STUDIES IN ATMOSPHERIC ELECTRICITY 



149 



compute the other. On the one hand, to determine hy 

 one must resort to the use of average values of i^, such 

 as are obtained from figure 15 for use with each group 

 of data of table 3, together with average values of iu 

 from table 3, thereby deriving average values of h^. On 

 the other hand, if suitable average values of hu are as- 

 sumed for each of the three groups in table 3, one may 

 compute individual values of in for each of the thirty- 

 one days comprising the three groups, and the values of 

 hu chosen must be such that, when the values of ij, are 

 averaged for each group, these average values will agree 

 with those required by figure 15. Comparison of the two 

 methods of treatment will be made later, after develop- 

 ing the formula from which either i^ or hu may be de- 

 rived. 



It was stated earlier that the relation between the 

 air-earth current density, i, the columnar resistance, R, 

 and the total potential between the earth and the upper 

 conducting region, E, would be taken as 



E = iR 



(1) 



The current density, i, is obtained from the total 

 conductivity, \+ + \-, multiplied by the potential-gra- 

 dient, G, or 



i =G(A+ + X-) =G/p 



(2) 



where p, the resistivity, is the reciprocal of the total 

 conductivity. 



Now if we assume that the columnar resistance, R, 

 is divided into two parts, r' and r", and assume further 

 that the lower part, r', is given by ph, where h is the 

 height of the lower region, we have 



R = r' + r", and r' = ph 

 whence 



R = yoh + r" 

 Then from (1), (2), and (5), we have 

 G{ph + r") 



(3), (4) 

 (5) 



(6) 



If, now, we use the subscript n to indicate normal, 

 least-disturbed conditions and the subscript u to repre- 

 sent unusual conditions, we may write for unusual condi- 

 tions 



Eu = 



Gu(puhu + r(i') 



and for normal conditions 



^n - in^n 



(7) 



(8) 



If, further, the unusual conditions are local, or limit- 

 ed in horizontal extent, they are not likely to affect E, so 

 that Eu = En, and we may then write, from (7) and (8) 



^u^n 



(9) 



Finally, adopting the view that r" does not change 

 when the unusual conditions occur, so that r^' = rj,' and 



hu = hn, and in (9) we replace t{{ by Rn-Pn*Hi> then we 

 obtain 





'n '-'u 



^n Pu Pu^n 



(10) 



Substituting iu for Gu/pu in (10) 



in = Guhu/Rn + iu(l-/Onhu/Rn) (H) 



Restating (11) as an expression for hu, we have 



hu = (in-iu)^n/(Pu-Pn)iu 



(12) 



In computing, now, values of hu from equation (12), 

 for each of the three groups of data in table 3, the value 

 of Rn is taken as 1.11 x 10^ esu, a figure discussed 

 earlier, and the value of Pn as 0.476 x 104 esu, this 

 value being the reciprocal of the average value of total 

 conductivity, 2.10 x 10-4 esu, found for least disturbed 

 conditions in the Pacific Ocean from the table in sectior 

 V. For in the three values are 11.8, 12.7, and 13.3 x 

 10-7 esu as stated on page 147. The values of hu found 

 from this computation for the three groups 3a, 3b, and 

 3c, are 1.1 km, 1.8 km, and 0.2 km, respectively. When 

 the attempt is made to compute individual values of hu, 

 however, particularly for the eleven days in 3b, impos- 

 sibly large values of hu are obtained for some of the 

 days, indicating that this method cannot be used in this 

 detailed manner. 



Equation (11), on the other hand, lends itself to the 

 computation of in in a detailed manner, for the data in 

 table 3. The thirty-one individual values of in in the 

 last colunm of table 3 were computed from this equation. 

 For this computation, the same values of Rn and Pn 

 were used as in the preceding computation. An average 

 value of hu was chosen for each of the three groups, of 

 such magnitude that, when used in the computation of the 

 individual values of in, an average value of in was ob- 

 tained for each group which closely approximated that 

 called for in figure 15 for the proper latitude. That is, 

 the height of the fog, mist, or haze in the atmosphere 

 was taken as 1.0 km, 1.3 km, and 0.1 km for groups 3a, 

 3b, and 3c, respectively, and individual values of in 

 computed, from which average values of in of 11.8, 12.6, 

 and 13.3 x 10-7 esu were obtained for the three groups 

 to meet the requirement of 11.8, 12.7, and 13.3 x 10-7 

 esu of figure 15. The fact that the scatter of individual 

 values of in around the required average values is es- 

 sentially the same as that exhibited by the day to day 

 values in table 2 would point to the use of equation (11) 

 as a satisfactory procedure. The differences shown be- 

 tween the values of hu obtained by the two methods, 

 namely 0.1 km, 0.5 km, and 0.1 km for the three groups, 

 are not significant except perhaps in the case of the 0.5 

 km for group 3b. The scatter of the observed data in 

 this group no doubt is responsible for divergent results 

 from the two computations, and in this connection it would 

 appear that the method using equation (11) gives a more 

 reliable valueof the average height, hu, than the other. 



Accepting, therefore, the results obtained with equa- 

 tion (11), it appears that during the hazy period between 

 June 2 and July 2, in which interval the conductivity was 

 less than half of normal value, the height of the haze was 

 about 1.0 km. After the region of cold surface water was 

 reached, with its attendant thick fog or mist, the height 



