593 



(2qit)2 (2qi)^bi 

 D = : + 



1 



h= (2qit)2-+ 



n2 



i i i 



Ab = (2qit)2aN + {2cii)^h^n^ 



where 



(57) 



1 + 1 + 



aq \ 2 



ait 



^ ^ a(a + 2) ^ = 1 

 aij 1 ' ai^ 3 

 22 





(63) 



which has the same form as the first term in (52) 

 or (60) . The present theory should not, however, 

 be regarded as an extension or modification of the 

 Tennekes ' theory because , as we have noted in sev- 

 eral places, the two theories differ fundamentally. 

 This is also evident in the difference in the nature 

 of the two constants of proportionality for the 

 Ri~l term in the two theories. The aj in (63) may 

 be identified physically as the ratio |q2/qi| which 

 is a universal constant in the Tennekes' theory. 

 The constant, a, in (52) or (60), however, is a 

 universal constant equal to the asymptotic value 

 of the ratio of the inversion layer thickness to 

 the thickness of the mixed layer. Tennekes assumed 

 a value of 0.2 or so for aj and it is a coincidence 

 that this is also a reasonable choice for a. 



We may attempt to estimate the constants in the 

 expressions 



a2 



*!•' 



b3 



ait 



2^a 



(58) 



— = aRi~^ + cRi 



q2 

 qi 



yRi 



(64) 



Using the relationship 



Nt 



Ri'^ 



2a-^ 



-T^(^*^)- 



we obtain for the entrainment velocity 



„ . 13 7 

 ^3 = 



(59) 



(60) 



using the data of Willis and Deardorff (1974)*. 

 Approximate estimates for the two cases : 



SI: D = 58 cm, h = 9 cm, AT = 1.7°C, 



Ab = 0.39 cm/sec2, Qq = 0.18°C cm/sec, 

 w^ = 1.3 cm/sec, Ri = 13.5, a = 0.16, 

 c = 1.09, Y = 1-61 



S2: D = 55cm, h = 8.5 cm, AT = 3°C, 



Ab = 0.69 cm/sec^, Qq = 0.22°C cm/sec, 

 w^ = 1.4 cm/sec, Ri = 20, a = 0.15, 

 c = 1.05, Y = 1.05 



The ratio of fluxes is 



q 13-3 

 l-^l = 22b,a'^Ri '^ 



'qi 



>1' 



(61) 



Notice that Ab/h ->■ N'^ as t -> " so that the IL be- 

 comes indistinguishable from the upper layer as the 

 turbulence in it weakens (becomes more intermittent) , 

 This contrasts with MISF in which the stability in 

 the IL is several times larger than the stability 

 in the upper fluid. Notice also that s -+ 1 implies 

 a2--.ci5 are universal constants. More accurately. 



Ab 

 hP 



= 1 + 04 



(62) 



We see from (32) that at, > so that the buoyancy 

 gradient in the IL is more stable than in the air 

 above. These results suggest that an inter facial 

 layer will be difficult to identify when there is 

 a stable buoyancy gradient aloft. This is certainly 

 the case in the experiments of Deardorff, Willis, 

 and Lilly (19^9) and Willis and Deardorff (1974) . 



DISCUSSION 



We may also attempt to compare with atmospheric data. 

 For example, using the 1200 observation on Day 33 

 for the Wangara data, [Zeman and Tennekes (1977)], 

 we obtain 



Ab 



1.1 10 ^cm, h 

 =13 cm/sec^ 



Q, 







< 10^ cm, Ae s 4°C, 

 20 "C cm/sec. 



194 cm/sec, Ri = 38 , a 



0.2, 



= 2.2 



These computations indicate that the two terms in 

 the expression for Ug in Eq. (64) are roughly 

 similar in magnitude for atmospheric and laboratory 

 conditions. 



It is interesting to compare the theory of the 

 erosion of a linear buoyancy field with a numerical 

 experiment of Zeman and Lumley (1977) using a 

 second-order closure model. The niomerical calcula- 

 tion began from an initial instant, tp, at which 



time D = Dg, w^ = 

 theory at time t. 



D , T 



= 1 + — + 



Do So 



'*0 

 = t 



(qjDp) 1/3. The present 

 tg is 



where we have assumed that (toN)!^ is large. The 

 numerical curves [Figure 1 of the paper of Zeman 

 and Lumley] are nearly linear after t exceeds 2 or 

 so although, as (57) would indicate, D/Do increases 

 somewhat more slowly after considerable time. The 



We have already contrasted the theory of this paper 

 with that of Tennekes (1973). He obtains 



Supplemented by information in a personal communication from 

 Dr. Willis. 



