590 



high Reynolds numbers will lie in the inertial sub- 

 range. They will have a spectrum function 



Ei(k) 



2 



e 3 k 

 so 



k ~ C 



(23) 



where k is the wave number so that the contribution 

 to the vertical velocity is EgO C • This is much 

 larger than the contribution from the flattened 



eddies so that w„ 



UH^/^ or w,~el/3 ^l/\ as 



-SO 



derived rigorously by Hunt and Graham (1977) . 



In the mixing experiments the surface at z = D 

 is not rigid but is agitated by disturbances of 

 amplitude 62- Assuming that eddies of this size 

 are in the inertial subrange, we get vertical veloc- 

 ities of order el/Sgl/B ^^^ again these, rather than 

 the eddies of size D, contribute most to the rms. 

 Then W2 - £"^'^52^^^ ■ Since e - K^/d'* , we get, as 

 in MISF, 



_3 1_ 1 



(26) 



(27) 



~- Bliqil2 = - D ^ + n2d ^^f- (D + h) - qj (28) 



_d 

 dt 



-a„q 



2^1 



, h N dAb Ab dh 1 ., dD 

 ■5' ^1°^ 3J^^T^+ 2 ^^iT 



1^, d (D + h) 



n2{D + |h) ^^ 

 2 dt 



(29) 



3 4 



62K 



I=\ 



3 3 

 '^'^^ where 0.2 = A Bjj/B3 . 



The problem of the present paper is somewhat more 

 complicated but the distorting effect of the inter- 

 face should be the same since the buoyancy varia- 

 tions in the mixed layer are very small. The air 

 in the main portion of the mixed layer has velocities 

 of order (qiD) ^/^ rather than K/D and in R2 the 

 buoyancy flux is similar to that in the case of 

 mechanical stirring. Equation (24) takes the form 



3 

 W2 



= Bnqi 



(25) 



This result, together with (19) , implies W2 ~ 

 w*Ri~^(h/D)^ , where Ri = DAb/w^ , and differs 

 fundamentally from that of Tennekes (1973b) who 

 assumed W2 ~ w^ by arbitrarily equating the buoy- 

 ancy flux and the energy flux divergence. Tennekes 

 has acknowledged [Zeman and Tennekes (1977)] the 

 inadequacy of this assumption. 



The drop-off of w as the interface is approached 

 is revealed in the data of Willis and Deardorff 

 (1974) . As shown by Hunt and Graham (1977) , the 

 total kinetic energy is the same near the distorting 

 surface as it is far away so that the horizontal com- 

 ponents of rms velocity should increase toward the 

 interface. There is an indication of this also in 

 the data of Willis and Deardorff. 



It is also interesting that we may predict the 

 same type of behavior near the lower heated surface. 

 In fact, earlier data of Deardorff and Willis (1967) 

 as well as the more recent data of Willis and Dear- 

 dorff (1974) show that the vertical velocity near 

 the heated plate increases with height, roughly in 

 accordance with similarity theory [Prandtl (1932)], 

 but that the horizontal velocity decreases with 

 height. Thus, it is possible to apply similarity 

 theory to obtain the vertical component, w, but not 

 to obtain the horizontal components, u and v. The 

 dimensional analysis for the horizontal components 

 at large Rayleigh number must include D as well as 

 qj and z no matter how small the ratio, z/D! There 

 are experimental indications that the classical 

 arguments of "localness" are also incorrect in prob- 

 lems of turbulent shear flow [Tritton (1977, p. 283)]. 



Using (25), the relations in (18)-(20) and the 

 expression for W2 are 



5. DIFFERENTIAL EQUATIONS 



Equation (29) is a single differential equation in 

 three unknowns, D, h, Ab. Let us now seek additional 

 information. The quantity, Uo/^s, is the dissipa- 

 tion in the turbulent patches in the interfacial 

 layer. We have seen that it is independent of Ri 

 in the lower portions of the layer. Obviously it 

 will vary continuously with C (now defined as z - D) 

 in the layer and, to the first order, will remain 

 independent of Ri although it may vary with the 

 quantity s = N h/Ab when the upper fluid has a 

 linear buoyancy field. We may therefore write 



3 

 "3 



qi i 



§- 



or using (14) 



-3 3, .3 

 k 2/ h V f i; 



'3 ^A^J nU' 



(30) 



(31) 



We may obtain another expression for U3 by in- 

 tegrating the energy equation over the interfacial 

 layer. We have already seen that |e3|--|q3| and 

 assuming that the energy flux is proportional to 

 U3 in this layer*, we have from the energy equation 



. 3 

 9u3 



3C 



Bl2q3 



(32) 



Using (8) and integrating, we get 



,3 = 



Wo + B 



12 



dAb f C,^ ^3 



'^^'^ ^^z\—--^ 



.^ , C^ dh (;2 (jp 



C^ d 

 n2|- (D + h) 



(33) 



We have seen that the energy flux at the bottom of the layer 

 is proportional to u^q. To the first order it should be pro- 



portional to U5 in the rest of the layer, i.e. 

 Ri. 



independent of 



