The Gill-Matsuno: Is It a Good Model for the MJO?

The Matsuno-Gill model for the atmospheric steady-state response to an imposed equatorial heat source is perhaps the most successful set of reduced equations in the theory of tropical dynamics. While it was originally designed to explain some key features of the tropical climatology such as the Walker and Hadley circulations, it is increasingly used for other transient tropical disturbances such as the Madden-Julian oscillation (MJO).
Here, we provide a semi-analytic solution to the time dependent Matsuno-Gill equations, with periodic boundary conditions and without the long-wave approximation, based on a meridional expansion using the parabolic cylinder functions combined with the method of characteristics and Fourier series in the zonal direction. Our method of solution leads to an ODE system which is solved numerically. We use this transient solution to investigate the conditions under which the transient response convergences to the steady state and study the effect of a moving heat source on the structure of the response and speculate about its resemblance or not to the MJO structure. In particular, we look at the sensitivity of the long time behavior of the solution with respect to the model parameters, namely, the imposed dissipation rate and the speed of the moving heat source as well as the form of the heating itself.