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A Spectral Budget Model for the Longitudinal Turbulent Velocity in the Stable Atmospheric Surface Layer

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  • 1 Nicholas School of the Environment, Duke University, Durham, North Carolina
  • | 2 Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey
  • | 3 Department of Geography, National Taiwan University, Taipei, Taiwan
  • | 4 Nicholas School of the Environment, and Department of Civil and Environmental Engineering, Duke University, Durham, North Carolina
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Abstract

A spectral budget model is developed to describe the scaling behavior of the longitudinal turbulent velocity variance with the stability parameter and the normalized height in an idealized stably stratified atmospheric surface layer (ASL), where z is the height from the surface, L is the Obukhov length, and δ is the boundary layer height. The proposed framework employs Kolmogorov’s hypothesis for describing the shape of the longitudinal velocity spectra in the inertial subrange, Heisenberg’s eddy viscosity as a closure for the pressure redistribution and turbulent transfer terms, and the Monin–Obukhov similarity theory (MOST) scaling for linking the mean longitudinal velocity and temperature profiles to ζ. At a given friction velocity , reduces with increasing ζ as expected. The model is consistent with the disputed z-less stratification when the stability correction function for momentum increases with increasing ζ linearly or as a power law with the exponent exceeding unity. For the Businger–Dyer stability correction function for momentum, which varies linearly with ζ, the limit of the z-less onset is . The proposed framework explains why does not follow MOST scaling even when the mean velocity and temperature profiles may follow MOST in the ASL. It also explains how δ ceases to be a scaling variable in more strongly stable (although well-developed turbulent) ranges.

Current affiliation: Atmospheric Environmental Research, Institute of Meteorology and Climate Research, Karlsruhe Institute of Technology, Garmisch-Partenkirchen, Germany.

Corresponding author address: Tirtha Banerjee, IMK-IFU, Karlsruhe Institute of Technology, Kreuzeckbahnstr. 19, 82467 Garmisch-Partenkirchen, Germany.E-mail: tirtha.banerjee@kit.edu

Abstract

A spectral budget model is developed to describe the scaling behavior of the longitudinal turbulent velocity variance with the stability parameter and the normalized height in an idealized stably stratified atmospheric surface layer (ASL), where z is the height from the surface, L is the Obukhov length, and δ is the boundary layer height. The proposed framework employs Kolmogorov’s hypothesis for describing the shape of the longitudinal velocity spectra in the inertial subrange, Heisenberg’s eddy viscosity as a closure for the pressure redistribution and turbulent transfer terms, and the Monin–Obukhov similarity theory (MOST) scaling for linking the mean longitudinal velocity and temperature profiles to ζ. At a given friction velocity , reduces with increasing ζ as expected. The model is consistent with the disputed z-less stratification when the stability correction function for momentum increases with increasing ζ linearly or as a power law with the exponent exceeding unity. For the Businger–Dyer stability correction function for momentum, which varies linearly with ζ, the limit of the z-less onset is . The proposed framework explains why does not follow MOST scaling even when the mean velocity and temperature profiles may follow MOST in the ASL. It also explains how δ ceases to be a scaling variable in more strongly stable (although well-developed turbulent) ranges.

Current affiliation: Atmospheric Environmental Research, Institute of Meteorology and Climate Research, Karlsruhe Institute of Technology, Garmisch-Partenkirchen, Germany.

Corresponding author address: Tirtha Banerjee, IMK-IFU, Karlsruhe Institute of Technology, Kreuzeckbahnstr. 19, 82467 Garmisch-Partenkirchen, Germany.E-mail: tirtha.banerjee@kit.edu
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