5 6> 



Tables 698-700 



TABLE 698. — Geopotential, Dynamic Heights 



The "geopolentiar' or "gravity potential" of a point is its potential energy relative to sea- 

 level of a unit-mass situated at the point. 



In comparisons of vertical positions in dynamical meteorology, advantages result by giving 

 the heights above sea-level in terms of the potential energy possessed by a unit-mass at 

 these positions. The use of geopotential for heights is better realized in that surfaces of 

 equal geopotential are identical with horizontal or level surfaces, and, due to the geographi- 

 cal variation of gravity, are not surfaces equally distant from sea-level. 



Heights measured thus are called "dynamic heights." Defined more precisely, geopoten- 

 tial is fh 



r = - \gdh . , 



J o (i) 



where T = geopotential in absolute units, g = acceleration of gravity in meters, h = 

 geometric height above sea-level in meters. 



T has the dimensional formula [L 2 T~ 2 ], and is expressed in absolute units, the "geodecimeter" 

 when g is expressed in m/sec. 2 , and h in meters. The derived unit adopted by the Commission 

 Internationale de la Haute Atmosphere is the "dynamic meter" Hd, io m 2 /sec. 2 , after 

 Prof. V. Bjerknes 1 . Then Hd = dynamic height (geopotential in dynamic meters) is 



H d = -(i/io) 



n 



dh 



(2) 



Helmert's equation (3) is substituted in (2), 



g = — (g<t> — 0.000003086 h), where, (3) 



g<j> = g at latitude sea-level, below given point (in m/sec. 2 ), g = acceleration of gravity 

 at point in m/sec. 2 , h = geometric height of point above sea-level in meters. (2) may then 

 be integrated, giving 



H d = [g<t,/io]h - 1.543 X io- 7 /* 2 . (4) 



The following table results from (4) using g$ computed from the U. S. Coast and Geodetic 

 Survey formula: 



gd> = 9.78039 (r + 0.005294 sin 2 <f> — 0.000007 sin 2 2 <f>) 

 Neglecting the h 2 term, Hd = 0.98 h, approximately, whence, h = 1.02 H d , approximately; 

 substituting this in (4) for the h 2 term we have h = (io/gd>)Hd + (io/g^,)i. 543(1. 02) 2 io- 7 H 2 d- 

 For simplification, 9.8062, the mean value of g at lat. 45 and sea-level is substituted for g<p 

 in the second term and then approximately, 



h = (10 /g^Hd + 1.637 X 10- 7 H 2 d (5) 



Table 699 is computed from (5) and values of g obtained as before. 



References: Dynamical Meteorology and Hydrography, V. Bjerkness and collaborators, 

 Carnegie Institution, 1910; Avant-propos of the C. R. des Jours internationaux 1923, Com- 

 mission Internationale de la haute atmosphere, 1927, Secretary of the commission, c/o Royal 

 Meteorological Society, London. 



TABLE 699. 



-Equivalents, in Geodynamic Kilometers, of Geometric Heights in Kilometers 

 for Various Latitudes 



TABLE 700.- 



-Equivalents, in Geometric Kilometers, of Dynamic Heights in Geodynamic 

 Kilometers for Various Latitudes 



Smithsonian Tables 



