589 



'3 "^Ab; "2 



(19) 



^Fl ^ 2 /Ab 

 TT '2"2 IfT 



-D ^ + n2d ^ (D + h) - qi (20) 



where the subscript "2" denotes values at a level 

 just above z = D. Equation (19) , which follows from 

 Eq. (18) , is consistent with the assumption that the 

 pressure fluctuations in eddies in region R2 of fre- 

 quency W2/62 of order of the natural frequency 

 (Ab/h)'5 are generating the brea)cing waves by reso- 

 nance. 



(^ o 



/ / y / / ^ /'/'/' / yl / / // //////^/ ^Ve^ ////// 



8v' 



FIGURE 3. Turbulence near a wall. 



4. TURBULENCE IN THE MIXED LAYER 



According to (17) the vertical turbulence velocity 

 in R2 is related to the average buoyancy flux in 

 the interfacial layer. The latter is related to 

 the entrainment velocity so that it is essential to 

 relate W2 to turbulence in the main portion of the 



mixing layer, or to w^ 



(qiD) 



1/3 



This is often 



called the convective velocity. A great deal of 

 confusion has arisen regarding this problem because 

 of two explicit or implicit asstunptions often made: 

 (1) that the turbulence near the interface is quasi- 

 isotropic, i.e., U2 ~ V2 - W2 , and (2) that W2 - w* . 

 We will try to show that both of these assumptions 

 are incorrect*. 



In laboratory experiments with mechanical mixing, 

 measurements indicate that the mean buoyancy gradi- 

 ent in the mixed layer is very weak and, in fact, 

 approaches zero as the Richardson number increases 

 [Wolanslci (1972) ] . Instantaneously, the lower sur- 

 face of the interfacial layer is very sharp (perhaps 

 a discontinuity for infinite Reynolds numbers!). 

 This surface is agitated by the disturbances of the 

 mixed layer so that the mean buoyancy curve varies 

 continuously, although rapidly in the region, R2 . 

 It seems quite safe, however, to neglect effects of 

 buoyancy on the turbulence of the instantaneous mixed 

 layer. Let us do this tentatively although we will 

 return to this point later. Since, for the highly 

 stable conditions of this paper, the interface dis- 

 turbances will be very small, the inversion will act 

 like a 'rigid lid' with slip^" and the turbulence will 

 be similar to turbulence between a rigid heated plate 

 at z = and a rigid plate a z = D. The first ques- 

 tion to face, then, is the nature of the turbulence 

 at some level t; = D - z near the upper "plate." To 

 do this, we first consider the findings in two recent 

 papers by Hunt (1977) and Hunt and Graham (1977) re- 

 garding the distorting effect of a rigid plane on 

 homogeneous turbulence. The corresponding labora- 

 tory experiment is produced by passing air through a 

 grid in a wind tunnel. The rigid plane is a moving 

 belt along one wall of the wind tunnel with speed 

 equal to the mean wind. This serves to eliminate 

 the shear near the wall and the corresponding energy 

 source. The wall causes two boundary layers (Fig- 

 ure 3) . One is a very thin viscous layer of thick- 

 ness 6v near the wall in which all three components 

 of velocity go to zero, and the other, called a 

 source layer of thickness 63, extends from the vis- 



We mean by A ~ B that A/B is finite and non-zero in the limit 

 as Ri -^ °o. 

 This xs the opinion also of Zeman and Tennekes (1977). 



cous layer to a level at which the disturbing ef- 

 fects of the wall are negligible. The vertical 

 velocity must decrease throughout the source layer 

 because it is very small at the top of the viscous 

 layer, but there is no obvious reason for a decrease 

 of the horizontal velocity components in the source 

 layer. This is confirmed by experiment and by the 

 mathematical analysis by Hunt and Graham who derive 

 the following results of interest in the present 

 problem: The rms vertical velocity in the lower 

 portions of the source layer is W2 = B(ec;)-'/3, where 

 B is a universal constant and e is the dissipation 

 function far from the wall, and the rms horizontal 

 velocities are of the same order as those far from 

 the wall although somewhat larger. It is useful to 

 obtain these and other results more intuitively. 

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

 has shown that turbulence at high Reynolds number in 

 a wind tunnel far from a wall is determined com- 

 pletely by two quantities, K and u/x, where K is a 

 quantity of dimensions l2t~1 characteristic of the 

 grid and proportional to uJl. u is the mean velocity 

 and x is distance downstream from the grid (or more 

 accurately from a virtual energy source replacing 

 the grid). For example, the dissipation function 

 far from the wall is e - Ku2/x2 , the rms velocity 

 is u - (Ku/x)^, and the integral length scale is 

 H - (Kx/u)%. 



Obviously the source layer thickness is 63 - H 

 [Hunt (1977) ] and the dissipation in the source 

 layer is 



ef 





(21) 



Just outside of the viscous layer, E is e , or 



s sO 



6vu^ 



E „ = Ef — i — r- 



sO h k 



(22) 



As V ->- 0, S^ -^ , and, since E must be independent 

 of viscosity for high Reynolds number turbulence, 



^so ~ e- 



At small C, eddies of length much less than t, 



will not feel the distorting effect of the surface 



and will be isotropic. Eddies of length much greater 



than 5 will feel the surface very strongly and will 



be strongly flattened. Eddies of length of order 



^ << i?. will feel the surface but will remain quasi- 



isotropic. From the equation of continuity the 



large flattened eddies of horizontal dimensions D 



yield vertical velocities of order ujC/D - K?/d2. 



The quasi-isotropic eddies are much smaller and for 



