Boundary-Layer Meteorology

, Volume 150, Issue 3, pp 515–523

Observational Support for the Stability Dependence of the Bulk Richardson Number Across the Stable Boundary Layer


    • Department of Marine, Earth, and Atmospheric SciencesNorth Carolina State University
  • A. A. M. Holtslag
    • Meteorology and Air Quality SectionWageningen University
  • L. Caporaso
    • ICTP
  • A. Riccio
    • Department of Applied ScienceUniversity of Naples ‘Parthenope’
  • G.-J. Steeneveld
    • Meteorology and Air Quality SectionWageningen University
Research Note

DOI: 10.1007/s10546-013-9878-y

Cite this article as:
Basu, S., Holtslag, A.A.M., Caporaso, L. et al. Boundary-Layer Meteorol (2014) 150: 515. doi:10.1007/s10546-013-9878-y


The bulk Richardson number (\(Ri_{Bh}\); defined over the entire stable boundary layer) is commonly utilized in observational and modelling studies for the estimation of the boundary-layer height. Traditionally, \(Ri_{Bh}\) is assumed to be a quasi-universal constant. Recently, based on large-eddy simulation and wind-tunnel data, a stability-dependent relationship has been proposed for \(Ri_{Bh}\). In this study, we analyze extensive observational data from several field campaigns and provide further support for this newly proposed relationship.


Boundary-layer heightStable boundary layerTurbulence

1 Introduction

The bulk Richardson number, \(Ri_\mathrm{{B}}\left( z\right) \), is commonly defined as
$$\begin{aligned} Ri_\mathrm{{B}}\left( z\right) = \left( \frac{g}{\varTheta _{vS}}\right) \frac{\varTheta _v(z) -\varTheta _{vS}}{U(z)^2 + V(z)^2} z, \end{aligned}$$
where \(g\) is the acceleration due to gravity, \(z\) is the height above ground level, \(\varTheta _v(z)\) is the virtual potential temperature at \(z\), \(\varTheta _{vS}\) is the virtual potential temperature near the surface, and \(U(z)\), \(V(z)\) are the horizontal components of velocity at \(z\).

Over the years, Golder (1972), Zoumakis and Kelessis (1991), Launiainen (1995), Grachev and Fairall (1997), de Bruin et al. (2000), for example, have documented various relationships of \(Ri_\mathrm{{B}}\left( z\right) \) with other stability parameters (e.g., \(z/L\); where \(L\) is the Obukhov length) within the atmospheric surface layer. A few of these studies (e.g., Grachev and Fairall 1997; de Bruin et al. 2000) also commented on the critical bulk Richardson number, which demarcates the transition between (fully) turbulent and (quasi) laminar flows.

In parallel with the surface-layer studies, past workers have focused on the practical application of the bulk Richardson number for the entire atmospheric boundary layer (ABL). For \(z = h\), where \(h\) is the height of the ABL, Eq. 1 becomes
$$\begin{aligned} Ri_{Bh}= \left( \frac{g}{\varTheta _{vS}}\right) \frac{\varTheta _v(h) -\varTheta _{vS}}{U(h)^2 + V(h)^2} h, \end{aligned}$$
where \(Ri_{Bh}\) is the bulk Richardson number of the ABL (e.g., Arya 1999). Based on the reported findings of the surface-layer studies, one would expect that \(Ri_{Bh}\) should also be related to other bulk stability parameters (e.g., \(h/L\)). For instance, Arya (1999) wrote1:

[\(Ri_{Bh}\)] may be used as a substitute for \(h/L\). Note that \(Ri_{Bh}\) does not have the limitations of the local bulk or the gradient Richardson number and is well defined for the stable, unstable, or convective boundary layer. Both \(h/L\) and \(Ri_{Bh}\) can be considered as good measures of turbulence in the ABL, as well as of the relative effects of buoyancy and shear on turbulent mixing and diffusion.

Interestingly, the aforementioned view on \(Ri_{Bh}\), has not been shared by the meteorological community at large. As a matter of fact, it is customary to assume \(Ri_{Bh}\) to be a constant rather than a function of stability (e.g., Mahrt 1981; Troen and Mahrt 1986; Holtslag et al. 1990; Heinemann and Rose 1990; Holtslag and Boville 1993; Vogelezang and Holtslag 1996; Sørensen et al. 1996; Seibert et al. 1998; García et al. 2002; Zilitinkevich and Baklanov 2002; Gryning and Batchvarova 2003; Jeričević and Grisogono 2006; Hong 2010; Esau and Zilitinkevich 2010). More surprisingly, \(Ri_{Bh}\) is often termed the critical bulk Richardson number (\(Ri_{Bc}\)). We would like to note that, in contrast to the surface-layer definition, \(Ri_{Bc}\) defined over the entire ABL does not signify a transition of the entire ABL from (fully) turbulent to (quasi) laminar state. On the contrary, it is defined for all possible states of the ABL. The primary usage of \(Ri_{Bc}\) is in the estimation of the ABL height. We believe that the usage of the term ‘critical’ in the entire ABL context causes considerable confusion. For this reason, we have opted not to use the term ‘critical’ herein. To be consistent, we also use the symbol \(Ri_{Bh}\) in lieu of the commonly used \(Ri_{Bc}\).

