Improving Stable Boundary-Layer Height Estimation Using a Stability-Dependent Critical Bulk Richardson Number

H. Richardson; S. Basu; A. A. M. Holtslag

doi:10.1007/s10546-013-9812-3

Boundary-Layer Meteorology

July 2013, Volume 148, Issue 1, pp 93–109 | Cite as

Improving Stable Boundary-Layer Height Estimation Using a Stability-Dependent Critical Bulk Richardson Number

Authors
Authors and affiliations

H. Richardson
S. BasuEmail author
A. A. M. Holtslag

Article

First Online: 21 March 2013

Received: 30 April 2012

Accepted: 23 February 2013

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12 Citations

Abstract

For many decades, attempts have been made to find the universal value of the critical bulk Richardson number (\(Ri_{Bc}\); defined over the entire stable boundary layer). By analyzing an extensive large-eddy simulation database and various published wind-tunnel data, we show that \(Ri_{Bc}\) is not a constant, rather it strongly depends on bulk atmospheric stability. A (qualitatively) similar dependency, based on the well-known resistance laws, was reported by Melgarejo and Deardorff (J Atmos Sci 31:1324–1333, 1974) about forty years ago. To the best of our knowledge, this result has largely been ignored. Based on data analysis, we find that the stability-dependent \(Ri_{Bc}\) estimates boundary-layer height more accurately than the conventional constant \(Ri_{Bc}\) approach. Furthermore, our results indicate that the common practice of setting \(Ri_{Bc}\) as a constant in numerical modelling studies implicitly constrains the bulk stability of the simulated boundary layer. The proposed stability-dependent \(Ri_{Bc}\) does not suffer from such an inappropriate constraint.

Keywords

Boundary-layer height Large-eddy simulation Low-level jet Resistance laws Stable boundary layer

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Notes

Acknowledgments

The authors are grateful to Larry Mahrt, Pal Arya, and Sethu Raman for useful discussions. The authors acknowledge 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 National Science Foundation.

Appendix A

A.1 Estimation of \(h\) from LES

There is a general consensus in the literature that the estimation of \(h\) from observational data is a challenging task. Over the years, several variables have been explored for this purpose, including (but not limited to): wind speed, wind-speed profile curvature, streamwise velocity variance, vertical velocity variance, momentum flux, buoyancy flux, and gradient Richardson number (see Vickers and Mahrt 2004; Steeneveld et al. 2007; Pichugina and Banta 2010 and references therein). Even though, in the case of observational data, different profiles could lead to significantly different estimates of \(h\), this is not the case for the LES-generated data. In Fig. 6, we plot vertical profiles of velocity components, (resolved) variances, (total) momentum fluxes and (total) sensible heat fluxes. It is clear that \(h\) estimated from most of these profiles is more-or-less similar. Sensible heat-flux profiles usually lead to marginally higher estimates of \(h\) (see also Basu and Porté-Agel 2006; Stoll and Porté-Agel 2008).

Fig. 6
The *panels* from *top* to *bottom* correspond to the vertical profiles of horizontal velocity components, variances, momentum fluxes, and sensible heat fluxes, respectively. The simulation shown on the *left panel* used \(G = 12\,\text{ m }\,\text{ s }^{-1}\) and \(C = 0.75\,\text{ K }\,\text{ h }^{-1}\), whereas the *right one* used \(G = 6\, \text{ m }\,\text{ s }^{-1}\) and \(C = 1.25\,\text{ K }\,\text{ h }^{-1}\). All these *panels* represent the last 6 h of the simulations. The 25th, 50th, and 75th percentiles are shown to depict the temporal variabilities. On *each panel*, \(h\) estimated from the peak of LLJ is overlayed

Similar to Fig. 2 (top-left panel) and Fig. 4, the dependencies of \(Ri_{Bc}\) on \(h/L\) are shown in Fig. 7. In these panels, the vertical profiles of momentum flux (left panel) and (resolved) horizontal velocity variance (right panel) are utilized for the estimation of \(h\). The locations where the momentum flux and (resolved) horizontal velocity variance decrease to less than 10 % of their respective surface value are denoted as \(h\). These estimates of \(h\) in conjunction with Eq. 2 are used for the calculation of \(Ri_{Bc}\). The scatter and the slopes of the linear regression in these figures are slightly larger than those in the top-left panel of Fig. 2. However, the trend is once again clear. Without any doubt, \(Ri_{Bc}\) increases with increasing stability.

