D. Vassiliadis

A description of the solar wind-magnetosphere coupling based on nonlinear filters

A nonlinear filtering method is introduced for the study of the solar wind — magnetosphere coupling and related to earlier linear techniques. The filters are derived from the magnetospheric state, a representation of the magnetospheric conditions in terms of a few global variables, here the auroral electrojet indices. The filters also couple to the input, a representation of the solar wind variables, here the rectified electric field. Filter-based iterative prediction of the indices has been obtained for up to 20 hours. The prediction is stable with respect to perturbations in the initial magnetospheric state; these decrease exponentially at the rate of 30 min−1. The performance of the method is examined for a wide range of parameters and is superior to that of other linear and nonlinear techniques. In the magnetospheric state representation the coupling is modeled as a small number of nonlinear equations under a time-dependent input.

The organized nonlinear dynamics of the magnetosphere

The linear prediction filters computed by Bargatze et al. [1985] have resulted in a turning point in the study of solar wind-magnetosphere coupling. The evolution of the filters with varying activity provides a clear demonstration that the coupling is nonlinear. The filters have thus brought about the end of one era of linear correlative studies and the beginning of a new era of nonlinear dynamical studies. Two separate, but complementary, approaches have emerged in these dynamical studies, analogue modeling and data-based phase space reconstruction. The reconstruction research has evolved from the original autonomous method studies, which were not conclusive, to the more recent input-output studies that are more appropriate for the solar wind-driven magnetosphere and have produced more reliable results. At present it appears that the modeling and reconstruction approaches may be merged in future attempts to produce analogue models directly from the results of the input-output data-based methods. If this can be accomplished, it will constitute a major step forward toward the goal of a low-dimensional analogue model of the magnetospheric dynamics derived directly from data and interpreted in terms of magnetospheric physics. These developments are reviewed in three sections: autonomous data analysis methods, analogue models, and input-output data analysis methods. The introduction provides sufficient information to read each of these sections independently.

Self-organized criticality in the substorm phenomenon and its relation to localized reconnection in the magnetospheric plasma sheet

Evidence is presented that suggests that there is a significant self-organized criticality (SOC) component in the dynamics of substorms in the magnetosphere. We assume that observations of bursty bulk flows, fast flows, localized dipolarizations, plasma turbulence, etc. show that multiple localized reconnection sites provide the basic avalanche phenomenon in the establishment of SOC in the plasma sheet. First results are presented from a study of this avalanche process based on this working assumption. A magnetic field reversal model is discussed. Resistivity, in this model, is self-consistently generated in response to the excitation of an idealized currentdriven instability. When forced by convection of magnetic flux into the field reversal region, the model yields rapid magnetic field annihilation through a dynamic behavior that is shown to exhibit many of the characteristics of SOC. Over a large range of forcing strengths, the annihilation rate is shown to self-adjust to balance the rate at which flux is convected into the reversal region. Several analogies to magnetotail dynamics are discussed: (1) It is shown that the presence of a localized criticality in the model produces a remarkable stability in the global configuration of the field reversal while simultaneously exciting extraordinarily dynamic internal evolution. (2) Under steady forcing it is shown that a loading-unloading cycle may arise that, as a consequence of the global stability, is quasi-periodic and, therefore, predictable despite the presence of internal turbulence in the field distribution. Indeed, it is shown that the global loading-unloading cycle is a consequence of the internal turbulence. (3) It is shown that under steady, strong forcing the loading-unloading cycle vanishes. Instead, a recovery from a single unloading persists indefinitely. The field reversal is globally very steady while internally it is very dynamic as field annihilation goes on at the rate necessary to match the strong forcing. From this result we speculate that steady magnetospheric convection events result when the plasma sheet has been driven close to criticality over an extended spatial domain. During these events we would expect to find localized reconnection sites distributed over the spatial domain of near criticality, and we would expect to find plasma sheet transport in that domain to be closely related to that of BBF and fast flow events.

Reconstruction of low‐dimensional magnetospheric dynamics by singular spectrum analysis

The low dimensionality of magnetospheric activity indicated by previous phase space reconstructions using the AE and AL data suffer from the limitations of these techniques. In this paper the singular spectrum analysis is used to study the global magnetospheric dynamics using the AE data and it yields a correlation dimension ≃2.5, thus confirming the low dimensionality published earlier. Further, this technique shows that the global magnetospheric dynamics can be described by 3 variables whose dynamical features are obtained from the AE data.

