Gravity current escape from a topographic depression

An inertial gravity current released within a topographic depression may climb up the incline from a lower to an upper plateau if it is sufficiently energetic and then continue to flow unsteadily away from the step while simultaneously draining back into the depression. This density-driven motion is investigated theoretically using the shallow-water equations to simulate the flow up a smooth step from a lower to an upper horizontal plane and to compute the volume of fluid that escapes from the depression. It is shown that it is possible for all of the fluid to drain back down the step and that the volume of the escaped fluid diminishes with a power-law dependence on time. This phenomenon is explained by analyzing the unsteady flow of a gravity current along a semi-infinite horizontal plane along which the front advances but simultaneously fluid drains from the rear edge of the plane. The dynamics of this motion at early times is calculated both numerically and analytically. The latter exploits the hodograph transformation of the shallow water equation, a technique which allows rapid and precise evaluation of flow features such as reflections and the onset of bores. The flow at later times becomes self-similar, and the self-similarity is of the second kind. It features an anomalous exponent that provides the power-law dependence of the volume of fluid on time, and this exponent is a function of the imposed Froude number at the front of the current. In this way the diminishing volume of fluid that escapes from a topographic depression is explained quantitatively. Eventually the flow may transition to a regime in which drag becomes non-negligible; the model of simultaneous propagation and draining is extended to this regime to show that the escaped volume of fluid also decays temporally and ultimately vanishes.