SECT. 1] THE EQUATIONS OF MOTION OF SEA-WATER 35 



In the ocean, the effect of all three gradients is undoubtedly small, but it is 

 not certain that the effect of any one is smaller than that of the others. This is 

 one of many unsolved problems. 



It should also be noted that (18) is a consequence of the assumption that the 

 Stokes-Navier and Fourier laws are correct. However, Onsager (1931; 1931a) 

 has shown that concentration gradients may cause heat flow; consequently 

 (16), and hence also (18), probably require modification. For present purposes, 

 the precise forms of (14), (16) and (18) are not essential, but may become 

 important in theories of oceanic turbulence, etc. 



One further consequence of the second law of thermodynamics is that, at 

 thermodynamic equilibrium, where Vu, s and h vanish, both Vd and Vyu, must 

 also vanish. If the equilibrium occurs in a gravitational field, the pressure 

 gradient will not vanish. Since /x is constant, but a function of p, 6 and S, 

 there must be salinity gradients to balance the pressure gradient. Actually, 

 the ocean is far from thermodynamic equilibrium. This conclusion is, therefore, 

 largely of theortical importance ; it raises the question of the quantitative 

 difference between the actual state of the ocean and the nearest equilibrium 

 state. 



5. Transformation of the Equations 



Returning to (10), (11) and (12); these may be transformed into others that 

 are sometimes more useful ; if they are multiplied by ~X, Y and K and then 

 added term by term, (2), (6) and (7) result in 



^ + pc2V.u = '^[G-V-h-Hj,,V.s]. (23) 



Hence, insofar as pressure changes are concerned, diffusion and heat conduc- 

 tion have similar effects, their equivalence ratio being the heat of diffusion, 

 Hpv, defined by (5). 



Using —Y,Z and L in the same way, it follows that 



^ + ^^Vu = -^[G-V-h-HevV-sl (24) 



Lft a ply vS 



so that, in respect to temperature changes, the equivalence ratio of diffusion 

 and heat conduction is the heat of diffusion at constant temperature and 

 volume, (6). 



Finally, a similar procedure yields 



^ + ?^//,„V.u = |5[6'-V.h-fl,„V.s] (25) 



for the rate of change of the potential of the fluid. 



If (8) is multiplied by u, and (10), (11), (12) by pp, p6 and p/x respectively, 

 and the results are added, the energy equation is obtained in the form 



p^^[iu2 + e + Vx] + V-(^u) = u-(V./)-V-h-/x V-s + <5, (26) 



