  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.
