This paper gives a simplified analytical description of spontaneous generation and finite amplitude saturation of sound in annular thermoacoustic engines, and also provides comparison with experiments. The model includes the precise description of thermoacoustic amplification of sound (induced by interaction between an heterogeneously heated stack of solid plates and resonant gas oscillations), which accounts for the details of the temperature distribution in the whole thermoacoustic device (i.e., which does not only account for the mean temperature gradient along the stack). The saturation of the acoustic wave amplitude is described by taking into account both the reverse influence of high amplitude acoustic field on temperature field, and the dissipation of acoustic energy due to higher harmonics generation and minor losses (vortex generation). From the comparison between simulation results and experiments, it is demonstrated that the dynamical behavior observed in our experimental device is predominantly controlled by the effects of acoustic streaming and acoustically enhanced thermal conductivity tending not only to reduce the externally imposed temperature gradient along the stack, but also to change the shape of the temperature field.