- 42 - THE COMPUTATION OF DIRECT INSOLATION ON A SLOPE by Atsumu Ohmura* 1. Introduction The contribution of direct insolation is of great importance to the microclimatological study of the heat budget at the surface of the ground, \ mainly for two reasons. Firstly, among the terms of the heat budget equation, heat exchange by radiation yields a large value and direct insolation is of primary importance among the five terms which comprise the radiation term. Secondly, direct insolation is more liable to be influenced by the topographic features of the ground than the other terms. Therefore, in order to study the heat budget on the actual surface of the ground, the evaluation of direct insolation must be the first element to be considered. Several studies of this kind have already been attempted, of which the works of Kondrat'yev (1965) and Lee (1963) are typical. The equation used by Kondrat'yev is only useful for the calculation of instantaneous values. To make the equation effective, consideration of variations in the relation between the direction of a slope and that of the sun must be added, and this is usually an enormous calculation. Lee (1963) used an equation devised by * Atsumu Ohmura is a graduate of Tokyo University. He is at present Research Assistant in Climatology at McGill University (Department of Geography).
- 43 - Okanoue to integrate direct insolation for a certain time period. However, the equation, unfortunately, does not take into account the existence of the atmosphere. Neither of these equations is, therefore, suitable for this study because the value wanted is the total for certain time periods under different atmospheric states. For this purpose an equation has been developed in which the transmissivity of the atmosphere is taken into account. 2. The Elements of the Equation by The instantaneous value of direct insolation on a slope can be expressed ( 1) where Is is the instantaneous value of direct insolation on the slope, ~ A -:; S A denotes a unit vector normal to the slope, is a unit vector expressing the direction of the sun, is a symbol denoting the angle between the two vectors, 10 is the solar constant = 2.00 1y min- 1 (Johnson, 1954), p is the transmissivity of the atmosphere, m is the optical airmass. To make the equation practical for computation, it is necessary to express equation (1) as a function of time. The co-ordinate system which -) will best express A is the one which is located at the site in question with the three axes directed to zenith, south and east. A system suitable for ~ the expression of S has the origin in the centre of the earth with three axes consisting of (a) the polar axis, (b) the axis of the direct line
