An Ocean Drum: quasi-geostrophic energetics from a Riemann geometry perspective

TL;DR

This paper introduces a geometric framework using Riemannian metrics to analyze the energetics of quasi-geostrophic flows, revealing new insights into ocean dynamics and potential diagnostic tools.

Contribution

It develops a spectral geometry approach with an effective metric for quasi-geostrophic flows, connecting stratification, topography, and flow energetics in a novel way.

Findings

Hyperbolic space $\\mathbb{H}^3$ as leading-order geometry for deep ocean mesoscale

Spectral geometry encodes non-local stratification and topography effects

Diagnostic tools proposed for numerical and observational analysis of flows

Abstract

We revisit the discussion of the energetics of quasi-geostrophic flows from a geometric perspective based on the introduction of an effective metric, built in terms of the flow stratification and the Coriolis parameter. In particular, an appropriate notion of normal modes is defined through a spectral geometry problem in the ocean basin (a compact manifold with boundary) for the associated Laplace-Beltrami scalar operator. This spectral problem can be used to systematically encode non-local aspects of stratification and topography. As examples of applications we revisit the isotropy assumption in geostrophic turbulence, identify (a patch of) the hyperbolic space $\mathbb{H}^3$ as the leading-order term in the effective geometry for the deep mesoscale ocean and, finally, discuss some diagnostic tools based on a simple statistical mechanics toy-model to be used in numerical simulations and/or observations of quasi-geostrophic flows.