Semi-analytical Footprint Model Compliant with Arbitrary Atmospheric Stratification: Application to Monin-Obukhov Profiles

A new model is proposed for the so-called scalar footprint and flux footprint in the atmospheric boundary layer. The underlying semi-analytical model allows computing the scalar concentration and flux fields related to turbulent diffusion of heat, water-vapor or to the dispersion of any scalar (e.g. passive pollutant) in the framework of K-theory. It offers improved capabilities regarding the representation of the gradual stratification in the boundary layer. In this model, the boundary layer is split in a series of sublayers in which the aerodynamic inertivity (a compound parameter aggregating wind-speed and eddy-diffusivity) is approximated by a sum of two power-law functions of a new vertical scale corresponding to the height-dependent downwind extension of the plume. This multilayer approach allows fitting with vanishing error any boundary-layer stratification, in particular those described by the Monin-Obukhov similarity theory (MOST) in the surface layer, while keeping the computation time of the footprint to low values. As a complement, a fully analytical surrogate model is presented for practical applications. For MOST profiles, the flux (resp. concentration) footprint is, to a RMS difference less than 1% (resp. 1.2%), equal (resp. equal to a constant multiplicative factor) to the inverse Gamma distribution. The optimal parameters of this distribution were evaluated for a broad range of atmospheric conditions and height. Regression formulas were also provided to compute the crosswind-integrated flux footprint distribution easily and with less than 1.6% RMS residual error. A comparison with the well-known footprint approximate model by Kormann and Meixner and the one by Hsieh, Katul and Chi has allowed quantifying their performances and limitations.