Ricci curvature for hydrodynamics on the sphere

TL;DR

This paper introduces a new finite-dimensional approach to defining Ricci curvature in 2D hydrodynamics on the sphere, providing insights into stability and long-term behavior of fluid flows.

Contribution

It proposes a novel Ricci curvature definition using Zeitlin models for 2D hydrodynamics, with formulas and numerical evidence supporting convergence to the infinite-dimensional case.

Findings

Finite-dimensional Ricci curvature approximations converge to the infinite-dimensional limit.

The limiting Ricci curvature indicates average instability in high-frequency modes.

Results help explain observed long-term mixing behavior in spherical hydrodynamics.

Abstract

The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas for Ricci curvature are often simpler than those for sectional curvature, which typically takes both signs, but the drawback is that Ricci curvature is rarely well-defined in infinite-dimensional spaces. Here we suggest a definition of Ricci curvature in the case of two-dimensional hydrodynamics, based on the finite-dimensional Zeitlin models arising in quantization theory, which gives a natural tool for renormalization. We provide formulae for the finite-dimensional approximations and give strong numerical evidence that these converge in the infinite-dimensional limit, based in part on four new conjectured identities for Wigner $6j$ symbols. The suggested limiting expression for (average) Ricci curvature is surprisingly simple and demonstrates an average instability for high-frequency modes which helps explain long-term numerical observations of spherical hydrodynamics due to mixing.