450 



Dr. H. Jeffreys on Periodic 



C.Gr.S. units are employed throughout, except where the 

 contrary is stated. 



Let fl be the angular velocity of the earth's rotation. 

 Then its component about the vertical at any point is 

 Q cos#, =&) say, where 6 is the co-latitude. So long as the 

 area considered is not too large, this may be taken as 

 constant over it, and then the problem to be considered 

 is that of a plane horizontal sheet of incompressible fluid 

 with a rotation about a vertical axis. Let this be the 

 axis of z, and let the axes of x and y be on the ground 

 and fixed relative to the earth. Let also the velocities 

 be u, v, w relative to these axes. Then the component 

 accelerations of a particle are 



du 9 dv dw 



-Yr-—zcov — (D*x, -y -4 Zcoii — coy, 



dt 



d . 



dt 



dt 



. . (1) 



where -j is the total derivative following the fluid, or 



dt ox ^y dz 



As the terms of the second order will always be neglected, 

 we can write ^j~dt for d\dt. 



Now let p and p denote the actual pressure and density 

 respectively at any point ; 

 p and p their values in the undisturbed state ; 

 k the kinematic coefficient of eddy viscosity and 

 the tbermometric eddy conductivity ; 



and 



"du "fry dw 

 "dx ~dy ^z' 



(2) 



Then the equations of motion are 



d* 



2cov — (o 2 x — k\/ 2 u = 



j^ip-hW), 



^- + 2am — wry — k V v 

 ot 



"dw 



Tt 



2,„ _ 



P 



p~dy 

 i ~d 



(p-ihpBi, y, . (3) 



-k\/ 2 w = —g — (p-±kp8), 



with the equation of continuity 



i ag = 



P ~dt ' 



W 



