554 

 3t3z 



3^u 3fw' 

 3€3z * '^ 3z2 



L_ 3^w 3c 

 k 3z3 3t 



ic _3fw ^ 

 k 3z3 3x 



(CD 



93 w , y , , , 3w'3c . , 3? 3w' 

 T— r- - s^^TTTj- + k-^cw' + lekCT- ^ - lek ^ t- 

 3x3t 3x3t 3z 3x 3t 3z 



3^ 

 3z2 



V2„ = p±'. k2w 



^., 3w' 3(; „2- 

 e2ik T- -r^ + V'^w 

 3z 3x 



(C.2) 



{C.3) 



3,.,. 



V-^u 



i 3^w 



k 3zJ 



3w' S_ H 3'*w ' 

 3z "*" k2 3x 3z'* 



3|_ 3fw' 

 3x 3z2 



+ V^u + U" + U'" C 



3 ? _ i^ d\i_ . 3^ ^ 3C_ 3fw ^fw 3|_ 

 3F " " k Sz"* " ^'^3z2 ^ k2 3x 3z5 + ^ 3z3 3x 



3z 



(V^u) + U"' 



{C.4) 



(C.5) 



^V2„ 

 3x 



ik(- 



3z2 



k'^w') + £3k 



3w'3<; 

 3z 3x 



3^w' 3£ 

 3z 3x 



3x 



{C.6) 



The equation for the small scale will contain 

 coefficients involving the mean flow expressed as 

 a function of z ' . 



U(z' ) = U(z) + U' (z) I 



and U"(z') = U"(z) + U"' (z) I {C.7) 



so that, for example, the term uV w, with only 

 linear terms retained, becomes 



i2,r. 



a^w' 



uV^w = UV'^w + [U + O'C + u] [ir-r - k^w'] 



+ u'v2w - e2ikU 1^^' 

 3x Sz 



(C.8) 



and 



- i 3 w' 9w ' 



wV2u = wU" + w- [U" + U'" C] + w[- ^r-T - ik-^ 



k 3z^ 3z 



+ w' V^u 



of these, v2w and 3 u/3x2 will be ignored. 

 The viscous term is manipulated as follows 



{C.9) 



iro2 3^ 



3u, 



i.[v^ ^ _ v^i ^: 



3w 



3u 



3x 



3z' 



[V^ -^ - V^ — 



3x 



5 2,., I 



^[ik(^' - k^W) 

 R 3z'^ 



3z 



2,.,. 



k dz"* "*■ ^^ 3z2 



REFERENCES 



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Gustavsson, L. H. (1977) . Private communication. 



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J-[3k2 ^ 

 R 3z 



3^w' 



1 3=w" 



3z2 r2 3z5 



(CIO) 



The fifth derivative is obtained from the Orr- 

 Sommerfeld equation. Some cancellation occurs 

 among these terms to yield the final result (75). 



