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arXiv:physics/0412063v1 [physics.space-ph] 10 Dec 2004

Scaling and a Fokker-Planck model for ?uctuations in geomagnetic indices and comparison with solar wind ? as seen by WIND and ACE

B. Hnat, S.C. Chapman, G. Rowlands Space and Astrophysics Group, University of Warwick Coventry, CV4 7AJ, UK February 2, 2008

Abstract The evolution of magnetospheric indices on temporal scales shorter than that of substorms is characterized by bursty, intermittent events that may arise from turbulence intrinsic to the magnetosphere or may re?ect solar wind-magnetosphere coupling. This leads to a generic problem of distinguishing between the features of the system and those of the driver. We quantify scaling properties of short term (up to few hours) ?uctuations in the geomagnetic indices AL and AU during solar minimum and maximum along with the parameter ? that is a measure of the solar wind driver. We ?nd that self-similar statistics provide a good approximation for the observed scaling properties of ?uctuations in the geomagnetic indices, regardless of the solar activity level, and in the ? parameter at solar maximum. This self-similarity persists for ?uctuations on time scales at least up to about 1 ? 2 hours. The scaling exponent of AU index ?uctuations show dependence on the solar cycle and the trend follows that found in the scaling of ?uctuations in ?. The values of their corresponding scaling exponents, however, are always distinct. Fluctuations in the AL index are insensitive to the solar cycle as well as being distinct from those in the ? parameter. This approximate self-similar scaling leads to a Fokker-Planck model which, we show, captures the probability density function of ?uctuations and provides a stochastic dynamical equation (Langevin equation) for time series of the geomagnetic indices.

1

Introduction

The Earth’s magnetosphere can be considered as non-linear, dissipative system which is driven by the time varying solar wind. Accumulated energy is ultimately dissipated, at least in part, through a system of currents generated in the auroral zones of the ionosphere. These currents produce small changes in the terrestrial magnetic ?eld which can be used to monitor magnetospheric activity. The complex behavior of the magnetosphere, as suggested by many observations (see, for example, [Horton et al.(1999), Lewis(1991), Sitnov et al.(2000), Takalo et al.(2000), Vassiliadis et al.(2000), V¨r¨s et al.(2002)]), could o o then be attributed either to intrinsic magnetospheric processes, the complex nature of its coupling with the solar wind and the ionosphere or both. Recent observations suggest that the multi-scale nature of this coupling is a fundamental aspect of magnetospheric dynamics (see, for example [Chang(1992), Chapman and Watkins(2001), Klimas et al.(1996), Ukhorskiy et al.(2003), Vassiliadis et al.(2003), Weigel et al.(2003)]). Evidence is provided by a variety of observations which exhibit statistical properties previously identi?ed as hallmarks of multi-scale systems. For example, bursty transport events have been reported in the magnetotail[Angelopoulos et al.(1992)] and their auroral signatures suggest self-similar statistics [Lui et al.(2000), Uritsky et al.(2001)]. The ?uctua1

1.2 1 0.8 ζ(m) 0.6 0.4 0.2 0 0 1 2 log (S )

m

20 15 10 5 6 3 Moment m log (τ [s])

