Time-averages of Fast Oscillatory Systems in Three-dimensional Geophysical Fluid Dynamics and Electromagnetic Effects

TL;DR

This paper analyzes how time-averaging oscillatory solutions in three-dimensional geophysical fluid dynamics and electromagnetic systems reveals emergent patterns like zonal flows, with rigorous proofs relating solutions to wave operator null spaces.

Contribution

It provides a PDE-based analysis showing that strong restoring forces cause time-averaged solutions to align with wave operator null spaces, revealing new insights into oscillatory geophysical systems.

Findings

Time-averaged solutions approximate null spaces of wave operators under strong forces.

Oscillatory solutions can be far from null spaces, despite averages being close.

Emergence of zonal flows from time-averaged oscillatory dynamics.

Abstract

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We will investigate, from the angle of partial differential equation analysis, some oscillatory geophysical fluid dynamics in three dimensions: Navier-Stokes equations in a fast rotating, spherical shell, and Magnetohydrodynamics subject to strong Coriolis and Lorentz forces. Upon averaging their oscillatory solutions in time, interesting patterns such as zonal flows can emerge. More rigorously, we will prove that, when the restoring forces are strong enough, time-averaged solutions stay close to the null spaces of the wave operators, whereas the solutions themselves can be arbitrarily far away from these subspaces.