- 44 - marking the junction of the meridian plane at solar noon and the equatorial plane, and (c) the axis which is normal to the previous two axes (fig. 1). We will call the first system the local system and the second system X' the global system. To complete x'..j-:=-2----:*-~ 2 the equation, the vectors must be Sll~...... discussed in a connnon system. It is mathematically equally possible either to transfer the local I X. system into the global system or the global system into the local Fig. 1 system. For the present purpose the latter procedure has been adopted, so that the local system is regarded as rotating once a day around the polar axis in the declination of (90 0 -f) where f is the latitude of the site. The result of the transfer enables one to expres& the contents of the brackets in equation (1) as a function of the azimuth and gradient of a slope, the declination of the sun, the latitude of the site, and time. The equation takes the following form: I = I plj. fi./t:os fo cos f sin w + sin fo sin S 0 f~ ~Sin a sin b) sin f sin w + (sin b cos a)cos w + cos b cos f sin w) cos fo + {- (sin a sin b)cos f + cos b sin f) sin fo (2)
- 45 - where fo is the declination of the sun, f w a b is the latitude of the site, is the hour angle measured in solar time from 6 o'clock, is the azimuth of the slope measured from east or west, negatively to north and positively to south, is the gradient of the slope measured in relation to the horizontal. If Id is the daily total direct insolation, then (3) where wi and w 2 are the times of sunrise and sunset respectively. Although the mathematical integration of the righ~ hand side of this equation is not possible, the total of the instantaneous values will give the same result as the integrated values if a sufficiently short time period is chosen. Thus, (4) where Aw is a sufficiently short time period to obtain the daily total. 3. The Computer Programming For practical purposes when computing the value for a certain time period such as a day, it is essential to know what is the maximum interval which will give the same value correct to the first decimal as is obtained when using a time interval of one minute. The desired time interval was found
TABLE ONE Table of Direct Insolation for Slopes of Different Angle and Azimuth in Latitude 4S~ under Selected Atmospheric Transmissivities TRANSMISSIVITY= 0.60 SUMMER SOLSTICE ANGLE OF AZIMUTH OF SLOPE SLOPE -90-80 -70-60 -50-40 -30-20 -LO 0 10 20 30 40 50 60 70 80 90 0.0 496.8 496.8 496.tl 496.8 496.8 496.8 496.8 496.8 496.8 496.8 496.8 496.8 496.8 496.8 496.8 "96.8 "96.8 "90.8 "96.8 10.0 469.5 469.8 470.7 "72.2 "H.l "71>.6 "79.4 482.5 485.8 489.3 492.7 496.0 49~.1 502.0 504.4 506.3 507.8 508.7 509.0 20.0 428.0 4.28.6 430.3 <'33.2 <'37.1 441.9 4.. 7.5 453.9 460.9 "b8.0 474.9 481.5 487.6 492.9 497... 501.0 503.6 505.2 505.7 30.0 373... 314.3 376.9 3S1.1 386.7 394.2 403.7 414.5 425.8,,36.8 "47.2 450.6 464.8 471.7 477." 481.8 484.9 486.7 487.4 40.0 301.6 308.7 3L2.0 3L 7.3 325.1 337.8 353.2 369.4 385.1 399.6 412.6 423.6 433.0 440.3 445.8 449.9 452.6 454.1.. 