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

A spectral budget model is developed to describe the scaling behavior of the longitudinal turbulent velocity variance sigma(2)(u) with the stability parameter zeta = z/L and the normalized height z/delta in an idealized stably stratified atmospheric surface layer (ASL), where z is the height from the surface, L is the Obukhov length, and delta 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 zeta. At a given friction velocity u(*), sigma(u) reduces with increasing zeta as expected. The model is consistent with the disputed z-less stratification when the stability correction function for momentum increases with increasing zeta linearly or as a power law with the exponent exceeding unity. For the Businger-Dyer stability correction function for momentum, which varies linearly with zeta, the limit of the z-less onset is zeta approximate to 2. The proposed framework explains why sigma(u) does not follow MOST scaling even when the mean velocity and temperature profiles may follow MOST in the ASL. It also explains how delta ceases to be a scaling variable in more strongly stable (although well-developed turbulent) ranges.