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Reduced-Order Modelling of Dispersion 247
2.1 Near-Field Solution
The first step is to model the solution at the outlet of the injection device used
to expand the phyto treatment in between rows. One important hypothesis
is to assume two different time scales based on the injection velocity and
the velocity at which the injection source moves. The injection velocity being
much higher, one assumes the local concentration at the outlet of the injection
device to be established instantaneously. This instantaneous local flow field
is devoted to vanish immediately and not to affect the overall atmospheric
circulation. This injection velocity is only designed to determine the part of
the pollutant leaving near-ground area and being candidate for transport over
large distances (see Section 2.2). These are strong hypotheses which seriously
reduce the search space for the solution.
One considers a cylindrical local reference frame where z indicates the
motion direction for the vehicle in the field. One looks for local injection
solutions of the form:
u l ∼ f 1 (r)g 1 (θ)(zh 1 (z)+(1− h 1 (z))r) and c l ∼ f 2 (r)g 2 (θ)h 2 (z), (1)
where the subscript l reads for local. r is a unit vector having its origin at the
injection point and visiting the unit circle around this point in the plan perpendicular
to z. This defines an instantaneous flow field around the injection
point. c l denotes the local distribution of a passive scalar. f i (r), i =1, 2, are
solutions of a control problem for the assimilation of experimental data by a
PDE based model obtained by dimension reduction of the Navier–Stokes and
transport equations [Fin00, RT81, Sum71, Bru06]. These experimental data
show that after injection both the flow velocity and phyto products concentration
drop to nearly zero after three rows of vegetation. g i (θ), i =1, 2, are
Gauss distributions describing the characteristics of the injection device and
are provided by the manufacturer. h i (z), i =1, 2, include the characteristics
of the vegetation by assimilation of experimental data and inform on how
the density of the vegetation deviates the flow horizontally. h 1 (z) ∈ [−1, 1] is
an erf function, odd and monotonic increasing, and h 2 (z) ∈ [0, 1] is a Gauss
distribution.
At this level, one includes compatibility conditions coming from the governing
equations. In particular, one aims for the conservation condition to
hold for the concentration of the passive scalar, the flow field to be divergence
free and both variables to verify an advection equation:
∫
∇·u l = u l ·∇c l =0 c l dv =given. (2)
R 3
To summarize, the coefficients in functions f i , g i , h i , i =1, 2, are a solution
of an assimilation problem for experimental data under the constraint (2)
[Bru06].
From now, one expresses the variables in a global Cartesian reference frame
where z denotes the vertical axis.