Predictability of forced Lorenz system

In recent years, the actual atmospheric predictability has attracted widespread attention. Improving our understanding of weather predictability is vital to developing numerical models and improving our forecast skill in weather and climate events. Given that the atmosphere is a complex and nonlinear system, taking the Lorenz system as an example is a better way to understand the actual atmosphere predictability. Up to now, some predictability problems of the Lorenz system have been investigated, such as the relative effects of the initial error and the model error. Previous advances in the research of predictability mainly focus on the relationship between the predictability limit and the initial error. As is well known, the external forcing can also result in the change of the predictability. Therefore, it is significant to investigate the predictability changing with the external forcing. The nonlinear local Lyapunov exponent (NLLE) is introduced to measure the average growth rate of the initial error of nonlinear dynamical model, which has been used for quantitatively determining the predictability limit of chaos system. Based on the NLLE approach, the influences of external forcing on the predictability are studied in the Lorenz system with constant forcing and Lorenz system with quasi-periodic forcing in this paper. The results indicate that for the Lorenz systems with constant and quasi-periodic forcings respectively, their predictability limits increase with forcing strength increasing. In the case of the same magnitude but different directions, the constant and quasi-periodic forcing both show different effects on the predictability limit in the Lorenz system, and these effects become significant with the increase of forcing strength. Generally speaking, the positive forcing leads to a higher predictability limit than the negative forcing. Therefore, when we consider the effects of positive and negative elements and phases in the atmosphere and ocean research, the predictability problems driven by different phases should be considered separately. In addition, the influences of constant and quasi- periodic forcings on the predictability are different in the Lorenz system. The effect of the constant forcing on the predictability is mainly reflected in the linear phase of error growth, while the nonlinear phase should also be considered additionally for the case of the quasi-periodic forcing. The predictability of the system under constant forcing is higher than that of the system under quasi-periodic forcing. These results based on simple chaotic model could provide an insight into the predictability studies of complex systems.