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S statistic to analyze AE data
Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1387–1397
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Using the R=S statistic to analyze AE data C.P. Price ∗ , D.E. Newman Physics Department, University of Alaska Fairbanks, 900 Yokon Drive, Suite 102, Fairbanks, AK 99775-5920, USA
Abstract Recently, it has been proposed that the dynamics of the global geomagnetic current system are consistent with a self-organized critical (SOC) state. Based on that idea, sandpile models of the auroral electrojet index AE have been constructed and compared with data. The Hurst R=S statistic is a powerful test for the presence of SOC, which is applied to AE time series. Those results are compared with the results of the R=S statistic for both the solar wind time series (Bz , vsw , etc.) and auroral oval magnetometer time series. We 7nd that the solar wind acts as if it was in a SOC state, and that the AE index time series indicates that AE is in a nearly identical SOC state. We cannot rule out the possibility that the AE results, which are the response to the solar wind, are a re8ection of the solar wind SOC state. However, the local measures (ground station magnetometer readings; c 2001 Elsevier ground station dynamical signals) give evidence of a di:erent type of SOC state within the magnetosphere. Science Ltd. All rights reserved. Keywords: Solar wind; Magnetosphere interactions; Nonlinear dynamics
1. Introduction The interaction between the solar wind and the magnetosphere is not understood in detail, in part because of the very large number of dynamical variables. Whatever the detailed mechanisms are, they must involve transport (and most likely turbulent transport). A new approach to understanding the transport, based on the idea of self-organized criticality (SOC) (Bak et al., 1987), has been suggested (Carreras et al., 1996) which seeks to describe the dynamics of transport without relying on speci7c local instabilities. Complex systems which self-organize, manifest states with no characteristic time (or length) scale, leading to scaling behavior in physical quantities related to the energy release of the system. Chang (1992) has proposed that the magnetosphere is such a self-organized system. Various researchers have demonstrated signatures of scale independence in the auroral electrojet index AE (Davis and Sugiura, 1966) time series. Tsurutani et al. (1990) showed a 1=f region in the power spectral density of AE. Takalo et al. (1993, 1994) showed self-aAnity in the AE time series. ∗ Corresponding author. Tel.: +1-907-474-6106; fax: +1-907474-6130. E-mail address: [email protected] (C.P. Price).
Takalo (1993), Takalo et al. (1999) and Consolini (1999) have demonstrated power law distributions in the lifetimes of substorm events. Consolini (1997, 1999) has shown a power law distribution in the energy of substorm events. However, since the magnetospheric system is not autonomous, but rather is strongly driven by the solar wind, those properties observed for the magnetosphere may simply be an echo of similar properties for the solar wind. Indeed, Freeman et al. (1999) have demonstrated that the distributions of burst lifetimes of the solar wind and of the AE index have power law components of similar form, and o:er arguments that the scale independence of the global output proxy AE are caused by the scale-free properties of the solar wind rather than by the magnetospheric dynamics. Noting that recent SOC models have demonstrated scale independence only in local (internal) output and not in global (external) output, they further argue that analysis of localized phenomena (re8ecting the internal output) should be pursued to determine whether the internal output of the magnetosphere has scale-free behavior. In this paper, we analyze the scaling properties of solar wind data, the AE index, and of individual auroral oval magnetometer station data. We 7nd evidence that suggests that there are two di:erent SOC processes in the solar wind, and that both global and local measures of the magnetospheric response have characteris-
c 2001 Elsevier Science Ltd. All rights reserved. 1364-6826/01/$ - see front matter PII: S 1 3 6 4 - 6 8 2 6 ( 0 0 ) 0 0 2 4 0 - 6
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tics consistent with the output of an SOC process. The scaling properties of AE and of the local dynamical signal are similar to the scaling properties of the solar wind speed v while the local station data has scaling properties more like those of the interplanetary magnetic 7eld (IMF) Bz component and the convective electric 7eld vBz . However, we cannot rule out the possibility that the geomagnetic output time series are only re8ecting the SOC state of the solar wind drivers. 2. The Hurst R=S statistic The rescaled range (R=S) analysis, pioneered by Mandelbrot and Wallis (1969) following Hurst (1951), can be used to demonstrate the scaling behavior associated with SOC
systems. That is, for sample size s, the ratio R=S of the sample sequential range R(s) to the square root of the sample sequential variance S 2 (s) varies as sH within the scaling region. A Hurst exponent H = 0:5 indicates a wholly stochastic process lacking correlation=coherence, while H = 1:0 indicates a very coherent process with perfect correlation, and H = 0:0 indicates a very coherent process with perfect anti-correlation. A Hurst exponent H between 0:5 and 1:0 indicates nonperiodic complex dynamics which are not inconsistent with SOC. This interpretation of the R=S analysis does not absolutely prove the presence of SOC: it is a strong (in a statistical sense) indication of the presence of SOC. The same range of H will be taken as consistent with SOC if the power spectrum of the time series shows a self-similar region of at least a decade in width with log–log slope − between 2:0 and 1:0. Note that for a time series which is a
Fig. 1. The time series, power spectral density and Hurst R=S plots for March 1979 IMF Bz component time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
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realization of fractional Brownian motion sequence, Voss (1990) has shown theoretically that one should have H = 1 + 12 (where = 2H − 1); in practice this relation is 2 only weakly satis7ed, and so cannot be used as a test for the presence of a fractional Brownian motion sequence. 3. Time series We will analyze a number of data sets, each with 1 min resolution, taken from the period 1978–1979. The behavior of the solar wind is represented by solar wind data from ISEE-3 (Frandsen et al., 1978; Bame et al., 1978), which measured the vector IMF and the solar wind speed, for the months from September 1978 to July 1979. The global geomagnetic response is represented by the auroral electrojet index AE (Davis and Sugiura, 1966), for the entire period 1978–1979. The geomagnetic response at a 7xed location is represented by the horizontal (‘H ’) components measured at the 11 auroral oval stations (‘station data’), and the short-term dynamical signals (‘signal data’, which are made from the station data by subtracting monthly quiet time averages), for the months from September 1978 to July 1979. (Note that AE is a nonlinear composition of the full set of auroral oval signals. As a result of the method by which AE is constructed, the AE time series may or may not share similar properties to any of the individual signals.) With the exception of the AE time series, these time series are not complete. There are 480; 960 possible data points for the months from September 1978 to July 1979; typically the coverage for the solar wind data and for the station data is in the range 80 –85%. In order not to introduce biases due to uneven station coverage, all stations are presumed to have missing
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data if the data is missing from any station. The data gaps are ignored: the sections of extant data are simply concatenated together. Tests on a set of surrogate time series, each composed from fractional Gaussian motion of a di:erent Hurst exponent into which random gaps eliminating 15% of the data are introduced, show that both the power spectrum analysis and the R=S analysis are robust when dealing with data set gaps in this way. 4. Analysis of solar wind data Fig. 1 shows the time series, the power spectrum and the R=S analysis for the IMF Bz component for the month of March 1979 (which is representative of the other 10 months of data for Bz ). The power spectral density plot shows a log–log 7t for frequencies f ¿ 1 day−1 . (Those frequencies greater than 0:001 min−1 have been binned, using 20 bins per decade.) If the power spectral density obeys a power law, |X˜ (f)|2 ˙ f− , then the log–log 7t yields the power spectral density index . The R=S plot shows a log–log 7t for s ¡ 1000 s (slope H¡ ) and a separate 7t for s ¿ 1000 s (slope H¿ ). We see for the IMF Bz component for the month of March 1979 that ≈ 1:5 and H¡ ≈ 0:95 and H¿ ≈ 0:65. The March 1979 Bz time series is self-similar for all time scales, highly coherent for time scales less than one day, and only slightly coherent for time scales greater than one day. However, we cannot rule out the possibility that self-scaling disappears for s greater than one month. Fig. 2 shows the Hurst exponents H¡ and H¿ versus month and the power spectral density index versus month, for the IMF Bz data. The Hurst exponent plot also shows
Fig. 2. The Hurst exponents H¡ and H¿ (see text) and power spectral indices for the monthly IMF Bz component time series versus month. Included are the exponents and indices for stochastic surrogate data sets.
