteorological Institute (Baas et al. 2017, manuscript submitted to Bound.-Layer Meteor.).
This analysis is used to answer the following questions: To what physical quantities is the rate at which the SBL evolves during the hours around sunset most sensitive? How may this (in)sensitivity be explained? And how similar are the results for three contrasting sites? A major part of the analysis is dedicated specifically to the wind speed dependence, since this quantity has been identified as a key parameter for the characterization of the SBL (Sun et al. 2012; van Hooijdonk et al. 2015; Monahan et al. 2015; Acevedo et al. 2016; van de Wiel et al. 2017).
This paper is organized as follows. In section 2 the important meteorological quantities in this study, the observations sites, and the numerical methods are discussed. Section 3 contains the results, which are then discussed in section 4. The paper is concluded in section 5.
2. Methods
a. Parameter definition
We investigate the temporal evolution of the near-surface temperature inversion 
First, time is defined relative to the time the net radiation 
To define 
(a) Time series of the temperature inversion between 40 and 1.5 m for two example cases at Cabauw: 24 Aug (crosses) and 29 Aug 2001 (circles). Solid lines are examples of a fit through the observations using Eq. (1). The vertical dotted lines indicate the approximate time interval on which the fitting procedure was based. (b) The normalized frequency of 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
Additionally, a temperature scale 
The quality of the fit greatly varied between nights owing to irregular time series. Therefore, we defined a measure for the quality of the fit, denoted by 
The characteristic values of the quantities introduced in section 2a. For each quantity listed in the first column, the minimum (min) and maximum (max) values, the mean μ, and standard deviation σ are listed as an indication of the observational spread of each quantity for each site. The final row contains the number of nights that had sufficient data and fit quality to be used in the analysis and two fractions (f1, f2); f1 indicates how many nights could be used as fraction of the total number of nights, and f2 indicates a similar fraction relative to the number of nights that had sufficient quality of the data (e.g., preselected on missing data points).
Since wind speed has been identified as a key parameter for the SBL (Sun et al. 2012; van de Wiel et al. 2012), we characterize each night by the mean wind speed around (the net-radiation based) sunset 
The established SBL may be characterized further using the net radiative emission 
Because of the differences in the relative importance of radiative, turbulent, and soil heat fluxes in the two regimes, it is possible that the character of the growth of the inversion may differ fundamentally in the very stable boundary layer (VSBL) from that in the weakly stable boundary layer (WSBL). As such, we will investigate the growth of the inversion separately in these two regimes. To separate the two regimes in observations, we make use of a hidden Markov model (HMM) analysis [as in Monahan et al. (2015), who also provided an introductory example for this method]. This approach models the observed variable 
A benefit of the HMM approach is that the separation of states is determined by the intrinsic structure of the data, rather than relying on subjective thresholds (which may be impossible to define in high-dimensional datasets). In this study, we follow the approach of Monahan et al. (2015), distinguishing the VSBL and the WSBL using a two-state HMM applied to three-dimensional data from Cabauw (the wind speed at 10 and 200 m and the potential temperature difference between 2 and 200 m, all at 10-min resolution).
b. Observation sites
Data from three observational sites were used for this study. The main focus lies on the CESAR observatory, while the Karlsruhe and Dome C stations are used to verify consistency of results among sites. To provide insight in the climate of each site, several characteristic values of the quantities introduced in the previous section are listed in Table 1.
The CESAR observatory near Cabauw, the Netherlands (51.971°N, 4.927°E), and its surroundings are extensively described in van Ulden and Wieringa (1996). Further details on the measurement equipment and the tower configuration may be found at http://www.cesar-observatory.nl. The surface is mostly covered with short grass and the surrounding land is relatively flat. We use observations collected between January 2001 and October 2016. The wind speed and direction (cup anemometers) and the (potential) temperature (KNMI Pt500 Element) are measured at 10, 20, 40, 80, 140, and 200 m. Additionally, the temperature is measured at 1.5 (all years) and 0.1 m (since 2013). The relative humidity is derived from the air and dewpoint temperature (Vaisala HMP243) at 1.5 m. All measurements are based on 10-min averages. The net radiation 
The Karlsruhe Boundary Layer Measurement Tower (49.176°N, 8.425°E) is located in the eastern Rhine Valley. We use the observations of the temperature (Pt-100) at 2, 30, 60, and 100 m, and of the wind speed (cup anemometers) at 30, 60, and 100 m (all based on 10-min averages). The observational period is January 2001–December 2016. The valley is about 30 km wide and hills of about 250 m are found 10 km to the east. Larger hills are approximately 50 km to the east and north. The tower is placed in a small forest clearing, a distance of 10–25 m in each direction from the tree line. The canopy height of the forest is 30 m. Further details on the site and the instrumentation may be found in Kalthoff and Vogel (1992) and at http://imkbemu.physik.uni-karlsruhe.de/~fzkmast/. To the northeast the urbanized area of the Karlsruhe Institut für Technologie is located, with buildings higher than 30 m at about 200 m from the tower. Since the forest affects the radiation measurements during sunset and sunrise (M. Kohler 2017, personal communication), these data were not used in our analysis. The sunset was therefore estimated based on the temperature inversion. For our analysis at this site the exact time of sunset is not important.
