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We study a simplified coupled atmosphere-ocean model using the formalism ofcovariant Lyapunov vectors (CLVs), which link physically-based directions ofperturbations to growth/decay rates. The model is obtained via a severetruncation of quasi-geostrophic equations for the two fluids, and includes asimple yet physically meaningful representation of theirdynamical/thermodynamical coupling. The model has 36 degrees of freedom, andthe parameters are chosen so that a chaotic behaviour is observed. One findstwo positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteennear-zero LEs. The presence of many near-zero LEs results from the vasttime-scale separation between the characteristic time scales of the two fluids,and leads to nontrivial error growth properties in the tangent space spanned bythe corresponding CLVs, which are geometrically very degenerate. Such CLVscorrespond to two different classes of ocean/atmosphere coupled modes. Thetangent space spanned by the CLVs corresponding to the positive and negativeLEs has, instead, a non-pathological behaviour, and one can construct robustlarge deviations laws for the finite time LEs, thus providing a universal modelfor assessing predictability on long to ultra-long scales along suchdirections. It is somewhat surprising to find that the tangent space of theunstable manifold has strong projection on both atmospheric and oceaniccomponents. Our results underline the difficulties in using hyperbolicity as aconceptual framework for multiscale chaotic dynamical systems, whereas theframework of partial hyperbolicity seems better suited, possibly indicating analternative definition for the chaotic hypothesis. Our results suggest the needfor accurate analysis of error dynamics on different time scales and domainsand for a careful set-up of assimilation schemes when looking at coupledatmosphere-ocean models.

journal of physics. a, mathematical and theoretical

Statistical and Dynamical Properties of Covariant Lyapunov Vectors in a  Coupled Atmosphere-Ocean Model - Multiscale Effects, Geometric Degeneracy,  and Error Dynamics