| Abstract: | 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, and 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. |
