SECT. 2] LARGE-SCALE INTERACTIONS 191 



a. Momentum budget along a trade-wind air trajectory 



We consider first the momentum budget along the Pacific trade trajectory 

 of Fig. 31 ; the situation along the Caribbean trajectory of Fig. 25 is qualita- 

 tively similar. In the natural co-ordinate system, s is chosen along the two- 

 dimensional flow, u{s, z). The vertical axis, z, points upward and the normal 

 axis, n, completes a right-handed system. In steady state, the s-component of 

 the equation of motion is 



8u du I'l dp F\ , , 



The frictional force per unit volume, F, is well approximated by 



F = ^. ,34, 



where the shearing stress component, tsz, is the vertical transport of s momen- 

 tum by convective, turbulent and smaller scales of motion. 



Upon multiplying through by p and integrating over a volume bounded by 

 ds, dz and of unit thickness in the normal direction n, we have 



pu — ds dz+ \\ piv — ds dz+ \\ -^ ds dz— I — {tsz) ds dz = (35) 



which becomes 



puudz— puudz+ puwds— puwds+ {pn s —pds)dz 



Ju.s. jd.s. Jc Jb J 



[{Tsz)t-{Tsz)b]ds = 0, (36) 



-/■ 



where the subscripts "u.s," and "d.s." indicate upstream and downstream 

 respectively ; t and b refer to the top and bottom of the layer. 



In equation (36), we are considering the budget of large-scale momentum, 

 namely that of the trade-wind flow, u. The first four terms are thus import and 

 export of this scale momentum by u itself and w, the overall sinking motion 

 produced by the divergence field. The latter is almost entirely two-dimensional 

 and well approximated by dujds (Fig. 31d), so that w results from the down- 

 stream acceleration of the trade. The fifth term is the local production of 

 momentum by pressure forces, and the sixth term is the transport divergence by 

 the much smaller scales of motion which are parameterized here in terms of 

 the shearing stresses they produce. 



At the surface, the frictional term tszq = pCoUa^ (equation 17). In the Pacific 

 study of Riehl et al. (1951), this term was computed twice daily at each of the 

 four observation stations and its component along s was found by multiplica- 

 tion with the cosine of the angle between the individual wind direction and 

 s; Cd was taken as 3 x 10~3. The horizontal transport terms were evaluated 



