Hamiltonian reductions, scalings, and effective wave models in stratified fluids

TL;DR

This work develops a Hamiltonian framework for fully nonlinear interfacial waves in sharply stratified, 2D Euler fluids and derives reduced 1D long-wave models via Poisson and Dirac-type reductions. Using Marsden–Ratiu reduction on a two-layer channel, the authors obtain a canonical 1D formulation and recover the Serre–Green–Naghdi equations through a double scaling limit, while also establishing a deep-water, local model under an alternate scaling. They further cast the Miyata–Choi Camassa equations in a Hamiltonian form and perform a Dirac reduction to obtain a constrained two-variable system that coincides with the reduced MR structure. The results provide a geometric, canonical underpinning for a family of interfacial wave models and unify reductions from the full 2D Euler structure to well-known long-wave equations. The approach clarifies how canonical Hamiltonian structures persist under asymptotics and reductions, with potential implications for robust, structure-preserving simulations of stratified fluids.

Abstract

We apply Poisson reduction techniques to describe asymptotic fully nonlinear models of fluid wave motion in the Hamiltonian setting. We start by considering Zakharov and Benjamin Hamiltonian settings for a stably stratified $2D$ Euler fluid. We use a Marsden-Ratiu reduction scheme for sharply stratified fluids to obtain a canonical formulation of the stratified effective model in one space variable. The long-wave Serre-Green Naghdi (SGN) equations is then recovered by means of a suitable double scaling limit in the Hamiltonian function. We also consider the opposite double-scaling limit, which leads to a local model in the "large-lower layer" regime. Furthermore, applying the previous results on the canonical structure of the SGN equations, we provide the Miyata-Choi Camassa (CC) equations for fully non-linear waves in sharply stratified fluids with a natural Hamiltonian structure. We also study the reduced Hamiltonian system obtained taking the natural constraints of the CC equations into account. To this end, we perform a Dirac-type reduction on a suitable constrained submanifold of fluid field configurations.

Figures (4)

• Figure 1: The flow-chart of the paper
• Figure 2: Graphical representation of the two-layer configuration. The quantity $\eta_2$ (resp., $\eta_1$) is the total thickness of the lower, heavier (resp., upper, lighter) fluid. The interface $\zeta$ is measured from the quiescent state $z=0$. ${\bf U}_j(x,z,t)=(u_j(x,z,t), w_j(x,z,t))$ are the velocities vector fields in the two layers. The Hamiltonian variables are related with the interface values, i.e., the "traces" of the velocities, ${{\widetilde{u}}}_j(x, t)=u_j(x, \zeta(x,t), t)$, ${{\widetilde{w}}}_j(x,t)=w_j(x, \zeta(x,t), t)$.
• Figure 3: The Air-Water limit. The vertical scaling parameter is the asymptotic height $h_2$ of the lower fluid. The double scaling limit $h_1=\infty$, $\rho_1\to 0$ with the product $\rho_1 h_1\to 0$ is performed. As it will be shown in this Section (and can naturally be expected), the equations of the stratified model reduce to the Serre Green-Naghdi fully non-linear water wave equations
• Figure 4: The large lower layer ("deep water") limit. The vertical-length scaling parameter is now the asymptotic height of the upper fluid $h_1$. The asymptotics $h_2\to\infty$ is performed. The densities $\rho_2>\rho_1>0$ are constant parameters in the scaling. As it will be show in this Section, the model reduces to a water wave-type model for the upper layer, the presence of the lower layer being reflected in the nonlinearity and dispersion parameters of the wave model.

Theorems & Definitions (3)

• Proposition 3.1
• Proposition 4.1
• Proposition 5.1