The footprint concept in complex terrain

The use of footprint functions in complex flows has been questioned because of anomalous behaviour reported in recent model studies. We show that the concentration footprint can be identified with the Greenʹs function of the scalar concentration equation or the transition probability of a Lagrangian formulation of the same equation and so is well behaved and bounded by 0 and 1 in both simple and complex flows. The flux footprint in contrast is not a Greenʹs function but a functional of the concentration footprint and is not guaranteed to be similarly well behaved. We show that in simple, homogeneous shear flows, the flux footprint, defined as the vertical eddy flux induced by a unit point source, is bounded by 0 and 1 but that this is not true in more general flow fields. Analysis of recent model studies also shows that the negative flux footprints reported in homogeneous plant canopy flows are an artefact of reducing a canopy with a complex source–sink distribution in the vertical to a single layer but that in canopies on hilly topography, the problems are more fundamental. Finally we compare footprint inversion with the direct mass-balance method of measuring surface exchange. We conclude first that the flux footprint is an appropriate measure of the area influencing both eddy and advective fluxes on a tower but that the concentration footprint is the correct measure when the storage term is important. Second, we deduce that there are serious obstacles to inverting flux footprints in complex terrain.