  Coupled data assimilation (CDA) distinctively appears as a main concern in
numerical weather and climate prediction with major efforts put forward
worldwide. The core issue is the scale separation acting as a barrier that
hampers the propagation of the information across model components. We provide
a brief survey of CDA, and then focus on CDA using the ensemble Kalman filter
(EnKF). We consider first coupled equations with temporal scale difference and
deduce that: (i) cross components effects are strong from the slow to the fast
scale, but, (ii) intra-component effects are much stronger in the fast scale.
While observing the slow scale is desirable and benefits the fast, the latter
must be observed with high frequency otherwise the error will affect the slow
scale. Experiments are performed using the atmosphere-ocean model, MAOOAM. Six
configurations are considered, differing for the strength of the
atmosphere-ocean coupling and/or the number of model modes. A comprehensive
dynamical characterisation of the model configurations is provided by examining
the Lyapunov spectrum, Kolmogorov entropy and Kaplan-Yorke attractor dimension.
We also compute the covariant Lyapunov vectors and use them to explain how
model instabilities act on different model's modes according to the coupling
strength. The experiments confirm the importance of observing the fast scale,
but show also that, despite its slow temporal scale, frequent observations in
the ocean are beneficial. The relation between the ensemble size and the
unstable subspace dimension has been studied. Results largely ratify what known
for uncoupled system: the condition N>n0 is necessary for the EnKF to converge.
But the quasi-degeneracy of the Lyapunov spectrum of MAOOAM, with many
near-zero exponents, is potentially the cause of the smooth gradual reduction
of the analysis error observed for some model configurations, even when N>n0.
