On the dispersion of trace species in the atmospheric boundary layer: a re-formulation of the governing equations for the turbulent flow of the compressible atmosphere

Since especially the parameterisation of the vertical dispersion of trace species in the atmosphericboundary layer has controversially been discussed in the literature, the 1st-order balance equationsfor matter, momentum, and various energy forms were re-formulated with Hesselberg’sdensity-weighted averaging calculus to point out that this problem arises from averaging themacroscopic balance equations of matter, momentum and various energy forms in the sense ofReynolds, rather than from the parameterisation of the vertical dispersion by 1st-order closureprinciples, as this discussion seems to reflect. Results of the SANA field experiment “Eisd”presented here substantiate that in the case of chemically reactive trace constituents segregationeffects owing to turbulence cannot generally be neglected as usually performed in Eulerian airpollution models. Modelling such segregation effects, however, requires, at least, 2nd-orderclosureprinciples. Therefore, the 2nd-order balance equations for 2nd moments like the eddyflux densities of matter and momentum as well as covariances of scalar quantities were alsore-formulated by considering Hesselberg’s averaging procedure. This re-formulated set of governing1st-order and 2nd-order balance equations may be considered as most exact becausethe degree of simplification is reduced to a minimum. To distinguish between the Boussinesqapproximated equation set for the turbulent atmospheric flow, denoted as Boussinesq fluid, and our re-formulated one, the turbulent flow of the compressible atmosphere for which there-formulated governing balance equations are valid may be denoted as Hesselberg fluid. It isargued that averaging in the sense of Hesselberg reduces the risk to misinterpret turbulentatmospheric processes to a minimum. As exemplary shown on the basis of the balance equationsfor dry air, water vapour, and trace species, the so-called Webb correction will become insignificantif Hesselberg’s averaging calculus is considered. Based on the results obtained from the “Eisdorf” experiment and from sensitivity studies with a Seinfeld-type kinetic mechanism forphotochemical smog, it is argued that an evaluation and improvement of Eulerian air pollutionmodels require directly measured 2nd-order moments. Since the number of fast-responsephysico-chemical analysers for chemically reactive trace constituents is strongly limited, suchfast-response sensors have to be (further) developed to set-up a true platform for model evaluationthat implies not only a comparison of calculated and observed distributions of 1stmoments (necessary condition), but also a comparison of the calculated and observed distributionsof 2nd moments (sufficient condition).