Stochastic Euler-Poincar\'e reduction for central extension

TL;DR

This paper applies stochastic Euler-Poincaré reduction to central extensions of Lie groups to derive viscous quasi-geostrophic equations with stochastic perturbations, providing a new geometric framework for these fluid dynamics models.

Contribution

It introduces a stochastic reduction framework on central extensions of Lie groups and derives viscous QGS equations as critical points, incorporating stochastic perturbations.

Findings

Derived viscous QGS equations from stochastic Euler-Poincaré reduction.

Proved integrability of the Roger Lie algebra cocycle.

Established stochastic perturbations lead to solutions of viscous QGS equations.

Abstract

This paper explores the application of central extensions of Lie groups and Lie algebras to derive the viscous quasi-geostrophic (QGS) equations, with and without Rayleigh friction term, on the torus as critical points of a stochastic Lagrangian. We begin by introducing central extensions and proving the integrability of the Roger Lie algebra cocycle $\omega_\alpha$, which is used to model the QGS on the torus. Incorporating stochastic perturbations, we formulate two specific semi-martingales on the central extension and study the stochastic Euler-Poincar\'e reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\widehat{\mathfrak{g}} = \mathfrak{g} \rtimes_{\omega_\alpha} \mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.