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0254-6124/2010/30(4)-356–06 Chin. J. Space Sci. Li Xiaoyu, Zhang Tielong. Statistics of the interplanetary magnetic fields observed at 0.72 AU and 1 AU. Chinese Journal of Space Science, 2010, 30(4): 356-361 Statistics of the Interplanetary Magnetic Fields Observed at 0.72 AU and 1 AU∗ Li Xiaoyu1,2 Zhang Tielong3 1(State Key Laboratory of Space Weather, Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing, 100190, China) 2(Graduate University of Chinese Academy of Sciences, China) 3(Space Research Institute, Austria Academy of Sciences, Graz, Austria) Abstract This paper presents the Interplanetary Magnetic Field (IMF) observations at 0.72 AU measured by Venus Express (VEX) and 1 AU by Advanced Composition Explorer (ACE) in 2007. The distributions of daily averages of B are lognormal in both locations. The multiscale structure of the magnetic field fluctuations was described by studying the increments of B over a range scales from 10 min to 21.3 hours. All the Probability Distribution Functions (PDFs) can be described quantitatively by Tsallis distribution function. On the ecliptic plane from 0.72 AU to 1 AU, the entropy index q increases with distance over all scales, indicating the intermittency of turbulence is growing. The widths of the PDFs at 0.72 AU are larger than those at 1 AU at all scales, which indicating the turbulence at 0.72 AU is more intense than that at 1 AU. This helps us understand the nature and development of the magnetic field fluctuations. Keywords 1 IMF, Tsallis distribution, Solar wind fluctuations Introduction The solar wind is a highly nonlinear, non-equilibrium process. Interplanetary Magnetic Field (IMF) changes with distance from the Sun. The Sun’s rotation, when coupled with the radial solar wind flow, winds up the field lines to form Archimedean spirals, now commonly referred to as “Parker’s model” (Parker, 1958). Parker’s model is supported by spacecraft observations, and the distribution of B in the supersonic solar wind is generally lognormal (Wang et al., 2003), even though there are nonParker fields (Zhang et al., 2008). Spacecraft observed large-amplitude fluctuations in the magnetic field strength B with very complex profiles, which has fractal and multi-fractal structures over a wide range of scales. The fluctuations include “kinetic∗ Supported scale” features (with sizes of the order of 10–100 gyroradii), such as isolated magnetic holes and humps, and trains of magnetic holes and humps (Zhang et al., 2008; Burlaga, 2006). They also include “microscale” features (> 100 proton gyroradii) which can be described by Magneto Hydrodynamics (MHD) theory (Burlaga et al., 2004, 2009). One way to describe the variability of B(t) as a function of scale is to analyze the fluctuations of the increments of B, viz., dBm (t) ≡ B(t + τm ) − B(t) on scales τm . The Probability Distribution Functions (PDFs) for the magnetic field strength differences were known to be non-Gaussian and have “tails”. No attempt were made to describe them quantitatively until Burlaga et al. (Burlaga et al., 2004, 2007) firstly applied Tsallis distribution to describe the fluctuations of both magnetic field and velocity field in solar wind at 1 AU by the NNSFC (40921063) and CAS grant KJCX2-YW-T13 Received October 16, 2009; revised April 20, 2010 E-mail: [email protected] Li Xiaoyu, Zhang Tielong: Statistics of the Interplanetary Magnetic Fields Observed at 0.72 AU and 1 AU from scale of 64 s to 128 days. Recently, Burlaga et al. concluded that the increments of magnetic field strength can be described by Tsallis distribution on all scales (from minutes to months) at distances from 1 to 100 AU (Li et al., 2003), while Li et al. found that the magnetic field strength fluctuations have been described well by Tsallis distribution in the inner heliosphere at 0.72 AU over a time scale from 1 hour to 85.3 days (Thang et al., 2003). The Venus Express (VEX) spacecraft was launched on 9 November 2005, and placed in elliptical orbits about the Venus. It spends majority of its 24hour period in solar wind, thus it provides good opportunity to investigate the interplanetary magnetic field at 0.72 AU (Tsallis, 1988). In the meantime, ACE is continuously monitoring the solar wind at 1 AU. Therefore, we can compare the interplanetary magnetic field observations made by VEX and ACE Figure 1 357 to examine the distribution and evolution of IMFs from 0.72 to 1 AU. 2 Overview of the IMFs at 0.72 AU and 1 AU Our statistical analysis