Partial Differential Equations - Modelling and ... - ResearchGate

250 J.-M. Brun and B. Mohammadi
one would like for the transport. As an example, the flow will be described
probably by less than one point by several square kilometers. We consider
the near to ground flow field built from observation data as solution of the
following system:
u = ∇φ, −∆φ = ∑
i=1,..,n obs
‖∇φ(x i ) − u obs (x i )‖, (5)
where φ is a scalar potential and n obs the number of observation points. The
observations are close to the ground at z = H and this construction gives a
map of the flow near the ground. This is completed in the vertical direction
using generalized wall functions for turbulent flows [MP94, MP06]:
(u · τ ) + =(u · τ )/u τ = f(z + )=f(zu τ /ν),
where τ = u H /‖u H ‖ is the local tangent unit vector to the ground in the
direction of the flow and we assume (u · n(z = H) =0)ifn is the normal
to the ground. This is a non-linear equation giving u τ , the friction velocity,
knowing (u · τ ) H and is used, in turn, to define the horizontal velocity u · τ =
u τ f(z + )forz>H. This construction gives two components of the flow and
the divergence free condition implies the third component is constant and,
therefore, it vanishes as it is supposed zero at z = H. This construction can
be improved but we find it sufficient for the level of accuracy required. In
presence of ground variations, the flow is locally rotated to remain parallel to
the ground (see also Section 2.2 for ground variation modelling).
Calculation of Migration Times
As we said, our approach aims to provide the solution at a given point without
calculating the whole solution. Being in point B, one needs an estimation
of the migration time from the source in A to B using the construction in
Section 2.2.
We avoid the construction of characteristics using an iterative polynomial
definition for a characteristic s(t) =(x(t),y(t),z(t)), t ∈ [0, 1], starting from
a third-order polynomial function verifying for each coordinate:
P n (0) = x A , P n (1) = x B , P ′ n(0) = u 1 A, P ′ n(1) = u 1 B (same for y and z).
If P ′ n(ζ) ≠ u 1 (x = P n (ζ)) this new point should be assimilated by the construction
increasing by one the polynomial order. ζ ∈]0, 1[ is chosen randomly.
The migration time is computed over this polynomial approximation of
the characteristic. Here we make the approximation B ⊥ = B which means
the characteristic passing by A passes exactly by B which is unlikely. In a
uniform flow, this means we suppose the angle between the central axis and
AB is small (cosine near 1). One introduces, therefore, a correction factor of
2/3 =0.636 on the calculated times. This is the stochastic averaged cosine