December 24, 1920] 



SCIENCE 



611 



SPECIAL ARTICLES 



LONG-TIME TEMPERATURE PREDICTION 



Ak approximate solution is here given for 

 the probable temperature at any desired place, 

 e. g.. Phoenix, Arizona, at any hour of the 

 day such as 10 a.m. on any desired day, 

 e. g., August 12. 



It is well known that the air gets warmer 

 as the day advances and cools off during the 

 night, repeating this rather regularly — also 

 that it gets warmer in the spring and cools 

 off in the fall. The normal air temperature 

 therefore is a periodic function of the time 

 with a 24-hour period and an annual period. 



The following equation expresses the tem- 

 perature T as a function of the time of the 

 year t and the time of the day 0- 



T = Ma + -^cosl -{--^ cos B. 



(1) 



2 " ' 2 



It is empirical and assumes that the annual 

 march of the temperature can be represented 

 by a simple cosine function, that the daily 

 march can also be so represented and that the 

 daily range does not appreciably change with 

 the season. 



The constants are readily obtained from the 

 U. S. Weather Bureau for any desired local- 

 ity. The first one. Ma is simply the mean 

 annual temperature of the place in question, 

 the second Ea/2 is one half of the Range of 

 the annual march, or the difference between 

 the mean daily temperatures of the hottest 

 and coldest days of the year and the third 

 constant Ed/2 is one half of the range of the 

 daily march or the difference between the 

 maximum and minimum temperatures for 

 the day. Ed remains approximately constant 

 for the United States, except for the arid 

 west. For example the equation becomes for 

 Chicago : 



T = 48 -f- 25 cos ( -f- 7 cos 6. 



Neither of the two marches exactly follow 

 the cosine law. The minimtun temperature 

 does not occur exactly half way between the 

 two maximums. As an average condition for 

 the United States it is but 9 hours from the 

 minimum (6 a.m.) to the maximum (3 p.m.), 

 but 15 hours from the maximum to the fol- 



lowing minimum through the evening. This 

 discrepancy can be almost entirely eliminated 

 by correcting the time from the nearest maxi- 

 mum on the curve in the process of changing 

 tJie days and hours into degrees, i. e., in the 

 winter time consider the coldest day of the 

 year 180° (about January 15) and February 

 15 would be 180 -|- 31° rather than counting 

 all the way from the maximum (about August 

 1), 7 A.M. would be 180+20° and 5 a.m. 

 180—12°, 2p.m. 360°— 20°. Thus the normal 

 temperature at Chicago February 15 at 2 p.m. 

 would be 



r = 48 -f 25 cos 211° -I- 7 cos 340°. 



This formula applied to the various parts 

 of the United States for various days of the 

 year and hours of the day gave a mean error 

 of 2j° F. This error was due largely to the 

 variable time of sunrise and could be cor- 

 rected if one knew even approximately the 

 time of sunrise on the day in question. 



In the arid west the daily range in tem- 

 perature is not constant but is a periodic 

 function of the time, being a maximum in the 

 smnmer time and a minimum in the winter 

 time. The reason for this is that in the 

 siunmer time heat is being received fast and 

 thus the maximum temperature attained would 

 be larger for the same time interval than in 

 the winter when the rate of absorption of heat 

 is slow. In this dry area the daily range is 

 very approximately 15° in winter and 25° in 

 summer. Assuming this range to vary as a 

 cosine function, which it does very approxi- 

 mately, the equation for the arid west be- 

 comes through the addition of one more term 



T = Ma + ^ cos < -f ^ cos » + ■^' cos e cos t. (2) 



The mean difference between the actual 

 normal hourly temperatures and those ob- 

 tained through using this equation was 

 2.75° F. 



A careful distinction should be made be- 

 tween the determination of the normal tem- 

 perature and the determination of the actual 

 temperature. The above formula gives nor- 

 mal hourly temperature and the errors are 

 almost always less than 5° F. and the mean 

 error only 2.5° F. 



