On the inviscid limit of stationary measures for the stochastic system of the Lorenz model for a baroclinic atmosphere

TL;DR

This paper investigates the behavior of stationary measures in a stochastic Lorenz model for a baroclinic atmosphere as the viscosity approaches zero, establishing conditions for convergence and properties of the limiting measure.

Contribution

It provides new conditions ensuring the convergence of stationary measures in a stochastic baroclinic atmosphere model as viscosity tends to zero.

Findings

Existence of a limiting measure as viscosity approaches zero.

Conditions on randomness and parameters for measure convergence.

Integral properties of the limiting measure proved.

Abstract

The paper is concerned with a nonlinear system of partial differential equations with parameters and the random external force. This system describes the two-layer quasi-solenoidal Lorenz model for a baroclinic atmosphere on a rotating two-dimensional sphere. The stationary measures for the Markov semigroup defined by the solutions of the Cauchy problem for this problem is considered. One parameter of the system is highlighted - the coefficient of kinematic viscosity. The sufficient conditions on the random right-hand side and the other parameters are derived for the existence of a limiting nontrivial point for any sequence of the stationary measures for this system when any sequence of the kinematic viscosity coefficients goes to zero. As it is well known, this coefficient in practice is extremely small. A number of integral properties are proved for the limiting measure. In addition, these results are obtained for one similar baroclinic atmosphere system.