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On Axisymmetric Intrusive Gravity Currents: The Approach to Self-Similarity Solutions of the Shallow-Water Equations in a Linearly Stratified Ambient

Tamar Zemach and Marius Ungarish
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 463, No. 2085 (Sep. 8, 2007), pp. 2165-2183
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/20209304
Page Count: 19
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On Axisymmetric Intrusive Gravity Currents: The Approach to Self-Similarity Solutions of the Shallow-Water Equations in a Linearly Stratified Ambient
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Abstract

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. Attention is focused on the development of self-similar propagation. The shallow-water 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 with a fixed ratio of inner to outer radii; the inner domain contains clear ambient fluid. To verify the similarity analytical prediction, a long-time finite-difference solution with realistic initial conditions was performed. To avoid accumulation of numerical errors, the problem was reformulated in terms of new variables. It is shown that the numerical solution has a 'tail-ring' shape. The 'tail' decays like t⁻² and the 'ring' tends to the analytical similarity prediction. The initial geometry of the lock does not influence this result. Comparison with the non-stratified case is also presented. It was found that for the non-stratified case, there is a stage of propagation in which the intrusion has a similar 'tail-ring' form; however, this stage is only a transient to a self-similar shape which is different from that obtained for the stratified ambient.

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