IV-24 



If the fluctuations of the index of refraction are stationary, we can intro- 

 duce the usual correlation function: 



<u(x) u(x+r)> 

 The correlation function is related to the structure function according to: 



Bjr) - < (v (x) - u (x+r)V>= 2 <'u^> fl- R^(rA (IV- 54) 



We note that the mean square fluctuation of the index of refraction is independent 

 of position in the stationary case; thus, we may just write <u^>. 



The original experimental studies of the correlation of the temperature 

 fluctuations in the ocean were made around 1948 by Urick and Searfoss, and were 

 followed in 1951 by further experiments by Liebermann. Some of Urick and 

 Searfoss' temperature recordings have already been presented in Figure IV-5. 

 In Figure IV- 8 we present a temperature recording near the surface made by 

 Liebermann, together with its corresponding correlation function. The points of 

 Figure IV- 8 suggest that the use of an exponential correlation function may be quite 

 appropriate for correlation distances between a few centimeters and a few meters: 



\,(r) = e"""/^ (IV- 55) 



Because of the heat conductivity of the water, the use of an exponential 

 correlation function is certainly not appropriate for very small correlation 

 distances, i.e., a few centimeters or less. We shall see later that the slope 

 of the correlation function must go to zero as r approaches zero if the micro- 

 structure is to be continuous; clearly the thermal conductivity will prevent any 

 discontinuity in the temperature from occurring. Whenever the fine structure of 

 the patches is important (i.e. , in the range below a few centimeters), it may be 

 preferable to use a Gaussian form of the correlation function: 



R (r) - e'^ '^ (IV-56) 



jarthur Sl.little.Ilnt. 



S-7001-0307 



