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

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 t1/3 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 (where rN is the radius of propagation, scaled with the initial value).