Triggered convection, gravity waves, and the MJO: A shallow water model
  The MaddenJulian oscillation (MJO) is the dominant mode of intraseasonal variability in the tropics. Despite its primary importance, a generally accepted theory that accounts for fundamental features of the MJO, including its propagation speed, planetary horizontal scale, multi-scale features, and quadrupole structures, remains elusive. In this study, we use a shallow water model to simulate the MJO. In our model, convection is parameterized as a short-duration localized mass source, and is triggered when the layer thickness falls below a critical value. Radiation is parameterized as a steady uniform mass sink. Slowly eastward propagating (MJO-like) signals and red noise spectra are observed in our simulations. In the time-longitude domain, MJO-like signals with multi-scale structures are observed. In the Fourier domain, spectral peaks associated with the MJO-like signals are observed. In the longitude-latitude map view, quadrupole vortex structures associated with the MJO-like signals are observed. We propose that the simulated MJO signal is an interference pattern of westward and eastward inertia-gravity(WIG and EIG) waves. Its propagation speed is one half of the speed difference between the WIG and EIG waves. The horizontal scale of its large-scale envelope is determined by the bandwidth of the excited waves, and the bandwidth is controlled by number density of convection events. Our results suggest that the MJO perhaps is not a large-scale low-frequency wave, in which convection acts as quasi-equilibrium adjustment. Small-scale high-frequency waves might be important. 
Da Yang and Andrew P. Ingersoll
physics.ao-ph