V-22 



This function measures the mean square difference of the index of refraction at 

 two fixed points. The mean must be taken over the ensemble of different micro- 

 structures which occur in the course of time. The structure function is, therefore, 

 in general a function of the coordinates of the two points involved, or as in (IV-52), 

 a function of the position of one of the points and of the vector connecting the two 

 points . Note further that the subscript u on the structure function indicates that 

 this is the structure function of the index of refraction. Later in this chapter, it 

 will be desirable to deal with the statistical properties of the fields of turbulent 

 velocity (u) or of the field of convected temperature (T). The current discussion 

 is equally applicable to the u and T fields. The field being considered will always 

 be indicated as the subscript of the structure function, correlation function, 

 spectrum, etc. 



The micro- structure of the ocean is clearly not stationary in space. In 

 other words, the structure function of (IV-52) can certainly be expected to depend 

 on the depth of the observation X3 . However, the structure function will usually 

 be stationary in the horizontal plane. For a particular depth, we may therefore 

 write: 



B^(x, r) = B (X3, r) (IV- 52a) 



If, for reasons of convenience, we desire to ignore the dependence of the structure 

 function on the depth of the observations, we shall simply suppress the jqb depend- 

 ence and write B (r). 



The structure function is also not isotropic, in that the mean square 

 fluctuations in the vertical direction may be expected to be somewhat different 

 from those in the horizontal direction. Thus, if we introduce cylindrical coordinates 

 for the vector r(r = (r, 9, V3 )J , the structure function may be expected to depend 

 both on r and on V3 . This is another way of stating that the patches, instead of being 

 spherical on the average, may have a tendency to be lenticular. In general, therefore, 

 the structure function depends on the three variables X3 , r, and ra : 



B^(x, r) = B (xi3, r, ra) (IV- 52b) 



Unfortunately, not enough is known about the vertical anisotropy and nonstationarity 

 of the micro- structure. From the little calculation that has been done, it is pretty 

 clear that their effect on scattering is substantial. However, we shall usually ignore 

 both effects and deal with a structure function B (r) which depends on the distance of 

 the two points alone. 



:artbur Sl.littleJnir. 



s-7001-0307 



