3G EOKART fCHAP. 2 



use having been made of the equation 



De Dv ^ Dri DS 



which is a consequence of (1). 



Inspection of (26) shows that, insofar as energy is concerned, the equivalence 

 ratio of heat flow and diffusion is /z, the chemical potential of the solution. It is 

 most unfortunate that no numerical values are known for any of these equiva- 

 lence ratios. Until such values become available, it will be almost impossible 

 to give any account of the convection and mixing processes in the ocean. 



Of the seven equations (10), (11), (12), (23), (24), (25) and (26), only three 

 are independent; for the following purposes, (11), (12) and (23) are the most 

 convenient independent set. 



6. The Zeroth Approximation 



The hydrodynamical equations are too elaborate to be solved except by 

 methods of successive approximation. One such systematic method will be 

 outlined here. It has been developed elsewhere in more detail (Eckart, 1960). It 

 begins with a rough approximation, variously known as the static or zeroth 

 approximation. This assumes that V -/i, Vh — G^ and Vs are all zero, and that 

 all time derivatives vanish. In that case, (9) reduces to 



Vp + pgVx = 0, (28) 



and (11), (12) and (13) are identically satisfied. The equations (16) and (18) 

 would be additional restrictions on d and /x, but these will be ignored ; the 

 equation (1) will be retained, however. 

 The equation (28) has the solution 



P = Po{x)> P = Po'ix)l9 = Po(x)> (29) 



po being an arbitrary positive, monotonically decreasing function of the 

 gravitational altitude, x, and the accent indicating differentiation with respect 



tox- 



If the entropy is eliminated between the equations (cf. (1)) 



= -— 6 = — 



dv bi] 



the result is the equation of state : 



p = iHv) = F{p,d,S). (30) 



When (29) is substituted into this, it becomes 



Po'{x) + 9nPo'{xhdo,So] = 0, (31) 



the subscript on ^o and So indicating that these quantities also refer only to 

 the zeroth approximation. This approximation, therefore, involves only two 



