By Susan Friedunder (Eds.)
Friedlander S. An creation to the mathematical idea of geophysical fluid dynamics (NH Pub. Co., 1980)(ISBN 0444860320)
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Additional info for An Introduction to the Mathematical Theory of Geophysical Fluid Dynamics
5 ) . (Again f o r convenience we w i l l drop t h e s t a r s ) . Let us consider t h e case where the p e r t u r b a t i o n from r i g i d r o t a t i o n i s very small s o t h a t we can l i n e a r i z e t h e equation by s e t t i n g E = 0. 9 = + 2 Eo 9. 0. (5- 2 ) We w i l l now perform a l i t t l e v e c t o r manipulation of these equations t o reduce t h e system t o a s i n g l e equation f o r the pressure P. 3 = 0 the expression i . 3) and ( 5 . 6) combine to give ($ - ,,+4q =o.
3. wo = 0 . and Also uo + uo, vo and wo = 0. a r e independent of z and t h e boundary conditions t h e r e f o r e imply n uo = Q1 Go = Q2 wo = 0 - Uo(X,Y) - V,(XJY) at 5=0 everywhere. 13), can be determined t o g i v e The Ekman l a y e r S u c t i o n c o n d i t i o n 42 Now wo = 0, hence t h e divergence e q u a t i o n i m p l i e s uox + v - 0 . OY And f o r c o m p a t i b i l i t y , t h e imposed h o r i z o n t a l v e l o c i t y must a l s o s a t i s f y Q1 + X Since Qz Y = 0. 18) g i v e s wo = 0, t h e h i g h e s t o r d e r boundary c o n d i t i o n on w becomes i;, + w1 = o a t S=O, Z=O.
W with be zero a t Thus t h e r e can be no flow over the b a l l , and the same argument a p p l i e s a t any l e v e l of z. Hence t h e column of f l u i d above the b a l l moves as though i t were r i g i d l y attached. As an i n t e r e s t i n g v a r i a t i o n of t h e experiment, consider moving t h e b a l l impulsively along t h e bottom through t h e fluid. ( a ) I f t h e f l u i d i s not r o t a t i n g (n = 0 ) t h e b a l l moves i n a straight l i n e . slowly, s o t h a t E ( b ) I f the f l u i d i s r o t a t i n g very i s not small and geostrophic balances does not hold, then t h e t r a j e c t o r y of t h e b a l l w i l l be d e f l e c t e d by t h e r o t a t i o n .