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.

Figures (10)

• Figure 1: Illustration of the L96 model (\ref{['eq:L96']}) and the typical solution with $J=40$ and $F=4$.
• Figure 2: Solution of the first-order asymptotic equation (\ref{['eq:eqn_1st']}) with negative initial data.
• Figure 3: Solution of the first-order asymptotic equation (\ref{['eq:eqn_1st']}) with positive initial data.
• Figure 4: Time-series of the key model integrals and the evolution of the period-two solution from smooth initial state in the inviscid L96 model \ref{['eq:l96_num']}.
• Figure 5: States $u,v,w$ at several different time instants during the development of break down of period-two solution.

Theorems & Definitions (19)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark
• Lemma 3
• Lemma 4
• Remark
• Theorem 5
• proof