Boundary-Layer Meteorology

, Volume 131, Issue 2, pp 293–307 | Cite as

Improvement of Vertical Diffusion Analytic Schemes Under Stable Atmospheric Conditions

  • Amela JeričevićEmail author
  • Željko Večenaj
Open Access


Based on gradient transport theory or K-theory, turbulent transport in the atmosphere has long been parameterized using the eddy diffusivity. Due to its simplicity, this approach has often been applied in many numerical models but rarely tested with observations. Here, the widely used O’Brien cubic polynomial approach has been validated together with an exponential approach against eddy diffusivity profiles determined from measurements and from large-eddy simulation data in stable conditions. Verification is completed by analyzing the variability effects on pollutant concentrations of two different vertical diffusion (K(z)) schemes incorporated in an atmospheric chemical model. It is shown that the analytical, exponential solution agrees better with observations than the O’Brien profile and should be used henceforth in practical applications.


Air quality models K-theory Linear exponential approach O’Brien profile 



Authors are grateful to Branko Grisogono for many valuable advises and suggestions; furthermore, we thank the reviewers for several important recommendations. This work is supported by the EMEP4HR project under number 175183/S30 funded by the Research Council of Norway, and by projects BORA 119-1193086-1311 and 004-1193086-3036 of the Croatian Ministry of Science, Education and Sport.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. Andren A, Brown A, Graf J, Moeng CH, Mason PJ, Nieuwstadt FTM, Schumann U (1994) Large-eddy simulation of neutrally-stratified boundary layer: a comparison of four computer codes. Q J Roy Meteorol Soc 120:1457–1484. doi: 10.1002/qj.49712052003 CrossRefGoogle Scholar
  2. Banta RM, Mahrt L, Vickers D, Sun J, Balsley BB, Pichugina YL, Williams EJ (2007) The very stable boundary layer on nights with weak low-level jets. J Atmos Sci 64:3068–3090. doi: 10.1175/JAS4002.1 CrossRefGoogle Scholar
  3. Berge E, Jacobsen HA (1998) A regional scale multi-layer model for the calculation of long-term transport and deposition of air-pollution in Europe. Tellus 50:205–223. doi: 10.1034/j.1600-0889.1998.t01-2-00001.x CrossRefGoogle Scholar
  4. Best MJ, Hopwood WP (2001) Modelling the local surface exchange over grass-field site under stable conditions. Q J Roy Meteorol Soc 127:2033–2052. doi: 10.1002/qj.49712757610 CrossRefGoogle Scholar
  5. Biswas J, Rao T (2000) Uncertainties in episodic ozone modelling stemming from uncertainties in the meteorological fields. J Appl Meteorol 40:117–136. doi: 10.1175/1520-0450(2001)040<0117:UIEOMS>2.0.CO;2 CrossRefGoogle Scholar
  6. Bjorge D, Skalin R (1995) PARLAM—The parallel HIRLAM version of DNMI. Research Report 27: ISSN: 0332-9879. Norwegian Meteorological Institute, Oslo, Norway, 49 ppGoogle Scholar
  7. Blackadar AK (1979) Modelling pollutant transfer during daytime convection. In: Fourth symposium on atmospheric turbulence diffusion and air quality. American Meteorological Society, Reno, NV, pp 443–447Google Scholar
  8. Deardorff JW (1970) Convective velocity and temperature scales for the unstable planetary boundary layer and for Raleigh convection. J Atmos Sci 29:91–115. doi: 10.1175/1520-0469(1972)029<0091:NIONAU>2.0.CO;2 CrossRefGoogle Scholar