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Fig. 3. The power spectral density and Hurst R=S plots for the concatenated IMF Bz component time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
Fig. 4. The power spectral density and Hurst R=S plots for the concatenated solar wind speed v time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
the exponents that are calculated from a surrogate data set composed by a random reshuOe of the original data set. For a random (stochastic) set, we expect H ≈ 0:5. Thus, the di:erence in values of the Hurst exponents for the surrogate data sets from 0.5 gives a measure of the uncertainty in the values of the Hurst exponents. The power spectral index plot also shows the indices that are calculated from the same surrogate data set. For a random set, we expect ≈ 0, and so the di:erence in values of the indices for the surrogate data sets from 0 gives a measure of the uncertainty in the values of the indices. We see no clear trend for ei-
ther the Hurst exponents or the power spectral indices as a function of the season, although there is a fair amount of variability in the long span Hurst exponent H¿ . The lack of an annual trend is not unexpected: the IMF measured upstream from the Earth should be ignorant of the Earth’s cycles. The lack of obvious trending in the monthly Bz results, and the apparent self-similarity of Bz for longer times, prompts us to analyze a concatenation of the set of month-long data sets. Fig. 3 shows the power spectrum and the R=S analysis for the concatenated IMF Bz component time series. We see that
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Fig. 5. The time series, power spectral density and Hurst R=S plots for the month of March 1979 auroral electrojet index AE. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
H¡ ≈ 0:95 and H¿ ≈ 0:65: the concatenated Bz time series has the same self-scaling properties as the monthly Bz time series. On the power spectrum plot, we have power-law 7ts for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The shorter period ≈ 1:3, is not appreciably di:erent from the power spectral index for the monthly Bz time series. The longer period power spectral index is nearly zero, as would be expected for a random (stochastic) time series. This reinforces the conclusions given for the month of March 1979 Bz time series: the Bz time series is self-similar for all time scales, highly coherent for time scales less than one day, and only slightly coherent for time scales greater than one day. We note that this time scale is only coincidentally that of the Earth’s rotational period; it could be related to either the solar wind transport time scale (structure size over solar wind speed) or to the turbulent decorrelation time scale of
turbulence in Bz . These results indicate that the IMF component Bz is the result of a SOC process, or more precisely, one may say that the results are not inconsistent with Bz resulting from a SOC process. We have also analyzed the monthly time series for the solar wind speed v and for the product of the solar wind speed v and the IMF Bz component (the convective electric 7eld). Both sets of monthly time series have no annual trends for either the Hurst exponents or the power spectral indices. For the monthly solar wind speed v time series, we 7nd a power spectral index in the range [1:9; 2:0] and Hurst exponents H¡ and H¿ ≈ 1. For the monthly convective electric 7eld vBz time series, we 7nd power spectral indices and the Hurst exponents are nearly identical to those for the corresponding monthly Bz time series, e.g., in the range [1:3; 1:6], H¡ ≈ 1 and H¿ in the range [0:6; 0:8].
