Ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analyzed. 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 Hölder continuous functions. The approach provides a general framework for generalized 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.

分析了大气和海洋动力学的随机介质复杂性模型的遍历特性。更具体地说，研究了用于地球物理流动的两层准地转模型，其中上层受到加性噪声的扰动。该模型在地球科学中很流行，例如研究随机风力对海洋的影响。然而，严格的数学分析遇到了挑战，即在所研究的模型中，噪声配置在空间上退化，因为随机强迫仅作用于顶层。建立了不变量测度的解法则的指数收敛，这意味着关联马尔可夫半群在 Hölder 连续函数空间上的谱间隙。该方法为适用于耗散 SPDE 应用的广义耦合技术提供了一个通用框架。在双层准地转模型的情况下，结果要求第二层服从一定的被动条件。