Several ‘universal’ values for \(Ri_{Bh}\) have been proposed in the literature (see Zilitinkevich and Baklanov 2002, and the references therein). In a recent work, Richardson et al. (2013) analyzed a large-eddy simulation database (comprising more than 80 runs) and various published wind-tunnel data. They found that \(Ri_{Bh}\) is not a constant, rather it strongly depends on the stability parameter \(h/L\). They proposed the following empirical relationship
$$\begin{aligned} Ri_{Bh} = \alpha \frac{h}{L}, \end{aligned}$$
where \(\alpha \) is a proportionality constant. In some sense, this proposed stability-dependency of \(Ri_{Bh}\) was a ‘reinvention’ of earlier formulations2 by Melgarejo and Deardorff (1974) and Nieuwstadt (1985).

One of the major limitations in Richardson et al. (2013) is that the analyzed datasets were not truly representative of ‘real-world’ SBL turbulence. The LES runs did not include, (i) the effects of natural topography and land-surface heterogeneities; (ii) baroclinicity, large-scale advection, and subsidence effects; and (iii) interactions between several physical processes—e.g., turbulence, radiative transfer, and cloud microphysics. Furthermore, the analyzed wind-tunnel data did not represent the very high Reynolds numbers of atmospheric flows. For this reason, in the present study, we re-examine the validity of Eq. 3 utilizing a diverse set of field observational data.

2 Description of Observational Data

Observational datasets (total sample size: 406) from five field sites around the world are used in this study (in the ascending order of surface roughness):
  • the Coupled Boundary Layers Air–Sea Transfer (CBLAST) experiment, Atlantic Coast, USA (Vickers and Mahrt 2004)

  • the Fluxes over Snow Surfaces (FLOSS) experiment, Colorado, USA (Vickers and Mahrt 2004)

  • the Cooperative Atmosphere–Surface Exchange Study (CASES-99), Kansas, USA (Vickers and Mahrt 2004)

  • San Pietro Capofiume (SPC), Po Valley, Italy (Caporaso et al. 2013)

  • Cabauw, The Netherlands (Vogelezang and Holtslag 1996; Steeneveld et al. 2007)

Vogelezang and Holtslag (1996), Vickers and Mahrt (2004), Steeneveld et al. (2007), and Caporaso et al. (2013) provided in-depth discussions of various facets of these datasets (including site characterization, data collection, quality control, variable estimation). These descriptions will not be repeated here for brevity. Sample sizes, aerodynamic roughness lengths3 (\(z_\circ \)), and relevant meteorological information of these datasets are provided in Table 1. It can be inferred from this Table that, in addition to geographic diversity, each dataset represents various SBL regimes. The aggregated dataset covers near-neutral to very stable regimes.
Table 1

Characteristics of observational datasets

Field campaign

# Samples

\(z_\circ \) (m)

\(h\) (m)

\(L\) (m)





\(2\times 10^{-5}\)







\(4\times 10^{-3}\)


























3 Results

In Fig. 1, \(Ri_{Bh}\) values are plotted as a function of \(h/L\). For most of the datasets (with the exception of the CBLAST dataset), the stability dependence of \(Ri_{Bh}\) is clearly evident. In the case of the SPC dataset, the trend is remarkable and valid for an extensive range of \(h/L\) values. In order to probe if the dependence is simply due to self-correlation (since \(h\) appears in both the independent and dependent variables), we opt for a visual approach (see Klipp and Mahrt 2004; Baas et al. 2006 for detailed discussions on self-correlation). We colour each data point based on its value of \(L\). Using this representation, one finds that the large values of \(Ri_{Bh}\) are associated with small values of \(L\), and vice versa. In other words, \(Ri_{Bh}\) has an inverse relationship with \(L\) (and not just a positive correlation with \(h\)). Similar results were reported by Richardson et al. (2013).
Fig. 1

\(Ri_{Bh}\) as a function of \(h/L\). The data points are coloured based on the value of \(L\). The individual panels represent the observational data from CBLAST (top-left), FLOSS (top-right), CASES-99 (middle-left), Po Valley (middle-right), and Cabauw (bottom-left) field campaigns. The bottom-right panel represents the aggregated dataset