Phase transition‐like behavior of the magnetosphere during substorms

The behavior of substorms as sudden transitions of the magnetosphere is studied using the Bargatze et al. [1985] data set of the solar wind induced electric field vBs and the auroral electrojet index AL. The data set is divided into three subsets representing different levels of activity, and they are studied using the singular spectrum analysis. The points representing the evolution of the magnetosphere in the subspace of the eigenvectors corresponding to the three largest eigenvalues can be approximated by two-dimensional manifolds with a relative deviation of 10–20%. For the first two subsets corresponding to small and medium activity levels the manifolds have a pleated structure typical of the cusp catastrophe. The dynamics of the magnetosphere near these pleated structures resembles the hysteresis phenomenon typical of first-order phase transitions. The reconstructed manifold is similar to the “temperature-pressure-density” diagrams of equilibrium phase transitions. The singular spectra of vBs, AL, and combined data have the power law dependence typical of second-order phase transitions and self-organized criticality. The magnetosphere thus exhibits the signatures of both self-organization and self-organized criticality. It is concluded that the magnetospheric substorm is neither a pure catastrophe of the low-dimensional system nor a random set of avalanches of different scales described by the simple sandpile models. The substorms behave like nonequilibrium phase transitions, with features of both first- and second-order phase transitions.

Substorm recurrence during steady and variable solar wind driving: Evidence for a normal mode in the unloading dynamics of the magnetosphere

Farrugia et al. (1993) have recently studied substorm activity driven by the passage of an interplanetary magnetic cloud during which the interplanetary magnetic field turned southward for approximately 18 hours. It was shown that both the ϵ and the V Bs parameters varied slowly on the timescale of a substorm but changed considerably over the interval as a whole. The substorm occurrence rate did not reflect the variation in magnetospheric energy loading rate as measured by these parameters but, rather, remained roughly constant with a 50-min average period. Klimas et al. (1992) showed that the Faraday loop analogue model of geomagnetic activity predicts this single unloading rate under various constant loading rates. However, various model parameters were adjusted to yield a 1-hour unloading period in agreement with the Bargatze et al. (1985) linear prediction filters and in approximate agreement with the Farrugia et al. (1993) results. It has since been found necessary to add a slow relaxation mechanism to the Faraday loop model to allow for its approach to a ground state during long periods of inactivity. It is proposed that the relaxation mechanism is provided by slow convection of magnetic flux out of the magnetotail to the dayside magnetosphere. In addition, a rudimentary representation of magnetotail-ionosphere coupling has been added to enable comparison of model output to measured AL. The present study is of the modified Faraday loop model response to solar wind input from the Bargatze et al. data set with comparison of its output to concurrent AL. This study has removed the degree of freedom in parameter choice that had earlier allowed adjustments toward the 1-hour unloading period and has, instead, yielded the 1-hour unloading period under various constant loading rates. It is demonstrated that the second peak of the bimodal Bargatze et al. linear prediction filters at ≈ 1-hour lag and the approximately constant substorm recurrence rate observed by Farrugia et al. can be interpreted as both being due to the existence of a normal unloading recurrence period in the dynamics of the magnetosphere.

Dst index prediction using data-derived analogues of the magnetospheric dynamics

The method of Klimas et al. [1997] for constructing dynamical analogues of physical input-output systems is generalized. Higher-order analogues with sensitivity to more features of the input data are derived. Solar wind V Bs parameter data are used for input and Dst index data are used for output to construct analogues of the magnetospheric dynamics responsible for Dst storms. A detailed study of the dynamics involved in a single storm is presented. It is shown that the relationship between V Bs input and Dst output for this storm can be described in the context of the model of Burton et al. [1975] but with variable decay time and strength of coupling to the solar wind V Bs parameter. During the storm recovery it is found that the decay time varies from ≈ 4 hours at the storm maximum to ≈ 20 hours midway in the recovery and then back to ≈ 10 hours. There appears to be nothing in the simultaneous solar wind data to explain this reversal in the evolution of the decay time. It is shown that the strength of coupling to the solar wind V Bs parameter varies considerably. The coupling strength peaks strongly at the time of the storm maximum and decays to low values during the storm recovery. Solar wind V Bs input during the storm recovery does not affect the recovery rate. Using this storm data, empirical nonlinear analogues are constructed. These analogues are tested out of sample for their prediction effectiveness. Comparisons with the predictions of the model due to Burton et al. are given. It is shown that these analogues are promising prediction tools, but their lack of sensitivity to solar wind dynamic pressure must be corrected.

