Gravity currents: entrainment, stratification and self-similarity

We present new experiments of the motion of a turbulent gravity current produced by the rapid release of a finite volume of dense aqueous solution from a lock of length $L$ into a channel $x>0$ filled with a finite depth, $H$, of fresh water. Using light attenuation we measure the mixing and evolving density of the flow, and, using dye studies, we follow the motion of the current and the ambient fluid. After the fluid has slumped to the base of the tank, there are two phases of the flow. When the front of the current, $x_{n}$, is within the region $2L<x_{n}<7L$, the fluid in the head of the current retains its original density and the flow travels with a constant speed. We find that approximately $0.75(\pm 0.05)$ of the ambient fluid displaced by the head mixes with the fluid in the head. The mixture rises over the head and feeds a growing stratified tail region of the flow. Dye studies show that fluid with the original density continues to reach the front of the current, at a speed which we estimate to be approximately $1.35\pm 0.05$ times that of the front, consistent with data of Berson (Q. J. R. Meteorol. Soc., vol. 84, 1958, pp. 1–16) and Kneller et al. (J. Geophys. Res. Oceans, vol. 104, 1999, pp. 5281–5291). This speed is similar to that of the ‘bore’, the trailing edge of the original lock gate fluid, as described by Rottman & Simpson (J. Fluid Mech., vol. 135, 1983, pp. 95–110). The continual mixing at the front leads to a gradual decrease of the mass of unmixed original lock gate fluid. Eventually, when the nose extends beyond $x_{n}\approx 7L$, the majority of the lock gate fluid has been diluted through the mixing. As the current continues, it adjusts to a second regime in which the position of the head increases with time as $x_{n}\approx 1.7B^{1/3}t^{2/3}$, where $B$ is the total buoyancy of the flow per unit width across the channel, while the depth-averaged reduced gravity in the head decreases through mixing with the ambient fluid according to the relation $g_{n}^{\prime }\approx 4.6H^{-1}B^{2/3}t^{-2/3}$. Measurements also show that the depth of the head $h_{n}(t)$ is approximately constant, $h_{n}\sim 0.38H$, and the reduced gravity decreases with height above the base of the current and with distance behind the front of the flow. Using the depth-averaged shallow-water equations, we derive a new class of self-similar solution which models the lateral structure of the flow by assuming the ambient fluid is entrained into the current in the head of the flow. By comparison with our data, we estimate that a fraction $0.69\pm 0.06$ of the ambient fluid displaced by the head of the current is mixed into the flow in this approximately self-similar regime, and the front of the current has a Froude number $0.9\pm 0.05$. We discuss the implications of our results for the evolution of the buoyancy in a gravity current as a function of the distance from the source.