The front speed of intrusive gravity currents

Abstract

This paper describes an experimental and numerical study of an intrusion propagating along the interface of a two-layer fluid in a channel. We find that when the density of the intrusion is the depth-weighted mean of the layer densities the interface ahead of the intrusion is undisturbed, but for other densities the interface ahead of the intrusion is displaced vertically. We find that this vertical displacement, which takes the form of an upstream-propagating long wave, depends on the properties of the intrusion and not only on the relative depths of the two layers. For the case when the interface is undisturbed the intrusion propagation speed is a minimum. We develop an energy argument that describes the observed variation of the intrusion speed from this minimum speed as a function of the intrusion and layer densities and the ratio of the layer depths. We also show that if, and only if, the layer depths are equal, the speed of the intrusion is independent of the density of the intrusion.

Intrusive gravity currents propagating into two-layer stratified ambients: Vorticity modeling

“…Within the present investigation, we conducted simulations for * = 0.1, 0.2...0.9, as well as various * -values, while only those of * = 0.5 are presented here. The figure shows that the vorticity model predictions are close to those of the earlier models by [3] and [4], and over a remarkable range of * fall in between these two models. Within its narrow range of validity between the vertical dotted lines, the model of [5] yields predictions in very close agreement with those of the other models, so that we do not show them in this figure.…”

“…ℎ * = * /2 and ℎ * = * /2. These results are fully consistent with [1], [2], [3] and [4]. For the non-symmetric intrusion, a closed-form solution is not attainable, due to the complexity and the nonlinearity of the system of the equations, but a numerical solution can be obtained via Newton iteration or other numerical methods.…”

“…This is consistent with the fact that for these limits, the single-layer gravity current case will be recovered, where we do not expect the emergence of any internal bore. the results presented by [4] and discrete circles and crosses show the DNS results of the present study and the experimental data of [3], respectively. Furthermore, the vertical dotted lines in 4a demonstrate the range of validity of the investigation by [1].…”

“…Various theoretical models have been proposed to predict the features of the intrusions, such as their thickness and front velocity ( [1], [2], [3], [4]). Unanimously, all these earlier models are based on some form of empirical energy assumptions, e.g.…”

“…Unanimously, all these earlier models are based on some form of empirical energy assumptions, e.g. they may assume zero-headloss along certain streamlines, while they employ mass and horizontal momentum conservation equations ( [1], [2], [4]), or they consider non-dissipative conversion of all the available potential energy to kinetic energy of the intrusion and return flows, in the absence of mixing ( [3]). In the present investigation, we aim to avoid such empirical assumptions, by using horizontal and vertical momentum conservation simultaneously in the form of vorticity equations, in addition to the mass conservation equations.…”