  We introduce a new class of Green-Naghdi type models for the propagation of
internal waves between two (1+1)-dimensional layers of homogeneous, immiscible,
ideal, incompressible, irrotational fluids, vertically delimited by a flat
bottom and a rigid lid. These models are tailored to improve the frequency
dispersion of the original bi-layer Green-Naghdi model, and in particular to
manage high-frequency Kelvin-Helmholtz instabilities, while maintaining its
precision in the sense of consistency. Our models preserve the Hamiltonian
structure, symmetry groups and conserved quantities of the original model. We
provide a rigorous justification of a class of our models thanks to
consistency, well-posedness and stability results. These results apply in
particular to the original Green-Naghdi model as well as to the Saint-Venant
(hydrostatic shallow-water) system with surface tension.
