Global and multi-scale features of solar wind-magnetosphere coupling:

From modeling to forecasting

A. Y. Ukhorskiy,1 M. I. Sitnov,2 A. S. Sharma,2 and K. Papadopoulos2

Received 27 October 2003; revised 8 March 2004; accepted 17 March 2004; published 16 April 2004.

[1] The Earth’s magnetosphere is a spatially extendednonlinear system driven far from equilibrium by the turbulentsolar wind. During substorms it exhibits both global andmulti-scale features which reconciliation has been a longstanding issue. This paper presents a data-derived model ofthe solar wind-magnetosphere coupling that combines anonlinear dynamical description of the global features with astatistical description of the multi-scale aspects. Thisapproach yields deterministic predictions of the globalcomponent of magnetospheric dynamics and probabilisticpredictions of its multi-scale features. INDEX TERMS: 2788

Magnetospheric Physics: Storms and substorms; 2784

Magnetospheric Physics: Solar wind/magnetosphere interactions;

3210 Mathematical Geophysics: Modeling; 3220 Mathematical

Geophysics: Nonlinear dynamics; 3240 Mathematical Geophysics:

Chaos. Citation: Ukhorskiy, A. Y., M. I. Sitnov, A. S. Sharma,

and K. Papadopoulos (2004), Global and multi-scale features of

solar wind-magnetosphere coupling: From modeling to

forecasting, Geophys. Res. Lett., 31, L08802, doi:10.1029/

2003GL018932.

1. Introduction

[2] The coupling of the solar wind mass, momentum andenergy to the magnetosphere is the main driver of geomag-netic activity observed on the Earth and in space. Thesouthward turning of the interplanetary magnetic field(IMF) leads to the accumulation of magnetic energy in themagnetotail which is subsequently released in the explosiveprocess known as magnetospheric substorm. The substormsare episodic in nature and have time scales of an hour or so,and are associated with global reconfiguration of the mag-netosphere. Substorms also exhibit phenomena on smallerspatial and temporal scales, such as turbulence, bursty bulkflows [Angelopolous et. al., 1999], and fluctuations in thenear-Earth current sheet [Ohtani et al., 1995] and auroralluminosity [Lui et al., 2000]. The substorm activity is oftenquantified with the AL index, which measures the magneticfield disturbances produced by the substorm current systemclosing through the ionosphere.[3] Due to the wide range of spatial and temporal scales

involved in the solar wind-magnetosphere interaction, de-veloping first principles models that encompass all therelevant scales has been a challenge. Recognizing thecapability of nonlinear dynamical techniques to reconstruct

behavior of the system independent of modeling assump-tions, long time series data of geomagnetic indices havebeen used to develop models in terms of a small number ofdynamical variables and these have led to space weatherforecasting tools based on local-linear filters [Vassiliadis etal., 1995]. However, the low-dimensional models do notdescribe all aspects of magnetospheric dynamics. In partic-ular, it was noted that the correlation dimension of themagnetosphere as a dynamical system does not converge onsmall scales. Moreover, the geomagnetic indices are foundto exhibit multi-scale properties more similar to bicolorednoise than to the time series generated by a low-dimensionalchaotic system [Takalo et al., 1994].[4] The multi-scale properties of magnetospheric dynam-

ics have been interpreted in terms of multi fractal behaviorof intermittent turbulence and self-organized criticality[Consolini, 1997], motivated by the observed power-lawbehavior [Tsurutani et al., 1990]. However, a state of self-organized criticality corresponds to the limit of vanishinglysmall control parameters [Vespignani and Zapperi, 1998],thus making the system effectively autonomous and thusquestioning the validity of such models to the magneto-sphere, whose dynamics is, to a large degree, driven by thesolar wind. An analysis of observational data indicates thatits dynamics resemble non equilibrium phase transitions[Sitnov et al., 2000]. In particular, it was shown that thelarge-scale features of solar wind-magnetosphere couplingare organized in a manner similar to first order phasetransitions. It was also suggested that the scale-free proper-ties of the data result from the dynamics in the vicinity of acritical point, as in second order phase transitions, and acritical exponent relating the output fluctuations to the solarwind input was obtained in a form similar to the criticalexponent b [Sitnov et al., 2001]. The phase transitionanalogy provides a conceptual framework for combiningthe global and multi-scale aspects of substorm dynamics ina unified model. In this paper the development of such amodel directly from data by combining nonlinear dynamicaltechniques with the conditional probability approach ispresented. The solar wind convective electric field givenby the product vBS of the solar wind speed v and thesouthward IMF component BS is the input of the model.The magnetospheric response to the solar wind activity isrepresented by the AL index. The model yields accuratepredictions of the average level of AL and also ranks theprobabilities of the discrepancy between the observed andpredicted values of AL.

