By Susan Friedunder (Eds.)
Friedlander S. An advent to the mathematical conception of geophysical fluid dynamics (NH Pub. Co., 1980)(ISBN 0444860320)
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Extra info for An Introduction to the Mathematical Theory of Geophysical Fluid Dynamics
V -m ] v2P+ 4 2 az 9 = 0. I n t h e case of the geostrophic approximation E = 0, and z. 7) reduces t o which is consistent w i t h t h e r e s u l t t h a t of 5 P i s independent The reduced i n v i s c i d equation i s n a t u r a l l y of lower 37 The Ekman l a y e r order and hence can not be used t o s a t i s f y boundary conditions on t h e t a n g e n t i a l , as well a s , the normal components of velocity. A s i s customarily the case i n a problem i n f l u i d dyna- mics, f r i c t i o n a l e f f e c t s must be included i f t a n g e n t i a l veloc i t y conditions a r e t o be s a t i s f i e d a t t h e boundary.
Illustrate t h i s statement by considering t h e following mathematical problem. Consider a Taylor Column bounded between h o r i z o n t a l p l a t e s a t z = 0 and Neglect t h e e f f e c t s of boundaries i n the y z = 1. x and directions. Show t h a t the v e l o c i t y of the f l u i d i n the Taylor Column moving i n a s t r a i g h t l i n e s a t i s f i e s the l i n e a r geostrophic equations. Determine a n O(E) c o r r e c t i o n term t o the v e l o c i t y t h a t s a t i s f i e s an Oseen approximation t o t h e nonl i n e a r momentum equation.
I n c y l i n d r i c a l co-ordinates the Ekman l a y e r equations a r e again I f we consider the case where the boundary condition i s 2 = rO at 5=0 and assume t h a t t h e i n t e r i o r flow is un- a f f e c t e d , then t h e s o l u t i o n I n the Ekman l a y e r i s 5 + iG $ = e-5 thus - = i r e ( l+i 1t , sin 5, $ = e-5 Hence a hodograph plane diagram i n which n against V 5 cos 5 . is plotted gives the c h a r a c t e r i s t i c Ekman l a y e r s p i r a l , which is sketched i n Figure 8.