Results and discussion Directory UMM :Data Elmu:jurnal:A:Agricultural & Forest Meterology:Vol104Issue3Sept2000:

J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 221 Fig. 5. Probability histogram of leaf length a and leaf area b, fitted by a lognormal distribution, measured by Ross in 1997 and 1998. tics of leaf, shoot, plant and canopy architecture. The first two factors determine the pathlength of the direct solar radiation beam in the plant canopy, τ =Lsin h. Hereafter, sunfleck length characteristics will be con- sidered as functions of pathlength τ , assuming that it is the key factor determining the absorption pattern of direct sunlight.

5. Results and discussion

5.1. Sunfleck length distribution function As an example, Fig. 6 expresses three sunfleck length distribution functions n S l S and cumulative sunfleck fractional area functions F S l S for 14 September 1998 at different relative heights zz U 0.85 a, 0.65 b and 0.36 c, when coppice height was z U = 310 cm, and n S l and F S l S are calculated using the recordings in Fig. 2. From Figs. 2 and 6, we can draw the following conclusions: 1. The averaging length 6+6 m corresponding to the track length of the sunfleck sensor seems to be too short to obtain the sunfleck distribution function, especially at large l S . As the distribution function n S l S is not continuous, we will use it together with the cumulative fractional area function F S l S . 2. The character of sunfleck distribution statistics de- pends strongly on the pathlength τ . It is practically impossible to fit the distribution function n S l S by some theoretical distribution given in literature. Therefore, we will characterise the sun- 222 J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 Fig. 6. Sunfleck length distribution function n S l S and sunfleck fractional area function F S l S calculated using data in Fig. 2a–c for relative heights of 0.84, 0.65 and 0.36, respectively. A SS and A SL are the the intervals of short and long sunflecks, respectively. J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 223 fleck length distribution n S l S by means of different numerical parameters. Since the character of n S l S depends mostly on the pathlength of the solar radia- tion beam in coppice, τ , we will treat these numerical parameters as functions of τ . In dense lower layers, the number of sunflecks per scan may be so small that determination of n S l S and F S l S proves impossible Fig. 6c; below a canopy with L4, direct solar radiation penetrates only in penumbra, but no sunflecks exist. 5.2. Vertical distribution of sunfleck length Fig. 7a–c expresses three sunfleck length character- istics: mean sunfleck length hl S i a, maximum sun- fleck length l S max b and the maximum length of short sunflecks l SS c as the functions of the pathlength τ . The data are collected from measurements during the growth period of 1996–1998. Values of hl S i and l S max decrease with τ exponen- tially. The maximum length of short sunflecks Fig. 7c decreases with τ linearly but not exponentially, with a small correlation coefficient R 2 = 0.52. In the S. viminalis coppice, the maximum length of sunflecks in the upper cylindrical layer take the values of 2.0–3.0 m, the length of short sunflecks in lower lay- ers, only 0.01–0.03 m. The maximum sunfleck length l S max and the mean sunfleck length hl S i are statistically interdependent Fig. 7a and b: the mean length hl S i increases logarithmically with increasing l S max . Theo- retically, if τ →0, hl S i→∞ ; if τ →∞, hl S i→ 0. 5.3. Vertical distribution of the number of sunflecks N S Fig. 8 shows the number of sunflecks, N S per unit scan length a, number of short sunflecks N SS b and the share of short sunflecks in the total number of sunflecks N SS N S c versus the pathlength τ . The total number of sunflecks as well as the number of short sunflecks increases with τ , reaches a maxi- mum value of 10–12 sunflecks m − 1 at τ =2.5 and then decreases slowly to zero at τ =6. As the function of τ , N S has been fitted by the formula N S = 11.5τ exp−0.13τ 2 , R 2 = 0.71 and N SS by the formula N S = 10.0τ exp−0.13τ 2 , R 2 = 0.65. Fig. 8c demonstrates that the number of short sun- flecks dominates in all layers, while in lower layers, starting from τ =2.5, practically all sunflecks are ‘short’ l S .6 cm. The function N S τ is different for the 1-year-old 1998 and for the several-years-old coppice 1995– 1996. The number of sunflecks is greater in the for- mer, which is evidently due to its better expressed cylindrical vertical structure that enhances penetration of direct solar radiation. Large deviations from the mean value indicate the existence of specific random configurations in willow coppice where the number of sunflecks is extremely large. 