SECT. 1] THE EQUATIONS OF MOTION OF SEA- WATER 33 



3. Hydrodynamics 



The Elder equation may be written 



P^ + V;) + /5SrVx + pX2xu = V-/ (8) 



where u = velocity of the water, {DIDt) = {dldt)+u-V, gr = acceleration of 

 gravity, gr^ = gravitational potential, -1^ = angular velocity of the earth's 

 rotation, /^ = stress tensor of molecular viscosity, and the other symbols have 

 the same meaning as above. 



The conservation of matter is expressed by 



^ + V.(^u) = 0, (9) 



Avhich may be transformed into 



Dv „ 



-^^ = vV-u. (10) 



Let h erg/cm 2 sec be the heat flow and G erg/cm 3 sec the heat generated by 

 irreversible processes such as friction, etc. ; then the rate of change of entropy 

 is given by 



^^= {vie){G-V.h); (11) 



G — Vh is the net accession of heat. Finally, let s g/cm^ sec be the flux of 

 salinity ; then the conservation of salinity is given by 



DS 



^=-.V.s. (12) 



— V • s is the net accession of salinity. 



4. The Irreversible Processes 



It remains to give expressions for G, h and s. It can be shown (Jaumann, 

 1911, 1918; Eckart, 1940; Meixner and'Reik, 1959) that 



G = (/•V)-u-(l/^)h-V^-s-V/x. (13) 



The second law of thermodynamics states that the laws of viscosity, heat 

 flow and diffusion are such that G is never negative. 



It is also necessary to specify the viscous stresses, /t, the heat flow, h, and the 

 diffusion flux of salinity, s. The Stokes-Navier laws of viscosity are expressed 

 by the equations 



/ixx = 2v'{du^l dz) + {v" - 2v' /3)V -u 



(14) 

 /^xy = v'[{^f^xlSy) + {dUyl8x)] 



etc., 



