TURBULENT TRANSPORT 341 



The mathematical relation on the right shows the total transport 

 of mass to the north in the upper layer {T^f) for a geostrophic flow 

 of arbitrary depth profile. In this relation g is the acceleration of 

 gravity, a the coefficient of thermal expansion of water, and / is 

 the Coriolis parameter. For continuity in our upper layer T^ must 

 equal the total mass transport to the south induced by the wind 

 stress over the entire layer. The mathematical relation on the left is 

 Sverdrup's (1947) famous deduction relating the southerly mass 

 transport to the wind stress and to j8, the variation of /with latitude. 



How^ever, the idealization requires that this Gulf Stream, or at 

 least a significant fraction of it, mix with the colder waters of the 

 north. Therefore conservation of mass requires that cold water be 

 reincorporated into the upper layer. The baroclinic energy sources 

 for this Gulf Stream fragmentation require that, on the average, 

 warm water rise and colder water sink. Hence it is unlikely that 

 more than a fraction of the cold water will be reincorporated into 

 the upper layer by horizontal transfer at the northern boundary of 

 the upper layer. The alternative is the reincorporation of cold 

 water into the upper layer by a general ascent under the whole 

 tropical ocean. The conservation of heat requires that this rein- 

 corporated cold water be warmed by a heat flow down from the 

 surface and also that the warm Gulf Stream fragments in the 

 northern sea be cooled by a heat flow up through the surface to the 

 atmosphere and to space. 



The integral consequences of these continuity requirements are: 



H = WAT ^ K^ 

 K = DW 



The total kinematic heat flux to the north is TxAT. As this is lost 

 to the upper layer, it must be resupplied by an average local heat 

 flux // equals to TnAT/A down from the tropical surface, where ^ 

 is the area of the upper layer and AT" is equal to Th — Tc- The 

 average vertical velocity of the cold water reincorporated into 

 the upper layer is Tn/A. Therefore, H equals IFAT. 



