Boundary-Layer Meteorology

, Volume 154, Issue 2, pp 171–187 | Cite as

Revisiting Surface Heat-Flux and Temperature Boundary Conditions in Models of Stably Stratified Boundary-Layer Flows

  • Jeremy A. Gibbs
  • Evgeni Fedorovich
  • Alan Shapiro


Two formulations of the surface thermal boundary condition commonly employed in numerical modelling of atmospheric stably stratified surface-layer flows are evaluated using analytical considerations and observational data from the Cabauw site in the Netherlands. The first condition is stated in terms of the surface heat flux and the second is stated in terms of the vertical potential temperature difference. The similarity relationships used to relate the flux and the difference are based on conventional log-linear expressions for vertical profiles of wind velocity and potential temperature. The heat-flux formulation results in two physically meaningful values for the friction velocity with no obvious criteria available to choose between solutions. Both solutions can be obtained numerically, which casts doubt on discarding one of the solutions as was previously suggested based on stability arguments. This solution ambiguity problem is identified as the key issue of the heat-flux condition formulation. In addition, the agreement between the temperature difference evaluated from similarity solutions and their measurement-derived counterparts from the Cabauw dataset appears to be very poor. Extra caution should be paid to the iterative procedures used in the model algorithms realizing the heat-flux condition as they could often provide only partial solutions for the friction velocity and associated temperature difference. Using temperature difference as the lower boundary condition bypasses the ambiguity problem and provides physically meaningful values of heat flux for a broader range of stability condition in terms of the flux Richardson number. However, the agreement between solutions and observations of the heat flux is again rather poor. In general, there is a great need for practicable similarity relationships capable of treating the vertical turbulent transport of momentum and heat under conditions of strong stratification in the surface layer.


Boundary conditions Flux-profile relationships Monin–Obukhov similarity Numerical models Stable boundary layer 



We acknowledge the Royal Netherlands Meteorological Institute (KNMI) and F. Bosveld (KNMI) for making the Cabauw data available.


  1. Abramowitz M, Stegun IA (1965) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York 1046 ppGoogle Scholar
  2. Arya SPS (1972) Comment on the paper by P. A. Taylor: ‘A note on the log-linear velocity profile in stable conditions’. Q J R Meteorol Soc 98:460–461CrossRefGoogle Scholar
  3. Baas P, Steeneveld GJ, van de Wiel BJH, Holtslag AAM (2006) Exploring self-correlation in fluxgradient relationships for stably stratified conditions. J Atmos Sci 63:3045–3054CrossRefGoogle Scholar
  4. Basu S, Moene AF, Holtslag AAM, Steeneveld G-J, van de Wiel BJH (2008) An inconvenient “truth” about using sensible heat flux as a surface boundary condition in models under stably stratified regimes. Acta Geophys 56(1):88–99CrossRefGoogle Scholar
  5. Bosveld FC (2012) Cabauw observational program on landsurface-atmosphere interaction (2000-today). Online at Accessed 18 Feb 2014
  6. Businger JA, Wyngaard JC, Izumi Y, Bradley EF (1971) Flux-profile relationships in the atmospheric surface layer. J Atmos Sci 28:181–189CrossRefGoogle Scholar
  7. Garratt JR (1992) The atmospheric boundary layer. Cambridge University Press, Cambridge 316 ppGoogle Scholar
  8. Hicks BB (1976) Wind profile relationships from the Wangara experiment. Q J R Meteorol Soc 102:535–551Google Scholar
  9. Russchenberg H, Bosveld F, Swart D, ten Brink H, de Leeuw G, Uijlenhoet R, Arbesser-Rastburg B, van der Marel H, Ligthart LP, Boers R, Apituley A (2005) Ground-based atmospheric remote sensing in the Netherlands: European outlook. IEICE Trans Commun 88-B(6):2252–2258Google Scholar
  10. Sorbjan Z (1989) Structure of the atmospheric boundary layer. Prentice Hall, New Jersey 317 ppGoogle Scholar
  11. Taylor PA (1971) A note on the log-linear velocity profile in stable conditions. Q J R Meteorol Soc 97:326–329CrossRefGoogle Scholar
  12. van de Wiel BJH, Moene AF, Steeneveld G-J, Hartogensis OK, Holtslag AAM (2007) Predicting the collapse of turbulence in stably stratified boundary layers. Flow Turbul Combust 79(3):251–274CrossRefGoogle Scholar
  13. Webb EK (1970) Profile relationships: the log-linear range, and extension to strong stability. Q J R Meteorol Soc 96:67–90CrossRefGoogle Scholar
  14. Zilitinkevich S, Esau I (2007) Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layer. In: Baklanov A, Grisogono B (eds) Atmospheric boundary layers. Springer, New York, pp 37–49Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Jeremy A. Gibbs
    • 1
  • Evgeni Fedorovich
    • 1
  • Alan Shapiro
    • 1
  1. 1.School of MeteorologyUniversity of OklahomaNormanUSA

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