An idealized semi-empirical framework for modeling the Madden-Julian oscillation

We present a simple semi-empirical model for MJO studies in which it is assumed that the MJO is a moisture mode destabilized by surface flux and cloudradiative feedbacks. The model is one-dimensional in longitude; vertical and meridional structure are entirely implicit. The only prognostic variable is column water vapor, W. The zonal wind field is an instantaneous, diagnostic function of the precipitation field. The linearized version of the model has only westward-propagating (relative to the mean flow) unstable modes, because wind-induced surface latent heat flux anomalies occur to the west of precipitation anomalies. The maximum growth rate occurs at a synopticto planetary-scale wavelength at which the correlation between precipitation and surface latent heat flux is maximized; this wavelength is proportional to the horizontal scale associated with the assumed diagnostic wind response to precipitation anomalies. The nonlinear version of the model has behavior that can be qualitatively different from the linear modes, and is strongly influenced by horizontal advection of moisture. The nonlinear solutions are very sensitive to small shifts in the phasing of wind and precipitation. Under some circumstances nonlinear eastward-propagating disturbances emerge on a state of mean background westerlies. These disturbances have a shock-like discontinuous jump in humidity and rainfall at the leading edge; humidity decreases linearly and precipitation decreases exponentially to the west. Use of a wind structure consistent with off-equatorial heating makes this nonlinear mode more likely to emerge. 1