Hourly Variation of the Total Insolation Incident Since Sunrise by Months for the Northern Hemisphere Citation Hales, James Vern (1941) Hourly Variation of the Total Insolation Incident Since Sunrise by Months for the Northern Hemisphere. Master's thesis, California Institute of Technology. doi:10.7907/x1aa-3826. https://resolver.caltech.edu/CaltechTHESIS:02252025-191822008 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Meteorology) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Meteorology Thesis Availability: Public (worldwide access) Research Advisor(s): Krick, Irving Parkhurst (advisor) Stewart, Homer Joseph (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1941 Record Number: CaltechTHESIS:02252025-191822008 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02252025-191822008 DOI: 10.7907/x1aa-3826 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17026 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 27 Feb 2025 17:29 Last Modified: 27 Feb 2025 19:14 Thesis Files PDF - Final Version See Usage Policy. 16MB Repository Staff Only: item control page

Hourly Varia tion of the Tot al Insola tion Incident Since Sunris e by Months for the Northern He mL .)here Thesis by J. Vern Hales In Partial Fulfilment of the Re quirements Master of Science in Me teorolo 6y Cal ifornia Institute of Technolo gy Pas adena , Ca lifornia 1941 S UII1."11,:. ry There has long been a ne'3d for da ta s hov;ing the diurnal v~ri a tion of the tot a l insolat ion for all l a titudes and seasons. Thes e d~t a have oeen prep ~red, Ernd 2: re presented here with a br i ef surrill12ry of re l a ted previous wo rk, methods of calcul a~ i on j ~nd of ~pplic ~tions of the da t a to fore0&sting probl effis . To Dr. Irving P. Krick, Dr. H. J. Stewart , and Dr. ~alt er M. Elsasse r at California Ins tit ute of Technolo f; y; Mr . L. H. Da.in;;erfield, c:.nd Jl.lir . Joe E. Fulks of the United Sta tes Weather Bure au, I wish to express my appreciation for t he v&lua bl e suggest ions and help given me. History: The first evaluation of solar radiation for meteorological purposes was done by Angot¾ who calculated the total amount of heat incident per square centimeter 0 at each 10 parallel of latitude during each month of the year. This work has been very helpful in under- standing the general circulation of the atmosphere, and was used by Simpson19 in his investigations of the problem of mean temperature balance; but it is not applicable to forecasting problems. Extensive measurements of incident solar radiation have been made by the Smithsonian Institution, by the United States We a ther Bureau and various cooper ating observa tories, and by observatories in several foreign countYes. The Smithsonian Institution's observa tions have been directed primarily toward evaluating the Solar Constant. Their value of 1.94 calories per square centimeter per minute is universally a ccepted as the best so far obtained. Radiation observati:::ms by the Weather Bureau, on the other hand, have been made for the primary purpose of evaluating the energy availa ble for a~mospheric warming. Hence these observations include sky radia- tion plus that part of the solar beam which reaches the surfac.e . * A summary and discussion of these obser- Numbers refer to bibliography 2. vations has been published by Hand?. The effect of different a ir massfs is not considered in computing the data published, nor is the variation in local atmospheric polution. As summari zed the observatLms re- present a mean summation of the resultant energy available at each station, and should be very useful for forecasts in the vicinity of the observatLm stations.