1. Introduction
The stably stratified boundary layer (SBL) presents ongoing challenges to observational characterization and physical modeling (e.g., Poulos et al. 2002; Fernando and Weil 2010; Holtslag et al. 2013; Mahrt 2014). Understanding the behavior of the lower atmosphere under conditions of stable stratification is important for accurate simulations of nocturnal near-surface temperatures, including the occurrence of fog and frost (e.g., Walters et al. 2007; Holtslag et al. 2013), modeling of lower-atmospheric wind variability and extremes (e.g., He et al. 2010; Monahan et al. 2011; He et al. 2012), and characterization of pollutant dispersal and air quality (e.g., Arya 1998; Salmond and McKendry 2005).
When large-scale pressure gradients are moderate or strong, or in overcast conditions, the stratification is generally relatively weak and mechanically driven turbulence remains sustained. The classical model of turbulent quantities decreasing upward from a surface layer characterized by Monin–Obukhov similarity theory generally holds in this situation (e.g., Mahrt et al. 1998; Pahlow et al. 2001; Grachev et al. 2005; Mahrt 2014). Under clear skies and relatively weak pressure gradient forces, the turbulence can collapse and the near-surface flow can become decoupled from that above (e.g., Derbyshire 1999; Banta et al. 2007; Williams et al. 2013; Mahrt 2014). The collapse of turbulence can occur abruptly as the mechanical driving of the boundary layer is reduced (Sun et al. 2012; Van de Wiel et al. 2012a,b; Liang et al. 2014; van Hooijdonk et al. 2015).
In this collapsed state, turbulence tends to be dominated by intermittent bursts (Kondo et al. 1978; Van de Wiel et al. 2002a,b, 2003; Acevedo and Fitzjarrald 2003; Nakamura and Mahrt 2005; Ohya et al. 2008; Medeiros and Fitzjarrald 2014; Mahrt 2014; Vercauteren and Klein 2015), which are responsible for most of the turbulent transport (Nappo 1991; Coulter and Doran 2002; Doran 2004; Basu et al. 2006; Acevedo et al. 2006; Williams et al. 2013). These bursts tend to be localized both horizontally and vertically (Acevedo and Fitzjarrald 2003; Nakamura and Mahrt 2005). No single mechanism appears to be responsible for these turbulent bursts. They have been associated with a range of processes, including passing density currents (e.g., Sun et al. 2002); propagating solitary and internal waves (e.g., Sun et al. 2004; Meillier et al. 2008); Kelvin–Helmoltz-like shear instabilities (e.g., Blumen et al. 2001); and interactions between the lower atmosphere and the underlying surface (e.g., Van de Wiel et al. 2003). It has long been recognized that such bursts of turbulence can break down the near-surface inversion (e.g., Durst 1933; Gifford 1952; Geiger 1965; Businger 1973; Nappo 1991; Acevedo and Fitzjarrald 2003; White 2009). If they are sufficiently strong, these turbulent mixing events can (but do not necessarily) drive transitions from strong to weak stable stratification.
