IV -29 



We desire the spectra of the fluctuations of the temperature and the 

 index of refraction rather than of the velocity field. It turns out, however, that 

 one may again expect a range of frequencies for which a universal spectrum 

 decaying like the -5/3 power of the frequency is applicable: 



E ~k-^/^ (^<k<^^^^) (IV-68) 



cond 



The meaning of -t and L is the following: 



In the case of sea water, the heat conduction cutoff of the spectrum occurs at lower 

 wave numbers than the viscous cutoff; in other words, the viscosity of the water 

 can support smaller eddies than the thermal conductivity. The range of validity 

 of (IV-68) is therefore cutoff at the high wave number end by the scale determined 

 by heat conduction. At low wave numbers, i.e. , for large patches, we would cer- 

 tainly not expect a universal law of turbulence for patch diameters of the order of 

 magnitude of the layer thickness of the originally layered structure which was 

 broken up by the turbulence. TTie outer scale L of the thermal micro- structure 

 will therefore be of the order of twice* the diameter of the largest patches, which 

 has been shown earlier (see Figure IV- 6) to have a diameter of the order of twice 

 the depth. We would, therefore, expect the outer structure L to be of the order of 

 four times the depth. For wave lengths of the order of magnitude of the largest 

 patches, the details of the original layered structure become important. We show 

 in Figure IV-9 the general shape of spectrum that is to be expected. For very small 

 wave numbers, the spectrum starts at zero and increases quadratically. For slightly 

 larger wave numbers, the spectrum is more or less flat, going through a maximum 

 corresponding to the dominant patches resulting directly from the breaking up of the 

 original layers. For the intermediate range of wave numbers, corresponding to 

 wave lengths between the depth of the measurement and a few centimeters, the 

 spectrum obeys the Kolmogoroff law. Finally, for very large wave numbers in the 

 conduction range, the spectrum decreases very rapidly. 



We may also obtain the structure function corresponding to the Kolmogoroff - 

 spectrum. According to (IV- 54) and (IV- 61b), the structure function is given by: 



CO 



R^(r)] = 2Jc 



B (r) - 2 < u^> ^1 - R (r)l = 2 f dk (1 - ^^^) E (k) ' (IV-69) 



u L u J 1 kr 



*"twice" because L is a full wavelength, containing both a patch with u > and a 

 patch with \J < 0. 



Arthur B.llittlpJnf. 



S-7001-0307 