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Fig. 6. The time series, power spectral density and Hurst R=S plots for the concatenated auroral electrojet index AE time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
The lack of trending in the monthly results leads us to analyze the concatenations of the sets of monthly time series. Fig. 4 shows the power spectrum and R=S analysis for the concatenated solar wind speed v time series. (The results for the concatenated convective electric 7eld vBz time series are nearly identical to the results for the concatenated IMF component Bz time series.) We see that H¡ ≈ 0:95 and H¿ ≈ 0:85: the concatenated v time series is both self-similar and highly coherent for time scales from several minutes to several months. The shorter period power spectral index ≈ 1:4, comparable to the values found for the monthly v time series. The longer period power spectral index ≈ 0:75: the solar wind speed has a nearly 1=f region for periods between 2 days and 2 months. All of these features are consistent with dynamics exhibited by SOC systems. We have found that, although the appearance of the time series of the di:erent solar wind quantities is di:erent, those concatenated time series have similar short time scale self-similarity and coherence properties: the Hurst exponent H is approximately 1 for time spans less than about one day. The main di:erence in the R=S analyses is that the solar wind speed v time series is highly coherent for very long time scales (H¿ ≈ 0:85), while the IMF Bz component time series and the convective electric 7eld vBz time series are only slightly coherent for time scales greater than one day (H¿ ≈ 0:65). The time series of the three solar wind quantities also have similar high-frequency (f ¿ 1 day−1 ) power spectra, power spectral index ∈ [1:5; 2:0]. For lower frequencies (0:5 month−1 ¡ f ¡ 0:5 day−1 ), the concatenated Bz and vBz time series have 8at power spectra while concatenated v time series has a 1=f power spectrum. These results indicate that the solar wind could be the result of a SOC system; that is, these results are not inconsistent
with the solar wind state being the result of one or more SOC processes. 5. Analysis of auroral electrojet index AE data Fig. 5 shows the time series, the power spectrum and the R=S analysis for the auroral electrojet index AE for the month of March 1979 (which is a representative of the other 23 months of data). We see that the AE time series is self-similar. For s ¡ 1000, we 7nd the exponent H¡ ≈ 0:95; the AE time series is remarkably coherent for timescales less than a day. For s ¿ 1000, we 7nd H¿ ≈ 0:80, still evidence for a coherent series. The AE time series also has a power law power spectrum, with ≈ 1:9. We also analyze a concatenation of all 24 of the month-long AE data sets. Fig. 6 shows the power spectrum and R=S plots for the two year sequence of AE data. We see that the AE time series is self-similar and very coherent for spans up to one day in length, and self-similar if less coherent for spans of several months, and possibly spans up to a year: H¡ ≈ 0:95 and H¿ ≈ 0:75. The power spectrum has broad peaks at f approximately 1 year −1 (2:0 × 10−6 min−1 ) and approximately 1 month−1 (2:3 × 10−5 min−1 ), and a narrow but statistically signi7cant peak at f = 1 day−1 . For higher frequencies (f ¿ 1 day−1 ), the power spectrum exponent is ≈ 1:85; for intermediate frequencies (1 day−1 ¿ f ¿ 1 month−1 ), we have ≈ 0:80. The three peaks can be understood in terms of the annual and diurnal motions of the geomagnetic pole, and the monthly nature of the quiet-time baseline signal which underlies the electrojet indices. The AE results are more similar to those of the solar wind speed v than they are to those of either Bz or vBz , especially in the appearance of the 1=f region for time scales between several days and several months.
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Fig. 7. The time series, power spectral density and Hurst R=S plots for the month of March 1979 College station H component. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
6. Analysis of ground station data Fig. 7 shows the time series, the power spectrum and the R=S analysis for the horizontal (H ) component of the magnetic 7eld as measured at the College station for the month of March 1979 (which is fairly representative of the other 10 months of data for this station, and for the data from the other stations). Those plots show that the College station H component time series is self-similar and very coherent for time spans s up to about 1 day, with the Hurst exponent H¡ ≈ 0:95. For greater time spans, the Hurst exponent H¿ ≈ 0:55, which would indicate that for time spans greater than 1 day, the time series is stochastic or slightly coherent. The College station time-series power spectral density has a power law behavior of f ¿ 1 day−1 , with spectral index ≈ 2, and a statistically signi7cant peak at f = 1 day−1 . These results imply strongly that the ground station time
series is the result of stochastic process with a periodicity of about one day. The results for the other months show no clear trend as a function of the season, which is somewhat surprising since one would expect some modulation on an annual scale. Since there is no strong dependence on season, we have analyzed a concatenation of all 11 of the monthly College station data sets. Fig. 8 shows the power spectrum and R=S plots for the concatenation of the College station H component data. We see that the concatenated station data time series has the usual strong peak in the power spectrum at f = 1 day−1 , and that the power spectral density falls o: as f−2 for frequencies greater than 1 day−1 , with a slight de7cit of power for f ¿ 0:1 min−1 . In addition, it has a strong peak in the power spectrum in the vicinity of f = 1 month−1 , and the power spectrum is relatively 8at for 1=f between 2 days and 2 months. The R=S analysis shows
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Fig. 8. The power spectral density and Hurst R=S plots for the concatenated College station H component time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
that the concatenated station data time series is also very coherent on time scales less than one day (H¡ ≈ 0:95), and is also less coherent but still self-similar for longer time scales (H¿ ≈ 0:65). So, the analysis suggests that the concatenated station data time series is the result of a noisy process with periodicities of about one day and about one month. The diurnal component can be understood from the daily motion of the station relative to the geomagnetic pole. Presumably, the monthly component results in some way from the (27-day period) solar rotation. The ground station results are more similar to the results for Bz or vBz than they are to those of either AE or the solar wind speed v. 7. Analysis of ground station signals Fig. 9 shows the time series, the power spectrum and the R=S analysis for the H component dynamical signal at the College station for the month of March 1979 (which is fairly representative of the other 10 months of data for this station, and for the signal data from the other stations). Those plots show that the dynamical signal extracted from the College station H component time series is self-similar and very coherent for time spans s up to about one day, with the Hurst exponent H¡ ≈ 0:95. For greater time spans, the Hurst exponent H¿ ≈ 0:45, which would indicate that for time spans greater than one day, the time series is stochastic or slightly anti-coherent. The College station dynamical signal time series also has a high-frequency power-law power spectral density, with = 1:9. (Close inspection shows that there may well be two scaling regions in the power spectral density. For f ¿ 1=100 min, we have ≈ 2:0, and for periods between 100 min and one-half day we have ≈ 0:5. However, there is no corresponding break in R=S at a period
