  We study a simplified coupled atmosphere-ocean model using the formalism of
covariant Lyapunov vectors (CLVs), which link physically-based directions of
perturbations to growth/decay rates. The model is obtained via a severe
truncation of quasi-geostrophic equations for the two fluids, and includes a
simple yet physically meaningful representation of their
dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and
the parameters are chosen so that a chaotic behaviour is observed. One finds
two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen
near-zero LEs. The presence of many near-zero LEs results from the vast
time-scale separation between the characteristic time scales of the two fluids,
and leads to nontrivial error growth properties in the tangent space spanned by
the corresponding CLVs, which are geometrically very degenerate. Such CLVs
correspond to two different classes of ocean/atmosphere coupled modes. The
tangent space spanned by the CLVs corresponding to the positive and negative
LEs has, instead, a non-pathological behaviour, and one can construct robust
large deviations laws for the finite time LEs, thus providing a universal model
for assessing predictability on long to ultra-long scales along such
directions. It is somewhat surprising to find that the tangent space of the
unstable manifold has strong projection on both atmospheric and oceanic
components. Our results underline the difficulties in using hyperbolicity as a
conceptual framework for multiscale chaotic dynamical systems, whereas the
framework of partial hyperbolicity seems better suited, possibly indicating an
alternative definition for the chaotic hypothesis. Our results suggest the need
for accurate analysis of error dynamics on different time scales and domains
and for a careful set-up of assimilation schemes when looking at coupled
atmosphere-ocean models.
