A numerical shallow-water model for gravity currents for a wide range of density differences

Gravity currents with various contrasting densities play a role in mass transport in a number of geophysical situations. The ratio of the density of the current, ρc, to the density of the ambient fluid, ρa, can vary between 100 and 103. In this paper, we present a numerical method of simulating gravity currents for a wide range of ρc/ρa using a shallow-water model. In the model, the effects of varying ρc/ρa are taken into account via the front condition (i.e., factors describing the balance between the driving pressure and the ambient resistance pressure at the flow front). Previously, two types of numerical models have been proposed to solve the front condition. These are referred to here as the Boundary Condition (BC) model and the Artificial Bed (AB) model. The front condition is calculated as a boundary condition at each time step in the BC model, whereas it is calculated by setting a thin artificial bed ahead of the front in the AB model. We assessed the BC and AB models by comparing their numerical results with the analytical results for a simple case of homogeneous currents. The results from the BC model agree well with the analytical results when ρc/ρa≲102, but the model tends to overestimate the speed of the front position when ρc/ρa≳102\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}$\end{document}. In contrast, the AB model generates good approximations of the analytical results for ρc/ρa≳102\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}$\end{document}, given a sufficiently small artificial bed thickness, but fails to reproduce the analytical results when ρc/ρa≲102. Therefore, we propose a numerical method in which the BC model is used for currents with ρc/ρa≲102 and the AB model is used for currents with ρc/ρa≳102\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}$\end{document}.