There is a general consensus in the literature that the estimation of \(h\) (and associated variables such as \(Ri_{Bh}\)) from observational data is a challenging task, especially for near-neutral and very stable conditions. So, we do not attempt to fit Eq. 3 for the entire range of stabilities. Instead, we use two thresholds. Data points with \(L > 500\) m (near-neutral condition) and \(L< L_\mathrm{{min}}\) (very stable conditions) are not included for regression analysis. \(L_\mathrm{{min}}\) corresponds to the so-called heat flux minimum (Malhi 1995; Mahrt 1998; Basu et al. 2008),
$$\begin{aligned} L_\mathrm{{min}} = \frac{2\beta z}{\ln \left( z/z_\circ \right) } \end{aligned}$$
where \(z\) is the height above the surface. Since this expression is valid only in the surface layer, we use \(z = 0.1 h\), and we assume \(\beta = 4.7\) following Businger et al. (1971) and Dyer (1974). These thresholds had no impact on the CBLAST and FLOSS datasets as they represent only weakly and moderately stable conditions. However, they excluded 19, 33, and 72 samples for CASES-99, SPC, and Cabauw datasets, respectively.

Given the scatter in the observational data, we employ three rigorous statistical approaches for the estimation of \(\alpha \): least median of squares (LMS) regression, iteratively reweighted least squares (IRLS) regression, and bootstrapping. Each approach has its strengths and weaknesses. However, they are all less prone to outliers than the traditional linear regression approach using the ordinary least squares (OLS) method (Wilcox 2012). Furthermore, used in a complementary manner, these approaches increase the reliability of our findings.

In OLS-based regression, one minimizes the sum of the squared residuals; in contrast, in LMS-based regression, the median of the squared residuals is minimized. By definition, the LMS estimator can resist 50 % contamination of data (Rousseeuw 1984; Massart et al. 1986; Rousseeuw and Leroy 2003; Wilcox 2012). The LMS-based slopes of the \(Ri_{Bh}\)-\(h/L\) relation are denoted by \(\alpha _\mathrm{{M}}\) in this study.

In the IRLS approach, smaller weights are assigned to the outliers (or influential points), which leads to the reliable estimation of the regressed coefficients (Holland and Welsch 1977; Street et al. 1988). An IRLS algorithm with a bisquare weighting function (Mathworks 2012) is utilized here. The IRLS-based estimates of \(\alpha \) are denoted by \(\alpha _\mathrm{{R}}\).

Bootstrapping is a popular resampling technique (Efron 1982; Mooney and Duval 1993; Good 2006). In this approach, \(N\) number of resamples are randomly drawn with replacement from an original sample of size \(N\). This drawing operation is repeated numerous times (a Monte-Carlo procedure)—in this study, we use 10,000 random drawings for each dataset. From each drawing, one value of \(\alpha \) is calculated using OLS—henceforth denoted as \(\alpha _\mathrm{{B}}\). Thus, from 10,000 drawings one histogram of \(\alpha _\mathrm{{B}}\) can be constructed (see Fig. 2) and different percentiles (e.g., median) can be calculated.
Fig. 2

Frequency distribution of \(\alpha _\mathrm{{B}}\) estimated via bootstrapping. Only data points within the range of \(L_\mathrm{{min}} \le L \le 500 \) m are considered. The individual panels represent the observational data from CBLAST (top-left), FLOSS (top-right), and Cabauw (bottom-left) field campaigns. The bottom-right panel represents the aggregated dataset

In the presence of several outliers, the bootstrap-based estimates could be biased. In the statistics literature, several approaches are proposed to confront this problem (e.g., Singh 1998; Srivastava et al. 2009; Wilcox 2012). In this study, prior to the bootstrapping operation, we use the Cook’s distance to detect and reject outliers in an automated fashion (Cook 1977; Fox 1991; Chatterjee and Hadi 2012). These outliers are marked with star symbols in Fig. 3. Please note that this automated outlier rejection approach is only used for \(\alpha _\mathrm{{B}}\) estimation and not for the estimations of \(\alpha _\mathrm{{M}}\) and \(\alpha _\mathrm{{R}}\).
Fig. 3

