587 



'Heated Surface 



FIGURE 2. Model of entrainment at an interface by 

 heating from below. The curve on the left is the mean 

 buoyancy, b, with an assumed linear profile above the 

 interfacial layer. The curve for buoyancy flux, q, is 

 on the right. The superadiabatic layer near z = is 

 not shown . 



is proportional to the square of the convective 

 velocity, (qjD)^'^, so that the ratio of the time 

 rate-of-change term to the other terms in Eq. (3) is 

 Ug/w*. This ratio is of order one if the convective 

 motions are spreading upward at a speed, Ug, in 

 initial conditions of neutral stability . Even a 

 fairly weak inversion will cause a great slowdown 

 and Ug/w^ will be small. Similar remarks apply to 

 the IL and we are assured that the time dependence 

 is negligible in the stable conditions of the paper, 

 although it has received some attention in considera- 

 tions of the real atmosphere [Zilitinkevich (1975)]. 

 We may conclude this introduction with reference 

 to work on penetrative convection in the atmosphere 

 and oceans including atmospheric observations: 

 Lettau and Davidson (1957), Ball (1960), Veronis 

 (1963), Izimi (1964), Summers (1965), Deardorff 

 (1967), Kraus and Turner (1967), Lilly (1968), Dear- 

 dorff (1972) , Betts (1973) , Carson (1973) , Stull 

 (1973), Tennekes (1973a, b, and 1975), Adrian (1975), 

 Farmer (1975) , Zilitinkevich (1975) , Kuo S Sun (1976) , 

 Stull (1976a, b, c) , and Zeman and Tennekes (1977). 

 Related experiments have been run by Deardorff, 

 Willis, and Lilly (1969), Willis and Deardorff (1974), 

 and Hedit (1977) . A second-order closure model has 

 been given by Zeman and Lumley (1977) . More recent 

 field observations have been made by Kaimal, et al . 

 (1976) . Mixed layer deepening in the upper layers 

 of the ocean, which is almost always associated with 

 wind stirring has been discussed by Niiler and Kraus 

 (1977) . 



2. 



RELATION OF FLUXES TO THE BUOYANCY JUMP AND TO 

 MIXING LAYER AND INTERFACIAL LAYER THICKNESSES 



In the theory of the paper we ignore rotation, radia- 

 tive heating, water vapor, and horizontal variations 

 of mean quantities . The model is shown in Figure 2 

 which contains curves for the mean buoyancy and buoy- 



ancy flux. The mean buoyancy curve above the IL is 

 assumed to be linear with buoyancy gradient N^ . In 

 one case we assume that N^ = so that the inversion 

 rises and weakens, eventually disappearing. When 

 N^ ^ we assume that the air was at rest with uni- 

 form buoyancy gradient when heating began. Then the 

 inversion strength increases with time. Since the 

 theory of this paper is concerned with very stable 

 conditions, the solutions hold for large values of 

 the Richardson number. 



The buoyancy flux curve is derived below from the 

 assumed buoyancy distribution. The latter is assimied 

 to be linear in the IL (region R3) . This is an ex- 

 cellent approximation* in certain circumstances at 

 least, for example in the mechanical stirring ex- 

 periments of Wolanski and Brush (1975) . Observa- 

 tions in the mixed layer [Willis and Deardorff 

 (1974)] indicate that there is very little mean 

 buoyancy variation in this layer except for some 

 indication of a stable mean gradient near the heated 

 plate. If we ignore these gradients for the moment, 

 the equation 



3t 



dq 

 3z 



(4) 



indicates that q is a linear function of z. In fact, 

 experiments show that q is nearly linear [Willis and 

 Deardorff (1974) ] so that the neglect of mean buoy- 

 ancy variations in the mixed layer in the model of 

 Figure 2 seems reasonable. The lo wer s urface is 

 heated and the buoyancy flux q = -w'b' (proportional 

 to the heat flux) is held constant at the lower sur- 

 face where it is denoted by qj . The mean buoyancy 

 in the mixed layer is 



bjj = n2(D + h) + Ab 



(5) 



where Ab is the buoyancy jump across the interfacial 

 layer and bgo is constant equal to the buoyancy at 

 the surface if the linear gradient above is extra- 

 polated down to the surface. Integrating (4) over 

 the mixed layer, we get the flux, q2 , just below the 

 IL. It is 



dAb 7 d , 



52- = qi + D —J-;: - N-^D 3^ (D + h) 



dt 



dt 



(6) 



On physical grounds q2 must be negative (Figure 2) 

 and this is confirmed by laboratory measurements 

 [Willis and Deardorff (1974)]. In the IL, the mean 

 buoyancy is 



ib - ^ (z - D) + %0 



N'^(D = h) 



(7) 



Integrating (4) , we get the flux at a given level 

 in the interfacial layer 



dAb / K^\ ,, ( C^ dh c dD 



- N^C 



dD dh 

 dt '*' dt 



(8) 



where C=z-D. Atz=D+h, the buoyancy flux 

 is zero so that 



Even when the approximation is only fair, the error in as- 

 suming a linear profile is small. We discuss this in Section 

 6. 



