Stochastic Geometric Mechanics for Fluid Dynamics

TL;DR

This paper develops stochastic geometric mechanics (SGM), a mathematical framework that models fluid dynamics and wave behavior while preserving key physical properties, with applications in climate modeling and uncertainty quantification.

Contribution

It introduces a new stochastic geometric mechanics framework based on Lie group-invariant variational principles for fluid flow and wave dynamics.

Findings

Preserves advective transport properties of ideal fluids

Provides a mathematical basis for uncertainty quantification in climate models

Enhances modeling of ocean and atmospheric dynamics

Abstract

Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. The present chapter describes the mathematical development of the framework of stochastic geometric mechanics in the context of fluid flow and wave dynamics obtained from Lie group-invariant variational principles.