368 FOFONOFF [sect. 3 



where S is salinity and D{i) is either the horizontal eddy diffusivity, Dn, or 

 the vertical eddy diffusivity, 7)^, depending on whether i is different from 

 or equal to 3.^ 



Neglecting internal sources of heat due to dissipation of kinetic energy of the 

 motion, we obtain the conservation equations for heat and salt by sub- 

 stituting (153) and (154) for the flux in the general conservation equation (4). 

 Expressing the heat in terms of temperature and using the continuity equation 

 (3), we obtain the conservation equations in the form 



f:+„,£^ = ^v.r+:^|!^, ,156) 



Ct CXj pCp pCp CXz^ 



where Cp is specific heat, and 



^ + Uj^ = V^S + —— 156 



Ot CXj p p 0x3^ 



We have assumed that Kh,Kv and Dh,Dv are constant in deriving (155) 

 and (156). However, as density depends both on temperature and salinity, it is 

 likely that turbulence will be modified at some length scales by the presence of 

 temperature and salinity gradients. Hence, the eddy coefficients would depend 

 on the gradients of the properties and the intensity of the mean flow. However, 

 because of the complexity of the complete set of conservation equations, we are 

 unable to cope with more than the simplest form of the equations. 



If we assume that the density can be approximated by a linear equation of 

 the form 



p - pQ\i-a{T-To) + h{S-So)'\ (157) 



and that 



KnlpCp = Duip = kH, KvjpCp = Dvjp = ky, (158) 



we can reduce (155) and (156) to the single equation 



|,„,g..„VV...^, (159, 



From its form, we may interpret (159) as an equation for the "diffusion" of 

 density. Clearly, the simplification of the conservation equations to (159) can 

 only be done in the linear approximation to the equation of state. ^ 



We can make an alternar^ive interpretation of the diffusion equation by 

 introducing an apparent temperature T* defined by 



T*_To* = T-To-{bla){S-So). (160) 



1 The reader should compare the assumed form of the turbulent flux equations with 

 those for molecular transports given in Chapter 1, eqns. (68) and (69), page 26, and in 

 Chapter 2, eqns. (16) and (19), page 34. 



2 Fofonoff (1956) has suggested that non-linearities in the equation of state of sea-water 

 may have a significant influence on the density structure of Antarctic bottom water. 



