# Axisymmetric gravity currents in two-layer density-stratified media

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## Abstract

Numerous studies have considered the flow of a rectilinear, high Reynolds number, Boussinesq gravity current through a two-layer stratified ambient, however, far less is known concerning the analogue axisymmetric problem. Whereas in both instances there is the possibility of a dynamic coupling between the gravity current front and the waves that are excited by its forward advance, axisymmetric gravity currents entail the added complexity of a radially-diverging flow. Because a steady-state formulation cannot then be developed, we instead present a one-layer shallow water model that describes the flow evolution for various initial conditions and ambient stratifications. We also report upon \(>\!\!30\) full- and partial-depth lock release laboratory experiments that span a densitometric range \(0 \le S < 0.8868\) where \(S=(\rho _1-\rho _2)/(\rho _c-\rho _2)\) in which \(\rho _c\), \(\rho _1\) and \(\rho _2\) denote, respectively, the densities of the gravity current and lower and upper ambient layers. Of principal interest is the initial front speed of the gravity current for which good agreement is observed between laboratory measurement and shallow water numerical simulation, despite the limiting assumptions of the latter. The horizontal distance over which the initial front speed is maintained may span several lock-lengths, however, this depends on whether or not the gravity current is substantially impacted by the interfacial wave(s). For example, when the lower ambient layer is moderate and \(S\) is large, the transfer of momentum from the gravity current front to the wave may lead to a deceleration so severe that gravity current fluid is swept in the \(-r\) direction. The connection between our analysis and problems of pollution dispersion is briefly outlined.

## Keywords

Gravity currents Ambient stratification Interfacial waves Shallow water theory## Notes

### Acknowledgments

Financial support was generously provided by NSERC through the Discovery, USRA and RTI Grant programs. The tank used in the experimental portion of this study was fabricated in the Dept. of Mech. Eng. Machine Shop at the Univ. of Alberta. The ESM videos were produced with the kind assistance of Mr. Esmatullah Naikyar.

## Supplementary material

## References

- 1.Maxworthy T, Leilich J, Simpson J, Meiburg EH (2002) The propagation of a gravity current in a linearly stratified fluid. J Fluid Mech 453:371–394CrossRefGoogle Scholar
- 2.Ungarish M, Huppert HE (2002) On gravity currents propagating at the base of a stratified fluid. J Fluid Mech 458:283–301CrossRefGoogle Scholar
- 3.Ungarish M (2006) On gravity currents in a linearly stratified ambient: a generalization of Benjamin’s steady-state propagation results. J Fluid Mech 548:49–68CrossRefGoogle Scholar
- 4.White BL, Helfrich KR (2008) Gravity currents and internal waves in a stratified fluid. J Fluid Mech 616:327–356CrossRefGoogle Scholar
- 5.Goldman R, Ungarish M, Yavneh I (2014) Gravity currents with double stratification: a numerical and analytical investigation. Environ Fluid Mech 14:471–499CrossRefGoogle Scholar
- 6.Holyer JY, Huppert HE (1980) Gravity currents entering a two-layer fluid. J Fluid Mech 100:739–767CrossRefGoogle Scholar
- 7.Rottman JW, Simpson JE (1989) The formation of internal bores in the atmosphere: a laboratory model. Q J R Meteorol Soc 115:941–963CrossRefGoogle Scholar
- 8.Flynn MR, Ungarish M, Tan AW (2012) Gravity currents in a two-layer stratified ambient: the theory for the steady-state (front condition) and lock-released flows, and experimental confirmations. Phys Fluids 24:026601CrossRefGoogle Scholar
- 9.White BL, Helfrich KR (2012) A general description of a gravity current front propagating in a two-layer stratified fluid. J Fluid Mech 711:545–575CrossRefGoogle Scholar
- 10.Tan AW, Nobes DS, Fleck BA, Flynn MR (2011) Gravity currents in two-layer stratified media. Environ Fluid Mech 11(2):203–224. doi: 10.1007/s10652-010-9174-z CrossRefGoogle Scholar
- 11.Faust KM, Plate EJ (1984) Experimental investigation of intrusive gravity currents entering stably stratified fluids. J Hydraul Res 22(5):315–325CrossRefGoogle Scholar
- 12.Ungarish M (2005) Intrusive gravity currents in a stratified ambient—shallow-water theory and numerical results. J Fluid Mech 535:287–323CrossRefGoogle Scholar
- 13.Wu J (1969) Mixed region collapse with internal wave generation in a density stratified medium. J Fluid Mech 35:531–544CrossRefGoogle Scholar
- 14.Sutherland BR, Kyba PJ, Flynn MR (2004) Intrusive gravity currents in two-layer fluids. J Fluid Mech 514:327–353CrossRefGoogle Scholar
- 15.Flynn MR, Linden PF (2006) Intrusive gravity currents. J Fluid Mech 568:193–202CrossRefGoogle Scholar
- 16.Simpson JE (1997) Gravity currents, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
- 17.Holdsworth AM, Barrett KJ, Sutherland BR (2012) Axisymmetric intrusions in two-layer and uniformly stratified environments with and without rotation. Phys Fluids 24:036603CrossRefGoogle Scholar
- 18.Ungarish M (2009) An introduction to gravity currents and intrusions. CRC Press, Boca RatonCrossRefGoogle Scholar
- 19.Tan AW (2010) Gravity currents in two-layer stratified media. Master’s thesis, Univ. of AlbertaGoogle Scholar
- 20.Benjamin TB (1968) Gravity currents and related phenomena. J Fluid Mech 31:209–248CrossRefGoogle Scholar
- 21.Ungarish M (2007) Axisymmetric gravity currents at high Reynolds number—on the quality of shallow-water modeling of experimental observations. Phys Fluids 19:036602CrossRefGoogle Scholar
- 22.Huppert HE, Simpson JE (1980) The slumping of gravity currents. J Fluid Mech 99:785–799CrossRefGoogle Scholar
- 23.McMillan JM, Sutherland BR (2010) The lifecycle of axisymmetric internal solitary waves. Nonlinear Process Geophys 17:443–453CrossRefGoogle Scholar