R. Leuning et al. Agricultural and Forest Meteorology 104 2000 233–249 235
above the canopy as reported in a companion paper by Miyata et al. 2000.
2. Theory
2.1. Localised near-field dispersion The scalar concentration field within a canopy can
be viewed as the linear superposition of contributions from all sources. Raupach 1987, 1989a,b divided the
concentration field into a ‘near-field’ part, C
n
z, dom- inated by the contribution of nearby sources, and a
‘far-field’ part, C
f
z, resulting from the contribution of distant sources. The near-field is defined as the dis-
tance from the source that fluid particles move in a time τ
L
, the characteristic Lagrangian time scale of the turbulence, whereas the far-field is the distance
from the source a fluid particle travels in times ≫τ
L
. Atmospheric turbulence causes transport of the
scalar from the source to the observation point. In the near-field, transport is dominated by coherent eddies,
while in the far-field transport is essentially diffu- sive. The LNF theory of Raupach 1987 assumes:
1 that the canopy is horizontally homogeneous at each height so that net transport occurs only in the
vertical direction; 2 that near-field transport can be described as if it occurred in homogeneous tur-
bulence with standard deviation of vertical velocity σ
w
z and Lagrangian time scale τ
L
z equal to that at the source height; and 3 the contribution to the
concentration field from distant far-field sources obeys the diffusion equation F
f
z=−K
f
z dC
f
zdz. The concentration field at any height is the average
across an area extending over many canopy elements Kaimal and Finnigan, 1994.
In the discrete form of the LNF analysis, the canopy is divided into m horizontally homogeneous layers
with thickness 1z
j
and source strength S
j
. By sum- ming the emissions from all source layers j j=1, m,
the concentration C
i
at level i i=1, n is given by C
i
− C
R
=
m
X
j = 1
D
ij
S
j
1z
j
1 where C
i
= C
ni
+ C
fi
, the sum of near-field and far-field contributions, and C
R
is the concentration at a refer- ence height above the canopy. The coefficients D
ij
of the dispersion matrix, D
D D
, describe the turbulent trans- port within and above the canopy. Once the D
ij
are known, the set of linear equations Eq. 1 can be in-
verted to solve for the source profile S
j
. The LNF the- ory, which solves the forward problem of predicting
concentration profiles from known profiles of sources and turbulence within the canopy, is used to calculate
the elements of the matrix D D
D as follows.
The near-field concentration profile is determined by the vertical distribution and strength of the sources,
and the vertical profiles of σ
w
and τ
L
. Designating the height of a given source as z
s
, the near-field con- centration profile is given by
C
n
z= Z
∞
Sz
s
σ
ws
k
n
z − z
s
σ
ws
τ
Ls
+ k
n
z + z
s
σ
ws
τ
Ls
dz
s
2 where σ
ws
= σ
w
z
s
, τ
Ls
= τ
L
z
s
and k
n
is a ‘near-field kernel’ whose analytical approximation is given by
Raupach 1989a as k
n
ζ = − 0.3989 ln1 − e
−| ζ |
− 0.1562 e
−| ζ |
3 An image source and turbulent flow are introduced
into Eq. 2 to ensure a zero-flux boundary condition at the ground for the scalar Raupach, 1989a.
