Local dimension and recurrent circulation patterns in long-term climate simulations

Abstract: With the recent advent of a sound mathematical theory for extreme events in dynamical systems, new ways of analyzing a system's inherent properties have become available: Studying only the probabilities of extremely close Poincar\'{e} recurrences, we can infer the underlying attractor's local dimensionality -- a quantity which is closely linked to the predictability of individual configurations, as well as the information gained from observing them. This study examines possible ways of estimating local and global attractor dimensions, identifies potential pitfalls and discusses conceivable applications. The Portable University Model of the Atmosphere (PUMA) serves a test subject of intermediate complexity between simple mathematical toys and truly realistic atmospheric data-sets. It is demonstrated that the introduction of a simple, analytical estimator can streamline the procedure and allows for additional tests of the agreement between theoretical expectation and observed data. We furthermore show how the newly gained knowledge about local dimensions can complement classical techniques like principal component analysis and may assist in separating meaningful patterns from mathematical artifacts.