Boundary-Layer Meteorology

, Volume 147, Issue 1, pp 51–82 | Cite as

The Critical Richardson Number and Limits of Applicability of Local Similarity Theory in the Stable Boundary Layer

  • Andrey A. Grachev
  • Edgar L Andreas
  • Christopher W. Fairall
  • Peter S. Guest
  • P. Ola G. Persson


Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin–Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson number, Ri, and the flux Richardson number, Rf, exceed a ‘critical value’ of about 0.20–0.25, the inertial subrange associated with the Richardson–Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson–Kolmogorov energy cascade weakens; therefore, the applicability of local Monin–Obukhov similarity theory in stable conditions is limited by the inequalities RiRi cr and RfRf cr. However, it is found that Rf cr  =  0.20–0.25 is a primary threshold for applicability. Applying this prerequisite shows that the data follow classical Monin–Obukhov local z-less predictions after the irrelevant cases (turbulence without the Richardson–Kolmogorov cascade) have been filtered out.


Flux–profile relationships Critical Richardson number Monin–Obukhov similarity theory Nieuwstadt local scaling Non-Kolmogorov turbulence Richardson–Kolmogorov cascade SHEBA Stable boundary layer z-less similarity 


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  1. Andreas EL (2002) Parameterizing scalar transfer over snow and ice: a review. J Hydrometerol 3: 417–432CrossRefGoogle Scholar
  2. 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
  3. Andreas EL, Horst TW, Grachev AA, Persson POG, Fairall CW, Guest PS, Jordan RE (2010a) Parametrizing turbulent exchange over summer sea ice and the marginal ice zone. Q J R Meteorol Soc 136(649B): 927–943CrossRefGoogle Scholar
  4. Andreas EL, Persson POG, Jordan RE, Horst TW, Guest PS, Grachev AA, Fairall CW (2010b) Parameterizing turbulent exchange over sea ice in winter. J Hydrometeorol 11(1): 87–104CrossRefGoogle Scholar
  5. Arya SPS (1972) The critical condition for the maintenance of turbulence in stratified flows. Q J R Meteorol Soc 98(416): 264–273CrossRefGoogle 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(11): 3045–3054CrossRefGoogle Scholar
  7. Baas P, de Roode SR, Lenderink G (2008) The scaling behaviour of a turbulent kinetic energy closure model for stably stratified conditions. Boundary-Layer Meteorol 127(1): 17–36CrossRefGoogle Scholar
  8. Banta RM, Pichugina YL, Brewer WA (2006) Turbulent velocity-variance profiles in the stable boundary layer generated by a nocturnal low-level jet. J Atmos Sci 63(11): 2700–2719CrossRefGoogle Scholar
  9. 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(9): 3068–3090CrossRefGoogle Scholar
  10. Basu S, Porté-Agel F, Foufoula-Georgiou E, Vinuesa J-F, 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. Boundary-Layer Meteorol 119: 473–500CrossRefGoogle Scholar
  11. Baumert H, Peters H (2004) Turbulence closure, steady state, and collapse into waves. J Phys Oceanogr 34(2): 505–512CrossRefGoogle Scholar
  12. Baumert HZ, Peters H (2009) Turbulence closure: turbulence, waves and the wave-turbulence transition – Part 1: Vanishing mean shear. Ocean Sci 5: 47–58CrossRefGoogle Scholar
  13. Businger JA (1988) A Note on the Businger-Dyer profiles. Boundary-Layer Meteorol 42: 145–151CrossRefGoogle Scholar
  14. Businger JA, Wyngaard JC, Izumi Y, Bradley EF (1971) Flux–profile relationships in the atmospheric surface layer. J Atmos Sci 28: 181–189CrossRefGoogle Scholar
