Agricultural and Forest Meteorology 104 2000 215–231
Statistical treatment of sunfleck length inside willow coppice
J. Ross
∗
, M. Mõttus
Tartu Observatory, 61602 Tõravere, Tartumaa, Estonia Received 4 January 2000; received in revised form 11 April 2000; accepted 12 April 2000
Abstract
Different sunfleck characteristics, length and number of sunflecks per metre and sunfleck fractional area, were measured at different depths inside willow Salix viminalis canopy. All these characteristics depend on the pathlength of the direct
solar radiation beam into the canopy, τ =Lsin h, where L is the downward cumulative leaf area index and h is the solar elevation. The mean sunfleck length hl
S
i decreases exponentially with τ . The maximum length of short sunflecks length
6 cm decreases linearly with τ , but not exponentially. Like the number of short sunflecks, the total number of sunflecks increases with τ , reaches a maximum value of 10–12 sunflecks m
− 1
at τ =2.5 and then decreases slowly to zero at τ =6. The vertical profiles of the sunfleck fractional area are different for long length 6 cm and short sunflecks. The fractional
area of long sunflecks decreases with τ exponentially; the fractional area of short sunflecks, k
SS
, can be approximated by the formula k
SS
= 0.19τ exp−0.15τ
2
, and the fractional area of all sunflecks k
S
, by the formula k
S
= exp−0.18τ
2
. Within the canopy, direct solar radiation is present in two areas — in sunflecks and in penumbra. The total flux of direct solar radiation
in penumbra is smaller than it is in sunflecks and reaches its maximum value of 50 at the pathlength τ =2. There exists a good correlation with R
2
= 0.94 between the fractional area of umbra, k
U
, and the fractional area of sunflecks, k
S
, which can be fitted by the exponential formula k
U
= 0.63 exp−2.30k
S
. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Sunflecks; Willow coppice; Radiation statistics in plant canopy
1. Introduction
If the solar discs were a source of point radiation, as it has been assumed in many earlier models of ra-
diation transfer in a plant canopy, two types of areas would form inside the canopy:
1. sunflecks, where the area is illuminated by a
parallel beam of direct solar radiation with the irradiance S
W m
− 2
, and 2. umbra shade, where the area is shaded by over-
∗
Corresponding author. Tel.: +372-7-410-278; fax: +372-7-410-205.
E-mail address: rossaai.ee J. Ross.
lying leaves or other phytoelements and where the irradiance of direct solar irradiation S=0.
In fact, the solar disc has a finite angular diameter, and hence a third type of area, penumbra, is formed
inside the canopy. In penumbra, the solar disc is partly covered by leaves and S may vary from 0 to S
. Exis- tence of penumbra complicates greatly the treatment
of radiation transfer in canopies. Due to the extremely complicated geometrical aspect of the problem, its
comprehensive theoretical treatment is impossible. In our earlier work Ross and Mõttus, 2000, we
presented an overview of the relevant literature.
As a first attempt, our statistical treatment of sun- flecks, umbra and penumbra may suggest some new
clues to a solution of the problem. Statistical treatment
0168-192300 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 0 0 0 0 1 6 2 - 3
216 J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231
of sunflecks, umbra and penumbra is closely related to the progress of new comprehensive computer mod-
els of canopy photosynthesis and evapotranspiration, which require detailed 3D characteristics of different
radiation types for input. As the interrelationship be- tween PAR and photosynthesis is nonlinear, the mean
values of radiation do not yield correct results. In our opinion, for further calculation of canopy photosyn-
thesis, leaves in plant canopies should be divided into three groups according to the type of direct sunlight
they receive: sunflecks, umbra and penumbra; and photosynthesis should be calculated separately for
each group. Therefore, it is necessary to know the area of sunflecks, its vertical distribution and tempo-
ral dynamics in order that different data for umbra, penumbra and sunflecks could be used in modern
radiation transfer models.
Ross et al. 1998 proposed a statistical treatment of PAR variability and its application to willow cop-
pice. In this treatment, statistical distribution of PAR global irradiance, obtained with a LI-COR quantum
sensor, was approximated by a normal distribution in sunfleck and umbra areas, and the fractional area of
penumbra was approximated by the Beta-distribution and considered to be function of pathlength τ and
solar elevation h.
This paper is a continuation of the previous work by Ross and Mõttus 2000 on a statistical treatment of
umbra length and is focused on the statistical treatment of sunfleck length inside a willow coppice. Experi-
mental data were obtained with a new instrument — the sunfleck sensor constructed by Sulev; the method-
ology of statistical data processing is analogous to that used in our previous paper Ross and Mõttus, 2000.
Sunfleck’s different statistical characteristics, such as sunfleck length distribution, sunfleck fractional
area, number of sunflecks, mean sunfleck length, etc. in different canopy layers will be studied as the func-
tions of the pathlength of the direct solar radiation
1
τ =Lsin h, where Lz = R
z
U
z
u
L
z dz is the down- ward cumulative leaf area index and u
L
z is the leaf area density.
1
In our previous paper Ross and Mõttus, 2000, we used the term ‘optical pathlength’ for this quantity. However, to avoid
any confusion that might arise from this general term, we use a more comprehensive term, ‘pathlength of the direct solar radiation
beam’, in this paper.
2. Materials and methods
