Ricci curvature for fluid models on the torus via Zeitlin's quantization

Abstract: Ricci curvature measures the average stability of geodesics under transverse perturbations, but how it should be defined in infinite dimensions is often unclear. This paper proposes a definition of Ricci curvature on the space of Hamiltonian diffeomorphisms on the two-dimensional flat torus $\mathrm{HDiff}(\mathbb{T}<sup>2)$, the state space for ideal fluids. Our definition is based on Zeitlin's model, which approximates $\mathrm{HDiff}(\mathbb{T}<sup>2)$ by finite-dimensional Lie groups $\mathrm{SU}(N)$. We derive a formula for the Ricci curvature tensor on $\mathrm{SU}(N)$ and provide numerical evidence for its convergence in the large-$N$ limit to our conjectured finite value. Additionally, we explore potential applications for hydrodynamics through the Lyapunov stability of gravest wave modes and Arnold's tradewind estimates for long-term weather predictability. Our framework extends to a wide range of settings. We demonstrate this by introducing Ricci curvature on the state spaces of fluids on rectangular domains, the Lagrangian averaged Euler equation induced by the $H<sup>1$-Sobolev metric, and the quasi-geostrophic equation incorporating the Coriolis effect.