Communications in Mathematical Sciences

Volume 14 (2016)

Number 2

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

Pages: 571 – 592



Shengqian Chen (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Samuel N. Stechmann (Department of Mathematics, and Department of Atmospheric and Oceanic Sciences, University of Wisconsin, Madison, Wisc., U.S.A.)


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 analytical 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.


nonlinear traveling waves, solitary waves, tropical climate, tropical intraseasonal variability, partial differential equations

2010 Mathematics Subject Classification

35Q35, 74J30, 76B60, 76U05, 86A10

Published 14 December 2015