  15. Busch NE (1973) The surface boundary layer (Part I). Boundary-Layer Meteorol 4: 213–240CrossRefGoogle Scholar
  16. Busch NE, Panofsky HA (1968) Recent spectra of atmospheric turbulence. Q J R Meteorol Soc 94(400): 132–148CrossRefGoogle Scholar
  17. Canuto V, Cheng Y, Howard A, Esau I (2008) Stably stratified flows: a model with no Ri(cr). J Atmos Sci 65(7): 2437–2447CrossRefGoogle Scholar
  18. Caughey SJ (1977) Boundary-layer turbulence spectra in stable conditions. Boundary-Layer Meteorol 11(1): 3–14CrossRefGoogle Scholar
  19. Cheng Y, Brutsaert W (2005) Flux–profile relationships for wind speed and temperature in the stable atmospheric boundary layer. Boundary-Layer Meteorol 114(3): 519–538CrossRefGoogle Scholar
  20. Derbyshire SH (1990) Nieuwstadt’s stable boundary layer revisited. Q J R Meteorol Soc 116: 127–158CrossRefGoogle Scholar
  21. Dias NL, Brutsaert W, Wesely ML (1995) Z-less stratification under stable conditions. Boundary-Layer Meteorol 75(1-2): 175–187CrossRefGoogle Scholar
  22. Dyer AJ (1974) A review of flux–profile relationships. Boundary-Layer Meteorol 7: 363–372CrossRefGoogle Scholar
  23. Dyer AJ, Hicks BB (1970) Flux–gradient relationships in the constant flux layer. Q J R Meteorol Soc 96: 715–721CrossRefGoogle Scholar
  24. Ellison T (1957) Turbulent transport of heat and momentum from an infinite rough plane. J Fluid Mech 2: 456–466CrossRefGoogle Scholar
  25. Ferrero E, Quan L, Massone D (2011) Turbulence in the stable boundary layer at higher Richardson numbers. Boundary-Layer Meteorol 139(2): 225–240CrossRefGoogle Scholar
  26. Forrer J, Rotach MW (1997) On the turbulence structure in the stable boundary layer over the greenland ice sheet. Boundary-Layer Meteorol 85: 111–136CrossRefGoogle Scholar
  27. Galperin B, Sukoriansky S, Anderson PS (2007) On the critical Richardson number in stably stratified turbulence. Atmos Sci Lett 8: 65–69CrossRefGoogle Scholar
  28. Garratt JR (1992) The atmospheric boundary layer. Cambridge University Press, UK 316 ppGoogle Scholar
  29. Grachev AA, Fairall CW, Persson POG, Andreas EL, Guest PS (2005) Stable boundary-layer scaling regimes: the SHEBA data. Boundary-Layer Meteorol 116(2): 201–235CrossRefGoogle Scholar
  30. 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(3): 315–333CrossRefGoogle Scholar
  31. 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(2): 329–341CrossRefGoogle Scholar
  32. 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(1): 142–166CrossRefGoogle Scholar
  33. Grachev AA, Andreas EL, Fairall CW, Guest PS, Persson POG (2012) Outlier problem in evaluating similarity functions in the stable atmospheric boundary layer. Boundary-Layer Meteorol 144(2): 137–155. doi: 10.1007/s10546-012-9714-9 CrossRefGoogle Scholar
  34. Handorf D, Foken T, Kottmeier C (1999) The stable atmospheric boundary layer over an antarctic ice sheet. Boundary-Layer Meteorol 91(2): 165–186CrossRefGoogle Scholar
  35. Högström U (1988) Non-dimensional wind and temperature profiles in the atmospheric surface layer: a re-evaluation. Boundary-Layer Meteorol 42: 55–78CrossRefGoogle Scholar
  36. Hong J (2010) Note on turbulence statistics in z-less stratification. Asia-Pacific J Atmos Sci 46(1): 113–117. doi: 10.1007/s13143-010-0011-6 CrossRefGoogle Scholar
  37. Hong J, Kim J, Ishikawa H, Ma Y (2010) Surface layer similarity in the nocturnal boundary layer: the application of Hilbert-Huang transform. Biogeosciences 7: 1271–1278CrossRefGoogle Scholar
