and, since Pti and drjjdx are both very small, 



[ v dP d f * „ cte 6 f" dP a f » a Z6 



T-d2 = — Pdz + P b —, T~ dz = T\ Pdz + P b — (5a, b) 



J^ d* dx J z 3a: J* dy dy J* dy 



Next, define the potential energy of a column 





£„ = Pdz 



h b 



(6) 



so that when equation (6) is substituted in (5a, b) 

 f"f dz J-^ + P b ^ : p 



J* B a* ax d* j, 



¥-dz= d -^ + P b d -*- 



b dy dy dy 



From equation (lc) 



(7a, b) 



and on substitution into (6) 



dz = dP 



g 



1 f° 



E P = aPdP. 



gh- h 



(8) 



Letting the specific volume, a = a +8 wherein a is the specific volume of water of sa- 

 linity 35°/oo, temperature 0° C, and pressure P, 



E„ = - f "Pa dP + 



gin g. 



ip 

 gJo 



(9) 



Fofonoff (1962) defines the last term in (9) as the anomaly of potential energy, 



.if 



gJo 



P8dP. 



(10) 



Using the definition in the first term on the left of equation (7a), 



"1 /-/>„ 



dEp = dX , d 



dx dx dx 



rjo 



Pa dP 



and by application of the chain rule 



dE p ^dx | dP b d 

 dx dx dx dPi, 



gJo 



PaodP 



Since the bracketed portion of the last term is a function of the bottom pressure only. 



dE p _d X | dP b P b a u (P b ) 

 dx dx dx g 



20 