on the magnetic fluctuating field at 0.72 AU are based on the observations by Venus Express during the year 2007. The magnetometer provided the interplanetary magnetic field measurements. We computed daily averages from the averaged 10 min resolution data with measurements down stream of Venus bow shock removed. Our comparative study at 1 AU applies observations by Advanced Composition Explorer (ACE). We use 4second resolution magnetic field observed by ACE to calculate the 10 min or daily averages. Figure 1 shows the daily observations of B(t), elevation angle δ(t), Daily averages of VEX and ACE observations in 2007: magnetic field strength B, azimuthal angle λ and elevation angle δ. Chin. J. Space Sci. 358 and azimuthal direction λ(t) for both VEX at 0.72 AU (left) and ACE at 1 AU (right). The profiles of B(t) are highly variable, and the elevation angle δ are close to δ = 0◦ (the solar equatorial plane) for both locations. The average azimuthal angle is about 39.4◦ at 0.72 AU, and 45.7◦ at 1 AU, respectively, which is consistent with Parker’s model. The distributions of daily averages at 0.72 AU and 1 AU are shown in Figure 2(a) and (b), respectively. The distributions of both daily averages of B are lognormal, y = [A/( (2π)ωB)] × exp{−[ln(B/Bc )]2 /(2ω 2 )}, as shown by the solid curves. For VEX data, the parameters of the fit are A = 1.12 ± 0.03, Bc = 5.45 ± 0.06, and ω = 0.31 ± 0.01. The average magnetic field strength B = 4.96 nT and the standard deviation SD = 0.05 nT. For comparison, the parameters of the fit for ACE A = 1.12±0.05, Bc = 1.95±0.09, and ω = 0.77 ± 0.03. The parameter ω here in lognormal distribution is a measure of the width of the distribution, in the same units as B. The average magnetic field strength B = 2.88 nT and the standard deviation SD = 0.06 nT. The plasma at both locations was not fully thermalized, hence not Gaussian. Figure 2 3 2010, 30(4) Distributions of Increments of the MagneticField Strength We use 10 minute averages magnetic field B(t) to compute a set of Probability Distributions Functions (PDFs) describing magnetic field strength differences normalized to B, dB(t) = [B(t + τ ) − B(t)]/B, where B is the mean for the whole interval, and τ is the time scale (or lag) considered. In this work, τ = 2m hours, where m ranges from 0 to 8. Thus the time scale in consideration of ranges from 1×10 min to 28 × 10 min, namely about 42.7 hours. As the shapes of PDFs will be distinctly affected by the width of bins, we adopted the same bin size to compute all the PDFs. It is noted that length of time series can also affect the results. Thus one had better to use intervals of the same length. We fit these PDFs to Tsallis distribution (Tsallis et al., 2004; Bruno et al., 2004), which was derived by extremizing the nonextensive entropy Sq = k (1 − pqi )/(q − 1) where pi is the probability of the microstate, and q is a scale-dependent constant measuring the nonextensivity. The Tsallis q-distribution function is defined as: y(x) = A[1 + (q − 1)Cx2 ]−1/(q−1) . Distributions of daily averages of B at 0.72 AU (a) and 1 AU (b). Li Xiaoyu, Zhang Tielong: Statistics of the Interplanetary Magnetic Fields Observed at 0.72 AU and 1 AU Here x is the physical quantity such as the magnetic magnitude; A, C, and q are constant at a give scale. The parameter q (entropic index or nonextensivity parameter) is related to the size of the tail in the distribution, which provides a measure of the intermittency. As q increases, tails of the PDFs become fatter. A PDF with large tails, namely with large q, implies that extreme events become statistically more probable than if they were normally distributed and the fluctuations at this scale is intermittent and spiky In the limit q → 1, the statistical mechanics of Tsallis reduces to the usual Boltzmann-Gibbs mechanics, where the PDF is proportional to an exponential (Gaussian) distribution. The widths of the PDFs are related to the amplitudes of the fluctuations. In the limit of large x, the Tsallis PDF approaches the power law. Figure 3 shows the PDFs and their fits to Tsallis distribution on 9 representative scales 2m ×10 min, where m = 0, 1, 2, · · · , 8, i.e., 10 min, 20 min, 40 min, 1.3 h, 2.7 h, 5.3 h, 10.7 h, 21.3 h, and 42.7 h, respectively. They are plotted as the fraction of counts Figure 3 359 versus normalized dB in semi-log scales. Each distribution is plotted as a set of points, where each PDF is displaced a factor of 100 times above the one below it, for the sake of clarity. The results for 0.72 AU