2

2

16 5 6

4

Figure 1: Exponents of unconditioned generalized structure functions as a function of order m for ?uctuations in the ?(?), AU (?) and AL(△) index during solar maximum. The inset shows structure functions Sm of orders m = 1 ? 4 for the ? parameter. tions in the ground based measurements of the magnetic ?eld are non-Gaussian and also exhibit scaling [Consolini et al.(1996), Kov?cs et al.(2001), V¨r¨s et al.(1998)]. In the context of time series analysis, gea o o omagnetic indices are of particular interest as they provide a global measure of magnetospheric output and are evenly sampled over a long time interval. These indices also show non-Gaussian statistics of ?uctuations and anomalous scaling over the short time scales of up to few hours [Consolini and De Michelis(1998), Takalo et al.(1993), Hnat et al.(2003a), Stepanova et al.(2003), Tsurutani et al.(1990)]. The extent to which observed statistical features of the geomagnetic indices arise directly from those of the solar wind driver or the auroral currents is of fundamental interest. This is an example of the generic problem of distinguishing between features intrinsic to a driven system and those in the driver, when both show scaling. Some recent studies has focused on direct comparison of scaling properties of the driver with these found in the geomagnetic indices[Freeman et al.(2000), Uritsky et al.(2001), Hnat et al.(2003a)] to establish whether, to the lowest order, they are directly related. The di?culty with interpreting these observations arises from the fact that statistical features described above can be recovered from many existing models. Self-Organized Criticality (SOC) and turbulence have both been extensively used [Angelopoulos et al.(1999), Consolini and De Michelis(1998)], [Kozelov and Kozelova(2003), Uritsky and Pudovkin(1998)] in the past. Practically, one needs to obtain experimental constrains with which di?erent models with similar characteristics can be tested. In this paper we present one possible approach to characterizing the time series in the context of scaling that does not rely on a speci?c model of multi-scale systems[Sornette(2000), Hnat et al.(2003a)]. The aim is to obtain statistical scaling properties directly from the data. Here, we will examine the statistical properties of Akasofu’s ?[Perreault and Akasofu(1978)] parameter, which represents the energy input from the solar wind into the magnetosphere, and that of magnetospheric response as seen by the geomagnetic indices. Previously, scaling has been quanti?ed over a 10 year data set for the indices and a comparison between δ? and the indices included, but was not restricted to, the solar minimum (1984 ? 1987)[Hnat et al.(2003a)]. Here, we will perform this comparison over intervals of solar 2

30 25 20 log2(Sm) 15 10 5 6 8 10 12 log2(τ [s]) 14 16

Figure 2: Structure functions Sm of orders m = 1? 6 for ?uctuations in the ? parameter at solar maximum. minimum and maximum separately. The statistical description of the experimental data will be extended to 10 standard deviations of the ?uctuations. To facilitate the comparison of all considered quantities we will ?rst explore to what extend their ?uctuations exhibit approximate self-similar scaling for temporal scales of 1 ? 2 hours. The quality of this self-similar approximation combined with values of the scaling exponents obtained at the solar minimum and maximum can be used to characterize each quantity. We will see that values of scaling exponents on these temporal scales for the geomagnetic indices are di?erent from these found in the solar wind ? regardless of the phase of the solar cycle. Remarkably, the scaling exponent of the AL index is unchanged between solar minimum and maximum whereas the AU scaling exponent changes with the solar cycle. In this respect, the AU index seems to follow the trend found in the driver, ? i.e., the value of scaling exponent increases with increasing solar activity. We then construct a Fokker-Planck model for ?uctuations in the geomagnetic indices and the ? at solar maximum as these exhibit the most satisfactory self-similar scaling. This allows us to obtain analytically a functional form of the ?uctuation probability density function (PDF) which we can then check against the data. A stochastic dynamical model can then be formulated by considering the most general form of the Langevin equation and deriving functional forms of the coe?cients that are consistent with the Fokker-Planck equation (see, for example [Hnat et al.(2003b)]).

2

2.1

Data and Methods

Data sets

To facilitate this analysis we used multiple data sets that spanned over di?erent phases of the solar cycle. Two year intervals of data were selected centered on solar minimum and solar maximum. The solar wind data were obtained from WIND and ACE spacecraft observations. These were collected in the vicinity of the L1 point approximately 1 AU (Astronomical Units) from the Sun. The periods of coverage, ?nal sampling frequencies and number of samples are given in the Table 1. The geomagnetic indices and the corresponding spacecraft data sets are not contiguous. The calibrated geomagnetic data set, from which 3

30 25 log (S ) 20 15 10 5 8 10 12 log2(τ [s]) 14 16

m 2

Figure 3: Same as ?gure 2 for the AU index. intervals of interest has been selected, spans from January 1978 to December 1988 inclusive, while the spacecraft data are available starting from 1995 for WIND and 1998 for ACE. This available data coverage does not permit examination of successive solar cycles. We thus need to assume that the statistical properties of ?uctuations are invariant from one solar cycle to the next. In the case of the spacecraft data, these include slow and fast solar wind streams. The solar wind velocity measurements, provided by the SWE instrument on board of WIND and ACE spacecraft, have varying temporal resolutions. In the case of WIND this resolution is in the range of 75 ? 98 seconds while for the ACE spacecraft it changes between 60 ? 120 seconds. The magnetometer data sets, on the other hand, have ?xed temporal resolution of 46 seconds for WIND MFI instrument and 16 seconds in the case of the ACE magnetometer. The SWE data sets have been then re-sampled using linear interpolation to give uniform resolution of 92 seconds for WIND (twice the magnetometer resolution) and 64 seconds for the ACE spacecraft (four times magnetometer resolution). No other post processing, such as detrending or smoothing, was applied to data. The ? parameter is de?ned[Perreault and Akasofu(1978)] 2 in SI units (Watts) as ? = v(B 2 /?0 )l0 sin4 (Θ/2), where l0 ≈ 7RE and Θ = arctan(|By |/Bz ), and was calculated from WIND and ACE spacecraft key parameter databases. All techniques discussed here are based on di?erencing of the original time series over a range of temporal scales τ . This method is often used in turbulence studies to compare the properties of ?uctuations on di?erent spatio-temporal scales (see, for example, [Frisch(1995)]). For a given time series x(t) a set of di?erences δx(t, τ ) = x(t + τ ) ? x(t) will then capture ?uctuations on temporal scale τ . Here, we will examine the statistical properties of the PDF of ?uctuations δx(t, τ ). The τ parameter will be given in power law form such as τ = δtAU (1.2)n seconds, where δtAU is a sampling time of the AU time series (here, 1 minute) and n ≥ 1 is an integer. This choice of τ gives a uniform distance between points when plotted on the logarithmic scale while the small base of the power law (1.2) assures that the adequate number of temporal scales are explored. We stress that the di?erencing is performed only if both x(t + τ ) and x(t) exist and are separated by time interval τ .

4

30 25 20 15 10 5 8 10 12 log2(τ [s]) 14 16

log2(Sm)

Figure 4: Same as ?gure 2 for the AL index.

2.2

Statistical Methods

Generalized structure functions (GSF) Sm are widely used to characterize non-Gaussian processes [Frisch(1995), Hnat et al.(2003a)]. These functions can be de?ned for ?uctuations δx(t, τ ) as Sm (τ ) ≡ |δx|m , where m can be any real number, not necessarily positive. If Sm exhibits scaling with respect to the time lag τ we also have Sm ∝ τ ζ(m) . In this case a log-log plot of Sm versus τ should reveal a straight line for each m and the gradients correspond to values of ζ(m). Generally, ζ(m) can be a non-linear function of order m, however if ζ(m) = αm (α constant) then the time series is self-similar (or more precisely, self-a?ne) with single scaling exponent α. This special case leads immediately to a Fokker-Planck description [Hnat et al.(2003b)]. The di?culty with computing GSF for higher orders, say, m > 4 arises from the slow convergence of this method and its sensitivity to large statistical errors in extremal events in the tails of the distribution. These e?ects can, as we shall see, lead to large errors in the estimation of ζ(m) (see also [Horbury and Balogh(1997)] for the discussion of error estimation for structure functions). One possible approach is to eliminate these extreme events from the ?uctuation time series δx(t, τ ) in a way that is consistent with the growth of the self-similar ?uctuations’ range on each temporal scale. This process is referred to as conditioning. Previously, a similar technique based on the wavelet ?lters has been used to separate the intermittent parts of the signal from the homogeneous noise in the AE index data[Kov?cs et al.(2001)]. We will condition our GSFs by imposing a threshold A a on the ?uctuation size[Hnat et al.(2003a)]. The threshold will be based on the standard deviation of the di?erenced time series for a given τ , A(τ ) = 10σ(τ ). Under conditioning, the GSF can be expressed in term of the ?uctuation PDF as: < |δx|m >=

A ?A

|δx|m P (δx, τ )d(δx).

