A numerical ensemble-mean approach was employed to solve a nonlinear barotropic model with chastic basic flows to analyze the nonlinear effects in the formation of the North Atlantic Oscillation （NAO）. The nonlinear response to external forcing was more similar to the NAO mode than the linear response was, indicating the importance of nonlinearity. With increasing external forcing and enhanced low-frequency anomalies, the effect of nonlinearity increased. Therefore, for strong NAO events, nonlinearity should be considered.
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.
By using the Jacobi elliptic-function method, this paper obtains the periodic solutions for coupled integrable dispersionless equations. The periodic solutions include some kink and anti-kink solitons.