2. Combined Description of Global andMulti-Scale Features

[5] The model parameters are derived from the data setcomprised by 34 isolated intervals representing different

GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L08802, doi:10.1029/2003GL018932, 2004

1Applied Physics Laboratory, Johns Hopkins University, Laurel,Maryland, USA.

2Department of Astronomy, University of Maryland, College Park,Maryland, USA.

Copyright 2004 by the American Geophysical Union.0094-8276/04/2003GL018932$05.00

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levels of geomagnetic activity. It contains about 4 � 104

points sampled at 2.5 min intervals [Bargatze et al., 1985].In order to use both vBS and AL data in a joint input-outputphase space their time series are normalized by the respec-tive standard deviations: It = vBS(t)/svBS

and Ot = AL(t)/sAL(svBS

= 976 � nT km/s, sAL = 171 nT).[6] To reconstruct the phase space underling the observed

scalar time series we use the delay embedding techniquedeveloped for nonlinear dynamical systems [Casdagli,1992]. The set of delay vectors given by

ITt ;OTt

� �¼ It; . . . ; It� M�1ð Þ;Ot; . . . ;Ot� M�1ð Þ

� �ð1Þ

where M is the total number of delays, yields the trajectoryof the system in the reconstructed space where thedirections of the largest variances reveal the dominantdynamical variables. These directions are derived from aprincipal component analysis [Broomhead and King, 1986]and the dominant variables of the solar wind-magnetospherecoupling are obtained by projecting the input-output delayvectors on the first D principal components (v1, . . ., vD) ofvBS � AL time series as

xt ¼ ITt ;OTt

� �� v1; . . . ; vDð Þ ð2Þ

The multi-scale properties of vBS � AL time series havedifferent origins compared to those of low-dimensionalchaotic systems, such as the Lorenz system, in which thescale-invariance is due to the fractal nature of its attractor.The magnetosphere exhibits a wide range of scales withproperties of high dimensional colored noise, presumablyrelated to the scale-invariance of its solar wind driver[Ukhorskiy et al., 2003]. This leads to complicatedtrajectories in the phase space on shorter scales, and as aresult, a deterministic model alone does not provide anadequate description of the system [Ukhorskiy et al., 2002].[7] To reveal the global dynamics in the presence of

multi-scale features an approach similar to the mean-fieldmodels in statistical physics is used. It was developed as thegeneralization of the false nearest neighbors technique ofKennel et al. [1992] for the case of stochastic dynamicalsystems [see Ukhorskiy et al., 2003]. In this approach theevolution of the system at time t is determined by theevolution of the fixed number NN of its nearest neighbors,viz. the states that are closest to it as measured by thedistance in the reconstructed space. These nearest neighborsare used to compute the center of mass xt

cm as their averageand the state of the system at the next time step is thenwritten as a function of the center of mass:

Omftþ1 ¼ F xcmt

� �¼ hOkþ1i; xk 2 NN ð3Þ

However, in the case of many multi-scale systems, theaveraging procedure may not be well defined for chosenvalues of the dimension and the number of nearest neigh-bors. Thus the model requires an additional criterion

if k xcmk � xcmn k ! 0 then jOmfkþ1 � O

mfnþ1j ! 0; ð4Þ

which ensures that the mean-field function F is well definedfor the given pair of parameters (D, NN). Then, for a given

NN and the accuracy e, a local minimum dimension Dt iscomputed such that the condition jOt+1

mf � Ot+1mf j < e holds for

each point of the data set, where Ot+1mf is calculated with

including the current state at time t in the set of nearestneighbors. The distribution of Dt is then calculated fordifferent embedding dimensions. If for a chosen value ofNN the mean-field function is well defined in some finitedimension, the distribution function drops suddenly at thisvalue. In the absence of such a characteristic behavior NN isincreased and the whole procedure is repeated. For differentvalues of parameters (D, NN) the dynamical model(equation (3)) is computed to obtain the optimal pair ofvalues corresponding to the highest accuracy. This value ofD represents the dimension of the space with the averagingdefined by NN and is referred to as the mean-fielddimension (MFD).[8] To elucidate the MFD analysis in the case of solar

wind-magnetosphere system the case of Lorenz attractor, awell known low dimensional dynamical system, is studied.Figure 1a shows probability density function P(Dt) calcu-lated for the x-component of Lorenz attractor with e = 0.1.Function P(Dt) calculated with NN = 3 drops abruptly at Dt

