A simple semi-empirical MJO model

We present a simple semi-empirical framework for constructing MJO theories. We assume that the MJO is a moisture mode destabilized at least in part by surface flux and cloud-radiative feedbacks. The model is one-dimensional in longitude; vertical and meridional structure are entirely implicit. The only prognostic variable is column water vapor, W. Precipitation is a function of W (here chosen exponential). The diagnosed zonal wind field is instantaneously tied to precipitation via a Greens function-like projection operator that can be derived either theoretically from a steady, linear dynamics (i.e., the Gill model) or empirically. The gross moist stability and other key parameters similarly can either be derived from explicit theoretical assumptions, or empirically from observations or numerical simulations. We present results only for the simplest configuration, allowing much room for future exploration.

Under the configuration we present, assuming background westerlies the linearized version of the model has only westward-propagating unstable modes. The maximum growth rate occurs at a wavelength which maximizes the correlation between precipitation and surface latent heat flux. Surface flux lags rainfall, as in observations, but by less than 90 degrees so that the correlation is positive; the lag is responsible for the westward growth.

The nonlinear version of the model has behavior that is very sensitive to small shifts in the phasing of the wind and precipitation. If the wind is shifted a few hundred km eastward of that computed from the Gill model, 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.