Vorticity Generation in the Shallow-Water Equations as Applied to Hydraulic Jumps

Abstract

The authors attempt to find a bridge between the vorticity dynamics of a finite cross-stream length hydraulic jump implied by the Navier-Stokes equations and that given by the shallow-water approximation (SWA) with the turbulence of the hydraulic jump parameterized. It is established that, in the actual hydraulic jump, there is horizontal vorticity associated with the time-mean flow in the fluid interior, and that this vorticity has been fluxed down by turbulent eddies from the upper part of the fluid layer. The authors then point out that this vertical flux of cross-stream vorticity component is (minus) the cross-stream flux of vertical vorticity component. The divergence of the latter at the lateral edges of a hydraulic jump of finite cross-stream extent produces time-mean vertical vorticity.) Hence, the line of inquiry devolves to a search for the source of the cross-stream vorticity that is being fluxed downward. For a hydraulic jump in the Ice of a submerged obstacle, the authors argue that that source is the baroclinic production 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 vertical vorticity has been discussed. On the understanding developed herein of the actual hydraulic-jump vorticity dynamics and the SWA version, the authors describe the relation between the vorticity distributions found in shallow-water models with paramerized turbulence and that in a continuously stratified model of flow past an obstacle.