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

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

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7 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)73-89
Number of pages17
JournalBoundary-Layer Meteorology
Issue number1
StatePublished - Jul 29 2015
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Atmospheric Science


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


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