of 100 min.) There is also a statistically signi7cant peak in the power spectral density at f = 1 day−1 . These results imply strongly that the ground station dynamical signal time series is the result of stochastic process with a periodicity of about one day. As with the ground station data, there is no clear trend as a function of the season, where one might expect some modulation on an annual scale. Given the absence of a yearly trend in the monthly dynamical signal time series, we have analyzed a concatenation of all 11 months of the various station dynamical signal time series. In Fig. 10, we display the power spectrum and R=S plots for the concatenation of College station dynamical signals. The R=S analysis shows that the concatenated dynamical signal time series is also very coherent on time scales less than one day (H¡ ≈ 0:95), but is only slightly coherent (but still approximately self-similar) for longer time scales (H¿ ≈ 0:65). We see that the concatenated dynamical signal power spectral density index for frequencies greater than 1 day−1 is ≈ 1:75; there is an excess of power for f ¿ 0:1 min−1 . The concatenated dynamical signal time series has the usual peak in the power spectrum at f=1 day−1 , which can be understood from the daily motion of the station relative to the geomagnetic pole. There is also a power-law region for periods of several days to several months with an index of roughly = 1. There is no signi7cant peak in the power spectrum in the vicinity of f = 1 month−1 , although we might presumably expect a monthly component as a result of the monthly nature of the quiet-time baseline signal which enters the construction of the dynamical signal. We are unable to advance an explanation for the scaling region for periods of several days to several months, except to say that feature, as well as the others cited, are consistent with dynamics exhibited by SOC systems. The dynamical signal results are more similar to the results for AE and the solar
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Fig. 9. The time series, power spectral density and Hurst R=S plots for the month of March 1979 College station dynamical signal. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
wind speed v than they are to those of the station data or the IMF component Bz or the convective electric 7eld vBz . 8. Conclusions Based on the Hurst R=S analysis and associated power spectral density calculations, we have shown that the solar wind data are consistent with SOC dynamics, in agreement with previous work. In fact, our results suggest that there may be two di:erent SOC states, one involving the IMF and the other involving the solar wind. The solar wind state appears to be fully developed (continuous transport at all the scales which we have analyzed) while the IMF seems to be only partially developed (reduced organization for larger scales). Given that there is strong turbulent transport in the solar wind, the former result might have been anticipated,
but we advance no hypothesis about the IMF state. The turbulent state of the solar wind driver complicates the analysis of the dynamical nature of the global geomagnetic system, since that systemic response to the solar wind driver will re8ect some of the characteristics of the driver. The auroral electrojet index AE is a commonly used global measure of magnetospheric activity. The Hurst R=S analysis and associated power spectral density calculations for AE are not inconsistent with the results of a SOC process. However, since the drivers (solar wind and IMF) have SOC characteristics, we cannot rule out the possibility that AE is only re8ecting the SOC characteristics of the drivers, as Freeman et al. (1999) have pointed out. In fact, the scaling properties of AE are similar to those of the solar wind speed v. The similarity possibly indicates that the global magnetospheric output is mostly a passive response (with a 7lter function) to the solar wind; this would follow if the system
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Fig. 10. The power spectral density and Hurst R=S plots for the concatenated College station dynamical signal time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
is mostly being swamped by the driver, so that only a small fraction of the global system response is due to internal SOC dynamics. Freeman et al. (1999) have argued that local measures of geomagnetic activity will re8ect internal output. Thus analysis of such local measures might give insight into magnetospheric SOC processes. We have analyzed the H component magnetometer data from the 11 auroral oval stations. These time series are local in space and in time (point measurements). We 7nd that the station data time series are not inconsistent with the results of a SOC process. The station data Hurst exponents and spectral indices are more like those of Bz and vBz than they are like those of AE and v. The di:erence between the AE results and the station data results could be just the di:erence between global output and internal output. Previous results (Takalo et al., 1993, 1995; Takalo and Timonen, 1994) have found lower values for the scaling exponent of AE (from self-aAnity analyses). From basic considerations, those values should be comparable. Takalo has argued (Takalo et al., 1994) that AE and the other response series (the ground station data and extracted signals) are the result of cumulative driving, as indicated by the strong autocorrelation in those series, and as a result, he has examined the self-aAnity of the di4erences (increments) in the AE and related series. When we calculate the Hurst index for the di:erences in the AE and related series, we get values which are consistent with those of Takalo and co-workers, that is, about half the values of which we are reporting for AE. However, given that a priori one does not know if an arbitrary series is incremental or cumulative, it is most prudent to treat each as if they were cumulative, which has been our approach here.
The dynamical signal from each of the stations was also analyzed. These time series are local in space but nonlocal in time, since they are composed of the di:erence between the station data and the average quiet time pattern for that station. We 7nd that the signal data time are not inconsistent with the results of a SOC process. The signal data Hurst exponents and spectral indices are more like those of AE and v than they are like those of the station data and Bz and vBz . This result probably re8ects the composition of AE from the envelope of the dynamical signals of the 11 auroral oval stations. We have found evidence for SOC dynamics in the solar wind, in the IMF, in the global measure of geomagnetic response, and in local measures of the geomagnetic response. The analysis is most informative when large, extended (several months or greater) data sets are used. However, the analysis of the geomagnetic response data do not point to an unambiguous SOC process in the magnetosphere, and may well simply re8ect the SOC states of the solar wind and IMF drivers. Based on our analysis, our best estimate is that the magnetosphere acts as a dynamically self-organized system, and the system’s output is mostly a passive response to an overly strong driver with hints of the internal magnetospheric SOC processes. Acknowledgements We thank T. Chang, N. Watkins, S. Chapman, J. Takalo and the referees for useful discussions. We thank NSSDC for assistance in acquiring the ISEE-3 data, and acknowledge S.J. Bame (plasma) and E.J. Smith (magnetometer) as Principal Investigators on those ISEE-3 instruments.
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References Bak, P., Tang, C., Weisenfeld, W., 1987. Self-organised criticality: an explanation of 1=f noise. Phys. Rev. Lett. 59, 381–384. Bame, S.J., Asbridge, J.R., Felthauser, H.E., Gore, J.P., Hawk, H.L., Chaves, J., 1978. ISEE-C solar wind plasma experiment. IEEE Transactions on Geoscience Electronics GE-16, 160–162. Carreras, B.A., Newman, D., Lynch, V.E., Diamond, P.H., 1996. A model realization of self-organized criticality for plasma con7nement. Physics of Plasmas 3, 2903–2913. Chang, T.S., 1992. Low-dimensional behavior and symmetry breaking of stochastic systems near criticality — can these e:ects be observed in space and in the laboratory? IEEE Transactions on Plasma Science 20, 691–695. Consolini, G., 1997. Sandpile cellular automata and magnetospheric dynamics. In: Aiello, S., Iucci, N., Sironi, G., Treves, A., Villante, U. (Eds.), Proceeding Cosmic Physics in the Year 2000, Vol. 58, Societa Italiana di Fisica. Consolini, G., 1999. Avalanches, scaling and 1=f noise in magnetospheric dynamics. Physical Review Letters, submitted for publication. Davis, T.N., Sugiura, M., 1966. Auroral electrojet activity index AE and its universal time variations. Journal of Geophysical Research 71, 785–801. Frandsen, A.M.A., Connor, B.V., Van Amersfoort, J., Smith, E.J., 1978. ISEE-C vector helium magnetometer. IEEE Transactions on Geoscience Electronics GE-16, 195–197. Freeman, M.P., Watkins, N.W., Riley, D.J., 1999. Evidence for a solar wind origin of the power law burst lifetime distribution of the AE indices. Geophysical Research Letters, submitted for publication.