Based on the observational results discussed above, the SBL is often classified into weakly and very stably stratified regimes (respectively denoted the wSBL and vSBL; cf. Okamoto and Webb 1970; Kondo et al. 1978; Mahrt 1998; Acevedo and Fitzjarrald 2003; Shravan Kumar et al. 2012; Liang et al. 2014; Mahrt 2014; van Hooijdonk et al. 2015; Vercauteren and Klein 2015). Other regime classifications have been suggested. For example, some studies consider a third transitional regime between the wSBL and the vSBL (e.g., Mahrt et al. 1998; Conangla et al. 2008), or separate the vSBL into very weak (bottom up) and moderate (top down) turbulence regimes (e.g., Sun et al. 2012). Grachev et al. (2005) separate the stable ABL into four regimes based on the character of turbulence and the relative importance of the Coriolis force and turbulent fluxes in the momentum budget, while Williams et al. (2013) define five regimes distinguished by stability and degree of mixing. Even more complicated classification schemes are those of Kurzeja et al. (1991) or the Pasquill stability classes (Arya 1998). Another class of SBL regime classifications uses external parameters, such as the geostrophic wind and radiative forcing, rather than internal variables, such as turbulence, stratification, and shear (e.g., Van de Wiel et al. 2002b, 2003; Bosveld and Beyrich 2004). While some studies have defined the regimes in terms of a single metric (such as
Early studies associated the sudden collapse of sustained turbulence for increased stability with the existence of a critical Richardson number above which turbulence cannot be maintained (Okamoto and Webb 1970; Businger 1973; Blackadar 1979; ReVelle 1993). Subsequent studies have cast doubt on the existence of such a critical Richardson number (e.g., Fernando and Weil 2010). More recent analyses interpret this collapse in terms of the existence of a maximum sustainable downward heat flux for intermediate stability (de Bruin 1994; Malhi 1995; Mahrt et al. 1998; Pahlow et al. 2001; Basu et al. 2006; Van de Wiel et al. 2007; Conangla et al. 2008; Sun et al. 2012; Van de Wiel et al. 2012a,b; van Hooijdonk et al. 2015). Increasing stability beyond this maximum reduces heat fluxes; the resulting positive feedback further enhances the stratification and suppresses turbulence until a limiting state of (on average) weak turbulence is reached. This local maximum in turbulent heat flux has been used to distinguish the wSBL from the transitional and vSBL regimes (e.g., Mahrt et al. 1998; Conangla et al. 2008; van Hooijdonk et al. 2015).
Idealized physical models of the SBL have displayed two kinds of regime behavior: parameter regions with multiple equilibria bounded by regions with only a single equilibrium (e.g., McNider et al. 1995; Derbyshire 1999; Walters et al. 2007) or parameter regions of oscillatory (limit cycle or chaotic) variability bounded by regions with a single steady state (e.g., ReVelle 1993; Van de Wiel et al. 2002a,b). Multiple equilibria and the wSBL-to-vSBL transition phase of modeled oscillations are produced by the positive feedbacks described above, while the reverse transition results from shear-driven turbulence (e.g., through formation of low-level jets) in the vSBL. This second feedback mechanism appears to be insufficiently strong in those models displaying multiple equilibria to generate sustained oscillatory behavior, although these systems can display transient oscillations (ReVelle 1993). In other models, the collapsed state is transient: after an initial reduction of turbulence intensity, the modeled SBL adjusts to a unique steady state without oscillations (Van de Wiel et al. 2012a).
Further evidence for the existence of two distinct states of Reynolds-averaged near-surface flow and stratification in the late-summer (July–September) nocturnal boundary layer at Cabauw, Netherlands, was presented in Monahan et al. (2011). This study demonstrated that the probability density function (PDF) of the 10-min-average potential temperature difference between 40 and 2 m has a long tail toward strongly stable stratification. When conditioned on values of this potential temperature difference above or below a subjectively determined value of 1 K, the joint distribution of 10-min-averaged wind speeds at 10 and 200 m separated into two distinct populations. In the first of these, associated with strongly stable stratification, the speeds at 200 m are much faster than those at 10 m. In the second regime, with a near-surface stratification in the weakly stable to weakly unstable range, the speeds at 10 m are closer to those at 200 m. There is no evidence for distinct regimes during the day. The distinction between the nocturnal regimes in Monahan et al. (2011) was based entirely on Reynolds-averaged quantities and made no use of turbulence information.
Similar to the calculations in Monahan et al. (2011), we estimated the joint and marginal PDFs of wind speed at 10 and 200 m (respectively,
Probability density functions of wind speed and stratification at Cabauw, for (top) daytime (0800–1600 UTC) and (bottom) nighttime (2000–0500 UTC) in the JAS season. (left) Joint distribution of wind speed at 10 and 200 m. (middle left) Distribution of 10-m wind speed. (middle right) Distribution of 200-m wind speed. (right) Distribution of potential temperature difference between 200 and 2 m. These distributions were estimated using data from 2001 through 2012.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
As was found in Monahan et al. (2011), the daytime joint distribution between
Three-dimensional scatterplots of the mean wind speed between 200 and 10 m, the wind speed difference between 200 and 10 m, and the potential temperature difference between 200 and 2 m for JAS nighttime. (left) All data points. (right) Data points in regimes R1 (blue) and R2 (red).
