Weakly nonhydrostatic effects in compositionally-driven gravity flows

In the study of compositionally-driven gravity currents it is customary to adopt the hydrostatic assumption for the pressure field which, in turn, leads to a depth-independent horizontal velocity field and significant simpilifications to the governing equations. The hydrostatic assumption is reasonable in, say, the case of a two-layer flow when the depth variations of the lower layer are small when considered as a function of space and time. However, for larger deflections of the interface (such as those caused by bottom topography) the flow will deviate in its behavior from the low aspect ratio, slowly varying purely hydrostatic flow because of the presence of vertical accelerations. In this paper we present an approach to capture the contribution of interface curvature to nonhydrostatic effects in fully time-dependent flows in two-fluid systems. Our approach involves expanding the relevant dependent variables in the form of an asymptotic expansion f = f⁽⁰⁾ + δ²f⁽¹⁾ + o(δ²), where 0 < δ ≪ 1 is the aspect ratio of the flow, and obtaining the first-order correction to hydrostatic theory. Numerical results and comparisions with the purely hydrostatic theory are included.