Revisiting the Turbulent Prandtl Number in an Idealized Atmospheric Surface Layer

Cospectral budgets are used to link the kinetic and potential energy distributions of turbulent eddies, as measured by their spectra, to macroscopic relations between the turbulent Prandtl number (Prt) and atmospheric stability measures such as the stability parameter ζ, the gradient Richardson number Rg, or the flux Richardson number Rf in the atmospheric surface layer. The dependence of Prt on ζ, Rg, or Rf is shown to be primarily controlled by the ratio of Kolmogorov and Kolmogorov–Obukhov–Corrsin phenomenological constants and a constant associated with isotropization of turbulent flux production that can be independently determined using rapid distortion theory in homogeneous turbulence. Changes in scaling laws of the vertical velocity and air temperature spectra are also shown to affect the Prt–ζ (or Prt–Rg or Prt–Rf) relation. Results suggest that departure of Prt from unity under neutral conditions is induced by dissimilarity between momentum and heat in terms of Rotta constants, isotropization constants, and constants in the flux transfer terms. A maximum flux Richardson number Rfm predicted from the cospectral budgets method (=0.25) is in good agreement with values in the literature, suggesting that Rfm may be tied to the collapse of Kolmogorov spectra instead of laminarization of turbulent flows under stable stratification. The linkages between microscale energy distributions of turbulent eddies and macroscopic relations that are principally determined by dimensional considerations or similarity theories suggest that when these scalewise energy distributions of eddies experience a “transition” to other distributions (e.g., when Rf is increased over Rfm), dimensional considerations or similarity theories may fail to predict bulk flow properties.