Convection Currents in the Atmosphere. 455 



The ratio of the term y(ps+p±)/9 to /*3 2 h(p 3 + p 4 )ly is 

 of order (4o> 2 -y 2 )\\ 2 gh, or about lb~ 17 \- 2 . Thus, if the 

 radius of the region considered is large compared with 

 10,000 km., the ratio is large ; similarly, the ratio of 

 the two terms in gabjv is of order (4go 2 — y 2 ) vj\ 2 g, which 

 is of the same order but somewhat larger. Hence for 

 large areas we can neglect the last two terms of (35), and 

 we have approximately 



p3+Pi = 9*I>lv, pi=p 2 = 0. . . . (36) 

 Similar results will hold for the term in e~ mz . 

 Hence the pressure variation is practically given by 



U = ^(l-e-")+^(l-e-™ ).. . . (37) 

 v m 



Thus the pressure variation in the upper air has the 

 same sign as the temperature variation ; if v is real 

 the phase of the first term is the same, but the term 

 in e~ mz would give rise bv itself to a lag of 45°. There 

 is no pressure variation at the surface. The vertical 

 velocity can be neglected in the equation of vertical 

 motion, as the terms depending on it are of the order 

 of 10 ~ 15 of the others. 



If the radius is less than 1000 km., and it is justifiable 

 to neglect the frictional terms in (35), we should have 



p z -\-p± = goch(l — e~ vk )/v 2 h + a similar term, 



while p ± and p 2 are of order (kX 2 j(o)(p z +p^). Then, as 

 these are small, we have approximately 



n = £?(ir£L,-\ + £?(iz*^_,-A (38) 



v \ vii J m \ mil J 



The pressure variation in the upper air then has the same 

 sign as the temperature variation, and that at the surface 

 the opposite. As before, the velocity can be neglected in 

 the equation of vertical motion. In the extreme case when 

 vh is very small, the pressure variation at the free surface 

 arising from the first bracket is given by W — \gabh and 

 that on the ground by —\gotbh, vanishing at a height 

 z=±h. 



The neglect of second order terms needs justification. In 

 place of the accurate operator 



dt~ bt^ dr^ r B^ + M 'V 



we have always used _— , and it is necessary to show that 

 ot 