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Hurst, H.E., 1951. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineering 116, 770–799. Mandelbrot, B.B., Wallis, J.R., 1969. Robustness of the rescaled range R=S in the measurement of noncyclic long run statistical dependence. Water Resources Research 5, 967–988. Takalo, J., 1993. Correlation dimension of AE data. Ph. Lic. Thesis, Laboratory Report 3, Department of Physics, University of JyvUaskylUa. Takalo, J., Timonen, J., Koskinen, H., 1993. Correlation dimension and aAnity of AE data and bicolored noise. Geophysical Research Letters 20, 1527–1530. Takalo, J., Timonen, J., 1994. Characteristic time scale of auroral electrojet data. Geophysical Research Letters 21, 617–620. Takalo, J., Timonen, J., Koskinen, H., 1994. Properties of AE data and bicolored noise. Journal of Geophysical Research 99, 13 239–13 250. Takalo, J., Lohikoski, R., Timonen, J., 1995. Structure function as a tool in AE and Dst series analysis. Geophysical Research Letters 22, 635–638. Takalo, J., Timonen, J., Klimas, A., Valdivia, J., Vassiliadis, D., 1999. Nonlinear energy dissipation in a cellular automaton magnetotail 7eld model. Geophysical Research Letters 26, 1813–1816. Tsurutani, B.T., Sugiura, M., Lyemori, T., Goldstein, B.E., Gonzalez, W.D., Akasofu, S.-I., Smith, E.J., 1990. The nonlinear response of AE to the IMF Bs driver: a spectral break at 5 hours. Geophysical Research Letters 17, 279–282. Voss, R., 1990. Self-aAnity in noise, music, mountains and clouds. In: Feder, J., Aharony, A. (Eds.), Fractals in Physics. North-Holland, Amsterdam.
www.elsevier.com/locate/jastp
Using the R=S statistic to analyze AE data C.P. Price ∗ , D.E. Newman Physics Department, University of Alaska Fairbanks, 900 Yokon Drive, Suite 102, Fairbanks, AK 99775-5920, USA
Abstract Recently, it has been proposed that the dynamics of the global geomagnetic current system are consistent with a self-organized critical (SOC) state. Based on that idea, sandpile models of the auroral electrojet index AE have been constructed and compared with data. The Hurst R=S statistic is a powerful test for the presence of SOC, which is applied to AE time series. Those results are compared with the results of the R=S statistic for both the solar wind time series (Bz , vsw , etc.) and auroral oval magnetometer time series. We 7nd that the solar wind acts as if it was in a SOC state, and that the AE index time series indicates that AE is in a nearly identical SOC state. We cannot rule out the possibility that the AE results, which are the response to the solar wind, are a re8ection of the solar wind SOC state. However, the local measures (ground station magnetometer readings; c 2001 Elsevier ground station dynamical signals) give evidence of a di:erent type of SOC state within the magnetosphere. Science Ltd. All rights reserved. Keywords: Solar wind; Magnetosphere interactions; Nonlinear dynamics
1. Introduction The interaction between the solar wind and the magnetosphere is not understood in detail, in part because of the very large number of dynamical variables. Whatever the detailed mechanisms are, they must involve transport (and most likely turbulent transport). A new approach to understanding the transport, based on the idea of self-organized criticality (SOC) (Bak et al., 1987), has been suggested (Carreras et al., 1996) which seeks to describe the dynamics of transport without relying on speci7c local instabilities. Complex systems which self-organize, manifest states with no characteristic time (or length) scale, leading to scaling behavior in physical quantities related to the energy release of the system. Chang (1992) has proposed that the magnetosphere is such a self-organized system. Various researchers have demonstrated signatures of scale independence in the auroral electrojet index AE (Davis and Sugiura, 1966) time series. Tsurutani et al. (1990) showed a 1=f region in the power spectral density of AE. Takalo et al. (1993, 1994) showed self-aAnity in the AE time series. ∗ Corresponding author. Tel.: +1-907-474-6106; fax: +1-907474-6130. E-mail address: [email protected] (C.P. Price).
Takalo (1993), Takalo et al. (1999) and Consolini (1999) have demonstrated power law distributions in the lifetimes of substorm events. Consolini (1997, 1999) has shown a power law distribution in the energy of substorm events. However, since the magnetospheric system is not autonomous, but rather is strongly driven by the solar wind, those properties observed for the magnetosphere may simply be an echo of similar properties for the solar wind. Indeed, Freeman et al. (1999) have demonstrated that the distributions of burst lifetimes of the solar wind and of the AE index have power law components of similar form, and o:er arguments that the scale independence of the global output proxy AE are caused by the scale-free properties of the solar wind rather than by the magnetospheric dynamics. Noting that recent SOC models have demonstrated scale independence only in local (internal) output and not in global (external) output, they further argue that analysis of localized phenomena (re8ecting the internal output) should be pursued to determine whether the internal output of the magnetosphere has scale-free behavior. In this paper, we analyze the scaling properties of solar wind data, the AE index, and of individual auroral oval magnetometer station data. We 7nd evidence that suggests that there are two di:erent SOC processes in the solar wind, and that both global and local measures of the magnetospheric response have characteris-
c 2001 Elsevier Science Ltd. All rights reserved. 1364-6826/01/$ - see front matter PII: S 1 3 6 4 - 6 8 2 6 ( 0 0 ) 0 0 2 4 0 - 6
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tics consistent with the output of an SOC process. The scaling properties of AE and of the local dynamical signal are similar to the scaling properties of the solar wind speed v while the local station data has scaling properties more like those of the interplanetary magnetic 7eld (IMF) Bz component and the convective electric 7eld vBz . However, we cannot rule out the possibility that the geomagnetic output time series are only re8ecting the SOC state of the solar wind drivers. 2. The Hurst R=S statistic The rescaled range (R=S) analysis, pioneered by Mandelbrot and Wallis (1969) following Hurst (1951), can be used to demonstrate the scaling behavior associated with SOC
systems. That is, for sample size s, the ratio R=S of the sample sequential range R(s) to the square root of the sample sequential variance S 2 (s) varies as sH within the scaling region. A Hurst exponent H = 0:5 indicates a wholly stochastic process lacking correlation=coherence, while H = 1:0 indicates a very coherent process with perfect correlation, and H = 0:0 indicates a very coherent process with perfect anti-correlation. A Hurst exponent H between 0:5 and 1:0 indicates nonperiodic complex dynamics which are not inconsistent with SOC. This interpretation of the R=S analysis does not absolutely prove the presence of SOC: it is a strong (in a statistical sense) indication of the presence of SOC. The same range of H will be taken as consistent with SOC if the power spectrum of the time series shows a self-similar region of at least a decade in width with log–log slope − between 2:0 and 1:0. Note that for a time series which is a
