Nonlinear Traveling Internal Waves in Depth-Varying Currents

TL;DR

This paper investigates nonlinear traveling internal waves in stratified fluids with depth-varying shear currents, deriving equations at the interface, numerically computing solutions, and analyzing their stability influenced by shear strength and direction.

Contribution

It extends existing models to include depth-varying shear and interface density differences, providing new insights into wave stability under complex shear conditions.

Findings

Shear strength affects wave stability.

Opposing shears can amplify or suppress instabilities.

Numerical continuation reveals bifurcation behavior.

Abstract

In this work, we study the nonlinear traveling waves in density stratified fluids with depth varying shear currents. Beginning the formulation of the water-wave problem due to [1], we extend the work of [4] and [18] to examine the interface between two fluids of differing densities and varying linear shear. We derive as systems of equations depending only on variables at the interface, and numerically solve for periodic traveling wave solutions using numerical continuation. Here we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier-Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding traveling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.