VORTICITY GENERATION IN THE SHALLOW-WATER EQUATIONS AS APPLIED TO HYDRAULIC JUMPS

The authors attempt to find a bridge between the vorticity dynamics of a finite cross-stream length hydraulic jump implied by the Navier-Sto kes equations and that given by the shallow-water approximation (SWA) with the turbulence of the hydraulic jump parameterized. It is establi shed that, in the actual hydraulic jump, there is horizontal vorticity associated with the time-mean flow in the fluid interior, and that th is vorticity has been fluxed down by turbulent eddies from the upper p art of the fluid layer. The authors then point out that this vertical flux of cross-stream vorticity component is (minus) the cross-stream f lux of vertical vorticity component. (The divergence of the latter at the lateral edges of a hydraulic jump of finite cross-stream extent pr oduces time-mean vertical vorticity.) Hence, the line of inquiry devol ves to a search for the source of the cross-stream vorticity that is b eing fluxed downward. For a hydraulic jump in the lee of a submerged o bstacle, the authors argue that that source is the baroclinic producti on of vorticity at the free surface. It is shown that the SWA version of the flow through the jump requires that the vertical flux of cross- stream vorticity component be independent of depth (but not zero), and that previously only its role as (minus) the cross-stream flux of ver tical vorticity has been discussed. On the understanding developed her ein of the actual hydraulic-jump vorticity dynamics and the SWA versio n, the authors describe the relation between the vorticity distributio ns found in shallow-water models with parameterized turbulence and tha t in a continuously stratified model of flow past an obstacle.