A Relationship between Local Error Growth and Quasi-stationary States: Case Study in the Lorenz System

Abstract

Properties of the local predictability in the Lorenz system of three variables are investigated as a first step to develop a dynamical method for the skill prediction in the numerical weather forecasts instead of conventional statistical and empirical methods. As a measure of the local predictability, we adopt Lorenz's index which gives the amplification rate of the root-mean-square error during a prescribed time interval. In particular, we exert ourselves to understand a role of the quasi-stationary state in determining the variation of the Lorenz index. In an intermittent chaos regime, the Lorenz index determined for a time interval of the one return in the Poincaré section has a minimum value at the onset of the laminar phase, gradually increases during the laminar phase, and abruptly attains a large value at the break of the laminar phase. If we consider the laminar phase as a quasi-stationary state generated by a local minimum point in the one-dimensional Poincaré map, this characteristic evolution of the Lorenz index is directly connected with the local dynamics of the local minimum point. The fine phase?spatial distribution of the Lorenz index for a short time interval on the Lorenz attractor is also discussed in connection with the role of the unstable stationary point in organizing the local predictability.