(1)

This procedure is then consistent with scaling ζ(m) = mα if it is present in the data, but for threshold A su?ciently large it does not enforce it on the data. The PDF rescaling technique is a generic and model independent method of testing for statistical selfsimilarity in the data set. If the data is self-similar, then a single argument representation of the ?uctuation 5

2.5 2 1.5 1 0.5 0 0 1 2 3 Moment m 4 5 6

Figure 5: Exponents of conditioned generalized structure functions as a function of order m for ?uctuations in the ?(?), AU (?) and AL(△) index during the solar maximum.

ζ(m)

log (S )

25

20

m 2

15

10

5

6

8

10

12 log (τ [s])

2

14

16

Figure 6: Structure functions Sm of orders m = 1 ? 6 for ?uctuations in the ? parameter at solar minimum.

6

30

25

20 log2(Sm)

15

10

5 8 10 12 log2(τ [s]) 14 16

Figure 7: Same as ?gure 6 for the AU index. PDF, Ps (δxs ), can be found in the form: P (δx, τ ) = τ ?α Ps (δxτ ?α ), (2)

where α is the rescaling exponent. Substituting the rescaled quantities Ps and δxs = δxτ ?α into the GSF de?nition given by (1) we obtain: < |δx|m >= τ mα

As ?As

|δxs |m P (δxs )d(δxs ) ∝ τ ζ(m) ,

(3)

where the integral now has no explicit dependence on temporal scale τ . This then immediately relates the PDF rescaling to self-similar scaling of the GSF with ζ(m) = mα. In this approach PDFs are generated using non-overlapping intervals of the original data, i.e., δx(t, τ ) = x[mτ ] ? x[(m ? 1)τ ]. The method assures that ?uctuations are not temporally correlated–an important assumption for a Fokker-Planck model we will consider later. These two methods are thus complementary as one provides a scaling exponent while the other gives an underlying probability distribution of ?uctuations.

3

3.1

Results and discussion

GSF analysis

We will ?rst present scaling properties of the GSFs for the indices and the ? parameter during solar minimum and solar maximum. To illustrate the e?ect of conditioning, we ?rst show, in the inset of ?gure 1, a log-log plot of structure functions Sm obtained for ?uctuations in the raw time series of the AU index at solar maximum for orders 1 ≤ m ≤ 6. We see that, for orders m > 3 there is no clear evidence of scaling–the points do not lie on straight lines. Similar lack of scaling was also found for ?uctuations in the AL index and the ? parameter. The main panel of ?gure 1 shows exponents ζ(m) obtained by performing linear ?ts to logarithms of moments log[Sm (log(τ ))]. We see that the curves ζ(m) are not monotonic functions of 7

30 25 20 15 10 5 8 10 12 log2(τ [s]) 14 16

log2(Sm)

Figure 8: Same as ?gure 6 for the AL index. m, excluding the possibility of multi-fractal scaling. We now condition this data set as discussed above, to check if true scaling properties are not obscured by poor statistics of extreme and very rare events. Figures 2-4 show a log-log plot of structure functions Sm for the δ?, δ(AU ) and δ(AL) indices at solar maximum and for order m from 1 to 6. The main indication of successfully recovered scaling after the conditioning process is the quality of the linear ?t to log[Sm (τ )] versus log(τ ). We clearly recover a family of straight lines with slopes ζ(m), up to order m = 6 for ?uctuations in AU and AL indices and in ?. This scaling extends up to temporal scales of ? 1 to ? 2 hours in good agreement with these reported earlier[Takalo et al.(1993), Tsurutani et al.(1990), Hnat et al.(2003a)]. Figure 5 shows that ?uctuations in all quantities, at solar maximum, exhibit approximate self-similar scaling to within statistical errors. The size of these error bars combined with the convex shape of the function ζ(m) for the AL index also allows a weakly multi-fractal interpretation of the scaling. We stress, however, that the error bars in ?gure 5 do not include many other uncertainties (not statistical) that are di?cult to estimate. For example, the WIND spacecraft magnetometer data has absolute accuracy of about 0.1nT and the indices data have integer values (also in units of nT). Such discreteness in the time series may lead to erroneous estimates of low order moments while the ?nite size of the data could alter true scaling of the high order moments. Independent of any given choice of a model for the functional form of ζ(m) we can perform a direct comparison between the ζ(m) measured for the di?erent quantities at solar minimum and maximum. In order to develop a Fokker-Planck approach we will then make a further step and assume that a reasonable approximation is given by ζ(m) = mα, that is, self-similar scaling. Figures 6-8 show structure functions Sm for all quantities at solar minimum and with order m varying again from 1 to 6. We see that moments of ?uctuations for the geomagnetic indices show satisfactory scaling up to temporal scales of 1 ? 2 hours. In the case of ? at solar minimum there is a departure from a single set of scaling exponents ζ(m) for the smallest time scales τ < 12 minutes. To facilitate a comparison with conditions at solar maximum and with the indices we will ?t straight lines to obtain ζ(m) for τ = [12, 90] minutes bearing in mind that this does not capture the behavior of ?uctuations on the smallest time scales. This change in scaling properties for ? may re?ect di?erences between solar wind evolution at solar minimum and maximum related to physical properties of slow and fast wind components[Pagel and Balogh(2001)]. 8