Fig. 1. The time series, power spectral density and Hurst R=S plots for March 1979 IMF Bz component time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
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realization of fractional Brownian motion sequence, Voss (1990) has shown theoretically that one should have H = 1 + 12 (where = 2H − 1); in practice this relation is 2 only weakly satis7ed, and so cannot be used as a test for the presence of a fractional Brownian motion sequence. 3. Time series We will analyze a number of data sets, each with 1 min resolution, taken from the period 1978–1979. The behavior of the solar wind is represented by solar wind data from ISEE-3 (Frandsen et al., 1978; Bame et al., 1978), which measured the vector IMF and the solar wind speed, for the months from September 1978 to July 1979. The global geomagnetic response is represented by the auroral electrojet index AE (Davis and Sugiura, 1966), for the entire period 1978–1979. The geomagnetic response at a 7xed location is represented by the horizontal (‘H ’) components measured at the 11 auroral oval stations (‘station data’), and the short-term dynamical signals (‘signal data’, which are made from the station data by subtracting monthly quiet time averages), for the months from September 1978 to July 1979. (Note that AE is a nonlinear composition of the full set of auroral oval signals. As a result of the method by which AE is constructed, the AE time series may or may not share similar properties to any of the individual signals.) With the exception of the AE time series, these time series are not complete. There are 480; 960 possible data points for the months from September 1978 to July 1979; typically the coverage for the solar wind data and for the station data is in the range 80 –85%. In order not to introduce biases due to uneven station coverage, all stations are presumed to have missing
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data if the data is missing from any station. The data gaps are ignored: the sections of extant data are simply concatenated together. Tests on a set of surrogate time series, each composed from fractional Gaussian motion of a di:erent Hurst exponent into which random gaps eliminating 15% of the data are introduced, show that both the power spectrum analysis and the R=S analysis are robust when dealing with data set gaps in this way. 4. Analysis of solar wind data Fig. 1 shows the time series, the power spectrum and the R=S analysis for the IMF Bz component for the month of March 1979 (which is representative of the other 10 months of data for Bz ). The power spectral density plot shows a log–log 7t for frequencies f ¿ 1 day−1 . (Those frequencies greater than 0:001 min−1 have been binned, using 20 bins per decade.) If the power spectral density obeys a power law, |X˜ (f)|2 ˙ f− , then the log–log 7t yields the power spectral density index . The R=S plot shows a log–log 7t for s ¡ 1000 s (slope H¡ ) and a separate 7t for s ¿ 1000 s (slope H¿ ). We see for the IMF Bz component for the month of March 1979 that ≈ 1:5 and H¡ ≈ 0:95 and H¿ ≈ 0:65. The March 1979 Bz time series is self-similar for all time scales, highly coherent for time scales less than one day, and only slightly coherent for time scales greater than one day. However, we cannot rule out the possibility that self-scaling disappears for s greater than one month. Fig. 2 shows the Hurst exponents H¡ and H¿ versus month and the power spectral density index versus month, for the IMF Bz data. The Hurst exponent plot also shows
Fig. 2. The Hurst exponents H¡ and H¿ (see text) and power spectral indices for the monthly IMF Bz component time series versus month. Included are the exponents and indices for stochastic surrogate data sets.
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Fig. 3. The power spectral density and Hurst R=S plots for the concatenated IMF Bz component time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
Fig. 4. The power spectral density and Hurst R=S plots for the concatenated solar wind speed v time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
the exponents that are calculated from a surrogate data set composed by a random reshuOe of the original data set. For a random (stochastic) set, we expect H ≈ 0:5. Thus, the di:erence in values of the Hurst exponents for the surrogate data sets from 0.5 gives a measure of the uncertainty in the values of the Hurst exponents. The power spectral index plot also shows the indices that are calculated from the same surrogate data set. For a random set, we expect ≈ 0, and so the di:erence in values of the indices for the surrogate data sets from 0 gives a measure of the uncertainty in the values of the indices. We see no clear trend for ei-
ther the Hurst exponents or the power spectral indices as a function of the season, although there is a fair amount of variability in the long span Hurst exponent H¿ . The lack of an annual trend is not unexpected: the IMF measured upstream from the Earth should be ignorant of the Earth’s cycles. The lack of obvious trending in the monthly Bz results, and the apparent self-similarity of Bz for longer times, prompts us to analyze a concatenation of the set of month-long data sets. Fig. 3 shows the power spectrum and the R=S analysis for the concatenated IMF Bz component time series. We see that
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Fig. 5. The time series, power spectral density and Hurst R=S plots for the month of March 1979 auroral electrojet index AE. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
H¡ ≈ 0:95 and H¿ ≈ 0:65: the concatenated Bz time series has the same self-scaling properties as the monthly Bz time series. On the power spectrum plot, we have power-law 7ts for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The shorter period ≈ 1:3, is not appreciably di:erent from the power spectral index for the monthly Bz time series. The longer period power spectral index is nearly zero, as would be expected for a random (stochastic) time series. This reinforces the conclusions given for the month of March 1979 Bz time series: the Bz time series is self-similar for all time scales, highly coherent for time scales less than one day, and only slightly coherent for time scales greater than one day. We note that this time scale is only coincidentally that of the Earth’s rotational period; it could be related to either the solar wind transport time scale (structure size over solar wind speed) or to the turbulent decorrelation time scale of
turbulence in Bz . These results indicate that the IMF component Bz is the result of a SOC process, or more precisely, one may say that the results are not inconsistent with Bz resulting from a SOC process. We have also analyzed the monthly time series for the solar wind speed v and for the product of the solar wind speed v and the IMF Bz component (the convective electric 7eld). Both sets of monthly time series have no annual trends for either the Hurst exponents or the power spectral indices. For the monthly solar wind speed v time series, we 7nd a power spectral index in the range [1:9; 2:0] and Hurst exponents H¡ and H¿ ≈ 1. For the monthly convective electric 7eld vBz time series, we 7nd power spectral indices and the Hurst exponents are nearly identical to those for the corresponding monthly Bz time series, e.g., in the range [1:3; 1:6], H¡ ≈ 1 and H¿ in the range [0:6; 0:8].
