Oscillatory solutions at the continuum limit of Lorenz 96 systems

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.