2.5

2

ζ(m)

1.5

1

0.5

0 0 1 2 3 Moment m 4 5 6

Figure 9: Same format as ?gure 5 for data around solar minimum. Figure 9 is constructed identically to ?gure 5 and shows scaling exponents ζ(m) for solar minimum. To make a comparison between behavior at maximum and minimum we plot, in ?gure 10, exponents ζ(m) at solar minimum and maximum overplotted for AL, AU and ? respectively. Examining these ?gures we conclude that the scaling properties of the AL index ?uctuations are remarkably insensitive to the change in solar activity. The values of ζ(m) and the corresponding scaling exponents are the same, to within the statistical error for solar minimum and maximum. On the other hand the scaling exponents of ?uctuations in both ? and the AU index vary with the solar cycle. The scaling of δ(AU ) is distinct from these of δ(AL) and δ? but follows the trend of δ?. A possible interpretation of these observations is that the AL index ?uctuations more closely re?ect the internal dynamics of the magnetotail and are insulated from solar cycle related changes in the solar wind. In contrast, the AU index is more strongly coupled to solar cycle associated changes in the solar wind driver. This is consistent with our understanding of the global roles of these indices (eg., [Baumjohann and Treumann(1996)]). The ?uctuations in AU , however, have values of scaling exponents di?erent from that observed for the driver ? at solar minimum and maximum, which may suggest that (i) ? does not completely capture all relevant information about the driver, (ii) the indices do not fully capture the magnetospheric response or (iii) the di?erence re?ects the non-linear nature of the solar wind-magnetosphere coupling. If we compare the scaling exponents of δ(AU ) and δ? during solar minimum and maximum we see that both quantities follow a similar trend. The exponents have values closer to that of Brownian motion (0.5) during solar maximum as compared to minimum. Closer examination of scaling exponents for ?uctuations in the ? and the AU index reveals that the di?erence αAU ? α? ≈ 0.06 is almost identical for solar minimum and solar maximum period, to within the statistical error. This could indicate that the “conversion rate” of ?uctuations in the driver to those in the AU index is nearly constant and independent of the strength of the driver. All scaling exponents α derived by ?tting ζ(m) = mα are given in the Table 2 together with the approximate maximum temporal scale τmax for which self-similarity can be identi?ed in the di?erenced time series. These temporal scales have been derived using R2 goodness of ?t analysis for moment with m = 2. We have also veri?ed that GSF analysis of combined time series over solar minimum and maximum 9

2 ζ(m) 1 0

δ(AL) 2

δ(AU)

2

δε

1 1 0 0 2 4 6 0 2 4 Moment m 6 0 0 2 4 6

Figure 10: Comparison of functions ζ(m) during solar minimum and maximum for ?uctuations in all three quantities: (a)-AL index, (b)-AU index and (c)-?. recovers results presented in our previous work[Hnat et al.(2003a)].

