Boundary-Layer Meteorology

, Volume 135, Issue 3, pp 385–405 | Cite as

An Evaluation of the Flux–Gradient Relationship in the Stable Boundary Layer



Data collected during the SHEBA and CASES-99 field programs are employed to examine the flux–gradient relationship for wind speed and temperature in the stably stratified boundary layer. The gradient-based and flux-based similarity functions are assessed in terms of the Richardson number Ri and the stability parameter z*, z being height and Λ* the local Obukhov length. The resulting functions are expressed in an analytical form, which is essentially unaffected by self-correlation, when thermal stratification is strong. Turbulence within the stably stratified boundary layer is classified into four regimes: “nearly-neutral” (0 < z* < 0.02), “weakly-stable” (0.02 < z* < 0.6), “very-stable” (0.6 < z* < 50), and “extremely-stable” (z* > 50). The flux-based similarity functions for gradients are constant in “nearly-neutral” conditions. In the “very-stable” regime, the dimensionless gradients are exponential, and proportional to (z*)3/5. The existence of scaling laws in “extremely-stable” conditions is doubtful. The Prandtl number Pr decreases from 0.9 in nearly-neutral conditions and to about 0.7 in the very-stable regime. The necessary condition for the presence of steady-state turbulence is Ri < 0.7.


CASES-99 data Flux-based scaling Flux–gradient relationship Gradient-based scaling Monin–Obukhov similarity SHEBA data Stable boundary layer 


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  1. Anderson PS (2009) Measurement of Prandtl number as function of Richardson number avoiding self-correlation. Boundary-Layer Meteorol 131: 345–362CrossRefGoogle Scholar
  2. Andreas EL, Hill RJ, Gosz JR, Moore DI, Otto WD, Sarma AD (1998) Statistics of surface-layer turbulence over terrain with metre-scale heterogeneity. Boundary-Layer Meteorol 86: 379–408CrossRefGoogle Scholar
  3. Andreas EL, Fairall CW, Guest PS, Persson POG (1999) An overview of the SHEBA atmospheric surface flux program. In: 13th symposium on boundary layers and turbulence. American Meteorological Society, Dallas, Proceedings, pp 550–555Google Scholar
  4. Andreas EL, Fairall CW, Grachev AA, Guest PS, Horst TW, Jordan RE, Persson POG (2003) Turbulent transfer coefficients and roughness lengths over sea ice: the SHEBA results. In: Seventh conference on polar meteorology and oceanography and joint symposium on high-latitude climate variations, 12–16 May 2003. American Meteorological Society, Hyannis, AMS Preprint CD-ROMGoogle Scholar
  5. Andreas EL, Claffey KJ, Jordan RE, Fairall CW, Guest PS, Persson POG, Grachev AA (2006) Evaluations of the von Kármán constant in the atmospheric surface layer. J Fluid Mech 559: 117–149CrossRefGoogle Scholar
  6. Baas P, Steeneveld GJ, van de Wiel BJH, Holtslag AAM (2006) Exploring self-correlation in flux–gradient relationships for stably stratified conditions. J Atmos Sci 63: 3045–3054CrossRefGoogle Scholar
  7. Banta RM (2008) Stable-boundary-layer regimes from the perspective of the low-level jet. Acta Geophys 56: 58–87CrossRefGoogle Scholar
  8. Beare RJ, Edwards JM, Lapworth AJ (2006) Simulations of the observed evening transition and nocturnal boundar layers: large-eddy simulation. Q J Roy Meteorol Soc 132: 81–99CrossRefGoogle Scholar
  9. Beljaars ACM, Holtslag AAM (1991) Flux parameterization over land surfaces for atmospheric models. J Appl Meteorol 30: 327–341CrossRefGoogle Scholar
  10. Blackadar AK (1962) The vertical distribution of wind and turbulent exchange in neutral atmosphere. J Geophys Res 67: 3095–3103CrossRefGoogle Scholar
  11. Blumen W, Banta RM, Burns SP, Fritts DC, Newsom R, Poulos GS, Jielun Sun J (2001) Turbulence statistics of a Kelvin–Helmholtz billow event observed in the night-time boundary layer during the Cooperative Atmosphere-Surface Exchange Study field program. Dyn Atmos Oceans 34: 189–204CrossRefGoogle Scholar