54.6 50.0 232.3 233.7 237.6 2"".0 259.2 280.2 301.9 322.8 341.8 358.6 372.8 384.4 393.4 399.9 404.4 407.1 408.4 408.6 408.8 60.0 150.0 151.5 156.0 L73.3 200.8 227.9 253.6 277.1 297.7 3L5.2 329.2 340.1 341.7 352.3 354.4 354.5 353.5 352.0 351.4 10.0 66.7 73.2 94.2 124.1 154.2 183.1 209.6 233.3 253.8 270.4 283.2 292.2 297.3 299.0 297.8 294.6 290.2 286.3 284.7 80.0 35.L 42.b 63.L 90.1 117.9 1"4.8 169.9 192.0 210.7 225.3 236.0 242.3 244.3 242.5 237.3 230.0 221.3 214.0 211.3 90.0 2L.l 27.2 43.5 65.5 89.0 ll2.3 134.1 153.3 169.2 181.1 189.0 192.2 L90.9 185.3 176.0 163.9 150.2 139.1 134.7 TRANSM ISS1 lilly: 0.80 WINTER SOLSTICE ANGLE OF AZIMUTH OF SLOPE SLOPE -90-80 -70-60 -50 -.. 0-30 -20-10 0 10 20 30 40 50 60 70 80 90 0.0 109.4 109.4 L09.4 L09.4 109.4 L09.4 L09.4 L09.4 109.4 L09.4 109.4 L09.4 109.4 109.4 109.4 109.4 109.4 109.4 109.4 10.0 52.6 53.5 56.1 60.3 65.9 12.8 80.6 89.3 98.4 L07.9 117.4 126.7 135.4 L43.3 150.1 155.6 159.7 162.2 163.0.po. 20.0 3.1 4.7 9.8 18.2 28.8 41.2 55.3 -,o.8 87.5 105.0 122.8 140.5 157.4 172.8 L86.3 197.2 205.2 210.1 2L1. 8 0\ 30.0 0.0 0.0 0.0 0.0 7.8 21.L J7.7 57.1 78.0 LOL.9 126.3 150.9 114.8 197.2 216.8 232.7 2"".5 251.6 254. L 40.0 0.0 0.0 0.0 0.0 1.5 1l.1 26.9 47.3 71.4 98.4 127.4 157." 187.2 215.6 240.7 261.2 276.3 285.5 288.6 50.0 0.0 0.0 0.0 0.0 0.3 6.5 20.2 40.0 65.0 94.1 126.1 159.9 194.4 227.5 257.3 281.7 299.7 310.7 314.5 bo.o 0.0 0.0 0.0 0.0 O.L 4.3 15.8 34.4 59.0 8B.6 122.2 158.3 195.7 232.5 266.1 293.7 314.0 326.5 330.7 10.0 0.0 0.0 0.0 0.0 0.1 3.0 12.5 29." 52.8 81.9 ll5.6 152.6 19 L.6 230.7 266.8 296.8 318.8 332.3 33b.9 80.0 0.0 0.0 0.0 0.0 0.0 2.1 10.0 24.9 "6.4 73.9 106.4 142.9 181.9 221.8 259.4 290.6 313.9 328.1 332.9 90.0 0.0 0.0 0.0 0.0 0.0 1.5 7.9 20.b 39.6 64.7 95.0 129.5 107.3 206.4 244.2 276.0 299.5 313.9 318.7 TRANSM.I SS1 II ny= 0.75 EQUINOXES ANGLE OF AZIMUTH OF SLOPE SLOPE -90-80 -70-60 -50-40 -30-20 -LO 0 10 20 30 "0 50 60 70 do 90 0.0 369.6 369.6 369.b 369.6 369.6 369.0 369.6 369.6 369.6 369.6 369.6 369.6 369.6 369.6 3b9.6 369.;' 3;'9.6 3b9.6 3b9.6 10.0 299.8 300.1 303.6 308." 31".8 322.8 332.1 342.3 353.1 364.2 315.4 386.1 396.2 405.3 413.2 419.6 42".3 "27.2 428.2 20.0 220'.8 222.8 228.7 238.6 252.3 269.0 287.9 308.3 329.6 351.2 372.4 393.0 412.1 429.5 "44.5 457.0 466.2 471.9 473.8 30.0 135.2 138.3 149.1 166.9 189.7 215.7 244.0 273.7 304.0 33".0 363.6 391. B 418.2 442.1 463.2 4BO.7 493.9 502.2 505.0 40.0 45.4 54.4 17.1 105.1 136.3 170.0 205.2 241.5 277.9 314.0 349.3 363.0 414.6 443.4 468.8 "90.3 506.8 517.3 52Cl.8 50.0 0.0 9.3 33.6 63.8 97.5 134.0 172.4 2L2.0 25 L.9 291.4 330.0 366.9 401.5 433.2 461.4 485.5 504.3 516.6 520.9 60.0 0.0 2.4 16.4 40.7 7L.b 106.8 L45.1 184.8 225.6 266.2 305.7 343.5 379.2 412.0 441.2 "66.5 486.8 5JO.2 505.1 70.0 0.0 1.0 9.3 27.6 53.7 85.5 121." 159.5 198.9 238.2 276.8 313.8 348.2 360.4 "09.1 434.0 454.5 468.7 473.9 80.0 0.0 0.5 6.0 19.5 40.9 68.4 100.2 135.3 171.3 208.1 243.9 278.2 310.2 339.1 366.2 389.4 "09.0 423.0 428.4 90.0 0.0 0.3 3.9 13.9 30.6 53.6 81.4 111.7 L44.0 176.1 208.2 236.2 266.3 291.5 314.1 334.2 351.4 364.5 369.8 Note: The azimuth of the slope is measured from east, negative to north and positive to south.
