Quantifying Predictability Variations in a Low-Order Ocean–Atmosphere Model: A Dynamical Systems Approach

A dynamical systems approach is used to quantify the predictability of weather and climatic states of a low-order, moist general circulation model. The effects on predictability of incorporating a simple oceanic circulation are evaluated. The predictability and structure of the model attractors are compared using Lyapunov exponents, local divergence rates, and the correlation and Lyapunov dimensions. Lyapunov exponents quantify global, or time-averaged predictability, by measuring the mean rate of growth of small perturbations on an attractor, while local divergence rates quantify temporal variations of this error growth rate and thus measure local, or instantaneous, predictability. Activating an oceanic circulation increases the average error doubling time of the atmosphere and the coupled ocean–atmosphere system by 10% while decreasing the variance of the largest local divergence rate by 20%. The correlation dimension of the attractor decreases slightly when an oceanic circulation is activated, while the Lyapunov dimension decreases more significantly because it depends directly on the Lyapunov exponents. The average predictability of annually averaged states is improved by 25% when an oceanic circulation develops, and the variance of the largest local divergence rate also decreases by 25%. One-third of the yearly averaged states have local error doubling times larger than 2 years, indicating that annual averages may, at times, be predictable, even without predictable variations in external forcing. The dimensions of the attractors of the yearly averaged states are not significantly different than the dimensions of the attractors of the original model. Arguably the most important contribution of this article is the demonstration that the local divergence rates provide a concise quantification of the variations of predictability on attractors and an efficient basis for comparing their local predictability characteristics. From a practical standpoint, local divergence rates might be computed to provide a real-time estimate of local predictability to accompany an operational forecast.