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
Inspection of Fig. 2 indicates that the two regimes of the Reynolds-averaged flow and stratification are not separated by a simple stratification threshold such as that used in Monahan et al. (2011). Furthermore, it is evident that these regimes are multidimensional structures in the state space of flow and stratification at Cabauw and are therefore likely to overlap even in the three-dimensional projection displayed in Fig. 2. To distinguish the regimes in the data and allow classification of individual observations without having to subjectively decide on a dividing surface by visual inspection, we will make use of hidden Markov model (HMM) analysis (Rabiner 1989; Murphy 2012). In an HMM analysis, the data under consideration are characterized by an unobserved Markov process, which takes a set of discrete values or hidden states. Within each hidden state, the observations are assumed to be drawn from a hidden-state-dependent probability distribution. Given a specific dataset, the HMM algorithm estimates the parameters of the distributions within each hidden state, the transition matrix of the Markov process, and the most likely state occupied by each given data point. Hidden Markov model analysis has the considerable advantage relative to clustering techniques, such as mixture models, that it estimates not simply the geometry of the distributions within each regime but classifies observations into these regimes and provides an empirical dynamics of transitions between them even when the regimes overlap.
The utility of HMMs in the diagnosis of metastable regimes of large-scale atmospheric flow was demonstrated by Majda et al. (2006) and Franzke et al. (2008, 2011). Yoo et al. (2010) used HMM analysis to study the Asian summer monsoon intraseasonal oscillation and its modulation by El Niño–Southern Oscillation. Hidden Markov model–based analysis and downscaling of precipitation has been considered in Robertson et al. (2004), Greene et al. (2008), and Fu et al. (2013). The present study is the first to apply HMMs to the problem of SBL regimes [although Vercauteren and Klein (2015) have recently applied a similar analysis to study the interaction between submesoscale motions and turbulence in the SBL]. The HMM analysis is applied to long time series of 10-min-averaged nighttime wind and potential temperature observations from the Cabauw tower, and the classification scheme is used to assess the marginal distributions of these variables conditioned by regime. We also consider the properties of observed turbulence kinetic energy and fluxes in these regimes, as well as the covariability of regime occurrence with large-scale forcing (geostrophic wind and cloud cover).
The data to be considered are discussed in section 2, followed by a brief introduction to HMMs in section 3. Consideration of Reynolds-mean and turbulence fields conditioned on HMM state are presented in sections 4 and 5, respectively. A discussion including consideration of the physical mechanisms responsible for the regimes is presented in section 6. Brief conclusions follow in section 7.
2. Data
The primary dataset considered in this analysis consists of 10-min-average wind speed, wind direction, and air temperature at the 213-m tower at Cabauw in the Netherlands (51.971°N, 4.927°E) maintained by the Cabauw Experimental Site for Atmospheric Research (CESAR; cf. van Ulden and Wieringa 1996). Validated observations for the period from 1 January 2001 through 31 December 2012 are used. Wind speed and direction are measured at altitudes of 10, 20, 40, 80, 140, and 200 m. Temperature is also measured at 2 m. Surface pressure measurements are used with the temperature observations to calculate potential temperatures, assuming hydrostatic equilibrium and a dry atmosphere. CT75 ceilometer cloud cover observations at 30-second resolution were combined into 10-min-resolution cloud occurrence data (an integer between 0 and 20), as described in He et al. (2013). These cloud data are available from 1 July 2007 to 30 September 2011. The data and current information about CESAR are available online (http://www.cesar-database.nl/).
Hourly geostrophic vector winds from 1 January 2001 to 31 December 2012 were provided by Fred Bosveld of The Royal Netherlands Meteorological Institute (KNMI). These data are computed using a two-dimensional polynomial fit to surface pressure observations within 75 km of Cabauw, from which the gradient is calculated.
One year (1 July 2007–30 June 2008) of 10-min-resolution turbulence quantities (variances and fluxes of momentum and temperature) at altitudes of 5, 60, 100, and 180 m were also made available by Fred Bosveld. Because of mast interference, turbulence data for winds coming from 280° to 340° are unreliable and are therefore neglected. A detailed description of the turbulence data is available online (http://www.knmi.nl/~bosveld).
