The Madden-Julian oscillation is the largest planetary scale disturbance on our planet; its wavelength is on the order of the circumference of the Earth. Another planetary scale disturbance that hasn’t been so extensively studied as the MJO is the convectively coupled equatorial Rossby wave.

A simple linear analytical model on an equatorial beta plane from Fuchs and Raymond, 2017, models two unstable modes that have their largest instabilities at long wavelengths. One of them moves eastward with a phase speed of about 15 m/s for planetary wavenumber l=1, however when it interacts with the Indo-Pacific warm pool, it slows down to 5 m/s. The other mode moves westward with phase speed of about 5 m/s for a planetary wavenumber of l=3. Both of the modes are unstable due to the interplay between moisture and surface fluxes (globally mean easterlies). Cloud-radiation interactions increase the instability.

We believe that based on the modeled characteristics (phase speed, largest instability at long wavelengths and their x-y-z structure) of the eastward and westward modeled modes (WISHE-moisture modes), this simple analytical model provides the zeroth order mechanism that explains the Madden-Julian oscillation and convectively coupled equatorial Rossby wave.

Other effects such as coupling with other waves, the influence of the extratropics as well as nonlinearities are further required to fully grasp the observed structures in finer detail.