Attractor radius and global attractor radius and their application to the quantification of predictability limits

Abstract

Quantifying the predictability limits of chaotic systems and their forecast models is an important issue with both theoretical and practical significance. This paper introduces three invariant statistical properties of attractors, namely the attractor radius, global attractor radius (GAR), and the global average distance between two attractors, to define the geometric characteristics and average behavior of a chaotic system and its error growth. The GAR is 2 times the attractor radius. These invariant quantities are applied to quantitatively measure the global and local predictability limits (both have practical and potential predictability limits, which correspond to the attractor radius and GAR, respectively) of both global ensemble average forecasts and one single initial state, respectively. Both the attractor radius and GAR are intrinsic properties of a chaotic system and independent of the forecast model and model errors, and thus provide more accurate, objective metrics to assess the global and local predictability limits of forecast models compared with the traditional error saturation or asymptotic value (AV). Both the Lorenz63 model and operational forecast data are used to demonstrate the theoretical aspects of these geometric characteristics and evaluate the feasibility and effectiveness of their application to predictability analysis.