Initial lock ratio effects on the dynamics of constant-volume density currents

The behaviour of non-Boussinesq constant-volume density currents of density \textit{$\rho $}$_{c}$, released from a lock of height $h_{0}$ and length $x_{0}$ into a ambient of height $H$ and density \textit{$\rho $}$_{a}$, are considered. Two-dimensional Navier-Stokes simulations are used to cover a wide range of density ratio 10$^{-2}<$\textit{$\rho $}$_{c}$/\textit{$\rho $}$_{a}<$10$^{2}$ and initial lock ratio 0.5$\le $\textit{$\lambda $}$\le $18.75. The Navier-Stokes results are compared with predictions of a shallow-water model. In particular, we derive novel insights on the influence of the lock aspect ratio \textit{$\lambda $}=$x_{0}$/$h_{0}$ on the shape and motion of the current in the slumping stage. It is shown that a critical value exists, \textit{$\lambda $}$_{crit}$; the dynamics of the current is significantly influenced by \textit{$\lambda $} if below \textit{$\lambda $}$_{crit}$. We conjecture that this critical lock ratio depends on two characteristic time scales, namely the slumping time and the time of formation of the current's head. Comparison with space-time diagrams obtained from the Navier-Stokes simulations show a good agreement. We present a simple analytical model which support the observation that for a light current the speed of propagation is proportional to \textit{$\lambda $}$^{1/4}$ when \textit{$\lambda $}$<$\textit{$\lambda $}$_{crit}$.