An inertia 'paradox' for incompressible stratified Euler fluids

TL;DR

This paper investigates a paradox in stratified incompressible Euler fluids where horizontal momentum is not conserved due to the coupling of incompressibility and rigid lid constraints, and explains how this is resolved by removing inertia or stratification.

Contribution

It identifies the cause of the momentum non-conservation paradox and demonstrates how it is resolved by eliminating inertia or stratification, providing exact results and simulations.

Findings

Horizontal momentum is not conserved due to the rigid lid constraint and inertia.

Removing inertia or stratification restores momentum conservation.

Layer-averaged models and simulations confirm the theoretical analysis.

Abstract

The interplay between incompressibility and stratification can lead to non-conservation of horizontal momentum in the dynamics of a stably stratified incompressible Euler fluid filling an infinite horizontal channel between rigid upper and lower plates. Lack of conservation occurs even though in this configuration only vertical external forces act on the system. This apparent paradox was seemingly first noticed by Benjamin (J. Fluid Mech., vol. 165, 1986, pp. 445-474) 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,action at a distance by incompressibility, and inertia is illustrated by layer-averaged exact results, two-layer long-wave models, and direct numerical simulations of the incompressible Euler equations with smooth stratification.