Ergodic Dynamics of the Coupled Quasigeostrophic Flow-Energy Balance   System
  The authors consider a mathematical model for the coupled atmosphere-ocean system, namely, the coupled quasigeostrophic flow-energy balance model. This model consists of the large scale quasigeostrophic oceanic flow model and the transport equation for oceanic temperature, coupled with an atmospheric energy balance model. After reformulating this coupled model as a random dynamical system (cocycle property), it is shown that the coupled quasigeostrophic-energy balance fluid system has a random attractor, and under further conditions on the physical data and the covariance of the noise, the system is ergodic, namely, for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long. 
Aijun Du, Jinqiao Duan, Hongjun Gao, Tamay OzgOkmen