553 



v'-kc 

 P 





 b 



(B.4) 



(B.4) is an underdetermined set of equations 

 that will yield two solution vectors with arbitrary 

 coefficients. They are not unique and any linear 

 combination will also be a solution. Two such 

 solution vectors are 



^1 = 

 where 



{0,0,1,0} 



and 



w^ = (-c,U'-kc,0,6) (B.5) 



6 = - [Q(U'-kc)-cP]/b 



We now enforce the requirement that P(w,v) be 

 zero. This requires that certain linear combinations 

 of V, v' , v", v'" be zero and these are of course 

 the required adjoint boundary conditions. 



Consider the solution vector w, . For P(w,v) to 

 be zero 



P(w,v) = v 



[U] 



This requires that 



r ~\ 

 



-ll 

 

 



= 



(B.6) 



Consider the solution vector w . 

 be zero, 



(B.7) 

 For P(w,v) to 



P(w,v) = V • [U] -w. 



> 

 (B.8) 



-icRU' + a(U'-kc) + 



-c 



U' - kc 



C 



so that 



[-icRU'k+a(U'-kc) + B]v - cav' + (U'-kc)v" 



+ c v'" = 



Since v' =0, this term may be eliminated from this 

 relationship. Thus given 







Pw + Qw' + bw'" = 



for P(w,v) to be zero requires 



V' = 



Wj (w) = cw' + (U' - kc)w 



W^ (w) 



at z 



(B.9) 



Vj (v) 



V2 (v) = [-ikcRU' + a(U' - kc) + 6]v + (U' 



+ cv'" 



at z = 



kc)v" 

 (B.IO) 



It can be shown that if (B.9) and (B.IO) are used 

 to construct P(w,v), the result is identically zero. 

 Thus (B.IO) are the boundary conditions for the 

 adjoint problem. 



The solvability condition for a problem of the 

 form [Ince (1926)] 



L(w) = r 

 W. (w) = Yi 

 is that 



i = l,n 



00 



J vrdz = Y^ V2n + Y2 V^^. 



(B.ll) 



(B.12) 



where the V^j^'s are determined such that P{w,v) 



"^ ™1 ^2n ■*■ "2 ^2n-l "''■•• ^""^ v is a solution of 

 the adjoint system. 



L(v) = 



V^(v) =0 i = l,n (B.13) 



For the present problem, only y, and Yj are 

 non-zero. By standard techniques, we have deter- 

 mined the additional linear combination of w and 



V, 



w 



Wq (w) = - — 



^ c 



Wij (w) = - w" - aw 



V3 (v) = v/b 



V4 (v) = v"/c + (o/c - Q/bc)v (B.14) 



such that the bilinear concomitant (B.3) may be 

 written in the form 



P(w,v) = Wj V^ + W2 V3 + W3 V2 + W^ Vj 



where Wj and W2 are the boundary conditions from 

 (B.l) and Vj and V2 are the adjoint boundary con- 

 ditions from (B.IO). 



Thus the solvability condition for non-homogeneous 

 problem with non-homogeneous free water boundary 

 conditions is 



J rvdz = Yi V^ + Y2 V3 



(B.15) 



at z = 





 where 



Vij = v"/c + (a/c - Q/bc)v 



Vj = v/b 



and V is a solution of the adjoint system 



L(v) =0 



Vj (v(o)) = 



V2(v(0)) = 



v(oo) = v' (■») = 



(B.16) 



APPENDIX C. 



In this section we give the expressions for the 

 various terms in (69) for the assigned form of the 

 small scale (71) . The continuity equation (73) 

 has been used to express u' in terms of w' . 

 The results are as follows 