5.4. Vertical distribution of the sunfleck fractional area k S From Eq. 4, k S = N S h l S i , it follows that mean sun- fleck length and the number of sunflecks determine the fractional area of sunflecks, k S , which is an im- portant radiation characteristic. In Ross and Mõttus 2000, it was taken that the sunfleck sensor’s reading S F W m − 2 corresponds to sunfleck when it is larger than 0.95S S is the sunfleck sensor’s reading above the canopy, and the sunfleck sensor’s reading corre- sponds to umbra when it is smaller than 0.007S . In accordance with this definition, the irradiance of di- rect solar radiation in sunflecks equals almost exactly the irradiance above the canopy, S h. Fig. 9 expresses the vertical distribution of the frac- tional area of a long sunflecks, k SL τ and b short sunflecks, k SS τ , and shows that the distribution is different for each group. The decrease of fractional area k SL Fig. 9a is nearly exponential and can be fitted by the formula k SL = exp−0.51τ 32 , R 2 = 0.84. Unlike the fractional area of long sunflecks, the frac- tional area of short sunflecks increases almost linearly when the pathlength τ increases Fig. 9b, reaches a maximum at τ =1.5 and decreases then to zero at τ =∞. The function k SS τ was fitted by the formula k SS = 0.19τ exp−0.15τ 2 , R 2 = 0.74. 224 J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 Fig. 7. Logarithm of mean sunfleck length hl S i a, logarithm of maximum sunfleck length l S max b and maximum length of short sunflecks, l SS c, versus the pathlength of a direct solar radiation beam, τ . J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 225 Fig. 8. Vertical distribution of the number of sunflecks per unit scan length N S τ a, number of short sunflecks N SS τ b and fraction of short sunflecks in the total number of sunflecks, N SS τ N S τ c. 226 J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 Fig. 9. Fractional area of long sunflecks k SL a, short sunflecks k SS b and sunfleck fractional area k S in 1998 c as the function of the pathlength of the solar radiation beam τ . J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 227 Fig. 10. The mean value of the penetration function of the direct solar radiation a S τ versus pathlength of the direct solar radiation beam in coppice τ . As we can see in Fig. 9c, sunfleck fractional area k S τ decreases with τ nearly exponentially, the argument being τ 2 but not τ , and for the year 1998 data can be fitted by the formula k S τ = exp−0.18τ 2 , R 2 = 0.96. 5.5. Penetration function of direct solar radiation The mean value of one scan of the sunfleck sensor’s readings, hS F i , at the pathlength τ , divided by the ir- radiance above the canopy, S , is equal to the mean value of the penetration function, i.e. a S τ =hS F i S . Fig. 10 expresses the logarithm of a S as the function of the pathlength τ and shows that a S τ is not a simple exponential function of τ . It appears that the canopy can be divided into two layers: the upper canopy layer, where 0τ 1.5, and the lower layer, where τ 1.5. For these two layers, as a first approximation, a S τ can be expressed by the formula a S τ = exp−0.21τ , 0 τ 1.5, R 2 = 0.91, 2.2 exp−0.74τ , 1.5 τ, R 2 = 0.91. 8 As the transition between a clumped and a semireg- ular leaf distribution is continuous and measurements were carried out during several growth periods and at different solar elevations, the location of the surface separating these layers cannot be determined exactly. However, the division of a S τ is justifiable by vi- sual observation. The uppermost layer of the coppice Ross and Ross, 1998 contains only nearly vertical foliage cylinders, inside of which the leaf area density u L is high. The lower coppice layer consists of shorter shoots and branches with approximately random shoot orientation and semiregular orientation. The small ex- tinction coefficient of the upper layer, 0.21 0.5, in- dicates clumping, whereas in the lower layer, where extinction coefficient 0.74 0.5, leaf distribution must be semiregular. 5.6. Interrelationship between sunfleck area and penumbra area The mean value of one scan of the sunfleck sensor is h S F i = S k S + h S FU i k U + h S FP i k P , 9 where hS FU i and hS FP i are the mean values of direct solar irradiance, measured in shade and penumbra, re- spectively, k S , k U and k P are the fractional areas of sunflecks, shade and penumbra, respectively, with the condition k S + k U + k P = 1. 10 The irradiance in umbra area is practically zero, hS FU i 0.007S . So, neglecting the umbral irradiance, the mean value of direct solar irradiance in penumbra area from 9 228 J. Ross, M. Mõttus Agricultural and Forest Meteorology 104 2000 215–231 h S FP i = h S F i − k S S k P 11 and the penetration function of direct solar radiation in penumbra area, a SP τ = hS FP i S , in accordance with 11, is a SP τ = a S τ − k S τ k P τ . 12 Thus, the total mean penetration of direct solar radia- tion in the canopy at the pathlength τ consists of two parts a S τ = k S τ + k P τ a SP τ . 13 The first term determines the mean penetration in sunflecks and the latter, penetration in penumbra. Fig. 11. a Penumbra fractional area k P as the function of the pathlength of the direct solar radiation beam in coppice τ ; b penetration function for direct solar radiation in penumbra a SP as the function of the pathlength of the direct solar radiation beam in coppice τ . Measurements with the sunfleck sensor allow calcu- lation of all terms in formula 13. The fractional area of penumbra, k P τ , as the func- tion of pathlength Fig. 11a shows that the role of penumbra is the greatest in the medium layer where τ is between 2 and 4. The maximum values of k P are about 0.35. The function k P τ was approximated by the formula k P τ = 0.32τ 2 exp−0.69τ , R 2 = 0.60. 14 Fig. 11b expresses the penetration function a SP τ in penumbra area as the function of the pathlength τ and shows that a SP τ decreases exponentially not with τ , but with τ 34 . In the upper canopy layer, penetration in 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