-* For widespread or general applicatLm, some method would have to be worked out for reducing all the observations to some standa rd, and applying corrections for different ty pe s of air mass or local modification. \vork more nearly .like the present was done by Lesterl3 as a part of his investigation of the problem of forecasting the dissipation of fog in the San Francisco Bay region. He calculated hourly values for the insolation which would be applicable to stratus dissipation in that region. His contribution will be discus- sed further. Need for Data: Many forecasting problems demand that the fores caster make some commitment which presupposed a good knowledge of the amount of heat that will be applied to the air by a given hour of the day. more obvious of these pr oblems are: * See page 11. Some of the At what time will the fog (or stratus) be dissipated? imum temperature be? What will the max- At what time will the temperature reach, s ay, 80? All of these problems require a knowledge of several quantities besides solar radiation: some have been sol- ved rather well for given localities; but with all of them, it would seem that a general solution would be easier if one value--that is, the amount of sola r heat which will have been received by each hour of the day-were known. Once this total incident ener gy is known, corrections can be made for absorption and reflection of that ener gy; for the additional heat which will be received from the sky; and for complicating factors such as wind, wetness of the ground, type of air mass, polution, etc. As a step towc.rd making such forecasts more accura te, the hourly totals for the sol ar energy incident since sunrise at the outer limits of the atmosphere has been calculated for the fifteenth of each month through the ye ar, for each 10 de grees of latitude for the northern hemis phere. Method: The method for obt a ining the values is not difficult.8 It c an be shown that: 4. where: Cv is the amount of heat (cal./cm:) which re a ches the outer limits 9f the atmosphere from the sun between the times Ji,and fl~· is the solar constant . . f 1.94 cal/cm; min. is the rate of rotation of the W e arth. A is the hour angle. /4J the hour angle a t sunrise, 1vvas obtained by the formula: I (_,tr.), 11. / ,:: - ~ /~cf' is the le. ti tude. o is the angle of the sun's declination. D is the r adius vector of the earth (iorrection for distance to the sun). All times are apparent solar time. The insol a tion values are presented in fi gures Oto 9. Use of Data: In using the data the forecaster must make modifi- cations and corrections according to the situa tion with which he is dealing . ions is: One of t he most impor t ant quest- How much of this incident radiation can I assume will be spent in modifying the air over my st a tion? Whatever this fr a ction is, all the data considered must be multiplied by it Defore being used for the forecast. To facilitate this, a set of sc ales can be made to measure effective radia tion directly fro m the graphs. Such a scale is shown in figure 10. Table 1 contains a sugg ested list of albedoes (reflected fr a ction of incident r adiation). For some problems a second correction must be applied to a ccount for the heat lost by radiation from the ground, and for heat gained by radi~'t ion from the sky. These can be approximated by use of the emp iric al equations published by Brunt3, and Krickt 2 ; or for more ,, exact work, by using the r adia tion chart develo ped by Elsasser 5 , 6 . Other corrections will have to be made to include the amount and hei ght of cloudiness, evapor a tion of water, he at contribution or deduction by the grou.~d, advection of warmer or colder air, and perha ps other. The effect of these influenc e s is beyond the scope of this paper, but is considered by Lest e r 1 3 and by Neiber ger 1 5. An a pproa ch to the problem usin g the 6. methods outlined in their papers should yield satisfac tory results. Perhaps the most direct application of t he data given here is to the problem of timing the dissipation of stratus. By using methods developed a t Calif- ornia Institu:t;e of Technology, Lester was able to predict the time at which stratus at Oakland .Airport would break within ten minutes. As an example of the application of this method to a problem at ano ther station, consider the forecast for dissipation of the fog at •San Antonio, Texas in the early morning hours of May 7, 1941. The dat a available by 0535 EST included: Radiosonde Observa tion at San Antonio 01 EST 5-7-41 Elev. p T RH w 2 4 7 16 24 3.8 989 967 930 845 765 644 20 87 73 76 52 37 29 13.2 15.7 15.9 9.2 5.2 2.3 26 24 20 15 4 Pilot Balloon Observation at San Ant onio 23 EST 5-6-41 Alt. Thsd. Feet Direction Vel.Mph surface 120° 1 2 140d 160° 3 4 5 200° 270° 320° 6 320· 5 16 15 5 4 6 11 7. Hourly Observations at San Antonio May 7, 1941 ZN ZN ZN ZN ZN ZN ZN 0035 EST 0135 r.;~T 0235 EST 0335 EST 0435 EST 0458 EST 0535 EST C C C C 085/70/66"'-5/979 091/67 /65t,.