On the rate of convergence as $t\to+\infty$ of the distributions of solutions to the stationary measure 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. A unique stationary measure for the Markov semigroup defined by the solutions of the Cauchy problem for this problem is considered. An estimate for the rate of convergence of the distributions of all solutions in a certain class of this system to the unique stationary measure as $t\to+\infty$ is proposed. A similar result is obtained for the equation of a barotropic atmosphere and the two-dimensional Navier-Stokes equation. A comparative analysis with some of the available related results is given for the latter.
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