All data were stratified by season—January–March (JFM), April–June (AMJ), July–September (JAS), and October–December (OND)—and by time of day: daytime (0800–1600 UTC) and nighttime (2000–0500 UTC). As the HMM analysis can accommodate gaps between blocks of data in a time series but requires the data to be continuous within these, it is natural for the present analysis to divide the data by time of day rather than by a more physically motivated criterion (such as the sign of the surface heat flux).
3. Hidden Markov models









Given a set of observations











Results of HMM analysis of the synthetic system given by Eqs. (3)–(5). Simulations of the variables (top)
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
The time series that we will consider have gaps. Because we do not want to confuse statistical structure imposed by the regular, externally forced diurnal and annual cycle with that produced by internal dynamics, we apply HMM analysis to nocturnal data within individual seasons. Gaps in the time series arise because the end of one night is followed by the beginning of the next night without the intervening time points. At the end of a season, the next point in the time series is separated by a gap of 9 months. Rabiner (1989) describes how the HMM algorithm is extended to account for time series with such gaps.
4. Regimes in near-surface wind and stratification
The nighttime distributions of
PDFs of nighttime wind speed and stratification at Cabauw for all four seasons: (top) distribution of
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1




The first regime R1 is associated with the strongest stable stratifications and relatively small
A striking degree of consistency in the regime structure across seasons is evident in the PDFs of
Principal axes of the nighttime conditional joint distributions of
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
The probabilities of occurrence of the two regimes covary with changes in the large-scale forcing. Figure 6 displays the mean hidden-state number conditioned on the geostrophic wind speed and the fractional cloud cover for the JAS season. When the large-scale pressure gradient force is weak, R1 dominates, except under overcast conditions. The likelihood of R2 increases with both increasing geostrophic wind speed and cloud cover. These covariations suggest a modulation of the regime occupation statistics by variations in large-scale forcing. All other factors being equal, larger values of the geostrophic wind result in a higher rate of mechanical generation of turbulence kinetic energy (TKE), enhanced mixing, and reduced stratification. Increased cloud cover reduces the radiative cooling of the surface and inhibits the formation of a strongly stratified nocturnal boundary layer. These covariations of the regime occupation statistics are consistent with their seasonal evolution (Table 1). In JFM and OND, the two regimes are equally likely, while in AMJ and JAS, the stratified regime R1 is considerably more likely than R2. At this midlatitude location, relatively strong geostrophic wind speed and extensive cloud cover are more common in winter than in summer.
Mean state number conditioned on the geostrophic wind speed and fractional cloud cover for the JAS season.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
Regime occupation probabilities for each season:
The estimated Markov chain transition probabilities for each season are displayed in Table 2. These transition probabilities indicate a strong degree of persistence in the hidden-state time series: the probability of changing from one hidden state to another over any 10-min period is much smaller than remaining in that state. This fact indicates that transitions from one regime to another during the night are relatively rare. The observed probability of remaining in a single regime during an entire night varies with regime and season from just over 50% to 75% (Table 3). Note that if the hidden-state sequence was truly Markovian, the probability of remaining in state j for an entire night would be
Instantaneous regime transition probabilities for each season.
(first and third columns) Probabilities of remaining within a given regime for an entire night estimated directly from hidden-state time series and (second and fourth columns) the associated probabilities if transitions between regimes were truly conditionally independent.
5. Regimes in near-surface turbulence
The HMM regimes that we have identified were determined by analysis of time series of 10-min-averaged potential temperature and wind speed. We will now investigate how these regimes relate to the observed distributions of TKE and turbulent fluxes, with a focus on JAS.
