Nonlinear shallow-water waves with vertical odd viscosity

TL;DR

This paper investigates how vertical odd viscosity influences nonlinear shallow-water waves, revealing new effects, modified wave equations, and the potential for compact solitary waves, with implications for geophysical fluid dynamics.

Contribution

It introduces the effects of vertical odd viscosity on shallow-water wave dynamics, deriving modified KdV, Ostrovsky, and KP equations, and predicts the emergence of compact solitary waves at high odd viscosity.

Findings

Odd viscosity induces new nonlinear effects in shallow-water waves.

Modified wave equations with altered dispersion properties.

Existence of compact two-dimensional solitary waves at high odd viscosity.

Abstract

The breaking of detailed balance in fluids through Coriolis forces or odd-viscous stresses has profound effects on the dynamics of surface waves. Here we explore both weakly and strongly non-linear waves in a three-dimensional fluid with vertical odd viscosity with and without the Coriolis effect. Our model describes the free surface of a shallow fluid composed of nearly vertical vortex filaments, which all stand perpendicular to the surface. We find that the odd viscosity in this configuration induces previously unexplored non-linear effects in shallow-water waves, arising from both stresses on the surface and stress gradients in the bulk. By assuming weak nonlinearity, we find reduced equations including Korteweg-de Vries (KdV), Ostrovsky, and Kadomtsev-Petviashvilli (KP) equations with modified coefficients. At sufficiently large odd viscosity, the dispersion changes sign, allowing for compact two-dimensional solitary waves. We show that odd viscosity and surface tension have the same effect on the free surface, but distinct signatures in the fluid flow. Our results describe the collective dynamics of many-vortex systems, which can also occur in oceanic and atmospheric geophysics.