- 47 - to be twenty minutes. Using this interval, tables of the daily total direct insolation at latitude 45 0 N for the equinoxes and solstices were prepared in steps of ten degrees for all slopes of all aspects and gradients, with the values for the transmissivity of the atmosphere ranging from 0.50 to 0.85 in steps of 0.05 (Table 1). The computation for the transmissivity at a lower value than 0.50 was not made because under such low transmissiyities, the contribution of the diffused insolation becomes too large for the estimation of direct solar radiation to be worthwhile. 4. The Application of the Equation To illustrate the application of the formula maps have been prepared to show how the distribution of direct insolation at Mont St. Hilaire, Quebec, is influenced by topographic irregularities (Figs. 3-5). These maps were constructed by using the values from Table 1 to plot radiation totals for each corner of a grid system in which each grid square is equivalent to 150 yards. The topographic map of the area (Fig. 2) was used as a base for the grid from which to obtain the appropriate slope angle and azimuth at each intersection. 5. The Features of the Distribution of the Direct Insolation on Slopes (i) The map for June 20, 1967, which approximates the summer solstice, shows the most uniform distribution over the rugged ground surface of Mont St. Hilaire. This is because firstly, the orbit of the sun in summer is fairly uniform for south and north slopes, and secondly, the relatively low intensities of a gently tilted north slope are compensated for by a longer
_"',... _,I~),'lS-O... II 48 GAULT DONATION MONT ST. HILAIRE, P.O. - Orchard boundary -..... Paved rood Unpaved rood Trail flnk'roil Blue lroil Yellow Ireil Red lrail Greoo Irail - -, -- =...,... ~....-'11111,.. ContOVl' interval- 50 feel Fig. 2. Contour Map of Mont St. Hilaire, Quebec.
- 49 - potential sunshine duration than that which exists on a south slope. This fact makes the difference in azimuth less important than the difference caused by gradient, the range of which is relatively narrow in nature. (ii) The topographic variation is effectively reflected in the map for September 19, 1967, representing the equinoxes. The enormous difference between north and south slopes is explained by the combined effect of longer sunshine duration and consistently larger instantaneous values on the south slope as compared with the north. Maximum values are found on a relatively steep south slope (40 0-500), while in summer the highest values appeared on a more gentle slope (100-15 0 ). For the east and west facing slopes, the azimuth of the slope is more effective than the gradient in causing radiation differences. (iii) At the winter solstice represented by the map for December 27, 1966, the difference between south and north slopes is as sharp as at the equinoxes if we take the ratio between the slopes, but since the maximum value of insolation is so small, the difference in the absolute values between north and south slopes does not appear as large as that found at the equinoxes. The gradient of the slope loses its significance for the east and west slopes and only the aspect of the slopes determines the major value of the daily total. 6. The Future Project To evaluate the contribution of insolation to the elements of the heat budget, it is not satisfactory to deal only with direct insolation. Both diffuse radiation and surface albedo need to be considered. To approach this
- 50 - () 500 N +-c--- JUNE 20, 1967 P = 0.60 Ya,d. SCALE 0 IUO~400 o V4 V2 1 Mill ----200 Longleys at 50 Iy. interval Fig. 3. Distribution of Direct Solar Radiation Totals at Mont St. Hilaire, June 20, 1967.
- 51 - N SEPTEMBER 19, 1967 p= 0.75 SCALE Yards 9 '$a Xjg. o Vq V2 31q I Mil. - 200 - Longleys ot lio Iy. intervals. Fig. 4. Distribution of Direct Solar Radiation Totals at Mont St. Hilaire, September 19, 1967.
- 52 - o '" ISO N... c- DECEMBER 27, 1966 P = 0.80 Yard. 0 too '200 :500400 b F+i Ed V4 SCALE V2 314 I Mill - 200 - Longleys a1 50 Iy. interval Fig. 5. Distribution of Direct Solar Radiation Totals at Mont St. Hilaire. December 27, 1966.
- 53 - aim, a larger programme is now being prepared in which Kondrat'yev's equation for sky-diffuse radiation will be adopted to compute the contribution from direct and diffuse radiation separately, so that mapping for overcast conditions is possible. In addition, the albedo for each point on the map will be put in to subtract the reflected part. Since the distribution of longwave radiation can be expected to be fairly homogeneous, a net short-wave radiation map of this type will help to estimate both the evaporation and the sensible heat flux at the surface of the ground. References Johnson, F. S., 1954: "The Solar Constant", Journ. Meteor., Vo1.,11, pp. 431-439. Kondrat'yev, K. Y., 1965: "Radiative Heat Exchange in the Atmosphere", (Trans. from the Russian), Pergamon Press. Lee, R., 1963: "Evaluation of Solar Beam Irradiation as a Climatic Parameter on Mountain Watersheds", Hydrology Papers, Colorado State University.