__ 4/981 091/68/65t~5/981 0 088/68/65t 8/980 1fD 1/z. r X 085/68/68 t«:. 6/979/803 Spl C ;~&r-- 088/69/68t8/980 J ED If"~ 088/69/68{8/980 N 0 0 0 50004 The synoptic map indicated nothing which would complicate the problem. The first step is to plot the tempera ture and dew point curves reported by the San Antonio Raob at 01 EST, before the fog formed. This is shown in figure 11. The structure of the lower levels of the air and the 0 C) fact that the dew point increased from 65 to 68 when fog was first reported ind$cate that the stratus was caused by turbulence as well as radiation. Since the moisture content is practically constant through the stratus this gives us some indication of its thickness. If a line of equal mixing ratio is drawn through the surface point (14.8 gr/kg.), the top of the fog will be at the height at which the low level increase in mixing ratio will be balanced by the assumed decrease just above. Once a reasonable value for the top of the fog has been found (in this c a se 962 mb.), the temperature curve on the chart can be modified to represent the struc::._ ture of the air during the fog. This is done by assum- 8. ing a pseudo-adiabatic lapse rate from the surface to the top of the fog, hence a steep inversion to the intersection with the sounding. The top of the stratus when it breaks will be at the intersection of this inversion line and the constant mixing ratio line as shown. From this intersection a dry adiabatic line is drawn to the surface, where it represents the critical breaking temperature for the stratus (72°F). In order to tell at what time 72°will be reached, the parallelogram AbCD is constructed as shown (AD and BC are adiabats; .AB and CD are isobars). This parallel- ogram represents the a mount of heat that is needed to warm each unit area column of air before the stratus will dissipa te. It may be evaluated by estimating its area and knowing how many calories of he a t the area represents on a particula r chart. A new method for ev- aluating this ener gy is outlined as an appendix to this thesis. Using this newer method, the amount of heat needed was found to be 19.6 calories per square centimeter. Reference to figure 3a using the 22% scale, and interpol a ting between curves for A_t-> ril 15 and May 15, we find that the 19.6 calories should be had by 0800 apparent solar time. San Antonio is 98 28t West Long- itude, so a correction of l hour 32 minutes is added to give 0932 EST a ~ the time of breaking, which is verified by the following reports: 9. Hourly Observations at San Antonio 5-7-41 ZN 0635 0735 0835 0935 1035 ZN ZN ZN :lN EST EST EST EST EST Spl Spl N 5,.(/l},q f:,t- C 0 N 5 ~ :z. {,,pN 7 .e ~ c,,F'~ C 70 cV 091/67/66f11 6/981 091/68/67/6/981 ThN SPOTS 102/69/67f~8/984 BRKS 095/72/68 fl3/982/~VG N 105/76/69t10/985/105 50094 Application to situations with greater complexity ~hould be made possible by the empirical calculation of corrections to be applied to one or another of the values used. These data can also be used in making maximum temperature foreca.sts. As an example , consider a f'orecast for the maximum temperature at Ely, Nevada May 16, 1941: The data available before sunrise indicates that no clouds would be expe cted, and that the air mass over the station would not change during· the day. The upper air sounding gave the following data: Radiodonde at Ely Nevada, ol EST May 16, 1941. Elev . 19 20 32 44 57 71 p T RH w 812 804 692 601 507 417 5 12 3 -7 -11 -23 58 55 42 52 39 37 4.0 5.6 2 .9 1.9 1.3 0.6 This sounding is shown plotted on an adiabatic chart in figure 12. If one assumes that under such conditions, 70% of 10. the total i nsolation will be appj_ ied for warming the air, and t hat t he maximum t emperature will be reached ne ar 3 p.m. local time, figure 4a (using the 70% scale) shows tha t 540 calories of heat will be available. It is possible by calculating the amount of heat needed to r a ise the temper a ture to several trial values to bring this amo unt of heat to 540 calories. In this way it is found that a maximum temperature of 76°F is likely. 