Extensive Chaos in the Lorenz-96 Model

TL;DR

This study investigates the high-dimensional chaotic behavior of the Lorenz-96 model, analyzing how system size and external forcing influence chaos, fractal dimension, and extensivity, revealing key scaling laws and dynamical regimes.

Contribution

It provides a detailed analysis of how chaos and fractal dimension scale with system parameters in the Lorenz-96 model, highlighting the relationship between system size, forcing, and chaos.

Findings

Chaos windows are extensive with ~20% deviation.

Chaotic dynamics occur for all sizes beyond a critical forcing.

Fractal dimension follows a power-law dependence on forcing.

Abstract

We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Edward Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and decreases non-monotonically with increasing system size. The length scale describing the deviations from extensivity and the natural chaotic length scale are approximately equal in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. As the forcing is increased at constant system size the fractal dimension exhibits a power-law dependence. The power-law behavior is independent of the system size and quantifies the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns.