Two‐layer Gravity Currents with Topography

Two‐dimensional and time‐dependent gravity currents involving the initial release of a fixed volume of heavy fluid over a gradually sloping bottom and underlying a layer of lighter fluid are considered. The equations which describe the resulting two‐layer flow are derived from the Navier–Stokes equations for a constant density, inviscid, nonrotating fluid, neglecting kinematic viscosity, surface tension, and entrainment between the layers. A new addition to the theory is introduced in the form of a forcing term in the lower layer horizontal momentum equation which is incorporated to produce the characteristic structure typical of such gravity currents in the laboratory. This delaying term is restricted to the front of the gravity current, and as such is shown to be valid under conventional shallow‐water scaling assumptions. The hyperbolic character of the equations of motion is shown, a simple numerical test for hyperbolicity is derived from theoretical considerations, and these results are related to the stability Froude number of the flow. Well‐posedness of the initial boundary value problem is proven via localization of the equations, and the discussion is extended to a two‐point boundary value problem with examples of steady‐state and traveling wave solutions given for a bottom surface of constant slope. Numerical results are obtained by using a recently developed finite‐difference relaxation scheme for conservation laws, sufficiently modified herein to include spatial variability and forcing terms, which approximates the material interface at the front of the lower fluid layer as a shock. The effects of slope and the delaying force are investigated numerically to determine their theoretical importance, and the range of expected values is compared to published experimental results. Some calculations for the temporal evolution of the flow are produced that display the phenomenon of rear wall separation for nonzero slopes.