A Hamiltonian Formulation for Long Internal Waves

TL;DR

This paper introduces a new Hamiltonian framework for modeling long internal waves in rotating fluids, accounting for vorticity and shear, and derives a spectral energy evolution equation with a spectrum similar to oceanic observations.

Contribution

It develops a novel Hamiltonian formalism for internal waves that includes vorticity and shear effects, providing new insights into wave dynamics in rotating environments.

Findings

Derived a Hamiltonian structure for internal waves with vorticity and shear.

Obtained a kinetic equation for spectral energy evolution.

Found a stationary spectrum close to the Garrett--Munk spectrum.

Abstract

A novel canonical Hamiltonian formalism is developed for long internal waves in a rotating environment. This includes the effects of background vorticity and shear on the waves. By restricting consideration to flows in hydrostatic balance, superimposed on a horizontally uniform background of vertical shear and vorticity, a particularly simple Hamiltonian structure arises, which can be thought of as describing a nonlinearly coupled infinite collection of shallow water systems. The kinetic equation describing the time evolution of the spectral energy of internal waves is subsequently derived, and a stationary Kolmogorov solution is found in the high frequency limit. This is surprisingly close to the Garrett--Munk spectrum of oceanic internal waves.