The joint PDFs of nighttime TKE at 5 and 180 m are shown in Fig. 7, along with the joint PDFs of the magnitude of the vertical eddy momentum flux
Joint distributions of base-10 logarithms of nighttime turbulent quantities at 5 and 180 m for the JAS season. (left) Turbulent kinetic energy
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
When conditioned by regime, two distinct populations in these turbulence quantities become evident (Fig. 7). Regime R1 is associated with lower values of TKE at both 5 and 180 m, while in R2 the TKE levels are higher at both altitudes. The correlation of TKE between 5 and 180 m is considerably larger in R2 than in R1. Fluxes in R1 are in the range of smaller values (particularly at 180 m) and weak correlation between altitudes discussed above. In contrast, flux values in R2 are relatively high at both 5 and 180 m, as is the correlation of the fluxes at these altitudes.
One possible reason for the difference in behavior of the TKE and the stresses is the presence of nonturbulent motions (e.g., meandering motions or gravity waves) in the observed eddy velocity components, which contribute to TKE but not stress (e.g., Mahrt 1998). The joint distribution of TKE and stress at 5 m (Fig. 7) demonstrates the ratio of TKE to stress is often larger in R1 than it is in R2. This observation suggests the presence of variability in R1 that contributes to variances but not to fluxes. No such evidence of nonturbulent motions is apparent at 180 m (not shown).
Considering the marginal distributions of the along-wind component of the turbulent momentum fluxes at 5 and 180 m (Fig. 8), we see that at both altitudes the distribution is strongly skewed toward negative (downward) fluxes, with a most likely value near zero. Conditioned by regime, it is evident that the peak of the distribution is primarily associated with R1, and that the corresponding conditional distribution takes small values distributed approximately symmetrically around the mean. In contrast, the entire long tail of strong negative momentum fluxes
Distributions of turbulent quantities at (left) 5 and (right) 180 m for the JAS season. Shown are the full PDF (black) and the PDFs conditioned on states R1 (blue) and R2 (red): (top)
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
As with the along-wind turbulent momentum flux, the turbulent temperature flux is negatively skewed with a distribution that peaks near a flux of 0 K m s−1 (Fig. 8). At 180 m, conditioning on regime occupation results in distinct populations. The peak near zero is dominated by R1, with a relatively narrow conditional distribution that is nearly symmetric. The long tail associated with strong downward temperature fluxes is associated with R2. In contrast, the two populations are not separated at 5 m. Both R1 and R2 contribute to strong downward transport of temperature at this altitude.
We interpret the results presented in Figs. 7 and 8, as follows. The intensity of turbulence and turbulent momentum fluxes is weak in R1, which is characterized by strongly stable stratification and relatively weak 10-m winds. While turbulent temperature fluxes in this regime are relatively weak at 180 m, this is not the case at 5 m. The persistence of relatively strong heat fluxes at 5 m indicates that the temperature gradients in R1 are sufficiently strong to maintain these fluxes even at low TKE values or that the turbulent Prandtl number becomes quite small (e.g., Grachev et al. 2007; Sorbjan and Grachev 2010; He et al. 2012). In R1, variations in TKE are only weakly coupled between 5 and 180 m, and variations of turbulent stresses at these altitudes are essentially uncoupled. This structure is consistent with a shallow turbulent layer at the surface overlaid with a quiescent layer (e.g., Banta 2008). In contrast, the TKE, turbulent fluxes (both momentum and temperature), and 10-m wind speed are relatively strong in the weakly stratified, relatively well-mixed R2. In this regime, variations in turbulence intensity and fluxes are more strongly coupled in the vertical than in R1 and with PDFs similar to those observed during the day (Figs. 9 and 10). Although the daytime TKE values and momentum fluxes are somewhat larger than those in R2 and the turbulent temperature fluxes are generally upward (as the stratification is, in general, unstable), the similarity between the turbulence distributions in R2 and those during the day is striking. In contrast, the daytime distributions of TKE and turbulent momentum fluxes have no features corresponding to R1.
As in Fig. 7 (left),(center), but for daytime (0800–1600 UTC) TKE and
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
As in Fig. 8, but for daytime (0800–1600 UTC) turbulent fluxes at 5 and 180 m for the JAS season. Conditional distributions are not shown, as there is only one daytime regime.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
The HMM regimes R1 and R2 separate the nocturnal near-surface TKE and turbulent momentum flux into two populations with distinct features. As was the case with the eddy-averaged populations, one of these (R2) is similar to the daytime population while the second (R1) has no daytime analog. A particularly striking aspect of this separation of the turbulence populations is that the regime classification was based on Reynolds-averaged time series of stratification and wind, and made no use of turbulence observations.
