586 



negligible if the mixed layer depth is much greater 

 than the Monin-Obukhov length, L = -uj/qj , [Monin 

 and Yaglom (1971, p. 427) ] where u* is the friction 

 velocity. Thus (-L/D) l/3 is proportional to the 

 ratio, u*/w«, of the turbulent velocity in the mixed 

 layer associated with shear to the turbulent veloc- 

 ity associated with convection, w* = (qiD)l/3. The 

 shear effect becomes less important as this ratio 

 decreases. Lenschow (1970, 1974) presents aircraft 

 measurements, which appear to confirm the unimpor- 

 tance of energy production by the shear for the 

 turbulence near the inversion if |l/d| is small 

 enough. 



The purpose of this paper is to construct a 

 theory for the rise of an inversion in the atmo- 

 sphere neglecting the effect of shear. The analysis 

 is similar in some respects to that in a recent paper 

 by the first author, [Long (1977b) , hereinafter 

 referred to as MISF] in which a theory is developed 

 for turbulence in a stably stratified liquid, as 

 for example in the experiments of Rouse and Dodu 

 (1955), Turner (1968), Wolanski (1972), Linden (1973), 

 Crapper and Linden (1974) , Linden (1975) , Thompson and 

 Turner (1975) , Wolanski and Brush (1975) , and Hop- 

 finger and Toly (1976) . In these experiments a 

 stably stratified fluid is agitated by a grid oscil- 

 lating up and down near the bottom of the vessel 

 (Figure 1) . A growing mixed layer of depth, D, 

 appears in the lower portion of the fluid separated 

 from the non-turbulent fluid above, in which the 

 buoyancy gradient is given, by an IL of thickness, 

 h. Observations indicate that the lower mixed layer 

 has a very weak mean buoyancy gradient. The buoyancy 

 difference across the IL is relatively large and is 

 denoted by Ab. 



As indicated by the experiments of Thompson and 

 Turner and Hopfinger and Toly, and derived by the 

 first author in a recent paper [Long (1977a)], the 

 turbulence generated by the grid in a homogeneous 

 fluid is nearly isotropic, and if u is the rms veloc- 

 ity and a is the integral length scale, the quantity, 

 uS. (proportional to eddy viscosity), is constant with 

 height. When there is stratification, the mixed 

 layer is nearly homogeneous and ui = K is again con- 

 stant near the grid [Hopfinger and Toly (1976)]. 

 Since I is proportional to the depth, D, the veloc- 

 ity, uj s K/D, is characteristic of the turbulent 

 velocities in the mixed layer. The quantity, K, 

 may be taken to be characteristic of the "action" 

 of the energy source (grid) . 



On the basis of observations, experimenters have 



proposed that the entrainment velocity Ug 

 is expressible in the form 



fS 



Rl 



-3/2 



Ri 



DAb 

 f2s2 



dD/dt 



(1) 



where Ri is the overall Richardson number, f is the 

 frequency, and S is the stroke of the grid. The 

 measurements correspond to large values of Ri so 

 that attention is confined to the usual situation 

 in nature in which the Richardson number is large. 

 In terms of the "action" K of the grid, another 

 Richardson number is 



Rl 



DAb 



2 

 "1 



K/D 



(2) 



This is very similar to the number Ri = HAb/u^ 

 proposed by Turner (1973) , where I and u are the 

 integral length scale and rms velocity measured at 

 the level z = D in a homogeneous fluid agitated by 

 the same grid at the same grid frequency and stroke. 



In MISF and in the present paper, the role of the 

 IL separating the mixed layer from the non-turbulent 

 fluid above is essential. This contrasts with ear- 

 lier theories in which h is neglected despite ex- 

 perimental evidence [Linden (1975)] that h is 

 proportional to D and is not particularly small 

 (h/D = 1/4). Observations [for example, Wolanski 

 and Brush (1975)] indicate that the IL with its 

 large density gradient is typified by wave motion. 

 Wolanski and Brush found that the frequency of dis- 

 turbances in this layer was proportional to the 

 Brunt-Vaisala frequency (Ab/h)'5 although numerically 

 one order of magnitude smaller. Certainly turbulence 

 of some kind exists in the IL and since the density 

 gradient there is strong rather than weak as in the 

 mixed layer, it is reasonable to assiraie that the 

 turbulence in the IL is intermittent and that this 

 intermittent, weak turbulence transfers the buoyancy 

 in the layer. In MISF the intermittency factor de- 

 creases with increase of stability so that for the 

 large Richardson numbers of the asymptotic theory 

 the layer is, for the most part, in laminar wave 

 motion with occasional breaking waves in the interior 

 and at the lower surface of the interface. 



Similar ideas may be applied to the present prob- 

 lem in which the turbulence in the mixed layer is 

 caused by heating at the lower surface. The princi- 

 pal differences are the effect of heating in causing 

 h to decrease and D to increase, and the differences 

 in the sources of turbulence kinetic energy. The 

 energy equation is 



= 



3_ 

 8z 



Pi 



:v^ 



t'2 



~ri 



-w'b' - e (3) 



FIGURE 1. Oscillating grid experiment. (S = stroke of 

 the grid.) 



where the first term is the energy flux divergence; 

 u', v' , w' are the instantaneous velocities, p' is 

 the pressure, Pj, is a reference density, q = -w'b' 

 ■is the buoyancy flux, and e is the energy dissipa- 

 tion. In the present problem the buoyancy flux 

 term, -w'b' , is of basic importance and corresponds 

 to the conversion of potential energy to kinetic 

 energy. This effect is missing of course, in the 

 case of mechanical stirring in a homogeneous fluid. 

 Equation ( 3) omits the local time rate-of-change 

 of kinetic energy although, in fact, the inversion 

 is rising and conditions are therefore unsteady. 

 With respect to the mixed layer, the kinetic energy 



