In this paper, the two-layer viscous shallow-water equations are derived from the threedimensional Navier-Stokes equations under the hydrostatic assumption. It is noted that the combination of upper and lower equations in the two-layer model produces the classical one-layer equations if the density of each layer is the same. The two-layer equations are approximated by a finite element method which follows our numerical scheme established for the one-layer model in 1978. Finally, it is numerically demonstrated that the interfacial instability generated when the densities are the same can be eliminated by providing a sufficient density difference.
"A Finite Element Scheme for Two-Layer Viscous Shallow-Water Equations." Japan J. Indust. Appl. Math. 23 (2) 163 - 191, June 2006.