40 ECKART [CHAP. 2 



9. The Field Equations 



The equations of the last section can be brought into a form that makes it 

 apparent that they are self-adjoint, and is otherwise more convenient to work 

 with. For this purpose, the change of variables 



U = Ul(/^oCo)'/^ P = piipoco)'"''', Q = (Zi(poCo)'/s 



J = {CIEc){poc^y^, (54) 



is one possibility. Then (45), (49) and (50) become 



^+co(VP+rp)+iy^2gvx + ^xU = o, (55) 



ot 

 ^ + co(V . U - r . U) = coiN^^JIg), (56) 



^ + VVx=J, (57) 



where 



r = (l/2poCo) V(poco) + iglco^) Vx (58) 



is a known vector. 



If (55) is multiplied by U • , (56) by P and (57) by Q, the result is the quadratic 

 integral 



l_l(U2 + p2 + iV2g2) + v.(PU) = N^J{Plg + Qlco). (59) 



The quantity 



^ = -L (U2 + P2 + A^2g2) erg/cm3 (60) 



2co 



may be called the external energy density (Eckart and Ferris, 1956, and 

 Eckart, 1960). However, it may not be considered to be an approximation for 

 the true energy density, which is (cf. (26)) 



W = p{hu^+e + gx); 



the difference between W and E is discussed more fully in Eckart and Ferris 

 (1956) and Eckart (1960). 



The equations (55), (56) and (57) are essentially identical with those derived 

 in Eckart and Ferris (1956), and Onsager (1931) for a chemically pure fluid. 

 There is only one difference : in the case of a pure fluid, the vector T is 

 always vertical. The horizontal component of F is a consequence of possible 

 horizontal gradients of the salinity. If the salinity is, to the zeroth approxima- 

 tion, independent of latitude and longitude (cf. (32)) then the same will be true 

 of all zero-order quantities, including poCo ; equation (58) shows that T will then 



