MECHANICAL EQUIVALENT OF PRESSURE MARGULES 511 



(b.) Whirl on a revolving horizontal plane 



For the case of relative motions above a small area on the surface 

 of the earth we simplify the equations by the assumption of a con- 

 stant polar distance and thus attain an approximation to terrestrial 

 conditions that suffices for certain special cases. 



We consider the substratum of the atmosphere as a rotating plane. 

 In order that the atmospheric pressure in the condition of relative 

 rest may be a function of the altitude alone, we must also assume 

 a force directed toward the axis of rotation, opposite and equal to 

 the centrifugal force. The location of the foot of this axis may then 

 remain arbitrary. 



Now the distribution of pressure in a stationary cylindrical whirl 

 with vertical axis is given by the equation 



l d ± = 1 + jG (0 



/n dv r 



where G designates the velocity of the air relative to the rotating 

 surface of the earth and is positive when the direction of rotation 

 of the whirl agrees with that of the earth [as in cyclones]. The con- 

 stant factor of the deflecting force due to the earth's r tation is 



j = 2 v sin <p 



where v is the angular velocity of rotation of the earth and <p is the 

 geographic latitude of the place. 



The potential energy of the distribution of pressure is not changed 

 by the rotation. But for positive G, the kinetic energy of the rela- 

 tive motion is smaller than it would have been for an equal pressure 

 gradient and j = o. Therefore the ratio K/A is also smaller. 



If Gjjr is very large then in the equation (/?) the first term on the 

 right-hand side exceeds the second. In the case of cyclones in low 

 latitudes whose horizontal extent is relatively small, we can esti- 

 mate the value of K/A approximately according to the example 

 just given, where K/A = 4/c. 



At latitude 15 and for r = 100 km. we have jr = 3.8 meters 

 per second. If now G is five times larger than this, then the right 

 hand of equation (/?) becomes (25 jr + jr) so that for such cases 

 the neglect of the second term will incur an error of 4 per cent or 

 less. In middle latitudes (40 to 50 ) the two terms are in general 

 of the same order of magnitude. 



