Oscillatory solutions at the continuum limit of Lorenz 96 systems

TL;DR

This paper investigates oscillatory solutions in Lorenz 96 systems, deriving modulation equations for different oscillation types, analyzing their stability, and confirming findings through numerical experiments, thus enhancing understanding of complex dynamics in these models.

Contribution

It introduces new modulation equations for oscillations in Lorenz 96 systems and analyzes their behavior, including the transition to chaos, with validation from numerical experiments.

Findings

Period-two oscillations are identified in the discrete L96 system.

Modulation equations describe the envelope evolution of oscillations.

Oscillations transition to chaos as amplitude increases.

Abstract

In this paper, we study the generation and propagation of oscillatory solutions observed in the widely used Lorenz 96 (L96) systems. First, period-two oscillations between adjacent grid points are found in the leading-order expansions of the discrete L96 system. The evolution of the envelope of period-two oscillations is described by a set of modulation equations with strictly hyperbolic structure. The modulation equations are found to be also subject to an additional reaction term dependent on the grid size, and the period-two oscillations will break down into fully chaotic dynamics when the oscillation amplitude grows large. Then, similar oscillation solutions are analyzed in the two-layer L96 model including multiscale coupling. Modulation equations for period-three oscillations are derived based on a weakly nonlinear analysis in the transition between oscillatory and nonoscillatory regions. Detailed numerical experiments are shown to confirm the analytical results.