J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 229
Table 1 Penetration function of direct solar radiation in sunfleck k
S
and penumbra k
P
a
SP
areas as the function of pathlength of the direct solar radiation beam in coppice τ
τ 0.5
1 2
3 4
6 8
k
S
1.00 0.98
0.71 0.22
0.07 0.02
k
P
a
SP
0.02 0.06
0.11 0.08
0.05 0.01
the penumbral area may increase total penetration by about 0.3–0.6. Penetration in penumbral area reaches
the deeper layers until τ =8–9 while penetration in sunfleck reaches the depth τ =4 only.
The irradiance of the direct solar radiation in the sunfleck area in units of S
is k
S
τ , and the ir- radiance of direct solar radiation in penumbra is
k
P
τ a
SP
τ . The difference k
S
τ −a
SP
τ k
P
τ shows how much irradiance is smaller in penumbra in units of S
than in sunflecks. Table 1 shows the energetical contribu-
tions of sunflecks k
S
τ and penumbra a
SP
τ k
P
τ to total penetrated direct solar radiation as the function
of the pathlength τ . In upper layers, the role of penum- bra is rather small but it increases rapidly between
τ =1 and 3 to dominate in lower layers where τ 3.
5.7. Correlation between sunfleck and umbra characteristics
Calculation shows that there is no correlation be- tween sunfleck length and umbra length, and between
Fig. 12. Interrelationship between umbra fractional area k
U
and sunfleck fractional area k
S
.
the number of sunflecks and umbrae. However, there exists a quite good correlation with R
2
= 0.94 between
their products — sunfleck fractional areas k
S
= N
S
h l
S
i and umbra fractional area k
U
= N
U
h l
U
i , respectively
Fig. 12. This correlation was fitted by the exponen- tial formula
k
U
= 0.63 exp−2.30k
S
, R
2
= 0.94,
15 which shows that sunfleck fractional area increa-
ses exponentially with decreasing umbra fractional area.
There exists a specific layer inside willow coppice between τ =2 and 4, where the total number of short
sunflecks, umbra fractional area as well as the num- ber of umbrae Fig. 7, Ross and Mõttus, 2000 reach
their maxima. In this layer, the maximum number of sunflecks reaches 17, while the maximum number of
umbrae is 18, i.e. in this layer, the number of umbrae and sunflecks are equal. Penumbrae Fig. 11a too ex-
ert a maximum effect in this layer. While sunflecks dominate above this layer, then umbrae together with
penumbrae dominate below it.
6. Concluding remarks
Using a special measurement system to deter- mine sunfleck length within willow coppice, it was
demonstrated that such sunfleck characteristics as length, number of sunflecks per unit length and their
230 J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231
fractional area are highly variable and require statisti- cal treatment. Variability of sunflecks is mainly caused
by architectural coppice characteristics at leaf, shoot, stem and coppice levels. In willow coppice, the prob-
lem is more complicated because of the great, about 100-fold, variability of the area of a single leaf.
Besides structural parameters, there are two other factors which determine the mean values of sunfleck
characteristics. One is solar elevation — different in- clinations of solar rays correspond to different path-
lengths through the plant canopies, which causes dif- ferences in the distribution of sunflecks. The other
factor is depth inside canopy, which is characterised by the downward cumulative leaf area index L. It is
expedient to merge these two factors into one factor τ =Lsin h, i.e. pathlength of the direct solar radiation
beam in the canopy. Introduction of τ facilitates anal- ysis of the problem.
An important methodological problem is how to perform the statistical data processing. Our measure-
ment system enables us to use a maximum averaging length of 12 m. However, Fig. 6a–c shows that the
obtained sample size is too small for sufficiently exact determination of probability density distribution char-
acteristics. Calculations not given here show that for willow coppice, an averaging length of about 50–70 m
would be required. Since this is impossible to realise technically, only mean values and standard deviations
can be used in data processing, which, however, does not allow us to study distribution functions in great
detail.
Existence of long sunflecks in the uppermost cop- pice layers and their rapid exponential decrease to
zero near τ =4 is typical of the S. viminalis coppice. The bulk of registered sunflecks have a length of
1–2 cm. The longest sunflecks measured within our willow coppice had a length of 300 cm, but their
number decreases exponentially with depth.
The number of sunflecks, N
S
, changes with depth Fig. 8a. N
S
increases with depth up to a maximum at τ =1–2 and then decreases slowly with further increase
in τ . It should be noted that the sunfleck length l
S
and number N
S
are very variable characteristics and do not characterise separately the regime of direct solar
radiation effectively. A more appropriate characteristic is their product k
S
=h l
S
i N
S
, i.e. the fractional area. As is evident from Fig. 9a–c, vertical change is different
for short, long and total sunflecks: the fractional area of long sunflecks, k
SL
, decreases most rapidly. Experimental data have been fitted with the for-
mula kτ =Aτ
2
exp−Bτ
b
. At small values of τ , exp−Bτ
b
≈1 and kτ ∼τ
2
; in lower canopy layers where τ 2–3, the term exp−Bτ
b
is more important. Thus, in upper canopy layers, kτ decreases propor-
tionally with τ
2
, at longer pathlengths, kτ decreases exponentially, which may be the result of manifold
shading in lower layers. We have also studied the interrelationship between
sunflecks and penumbra. When the vertical profile de- pendence on z of sunflecks is exponential, the frac-
tional area of penumbra k
P
τ is similar to that of short sunflecks k
SS
τ : the existence of short sunflecks in- creases the fractional area of penumbra.
7. Nomenclature