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Fig. 6. The time series, power spectral density and Hurst R=S plots for the concatenated auroral electrojet index AE time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
The lack of trending in the monthly results leads us to analyze the concatenations of the sets of monthly time series. Fig. 4 shows the power spectrum and R=S analysis for the concatenated solar wind speed v time series. (The results for the concatenated convective electric 7eld vBz time series are nearly identical to the results for the concatenated IMF component Bz time series.) We see that H¡ ≈ 0:95 and H¿ ≈ 0:85: the concatenated v time series is both self-similar and highly coherent for time scales from several minutes to several months. The shorter period power spectral index ≈ 1:4, comparable to the values found for the monthly v time series. The longer period power spectral index ≈ 0:75: the solar wind speed has a nearly 1=f region for periods between 2 days and 2 months. All of these features are consistent with dynamics exhibited by SOC systems. We have found that, although the appearance of the time series of the di:erent solar wind quantities is di:erent, those concatenated time series have similar short time scale self-similarity and coherence properties: the Hurst exponent H is approximately 1 for time spans less than about one day. The main di:erence in the R=S analyses is that the solar wind speed v time series is highly coherent for very long time scales (H¿ ≈ 0:85), while the IMF Bz component time series and the convective electric 7eld vBz time series are only slightly coherent for time scales greater than one day (H¿ ≈ 0:65). The time series of the three solar wind quantities also have similar high-frequency (f ¿ 1 day−1 ) power spectra, power spectral index ∈ [1:5; 2:0]. For lower frequencies (0:5 month−1 ¡ f ¡ 0:5 day−1 ), the concatenated Bz and vBz time series have 8at power spectra while concatenated v time series has a 1=f power spectrum. These results indicate that the solar wind could be the result of a SOC system; that is, these results are not inconsistent
with the solar wind state being the result of one or more SOC processes. 5. Analysis of auroral electrojet index AE data Fig. 5 shows the time series, the power spectrum and the R=S analysis for the auroral electrojet index AE for the month of March 1979 (which is a representative of the other 23 months of data). We see that the AE time series is self-similar. For s ¡ 1000, we 7nd the exponent H¡ ≈ 0:95; the AE time series is remarkably coherent for timescales less than a day. For s ¿ 1000, we 7nd H¿ ≈ 0:80, still evidence for a coherent series. The AE time series also has a power law power spectrum, with ≈ 1:9. We also analyze a concatenation of all 24 of the month-long AE data sets. Fig. 6 shows the power spectrum and R=S plots for the two year sequence of AE data. We see that the AE time series is self-similar and very coherent for spans up to one day in length, and self-similar if less coherent for spans of several months, and possibly spans up to a year: H¡ ≈ 0:95 and H¿ ≈ 0:75. The power spectrum has broad peaks at f approximately 1 year −1 (2:0 × 10−6 min−1 ) and approximately 1 month−1 (2:3 × 10−5 min−1 ), and a narrow but statistically signi7cant peak at f = 1 day−1 . For higher frequencies (f ¿ 1 day−1 ), the power spectrum exponent is ≈ 1:85; for intermediate frequencies (1 day−1 ¿ f ¿ 1 month−1 ), we have ≈ 0:80. The three peaks can be understood in terms of the annual and diurnal motions of the geomagnetic pole, and the monthly nature of the quiet-time baseline signal which underlies the electrojet indices. The AE results are more similar to those of the solar wind speed v than they are to those of either Bz or vBz , especially in the appearance of the 1=f region for time scales between several days and several months.
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Fig. 7. The time series, power spectral density and Hurst R=S plots for the month of March 1979 College station H component. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
6. Analysis of ground station data Fig. 7 shows the time series, the power spectrum and the R=S analysis for the horizontal (H ) component of the magnetic 7eld as measured at the College station for the month of March 1979 (which is fairly representative of the other 10 months of data for this station, and for the data from the other stations). Those plots show that the College station H component time series is self-similar and very coherent for time spans s up to about 1 day, with the Hurst exponent H¡ ≈ 0:95. For greater time spans, the Hurst exponent H¿ ≈ 0:55, which would indicate that for time spans greater than 1 day, the time series is stochastic or slightly coherent. The College station time-series power spectral density has a power law behavior of f ¿ 1 day−1 , with spectral index ≈ 2, and a statistically signi7cant peak at f = 1 day−1 . These results imply strongly that the ground station time
series is the result of stochastic process with a periodicity of about one day. The results for the other months show no clear trend as a function of the season, which is somewhat surprising since one would expect some modulation on an annual scale. Since there is no strong dependence on season, we have analyzed a concatenation of all 11 of the monthly College station data sets. Fig. 8 shows the power spectrum and R=S plots for the concatenation of the College station H component data. We see that the concatenated station data time series has the usual strong peak in the power spectrum at f = 1 day−1 , and that the power spectral density falls o: as f−2 for frequencies greater than 1 day−1 , with a slight de7cit of power for f ¿ 0:1 min−1 . In addition, it has a strong peak in the power spectrum in the vicinity of f = 1 month−1 , and the power spectrum is relatively 8at for 1=f between 2 days and 2 months. The R=S analysis shows
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Fig. 8. The power spectral density and Hurst R=S plots for the concatenated College station H component time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
that the concatenated station data time series is also very coherent on time scales less than one day (H¡ ≈ 0:95), and is also less coherent but still self-similar for longer time scales (H¿ ≈ 0:65). So, the analysis suggests that the concatenated station data time series is the result of a noisy process with periodicities of about one day and about one month. The diurnal component can be understood from the daily motion of the station relative to the geomagnetic pole. Presumably, the monthly component results in some way from the (27-day period) solar rotation. The ground station results are more similar to the results for Bz or vBz than they are to those of either AE or the solar wind speed v. 7. Analysis of ground station signals Fig. 9 shows the time series, the power spectrum and the R=S analysis for the H component dynamical signal at the College station for the month of March 1979 (which is fairly representative of the other 10 months of data for this station, and for the signal data from the other stations). Those plots show that the dynamical signal extracted from the College station H component time series is self-similar and very coherent for time spans s up to about one day, with the Hurst exponent H¡ ≈ 0:95. For greater time spans, the Hurst exponent H¿ ≈ 0:45, which would indicate that for time spans greater than one day, the time series is stochastic or slightly anti-coherent. The College station dynamical signal time series also has a high-frequency power-law power spectral density, with = 1:9. (Close inspection shows that there may well be two scaling regions in the power spectral density. For f ¿ 1=100 min, we have ≈ 2:0, and for periods between 100 min and one-half day we have ≈ 0:5. However, there is no corresponding break in R=S at a period
