Linear Theories — Viscous 



(Stommel, 1948), to show the important efifect on the transport hnes which 

 results from the variation of the Coriohs parameter with latitude. 



A rectangular ocean is envisaged, with the origin of a Cartesian- coordinate 

 system at the southwest corner (see fig. 54). The y-axis points northward, 

 the X-axis eastward. The shores of the ocean are at a: = 0, r, and y = 0,b. The 

 ocean is considered as a homogeneous layer of constant depth D when at 

 rest. When currents occur, as ia the real oceans, the depth differs from 

 D everywhere by a small variable 

 amount h. The quantity h is much 

 smaller than D. The total depth of 

 the water column is therefore D + h, 

 D being everywhere constant, and h 

 a variable yet to be determined. 



The winds over the ocean are the 

 trades over the equatorial half of 

 the rectangular basin, and prevaUing 

 westerhes over the poleward half. 

 An expression for the wind stress 

 acting upon a column of unit hori- 

 zontal area and depth D-\-h must 

 include this dependence upon y. 

 A simple functional form of the wind 

 stress is taken as — i^cos {nyjb). 



To keep the equations of motion as simple as possible, the component 

 frictional forces are taken as — Ru and — Rv, where R is the coefficient of 

 friction, and u and v are the x and y components of the velocity vector, 

 respectively. The Coriohs parameter / is also introduced. In general, it is 

 a function of y. 



The vertically integrated steady-state equations of motion, with the 

 inertial terms omitted, are written in the form 



Fig. 54. Coordinates and boundaries 

 of the rectangular ocean basin used by 

 Stommel (1948, fig. 1), but with notation 

 changes. The x-axis points toward the 

 east ; the y-axis to the north. The dimen- 

 sions of the rectangular basin are r and b. 



Try ^ ^ , Sh 



= fm + h)v-Fcos-^-Ru-g{D + h)^, 

 •^ ox 



0=-f(D + h)ti-Rv-g{D + h) 



dh 



dy' 



(1) 

 (2) 



The quantities u and v are taken to be independent of depth, an assump- 

 tion which simplifies the analysis, but requires regarding the wind as 

 essentially a body force instead of a surface stress. To these, the equation of 

 continuity must be added: 



8[iD + h)u] d[{D + h)v] 



dx 



+ 



dy 



= 0. 



(3) 



