632 The Tropospheric Circulation 



MT = r (pm2 + p) dz, (XIX.35) 



where z is counted upward from the bottom and where p is the water hydrostatic 

 pressure. Assuming that the mass transport in every infinitesimal isopycnic layer 

 remains constant during the variation process, then 



puz da = pUq Zq da = v{a) da, (XIX. 36) 



where the subscript indicates initial conditions. Here a is a new independent variable 

 which determines the vertical density distribution and i = dzjda. With these quantities 

 (XIX.35) gives 



MT = f /-^ + pz\ da. (XIX. 37) 



With the fundamental hydrostatic equation 



one obtains finally 



f = - gP^ (XIX.38) 



aa 



MT^ - \ i^ +-\ da. (XIX.39) 



The variation problem is the determination of the particular function p of a which 

 reduces MT to a minimum value for the given distribution of v with a. The variation of 

 p vanishes at the sea surface and it can be assumed that it also vanishes at great 

 depths. Under these circumstances the minimum value of MT is then given by 



8{MT) =[ \(^-^-^~-]8p-^ 8p] da = 0. (XIX.40) 



This is true for arbitrary values of 8p provided the function p satisfies Euler's equation 



gp da 



P^ gp. 



(XIX.41) 



which on substitution reduces to 



du^ = p da, (XIX.42) 



where a is the specific volume. 



To determine the final velocity distribution from the initial mass transport distri- 

 bution it is necessary to combine (XIX.42) with (XIX. 36) or 



pu dz = v{a) da. (XIX.43) 



Rossby has discussed several models with special density distributions according to 

 this principle; only those more or less directly concerned with the Gulf Stream will be 

 considered here. 



For a uniformly stratified current with speed Uq and depth D^ that is flowing on 

 top of a homogeneous bottom layer of density p^, in which the volume transport is 

 zero and that is allowed to readjust itself to a minimum momentum transfer current 