0 The maximum tempe rature reported was 73 F. The cause for the error (though slight) seems to have been the occurrence of cirro-stratus clouds by mid-morning with a gradual increase to four or five tent hs by 1630PST. If the problem be worked agai-n, assumi ng an average cloudiness of 2/10 of the sky, and hence the 60% applic at ion of the total insole.tion, the results are much better. This new a ss~mption shows 460 calories of heat av a il able, and a max imum temperature of 74 F--as close a s graphical methods justify. As a check on the calcul a tion, let us see by what time of day the temperature should ha~e re&.ched 66° F, the value repo rte d on the 1030 PST report. This is much easier to calcula te than the maximum temperature because there is no need for a trial and error solution. Dr awing an adiab 2. t through the surface pressure at 6~F, we find that 326 calories per square centimeter 11. are needed to warm the a ir to this value. Going ag ain to figure 4a and using the 60% scale, we f ind that this amount of he a t should be ava ilable by 1035 apparent solar time . This is equivalent to 1015 PST ---within a few minutes of the time at which the ther mometer should have been re ad. Neiber ger 15 has applied t he We a ther Bure a u radia tion summaries of Hand 7 to the pro blem of forecasting maximum tempe r a tures a t Chic a.1:=; o and Joliet, Illinois on c alm, cloudless days. He recommends the work of Thornthwaite and Holzman 17 , of Simpson20 , and of Homen1 6 as helpful in eva luating t he portion of the i nsolation applic a ble to his problems. His method of eva l ua tir1g the nw~ er of ca lories needed to warm t he air to an adiab a tic l ap se r a te eliminates the use of tria l and error solutions and is otherwise intere s ting . The dat a can a lso be a pplied to the solution of mor e co mplic a ted problems . As an example, let us con- sider a forec &st for the time at which cumulus clouds will form ove r St. Louis, Mo. on the morning of May 12, 1941: The data avail able by early mo rning hours included: Radiosonde Obs e rvat i on a t StA Louis, 01 EST, 5-12-41 Elev p T RH V\ i 2 998 980 855 12 13 3 65 62 51 5. 8 6.0 2 .9 3 14 I 12. s t. Louis Raci. ios onde Cont Td . Elev. 22 31 48 50 56 p T RH w 744 698 556 544 496 -4 -7 - 20 -20 - 24 72 53 43 44 39 2.7 1.8 0.7 0.6 0 .6 0 The minimum temperature was 45 F. l ess . The sky vvas clo ud- No change of a ir mc.ss or f ront al ac tion was ex- pec ted. The rad i osonde obs er va t i on is plotted on the ad i a bJ.tic char t s hown i n f i gure 13. The curve i s modi f ied ,. to show 45 Fas the surf a ce value, with the new invers i on drawn at a right an 6 le to the tempera ture curve . Ins pe ction shows tha t cumulus clouds may be expe cted by t he time the temperature reache s 62°F . The heat needed to raise the surface tempe r a ture to 6~F i s 136 Y' c a lories pew square c ent i met er. Fi gure 4a, with a see.le for ?O<f{ use of the max imum insolation, shows that cumulus should be observed by 0915 app G. rent solar t ime , or 1015 ES T (s t. Lo u is i s 90 28 1 ~ e st Long i tude) . A few cum\11 us wer e r e ported a t 103 5 EST , with r c:.p idly i ncreasinc: &mounts ciur i ng the· next t wo hours : Li:; LS LS LS 0935 EST 1035 ES T 1135 Et: T 1235 EST 200/60/47~7/0ll 196/63/45~ 11/011/102 193/65/ 45P 10/ttno 190/66/ 41~ 5/009 It is hopeci that in t he f uture ti1e use of these d~ta c ~n be ext ended to other problems such as the 10399 13. l ake cmd s ea bre eze s; i ns t a b i l ity cloudiness, a ir ma ss wE. rming , snow melting , r a tes of eva po r ation , a nd pe rh ap s othe rs. Table 1. A List of Suggested Albedo Values. Surface Albedo Source stratus Clouds Fresh Fallen Snow White -Paper White Sandstone Clay Ma rl Moist Earth Water 0.78 0.783 Aldrich 2 Zollner 11 Cloudless Sky 0.1 Cloudy 0.17 0.22 0.3 0. 4 11 0.32 ti 0. 37 0•5 o.6 ti ti · 0.9 1. 0 Water " " 11 11 11 ti 0.156 0.079 0.021 " 0.43 0.48 o. 54 Tl 0 •7 0. 8 0.700 0. 