Fig. 7
Variation of \(Ri_{Bc}\) with respect to \(h/L\) based on LES-generated samples. The *red lines* denote the best-fit linear regression line (with the constraint of passing through the origin). \(h\) is estimated from the profiles of (total) momentum flux (*left panel*) and (resolved) horizontal velocity variance (*right panel*), respectively

A.2 Sensitivity Experiments

In this appendix, we document the effects of grid resolution and longwave radiation on the LES-generated data. We consider three cases representing distinct stability regimes: (i) weakly stable (\(h/L \approx 3\)), (ii) moderately stable (\(h/L \approx 6\)), and (iii) very stable (\(h/L \approx 9\)). We perform several 6-h long simulations using a range of grid sizes (5–10 m). In line with the LES database, some simulations use a domain size of 800 m \(\times \) 800 m \(\times \) 790 m, while a few others use a smaller domain size of 400 m \(\times \) 400 m \(\times \) 400 m. In some simulations, a radiation scheme is utilized. Recently, we incorporated the Column Radiation Model (CRM; version 2.1.7) in our MATLES code. CRM is a stand-alone version of the radiation code utilized by the NCAR Community Climate Model (Kiehl et al. 1998). See Table 2 for the specifications of all the these simulations.

Table 2

List of sensitivity experiments

Run	\(G\)	\(C\)	Domain size	Resolution	Radiation
	(\(\text{ m } \,\text{ s }^{-1}\))	(\(\text{ K }\,\text{ h }^{-1}\))	(m \(\times \) m \(\times \) m)	(m \(\times \) m \(\times \) m)
W0	12	0.75	\(800\times 800\times 790\)	10.00 \(\times \) 10.00 \(\times \) 10.00	No
W1	12	0.75	\(400\times 400\times 400\)	6.25 \(\times \) 6.25 \(\times \) 6.35	No
W2	12	0.75	\(400\times 400\times 400\)	6.25 \(\times \) 6.25 \(\times \) 6.35	Yes
M0	12	2.00	\(800\times 800\times 790\)	10.00 \(\times \) 10.00 \(\times \) 10.00	No
M1	12	2.00	\(800\times 800\times 790\)	6.25 \(\times \) 6.25 \(\times \) 6.22	No
M2	12	2.00	\(400\times 400\times 400\)	6.25 \(\times \) 6.25 \(\times \) 6.35	No
M3	12	2.00	\(400\times 400\times 400\)	5.00 \(\times \) 5.00 \(\times \) 5.06	No
M4	12	2.00	\(800\times 800\times 790\)	10.00 \(\times \) 10.00 \(\times \) 10.00	Yes
V0	6	1.25	\(800\times 800\times 790\)	10.00 \(\times \) 10.00 \(\times \) 10.00	No
V1	6	1.25	\(400\times 400\times 400\)	6.25 \(\times \) 6.25 \(\times \) 6.35	No
V2	6	1.25	\(400\times 400\times 400\)	6.25 \(\times \) 6.25 \(\times \) 6.35	Yes

The results from the sensitivity experiments are reported in Table 3 and Fig. 8. All the profiles and statistics are computed from the last 30 min of the simulations. The following statements can be made based on these results:

for the weakly stable case, the surface fluxes are almost independent of grid resolution. For all other cases, the surface fluxes minutely decrease with increasing resolutions;
for all the cases, \(h\) decreases with increasing resolution;
for all the cases, the values of \(h/L\) estimated from the fine-resolution runs are within 10 % of their corresponding coarse-resolution values;
\(Ri_{Bc}\) decreases with increasing resolution. For the very stable case, \(Ri_{Bc}\) decreases by almost 25 %; whereas, in the case of the weakly stable case, it decreases by only \(\approx 8\) %;
\(\alpha \) varies with grid-resolution;
the overall impact of longwave radiation on the simulated statistics is marginal; and
most importantly, \(Ri_{Bc}\) increases with increasing stability. This trend is not masked by the effects of grid resolution and radiation.

