Nonlinear Thbobies — Inertial 105 



rossby's wake-stream theory 



Fourteen years before the development of the linear theories discussed in 

 the preceding chapter, Rossby (1936 a) worked out an interesting nonlinear 

 model of the Gulf Stream in which large-scale lateral mixing was involved. 

 The geostrophic relationship alone does not prescribe a transverse velocity 

 profile for the Stream ; and hence Rossby made use of the ideas of a turbulent 

 jet stream from the field of experimental fluid mechanics to define the 

 profile. The Straits of Florida were supposed by Rossby to act as a nozzle, 

 or jet; the Gulf Stream was therefore regarded as a purely inertial stream, 

 interacting with its environment by mixing. 



Rossby reduced the problem to one on a nonrotating system, by ex- 

 tracting from the equations the Coriolis-force terms and their associated 

 pressure gradients, and by assuming that the motion is essentially non- 

 divergent. The problem was thus reduced to the one that had already been 

 discussed by ToUmien (1926), who was able to obtain good agreement with 

 experimental results by assuming (i) that the pressure gradient along the 

 jet is small, and therefore that the total momentum transport of the jet is 

 the same at all sections downstream; and (ii) that the shearing stresses 

 acting upon the jet may be described in terms of a mixing length propor- 

 tional to the distance from the nozzle. 



ToUmien found that the mass transport and width of the jet increase 

 downstream. This requires that there be an inflow of water from the sur- 

 rounding medium into the jet. 



Although the mass transport of the Gulf Stream does increase after 

 leaving the Florida Straits, it does not increase after passing Cape Hatteras. 

 The angular spread of experimentaUy produced jets (Forthmann, 1934) 

 varies greatly, but Peters and Bicknell (1936) obtained spreads of between 

 8 and 14°. The spread of the actual Gulf Stream in its instantaneous form is 

 less than 1°, although the mean spread averaged over many different sets of 

 observations is greater (Stommel, 1951). 



After discussing the analogy to ToUmien's jet in a homogeneous ocean, 

 Rossby (1936a) made some qualitative studies of the effect of the stratifi- 

 cation of the sea, by considering an ocean made up of two layers of shghtly 

 diff"erent density, the lower layer being at rest. By the geostrophic relation- 

 ship, the difference in level of the interface at the two sides of the jet stream 

 is a measure of the total mass transport of the stream; therefore, if the mass 

 transport increases downstream, it soon becomes impossible to join the 

 interface inside the stream to the level of the undisturbed water masses 

 outside the stream, unless the interface is bent back (up on the right, down 

 on the left) to these levels. This requires, geostrophicaUy, a countercurrent 

 on each side of the jet. 