The Dome C observatory is located on Antarctica (75.06°S, 123.200°E). The station is located on a continental-scale dome (3233 m AGL), and the local slope is very weak (<1%). We use 30-min averages of the temperature (Campbell HMP155 thermohygrometers in mechanically ventilated shields) and the wind measurements (Young 05106 aerovanes) from a 45-m tower at z = [18, 25, 33, 41] m. Additionally, the surface skin temperature is estimated from longwave radiation measurements. Detailed information on the instrumentation and processing are given by Genthon et al. (2013) and Vignon et al. (2016) (see also http://www.institut-polaire.fr/ipev-en/infrastructures-2/stations/concordia/). The analysis is limited to nights with a clear diurnal cycle in the months November–February during the period November 2011–December 2016 (i.e., the Antarctic summer). During the Antarctic summer the sun never sets; that is, the shortwave incoming radiation is always nonzero. Nonetheless, the net radiation does become negative during the “night” owing to variations in the zenith angle. It should be noted that the diurnal cycle of 
c. Numerical models
This section provides an overview of the three numerical models that were used to aid the interpretation of the observations. These models have been described elsewhere, and our main interest is to what extent each model reproduces elements of the observations. As such, only a brief summary of each model is given here, and we do not discuss the merits and drawbacks of specific model choices. For detailed descriptions of the models, the reader is referred to the cited papers.
This model is a strong idealization of reality, especially since MOST may not be valid during the sunset transition (Sun et al. 2003). Nonetheless it is instructive to determine whether or not the model reproduces aspects of the observations. Equation (3) is numerically integrated for various combinations of 
The second model is an idealized SCM for the SBL (SBL-SCM). It solves the Reynolds-averaged equations for the two components of the horizontal wind vector profile (including the Coriolis force), the potential temperature profile, and the ground temperature as set up by Blackadar (1979). The model height is 5000 m and it uses a stretched grid with 100 vertical levels and the highest resolution near the surface. The turbulent diffusivities are modeled using the Prandtl mixing-length hypothesis and the standard Businger–Dyer relation to account for stable and slightly unstable conditions (Businger et al. 1971). The soil model is based on Blackadar (1976), and the radiation scheme is based on the effective atmospheric emissivity (Staley and Jurica 1972). Details may be found in A. M. Holdsworth and A. H. Monahan (2017, unpublished manuscript). Apart from an idealized treatment in the radiation scheme, the model neglects moisture effects.
RACMO is used with the aim to match the observations as closely as possible. The model was run daily for the period 2005–15 as a single-column model (RACMO-SCM) by Baas et al. (2017, manuscript submitted to Bound.-Layer Meteor.). The following model details are a summary of their work. RACMO-SCM is based on the ECMWF Integrated Forecast System (IFS), with specific parameterization details described in Baas et al. (2017, manuscript submitted to Bound.-Layer Meteor.). The main difference with the IFS model is that the first-order closure model was replaced with a prognostic turbulent kinetic energy (TKE) closure model, such that TKE is not forced to be in equilibrium (details in Lenderink and Holtslag 2004; Baas et al. 2017, manuscript submitted to Bound.-Layer Meteor.). The model has 90 vertical levels up to a height of 20 km. The grid spacing near the surface is approximately 6 m, with the lowest model level at 3 m. To investigate the effect of using a TKE scheme, the model was also run using the default IFS first-order closure for the years 2014–15. Further details on the computational details and the model physics may be found in ECMWF (2007). The model was initialized at 1200 UTC using the daily forecast output of the three-dimensional RACMO, version 2.1 (van Meijgaard et al. 2008). The total model run spanned 48 h, but only the second 24-h period was used for the analysis.
