588 



52 



- Ab 



dP 

 dt 



d_ 

 dt 



Ab 



o^ /dD dh 



Using (6) we get 



qi = 



dt 



[(D + ijh) Ab - J5n2(d + h)^] 



(9) 



(10) 



The integral of (10) is 



(D + %h) Ab - kN2(D + h) ^ = v^ - q t (11) 



where 



b3, U3 , and I3 may vary with height. The turbulence 

 is certainly strongly influenced by buoyancy in this 

 layer so that kinetic and available potential ener- 

 gies [Long (1977d) ] are of the same order not only 

 in the waves but in the turbulent patches, i.e., 



B 6 b =8 6^ 

 2 3 3 3 3 h 



Ab 



(14) 



where 63 is the order of the size of the disturbances 

 and because of the tendency for conservation of buoy- 

 ancy, we assiome b3 is proportional to 63(Ab/h). Us- 

 ing (14) , Eq. (13) becomes 



Vq = (I> 







'3h„) 



Abn - 







^'' 



(12) 



and the zero subscript denotes values at t = 0. 

 Tennekes {1973b) obtained (11) and (12) with h and 

 hg missing. As we have indicated, the interfacial 

 layer thickness h plays an important role in the 

 theory of this paper. The time tg = V?/qi is the 

 time for an initial buoyancy difference to disappear 

 when the upper air has a uniform potential tempera- 

 ture [Tennekes (1973b)]. 



^3 



Bo ""S 



(15) 



Let us now find the dissipation. This occurs only 

 in the turbulent patches and we assume that the 

 local dissipation ep = f(u3,63, b3) . Since u^ ~ 

 b36 3, we get £„ ~ U3/63 and 



£3 



Wi 



BitB3U3 



(16) 



3. THE INTERFACIAL LAYER (REGION R3) 



According to the discussion in Section 1, the IL 

 in our model is turbulent with intermittency factor, 

 I3, defined here as the ratio of the volume in tur- 

 bulent motion to the whole volume*. Much of the 

 layer is in wave motion in which all of the compo- 

 nents of the fluid velocity are of the same order, 

 i.e., the ratios W3/U3 , W3/V3 are independent of 

 the Richardson number. The intermittent turbulence 

 is caused by the intermittent breaking of these 

 waves. Since the wave amplitude is of the order of 

 the wave length when the wave breaks , we should have 

 U3 ~ V3 - W3 initially in the breaking waves as well 

 and we assume this. Of course the "homogeneous" 

 fluid in the breaking patch will tend to flatten 

 out and the vertical velocities in the patch will 

 decrease relatively as time goes on. In our model 

 we ignore the patch after a time of order (h/Ab)"^ 

 and consider that the local heat transfer has al- 

 ready been accomplished. In actual fact this trans- 

 fer is accomplished by the spreading of the patch 

 over a larger time interval and the ultimate trans- 

 fer by molecular processes. Since buoyancy flux 

 occurs only in the turbulent portions of this layer, 

 we get, at any level in the IL, 



53 



BlU3t>3l3 



(13) 



where b3 is the rms buoyancy fluctuation in the 

 interfacial layer. Bj is a universal constant''' but 



The introduction of intermittency may result in confusion 

 if one inadvertently thinks of the IL as a' surface or even 

 as a layer with thickness of the order of the amplitude of 

 the wave disturbances. The latter is not excluded as a 

 possibility in this section but, in fact, as we see in 

 Eq. (26) the wave amplitude is much smaller than the thick- 

 ness of the IL so that I is not the ratio of the times that 

 a fixed point is in the upper (non-turbulent) and lower (tur- 

 bulent) fluid. 



Twe use symbols Bj , B2 , . . . to denote universal constants. 

 Later, "constants" arise which, at first glance at least, 

 may be functions of s = N2/(Ab/h) , i.e., the ratio of the 

 stabilities of the upper "quiescent" layer and the inter- 

 facial layer. We denote these "constants" by Ai,A2,... . 



Equations (15) and (16) show that £3 - q3 . Since 

 these are both dissipative, it follows that they are 

 of the order of the energy flux divergence. At the 

 upper boundary of the IL, the kinetic energy of the 

 waves has been so reduced by losses to potential 

 energy and dissipation, that there can no longer be 

 wave breaking and turbulence. Thus h is the depth 

 of penetration of the turbulence. At the height z 

 = D + h, the energy flux is too weak to support tur- 

 bulence so that it has apparently decreased to a 

 value well below that at the bottom of the IL. 

 Therefore, the increment in energy flux over the IL 

 is proportional to the value at the bottom of the 

 IL. Integrating Eq. (3) between levels in the layer 

 near the upper and lower surface , we find that q3h 

 is of the order of the energy flux just below the 

 inversion where q3 is the average buoyancy flux in 

 R3. Since the interface is being distorted by the 

 vertical motions (inducing pressure fluctuations), 

 the energy flux should be proportional to wipj/po 

 We may write 



- w" in R2 



53^ 



3 

 A2W2 



(17) 



Equation (17) has a form superficially similar to 

 that proposed by others in a number of papers [for 

 example long (1975), Zeman and Tennekes (1977)] on 

 the basis of assumptions about the size of terms in 

 the mixed layer. In present notation, these authors 

 propose q2D - w^ and this leads rather directly to 

 the Ri~ law for the entrainment. Equation (17) is 

 really quite different. If the upper fluid is ho- 

 mogeneous, A2 should be a universal constant. How- 

 ever, when the upper layer is stratified, losses of 

 energy may occur by wave radiation and A2 may then 

 be a function of s = N2/(AbA') • 



using (6) , (8) , (14) , (15) , (17) , we get 



3 

 A2W2 dAb h dAb Ab dh 1 ,, dD 



+ qj - N^(D 



■in) 



dt 



(D + h) 



(18) 