The far-field concentration profile obeys the gradient-diffusion relationship
F z = − K
f
z dC
f
z dz
4 where K
f
= σ
2 w
zτ
L
z and where the source strength
is related to the flux at height z by F z = F
0 + Z
z
Sz dz
5 By integrating Eq. 4, C
f
is found to be C
f
z − C
f
z
R
= Z
z
R
z
F z K
f
z dz
6 in which
C
f
z
R
= Cz
R
− C
n
z
R
7 Now consider a scalar which is released uniformly in
layer j with source strength, s, but with zero strength in all other layers. The resulting partial concentration
profile c
i
defines the elements of the dispersion matrix
236 R. Leuning et al. Agricultural and Forest Meteorology 104 2000 233–249
for dispersion from layer j to concentration at height i
, i.e. D
ij
= c
i
− c
R
s 1z
j
8 Each element of D
D D
has a near-field and a far-field component because c
i
= c
ni
+ c
fi
. We assume a unit source strength in each layer and use Eqs. 2–6 to
estimate c
ni
and c
fi
, and hence, the coefficients D
ij
from Eq. 8. We can simplify the procedure considerably if we
assume that the turbulent structure and hence the D
ij
scales with u
∗
. In this case, once the scalar sam- pling heights and the normalised profiles for σ
w
z and τ
L
z have been adopted see Fig. 1, we need only calculate the dispersion coefficients once us-
ing the above integration procedures. From then on D
ij
with dimensions of resistance scale according to D
ij
= D
ij , ref
u
∗ ,
ref
u
∗
, where the D
ij , ref
have been computed using a reference value u
∗ ,
ref
. 2.2. Profiles for σ
w
, τ
L
and K
f
To evaluate the dispersion coefficients required in Eq. 1, the profiles of σ
w
z and τ
L
z within and above the canopy must be specified for a Lagrangian
Fig. 1. Normalised profiles of the standard deviation of verti- cal velocity, σ
w
u
∗
, the Lagrangian time scale, τ
L
u
∗
h
c
, and the far-field diffusivity K
f
u
∗
h
c
as a function of normalised height, z
h
c
. Theoretical profiles of the latter two quantities for neutral stratification in the inertial sublayer above the canopy are also
shown. The measured composite profile of σ
w
u
∗
and the standard deviations of the measurements obtained using a miniature sonic
anemometer are indicated by circles. Table 1
List of variables and parameter values used to describe normalised profiles of the Lagrangian time scale, τ
L
u
∗
h
c
, and the standard deviation of vertical velocity, σ
w
u
∗ a
z h
c
x y
θ a
b d
≥ 0.25
z h
c
− 0.8
τ
L
u
∗
h
c
0.98 0.256
0.40 +
1 0.25
4zh
c
τ
L
u
∗
h
c
0.98 0.850
0.41 −
1 0.8
z h
c
σ
w
u
∗
0.98 0.850
1.25 −
1
a
See Eq. 10.
moving coordinate frame, but measurements are made relative to fixed coordinates. Finding τ
L
z is difficult and will be considered below, but there is no
problem for σ
w
z, because it is the same in both co- ordinate systems and hence can be measured directly.
To describe the normalised profile y=σ
w
u
∗
, an ex- ponential function Eq. 9 was used up to zh
c
= 0.8,
and a non-rectangular hyperbola Eq. 10 above that height, i.e.
y = c
1
e
c
2
x
, z
0.8h
c
9 y =
ax + b + d p
ax + b
2
− 4θ abx
2θ ,
z ≥ 0.8h
c
10 where x=zh
c
is the normalised height, h
c
the canopy height, and θ is the parameter describing the curva-
ture of the hyperbola. Good agreement was obtained between direct observations and the fitted functions
Fig. 1 using the parameter values given in Table 1 and with c
1
= 0.2, c
2
= 1.5. The limiting value of
σ
w
u
∗
well above the canopy was set to 1.25, which is appropriate for the inertial sublayer under neutral
conditions Kaimal and Finnigan, 1994, even though this is a little higher than the mean of the measure-
ments. With the hyperbolic formulation, σ
w
u
∗
= 1.25
is not attained until zh
c
= 2.5, in conformity with
the summary of observations reported by Raupach 1989b, and consistent with the mixing-layer hy-
pothesis for canopy flow developed by Raupach et al. 1996. Note also that as z → 0 Eq. 9 results
in σ
w
0 → c
1
rather than σ
w
0 → 0. This was done to ensure that turbulent fluxes are not forced to zero
at the ground, and we discuss later the sensitivity of estimated sourcesink strengths to the choice of the
σ
w
profile close to the ground. The shape of the τ
L
u
∗
h
c
profile remains somewhat speculative because we do not have direct estimates
R. Leuning et al. Agricultural and Forest Meteorology 104 2000 233–249 237
of τ
L
z. We know that in the inertial sublayer which typically begins at zh
c
≈2, Kaimal and Finnigan, 1994 we are outside the near-fields of any canopy
sources and thus the diffusion equation Eq. 4 applies. In this region, turbulent transport in the neu-
trally stratified atmosphere is well described by the Monin–Obukhov similarity relationships Kaimal and
Finnigan, 1994, and K
f
= ku
∗
z−d. Matching this with K
f
= τ
L
σ
2 w
, the expected form of the Lagrangian time scale in the inertial sublayer is
τ
L
u
∗
h
c
= k
σ
w
u
∗ 2
z − d h
c
11 where d is the zero-plane displacement and k is the
von Karman constant 0.4. In the roughness sublayer zh
c
≈ 2, turbulent
transport is dominated by coherent eddies associated with the strong inflection in the mean velocity profile
at the top of the canopy Kaimal and Finnigan, 1994; Raupach et al., 1996. This produces a single dom-
inant turbulence length scale, L
s
∝ U h
c
U
′
h
c
, where Uh
c
is the mean velocity at h
c
and U
′
= dUdz.