Table 3

Basic characteristics of the simulated SBL

Run	\(h\)	\(u_*\)	\(\langle w^{\prime }\theta ^{\prime }\rangle \)	\(L\)	\({\displaystyle \frac{h}{L}}\)	\(Ri_{Bc}\)	\(\alpha \)
	(m)	(\(\text{ m }\,\text{ s }^{-1}\))	(\(\text{ K }\,\text{ m }\,\text{ s }^{-1}\))	(m)
W0	205.00	0.28	-0.025	65.36	3.14	0.12	0.039
W1	187.30	0.28	-0.026	64.90	2.89	0.11	0.040
W2	200.00	0.27	-0.023	66.99	2.99	0.12	0.041
M0	145.00	0.25	-0.050	23.30	6.22	0.23	0.037
M1	133.74	0.24	-0.049	20.92	6.39	0.21	0.032
M2	130.16	0.24	-0.047	21.00	6.20	0.20	0.033
M3	124.05	0.23	-0.046	20.51	6.05	0.19	0.032
M4	155.00	0.25	-0.053	21.26	7.29	0.23	0.031
V0	95.00	0.13	-0.016	10.97	8.66	0.37	0.043
V1	79.37	0.12	-0.016	8.78	9.04	0.28	0.031
V2	79.37	0.12	-0.016	9.11	8.71	0.27	0.031

Fig. 8
The *panels* from *top* to *bottom* correspond to the vertical profiles of wind speed, vertical profiles of potential temperature, time series of friction velocity, and time series of sensible heat fluxes, respectively. All the profiles are computed from the last 30 min of the simulations. The simulation shown on the *left panel* used \(G = 12\,\text{ m } \,\text{ s }^{-1}\) and \(C = 2.00\,\text{ K }\,\text{ h }^{-1}\), whereas the *right one* used \(G = 6\,\text{ m } \,\text{ s }^{-1}\) and \(C = 1.25\, \text{ K } \,\text{ h }^{-1}\). See Table 2 for the specifications of the simulations