  9. de Foy B, Lei W, Zavala M, Volkamer R, Samuelsson J, Mellqvist J, Galle B, Martinez AP, Grutter M, Molina L (2007) Modelling constraints on the emission inventory and on vertical diffusion for CO and SO2 in the Mexico city metropolitan area using solar FTIR and zenith sky UV spectroscopy. Atmos Chem Phys 7(3): 781–801Google Scholar
  10. Dias NL, Brutsaert W (1996) Similarity of scalars under stable stratification. Boundary-Layer Meteorol 80:355–373. doi: 10.1007/BF00119423 CrossRefGoogle Scholar
  11. Ding F, Arya SP, Lin YL (2001) Large-eddy simulations of the atmospheric boundary layer using a new subgrid-scale model. Part II: weakly and moderately stable cases. Environ Fluid Mech 1:49–69. doi: 10.1023/A:1011543715591 Google Scholar
  12. ENVIRON (1998) User’s guide to the comprehensive air quality model with extensions (CAMx) version 2.00. ENVIRON International Corporation, 101 Rowland Way, Suite 220, Novato, California, 265 ppGoogle Scholar
  13. Esau I, Zilitinkevich SS (2006) Universal dependences between turbulent and mean flow parameters is stably and neutrally stratified planetary boundary layers. Nonlinear Process Geophys 13: 122–144Google Scholar
  14. Garratt JR, Pielke RA (1989) On the sensitivity of mesoscale models to surface-layer parametrisation constants. Boundary-Layer Meteorol 48:377–387. doi: 10.1007/BF00123060 CrossRefGoogle Scholar
  15. Grisogono B, Oerlemans J (2002) Justifying the WKB approximation in pure katabatic flows. Tellus, Ser A, Dyn Meterol Oceanogr 54:453–462. doi: 10.1034/j.1600-0870.2002.201399.x Google Scholar
  16. Grisogono B, Kraljević L, Jeričević A (2007) The low-level katabatic jet height versus Monin-Obukhov height. Q J Roy Meteorol Soc, Part B 133(629): 2133–2136CrossRefGoogle Scholar
  17. Jeričević A, Grisogono B (2006) The critical bulk Richardson number in urban areas: verification and application in a numerical weather prediction model. Tellus, Ser A, Dyn Meterol Oceanogr 58:19–27. doi: 10.1111/j.1600-0870.2006.00153.x Google Scholar
  18. Kosovic B, Curry JA (2000) A quasi steady state of a stable stratified atmospheric boundary layer: a large-eddy simulation study. J Atmos Sci 57:1052–1068. doi: 10.1175/1520-0469(2000)057<1052:ALESSO>2.0.CO;2 CrossRefGoogle Scholar
  19. Lee HN, Larsen RJ (1997) Vertical diffusion in the lower atmosphere using aircraft measurements of 222Rn. J Appl Meteorol 36:1262–1270. doi: 10.1175/1520-0450(1997)036<1262:VDITLA>2.0.CO;2 CrossRefGoogle Scholar
  20. Mahrt L (1999) Stratified atmospheric boundary layers. Boundary-Layer Meteorol 90:375–396. doi: 10.1023/A:1001765727956 CrossRefGoogle Scholar
  21. Mahrt L (2007) The influence of nonstationarity on the turbulent flux-gradient relationship for stable stratification. Boundary-Layer Meteorol 125:245–264. doi: 10.1007/s10546-007-9154-0 CrossRefGoogle Scholar
  22. Mahrt L, Vickers D (2006) Extremely weak mixing in stable conditions. Boundary-Layer Meteorol 119:19–36. doi: 10.1007/s10546-005-9017-5 CrossRefGoogle Scholar
  23. Mauritsen T, Svensson G, Zilitinkevich S, Esau I, Enger L, Grisogono B (2007) A total turbulent energy closure model for neutral and stably stratified atmospheric boundary layers. J Atmos Sci 64:4113–4126. doi: 10.1175/2007JAS2294.1 CrossRefGoogle Scholar
  24. Mihailovic DT, Alapaty K (2007) Intercomparison of two K-schemes: local versus nonlocal in calculating concentrations of pollutants in chemical and air-quality models. Environ Model Softw 22:1685–1689. doi: 10.1016/j.envsoft.2007.03.002 CrossRefGoogle Scholar
  25. Monin AS, Obukhov AM (1954) Basic laws of turbulent moxing in the surface layer of the atmosphere. Trudy Geofiz Inst Akad Nauk SSSR 151: 163–187Google Scholar