3. Results
a. Inversion growth rate versus wind speed
Figure 2 shows the joint frequency distributions of 
Joint frequency distributions of the 40-m wind speed around sunset and (a) 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
It is not surprising that the absolute growth rate 
To gain a more systematic insight into the wind dependence of the inverse time scale, Fig. 3a shows the median value of 
(a) The median of 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
Figure 3a shows how 
Figure 3b shows that the results from Karlsruhe and Dome C are remarkably consistent (i.e., in the range 0.2–0.4) with the observations at Cabauw, considering the differences among the sites. At Karlsruhe the range of 
Figure 4 indicates that the results in Fig. 3 are not sensitive to the measurement height of 
(a) The median inversion growth rate as function of 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
b. Width of the distributions
The size of the error bars in Fig. 3b suggests that other parameters than 
The common factor of moisture, clouds, and season is that they are coupled (or at least correlated) to the evolution of 
First, we investigate if high or low values of 
(a) Ensemble-averaged time series of 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
A similar analysis was performed using several other quantities. We found no systematic relation with the wind direction, the absolute temperature, the friction velocity, the wind speed at 200 m, or the sensible heat flux (all before sunset). Therefore, we do not present results related to those quantities.
Next, we focus directly on how the evolution of the net radiation is related to 
(a) The median inversion growth rate as function of 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
At Dome C the diurnal cycle of 
Figure 6b shows essentially the same result based on 
c. Numerical models
The observations suggest that 
Given the idealized nature of the bulk model, its use for studying the effect of the wind speed provides qualitative insight only. Nonetheless, it would be highly advantageous if such a model is sufficient to explain the observations, since it provides the most direct mechanistic insight. As a first step, one could approach the SBL growth as a simple diffusion problem, which is governed by a time scale 
Figure 7a shows that, at a fixed 
(a) Temporal evolution of the normalized temperature inversion for various values of U (see legend) and zref = 40 m in the bulk model. (b) As in (a), but for varying combinations of U and zref (see legend) such that u* = 0.6 m s−1.
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
In the SBL-SCM both the wind speed and a vertical length scale are dynamic quantities that result from the imposed geostrophic wind speed, the surface properties, and the initial conditions. If the results would be consistent with the observations, this would imply that the compound effect of 
(a) Time series of the normalized temperature inversion for dry clay (circles), and fresh snow (crosses) for various values of geostrophic wind: 10 (black), 14 (blue), and 18 m s−1 (red) in the SBL-SCM. Note that for 10 m s−1, the fresh snow case shows signs of a collapse of turbulence. (b) Time series of the normalized temperature inversion for a geostrophic wind of 18 m s−1 for the same initial conditions as in (a) (black), and using the profile at t = 3 h (blue). Symbols are as in (a).
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
The qualitative disagreement with the observations may be explained partially by the invalidity of the equilibrium flux-gradient relations during the sunset transition. Moreover, the model is initialized right before the onset of the SBL, such that the SBL growth is very sensitive to the initial conditions (Fig. 8b). In fact, model simulations are more sensitive to the initial conditions than to the soil type. Thus, although the model provides useful insight in the relation between the model parameters and dynamics after the sunset transition, it appears less suitable to model the transition itself.
Since the idealized models do not provide qualitative similarities with the observations, we also investigate a more realistic model. The RACMO-SCM with a TKE scheme (as in Baas et al. 2017, manuscript submitted to Bound.-Layer Meteor.) does not rely on a first-order parameterization of turbulence. Additionally, this model is more elaborate than the SBL-SCM, in the sense that it models a full diurnal cycle with a simplified representation of horizontal advection and moisture schemes and more advanced radiation and soil schemes. Therefore, we may expect that the model provides a more realistic representation of the SBL transition.
