SECT. 1] THE EQUATIONS OF MOTION OF SEA-WATER 37 



arbitrary functions, which will be taken to be ^o(x) ^^^d Soix, ^' ^)^ ^^^^ ^ 

 being the horizontal co-ordinates, latitude and longitude. The equation (31) 

 can then be solved for ^o : 



^o(x, 0, A) = T[x, Soix, </>> A)]. (32) 



Having thus obtained ^o from po (or ^o) and >S^o, the spacial distribution of 

 any other thermodynamic variable, such as 17, /x, a, Cvs, Hpv, s, h, can be 

 obtained by using the equations of Section 2 ; these zeroth approximation 

 values will be indicated by a subscript zero — e.g. 170, Hpvo, So — and are to be 

 treated as empirically known functions of x, ^, A in all that follows. 



Relations between the gradients of the zero-order quantities can be calcu- 

 lated from (2) ; thus 



V^o = -Xo Vvo + Yo Vrjo + Ko VSq. (33) 



From (29), it follows that 



V^o = -pog^x^ Vvo = - (po'/po^) Vx ; 

 since Xq = poCo-, this becomes 



- ipog + po'c2) Vx = Fo Vr^o + A^o V*So. (34) 



Two quantities that will be useful in the work below are the Brunt-Vaisala 

 frequency, N, and the adiabatic temperature gradient, Oa' ; these are defined by 



N'lg = -{po'lpo)-{gM (35) 



and 



dA' = Sr(yo-l)/«oCo-. (36) 



With these definitions, (34) becomes 



m Vx = ^^'[V^o + Hpvo VSoldol (37) 



a result that will be needed in the following sections. 



The Brunt-Vaisala frequency is a measure of the stability of the oceanic 

 stratification. The accepted treatment of this problem is that of Hesselberg and 

 Sverdrup (1914). They give numerical tables for the calculation of Oa' and 

 E= — poN^lg, but these tables should be used with caution, since they are 

 reported to contain numerical errors. 



7. The First Approximation 



To obtain the next approximation, one sets p=po+pi, p = po + pi, u = ui, 

 etc., and substitutes into (8), (11), (12) and (23). Every term is then expanded 

 by Taylor's Theorem, and squares and products of quantities carrying the 

 subscript 1 are neglected. An exception to this rule are the quantities y^, h and s, 