6. Discussion
a. Physical mechanism of regimes
We have presented empirical evidence for the existence of two distinct regimes of 10-min-averaged flow and stratification at Cabauw and demonstrated that these can be separated through use of a two-state HMM. While these states are distinct, the distributions of the associated 10-min-averaged and turbulence quantities overlap (Figs. 4, 7, and 8). We now investigate the related questions of the existence of a dynamically based parameter (or parameters) separating these regimes and of the underlying physical mechanism.



Joint distribution of the logarithms of the numerator and denominator of the bulk Richardson number [Eq. (9); black contours] obtained using a reference temperature θ0 = 290 K. Contours denote the joint distribution conditioned on being in regime R1 (blue) or R2 (red). Gray lines denote constant bulk Richardson numbers, with values indicated on the right ordinate of the left panel.
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
When conditioned on the HMM regimes, the upper part of the distribution (including the lobe toward the upper left) corresponds to R1, while the lower lobe corresponds to R2. The overlap of these two conditional distributions is substantial. The inclined gray lines in Fig. 11 correspond to a range of values of Rib, no single value of which separates the two regimes. While the entire distribution in R1 falls above the Rib = 0.1 line, 84% of the points in R2 do as well: Rib does not provide a clear separation of R1 and R2. These results indicate that there is no critical value of Rib as defined by Eq. (9).
As the single bulk Richardson number derived from 10-min-averaged data does not cleanly separate the regimes, we investigate whether these are separated by a turbulent flux-based measure. As in Mahrt et al. (1998), we consider a scatterplot of the turbulent temperature flux and
(top) Scatterplots of 5-m friction velocity and turbulent temperature flux. (top middle) Scatterplot of 5 m
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1





Black curves show a plot of Eq. (10) describing the relationship between (normalized) steady-state momentum and heat fluxes for Couette flow. The solid and dashed curves correspond, respectively, to stable and unstable equilibria. Gray contours are kernel density estimate of 5-m normalized fluxes from nighttime JAS observations at Cabauw, using
Citation: Journal of the Atmospheric Sciences 72, 8; 10.1175/JAS-D-14-0311.1
Figure 13 displays joint distributions of
b. Relation to previously documented regimes
As discussed in the introduction, there have been many previous attempts to characterize distinct states of the SBL. While most of these classification approaches have been based on the state of the turbulence, the present classification was based entirely on Reynolds-averaged data.
Our regimes R1 and R2 strongly resemble the classification into weakly stable, transitional, and very stable boundary layers (Mahrt 1998; Mahrt et al. 1998; Conangla et al. 2008; Mahrt 2014). Specifically, R1 corresponds to the transition and very stable boundary layers, while R2 resembles the weakly stable boundary layer. In contrast with the classification used in Mahrt et al. (1998) and Conangla et al. (2008), R1 and R2 are not separated by a clear threshold in
In contrast with most previous regime classifications in the stable ABL, R1 and R2 are determined not only by the local characteristics of near-surface turbulence but by the bulk structure of the entire bottom 200 m of the atmosphere. The regimes discussed in Williams et al. (2013) are also characterized in terms of bulk boundary layer structure. Their near-neutral deep and shallow SBL classes clearly are included in our
The classification schemes of Van de Wiel et al. (2002b, 2003) and Bosveld and Beyrich (2004) are based on external parameters corresponding to the mechanical and radiative driving of the SBL. The regime diagram presented in Van de Wiel et al. (2003) qualitatively resembles the dependence of mean regime number on geostrophic wind speed and cloud cover shown in Fig. 6 if their radiative and turbulent regimes are related to our R1, and continuous turbulent regime with R1. In contrast with the results of Van de Wiel et al. (2002b, 2003), we do not find a sharp transition between R1 and R2 when occupation statistics are conditioned on these external parameters. For a broad range of radiative and mechanical forcing values, both regimes can be occupied, consistent with the suggestion that the SBL can display multiple equilibria (e.g., McNider et al. 1995; Derbyshire 1999; Walters et al. 2007).