�3–4, indicating that the ensemble averaging is welldefined and that any D � 4 is a good choice for theembedding space dimension. On the other hand in case ofvBS � AL time series, P(Dt) calculated with small level ofaveraging (NN = 3–10) with e = 0.1 have a power-lawshape in Dt = 1–100 range (Figure 1b). This indicates thatthe ensemble averaging is ill defined for this range ofparameters and the mean-field model is not meaningful.For increased levels of averaging (NN � 100) the functionsP(Dt) drop abruptly at finite values. However, unlike theLorenz attractor which exhibits the same dimension at allscales, in the case of coupled solar wind-magnetospheresystem the dimension is a function of the level of averaging.The higher the value of NN the wider the range of scaleswhich are smoothed away and the smaller the effectivedimension D of the averaged system, e.g., D = 8 for NN =100, and D = 3 for NN = 250.[9] The MFD approach can be effectively used to model

the global component of complex systems such as the solarwind-magnetosphere coupling. The phase portrait for themagnetosphere in the three dimensional space spanned bythe first three principal components of vBS � AL time seriesis shown in Figure 2. The technique used earlier [Sitnov et

Figure 1. MFD analysis. The probability density functionsof local dimension calculated with e = 0.1. (a) Lorenzattractor; D = 4 for NN � 3 (b) vBS � AL time series; D = 8for NN = 100, and D = 3 for NN = 250.

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al., 2000, 2001] for such studies involved cumbersomeprocedures such as the removal of the hysteresis loopsand smoothing of the resultant surface. The MFD approachdiscussed above resolves these problems by using the mostprobable states of the nearest neighbors whose number isconsistent with the embedding dimension and chosen accu-racy. The surface is shown in the x1 � x3 projection planedefined by the first (v1) and the third (v3) principal compo-nents. The component x1 corresponds to the time averagevalue of input (It) while x3 corresponds to its average timederivative [Sitnov et al., 2001]. The projection x2 of the dataon the v2 direction, which corresponds to the time averagevalue of the output (Ot), is color coded. The evolution of thesystem along this surface is shown by the two-dimensionalvelocity field. The substorm cycles are confined to a surfacewhich is broadly two-level, and the levels are associatedwith the growth phase (orange-yellow colors) and theexpansion phase (black-blue colors). The recovery phasecorresponds to the transition region with blue-orange-yellow color in the lower left quadrant of Figure 2. Thetypical substorm cycle starts with an increase in the averageinput x1 while the average output x2 is nearly constant orslowly decreasing, i.e. the system is in the growth phase,which may be viewed as the ground state of the system.During this phase the average input rate x3 first increasesand then decreases to small values. Then the output x2 fallsrapidly to negative values at almost constant values ofthe input and this corresponds to the expansion phase ofsubstorms, which may be viewed as a transition to theexcited state. The recovery of the system to its ground stateinvolves an increase of x2 while the magnitude of x3increases to near-zero values.[10] In the reconstructed phase space the multi-scale

features have been eliminated largely due to the averagingprocess. They appear as fluctuations around the smoothmanifold containing the trajectories of averaged system andtheir statistical properties depend on the state of the system,xt, characterized by the input as well as the output, and canbe described in terms of a conditional probability P(Ot+1jxt).At each time step the conditional probability can be esti-

mated using the nearest neighbors of the current state,defined in the MFD analysis, as

P Otþ1jxtð Þ � P Okþ1ð Þ; xk 2 NN ð5Þ

To analyze the nature of multi-scale dynamical features theevolution of conditional probability P(Ot+1jxt) is calculatedin the one and two-dimensional subspaces spanned by v1and (v1, v3) principal components of vBS � AL time seriesrespectively. The probability distribution functions P(Ot, x1)are shown in Figure 3a for different levels of solar windactivity, represented by the blue, red and yellow colors. Themarginal distribution P(Ot) for all levels of the solar windtaken together, is the black curve in the back panel of theplot, and has a power-law shape with a break correspondingto �AL �500 nT. The distribution functions correspondingto the medium (red) and high (yellow) activities have distinctmaxima and thus do not exhibit multi-scale behavior. On theother hand, the distribution function corresponding to thelow solar wind activity has a structure similar to the multi-scale marginal distribution. However, if the input space isexpanded from one to two dimensions, then the multi-scalefunctions P(Otjx1) break into a number of distributionfunctions P(Otjx1, x3) with pronounced peaks whose widthand position depend on both the input parameters (x1, x3)(Figure 3b). With an increase in the phase space dimensionthe width of the corresponding distribution functionsdecreases until it saturates at the mean-field dimension.This indicates that a large portion of the multi-scale distri-bution of AL is directly induced by the similar properties ofthe solar wind driver rather than being a result of theinherent complexity of magnetospheric dynamics.[11] The strong influence of the solar wind driver on the

multi-scale properties of the system provides a new under-standing of the magnetosphere and forms the basis for a

