On axisymmetric intrusive gravity currents in a stratified ambient – shallow-water theory and numerical results

Abstract

The intrusion of a fixed volume of fluid, which is released from rest and then propagates horizontally-radially at the neutral buoyancy level in a stratified ambient fluid, in a cylindrical geometry with a vertical axis (either fully axisymmetric or a wedge), is investigated. It is assumed that the density change of the ambient fluid is linear with height. A closed one-layer shallow-water Boussinesq inviscid formulation is presented. In general, the solution of the resulting hyperbolic system is obtained by a finite-difference scheme. However, for the large-time developed motion an analytical similarity solutions is derived. The self-similar result indicates radial expansion with t 1 / 3 but the shape is peculiar: the intruding fluid propagates like a ring with a fixed ratio of inner to outer radii; the inner domain (between the axis and the inner radius of the ring) contains clear ambient fluid. It is verified that the initial-value lock-release finite-difference solution indeed approaches the similarity predictions after an initial spread of the outer radius to about 2.5 times the initial radius. The shallow-water results are corroborated by numerical solutions of the full axisymmetric Navier–Stokes formulation. It is concluded that the shallow-water model is a versatile and accurate predictive tool, and that the peculiar ring-shape prediction reproduces an interesting physical property of the axisymmetric intrusion. The interaction between the internal gravity waves and the head is less significant than in the two-dimensional geometry. However, a practical limitation on the applicability of the inviscid model is imposed by the prediction that the ratio of viscous to inertia forces increases like r N 7 (where r N is the radius of propagation, scaled with the initial value).