On gravity currents of fixed volume that encounter a down-slope or up-slope bottom

We consider a gravity current released from a lock into an ambient fluid of smaller density, that, from the beginning or after some horizontal propagation X-1, propagates along an inclined (up- or down-) bottom. The flow (assumed in the inertial-buoyancy regime) is modeled by the shallow-water (SW) equations with a jump condition applied at the nose (front). The behavior of the current is dominated by the slope angle, theta, but is also affected by additional dimensionless parameters: the aspect ratio of the lock x(0)/h(0), the height ratio of the ambient to lock, H/h(0), and the distance of the backwall from the beginning of the slope, X-1/x(0). We show that the stability of the interface, reflected by the value of the bulk Richardson number, Ri, is essential in the interpretation and modeling. In the upslope flow, Ri increases and hence entrainment/mixing effects are unimportant. In the downslope flow, the current first accelerates and Ri decreases; this enhances entrainment and drag, which then decelerate the current. We show that the accelerating-decelerating downstream current is reproduced well by a SW model combined with a simple closure for the entrainment and drag. A comparison of the theoretical results with previously published experimental data for both upslope flow and downslope flow show fair agreement.