Figure 2. Left panel: phase portrait of the global featuresof solar wind - magnetosphere coupling during substorms;right panel: first three principal components of vBS � ALtime series.

Figure 3. Conditional probabilities of AL as a function ofsolar wind conditions. (a) P(Ot, x1): The yellow, red andblue curves correspond to strong (vBS > 9 mV), medium(0.6 < vBS < 9 mV) and low (vBS < 0.6 mV) solar windactivity levels, respectively. The floor shows all point in thedatabase corresponding to the marginal probability distribu-tion function shown in the back panel. (b) P(Otjx1, x3)computed for the points at the centers of the blue circlesalong the value of x1 corresponding to the low solar windactivity (blue curve in panel (a)). The circles indicate theclusters of equal number of nearest neighbors that yield theconditional probability functions.

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new approach to data-derived modeling. The global behav-ior can be described by a low-dimensional dynamicalmodel (equation (3)) based on the mean-field concept andthe high-dimensional multi-scale features on the otherhand are described in terms of conditional probabilities(equation (5)). The predictions from a model with thiscombination of dynamical and probabilistic descriptionsare shown in Figure 4. The AL data is shown in blue inFigure 4 (bottom) and the iterative predictions obtained withthe mean-field model are shown in red. The mean-fieldmodel effectively reproduces the local average of AL, yield-ing predictions of the average level of substorm activity. Itshould be noted that once the mean-field dimension isdefined following the procedure described above, the pre-diction process does not need any tuning of parameters,unlike the earlier local-linear techniques [Vassiliadis et al.,1995; Ukhorskiy et al., 2002]. Being regular and low-dimen-sional this component of AL is a manifestation of the globalconstituent of substorm dynamics. However, due to theinherent averaging the model can not capture the abruptvariations in the data and as a result it underestimates thesubstorm activity in many cases. Being a result of the multi-scale constituent of AL these discrepancies can be describedin terms of conditional probabilities (equation (5)), which areshown in Figure 4 (middle). Estimated at each time step,these probabilities are then used to compute the probabilityP(Ot+1 > Ot+1

mf jxt) of the deviations from the mean-fieldprediction, shown in Figure 4 (bottom). The probabilitydensity functions are computed at each time step using onlythose events that correspond to AL values greater than the

mean-field model output. Thus in Figure 4 (bottom) thedeviations are represented by the different shades of yellowabove the mean-field curve. The 100% contour includes alldeviations from the mean-field prediction, including thesharpest peaks and most of the peaks lie below the 70%contour. Thus the model yields not only the deviationsfrom the predictions of the mean-field model but also theirprobabilities.

3. Conclusions

[12] The new approach presented here for modeling thesolar wind-magnetosphere coupling is based on the recog-nition that the magnetospheric dynamics are neither clearlylow dimensional nor completely random, but exhibit com-binations of these two aspects. The modeling based on themean-field concept yields a low dimensional descriptionand the multi-scale aspects are modeled using conditionalprobabilities. Such a combined approach yields an im-proved and effective tool for forecasting space weather.

[13] Acknowledgments. The research is supported by NSF grantsATM-0001676, 0119196 and 0318629.

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�����������������������K. Papadopoulos, A. S. Sharma, and M. I. Sitnov, Department of

Astronomy, University of Maryland, College Park, MD 20742-3921, USA.A. Y. Ukhorskiy, Applied Physics Laboratory, Johns Hopkins University,

Johns Hopkins Road, Laurel, MD 20723-6099, USA. (aleksandr.ukhorskiy@jhuapl.edu)

Figure 4. Long-term predictions of AL index for the 29thinterval of Bargatze et al [1985] database. Upper panel:solar wind vBS data. Middle panel: Probability P(Ot+1jxt) ofAL computed with NN = 50; xt is calculated with the mean-field dimension D = 5. Bottom panel: the actual AL data(blue), the mean-field prediction (red); contours of prob-ability P(Ot+1 > Ot+1

mf jxt) of the AL deviation from the mean-field model output(yellow shades).

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