Double-averaging methodology and its application to turbulent flow in and above vegetation canopies

Double averaged equations for atmospheric boundary layer flows are introduced as natural extensions of single averaged Reynolds equations. We show that in circumstances where double averaged equations are needed, the two fundamental properties of Reynolds averaging are violated. First, we consider double-averaging in free air turbulence, where the aim is to separate coherent motions from background turbulence. We illustrate the different properties of the main operators that have been used and the physical meaning of the terms that result. Second, in canopy flows, the multiply connected nature of the canopy airspace leads to a different set of departures from the standard Reynolds equations. We establish the physical meaning of the extra terms that arise. Finally we briefly discuss the problems, both practical and theoretical, that arise when we use double averaged equations to interpret real data.