The Square Root Depth Wave Equations

TL;DR

The paper introduces the Square Root Depth equations, a new set of well-posed, Hamiltonian multilayer water wave equations that improve upon the Green-Naghdi equations by handling shear effects and preserving key wave solutions.

Contribution

It presents the Square Root Depth equations, a novel, well-posed, Hamiltonian model for multilayer water waves that retains key solutions of previous models while addressing their ill-posedness.

Findings

The new equations are well-posed in the presence of shear.

They preserve traveling wave solutions similar to the Green-Naghdi equations.

Numerical results demonstrate accurate modeling of multilayer wave interactions.

Abstract

We introduce a set of coupled equations for multilayer water waves that removes the ill-posedness of the multilayer Green-Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear they retain the same travelling wave solutions as MGN. We call the new model the Square Root Depth equations, from the modified form of their kinetic energy of vertical motion. Our numerical results show how the Square Root Depth equations model the effects of multilayer wave propagation and interaction, with and without shear.