The paper is devoted to an investigation of convective turbulence. A simplified approach is used for this purpose. It considers an isolated turbulent pulsation as the eigensolution to the corresponding equations of thermohydrodynamics. Turbulence is generated by nonlinear interaction of pulsations: not all interactions, but only the most probable of them are investigated. It is assumed that during convection these are interactions of cells located along the gravity vector, i.e. lying in a vertical line, and lateral interaction of the cells is ignored. This assumption allows one to consider the process of the evolution and interaction of cells as axially symmetric. It is also assumed that the vertical scales of convective cells are larger than their horizontal scales. Therefore, the Boussinesq equations simplified in accordance with the theory of vertical boundary layers can be used. The fact that buoyancy forces, in addition to diffusion, influence the increase of the vertical scales, serves as a basis for this assumption. These assumptions make it possible to obtain the analytical and numerical–analytical solutions, which qualitatively describe the evolution and interaction of convective cells of two essentially different scales: (i) centimetre-scale convective pulsations and (ii) thermals and convective clouds, and to reduce the problem to the solution of nonlinear equations (equations of the Burgers type). Two opposite tendencies are revealed, manifested in the interaction of convective cells. First, there is coagulation of cells and fine nonlinear effects associated with it, which are known from observations and supported by the theory. Secondly, there is destruction of a strong rising cell through its collision with a weak descending ‘cold’ cell. It is assumed that the destruction of cells corresponds to the absence of solutions, when some parameters reach their critical values. A numerical solution to a more accurate problem without simplifications of the vertical boundary layer serves as a basis for this hypothesis. It shows that at critical values of the parameters the process of ‘wave turnover’ begins. It is accompanied by entrainment of the motions of the cold surrounding air into a system of convection and fast dissipation of a cell. In the simplified model, this dissipation is considered to be instantaneous and is called destruction. When the cells are sufficiently strong vertically, weak random fluctuations in the fields of meteorological elements cause their destruction. These results make it possible to propose a hypothesis which relates the degree of instability of cells with the probability of their existence, and to construct functions of cell distributions.