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An Introduction to the Mathematical Theory of Geophysical by Susan Friedunder (Eds.)

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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Example text

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 .

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