Quantifying the predictive skill in long-range forecasting. Part II: Model error in coarse-grained Markov models with application to ocean-circulation regimes

Aninformation-theoretic framework is developed to assess the predictive skill and model error in imperfect climate models for long-range forecasting. Here, of key importance is a climate equilibrium consistency test for detecting false predictive skill, as well as an analogous criterion describing model error during relaxation to equilibrium. Climate equilibrium consistency enforces the requirement that long-range forecasting models should reproduce the climatology of prediction observables with high fidelity. If a model meets both climate consistency and the analogous criterion describing model error during relaxation to equilibrium, then relative entropy can be used as an unbiased superensemble measure of the model's skill in long-range coarse-grained forecasts. As an application, the authors investigate the error in modeling regime transitions in a 1.5-layer ocean model as a Markov process and identify models that are strongly persistent but their predictive skill is false. The general techniques developed here are also useful for estimating predictive skill with model error for Markov models of low-frequency atmospheric regimes.