On variational formulations and conservation laws for incompressible 2D Euler fluids

With the aim of presenting a unified viewpoint for the variational and Hamiltonian formalism of two-dimensional incompressible stratified Euler equations, we revisit some of the formulations currently discussed in the literature and examine their mutual relations. We concentrate on the example of two-layered systems and its one-dimensional reduction, and use it to illustrate general consequences of density stratification on conservation laws which have been partially overlooked until now. In particular, we focus on the conservation of horizontal momentum for stratified ideal fluid motion under gravity between rigid lids.