  26. Nowacki P, Samson PJ, Sillman S (1996) Sensitivity of urban airshed model (UAM-IV) calculated air pollutant concentrations to the vertical diffusion parametrisation during convective meteorological situations. J Appl Meteorol 35:1790–1803. doi: 10.1175/1520-0450(1996)035<1790:SOUAMI>2.0.CO;2 CrossRefGoogle Scholar
  27. O’Brien JJ (1970) A note on the vertical structure of the eddy exchange coefficient in the planetary boundary layer. J Atmos Sci 27:1213–1215. doi: 10.1175/1520-0469(1970)027<1213:ANOTVS>2.0.CO;2 CrossRefGoogle Scholar
  28. Oliviè DJL, van Velthoven PFJ, Beljaars ACM (2004) Evaluation of archived and off-line diagnosed vertical diffusion coefficients from ERA-40 with 222Rn simulations. Atmos Chem Phys 4(9/10): 2313–2336CrossRefGoogle Scholar
  29. Pahlow M, Parlange MB, Porté-Agel F (2001) On Monin-Obukhov similarity in the stable atmospheric boundary layer. Boundary-Layer Meteorol 99:225–248. doi: 10.1023/A:1018909000098 CrossRefGoogle Scholar
  30. Poulos GS, Burns SP (2003) An evaluation of bulk Ri-based surface layer flux formulas for stable and very stable conditions with intermittent turbulence. J Atmos Sci 60:2523–2537. doi: 10.1175/1520-0469(2003)060<2523:AEOBRS>2.0.CO;2 CrossRefGoogle Scholar
  31. Simpson D, Fagerli H, Jonson JE, Tsyro S, Wind P, Tuovinen JP (2003) Unified EMEP model description. Status Report 1, Part I, Oslo, Norway, 104 ppGoogle Scholar
  32. Smedman AS (1988) Observations of a multi-level turbulence structure in a very stable atmospheric boundary layer. Boundary-Layer Meteorol 44:231–253. doi: 10.1007/BF00116064 CrossRefGoogle Scholar
  33. Stull RB (1988) An introduction to boundary layer meteorology. Kluwer Academic Publishers, Dordrecht, p 666 ppGoogle Scholar
  34. Tarrasón L, Simpson D, Fagerli H, Jonson JE, Tsyro S, Wind P (2003) Transboundary acidification and eutrophication and ground level ozone in Europe, Unified EMEP model validation. Status Report 1, Part II, Oslo, Norway, 170 ppGoogle Scholar
  35. Wyngaard JC, Brost RA (1984) Top-down and bottom-up diffusion of a scalar in the convective boundary layer. J Atmos Sci 41:102–112. doi: 10.1175/1520-0469(1984)041<0102:TDABUD>2.0.CO;2 CrossRefGoogle Scholar
  36. Zhang Y, Pun B, Wu SY, Vijayaraghavan K, Seigneur C (2004) Application and evaluation of two air quality models for particulate matter for a southeastern U.S. episode. J Air Waste Manag Assoc 54: 1478–1493Google Scholar
  37. Zilitinkevich S, Calanca P (2000) An extended theory for the stably stratified atmospheric boundary layer. Q J Roy Meteorol Soc 126:1913–1923. doi: 10.1256/smsqj.56617 CrossRefGoogle Scholar
  38. Zilitinkevich S, Esau IN (2003) The effect of baroclinicity on the equilibrium depth of neutral and stable planetary boundary layers. Q J Roy Meteorol Soc 129:3339–3356. doi: 10.1256/qj.02.94 CrossRefGoogle Scholar
  39. Zilitinkevich SS, Esau IN (2007) Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layer. Boundary-Layer Meteorol 125:193–205. doi: 10.1007/s10546-007-9187-4 CrossRefGoogle Scholar
  40. Zilitinkevich SS, Baklanov A, Rost J, Smedman A-S, Lykosov V, Calanca P (2002) Diagnostic and prognostic equations for the depth of the stably stratified Ekman boundary layer. Q J Roy Meteorol Soc 128:25–46. doi: 10.1256/00359000260498770 CrossRefGoogle Scholar

Copyright information

© The Author(s) 2009

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.MHSC Meteorological ServiceZagrebCroatia
  2. 2.AMGI Department of Geophysics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

Personalised recommendations