On the statistical properties of the hydro dynamic models based on solutions to the Boussinesq equations

Using simplified Boussinesq equations as an example, we show that their solutions describing vortex structures (convective cells) are unstable perturbations of finite-amplitude. This property of the solutions makes it possible to advance a hypothesis concerning the mechanism of formation of the spectrum of the ensemble of convective cells and go from the hydrodynamic model to a statistical model. The results obtained earlier for the adiabatic atmosphere are generalized to a more general case of a polytropic atmosphere. This case includes the spontaneous convection that leads to formation of mesoscale convective ensembles. Such ensembles consisting of thermals and convective clouds play an important role in formation of the weather and climate of the planet.