  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.
