Reply to the Comments by J. C. Bergmann on the Paper: “Local Similarity in the Stable Boundary Layer and Mixing Length Approaches: Consistency of Concepts”

Abstract

In his comment J. C. Bergmann discusses the analysis of Van de Wiel et al. (2008; VDW08), acknowledges the conceptual merits of the work related to the introduced “apparent wall concept”, but identifies a factor of two error in the potential energy calculation. As a consequence (see Bergmann 2011; B11), the derived value (equal to 6.25) of the numerical constant (slope) in the Monin–Obukhov similarity function, respectively in its local-scaling equivalent (Nieuwstadt 1984), is reduced to half its value. The present authors fully agree with this re-analysis by B11, but wish to emphasise that the “apparent wall concept” is used to anticipate a plausible order of magnitude of the abovementioned constant, rather than predicting an exact value. In the following, we briefly reanalyse the comment of B11. Consider a linear density profile (the very stable limit considered in VDW08). On average, parcels at a reference level z′ = 0 with density ρ0 will reach their ‘maximum’ displacement zB when all their vertical kinetic energy is converted into potential energy (see Fig. 1). B11 refers to zB “as the root-mean-square displacement”. Thus we can find this length scale by computing the maximum work (in the mean sense: B11) that can be done by parcels, given their mean initial vertical kinetic energy (1/2)ρσ 2 w (per volume):