The results of the present study are closely related to those of van Hooijdonk et al. (2015), who provide evidence of two boundary layer regimes at Cabauw using an entirely different line of reasoning to ours. Defining a normalized shear measure denoted the shear capacity (which makes use of both flow and boundary condition information), they show that this measure clearly distinguishes between the two regimes. Consistent with our results, van Hooijdonk et al. (2015) show that these regimes are not distinguished by
c. Directions of future research
The basic tool used by this study to diagnose the regimes of near-surface flow and stratification and their occupation statistics was hidden Markov model analysis. This approach assumed that regime transitions are governed by an autonomous Markov chain with a specified number of hidden states and that, within any hidden state, the observations are independent and identically distributed (as multivariate Gaussian).
A general difficulty with clustering analyses is that the number of clusters must be specified initially, based on some other source of information. When a clustering algorithm is tasked to find N clusters, N clusters will be found irrespective of the degree to which the distribution of observations is made up of discrete populations gathered around distinct centers. Franzke et al. (2008) suggest that the eigenspectrum of the Markov chain stochastic matrix can be used to estimate the number of hidden states. If the underlying dynamics are associated with M hidden states with characteristic occupation time scales much longer than the time scales of variability within any hidden state, the stochastic matrix for
The presence of nontrivial dynamics within the hidden states can be accommodated by considering a more general statistical model. In particular, autoregressive HMMs fit the data within a hidden state not just to a specified probability distribution, but to a first-order autoregressive process [AR(1)] model (e.g., Chiang et al. 2008). Furthermore, it is evident from Fig. 6 that the stochastic matrix is not autonomous but is conditionally dependent on the externally varying, large-scale cloud cover and pressure gradient force. Hidden Markov model analysis can be extended to nonhomogeneous HMMs, which allow modulations of transition probabilities by external variables (e.g., Robertson et al. 2004; Yoo et al. 2010; Fu et al. 2013). The simple HMM analysis considered in this study does a reasonable job of separating the two regimes, as is evident in Figs. 2, 4, and 7. While it is not expected that generalizing the HMM analysis will change the regime classification obtained using two regimes, the dynamics of the estimated Markov chain may be more physically meaningful. Extending the present analysis by considering autoregressive and nonhomogeneous HMMs is an interesting direction of future study. The fact that the hidden-state process does not appear to be Markovian (Table 3) can also, in principle, be accommodated by extending the data into a higher-dimensional embedding space spanned by time-lagged copies of the data (e.g., Broomhead and King 1986; Horenko et al. 2008).
Another valuable direction of future research would be to apply the finite-element variational approaches recently introduced by Horenko (Franzke et al. 2009; Horenko 2010a,b; O’Kane et al. 2013; Risbey et al. 2015), particularly the finite-element, bounded-variation, vector autoregressive factor method (FEM-BV-VARX) that allows both autoregressive dynamics within individual regimes and modulation of regime variability by external variables. In a recent analysis, Vercauteren and Klein (2015) used FEM-BV-VARX to characterize the interaction of turbulence with submesoscale, nonturbulent motions. This work, which was carried out as the first step in the development of parameterizations of intermittent, submeso-induced turbulence, would naturally inform the improvement of the highly simplified parameterizations used in Monahan et al. (2011) and He et al. (2012).
The HMM analysis presented in this study modeled the data distribution within any hidden state as multivariate Gaussian. In fact, the conditional distributions are not Gaussian, as is evident in the marginal distributions shown in Fig. 4. The conditional wind speed PDFs, in particular, display distinct skewness (generally positive at 10 m and either positive or negative at 200 m). The use of Gaussian distributions in the analysis is therefore an approximation, justified a posteriori by the clear regime separation that is found. Another direction of future research would be to repeat the HMM analysis using a parametric distribution allowing for the evident non-Gaussian structure in the marginal distributions. It is not evident what form such a distribution would take, as no physically justified parametric distribution exists for wind speed [e.g., Monahan (2014); although the Weibull distribution is often used as an empirical fit], and the multivariate distribution would need to model both wind speeds at different altitudes and the potential temperature difference (all of which are correlated).
