Coordinate invariance in stochastic dynamical systems

Abstract

Stochastic dynamical systems have been used to model a broad range of atmospheric and oceanicphenomena. Previous work has focused on the stochastic differential equation formulation of thesesystems, has largely remained in a single coordinate system, and has highlighted the role of nonnormalityof the deterministic dynamics. Here, the coordinate independent properties of stochasticdynamical systems are studied. The properties previously attributed to non-normality, which can beremoved by a coordinate transformation, are more fundamentally seen to be coordinate-dependentmanifestations of violations of detailed balance. Systems violating detailed balance can both amplifyand rectify the random forcing.Newcoordinate-invariant measures of noise amplification are introducedand shown to achieve their lower bound when detailed balance is satisfied. Rectification results in acoherent phase space velocity which gives rise to a structured nonzero flux of all physically importantquantities such as energy and momentum. The qualitative and quantitative features of these fluxesprovide new predictions which can be used to further validate previously proposed stochastic modelsof geophysical systems.