AI 中文总结
本文针对理想流体状态空间$\text{HDiff}(\text{T}^2)$，基于Zeitlin模型提出里奇曲率定义，推导$\text{SU}(N)$上里奇曲率张量公式并验证其大$N$收敛性，还将框架扩展至多类流体相关方程，探索其在流体力学中的应用。
AI 中文摘要
里奇曲率衡量测地线在横向扰动下的平均稳定性，但其在无穷维空间中的定义往往尚不明确。本文针对理想流体的状态空间——二维平坦环面上的哈密顿微分同胚空间$\boldsymbol{\text{HDiff}(\boldsymbol{\text{T}}^2)}$，提出了里奇曲率的定义。该定义基于Zeitlin模型，该模型将$\text{HDiff}(\text{T}^2)$近似为有限维李群$\boldsymbol{\text{SU}(N)}$。我们推导了$\text{SU}(N)$上的里奇曲率张量公式，并提供了数值证据，证明当$N$趋于无穷大时，该曲率收敛到我们推测的有限值。此外，我们通过最低频波模的李雅普诺夫稳定性以及用于长期天气预报的Arnold信风估计，探索了其在流体力学中的潜在应用。我们的框架可扩展至多种场景，通过在矩形域流体状态空间、由$H^1$-索伯列夫度量诱导的拉格朗日平均欧拉方程、包含科里奥利效应的准地转方程上引入里奇曲率，对此进行了验证。
英文摘要
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 $\mathrm{HDiff}(\mathbb{T}^2)$, the state space for ideal fluids. Our definition is based on Zeitlin's model, which approximates $\mathrm{HDiff}(\mathbb{T}^2)$ 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 $H^1$-Sobolev metric, and the quasi-geostrophic equation incorporating the Coriolis effect.
