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

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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

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## Abstract

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.

### Keywords

Boundary-layer heightStable boundary layerTurbulence## 1 Introduction

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.

^{1}:

[\(Ri_{Bh}\)] 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 \(Ri_{Bc}\).

^{2}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

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)

^{3}(\(z_\circ \)), 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.

Characteristics of observational datasets

Field campaign | # Samples | \(z_\circ \) (m) | \(h\) (m) | \(L\) (m) | \(h/L\) | \(Ri_{Bc}\) |
---|---|---|---|---|---|---|

CBLAST | 42 | \(2\times 10^{-5}\) | 35–145 | 27.6–354.5 | 0.27–1.72 | 0.01–0.20 |

FLOSS | 109 | \(4\times 10^{-3}\) | 11–60 | 4.5–110.9 | 0.44–4.01 | 0.03–0.32 |

CASES-99 | 23 | 0.027 | 25–59 | 0.7–14.4 | 2.23–67.18 | 0.07–0.75 |

SPC | 39 | 0.05–0.11 | 89–609 | 0.3–3.6 | 0.25–426.98 | 0.04–93.77 |

Cabauw | 193 | 0.15 | 50–200 | 1.4–10,000 | 0.01–81.48 | 0.04–1.19 |

## 3 Results

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 \(\alpha _\mathrm{{R}}\).

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.

\(Ri_{Bh}\) 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.

## Acknowledgments

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).