All-101 ■ 80 



sphere. We then have 



p ar ^ ^-^^ 



M + u $L + 1 av ^ ___w_ _^ + uv . wicot__9 , Q^v sin Q cos 



at ar r ag r sin a9 F ^r — 



-2wQcos = - i I 9J2 + Ay2v + 1 .L(a^ 9.Y)(2) 

 prao ^ par3ar 



9^ + u9il + X 9w ^ _„w__ aw + wu _ vw cot Q , Pv Q ooq 

 at ar r aQ r sin acp r ^ ^v ^^ cos « 



- 2u Q sin Q=-l — .4^"^ ^ + AV^ w + 1 A (A, ^) (3) 



p r sm acp p ar 3 ar 



where g = g' _ J,.(|.Q2p2 ^^^2 g^ ^^^ ^j^^ apparent gravitational 

 force. The viscous terms for equations (2) and (3) are 



A V^v = A_ J eot 9X + £x + __i_ jHz _ v _ 2 cos Q .^aw \ 

 r^ [ 9® 902 sin^O 89^^ sin^© ' sin^Q acp J 



A v^w = A. Lot 9 ^11 + 9i^ + 1- ^ . _V - ^-2^^ ^^ 

 r"^ L °^ ae^ sin'^© ocp'^ sin^g sin © ^9 



Since the region of interest to us consists of a very 

 thin layer on the surface of the globe, we shall approximate 

 r by R, the mean radius of the earth, v/henever r appears in 

 undifferentiated form. At the same tine let us define a new 

 east-west coordinate by x ■- cpR sin 0, a north-south coordinate 

 by y = R(2 - ©) and a vertical coordinate by z = r. Then 

 equations (l)-(3) become 



1 IP. = g (1+) 



p az ^ 



