IV-25 



Both (IV- 55) and (IV- 56) are strictly empirical correlation functions, but both have 

 been used extensively in the literature. Neither reflect the turbulent mechanisms 

 which generate the thermal micro- structure of the ocean, and it was not until the 

 last five years that attempts have been made to base the scattering calculations 

 on a correlation function which does reflect the physical mechanisms responsible 

 for the ocean micro- structure. 



A substantial amount of effort has been devoted during the past 10 years 

 to the statistical investigation of turbulence, aimed both at representing the tur- 

 bulent velocity field and at describing the resulting micro- structure of the 

 distribution of the temperature and of the index of refraction. We shall merely 

 sketch the principal conclusions from this research; two excellent surveys with 

 considerably greater detail may be found in the book by Tatarski (Part I) and in 

 the paper by Batchelor. 



To describe the micro- structure resulting from turbulence, it is con- 

 venient to use a spectral representation of the fields involved. Consider the 

 representation of the covariance function of the index of refraction in a stationary 

 medium in terms of its Fourier transform: 



< V (x) V (x + r)> = < u^ > R^ (r) = f f fd k e' - ■ - S^ (k) (IV-57a) 



The function S, , is the Fourier transform of the covariance function, and it indi- 

 cates the amplitude of each spectral component present in the covariance function. 

 Hie spectrum itself is therefore given by: 



///■ 



S^ (k) = I I I dr e"^ ^ • ^ R^, (r) ^0- (IV-57b) 



The above merely reflects the Fourier transform theorem. If the turbulence is 

 isotropic, i.e., the correlation function depends on the distance r alone, the 

 spectrum S,j (k) will also be isotropic, i.e., depend only on the magnitude of the 

 wave number (k). To show this, we introduce spherical coordinates for the vector 

 r in (IV-57b) using (k) as the polar axis of the coordinate system: 



r = (r, 9, <J) such that k . r = k r cos 6 (IV-58) 



The element of volume in the integral now becomes d r = 2rT r^ sin 9 dr d9, and 

 (IV- 57b) may be rewritten as: 



j..i 



^M<^> = I dr 1 -(^7^r^sin9e-^^'^'^°'^R^(r)<L.^> (IV-59) 



:arthur Jl.HittlcJnir. 



S-7001-0307 



