An inertia paradox for incompressible stratified Euler fluids

The interplay between incompressibility and stratification can lead to non-conservation of horizontal momentum in the dynamics of a stably stratified Euler fluid filling an infinite horizontal channel between rigid upper and lower plates. Lack of conservation occurs even though gravity is the only (vertical) force acting on the system, and no lateral boundaries are present. This apparent paradox was seemingly first noticed by Benjamin (1986) in his classification of the invariants by symmetry groups with the Hamiltonian structure of the Euler equations in two dimensional settings, but it appears to have been largely ignored since. By working directly with the motion equations, the paradox is shown here to be a consequence of the rigid lid constraint coupling through incompressibility with the infinite inertia of the far ends of the channel, assumed to be at rest in hydrostatic equilibrium. Accordingly, when inertia is removed by eliminating the stratification, or, remarkably, by using the Boussinesq approximation of uniform density for the inertia terms, horizontal momentum conservation is recovered. This interplay between constraints, incompressibility-induced action-at-a-distance and inertia is illustrated by layer averaged exact results, two-layer long-wave models, and direct numerical simulations of stratified Euler equations with smooth stratification.