3.2

Probability Density Function (PDF) Rescaling

We now present the results of the PDF rescaling analysis which allows us to compare directly the PDFs of the studied parameters at solar minimum and maximum. Due to the rather poor scaling of the higher moments for the ? ?uctuations at solar minimum we will not apply this rescaling to their PDFs. We simply state that our previous work suggested that PDFs of ?uctuations in the geomagnetic indices and that of the ? ?uctuations di?ered signi?cantly when considered time interval spanned more then a solar minimum [Hnat et al.(2003a)]. Figures 11 and 12 show rescaled PDFs of ?uctuations in the AU , AL index respectively for solar minimum (empty symbols) and solar maximum (?lled symbols) while ?gure 13 shows these PDFs for the ? parameter but only at solar maximum. These PDFs correspond to the function Ps (δxs ) in equation (2). Each plot shows overplotted Ps (δxs ) at four temporal scales, τ = 10, 16, 26 and 42 minutes. These ?gures show data up to 10 standard deviation on any given temporal scale–consistent with the conditioning procedure described above. All rescaling exponents α used to construct these plots, are taken directly from the GSF analysis. We ?nd that, when solar minimum and maximum data sets are taken separately, these PDFs collapse on a single curve after rescaling (2) is applied. The quality of the collapse for the PDFs was checked using the Smirnov-Kolmogorov[Press et al.(1988)] test and the signi?cance of the null hypothesis (both curves drawn from the same distribution) was always found to be above the 0.975 ± 0.05 level. The rescaling con?rms what we have already found by applying GSF analysis, in that a single exponent α is su?cient to give close correspondence of the curves. As we have shown in equation (3) this is consistent with approximate self-similar scaling ζ(m) = mα from the GSF analysis. Once rescaled, using the values of exponents obtained separately for solar minimum and maximum, we see that the curves are distinct and the di?erence is most clear for the AU index shown in ?gure 11. We also compared the functional form of these curves by applying normalization to their respective standard deviation on a given temporal scale, σs (τ ). We found that the normalized PDFs for maximum and minimum are indistinguishable within the errors for AU and AL. Similar results were reported for ground based measurements of the magnetic ?eld [Weigel and Baker(2003)] where authors also found that the statistics of ?uctuations, when normalized to the standard deviation, is not sensitive to changing solar wind conditions. 10

0

log10(Ps(δ (AU)s) [sα/nT])

?1

?2

?3

?4

?5 ?30

?20

?10

0 10 δ (AU) [nT/sα]

s

20

30

Figure 11: Rescaled PDFs of the δ(AU ) during solar minimum (empty symbols) and maximum (?lled symbols). Symbols correspond to temporal scales τ = 10, 16, 26 and 42 minutes. The dashed line represents a solution of the Fokker-Planck model (6) with parameters given in Table 3.

0

?1 log10(Ps(δ (AL)s) [s /nT])

α

?2

?3

?4

?5 ?50 0 δ (AL)s [nT/sα] 50

Figure 12: Same format as ?gure 11 for the AL index ?uctuations.

11

4

The Fokker-Planck Model

The Fokker-Planck (F-P) equation provides an important link between statistical properties of the system and the dynamical approach expressed by the Langevin equation[van Kampen(1992)]. The F-P approach can be readily applied if ?uctuations are self-similar and statistically independent (uncorrelated) [van Kampen(1992)]. The above analysis suggests that self-similar scaling is a reasonable approximation to the data. The independent nature of increments is enforced by considering non-overlapping intervals for di?erencing, as discussed in Section II B. In the most general form the F-P equation can be written as: ?P = ?δx (A(δx)P + B(δx)?δx P ), ?τ (4)

where P ≡ P (δx, τ ) is a PDF for the di?erenced quantity δx that varies with time τ and A(δx) and B(δx) are transport coe?cients which vary with δx. It can be shown that, under the assumption of power law scaling A(δx) ∝ δx1?1/α and B(δx) ∝ δx2?1/α , a class of self-similar solutions of (4) can be found that also satis?es the rescaling relation (2)[Hnat et al.(2003b)]. These assumptions combined with the use of rescaled variables δxs = δxτ α and Ps lead to the following equation:

