On integral properties of stationary measures for the stochastic system of the Lorenz model describing a baroclinic atmosphere

The paper is concerned with a nonlinear system of partial differential equations with parameters which describes the two-layer quasi-solenoidal Lorenz model for a baroclinic atmosphere on a rotating two-dimensional sphere. The right-hand side of the system is perturbed by white noise. We give some upper bounds and a lower bound for some moments of these measures in terms of the set of parameters, an external force and numerical characteristics of white noise. These bounds show, in particular, that these moments are finite. We will prove a number of integral equalities, which can be considered as laws of conservation of these stationary measures. Under certain conditions, these estimates and equalities do not depend on the coefficient of kinematic viscosity $\nu>0$, which leads to the possibility of passing to the limit as $\nu \to 0$ and studing with their help the properties of limiting measures, which will be done in subsequent work. As it is well known, the coefficient of kinematic viscosity $\nu$ in practice is extremely small. In addition, these results are obtained for one similar baroclinic atmosphere system and the barotropic atmosphere equation.