  38. Howard LN (1961) Note on a paper of John W. Miles. J Fluid Mech 10: 509–512CrossRefGoogle Scholar
  39. Howell JF, Sun J (1999) Surface-layer fluxes in stable conditions. Boundary-Layer Meteorol 90: 495–520CrossRefGoogle Scholar
  40. Itsweire EC, Koseff JR, Briggs DA, Ferziger JH (1993) Turbulence in stratified shear flows: Implications for interpreting shear-induced mixing in the ocean. J Phys Oceanogr 23(7): 1508–1522CrossRefGoogle Scholar
  41. Jimenez MA, Cuxart J (2005) Large-eddy simulations of the stable boundary layer using the standard Kolmogorov theory: Range of applicability. Boundary Layer Meteorol 115(2): 241–261CrossRefGoogle Scholar
  42. Jordan RE, Andreas EL, Makshtas AP (1999) Heat budget of snow-covered sea ice at North Pole 4. J Geophys Res 104(C4): 7785–7806CrossRefGoogle Scholar
  43. Kaimal JC (1973) Turbulence spectra, length scales and structure parameters in the stable surface layer. Boundary-Layer Meteorol 4: 289–309CrossRefGoogle Scholar
  44. Kaimal JC, Finnigan JJ (1994) Atmospheric boundary layer flows: their structure and measurements. Oxford University Press, New York/Oxford, 289 ppGoogle Scholar
  45. Kaimal JC, Wyngaard JC, Izumi Y, Coté OR (1972) Spectral characteristics of surface-layer turbulence. Q J R Meteorol Soc 98(417): 563–589CrossRefGoogle Scholar
  46. King JC (1990) Some measurements of turbulence over an Antarctic shelf. Q J R Meteorol Soc 116: 379–400CrossRefGoogle Scholar
  47. Klipp CL, Mahrt L (2004) Flux–gradient relationship, self-correlation and intermittency in the stable boundary layer. Q J R Meteorol Soc 130(601): 2087–2103CrossRefGoogle Scholar
  48. Kondo J, Kanechika O, Yasuda N (1978) Heat and momentum transfers under strong stability in the atmospheric surface layer. J Atmos Sci 35(6): 1012–1021CrossRefGoogle Scholar
  49. Kouznetsov RD, Zilitinkevich SS (2010) On the velocity gradient in the stably stratified sheared flows. Part 2: Observations and models.. Boundary-Layer Meteorol 135(3): 513–517CrossRefGoogle Scholar
  50. Mahrt L (2007) The influence of nonstationarity on the turbulent flux–gradient relationship for stable stratification. Boundary-Layer Meteorol 125(2): 245–264CrossRefGoogle Scholar
  51. Mahrt L (2010a) Variability and maintenance of turbulence in the very stable boundary layer. Boundary-Layer Meteorol 135(1): 1–18CrossRefGoogle Scholar
  52. Mahrt L (2010b) Common microfronts and other solitary events in the nocturnal boundary layer. Q J R Meteorol Soc 136(652A): 1712–1722CrossRefGoogle Scholar
  53. Mahrt L (2011) The near-calm stable boundary layer. Boundary-Layer Meteorol 140(3): 343–360CrossRefGoogle Scholar
  54. Mahrt L, Sun J, Blumen W, Delany T, Oncley S (1998) Nocturnal boundary-layer regimes. Boundary-Layer Meteorol 88(2): 255–278CrossRefGoogle Scholar
  55. Mahrt L, Vickers D (2002) Contrasting vertical structures of nocturnal boundary layers. Boundary-Layer Meteorol 105(2): 351–363CrossRefGoogle Scholar
  56. Mahrt L, Vickers D (2006) Extremely weak mixing in stable conditions. Boundary-Layer Meteorol 119(1): 19–39CrossRefGoogle Scholar
  57. Mauritsen T, Svensson G (2007) Observations of stably stratified shear-driven atmospheric turbulence at low and high Richardson numbers. J Atmos Sci 64(2): 645–655CrossRefGoogle Scholar
  58. Mazellier N, Vassilicos JC (2010) Turbulence without Richardson–Kolmogorov cascade. Phys Fluids 22: 075101. doi: 10.1063/1.3453708 CrossRefGoogle Scholar
