Nonlinear traveling waves for the skeleton of the Madden-Julian oscillation

TL;DR

This paper presents nonlinear traveling wave solutions for the Madden-Julian Oscillation (MJO) model, revealing pulse-like convection patterns and amplitude-dependent dispersion relations that differ from linear wave predictions.

Contribution

The study introduces nonlinear traveling wave solutions for the MJO skeleton model, including an analytic dispersion relation and analysis of wave properties under weak forcing.

Findings

Nonlinear MJO waves have pulse-like convection patterns.

Amplitude affects wave frequency and speed.

Weak forcing leads to sech-squared waveforms.

Abstract

The Madden-Julian Oscillation (MJO) is the dominant component of intraseasonal (30-90 days) variability in the tropical atmosphere. Here, traveling wave solutions are presented for the MJO skeleton model of Majda and Stechmann. The model is a system of nonlinear partial differential equations that describe the evolution of the tropical atmosphere on planetary (10,000-40,000 km) spatial scales. The nonlinear traveling waves come in four types, corresponding to the four types of linear wave solutions, one of which has the properties of the MJO. In the MJO traveling wave, the convective activity has a pulse-like shape, with a narrow region of enhanced convection and a wide region of suppressed convection. Furthermore, an amplitude-dependent dispersion relation is derived, and it shows that the nonlinear MJO has a lower frequency and slower propagation speed than the linear MJO. By taking the small-amplitude limit, an analytic formula is also derived for the dispersion relation of linear waves. To derive all of these results, a key aspect is the model's conservation of energy, which holds even in the presence of forcing. In the limit of weak forcing, it is shown that the nonlinear traveling waves have a simple sech-squared waveform.