Boundary-Layer Meteorology

, Volume 156, Issue 1, pp 73–89 | Cite as

Approximate Analytical Solution to Diurnal Atmospheric Boundary-Layer Growth Under Well-Watered Conditions

  • J. R. Rigby
  • Jun Yin
  • John D. Albertson
  • Amilcare Porporato


Simplified numerical models of the atmospheric boundary layer (ABL) are useful both for understanding the underlying dynamics and potentially providing parsimonious modelling approaches for inclusion in larger models. Herein the governing equations of a simplified slab model of the uniformly mixed, purely convective, diurnal ABL are shown to allow immediate solutions for the potential temperature and specific humidity as functions of the ABL height and net radiation when expressed in integral form. By employing a linearized saturation vapour relation, the height of the mixed layer is shown to obey a non-linear ordinary differential equation with quadratic dependence on ABL height. A perturbation solution provides general analytical approximations, of which the leading term is shown to represent the contribution under equilibrium evaporation. These solutions allow the diurnal evolution of the height, potential temperature, and specific humidity (i.e., also vapour pressure deficit) of the mixed layer to be expressed analytically for arbitrary radiative forcing functions.


Analytical solution Atmospheric boundary layer Bowen ratio Equilibrium evaporation Evaporative fraction  Mixed-layer model Perturbation theory 



We gratefully acknowledge funding from National Aeronautics and Space Administration (NASA grant NNX09AN76G), National Science Foundation (NSF-CBET-1033467, NSF-EAR-0838301, NSF-EAR-1331846, and NSF-EAR-1316258, NSF FESD 1338694), the US DOE through the Office of Biological and Environmental Research, Terrestrial Carbon Processes program (DE-SC0006967), as well as the Agriculture and Food Research Initiative from the USDA National Institute of Food and Agriculture (201167003-30222). Data were obtained from the Atmospheric Radiation Measurement Program sponsored by the U.S. Department of Energy. The comments and useful criticisms of anonymous reviewers are gratefully acknowledged.


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Copyright information

© Springer Science+Business Media Dordrecht (outside the USA) 2015

Authors and Affiliations

  • J. R. Rigby
    • 1
  • Jun Yin
    • 2
  • John D. Albertson
    • 2
  • Amilcare Porporato
    • 2
  1. 1.USDA-ARS National Sedimentation LaboratoryOxfordUSA
  2. 2.Department of Civil and Environmental EngineeringDuke UniversityDurhamUSA

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