Another natural question raised by the present study is the extent to which the regime structure characterized at Cabauw generalizes to other locations. The character of turbulence and flow in the stable ABL is strongly influenced by local conditions (e.g., Acevedo and Fitzjarrald 2003), so it may be expected that this regime structure is specific to the location under consideration. In particular, although the vicinity of the Cabauw tower is relatively flat, the structure of turbulence is still affected by the presence of internal boundary layers (e.g., Optis et al. 2014, 2015, manuscript submitted to Wind Energy). While this local sensitivity is almost certainly true in regards to the details of the regimes, there is evidence that their existence and general character occur more broadly than at this one location. Most of the studies documenting multiple regimes of stable ABL structure discussed earlier were conducted using data collected at locations other than Cabauw. Beyond this, the extended tail in the PDF of
While the analysis presented in this study provides evidence regarding the feedback resulting in the separate regimes, it does not provide direct insight regarding transitions between these regimes. The transition from R2 to R1 resulting from the collapse of turbulence is expected to occur when the radiative loss of energy by the surface cannot be balanced by downward turbulent heat transport. The reverse transition is presumably driven by variations in geostrophic wind or cloud cover or by the intermittent turbulent events characteristic of the vSBL (e.g., Banta 2008; White 2009). These events can be driven by local processes, such as the increase in near-surface shear (e.g., Blumen et al. 2001; Van de Wiel et al. 2002a, 2012a) or the propagation of remote disturbances (e.g., Sun et al. 2002, 2004). Studies of intermittent turbulence in the vSBL indicate that there are often multiple such events during the night (e.g., Coulter and Doran 2002; Nakamura and Mahrt 2005). The fact that such events appear to be more common than transitions from R1 to R2 during a given night indicates that not all bursts of turbulence are sufficiently strong to break down the inversion (e.g., Acevedo and Fitzjarrald 2003). A more detailed consideration of the mechanisms responsible for regime transitions is a third interesting direction of future study.
7. Conclusions
This study used a two-state hidden Markov model to diagnose the structure of nocturnal boundary layer regimes from long observations of wind and temperature at the 213-m tower at Cabauw, Netherlands, and to classify individual 10-min observations into these two regimes. The regimes were found to correspond to a low–wind speed, strong-stratification, low-turbulence state (regime R1) and a high–wind speed, weak-stratification, high-turbulence state (regime R2). Furthermore, the regime structures separate two populations in the joint distribution of wind speed at 10 and 200 m. The probability distributions of both 10-min-averaged and turbulence variables in R2 are similar to those observed during the day. Regime occupation statistics were found to covary with external forcing such that R1 is more common under conditions of clear skies and low geostrophic wind speed, while R2 is more representative of cloudy conditions and strong geostrophic winds. While no single turbulence or Reynolds-averaged quantity was found to separate these two states, it was found that they correspond well to the stable and unstable branches of the idealized Couette flow model of Van de Wiel et al. (2007, 2012a).
Further work is required to characterize in more detail the mechanisms responsible for transitions between these regimes [it is anticipated that no single mechanism is at work (cf. Mahrt 2014)] and to extend this analysis to other locations. This study provides further insight into the physical mechanisms controlling the probability distribution of near-surface wind speeds over land (He et al. 2010; Monahan et al. 2011; He et al. 2012, 2013), as well as variations of near-surface temperature and air quality (e.g., Salmond and McKendry 2005; Holtslag et al. 2013). Furthermore, it has provided clear evidence of regimes that can be interpreted as more than “prototypes” (cf. Mahrt 1998, 2014) and which are clearly present in the geometric structure of the distribution of near-surface flow and stratification and the temporal variability of these quantities.
Acknowledgments
The authors gratefully acknowledge the provision of 10-min-averaged tower data for this study by the Cabauw Experimental Site for Atmospheric Research (CESAR) and the turbulence and geostrophic wind data by Fred Bosveld. We also thank Bas van de Wiel, Nikki Vercauteren, and one anonymous reviewer for their thoughtful comments, which improved this manuscript. AHM was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant Program. TR and YH were supported by the NSERC CREATE Training Program in Interdisciplinary Climate Science.
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