A new class of two-layer Green-Naghdi systems with improved frequency dispersion

Vincent Duchêne (1) , Samer Israwi (2) , Raafat Talhouk (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Université Libanaise, Faculté des Sciences Section (1)

We introduce a new class of Green-Naghdi 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 Green-Naghdi model, and in particular to manage high-frequency Kelvin-Helmholtz instabilities. 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.

Mots clés: Kelvin-Helmholtz; Green-Naghdi; asymptotic models; internal waves

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Duchêne, Samer Israwi, Raafat Talhouk. A new class of two-layer Green-Naghdi systems with improved frequency dispersion. 2015. ⟨hal-01127972v1⟩