1 α dPs b0 + Ps + (δxs ) α Ps = C, (δxs ) a0 d(δxs ) a0

(5)

where a0 , b0 , C are constants and α is the rescaling exponent derived, for example, from GSF analysis. The general solution of (5) is given by the sum of homogeneous and inhomogeneous solutions[Hnat et al.(2003b)]: Ps (δxs ) =

δxs

C a0 α2 exp ? (|δxs |)1/α b0 |δxs |a0 /b0 b0

α2 ′ 1/α b0 (δxs )

exp

×

0

(δx′ )1?a0 /b0 s

d(δx′ ) + k0 H(δxs ), s

(6)

where k0 is a constant and H(δxs ) is the homogeneous solution: H(δxs ) = 1 α2 exp ? (|δxs |)1/α . b0 |δxs |a0 /b0 (7)

The simple model described above assumes that self-similar scaling persists for all δx. This assumption is expected to hold for a physical system for a large but ?nite range of δx. In particular, it will break down as δxs → 0 giving a singularity in the solution Ps as δxs → 0.This singularity, however, is integrable so ∞ that ?∞ Ps d(δxs ) is ?nite. We have found that ?uctuations in the geomagnetic indices in solar minimum and maximum and these in ? at solar maximum exhibit self-similar statistics to a reasonable approximation. We will now show that the functional form of the PDF obtained from the F-P model (6) is a good approximation for the observed rescaled distribution Ps (δxs ) of ?uctuations shown in ?gures 11-13. On ?gures 11-13 we have overplotted solution (6), shown by thick dashed line, with α taken to be that obtained from the GSF analysis. Table 3 gives values of all parameters assumed for each of the plotted solutions. We see that all PDFs shown in ?gures 11-13 are well approximated by their F-P solutions. In the case of the geomagnetic indices some departures of the predicted curves from the observed distributions do occur and can be attributed to the asymmetry of these observed PDFs. 12

?8.5 ?9 log (P (δ ε )) [sα/Watt × 1015] ?9.5 ?10

?10.5 ?11

s

s

?11.5 ?12

10

?12.5 ?13 ?13.5 ?2 ?1 0 δ εs [Watt × 1015/sα] 1 2 10 x 10

Figure 13: Same format as ?gure 11 for the ? parameter ?uctuations at solar maximum. We note an obvious departure of our predicted curves from the measured PDF for the smallest ?uctuations, in all considered cases. This is due to the functional form of (7) where H(δxs ) → ∞ when δxs → 0 arising from the assumption that the self-similar scaling extends to arbitrary small ?uctuations. To model this part of the curve, we would need to include the scaling, or lack of thereof, introduced by the uncertainty in the measurements. We would expect such processes to be dominant for the smallest ?uctuations. For example, if we assume that the smallest ?uctuations are dominated by Normally distributed noise, then a di?usion model with a constant di?usion coe?cient D0 could, in principle, be used to tame this singular behavior. This stochastic approach can be extended to obtain the Langevin equation for the dynamics of the ?uctuations[Hnat et al.(2003b)]. The Langevin equation can be written in the most general form as: d(δx) = β(δx) + γ(δx)ξ(t), dt (8)

where the random variable ξ(t) is assumed to be δ-correlated. Equation 8 can be transform into purely additive noise form: dz β(z) = + ξ(t), (9) dt γ(z) where z = 0 1/γ(δx′ )d(δx′ ). It has been shown [Hnat et al.(2003b)] that one can obtain a functional form of coe?cients β(δx) and γ(δx) in terms of a0 , b0 (from equation 5) and the scaling exponent α. Such an equation provides a dynamical model for time series with the required statistical properties.

