By Susan Friedunder (Eds.)
Friedlander S. An creation to the mathematical thought 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
Hence, a s we would expect from t h i s system of equations where each term i s 0(1), the time dependence decays t o give a steady s o l u t i o n on a time s c a l e of O ( 1 ) . I n other words, the steady Ekman layer has developed a t t h e boundary a f t e r a couple of revolutions. The steady Exman l a y e r w i l l now modify t h e i n t e r i o r v e l o c i t y by means of the c i r c u l a t i o n induced by E K M n layer suction. 49 The Ekman l a y e r This process i s known a s spin-up ( o r spin-down i f t h e boundary v e l o c i t y i s decreased r e l a t i v e t o n), and t h e time taken f o r the i n t e r i o r v e l o c i t y t o reach a new steady s t a t e i s c a l l e d t h e spin-up time s c a l e .
Eo ‘L cos -z eo[e-e,l sin + .... 14) az We have therefore approximated the problem of flow i n an ocean basin by flow on a plane r o t a t i n g about an axis ponding t o ^r a t point (610,$o)), with A kt an angular v e l o c i t y which i s no longer constant, but v a r i e s l i n e a r l y with [Note increasing on the plane]. y (corresy. corresponds t o the northward d i r e c t i o n I n l a t e r chapters we w i l l discuss i n d e t a i l the very i n t e r e s t i n g e f f e c t t h a t t h i s l a t i t u d i n a l v a r i a t i o n has on the behavior of a r o t a t i n g f l u i d , and show how i t can be used t o explain observed oceanographic phenomena.
8) v cos 8 cos 8 everywhere -, 0. The narrow s h e l l assumption a l s o enables us t o approximate radius % of the sphere. 9), = -g. 11) a r e the geostrophic approxima- t i o n f o r a spherical s h e l l . The equations f o r geostrophic balance i n a s p h e r i c a l s h e l l a r e of course s i m i l a r t o the Cartesian equations ( 4 . 4 ) . " Consider an ocean, o r s e c t i o n of the atmosphere centered a t a c o - l a t i t u d e 8*: we then assume t h a t the l a t i t u d i n a l s c a l e of the problem is small enough t h a t we can neglect the curvature of the e a r t h and approximate the Geos t rophic flow geometry by a tangent plane centered a t E The tangent plane FIGURE 4 29 8,.