and 1 AU are show in Figure 3(a) and (b) respectively. All these PDFs have large tails. PDFs at both 0.72 AU and 1 AU fit by Tsallis distribution were over all scales discussed. Table 1 show the parameters’ best fit value, their 95% confidence intervals (Ci ) and R2 , the measurement of the goodness of fit. The fits hold high qualities with R2 > 0.90. Figure 4 shows the entropic index q as a function of scale at 0.72 AU (a) and 1 AU (b). The width of the PDFs was a function of scale at 0.72 AU (c) and 1 AU (d). At the small scales at 0.72 AU, 1.4 < q < 1.5, which is less than those at 1 AU, 1.5 < q < 1.6, corresponding to the observation of fewer “outliers”. At largest scales (10.7 h, 21.3 h and 42.7 h), 1.3 < q < 1.4 at 0.72 AU, while 1.4 < q < 1.5 at 1 AU. This difference expresses the observation that the PDFs of increments of B at largest scales are more non-Gaussian at 1 AU than those at 0.72 AU. The increase of q from (a) PDFs of increments of B, dBm (t), on scales τm = 2m × 10 min, where m = 0, 1, 2, · · · , 8, at 1 AU (a) and at 0.72 AU (b). (b) Curves are fits of the observed PDFs to the Tsallis distribution. Chin. J. Space Sci. 360 Table 1 2010, 30(4) Two parameters q, ω, with 95% Ci derived from fit and R2 measured the goodness of fit. VEX ACE w Ci of w q Ci of q R2 0.95 0.10 0.01 1.59 0.03 0.97 0.97 0.14 0.01 1.57 0.02 0.97 0.96 0.17 0.01 1.55 0.02 0.98 0.02 0.97 0.22 0.01 1.52 0.02 0.98 0.02 0.98 0.28 0.02 1.49 0.02 0.98 1.48 0.02 0.98 0.33 0.02 1.52 0.02 0.97 0.03 1.46 0.02 0.96 0.41 0.02 1.53 0.02 0.98 0.48 0.02 1.39 0.02 0.97 0.57 0.03 1.47 0.02 0.96 0.55 0.03 1.36 0.02 0.95 0.74 0.03 1.39 0.02 0.97 w Ci of w q 0.067 0.01 1.47 0.04 0.10 0.01 1.51 0.03 0.14 0.01 1.49 0.03 0.19 0.01 1.46 0.24 0.01 1.45 0.29 0.01 0.36 Figure 4 Ci of q R2 Entropic index q as a function of scale at 0.72 AU (a) and 1 AU (b). The width of the PDFs w as a function of scale at 0.72 AU (c) and 1 AU (d) from 0.72 AU to 1 AU might indicate the increasing intermittent turbulence, and perhaps also associated with development of shocks driven by ICMEs or IRs. The widths of the PDFs in Figure 4 are described quantitatively by the parameter w derived from the fits of the Tsallis distribution to the observations. The parameter is plotted as a function of scale at 0.72 AU (c) and 1 AU (d), respectively. The widths of the PDFs of both 0.72 AU and 1 AU increased nonlinearly with scale. Figure 4(c) and (d) show quantitatively that the widths of the PDFs at 0.72 AU are larger than those at 1 AU at all scales. The widths of the PDFs are related the amplitudes of the fluctuations for the increments of B. In this sense, the turbulence at 0.72 AU is more intense than that at 1 AU. 4 Summary We have examined the interplanetary magnetic field distribution and a set of PDFs derived from observations at 0.72 AU by VEX and 1 AU by ACE in the Li Xiaoyu, Zhang Tielong: Statistics of the Interplanetary Magnetic Fields Observed at 0.72 AU and 1 AU year of 2007. The distributions of daily averages of B are lognormal at both locations, indicating that the plasma at both locations was not fully thermalized, hence not Gaussian. The multiscale structure of the magnetic field fluctuations was described by studying the increments of B over a range scales from 10 minutes to 42.7 h. The time series of the increments of B are best described quantitatively and statistically by PDFs. All the PDFs are peaked and have large tails, which can be described quantitatively by Tsallis distribution function. The two parameters of Tsallis distribution, w and q, measure the width and tails of a PDF. They denote the amplitude and non-Gaussianity of the fluctuations respectively. On the ecliptic plane from 0.72 AU to 1 AU, q increases with distance over all scales, indicating the intermittency of turbulence is growing. The widths of the PDFs of both 0.72 AU and 1 AU increased nonlinearly with scale. The widths of the PDFs at 0.72 AU are larger than those at 1 AU at all scales, which indicating the turbulence at 0.72 AU is more intense than that at 1 AU. The turbulence may consist of both coherent (semi-deterministic) structures and random structures, but the exact nature and origin of the magnetic fluctuations will be investigated in our future work. Acknowledgments The magnetic field data are obtained from ACE. References Burlaga L F, Ness N F. 2009. Compressible “turbulence” observed in the heliosheath by Voyager 2 [J]. Astrophys. J., 703:311-324 Burlaga L F, Vinas F. 2004. 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