References

Anderson WC, Basu S, Letchford CW (2007) Comparison of dynamic subgrid-scale models for simulations of neutrally buoyant shear-driven atmospheric boundary layer flows. Environ Fluid Mech 7:195–215CrossRefGoogle Scholar
Andreas EL, Claffey KJ, Makshtas AP (2000) Low-level atmospheric jets and inversions over the western Weddell sea. Bound-Layer Meteorol 97:459–486CrossRefGoogle Scholar
Arya S (1975) Buoyancy effects in a horizontal flat-plate boundary layer. J Fluid Mech 68:321–343CrossRefGoogle Scholar
Arya SP (1999) Air pollution meteorology and dispersion. Oxford University Press, New YorkGoogle Scholar
Basu S, Porté-Agel F (2006) Large-eddy simulation of stably stratified atmospheric boundary layer turbulence: a scale-dependent dynamic modeling approach. J Atmos Sci 63:2074–2091CrossRefGoogle Scholar
Basu S, Porté-Agel F, Foufoula-Georgiou E, Vinuesa JF, Pahlow M (2006) Revisiting the local scaling hypothesis in stably stratified atmospheric boundary layer turbulence: an integration of field and laboratory measurements with large-eddy simulations. Bound-Layer Meteorol 119:473–500CrossRefGoogle Scholar
Basu S, Holtslag AAM, van de Wiel BJH, Moene AF, Steeneveld GJ (2008a) An inconvenient “truth” about using sensible heat flux as a surface boundary condition in models under stably stratified regimes. Acta Geophys 56:88–99CrossRefGoogle Scholar
Basu S, Vinuesa JF, Swift A (2008b) Dynamic LES modeling of a diurnal cycle. J Appl Meteorol Climatol 47:1156–1174CrossRefGoogle Scholar
Basu S, Holtslag AAM, Bosveld FC (2011) GABLS3 LES intercomparison study. In: ECMWF/GABLS workshop on “Diurnal cycles and the stable atmospheric boundary layer”, ECMWF, pp 75–82Google Scholar
Brost RA, Wyngaard JC (1978) A model study of the stably stratified planetary boundary layer. J Atmos Sci 35:1427–1440CrossRefGoogle Scholar
Brown AR, Derbyshire SH, Mason PJ (1994) Large-eddy simulation of stable atmospheric boundary layers with a revised stochastic subgrid model. Q J R Meteorol Soc 120:1485–1512CrossRefGoogle Scholar
Clarke RH, Dyer AJ, Brook RR, Reid DG, Troup AJ (1971) The Wangara experiment: boundary layer data. Technical paper no. 19. CSIRO, Division of Atmospheric Physics. Aspendale, Australia, p 362Google Scholar
Esau I, Zilitinkevich S (2010) On the role of the planetary boundary layer depth in the climate system. Adv Sci Res 4:63–69CrossRefGoogle Scholar
Esau IN, Zilitinkevich SS (2006) Universal dependences between turbulent and mean flow parameters instably and neutrally stratified planetary boundary layers. Nonlinear Process Geophys 13:135–144CrossRefGoogle Scholar
García JA, Cancillo ML, Cano JL (2002) A case study of the morning evolution of the convective boundary layer depth. J Appl Meteorol 41:1053–1059Google Scholar
Garratt JR (1992) The atmospheric boundary layer. Cambridge University Press, CambridgeGoogle Scholar
Gryning SE, Batchvarova E (2003) Marine atmospheric boundary-layer height estimated from NWP model output. Int J Environ Pollut 20:147–153Google Scholar
Hanna SR (1969) The thickness of the planetary boundary layer. Atmos Environ 3:519–536CrossRefGoogle Scholar
Heinemann G, Rose L (1990) Surface energy balance, parameterizations of boundary-layer heights and the application of resistance laws near an Antarctic ice shelf front. Bound-Layer Meteorol 51:123–158CrossRefGoogle Scholar
Holtslag AAM, Boville BA (1993) Local versus nonlocal boundary-layer diffusion in a global climate model. J Clim 6:1825–1842CrossRefGoogle Scholar
Holtslag AAM, de Bruijn EIF, Pan HL (1990) A high resolution air mass transformation model for short-range weather forecasting. Mon Weather Rev 118:1561–1575CrossRefGoogle Scholar
Holtslag AAM, Steeneveld GJ, van de Wiel BJH (2007) Role of land surface temperature feedback on model performance for stable boundary layers. Bound-Layer Meteorol 125:361–376CrossRefGoogle Scholar
Hong SY (2010) A new stable boundary-layer mixing scheme and its impact on the simulated East Asian summer monsoon. Q J R Meteorol Soc 136:1481–1496CrossRefGoogle Scholar
Jeričević A, Grisogono B (2006) The critical bulk Richardson number in urban areas: verification and application in numerical weather prediction model. Tellus 58A:19–27Google Scholar
Jiménez MA, Cuxart J (2005) Large-eddy simulations of the stable boundary layer using the standard Kolmogorov theory: range of applicability. Bound-Layer Meteorol 115:241–261CrossRefGoogle Scholar
