Nonlinear Theories — Inertial 115 



THE CRITICAL INTERNAL FROUDE NUMBER 



One of the important dimensionless numbers characterizing the flow of 

 homogeneous frictionless fluids in channels with a free surface is the well- 

 known Froude number, U^jgD, where U is the mean velocity of the fluid 

 along the channel, g is the acceleration due to gravity, and D is the distance 

 from the free surface to the bottom of the channel. According to ordinary 

 hydrauHc engineering practice, the flow is said to be subcritical, critical, or 

 supercritical, depending upon whether the Froude number is less than, 

 equal to, or greater than unity. In natural watercourses and engineering 

 works such as canals and aqueducts, the flow is almost always subcritical 

 except at certain control points (weirs and dams, for example), where the 

 flow is locally critical. Whereas small perturbations m water level (long 

 gravity waves) can move upstream in subcritical flow, they are brought to a 

 standstill at points along the watercourse where the flow is critical. This 

 explains the importance of points of critical flow in determining the water 

 level upstream. Supercritical flow in nature is comparatively rare; it 

 normaUy terminates abruptly in a hydraulic jump which converts the flow 

 back to subcritical. 



Considering the great depth of the ocean, and the very moderate velocities 

 of flow, it is clear that no ocean current even remotely approaches the 

 ordinary critical Froude number. However, in a density-stratifled fluid 

 there are other 'internal' Froude numbers associated with the speed of 

 propagation of long internal waves. In a system consisting of a very deep 

 dense layer at rest, and a less dense surface layer of depth D moving at 

 velocity U, the internal Froude number is JJ^jg'D, where g' = {^p/p) g. In 

 the stream of uniform potential vorticity discussed earlier in this chapter, 

 in the theories of Morgan and Charney outhned later in this chapter, and in 

 the simple theory of steady meanders presented in Chapter IX, parts of the 

 flow are supercritical with respect to the internal Froude number. In the 

 summer of 1955, in discussions at the Woods Hole Oceanographic Institu- 

 tion, Dr Rossby pointed out that such regimes are hkely to be unstable and 

 probably do not exist in nature. Therefore we should not expect transverse 

 proflles of velocity such as those given in equation (12) to hold for values of 

 DjDq less than that for which v = ^{g'D). An easy calculation, for the 

 stream of uniform vorticity, shows that this occurs at Z) = 0-38 D^. In view 

 of the fact that this simple model of the Gulf Stream fits the data from 

 observations so well, it is interesting to note that this is where the point of 

 inflection in the depth of the 10° C. isotherm actually occurs (see fig. 33). 

 Rossby (1951) has extended the concept of critical flow to flows with 

 arbitrary vertical density structure. 



Up to the present, no one has succeeded in working out a detailed Froude 



