Marc Aubinet. 2008. Eddy covariance CO₂ flux measurements in nocturnal conditions: an analysis of the problem. Ecological Applications 18:1368–1378.

Appendix A. Features and development of the model for convergence flows.

In Aubinet et al.(2005), we developed a simple model that described the horizontal evolution of tracer concentration in the case of a 2D drainage flow in a convergence situation. The model was based on the tracer conservation and on the continuity equations and assumed similarity of the vertical profiles of horizontal velocity and of tracer concentration, constant vertical velocity and constant tracer source intensity. After integration, they produced the following solution:

(A.1)

Where _r represents the tracer concentration difference between control volume top and a reference height in the control volume, x represents the distance along the slope, u_r represent the horizontal velocity taken at the reference height, the o index refers to the spatial origin (x = 0), w_h is the vertical velocity at the control volume top, F_s is the source intensity and I₁ and I₂ are coefficients that depend on the u and vertical profile shapes. For the computation details, refer to Aubinet et al. (2005).

This equation showed that the tracer concentration difference evolves from its starting value _ro to an asymptotical value given by:

(A.2)

This means that at a long distance from the edge, the tracer concentration difference (and thus the tracer concentration at reference height, if the tracer concentration at the control volume top is considered constant) tends towards a value that depends only on the source intensity and on the wind flow pattern. At shorter distances from the edge, the evolution and thus the horizontal advection sign can be predicted by deriving (A.1) according to x:

With the same notations, the horizontal advection writes:

(A.3)

The derivation of (A.1) and its introduction in (A.3) gives:

(A.4)

As w_h is negative, it is clear that the sign of V_h depends on the relative importance of the two terms of the numerator, namely, V_h is positive (negative) if is bigger (smaller) than _ro. If we consider that _ro corresponds to an asymptotic value that was reached before the air enters the control volume and if we also assume that w_h doesn’t change along the slope, i.e. :

(A.5)

we can conclude that the horizontal advection sign depends on the horizontal evolution of the source intensity: if it increases (decreases) with x, the horizontal advection will be positive (negative) but if it remains constant, the horizontal advection will be zero.

LITERATURE CITED

Aubinet, M., Berbigier, P., Bernhofer, Ch., Cescatti, A., Feigenwinter, C., Granier, A., Grünwald, Th., Havrankova, K., Heinesch, B., Longdoz, B., Marcolla, B., Montagnani, L., Sedlak, P., 2005. Comparing CO₂ storage and advection conditions at night at different CARBOEUROFLUX sites. Boundary-Layer Meteorol. 116:63–93.