Kiehl J, Hack J, Bonan G, Boville B, Williamson D, Rasch P (1998) The National Center for Atmospheric Research Community Climate Model: CCM3. J Clim 11(6):1131–1149CrossRefGoogle Scholar
Mahrt L (1981) Modelling the depth of the stable boundary-layer. Bound-Layer Meteorol 21:3–19CrossRefGoogle Scholar
Melgarejo JW, Deardorff JW (1974) Stability functions for the boundary-layer resistance laws based upon observed boundary-layer heights. J Atmos Sci 31:1324–1333CrossRefGoogle Scholar
Nieuwstadt FTM (1984) The turbulent structure of the stable, nocturnal boundary layer. J Atmos Sci 41(14):2202–2216CrossRefGoogle Scholar
Nieuwstadt FTM (1985) A model for the stationary, stable boundary layer. In: Hunt JCR (ed) Turbulence and diffusion in stable environments. Clarendon Press, Oxford, pp 149–179Google Scholar
Ohya Y (2001) Wind-tunnel study of atmospheric stable boundary layers over a rough surface. Bound-Layer Meteorol 98(1):57–82CrossRefGoogle Scholar
Ohya Y, Uchida T (2003) Turbulence structure of stable boundary layers with a near-linear temperature profile. Bound-Layer Meteorol 108(1):19–38CrossRefGoogle Scholar
Ohya Y, Neff D, Meroney R (1997) Turbulence structure in a stratified boundary layer under stable conditions. Bound-Layer Meteorol 83(1):139–161CrossRefGoogle Scholar
Pichugina Y, Banta R (2010) Stable boundary layer depth from high-resolution measurements of the mean wind profile. J Appl Meteorol Climatol 49(1):20–35CrossRefGoogle Scholar
Saiki EM, Moeng CH, Sullivan PP (2000) Large-eddy simulation of the stably stratified planetary boundary layer. Bound-Layer Meteorol 95:1–30CrossRefGoogle Scholar
Seibert P, Beyrich F, Gryning SE, Joffre S, Rasmussen A, Tercier P (1998) Mixing height determination for dispersion modelling, COST action 710-final report, harmonisation of the pre-processing of meteorological data for atmospheric dispersion models, L-2985. European Commission, Luxembourg, EUR 18195 EN (ISBN 92-828-3302-X)Google Scholar
Sørensen JH, Rasmussen A, Svensmark H (1996) Forecast of atmospheric boundary-layer height utilised for ETEX real-time dispersion modelling. Phys Chem Earth 21:435–439CrossRefGoogle Scholar
Steeneveld GJ, van de Wiel BJH, Holtslag AAM (2007) Diagnostic equations for the stable boundary layer height: evaluation and dimensional analysis. J Appl Meteorol Climatol 46(2):212–225CrossRefGoogle Scholar
Stoll R, Porté-Agel F (2008) Large-eddy simulation of the stable atmospheric boundary layer using dynamic models with different averaging schemes. Bound-Layer Meteorol 126(1):1–28CrossRefGoogle Scholar
Stull RB (1988) An introduction to boundary layer meteorology. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
Troen I, Mahrt L (1986) A simple model of the atmospheric boundary layer; sensitivity to surface evaporation. Bound-Layer Meteorol 37:129–148CrossRefGoogle Scholar
Vickers D, Mahrt L (2004) Evaluating formulations of stable boundary layer height. J Appl Meteorol 43: 1736–1749Google Scholar
Vinuesa JF, Basu S, Galmarini S (2007) The diurnal evolution of \(^{222}\) Rn and its progeny in the atmospheric boundary layer during the Wangara experiment. Atmos Chem Phys 7:5003–5019CrossRefGoogle Scholar
Vogelezang DHP, Holtslag AAM (1996) Evaluation and model impacts of alternative boundary-layer height formulations. Bound-Layer Meteorol 81:245–269CrossRefGoogle Scholar
van de Wiel BJH, Moene AF, Jonker HJJ, Baas P, Basu S, Donda JMM, Sun J, Holtslag AAM (2012) The minimum wind speed for sustainable turbulence in the nocturnal boundary layer. J Atmos Sci 69:3116–3127CrossRefGoogle Scholar
Yamada T (1976) On the similarity functions A, B and C of the planetary boundary layer. J Atmos Sci 33:781–793CrossRefGoogle Scholar
Zilitinkevich S, Baklanov A (2002) Calculation of the height of the stable boundary layer in practical applications. Bound-Layer Meteorol 105:389–409Google Scholar

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H. Richardson
- 1
S. Basu
- 1
Email author
A. A. M. Holtslag
- 2

1.Department of Marine, Earth, and Atmospheric SciencesNorth Carolina State UniversityRaleighUSA
2.Meteorology and Air Quality SectionWageningen UniversityWageningenThe Netherlands

Improving Stable Boundary-Layer Height Estimation Using a Stability-Dependent Critical Bulk Richardson Number

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