of 100 min.) There is also a statistically signi7cant peak in the power spectral density at f = 1 day−1 . These results imply strongly that the ground station dynamical signal time series is the result of stochastic process with a periodicity of about one day. As with the ground station data, there is no clear trend as a function of the season, where one might expect some modulation on an annual scale. Given the absence of a yearly trend in the monthly dynamical signal time series, we have analyzed a concatenation of all 11 months of the various station dynamical signal time series. In Fig. 10, we display the power spectrum and R=S plots for the concatenation of College station dynamical signals. The R=S analysis shows that the concatenated dynamical signal time series is also very coherent on time scales less than one day (H¡ ≈ 0:95), but is only slightly coherent (but still approximately self-similar) for longer time scales (H¿ ≈ 0:65). We see that the concatenated dynamical signal power spectral density index for frequencies greater than 1 day−1 is ≈ 1:75; there is an excess of power for f ¿ 0:1 min−1 . The concatenated dynamical signal time series has the usual peak in the power spectrum at f=1 day−1 , which can be understood from the daily motion of the station relative to the geomagnetic pole. There is also a power-law region for periods of several days to several months with an index of roughly = 1. There is no signi7cant peak in the power spectrum in the vicinity of f = 1 month−1 , although we might presumably expect a monthly component as a result of the monthly nature of the quiet-time baseline signal which enters the construction of the dynamical signal. We are unable to advance an explanation for the scaling region for periods of several days to several months, except to say that feature, as well as the others cited, are consistent with dynamics exhibited by SOC systems. The dynamical signal results are more similar to the results for AE and the solar
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Fig. 9. The time series, power spectral density and Hurst R=S plots for the month of March 1979 College station dynamical signal. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
wind speed v than they are to those of the station data or the IMF component Bz or the convective electric 7eld vBz . 8. Conclusions Based on the Hurst R=S analysis and associated power spectral density calculations, we have shown that the solar wind data are consistent with SOC dynamics, in agreement with previous work. In fact, our results suggest that there may be two di:erent SOC states, one involving the IMF and the other involving the solar wind. The solar wind state appears to be fully developed (continuous transport at all the scales which we have analyzed) while the IMF seems to be only partially developed (reduced organization for larger scales). Given that there is strong turbulent transport in the solar wind, the former result might have been anticipated,
but we advance no hypothesis about the IMF state. The turbulent state of the solar wind driver complicates the analysis of the dynamical nature of the global geomagnetic system, since that systemic response to the solar wind driver will re8ect some of the characteristics of the driver. The auroral electrojet index AE is a commonly used global measure of magnetospheric activity. The Hurst R=S analysis and associated power spectral density calculations for AE are not inconsistent with the results of a SOC process. However, since the drivers (solar wind and IMF) have SOC characteristics, we cannot rule out the possibility that AE is only re8ecting the SOC characteristics of the drivers, as Freeman et al. (1999) have pointed out. In fact, the scaling properties of AE are similar to those of the solar wind speed v. The similarity possibly indicates that the global magnetospheric output is mostly a passive response (with a 7lter function) to the solar wind; this would follow if the system
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Fig. 10. The power spectral density and Hurst R=S plots for the concatenated College station dynamical signal time series. The power spectral density plot shows the power-law 7t for f ¿ 1 day−1 and for 0:5 month−1 ¡ f ¡ 0:5 day−1 . The Hurst R=S plot shows the power-law 7ts for s ¡ 1 day and for s ¿ 1 day.
is mostly being swamped by the driver, so that only a small fraction of the global system response is due to internal SOC dynamics. Freeman et al. (1999) have argued that local measures of geomagnetic activity will re8ect internal output. Thus analysis of such local measures might give insight into magnetospheric SOC processes. We have analyzed the H component magnetometer data from the 11 auroral oval stations. These time series are local in space and in time (point measurements). We 7nd that the station data time series are not inconsistent with the results of a SOC process. The station data Hurst exponents and spectral indices are more like those of Bz and vBz than they are like those of AE and v. The di:erence between the AE results and the station data results could be just the di:erence between global output and internal output. Previous results (Takalo et al., 1993, 1995; Takalo and Timonen, 1994) have found lower values for the scaling exponent of AE (from self-aAnity analyses). From basic considerations, those values should be comparable. Takalo has argued (Takalo et al., 1994) that AE and the other response series (the ground station data and extracted signals) are the result of cumulative driving, as indicated by the strong autocorrelation in those series, and as a result, he has examined the self-aAnity of the di4erences (increments) in the AE and related series. When we calculate the Hurst index for the di:erences in the AE and related series, we get values which are consistent with those of Takalo and co-workers, that is, about half the values of which we are reporting for AE. However, given that a priori one does not know if an arbitrary series is incremental or cumulative, it is most prudent to treat each as if they were cumulative, which has been our approach here.
The dynamical signal from each of the stations was also analyzed. These time series are local in space but nonlocal in time, since they are composed of the di:erence between the station data and the average quiet time pattern for that station. We 7nd that the signal data time are not inconsistent with the results of a SOC process. The signal data Hurst exponents and spectral indices are more like those of AE and v than they are like those of the station data and Bz and vBz . This result probably re8ects the composition of AE from the envelope of the dynamical signals of the 11 auroral oval stations. We have found evidence for SOC dynamics in the solar wind, in the IMF, in the global measure of geomagnetic response, and in local measures of the geomagnetic response. The analysis is most informative when large, extended (several months or greater) data sets are used. However, the analysis of the geomagnetic response data do not point to an unambiguous SOC process in the magnetosphere, and may well simply re8ect the SOC states of the solar wind and IMF drivers. Based on our analysis, our best estimate is that the magnetosphere acts as a dynamically self-organized system, and the system’s output is mostly a passive response to an overly strong driver with hints of the internal magnetospheric SOC processes. Acknowledgements We thank T. Chang, N. Watkins, S. Chapman, J. Takalo and the referees for useful discussions. We thank NSSDC for assistance in acquiring the ISEE-3 data, and acknowledge S.J. Bame (plasma) and E.J. Smith (magnetometer) as Principal Investigators on those ISEE-3 instruments.
C.P. Price, D.E. Newman / Journal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1387–1397
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