Abstract/Summary

In this work, ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analysed. Specifically, we study a two– layer quasi–geostrophic (2LQG) model with the upper layer perturbed by additive noise for geophysical flows. This model is popular in the geosciences, for instance to study the effects of a random wind forcing on the ocean. Yet it is less studied in mathematics, especially if the stochastic perturbation is acting only on one of the layers. In this case the noise is effectively spatially degenerate, posing a significant challenge to the analysis. After showing the model well–posedness, we focus on its long time average behaviour and ergodic properties: existence and uniqueness of an invariant measure (ergodicity), exponential convergence of solutions laws to the invariant measure (exponential stability or spectral gap), differentiability or only Hölder continuity of the invariant measure with respect to system parameters (linear or fractional response). Existence of an invariant measure is shown with classic techniques. Its uniqueness is established using a recent technique from stochastic analysis called asymptotic coupling, to account for the noise spatial degeneracy. This is proved provided a certain passivity condition on the second layer holds. Under the same condition, exponential stability is shown by blending different recent approaches like the asymptotic coupling. An important application of spectral gaps is response theory. The only result on linear response applicable to a large class of SPDEs is the work by Hairer and Madja (2010). We modify their approach treating a class of less regular observables. In particular we give a toolkit for linear and fractional response for SPDEs with moderately degenerate noise using the strength of a deterministic forcing as parameter. We apply such a framework to the 2D stochastic Navier-Stokes equation as test model, and finally to the stochastic 2LQG model.