Breaker and O'Neil 



regions whose scale lies well within this range the velocity field 

 of the turbulence will (in consequence of what we mean by l^ and Lq) 

 be locally homogeneous and isotropic. That is, if R is such a 

 region, v(p) is the velocity vector at point p, and p, and p- lie 

 within R, then the probability distribution of velocity differences 

 v(p,)-v(p ) depends only upon the distance between the points 

 r= ip^-p-f and not at all on their positions within the region or 

 their orientation with respect to one another. Thus the mean square 

 velocity differences, or structure functions can be written, 



(v,(p,)-V,(P3)f'' 



\2 





(V^(p, )-V^{p,)f 



D , the longitudinal structure function, is taken along the line 

 connecting p and p» while D^*, the transverse structure function is 

 taken across the line. 



If the fluid can be regarded as incompressible then it can 

 be shown, from purely dimensional considerations, that 



,V» 



Drr(r) = C(6r)' 

 D,t(r) = -|c(€rf 



i 



0<<r« Lo 



/o«r«Lo 



where C is a dimensionless constant of order unity and e is the 

 energy dissipation rate. From these may be determined the statistics 

 of the concentration of a conservative (with respect to position 

 changes) and passive (with respect to the turbulence mechanism) 

 "additive" to the fluid, such as heat or salt. In fact, if 0(p) 

 the concentration of such an additive at point p, then 



IS 



D^{r) = Cg r'/. 



r »> 



(1) 



where Cg is a physical constant depending upon the fluid and the 

 additive. 



As one would expect, the index of refraction structure 

 also follows a two-thirds law. 



Dn(r) 



^n ' 



(2) 



C , the refraction-index structure constant, is (in the 

 ocean) essentially directly proportional in the usual case to Cj, the 

 temperature structure constant of equation (1). 



Finally, it has been found that such fluctuations in the 

 index of refraction will indeed give rise to significant fluctuations 

 in acoustic amplitudes. Suppose a wave, initially plane and 

 orthogonal to the x axis, is traveling along the x axis through a 

 body of seawater whose index of refraction structure is as described 

 above. If the wave's transit time is short with respect to the time 



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