By Susan Friedunder (Eds.)
Friedlander S. An advent to the mathematical concept of geophysical fluid dynamics (NH Pub. Co., 1980)(ISBN 0444860320)
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Extra resources for An Introduction to the Mathematical Theory of Geophysical Fluid Dynamics
B a l l a t the bottom of the c y l i n d e r . Set the cylinder on a uniformly r o t a t i n g t u r n t a b l e and allow enough time f o r t h e f l u i d t o achieve r i g i d body r o t a t i o n . Without d i s t u r b i n g the f l u i d , move the small b a l l slowly with v e l o c i t y 1 r e l a t i v e t o the r o t a t i n g f l u i d ( t h i s could possibly be done with a magnet if the b a l l were magnetized). physical parameters c),v,U,L such t h a t so t h a t the flow i s geostrophic.
2 0 ) which was d e r i v e d a s a boundary c o n d i t i o n h o l d i n g a t f o r a l l v a l u e s of and z=O, must i n f a c t hold z, s i n c e t h e q u a n t i t i e s involved, wl, go €&, a r e a l l independent of z. We w i l l g i v e a n example t o i l l u s t r a t e t h e power of t h i s r e s u l t i n d e t e r m i n i n g t h e s t e a d y flow i n a r o t a t i n g c y l i n d e r . n terms of t h e dimensionless para- meters, and in t h e r o t a t i n g c o - o r d i n a t e system, t h e boundary conditions a r e We write 9 4 = 9 = re^ a t 0 at Z=O z=1.
7), E V P + 4 5 = 0. az 2 namely (5-8) We note t h a t t h e f i r s t term represents t h e viscous f o r c e and t h e second term C o r i o l i s f o r c e . T’ The Ekman l a y e r co-ordinates FIGURE 5 38 The boundary l a y e r thiclzness I n most s p a t i a l regions 2 6 E v P is very small since E << 1, however i n a narrow region c l o s e t o the boundary, the derivat i v e s could be l a r g e . layer by s c a l i n g z: We examine the h o r i z o n t a l boundary write z =: EaS. 8) becomes Retaining only the dominant terms gives If the boundary layer i s t o be n o n t r i v i a l , there must be balance between t h e two terms i n equation ( 5 .
An Introduction to the Mathematical Theory of Geophysical Fluid Dynamics by Susan Friedunder (Eds.)