Figure 9a shows the median of 
(a) The median inversion growth rate as function of 
Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0084.1
The reasonable agreement between the RACMO-SCM and the observations could be explained by the more realistic prognostic representation of the TKE in this model. To test this hypothesis, a new set of runs was performed (spanning 2 years of observations) using the first-order IFS closure for turbulence of the ECMWF. Figure 9a shows that this severely reduces the model performance. The reduction of the performance is almost completely due to the poor prediction of 
In summary, the bulk model and the SBL-SCM appeared to be unable to provide qualitative agreement with the observations. This suggests that the physical processes of particular importance are either absent or poorly represented in these models. A first qualitative important improvement is observed when the RACMO-SCM with the IFS scheme is used, which we attribute to the fact that a full diurnal cycle was modeled, and thus that a realistic and coherent state of the boundary layer was known at the time of the onset of the SBL. Further improvement by using the TKE scheme suggests that the history of the boundary layer is important in a more general sense. This seems to be consistent with the notion that turbulence is not in local equilibrium (e.g., Blay-Carreras et al. 2014).
4. Discussion
In this paper, we investigated the relation between the growth of the SBL across the evening transition and other physical quantities. For this approach it was necessary to use a statistical fitting procedure, which neglects the large variety in how the temperature inversion may develop in individual cases.
Previous studies found strong relations between the wind shear and the temperature inversion (Sun et al. 2012; van Hooijdonk et al. 2015), which could be explained using a strongly idealized model (van de Wiel et al. 2012, 2017). Furthermore, evidence was found of at least two distinct regimes in the SBL (Grachev et al. 2005; Banta et al. 2007; van Hooijdonk et al. 2015; Monahan et al. 2015), of which the most stable regime seems incompatible with MOST-type scaling laws (Sorbjan 2010; Mahrt 2014). Considering the distinct behavior between the regimes later in the night, it would not have been surprising if signals of distinct development were detected at an earlier stage. However, once normalized by the maximum temperature inversion, it appears there is little evidence of distinct regimes (Fig. 3). A possible explanation is that the initial growth is dominated by the same processes (e.g., continuous turbulence or remnants of convective motions). At a later stage turbulence may be weak and intermittent, such that other processes (e.g., the ground heat flux) may be more significant (van de Wiel et al. 2017). In fact, the wind shear appeared to have only a very mild effect on 
Pattantyús-Ábrahám and Jánosi (2004) suggested that the tropospheric water content was the most important factor in the evolution of the nocturnal temperature. Our observations indicate that very low (absolute) values of 
To further complicate the problem, we found indications that the daytime evolution (e.g., how the CBL decays) may be an important factor that affects the SBL growth (Fig. 9). This would explain why no, or only weak, relations were found with many contemporaneously measured variables (e.g., wind speed, temperature, humidity). Since these quantities contain little information on the SBL growth, it seems unlikely that a simple physical model assuming instantaneous equilibrium of turbulent fluxes to the resolved state could qualitatively reproduce the early SBL growth, and how this growth depends on the various parameters.
Consistent with the present results, Edwards (2009) observed clear temporal differences depending on whether or not the full diurnal cycle was modeled. A plausible explanation is that when a full diurnal cycle is being modeled, the flow constitutes a physically realizable state at the time of the onset of the SBL. The archetypal neutral boundary layer (a logarithmic wind profile and no temperature gradients) may not be a realistic state of the boundary layer at any time around sunset. Consequently, they cannot reproduce important aspects of the transient SBL (Figs. 7 and 8). This suggests that at least the presunset history (i.e., the afternoon decay of convective turbulence) should be taken into account in models that aim to study the onset of the SBL.
The results of Jensen et al. (2016) also suggest that the presunset state is important for the dynamics at sunset. They investigated detailed flow characteristics during this presunset period, and they found that the (third order) budget terms of the sensible heat flux contain clues about the nature of the countergradient fluxes around sunset. However, it remains unclear if these budget terms qualitatively impact the growth rate of the temperature inversion, since the RACMO-SCM is able to reproduce the inversion growth with reasonable accuracy, even though it does not resolve third-order terms.