  12. Businger JA (1973) Turbulent transfer in the atmospheric surface layer, chapter 2. In: Haugen DA (ed) Workshop on micrometeorology. American Meteorological Society, Boston, pp 67–100Google Scholar
  13. Businger JA, Miyake M, Dyer AJ, Bradley F (1967) On the direct determination of the turbulent heat flux near the ground. J Appl Meteorol 6: 1025–1032CrossRefGoogle Scholar
  14. Churchill SW (2002) A reinterpretation of the turbulent Prandtl number. Ind Eng Chem Res 41: 6393–6401CrossRefGoogle Scholar
  15. Coulter RL, Doran JC (2002) Spatial and temporal occurrences of intermittent turbulence during CASES-99. Boundary-Layer Meteorol 105: 329–349CrossRefGoogle Scholar
  16. Cuxart J (2008) Nocturnal basin low-level jets: an integrated study. Acta Geophys 56: 100–113CrossRefGoogle Scholar
  17. Cuxart J, Yagüe C, Morales G, Terradellas E, Orbe J, Calvo J, Fernandez A, Soler MR, Infante C, Buenestado P, Espinalt A, Joergensen HE, Rees JM., Vila J, Redondo JM, Cantalapiedra IR, Conangla L (2000) Stable atmospheric boundary layer experiment in Spain (SABLES, 98): a report. Boundary-Layer Meteorol 96: 337–370CrossRefGoogle Scholar
  18. Duynkerke PG (1999) Turbulence, radiation and frog in Dutch stable boundary layers. Bounduary-Layer Meteorol 90: 447–477CrossRefGoogle Scholar
  19. Dyer AJ (1974) A review of flux profile relationship. Boundary-Layer Meteorol 7: 363–372CrossRefGoogle Scholar
  20. Edwards JM, Beare RJ, Lapworth AJ (2006) Simulations of the observed evening transition and nocturnal boundary layers: single-column modelling. Q J Roy Meteorol Soc 132: 61–80CrossRefGoogle Scholar
  21. Esau I, Grachev A (2007) Turbulent Prandtl number in stably stratified atmospheric boundary layer: intercomparison between LES and SHEBA data. e-WindEng 006: 01–17Google Scholar
  22. Galperin B, Sukoriansky S, Anderson PS (2007) On the critical Richardson number in stably stratified turbulence. Atmos Sci Letters, ASL.153Google Scholar
  23. Grachev AA, Fairall CW, Persson POG, Andreas EL, Guest PS (2005) Stable boundary-layer scaling regimes: the SHEBA data. Boundary-Layer Meteorol 116: 201–235CrossRefGoogle Scholar
  24. Grachev AA, Andreas EL, Fairall CW, Guest PS, Persson POG (2007a) SHEBA flux–profile relationships in the stable atmospheric boundary layer. Boundary-Layer Meteorol 124: 315–333CrossRefGoogle Scholar
  25. Grachev AA, Andreas EL, Fairall CW, Guest PS, Persson POG (2007b) On the turbulent Prandtl number in the stable atmospheric boundary layer. Boundary-Layer Meteorol 125: 329–341CrossRefGoogle Scholar
  26. Grachev AA, Andreas EL, Fairall CW, Guest PS, Persson POG (2008) Turbulent measurements in the stable atmospheric boundary layer during SHEBA: ten years after. Acta Geophys 56: 142–166CrossRefGoogle Scholar
  27. Hicks BB (1976) Wind profile relationship from Wangara experiments. Q J Roy Meteorol Soc 102: 535–551Google Scholar
  28. Högström U (1988) Non-dimensional wind and temperature profiles in the atmospheric surface layer: a re-evaluation. Boundary-Layer Meteorol 42: 55–78CrossRefGoogle Scholar
  29. Holtslag AAM, De Bruin FTM (1988) Applied modeling of night-time surface energy balance over land. J Appl Meteorol 27: 689–704CrossRefGoogle Scholar
  30. Kader BA, Yaglom AM (1990) Mean fields and fluctuation moments in unstably stratified turbulent boundary layers. J Fluid Mech 212: 637–662CrossRefGoogle Scholar
  31. King JC, Connolley WM, Derbyshire SH (2001) Sensitivity of modelled Antarctic climate to surface and boundary layer parameterizations. Q J Roy Meteorol Soc 127: 779–794CrossRefGoogle Scholar
  32. Klipp CL, Mahrt L (2004) Flux–gradient relationship, self-correlation and intermittency in the stable boundary layer. Q J Roy Meteorol Soc 130: 2087–2103CrossRefGoogle Scholar
