Appendix I. 



Physical Expressions for the Problem 



Sverdrup (1947), Munk (1950), and Fofonoff (1962) have developed expressions relat- 

 ing total transport to the curl of the wind stress. All three formulations were predicated 

 on a velocity structure such that velocity vanished well above the bottom. Thus, it was 

 not necessary to consider the effects of bottom topography on the flow. In the present 

 problem, portions of these existing solutions applicable to central regions are employed. 

 Approximations and assumptions are made pertaining to the effects of bottom topography 

 on the solution for transport. 



The development is based on Fofonoff s (1962) treatment of Munk's (1950) wind- 

 driven flow formulation which is in turn, developed from the momentum equations for 

 steady motion. In a coordinate system where x is positive eastward, y is positive north- 

 ward, and z is positive upward and assuming incompressibility and negligible molecular 

 stress terms, the following equations are obtained: 



+ ju, w V 2 u + fi v -^ 



dhi 



dz 2 



+ /U,//V 2 l> + p,\ T-~2 



dz 



(la, b, c) 



and continuity is expressed by 



d(pu) d{pv) | d(pw) = Q 



dx dy dz 



Assuming terms with vertical velocities are negligible, the vertically integrated equa- 

 tions of motion can be written as 



["dpu 2 , hdpuv, , f" , f v dP , , . f" (d z u^d 2 u\ , , f" A d 2 u . 



p B&, dz+ p ^ p p d^ dz + AH p g^+0) *+fW£ A, 



Jb dx J b dy Jb Ji, dy J b \dx- dy 2 ) Jb r dz~ 



where /is the Coriolis parameter, p is mean density, and A H and A v are the horizontal and 

 vertical kinematic eddy viscosities, respectively. 



Convenient terms for the integrals are derived by the following definitions and argu- 

 ments. First, define the components of horizontal momentum transport 



uv= puvdz, u 2 = pu 2 dz; (3a, b) 



Jz h Jz h 



and the components of mass transport, U and V, 



U= I pudz, V= I pvdz, 



)z b Jz b 



■■ f" pudz, V= P 



Jz b Jz b 



f*dP , d f" „ dry 326 ,. , , 



— dz = — Pdz-P-n- — \-Pb- ; (4a, b) 



J* 6 dx 3* J 2ft ax dx 



19 