Comparative study of dynamical critical scaling in the auroral electrojet index versus solar wind fluctuations

Based on an analysis of auroral electrojet (AE) index data, we demonstrate that the temporal evolution of magnetospheric perturbations exhibits non-trivial power-law relations consistent with the behavior of a general class of statistical physical models in the vicinity of a stationary critical point. We show that the ensemble average dynamics of the activity bursts in the AE index is essentially scale-free and can be characterized in terms of spreading critical exponents and δ. The lifetime T and the size S of the bursts show strong algebraic correlation that is approximated by the dependence S ∼ T 1+η+δ . For times shorter than 3.5 hours, we find scaling features in the AE index that are independent of the solar wind input, thus indicating the internal magnetospheric origin of the revealed effects.

The relation between the northern polar cap and auroral electrojet geomagnetic indices in the wintertime

The polar cap (PC) index is a measure of the high-latitude geomagnetic disturbances due to Hall and field-aligned currents. The index is well correlated with the auroral electrojet AL and AU indices (correlation with the PC index is 76% and 66%, resp.). Here we obtain several types of data-based models that relate the PC to the AL and AU indices in the wintertime, when the ionospheric conductivity is mostly due to the precipitating particles of the field-aligned currents. The new models predict AL and AU from PC with correlations much higher than those found by earlier studies. Thus linear moving-average filters reproduce the observed AL with a correlation of 88% (AU : 75%) while linear autoregressive moving-average (ARMA) models based on the PC index produce in-sample single-step predictions with 98% and 97% correlations with AL and AU respectively. For long-term, out-of-sample prediction, the linear ARMA prediction from the PC index has an asymptotic prediction error which is at least 25% more accurate than the prediction from solar wind input. Nonlinear models are slightly more accurate than their linear counterparts, indicating a weak nonlinearity in the relation between the polar cap and auroral zone indices. The prediction-observation correlations are sufficiently high that models based on the PC index can be used for specification of the auroral geomagnetic activity.

The Dst geomagnetic response as a function of storm phase and amplitude and the solar wind electric field

We examine the dependence of the Dst timescales on storm conditions and its implications for the storm-substorm relationship. The growth, decay and oscillation timescales are expressed as functions of the storm magnitude and phase, and the solar wind electric-field input VBs. Nonlinear, second-order autoregressive moving average (ARMA) models are fit to 5-min data and yield two timescales, an exponential decay with an average e-folding time τ1 = −4.69 hours (−7.26 hours for the pressure-corrected Dst(0)) and an inductive time τ2 = −0.81 hours (−0.05 hours for Dst(0)). Around these average values there is a systematic variation: (1) For most of the storm duration, τ1 is negative representing the rapid adjustment of the inner magnetosphere to the imposed electric field. (2) In the early main phase, however, τ1 = 5.29 hours (1.76 hours for Dst(0)), so the disturbance grows as a slow exponential. (3) During commencement and main phase, the timescales are complex conjugate and the response is oscillatory. Fast oscillations during storm commencement (period 1.13 hours: 8.48 min for Dst(0)) are a “ringing” response to interplanetary pressure enhancements. Slow oscillations in the main phase have an average period of 1.96 hours (1.55 hours for Dst(0)) and coincide with AL intensifications. The main phase can be separated into periods of oscillatory, fast decay (coincident with AL activity and probably due to injections) and monotonic slow decay (regular convection). (4) All timescales decrease with increasing interplanetary activity because high activity involves acceleration and loss of heavy ions with shorter lifetimes than protons. (5) Also, decay times are about twice as long during recovery than during main phase. (6) Similar dependences are found for the solar wind coupling coefficients. The models are similar to linear models in predictability and are stable with respect to perturbation in the initial conditions.