* s * ^ \ K* E(k)JK = '«-< I k l $ u (K)J If J 



TM No. 377 



(V-61) 



where -*i is the kinematic viscosity. 



The functions K E(k) and K Ofk) are termed dissipation spectra, and 

 describe the distribution (as a function of wave number) of the rate of decay 

 of the turbulent energy to heat. 



The validity of equation (V-6l) is predicated on the Kolmogoroff hypothesis 

 (Kolmogoroff, 19^-1): at sufficiently high wave numbers, the only parameters 

 affecting the energy density E(k) at wave number K are the rate of energy- 

 dissipation € and the kinematic viscosity ^ . Also, since it is assumed for 

 large wave numbers that the turbulence tends to be isotropic, measurement of a 

 single velocity component suffices to describe the total energy of the system. 



Estimates of & obtained by Stewart and Grant (1962) at various depths 

 and for various wind wave heights are listed in table V-6. Some of the tabulated 

 data were reported earlier (see Grant, Stewart and Moliiet, 1962). The values of 

 & are generally smaller than £ w , even when a 5 dyne cm~2 stress is used to 

 estimate £ w (see table V-5). A relatively small variation of €r occurs with 

 depth, and bears no resemblance to the distribution of kinetic energy with depth 

 described earlier in this chapter. Stewart and Grant (1962) suggest that the 

 observed turbulent velocity they measured is more associated with the "wind 

 driven drift current" than with the waves. 



This would tend to explain the large differences between £ and df w . 

 Thus, £ is associated with the high frequency or dissipative turbulent range 

 (as shown in figure V-38), whereas g w is associated with the inertial subrange 

 or region of wind wave frequencies. 



Since the kinetic energy is centered about the ambient wave frequencies, it 

 is plausible that a relatively large amount of energy transfer occurs from the 

 mean motion to the dominant wave frequencies - (see table V-5). Much of this 

 wave energy is transported over great distances to the coastal beaches. A 

 relatively small amount of wave energy would be dissipated directly to high 

 frequency turbulence in the area of wave generation, as indicated by the magni- 

 tudes of & . 



The dissipation relation £ w demonstrates how the large scale wave energy 

 is dependent upon the mean current environment. The ^^/&\ term can be 

 governed by several factors which control shear, such as tide, wind driven 

 surface current interacting with the bottom, or wind stresses. Thus, the 

 numerous factors which control the shear can have marked effects on the actual 

 waves produced. 



159 