  59. Miles JW (1961) On the stability of heterogeneous shear flows. J Fluid Mech 10: 496–508CrossRefGoogle Scholar
  60. Mellor G (1973) Analytic prediction of the properties of stratified planetary surface layers. J Atmos Sci 30: 1061–1069CrossRefGoogle Scholar
  61. Monin AS, Obukhov AM (1954) Basic laws of turbulent mixing in the surface layer of the atmosphere. Trudy Geofiz Inst Acad Nauk SSSR 24(151): 163–187Google Scholar
  62. Monin AS, Yaglom AM (1971) Statistical fluid mechanics: mechanics of turbulence, vol 1. MIT, Cambridge, Mass 769 ppGoogle Scholar
  63. Nieuwstadt FTM (1984) The turbulent structure of the stable, nocturnal boundary layer. J Atmos Sci 41: 2202–2216CrossRefGoogle Scholar
  64. Nilsson ED (1996) Planetary boundary layer structure and air mass transport during the International Arctic Ocean Expedition 1991. Tellus B 48: 178–196. doi: 10.1034/j.1600-0889.1996.t01-1-00004.x CrossRefGoogle Scholar
  65. Obukhov AM (1946) Turbulence in an atmosphere with a non-uniform temperature. Trudy Inst Teoret Geophys Akad Nauk SSSR 1:95–115 (translation in: Boundary-Layer Meteorol 1971, 2:7–29)Google Scholar
  66. Ohya Y, Nakamura R, Uchida T (2008) Intermittent bursting of turbulence in a stable boundary layer with low-level jet. Boundary-Layer Meteorol 126(3): 349–363CrossRefGoogle Scholar
  67. Okamoto M, Webb EK (1970) The temperature fluctuations in stable stratification. Q J R Meteorol Soc 96(410): 591–600CrossRefGoogle Scholar
  68. Pahlow M, Parlange MB, Porté-Agel F (2001) On Monin–Obukhov similarity in the stable atmospheric boundary layer. Boundary-Layer Meteorol 99: 225–248CrossRefGoogle Scholar
  69. 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(C10): 8045. doi: 10.1029/2000JC000705 CrossRefGoogle Scholar
  70. Peters H, Baumert HZ (2007) Validating a turbulence closure against estuarine microstructure measurements. Ocean Modell 19(3–4): 183–203CrossRefGoogle Scholar
  71. Richardson LF (1920) The supply of energy from and to atmospheric eddies. Proc R Soc Lond A 97: 354–373CrossRefGoogle Scholar
  72. Rohr JJ, Itsweire EC, Helland KN, Van Atta CW (1988) Growth and decay of turbulence in a stably stratified shear flow. J Fluid Mech 195: 77–111CrossRefGoogle Scholar
  73. Scotti RS, Corcos GM (1972) An experiment on the stability of small disturbances in a stratified free shear layer. J Fluid Mech 52: 499–528CrossRefGoogle Scholar
  74. Smedman A-S (1988) Observations of a multi-level turbulence structure in a very stable atmospheric boundary layer. Boundary-Layer Meteorol 44: 231–253CrossRefGoogle Scholar
  75. Sorbjan Z (1986) On similarity in the atmospheric boundary layer. Boundary-Layer Meteorol 34: 377–397CrossRefGoogle Scholar
  76. Sorbjan Z (1988) Structure of the stably-stratified boundary layer during the SESAME-1979 experiment. Boundary-Layer Meteorol 44: 255–260CrossRefGoogle Scholar
  77. Sorbjan Z (1989) Structure of the atmospheric boundary layer. Prentice-Hall, Englewood Cliffs, 317 ppGoogle Scholar
  78. Sorbjan Z (2008) Gradient-based similarity in the atmospheric boundary layer. Acta Geophys 56(1): 220–233CrossRefGoogle Scholar
  79. Sorbjan Z (2010) Gradient-based scales and similarity laws in the stable boundary layer. Q J R Meteorol Soc 136(650A): 1243–1254Google Scholar
  80. Sorbjan Z, Grachev AA (2010) An evaluation of the flux–gradient relationship in the stable boundary layer. Boundary-Layer Meteorol 135(3): 385–405CrossRefGoogle Scholar
  81. Stull RB (1988) An Introduction to boundary-layer meteorology. Kluwer, Boston, 666 ppGoogle Scholar