The studies of De Wekker and Whiteman (2006) and Pattantyús-Ábrahám and Jánosi (2004) shared similar approaches to determine a typical evolution of the SBL. These approaches were distinct from this study, but they are related in the sense that, in first-order approximation, a fixed functional form was used to fit the temporal evolution of a temperature-based quantity. For a set of 18 individual nights from 9 different locations, De Wekker and Whiteman (2006) used an exponential function to the normalized cumulative cooling during the whole night. The time scale that they found ranged from 3 to 7.5 h, corresponding to a rate of ~0.13–0.33 h−1. The related approach by Pattantyús-Ábrahám and Jánosi (2004) yielded values for the inverse time scale of 0.1–0.5 h−1, compared to the scale of 0.2–0.4 h−1 that we found at three distinct sites. As such, there appears to be some generality to the evolution rate of the SBL among studies, locations, measurement heights (Fig. 4), and a broad range of conditions. So despite the complexity of the dynamics, a simple ad hoc model for the evolution of the SBL (or at least the temperature inversion) may be sufficient to be useful in studies that do not require detailed insight in the boundary layer itself. The consistency among sites could be further verified by performing a similar analysis on a large number of sites [e.g., as in Monahan et al. (2011)]. Of course, such an analysis could not achieve the same level of detail per site, but it could reveal a systematic dependence on other parameters, such as soil characteristics.
Since at Dome C clouds are mostly absent and conditions are dry in general, the diurnal cycle of 
Finally, we found that an operation-level SCM results in reasonable agreement in terms of statistics. Unfortunately these models are typically complex and may behave unexpectedly if dynamical features (such as clouds) are turned off, which makes them less versatile than more idealized models. It may be possible, however, to investigate the effect of small perturbations of preexisting cases to gain insight in the effect on the growth of the temperature inversion.
5. Conclusions
A large number of observations from three distinct sites were used to study the growth of the SBL, as measured by the temperature inversion. The results show remarkable consistency of the normalized growth rate among these sites. The sensitivity to several parameters was tested and, similar to Pattantyús-Ábrahám and Jánosi (2004), most of these parameters did not indicate a systematic relation with the growth rate. Previous studies identified the wind speed as a key parameter for the structure of the established SBL (e.g., Sun et al. 2012; van de Wiel et al. 2012). Consistent with these studies, we observe that the absolute growth of the temperature inversion sharply increases once the wind is below a critical range. Once normalized by the ultimate inversion strength, the inverse time scale does not show clear evidence of multiple regimes. This suggests that the early development of the SBL is dominated by the same dynamical processes, and the qualitatively distinct regimes become apparent later. Moreover, the present results suggest at most a very weakly decreasing dependence of the normalized growth rate with increasing wind speed.
A clearer trend was found when the decay of the net radiation 
To further investigate the underlying mechanisms, numerical models were used. These showed that a simple bulk model, or the SBL-SCM, could not replicate the dependency on the wind speed, despite the qualitative success of similar models in describing the (quasi) steady state of the SBL (A. M. Holdsworth and A. H. Monahan 2017, unpublished manuscript; van de Wiel et al. 2017). The RACMO-SCM is capable of modeling the full diurnal cycle and it provided similar statistics to the observations when the TKE scheme was used, while the first-order IFS scheme resulted in qualitative similarities only. This suggests the importance of the history of the system in terms of TKE and the mean flow properties of the whole boundary layer for the sunset transition and the early SBL growth.
Acknowledgments
The authors thank Christoph Genthon for providing the Dome C meteorological data, the French polar institute (IPEV, CALVA program 1013) for the logistical support, and the WRMC-BSRN network, Angelo Lupi, and Christian Lanconelli for dissemination of the Dome C radiation data. We also thank Martin Kohler and the Institute for Meteorology and Climate Research of Karlsruhe Institute of Technology (KIT) for providing the Karlsruhe observations. Fred Bosveld and the Royal Dutch Meteorological Institute (KMNI) are thanked for the freely available Cabauw observations. The following funding agencies are acknowledged for their contribution: IvH is supported by the Netherlands Organisation for Scientific Research (NWO, ALW Grant 832010110); AHM, AH, and CA are supported by the Natural Sciences and Engineering Research Council Canada; and BvdW and PB are supported by an ERC Consolidator grant (648666). Finally, the University of Victoria is thanked for hosting IvH during several weeks for this research.
APPENDIX
Fitting Procedure
The fitting procedure was aimed at estimating a “typical” inversion growth rate for each night, measured by 
The procedure used 
First, the time was located when the stable boundary layer (SBL) starts—that is, when 
Third, the minimum value of 
The lsqcurvefit function of MATLAB was used to fit Eq. (1) to the shifted Δθ–time series of each day between 
Finally, for each night, the extremes were removed, i.e., the two highest and two lowest estimates of 
To determine 
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