Backward-Eulerian Footprint Modelling Based on the Adjoint Equation for Atmospheric and Urban-Terrain Dispersion

This study developed a backward-Eulerian footprint modelling method based on an adjoint equation for atmospheric boundary-layer flows. In the proposed method, the concentration footprint can be obtained directly by numerical simulation with the adjoint equation, and the flux footprints can be estimated using the adjoint concentration based on the gradient diffusion hypothesis. We first tested the proposed method by estimating the footprints for an ideal three-dimensional boundary layer with different atmospheric stability conditions based on the Monin–Obukhov profiles. It was indicated that the results were similar to the FFP method (Kljun et al. in Boundary-Layer Meteorol 112:503–523, 2004, 10.1023/B:BOUN.0000030653.71031.96; Geosci Model Dev 8:3695–3713, 2015, 10.5194/gmd-8-3695-2015) for convective conditions and the K–M method (Kormann and Meixner in Boundary-Layer Meteorol 99:207–224, 2001, 10.1023/A:1018991015119) for stable conditions. The proposed method was then coupled with the Reynolds averaged Navier–Stokes model to calculate the footprints for a block-arrayed urban canopy. The results were qualitatively compared to the results from the Lagrangian-Large-Eddy-Simulation (LL) method (Hellsten et al. in Boundary-Layer Meteorol 157:191–217, 2015, 10.1007/s10546-015-0062-4). It was shown that the proposed method reproduced the main features of footprints for different sensor positions and measurement heights. However, it is necessary to simulate the adjoint equation with a more sophisticated turbulence model in the future to better capture turbulent effects in the footprint modelling.