Logarithmic scaling in the longitudinal velocity variance explained by a spectral budget in a neutral and unstable atmosphere

A logarithmic scaling for the streamwise turbulent intensity $\sigma_u^2/{{u_*}^2}=B_1-A_1\,\ln\left({z}/{\delta}\right)$ was reported across several high Reynolds number laboratory experiments as predicted from Townsend's attached eddy hypothesis, where $u_*$ is the friction velocity and $z$ is the height normalized by the boundary layer thickness $\delta$. A phenomenological explanation for the origin of this log-law in the intermediate region is provided here based on a solution to a spectral budget where the production and energy transfer terms are modeled. The solution to this spectral budget predicts $A_1=C_o/{\kappa^{2/3}}$ and $B_1=(3/2) A_1$, where $C_o$ and $\kappa$ are the Kolmogorov and von K\'arm\'an constants. The spectral budget approach is then extended to explore the scaling behavior of $\sigma_u/{{u_*}}$ in the unstably stratified atmosphere. It is demonstrated with support from recent datasets that $\sigma_u/{{u_*}}$ does not only depend on $\delta/L$ but also depends on the atmospheric stability parameter $\zeta=z/L$. Thus, the proposed spectral budget shows how Townsend's attached eddy hypothesis, the $k^{-1}$ spectral law in low wavenumbers and the similarity arguments for a stratified atmosphere are all interconnected.