In this paper, taking the Lorenz system as an example, we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth. The results show that the geometric mean error （GME） has a smoother growth than the arithmetic mean error （AME） for the global average error growth, and the GME is directly related to the maximal Lyapunov exponent, but the AME is not, as already noted by Krishnamurthy in 1993. Besides these, the GME is shown to be more appropriate than the AME in measuring the mean error growth in terms of the probability distribution of errors. The physical meanings of the saturation levels of the AME and the GME are also shown to be different. However, there is no obvious difference between the local average error growth with the arithmetic mean and the geometric mean, indicating that the choices of the AME or the GME have no influence on the measure of local average predictability.
In this article,we address both recent advances and open questions in some mathematical and computational issues in geophysical fluid dynamics(GFD)and climate dynamics.The main focus is on 1)the primitive equations(PEs)models and their related mathematical and computational issues,2)climate variability,predictability and successive bifurcation,and 3)a new dynamical systems theory and its applications to GFD and climate dynamics.
Recent advances in the study of nonlinear atmospheric and climate dynamics in China （2003 2006） are briefly reviewed. Major achievements in the following eight areas are covered： nonlinear error dynamics and predictability; nonlinear analysis of observational data; eddy-forced envelope Rossby soliton theory; sensitivity and stability of the ocean＇s thermohaline circulation; nonlinear wave dynamics; nonlinear analysis on fluctuations in the atmospheric boundary layer; the basic structures of atmospheric motions; some applications of variational methods.
