Fluxes by eddy correlation over heterogeneous landscape: How shall we apply the Reynolds average?

Top-down estimates of carbon exchange across the earth's surface are implicitly an integral scheme, deriving bulk exchanges over large areas. Bottom-up estimates explicitly integrate the individual components of exchange to derive a bulk value. If these approaches are to be properly compared, their estimates should represent the same quantity. Over heterogeneous landscape, eddy-covariance flux computations from towers or aircraft intended for comparison with top-down approach face a question of the proper definition of the mean or base state, the departures from which yield the fluxes by Reynolds averaging. 1)≠Use a global base state derived over a representative sample of the surface, insensitive to land use. The departure quantities then fail to sum to zero over any subsample representing an individual surface type, violating Reynolds criteria. Yet fluxes derived from such subsamples can be directly composed into a bulk flux, globally satisfying Reynolds criteria. 2)≠Use a different base state for each surface type. satisfying Reynolds criteria individually. Then some of the flux may get missed if a surface's characteristics significantly bias its base state. Base state≠(2) is natural for tower samples. Base state≠(1) is natural for airborne samples over heterogeneous landscape, especially in patches smaller than an appropriate averaging length. It appears (1) incorporates a more realistic sample of the flux, though desirably there would be no practical difference between the two schemes. The schemes are related by the expression

w¯*a*)C - w¯'a¯')C = w¯'ã¯)C+ wtilde ¯a¯')C+ wtilde ¯ã¯)C

Here w is vertical motion, and a is some scalar, such as CO2. The star denotes departure from the global base state≠(1), and the prime from the base state≠(2), defined only over surface class≠C. The overbar with round bracket denotes average over samples drawn from class≠C, determined by footprint model. Thus a¯')C = 0 but a¯*)C ≠ 0 in general. The tilde denotes the departure of base-state≠(2) from base-state≠(1). It represents surface≠C's characteristic bias. The equation is defined only over class≠C. A similar equation applies to each surface class. The first and second righthand terms express interaction of the departure quantities with surface≠C's characteristic bias. These terms are zero if the base states are simple means. The third term becomes important if class C has a significant bias both in vertical motion and in its characteristic values of a. A practical example from 2005 June 18 at 1015 LST in Illinois is illustrative. Turbulence measurements were made by aircraft at 20≠m above ground along a 50≠km track approximately evenly divided between corn and soybean. Corn (type≠C) was growing quickly, increasing the mixing ratio of moisture (r) and reducing that of CO2 (a), relative to soybean. Soybean characteristically heated the air and favored updrafts. These biases were evident in r¯*)C, a¯*)C, θ¯*)C, and w¯*)C relative to their corresponding averages over soybean. In particular the bias in CO2 mixing ratio, negative over corn and positive over soybean, was about 20% of the standard deviation of a*. Nevertheless, neither surface type strongly favored vertical motion, giving the encouraging result that the two approaches do not differ by more than an insignificant few per cent. The theoretical analysis indicates care, however, where extensive areas of both bare soil and vegetated land may enhance the bias in vertical motion between different components of the landscape.