On axisymmetric intrusive gravity currents: The approach to self-similarity solutions of the shallow-water equations in a stratified ambient

The axisymmetric intrusion of a fixed volume of fluid, which is released from rest and then propagates radially at the neutral buoyancy level in a deep linearly stratified ambient fluid is investigated. The SW equations representing the high-Reynolds number motion are used. For the long-time motion an analytical similarity solution indicates propagation with t 1 / 3 , but the shape is peculiar: the intrusion propagates like a ring and the inner domain contains a thin tail of clear ambient fluid. To avoid accumulation of numerical errors the problem was reformulated in terms of new variables and solved by finite-difference scheme. It is shown that the initial-value problem tends to the similarity prediction. Comparison with the non-stratified case is presented. It was found that for the non-stratified case there is a similar “tail-ring” stage of propagation, however this stage is only a transient to a different self-similar shape.