Shallow-Water Flow past Isolated Topography. Part I: Vorticity Production and Wake Formation

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Abstract
The flow of a single layer of shallow water past high three-dimensional topography is studied in a nonrotating environment and in the absence of surface friction. The dimensionless parameters for this problem are the upstream Froude number, the dimensionless mountain height, and a dimensionless measure of the dissipation rate. In part I of this study, high-resolution numerical simulations are utilized to construct a regime diagram for steady left–right symmetric flow and for the domain of parameter space with subcritical upstream conditions. Three distinct regimes occur. They are characterized, respectively, by fore–aft symmetry, essentially inviscid dynamics, and entirely subcritical conditions (regime I); by transition to supercritical flow and the occurrence of a hydraulic jump over the lee slope (regime II); and by the inability of the flow to climb the mountain top resulting in flow separation (regime III). Regimes II and III are associated with a wake that entails significant potential vorticity features and sometimes reversed flow. Potential vorticity is produced by two related mechanisms. First, internal dissipation in the shallow-water system is generally not possible without potential vorticity production, even in an initially fully irrotational state and in absence of surface friction. The proof of this follows from a new theorem, which states that the steady-state Bernoulli function is the streamfunction of the total (i.e., advective and dissipative) vorticity flux. Second, flow separation in regime III can lead to the formation of contact discontinuities that are connected to the separation point and represent the inviscid limit of shearlines. Here potential vorticity production at the separation point is related to the joining of two streams of fluid with different values of the Bernoulli function.

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