Abstract
Abstract. The solar wind-magnetosphere coupling during substorms exhibits dynamical features in a wide range of spatial and temporal scales. The goal of our work is to combine the global and multi-scale description of magnetospheric dynamics in a unified data-derived model. For this purpose we use deterministic methods of nonlinear dynamics, together with a probabilistic approach of statistical physics. In this paper we discuss the mathematical aspects of such a combined analysis. In particular we introduce a new method of embedding analysis based on the notion of a mean-field dimension. For a given level of averaging in the system the mean-filed dimension determines the minimum dimension of the embedding space in which the averaged dynamical system approximates the actual dynamics with the given accuracy. This new technique is first tested on a number of well-known autonomous and open dynamical systems with and without noise contamination. Then, the dimension analysis is carried out for the correlated solar wind-magnetosphere database using vBS time series as the input and AL index as the output of the system. It is found that the minimum embedding dimension of vBS - AL time series is a function of the level of ensemble averaging and the specified accuracy of the method. To extract the global component from the observed time series the ensemble averaging is carried out over the range of scales populated by a high dimensional multi-scale constituent. The wider the range of scales which are smoothed away, the smaller the mean-field dimension of the system. The method also yields a probability density function in the reconstructed phase space which provides the basis for the probabilistic modeling of the multi-scale dynamical features, and is also used to visualize the global portion of the solar wind-magnetosphere coupling. The structure of its input-output phase portrait reveals the existence of two energy levels in the system with non-equilibrium dynamical features such as hysteresis which are typical for non-equilibrium phase transitions. Further improvements in space weather forecasting tools may be achieved by a combination of the dynamical description for the global component and a statistical approach for the multi-scale component.Key words. Magnetospheric physics (solar wind– magnetosphere interactions; storms and substorms) – Space plasma physics (nonlinear phenomena)
Highlights
The magnetospheric dynamics during substorms exhibits both globally coherent and multi-scale features
The solar wind-magnetosphere coupling during substorms exhibits dynamical features in a wide range of spatial and temporal scales
Low dimensional dynamical models effectively extract the time series constituent generated by the large-scale coherent behavior, but are unable to predict the features associated with high dimensional multi-scale dynamics
Summary
The magnetospheric dynamics during substorms exhibits both globally coherent and multi-scale features. In SOC models the multi-scale features of substorms were reproduced by a “sand-pile” or other non-equilibrium cellular automata which explicitly take into account the large number of degrees of freedom in the system and the interactions among them on different scales (Consolini, 1997; Uritsky and Pudovkin, 1998; Watkins et al, 1999, 2001; Takalo et al, 1999; Chapman and Watkins, 2001; Uritsky et al, 2002) It was shown (Vespignani and Zapperi, 1998) that in order for the system to achieve criticality, the fine tuning of control parameters is required. The phase transition analogy clearly provides a framework for understanding the magnetospheric dynamics in which the global and multi-scale processes coexist This led to a new approach to the data-derived modeling of the solar wind-magnetosphere coupling that combines the methods of nonlinear dynamics and statistical physics (Ukhorskiy et al, 2002b). The last section presents the main results of the paper and their implications to magnetospheric modeling
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