  33. Kukharets VP, Tsvang LR (1998) Atmospheric turbulence characteristics over a temperature-inhomogeneous land surface. Part I: Statistical characteristics of small-scale spatial inhomogeneities of land surface temperature. Boundary-Layer Meteorol 86: 89–101CrossRefGoogle Scholar
  34. Kustas W, Li F, Jackson J, Prueger J, MacPherson J, Wolde M (2004) Effects of remote sensing pixel resolution on modeled energy flux variability of croplands in Iowa. Remote Sens Environ 92: 535–547CrossRefGoogle Scholar
  35. Louis J-F (1979) A parametric model of vertical eddy fluxes in the atmosphere. Boundary-Layer Meteorol 17: 187–202CrossRefGoogle Scholar
  36. Mahli YS (1995) The significance of dual solutions for heat fluxes measured by the temperature fluctuation method in stable conditions. Boundary-Layer Meteorol 74: 389–396CrossRefGoogle Scholar
  37. Mahrt L (1998) Stratified atmospheric boundary layers and breakdown of models. J Theor Comp Fluid Dyn 11: 263–280CrossRefGoogle Scholar
  38. Mahrt L (1999) Stratified atmospheric boundary layers. Boundary-Layer Meteorol 90: 375–396CrossRefGoogle Scholar
  39. Mahrt L (2007) Weak-wind mesoscale meandering in the nocturnal boundary layer. Environ Fluid Mech 7: 331–334CrossRefGoogle Scholar
  40. Mahrt L, Vickers D (2006) Extremely weak mixing in stable conditions. Boundary-Layer Meteorol 119: 19–39CrossRefGoogle Scholar
  41. Mahrt L, Sun J, Blumen W, Delany T, Oncley S (1998) Nocturnal boundary-layer regimes. Boundary-Layer Meteorol 88: 255–278CrossRefGoogle Scholar
  42. Massman WJ (2000) A simple method for estimating frequency response corrections for eddy covariance systems. Agric For Meteorol 104: 185–198CrossRefGoogle Scholar
  43. Mellor GL, Yamada T (1982) Development of a turbulence closure model for geophysical fluid problems. Rev Geophys Space Phys 20: 851–875CrossRefGoogle Scholar
  44. Monin AS, Obukhov AM (1954) Basic laws of turbulence mixing in the surface layer of the atmosphere. Trudy Geof Inst AN SSSR 24: 163–187Google Scholar
  45. Moore CJ (1986) Frequency response corrections for eddy correlation systems. Boundary-Layer Meteorol 37: 17–36CrossRefGoogle Scholar
  46. Newsom KR, Banta RM (2003) Shear-flow instability in the stable nocturnal boundary layer as observed by Doppler lidar during CASES-99. J Atmos Sci 60: 16–33CrossRefGoogle Scholar
  47. Nieuwstadt FTM (1984) The turbulent structure of the stable, nocturnal boundary layer. J Atmos Sci 41: 2202–2216CrossRefGoogle Scholar
  48. Oncley SP, Foken T, Vogt R, Kohsiek W, de Bruin H, Bernhofer C, Christen A, van Gorsel E, Grantz D, Lehner I, Liebethal C, Liu H, Mauder M, Pitacco A, Ribeiro L, Weidinger T. (2007) The energy balance experiment EBEX-2000. Part I: Overview and energy balance. Boundary-Layer Meteorol 123: 1–28CrossRefGoogle Scholar
  49. Oyha YD (2001) Wind tunnel study of atmospheric stable boundary layers over a rough surface. Boundary-Layer Meteorol 98: 57–82CrossRefGoogle Scholar
  50. Oyha YD, Neff E, Meroney EN (1997) Turbulence structure in a stratified boundary layer under stable conditions. Boundary-Layer Meteorol 83: 139–161CrossRefGoogle Scholar
  51. Persson POG, Fairall CW, Andreas EL, Guest PS, Perovich DK (2002) Measurements near the atmospheric surface flux group tower at SHEBA: near-surface conditions and surface energy budget. J Geophys Res 107: 8045. doi:10.1029/2000JC000705 CrossRefGoogle Scholar
  52. Poulos GS, Blumen W, Fritts DC, Lundquist JK, Sun J, Burns SP, Nappo C, Banta R, Newsom R, Cuxart J, Terradellas E, Balsley BB, Jensen ML (2002) CASES-99: a comprehensive investigation of the stable nocturnal boundary layer. Bull Am Meteorol Soc 83: 555–581CrossRefGoogle Scholar
  53. Prandtl L (1932) Meteorologische Anwendungen der Stromungslehre. Beitr Phys Atmos 19: 188–202Google Scholar
  54. Savijärvi H (2009) Sable boundary layer: parameterizations for local and larger scales. Q J Roy Meteorol Soc 135: 914–921CrossRefGoogle Scholar
