609 



of jets and plumes and can be regarded as amply 

 justified. 



Comparing the first and last terms in Eq. (2), 

 then, we have 



Ui^ 



(10) 



4. THE TRANSFORMATIONS AND THE DIFFERENTIAL 

 SYSTEM TO BE SOLVED 



The transformations to be used to obtain similarity 

 solutions are already suggested by (12) , (13) , and 

 (16) and are 



in which S.^, and l^ are the length scales for the 

 X and z directions. Comparing the first term in 

 Eq. (2) with the term gO , we have 



U /G_ 

 3a ( 2 



1/3 



x-'*/3h(n,C), (18) 



Uw 



(11) 



{v,w) 



3a 



Gg 



1/3 



(V,W), 



(19) 



where 6 and w stand for the magnitudes of 9 and w, 

 rather than 6 and w rigorously, as they do also 

 in the following proportionalities. Equation (6) 

 gives, further, 



(12) 



US., 



''^'^' = (3a"l/2 <^5x2) ^/3{y,z). (20) 

 Then the equation of continuity (5) becomes 



V^ + W^ = 0, 

 and Eqs. (7) become 



if we take $,„ and l^ to be equal. From proportion- 

 alities (11) and (12) we have, after some rearrange- 

 ment. 



But surely 



Hence 



wll„ 



gGil 



U J., 



e - wS,,. 



(13) 



(14) 



V = Tj,, W = -'i^, (21) 



in which ^ is the dimensionless stream function 

 related to i|( by 



^ = (3atl/2o (GVx)l/3f(n,?) 

 Equation (9) now takes the form 



5 = w 



-n \ 





(22) 



(23) 



gG£ 

 X 



2 



From proportionalities (10) and (15) we have 



(15) 



z ^ 



gG 2 



3 ' 



u 



(16) 



since the S.^,, the scale of x, is just x. Thus (12) , 

 (13) , (14) , and (16) give 



I ~ x2/3, e - X 

 z 



1/3, w ~ x-1/^ e - x-^/3. 



These results are unaffected when other comparisons 

 are made between terms in either Eq. (1) , (2) , (4) , 

 or (5). 



From porportionalities (15) and (15) we have 



z =^(g2G2x)l/3 , 



(17) 



where a is a dimensionless constant to be determined 

 experimentally or estimated from known values of 

 £ in similar phenomena. We shall leave it free 

 throughout our analysis. Equation (17) gives the 

 form of e to be used in this paper. 



It seems strange at first sight that £ should 

 vary inversely as U. I believe that the interpre- 

 tation of e - U~l is that £ increases with the 

 time that is required for the wind to travel a unit 

 distance in the x direction, because turbulence 

 needs time to develop. 



where 5 is the dimensionless vorticity component in 

 the x direction. 



With the transformations (18), (19), and (20), 

 Eq. (4) becomes 



Lh 



X (Vh + Wh ) , 



where L is the linear operator defined by 

 a2 



3^ 32 3^3. 

 T-y + -rrr- + 2t]-- + 2?^- + 4, 

 3n2 3C2 3ri 3c; 



and 



.3, 



-1/2 



X = (3a-^) 

 Equation (8) now has the form 



(L - 1)5 = -h^ + X(VC^ + WC^). 



(24) 



(25) 



(26) 



(27) 



Equations (23), (24), and (27) are the final equa- 

 tions governing the dynamics of the plume in a 

 transverse wind. They are to be solved with the 

 boundary conditions 



0, and 'i^^ = at n = . 



(i) hr, = 0, S = 0, ' 



(ii) h = 0, C = 0, >l' = Oatn = ±<»orC 



Boundary conditions (i) correspond to symmetry with 

 respect to the c, axis, and conditions (ii) ensure 

 that there is no temperature variation and no 



