Exponential ergodicity for a stochastic two-layer quasi-geostrophic model

TL;DR

This paper proves exponential convergence to equilibrium for a stochastic two-layer quasi-geostrophic model, demonstrating spectral gap existence despite spatially degenerate noise, with implications for geophysical fluid dynamics.

Contribution

It introduces a novel framework using generalized coupling techniques to establish ergodicity in a complex, degenerate stochastic geophysical flow model.

Findings

Solutions converge exponentially to the invariant measure.

Spectral gap of the Markov semigroup is established.

Results depend on passivity condition of the second layer.

Abstract

Ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analysed. More specifically, a two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that in the model under study, the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws to the invariant measure is established, implying a spectral gap of the associated Markov semigroup on a space of H\"older continuous functions. The approach provides a general framework for generalised coupling techniques suitable for applications to dissipative SPDEs. In case of the two-layer quasi-geostrophic model, the results require the second layer to obey a certain passivity condition.