According to Raupach et al. 1996, the correspond- ing Lagrangian time scale is τ
L
≈ 0.7L
s
σ
w
. Using the observed value of σ
w
u
∗
= 0.9 at the top of the canopy
Fig. 1 and a typical value of L
s
h
c
= 0.5 Raupach
et al., 1996, we obtain τ
L
u
∗
h
c
= 0.4. At heights
z h
c
0.25, the turbulent time scale is influenced by the presence of the ground, and the normalised time
scale is allowed to decrease from 0.4 at zh
c
= 0.25 to
0 at zh
c
= 0. To avoid discontinuities at the transition
between the these regions, Eq. 10 was again used to describe the three sections of the profile. The ap-
propriate x variable and parameter values are given in Table 1, with the resultant profile shown in Fig. 1.
Except for the linear reduction in τ
L
u
∗
h
c
below z
h
c
= 0.25, this profile is a smoothed version of the
linear-piecewise form adopted by Raupach 1989b. Use of the Lagrangian time scale profile shown in
Fig. 1 leads to good qualitative agreement between the predicted shape of the K
f
profile Fig. 1 and the wind tunnel results of Brunet et al. 1994; Fig. 6 for
z h
c
2.5 wind tunnel results above this height are in doubt because the wind tunnel boundary layer was
insufficiently deep relative to canopy height. Finally, excellent matching of fluxes for sensible heat, water
vapour and CO
2
as measured using eddy covariance and those deduced from the inverse analysis see
below provides further indirect, a posteriori support for the τ
L
u
∗
h
c
profile adopted. 2.3. Number of source layers
Matrix algebra may be used to invert Eq. 1 to determine source strength profiles, S
j
, from mea- sured profiles of C
i
, and the calculated D
ij
. However, both C
i
, and D
ij
contain errors, and instabilities in the solution arise when we attempt to infer source
strengths for the same number of layers as the num- ber of heights at which concentrations are measured.
To overcome this problem, Raupach 1989b recom- mended that redundant concentration data be included
so that source densities S
j
in m layers are sought from n concentration measurements, with nm.
The maximum number of source layers that can be extracted from concentration measurements at n
heights is determined by the ‘information content’ of the dispersion matrix D
D D
, information which can be obtained using the technique of singular value
decomposition SVD, Press et al., 1992. This anal- ysis decomposes the n×m matrix D
D D
into the product of an n×m column-orthogonal matrix, U
U U
, an m×m diagonal matrix W
W W
with diagonal elements w
j
the singular value matrix, and the transpose of an m×m
orthogonal matrix V V
V according to
D D
D
n×m
= U
U U
n×m
W W
W
m×m
V V
V
T m×m
12 The singular values are positive and are arranged
down the diagonal of W W
W in decreasing magnitude
according to the index j, with w
j
≥ w
j + 1
. In terms of the SVD components, the solution of
Eq. 1 for the source vector S S
S is given by Press et
al., 1992 S
S S = V
V V
diag 1
w
j
U U
U
T
C C
C 13
Note that the singular values appear in the denomi- nator of Eq. 13, and thus, the sensitivity of the so-
lution to errors in w
j
i.e. to errors in the disper- sion coefficients increases with the index j due to the
increasingly small values of w
j
. The absolute mini- mum in the residuals of the solution Eq. 13 is ob-
tained by inclusion of all w
j
, but Press et al. 1992 recommend including only those equations such that
w
j smallest
w
j largest
α as a compromise between
238 R. Leuning et al. Agricultural and Forest Meteorology 104 2000 233–249
the best least-squares fit and sensitivity of the solution to errors in the matrix operator D
D D
. Using the criterion α=
0.01, six is the maximum number of source lay- ers that can be obtained from the eight concentration
measurement heights used in this study. However, the number of layers finally adopted was reduced to five
because errors in the measured scalar concentrations also contributed to instability of the inversion analysis.
3. Experimental site and methods
