Stationarity, Homogeneity, and Ergodicity in Canopy Turbulence

One of the defining syndromes of turbulence is nonlinear stochasticity. This view of turbulence motivated the development of statistical me- chanics theories that have served to connect the basic Navier-Stokes (NS) equations of motion to the statistical results of numerous field experiments. In general, the proper averaging operator for stochastic processes is ensemble averaging. Given the transient nature of flow boundary conditions in natural systems, field experiments are typically unable to capture a suitable ensemble, in a strict sense. Instead, field experiments typically focus on time averaged statistics. Stationarity and ergodicity are two central concepts (required conditions) used to link field measurements and the NS equations or field measurements to "boundary conditions" at the land-atmosphere interface. In this Chap- ter, we present an elementary review of these two concepts for the at- mospheric surface layer (ASL) and canopy sublayer (CSL) and proceed to show why the stable CSL tends to violate both conditions. A weaker form of these two conditions may be applicable to CSL flows that are only moderately stably stratified. Practical implications for nighttime CO2 flux corrections are also discussed.