IV -27 



In other words, the spectrum vanishes quadratically near k = 0, and the range 



for which this quadratic behavior is appropriate is somewhat less than — . 



R 



Because of the definition of the correlation function (IV- 53), the value 

 of the correlation function for zero correlation distance will be unity. It follows 

 from (IV-6Ib) that the mean square fluctuation of the index of refraction is just 

 the total area under the spectrum: 



00 



■I 



<u^> = I dk E (k) (IV-63) 



Just as we were able above to find the behavior for the spectrum for small values of k 

 based on the known behavior of the correlation function for large values of r, so 

 we may also determine the behavior of the correlation function for small values of 

 r from the known behavior of the spectrum for values of k larger than K. In fact, 

 for any correlation distance r substantially less than — , we may approximate 

 (IV- 61b) as follows: ^ 





< u^ > R (r) =- I dk (1 - ^^f^ . ■ • ) E (k) = < u^ > - ^ dk k^ E (k) (IV-64) 

 \jI 6 u 61 V 



We observe that the expansion of the correlation function near r = does not con- 

 tain a term proportional to r, which substantiates our earlier assertion that an 

 exponential correlation function cannot possibly be applicable for very small 

 values of r , the correlation distance, if there is a cutoff frequency of the spectrum 

 due to conduction. We may also obtain the structure function for very small values 

 of r through the substitution of (IV-64) in (IV-54): 



(r) = 2<u^>[l-R (r)]~r^ (IV-64a) 



u 

 For small values of r, the structure function therefore behaves like r^. 



The above argument will hold equally well for the spectrum and structure 

 function of the turbulent velocity (u) or of the temperature distribution (T). To find 

 the spectrum in the intermediate range of wave numbers, we must appeal to 

 Kolmogoroff's general theory of turbulence as described, for example, in Tatarski. 

 We shall content ourselves with a general dimensional argument providing some 

 justification for the final results . 



artbur ai.tittleJmr. 



S-7001-0307 



