592 



2 



/ Dl Di hi' 



(6a2 ^ - 6a3 j^ + 6at, + 605 — 



7 7 7 



k 1 1* 4 



hi (Dl + jhj) 6 



Did 



7 

 ax) 4 



Solutions are of the form 



h 

 ho 



1 ^it 

 (1 + -,.) -^ 



3 731 



4 14 "Si^D^t 

 (03 - 6053) a (1 + ja) — 'j- 



V 2 

 



2.2 

 1 2 <3it 



+ ^ a(2 + a) — 1 + . . . 



8 4 



Vo 



D 1 "^i^ 



^ = 1 + a(l + ia) ^- 



2°' v2 



(47) 



(48) 



The first term is of the same form as the non- 

 dimensional entrainment velocity of Tennekes (1973) 

 but, as already pointed out, the derivation and 

 physical mechanism are very different. It is easy 

 to trace the error in (52) arising from the simpli- 

 fication of Section 2 that the XL has a linear 

 buoyancy field. The error is proportional to 

 (Ug/w^) abg/Ab where bg is the maximum difference 

 between the actual buoyancy in the IL and the as- 

 sumed buoyancy. Since a is 1/6 or so and bg/Ab is 

 fairly small, this error is negligible. Notice also 

 that the theory concerns strongly stable conditions 

 so that (52) does not apply in the limit as Ri -* 0. 

 As Ri tends to order one Ug becomes of order w* as 

 one would expect. 



The ratio q2/qi is of interest. Using (6) , we get 



92 



qi 



1 + 



dAb 

 qi dt 



Using (50) and (52) , we get 



(53) 



+ 2ait + 3053! (54) 



The expressions (26) -(28) are 



+ (ao + 2ait)a'*(l + ^a) ** 



' li^^lS't 



2 ' 7 



1 



1 



B3' 



^U" 



3 3 

 '3^a^ 



Ri 



2 2 

 1 9 9 "Jit 

 i a2(2 + a)2 -J-— + 

 8 ■+ 



Vo 



(49) 



B2 



B,B 



1^11 



yRi 



(55) 



DgAb 



q^t 



2 + a 



7 



1 



3 1 



a\ >* 1i^°0^^ 

 j^ + 2a,+ + Sasa^ a ' I 1 + | ^ 



Vo 



These relations are identical to those in MISF. The 

 result that the disturbances in the IL are small 

 compared to the thickness of the IL is contrary to 

 speculation [Stull (1973) and Zeman and Tennekes 

 (1977)] that h is the depth of penetration of the 

 eddies into the stable region. 



2 2 

 a(2 + a) qjt 

 + + 



4 V^ 



(50) 



7. LINEAR BUOYANCY FIELD IN THE UPPER LAYER (N 7^ 0) 



where hg and Dq are related by the equation 



6a2 - 6033 + 6ai|a + 6a5a3 = 



(51) 



We consider initial conditions in which the fluid is 

 at rest initially with a linear buoyancy field, so 

 that D, h, Ab are zero at t = 0. Equation (11) be- 

 comes 



The entrainment velocity, Ug = dD/dt, may be ex- 

 pressed in terms of the Richardson number, Ri = 

 DAb/w^, by using 



(D + 5h)Ab = ^ (D 



h)- 



qit 



(56) 



we get 



6(1 + ja) - Ri-1 + 



Equations (56), (29), and (35)-(37) determine the 

 problem. The approximate solutions* are 



"e -.-1 ~ .k 

 — = aRi + cRi + 



c = (03 + 2a4)a (52) 



The solutions, as throughout the paper, are for strong 

 stability, which implies here that Nt is large. Then s -^ 1, 

 and a2, 03,0^,05 are independent of s. 