  82. Townsend A (1958) Turbulent flow in a stably stratified atmosphere. J Fluid Mech 3: 361–372CrossRefGoogle Scholar
  83. Vande Wiel BJH, Moene AF, Steeneveld GJ, Hartogensis OK, Holtslag AAM (2007) Predicting the collapse of turbulence in stably stratified boundary layers. Flow Turbul Combust 79: 251–274. doi: 10.1007/s10494-007-9094-2 CrossRefGoogle Scholar
  84. Vickers D, Mahrt L (1997) Quality control and flux sampling problems for tower and aircraft data. J Atmos Oceanic Technol 14(3): 512–526CrossRefGoogle Scholar
  85. Vickers D, Mahrt L (2004) Evaluating formulations of stable boundary layer height. J Appl Meteorol 43(11): 1736–1749CrossRefGoogle Scholar
  86. Webb EK (1970) Profile relationships: the log-linear range, and extension to strong stability. Q J R Meteorol Soc 96: 67–90CrossRefGoogle Scholar
  87. Woods JD (1969) On Richardson’s number as a criterion for laminar-turbulent-laminar transition in the ocean and atmosphere. Radio Sci 4: 1289–1298CrossRefGoogle Scholar
  88. Wyngaard JC (1973) On surface-layer turbulence. In: Haugen DA (ed) Workshop on micrometeorology. American Meteorology Society, Boston, pp 101–149Google Scholar
  89. Wyngaard JC (2010) Turbulence in the atmosphere. Cambridge University Press, New York, 393 ppGoogle Scholar
  90. Wyngaard JC, Coté OR (1972) Cospectral similarity in the atmospheric surface layer. Q J R Meteorol Soc 98: 590–603CrossRefGoogle Scholar
  91. Yagüe C, Maqueda G, Rees JM (2001) Characteristics of turbulence in the lower atmosphere at Halley IV Station, Antarctica. Dyn Atmos Ocean 34: 205–223CrossRefGoogle Scholar
  92. Yagüe C, Viana S, Maqueda G, Redondo JM (2006) Influence of stability on the flux–profile relationships for wind speed, φm, and temperature, φh, for the stable atmospheric boundary layer. Nonlinear Process Geophys 13(2): 185–203CrossRefGoogle Scholar
  93. Yamada T (1975) The critical Richardson Number and the ratio of the eddy transport coefficients obtained from a turbulence closure model. J Atmos Sci 32: 926–933CrossRefGoogle Scholar
  94. Zilitinkevich SS, Chalikov DV (1968) Determining the universal wind-velocity and temperature profiles in the atmospheric boundary layer. Izvestiya, Acad. Sci. USSR. Atmos Oceanic Phys 4:165–170 (English Edition)Google Scholar
  95. Zilitinkevich S, Baklanov A (2002) Calculation of the height of the stable boundary layer in practical applications. Boundary-Layer Meteorol 105(3): 389–409CrossRefGoogle Scholar
  96. Zilitinkevich SS, Elperin T, Kleeorin N, Rogachevskii I (2007) Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows Part I: steady-state, homogeneous regimes. Boundary-Layer Meteorol 125(2): 167–191CrossRefGoogle Scholar
  97. Zilitinkevich SS, Esau I, Kleeorin N, Rogachevskii I, Kouznetsov RD (2010) On the velocity gradient in the stably stratified sheared flows. Part 1: asymptotic analysis and applications. Boundary-Layer Meteorol 135(3): 505–511CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Andrey A. Grachev
    • 1
  • Edgar L Andreas
    • 2
  • Christopher W. Fairall
    • 3
  • Peter S. Guest
    • 4
  • P. Ola G. Persson
    • 1
  1. 1.NOAA Earth System Research Laboratory/Cooperative Institute for Research in Environmental SciencesUniversity of ColoradoBoulderUSA
  2. 2.North West Research Associates, Inc.LebanonUSA
  3. 3.NOAA Earth System Research LaboratoryBoulderUSA
  4. 4.Naval Postgraduate SchoolMontereyUSA

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