2 37 Tl II ti II Simpson20 after Angstrom &: Aldrich " " II ti 1l II 0.61 0.67 t1 0.74 11 Angle of Incidence 11 Albedo 40 0.01 8 0.022 0.065 80 89.5 0.333 0.721 0 60 Source Tyndall-10 " . 11 11 II i3IBLIOGRAPi.-lY 1. ABBTTI, GIORGIO. The Sun. New York, D. Van Nostr and Co., Inc., 1938. 350 pp . Illust. Biblio g . 2. ALDR IC11 , L. .di. fl The Reflecting PoVie r of Clouds". Smithsonian Mis c. Coll., Vol. 69 # 10, Washington D. C., 1919 (Publ. #2530). Also in:Ann. of As tro. Obs. Sm. In s t. Vol LV, P 375, Washingto n , D. C. 1922. 3. BRUNT, DAVID. Physic a l and Dynamic al Meteorolo gy. New York, The Macmillan Co., 1934. Sections on Radiation. 4. Page 100 contains da ta from Angotrs works originally published: Ann. Bur. Cent. Met., Paris 1883 , Part I, pp 136-61~ ELSASSER , VvALT.c.iR iVl. "An Atmos pher ic Radiation Chart And Its Use 11 • Quarterly Journal of the Royal Meteorologic al Society. Vol.LXVI Supplement 1940. 6. ELSASSER, WALTER M. "Rad i a tive Cooling in the Lower Atmosphere 11 • lVionthly V:e a ther Review. Vol.68 1940 (July) pp. 1 85 -188 . 7. HAND, I RVL.~G F . 8. HUlViPHREYS , v~ . J. Phys ics of The .Air. Book Co., Inc., New York. 1929. 9. KINlliALL H. . H. "Some Causes of The Varia bility of Earth Shine 11 • Monthly Wea ther Review May 19ol, P 20 9. Contains da t a f rom: 10. TYNDALL, JO HN . P 17. 11. LJOLLN ER , J. C. :F'. Photome trische Untersuchungen, Leipzig , 1 865. 12. KRICK, IRVING P. "Elements of Weat her Prediction". Ca lifornia Institute of Te chnolo gy, Pas adena. "Review of United Stat es We a ther Bureau Solar Radiation Investigations 11 • Monthly We a ther Review, Vol. 65. De c. 1937. pp. 415-41. Illust. McGr avv- Hill Six Lectures on Light. London 1 873, ii 13. LE0Tii;R, RAY1v10ND T. 1st Lieut., A . C. 11 Forec2.sting The Dis sipation of Fog in The San Fr anc i sco Bay Re gi on 11 • Un pub lished Thesi~, At California Institute of Technology, P2.s 2.dena- 1940. 14. 11 15. NEioERGER, M. "Insola tion and The Prediction of Maximum Temper a tures". Bull. Am. :Meteorol. Soc. Vol 22, M.arch 1941, pp 95-102 . . Contains data. from: 16. HOMEN , TH . Der Ta. glische W&rme umsa tz im Boden , Leipzig, 1896. 17. ThOHNTHVvAITb , C. Vv . and HOLZi:viAN, B. "The Determination of Evapor a tion f rom Land And Water Surf aces. 11 Monthly ·weathe r Review, Vol 67, pp 4-11. 18. Reducti 8n of Obs e rvation s to The Solar Constant. Annals of Smithsonian Astra. Obs. Vol 2, P 56; Vol 3, P 28. 19. SLviPSON, G. C., C . .t.:l., F . H. S., F., R., Met Soc. 11 Further Studies in Terrestrial Radiation". Me m. R. Me t. boc. Vol 3 #21. London. Edw . Stanford, Ltd., July 1928. 20. Silvi.PSON , "The Distribution of Terrestrial Radiation". Mem. R. Met. Soc., Vol 3, #23, London, Edw. Stanford, Ltd., April 1929. (13 fi gs --15 t ables). MetLod of Solar Const ant Obs e rv &.t i :Jns n. Annals of Smithson i an Astra. Obs. Vo l 5. pp 110-120. Ap pend i x If a small anount of he a t /Q be 2dded to a parcel of a ir, the temperatur e incr ea se Jr pro Quc e d i n the air will be given by t he equa t i on : vv'he re : Cpis the s pec i fic he& t of t he air a t const a nt Jressure . i s the ma ss of the a ir. ;V/ I f the parcel of a i r ha s a cross section a rea of one s quare centimeter it s ma ss will be given by: I:, -P /111::: -rJ where ? . is the pressure a t t he top of the column, and J f,, af -1-Ae /JqJ-e • is the a ccele ratio n due to gr avity. Our equa t i on then becomes : cl {j) ..,, T (!?-?) df a1/4rin ,.o-"r Cm.~ or if !:-? be in mill i ba rs , Cp ~ J,, ,9'/d C,,mJ_/AUc,3: / ~ . z J f (, : dt? ~ _2,1,1/(1?.-P)J'!; ,,.-,-- where ? and ~1"~: ~ refer t o the a ir be - fore the &.dd i tion of he a t, and T an d refer to it afterward. Q 2a . The qu a ntity Q - Q., i s the a mount of he a t v;e want to 't:.novv , a na" can -oe 1., ound.- i• i' ' - -t .ne va 1 u es o f r,O toC? 1 k:.n.own . 1,,,a nd- % "" 2.re In solving c:. s tr c:. tus dissipc. tion pro b lem, the difference betv!een the minimum and critic a l temperatures i s multiplied by t he d e pth of the l a yer a s mea sured in millibars a nd by 0 . 244 to obt a in the v a lue needed . Th e ev2.lu2. t i on i s e x2.ct a nd t a kes but c. mo ment on a slide rule .