An approach for analyzing the ensemble mean from a dynamic point of view

Abstract

Simultaneous ensemble mean equations (LEMEs) for the Lorenz model are obtained, enabling us to analyze the properties of the ensemble mean from a dynamical point of view. The qualitative analysis for the two-sample and n-sample LEMEs show the locations and number of stable points are different from the Lorenz equations (LEs), and the results are validated by numerical experiments. The analysis for the eigenmatrix of the stable points of LEMEs indicates that the stability of these stable points is similar to the LEs'. The eigenmatrix for non-stable points can be obtained too, but the eigenvalues depend not only on the value of the mean variable but also the other n-1 sample equation's variable, and thus for these points there may be different stabilities compared to the LEs'. The divergence of the LEMEs' flow has a negative value, which is the same as the LEs', and thus the trajectory in phase space approaches zero and the trajectory will be attracted to a low-level dimensional curved surface, i.e., the LEMEs have the attractor property, but the structure of the attractor is not the same as the LEs'.