Download Statistics of the Interplanetary Magnetic Fields Observed at 0.72 AU

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. Multi-scale probability distributions of solar wind speed fluctuations at 1 AU described
361
by a generalized Tsallis distributions [J]. Geophys. Res.
Lett., 31, L16807, doi:10.1029/2004GL020715
Burlaga L F, Vinas F. 2004. Multiscale structure of the magnetic field and speed at 1 AU during the declining phase
of solar cycle 23 described by a generalized Tsallis probability distribution function [J]. J. Geophys. Res., 109,
A12107, doi:10.1029/2004JA010763
Burlaga L F, Vinas F. 2007. Tsallis distribution funcitons
in the solar wind: Magnteic field and velocity observations [R]//AIP Conf. Proc., 965:259-266
Burlaga L F, Viñas A F, Ness N F, Acuna M H. 2006. Tsallis statistics of the magnetic field in the heliosheath [J].
Astrophys. J., 644:L83-L86
Bruno R, Carbone V, Primavera L, Malara F, Sorriso-Valvo
L, Bavassano B, Veltri P. 2004. On the probability distribution function of small-scale interplanetary magnetic
field fluctuations [J]. Ann. Geophys., 22:3751-3769
Li X Y, Wang C, Zhang T L. 2009. Tsallis distribution of
the interplanetary magnetic field at 0.72 AU: Venus Express observation [J]. Geophys. Res. Lett., 36, L11103,
doi: 10.1029/2009GL038395
Parker E N. 1958. Dynamics of the interplanetary gas and
magnetic field [J]. Astrophys. J., 128:664
Burlaga L F. 2001. Lognormal and multifractal distributions
of the heliospheric magnetic field [J]. J. Geophys.Res.,
106:15 917-15 927
Thang T L, et al. 2007. Little or no solar wind enters Venus’
atmosphere at solar minimum [J]. Nature, 450:654-656,
doi:10.1038/nature06026
Tsallis C. 1988. Possible generalization of Boltzmann-Gibbs
statistics [J]. J. Stat. Phys., 52:479
Tsallis C, Brigatti E. 2004. Nonextensive statistical mechanics: A brief introduction [J]. Contin. Mech. Thermodyn.,
16:223-235
Wang C, Richardson J D, Burlaga L F, Ness N F. 2003. On
radial heliospheric magnetic fields: Voyager 2 observation
and model [J]. J. Geophys. Res., 108:1205
Zhang T L et al. 2008. Characteristic size and shape of the
mirror mode structures in the solar wind at 0.72 AU [J].
35, L10106, doi:10.1029/2008GL033793
Zhang T L et al. 2008. Behavior of current sheets at directional magnetic discontinuities in the solar wind at
0.72 AU [J]. Geophys. Res. Lett., 35, L24102, doi:10.
1029/2008GL036120