δx

5

Summary

The response of the Earth’s magnetosphere to the solar cycle and, by implication, a changing character of solar wind activity, illuminates the interplay between intrinsic magnetospheric dynamics and solar windmagnetosphere coupling. Statistical studies provide a simple and yet unifying way to quantify this behavior in the context of models for intermittency. In this paper we considered scaling properties of the solar wind 13

driver, quanti?ed by the ? parameter, and geomagnetic indices during solar minimum and maximum. We ?nd that: 1. Fluctuations in the geomagnetic indices show approximate statistical self-similarity for a range of temporal scales. Fluctuations in the ? at solar minimum show departure from scaling for τ <? 10 minutes. The self-similar scaling emerges as a reasonable approximation for ?uctuations δ? at solar maximum. Fluctuations in the geomagnetic indices exhibit self-similar scaling on temporal scales between ? 2 minutes to ? 1 ? 2 hours. The ?uctuations in ? scales from ? 2 minutes to ? 1.5 hours, but only at solar maximum. 2. Fluctuations in the AL index exhibit scaling properties insensitive to the phase of the solar cycle. 3. The scaling exponent of δ(AU ) changes with the solar cycle and the trend follows that of the ? parameter 4. The value of the scaling exponents of indices and that of the ? parameter di?er from each other at both solar minimum and maximum. This di?erence between scaling exponents of δ(AU ) and the driver δ? is approximately the same at solar minimum and maximum. 5. A Fokker-Planck approach can be used to model the ?uctuation PDF for the geomagnetic indices in both phases of the solar cycle and the ? at solar maximum to a good approximation The approximate statistical self-similarity found for the indices for solar minimum and maximum and the ? at solar maximum is consistent with complex multi-scale processes such as turbulence or Self-Organized Criticality (SOC). The distinct values found for scaling exponents may re?ect physical di?erences in the solar wind and the magnetosphere but may also be due to the very di?erent way in which these quantities are derived. The ?uctuations in the AU index depend on the solar cycle but the scaling exponent is distinct from that of ? ?uctuations. Interestingly, the di?erence between scaling exponents of δ(AU ) and the driver δ? appears to be approximately constant. These observations, when combined together, suggest that the process involved in generating ?uctuations in the AU index is coupled to the solar wind driver, as seen in the solar cycle dependence. In contrast to the AU index ?uctuations, these in the AL index are nearly insensitive to the change in solar cycle implying that the AL index is a measure of activity intrinsic to the magnetosphere. This is consistent with the AU index more closely monitoring activity on the day-side and AL re?ecting activity in the magnetotail. The self-similar scaling of ?uctuations allows us to model their statistics using a Fokker-Planck approach. We obtained analytically a functional form of the ?uctuation PDF which approximates the measured PDF rather well. We stress that such an approach links the statistical features discussed here to dynamical modeling of the time series via stochastic Langevin equations.

6

Acknowledgments

B. Hnat acknowledges support from the PPARC, S. C. Chapman from the Radcli?e Institute and G. Rowlands from the Leverhulme Trust. We thank R.P Lepping and K. Ogilvie for provision of data from the NASA WIND spacecraft, the ACE SWEPAM instrument team and the ACE Science Center for providing the ACE data and the WDC for the geomagnetic indices data.

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References

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Quantity AL, AU AL, AU ? ?

Table 1: Data sets description Solar cycle dt[s] Dates N[mln] Min 60 01/85 ? 12/86 1.05 Max 60 01/79 ? 12/80 1.05 Min 92 08/95 ? 07/97 0.63 Max 64 01/00 ? 12/01 0.68

Source WDC STP WDC STP WIND ACE

Table 2: Scaling indices Quantity Solar cycle δAU Min δAU Max δAL Min δAL Max δ? Min δ? Max

derived from GSF analysis α from GSF τmax [hr] ?0.35 ± 0.01 ?1 ?0.43 ± 0.01 ?1 ?0.39 ± 0.02 ?2 ?0.37 ± 0.03 ?2 ?0.26 ± 0.02 ?2 ?0.32 ± 0.02 ?2

Table 3: Values of parameters used for F-P model solutions plotted in ?gures 11-13. Quantity Solar cycle b0 b0 /a0 k0 C [×10?5 ] δAU Min 170 1.875 0.28 32.5 δAU Max 16 1.875 0.20 26.4 δAL Min 2200 1.8 0.36 7.77 δAL Max 1000 1.8 0.32 6.66 δ? Max 4 × 1012 2.15 5.3 × 1010 2.84 × 10?10

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