\(Ri_{Bh}\) as a function of \(h/L\). Only data points within the range of \(L_\mathrm{{min}} \le L \le 500 \) are considered for calculations. The data points are coloured based on the value of \(L\). Data points with \(Ri_{Bh} > 1.25\) are not shown in these plots to enhance the readability; nonetheless, these data points are used in the estimations of \(\alpha _\mathrm{{M}}\) and \(\alpha _\mathrm{{R}}\). The solid magenta and green lines correspond to \(\alpha _\mathrm{{M}}\) and \(\alpha _\mathrm{{R}}\), respectively. The outliers (determined via Cook’s distance) are represented by star symbols and are not used in the bootstrapping approach. In top and bottom panels, the solid black line corresponds to the median value of \(\alpha _\mathrm{{B}}\). The upper and lower dashed lines correspond to the 75 and 25 percentiles of \(\alpha _\mathrm{{B}}\), respectively. These dashed lines are difficult to see due to their close proximity to the median lines. The values of \(\alpha _\mathrm{{M}}\), \(\alpha _\mathrm{{R}}\), and the median value of \(\alpha _\mathrm{{B}}\) are reported at the top-right corner. The individual panels represent the observational data from CBLAST (top-left), FLOSS (top-right), CASES-99 (middle-left), SPC (middle-right), and Cabauw (bottom-left) field campaigns. The bottom-right panel represents the aggregated dataset

In Fig. 3, we plot \(Ri_{Bh}\) as a function of \(h/L\) for the thresholded data. As discussed before, \(L = 500\) m and \(L = L_\mathrm{{min}}\) are used as upper and lower thresholds, respectively. Regression lines are overlayed on each panel, and as in Fig. 1 the data points are coloured based on the value of \(L\). The values of \(\alpha _\mathrm{{M}}\), \(\alpha _\mathrm{{R}}\), and the median value of \(\alpha _\mathrm{{B}}\) are reported at the top-right corner. Given the exceedingly small sample sizes of the CASES-99 and SPC datasets (after thresholding), the LMS and the bootstrap approaches do not allow for a reliable estimation of \(\alpha \) values. In these cases, only \(\alpha _\mathrm{{R}}\) values are calculated using the robust regression approach.

Based on Figs. 1, 2 and 3, we draw the following inferences:
  • \(Ri_{Bh}\) increases with increasing stability (\(h/L\)). This trend is not masked by the scatter in the observed data.

  • All the regression approaches lead to more-or-less similar estimates of \(\alpha \). However, one can notice the following trend: \(\alpha _\mathrm{{M}} < \alpha _\mathrm{{R}} < \alpha _\mathrm{{B}}\). For this reason, \(\alpha _\mathrm{{R}}\) values could be considered a ‘safe compromise’.

  • The slope, \(\alpha \), varies between 0.03 and 0.21. The estimate of \(\alpha _\mathrm{{R}}\) from the aggregated dataset is 0.07.

  • The scatter in the data for each individual site is likely due to omni-present atmospheric variabilities.

  • The dependence of \(\alpha \) on \(z_\circ \) is somewhat noticeable; the \(\alpha \) values from CBLAST (\(z_\circ = 2\times 10^{-5}\) m) and FLOSS (\(z_\circ = 4\times 10^{-3}\) m) are substantially lower than Cabauw (\(z_\circ = 0.15\) m). However, this trend is not observed for CASES-99 and SPC, which could be due to the limited sample size.

4 Conclusion

In the literature (see Zilitinkevich and Baklanov 2002) and in practical applications (e.g., the YSU boundary-layer scheme of the WRF model), \(Ri_{Bh}\) is commonly assumed to be a constant. Recently, Richardson et al. (2013) pointed out that by setting \(Ri_{Bh}\) as a constant, one implicitly constrains the bulk stability of the modelled boundary layer. As a better alternative, they recommended the use of a stability-dependent \(Ri_{Bh}\) formulation (Eq. 3) for practical applications (e.g., numerical weather prediction, dispersion modelling).

In this study, we re-examined the validity of Eq. 3 using extensive observational data. The results are in complete agreement with Richardson et al. (2013). More field observations and realistic LES datasets are, however, needed to better understand the sensitivity of \(\alpha \) with respect to surface characteristics (e.g., roughness) and/or atmospheric conditions (e.g., baroclinicity). In the absence of such dataset, a value of \(\alpha = 0.07\) should be used.


The symbols and acronyms have been slightly modified to be consistent with the rest of the paper.


These earlier formulations were largely overlooked by the meteorological community.


Mesoscale effects are included in the estimate of surface roughness at Cabauw. Please refer to Verkaik and Holtslag (2007) for details.



We are grateful to L. Mahrt and D. Vickers for sending us the CBLAST, FLOSS, and CASES-99 datasets. We thank the editor and two anonymous reviewers for constructive comments and criticisms. We also thank J. Craft for providing useful feedback on a draft of this paper. S. Basu acknowledges the financial support received from the National Science Foundation by way of Grant AGS-1122315. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the U.S. National Science Foundation. The contribution by G.-J. Steeneveld has partly been sponsored by the NWO contract 863.10.010 (Lifting the Fog).

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© Springer Science+Business Media Dordrecht 2013