Recent observational analysis of both individual realizations and statistical ensembles identifies moist convectively coupled Kelvin waves in the tropics with supercluster envelopes of convection. This observational analysis elucidates several key features of these waves including their propagation speed of roughly 15 m s$^{-1}$, spatial scale in the lower troposphere of order 1000-2000 km, and many aspects of their dynamical structure. This structure includes anomalously cold temperatures in the lower troposphere and warm temperatures in the upper troposphere (below 250 hPa) within and often leading the heating region and strong updrafts in the wave, and an upward and westward tilting structure with height below roughly 250 hPa. Other key features in the wave are that anomalous increases in CAPE and surface precipitation lead the wave while the trailing part of the supercluster is dominated by stratiform precipitation. The main result in this paper is the development of a simple model convective parametrization with nonlinear convectively coupled moist gravity waves which reproduce many of the features of the observational record listed above in a qualitative fashion. One key feature of the model convective parametrization is the systematic use of two vertical modes with one representing deep convective heating and the other stratiform heating. The other key feature in the model is the explicit parametrization of separate deep convective and stratiform contribution to the downdrafts which change equivalent potential temperature in the boundary layer. It is established here for the model convective parametrization that stratiform heating on the second baroclinic generates the anomalies in potential temperature which often tend to lead the wave. On the other hand, for the model convective parametrization, negative anomalies in the stratiform mass flux on the boundary layer are responsible for increasing the equivalent potential temperature and CAPE which leads the convectively coupled nonlinear wave while positive anomalies in this stratiform mass flux contribute to the largely stratiform precipitation in the trailing part of the wave. The effects of rotation on convectively coupled equatorial waves are also included through a suitable linear stability theory for the model convective parametrization about radiative convective equilibrium, reproducing various aspects of other convectively coupled tropical waves. The model is then coupled to a systematic coarse grained stochastic parametrization of convective inhibition (CIN). The basic idea is that the boundary layer is regarded as a 'heat bath' supplying energy for convection, so that CIN is represented by an order parameter with very natural stochastic interaction rules, and coupled to the large scale flow through an external potential. Numerical simulations with the stochastic CIN model included lead to quantitative and qualitative improvements for the convective and dynamical structures of the reproduced convective superclusters.