  55. Schwarz P, Law B, Williams M, Irvine J, Kurpius M, Moore D (2004) Climatic versus biotic constraints on carbon and water fluxes in seasonally drought-affected ponderosa pine ecosystems. Glob Biochem Cycl 18: 1029–1037Google Scholar
  56. Sorbjan Z (1986a) On similarity in the atmospheric boundary layer. Boundary-Layer Meteorol 34: 377–397CrossRefGoogle Scholar
  57. Sorbjan Z (1986b) On the vertical distribution of passive species in the atmospheric boundary layer. Boundary-Layer Meteorol 35: 73–81CrossRefGoogle Scholar
  58. Sorbjan Z (1986c) Local similarity of spectral and cospectral characteristics in the stable-continuous boundary layer. Boundary-Layer Meteorol 35: 257–275CrossRefGoogle Scholar
  59. Sorbjan Z (1988) Structure of the stably-stratified boundary layer during the Sesame-1979 experiment. Boundary-Layer Meteorol 44: 255–260CrossRefGoogle Scholar
  60. Sorbjan Z (1989) Structure of the atmospheric boundary layer. Prentice-Hall, Englewood Cliffs, 317 ppGoogle Scholar
  61. Sorbjan Z (2006a) Local structure of turbulence in stably-stratified boundary layers. J Atmos Sci 63: 526–537CrossRefGoogle Scholar
  62. Sorbjan Z (2006b) Comments on “flux–gradient relationship, self-correlation and intermittency in the stable boundary layer”. Q J Roy Meteorol Soc B 617(132): 1371–1373CrossRefGoogle Scholar
  63. Sorbjan Z, Balsley BB (2008) Microstructure of turbulence in the nocturnal boundary layer. Boundary-Layer Meteorol 129: 191–210CrossRefGoogle Scholar
  64. Taylor PA (1971) A note on the log-linear velocity profile in stable conditions. Q J Roy Meteorol Soc 97: 326–329CrossRefGoogle Scholar
  65. Thomas C, Martin JG, Goeckede M, Siqueira MBS, Foken T, Law BE, Loescher HW, Katul G (2008) Estimating daytime subcanopy respiration from conditional sampling methods applied to multi-scalar high frequency turbulence time series. Agric For Meteorol 148: 1210–1229CrossRefGoogle Scholar
  66. Tsvang LR, Kukharets VP, Perepelkin VG (1998) Atmospheric turbulence characteristics over a temperature-inhomogeneous Land Surface. Part II: The effect of small-scale inhomogeneities of surface temperature on some characteristics of the atmospheric surface layer. Boundary-Layer Meteorol 86: 103–124CrossRefGoogle Scholar
  67. Van de Wiel BJH, Moene A, Hartogenesis G, De Bruin HA, Holtslag AAM (2003) Intermittent turbulence in the stable boundary layeor over land. Part III. A classification for observations during CASES-99. J Atmos Sci 60: 2509–2522CrossRefGoogle Scholar
  68. Vickers D, Mahrt L (2003) The cospectral gap and turbulent flux calculations. J Atmos Ocean Technol 20: 660–672CrossRefGoogle Scholar
  69. Vickers D, Mahrt L (2006) A solution for flux contamination by mesoscale motions with very weak turbulence. Boundary-Layer Meteorol 118: 431–447CrossRefGoogle Scholar
  70. Wyngaard JC (1973) On surface-layer turbulence. In: Haugen DA (eds) Workshop on micrometeorology. American Meteorological Society, Boston, pp 101–148Google Scholar
  71. Yagüe C, Viana S, Maqueda G, Redondo JM (2006) Influence of stability on the flux–profile relationships for wind speed, φ m, and temperature, φ h, for the stable atmospheric boundary layer. Nonlinear Process Geophys 13: 185–203CrossRefGoogle Scholar
  72. Zilitinkevich SS, Chalikov DV (1968) On determination of the universal wind and temperature profiles in the surface layer of the atmosphere. Izv Acad Sci USSR Atmos Ocean Phys 4: 915–929Google Scholar
  73. Zilitinkevich S, Elperin T, Kleeorin N, Rogachevskii I, Esau I, Mauritsen T, Miles M (2008) Turbulence energetics in stably stratified geophysical flows: strong and weak mixing regimes. Q J Roy Meteorol Soc 134: 793–799CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of PhysicsMarquette UniversityMilwaukeeUSA
  2. 2.Institute of GeophysicsPolish Academy of SciencesWarsawPoland
  3. 3.Cooperative Institute for Research in Environmental SciencesUniversity of ColoradoBoulderUSA
  4. 4.NOAA Earth System Research LaboratoryBoulderUSA

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