Coherence Lengths of the Interplanetary Electric Field:
Solar Cycle Maximum Conditions
Charles J. Farrugia, Hiroshi Matsui, Roy B. Torbert
Space Science Center, University of New Hampshire, Durham, NH.
Abstract.
It is increasingly being realized that by affecting geoeffective scale lengths the interplanetary electric field (IEF) is a key
quantity in space weather discussions. In this work we derive and analyze statistically IEF coherence lengths in year 2000, i.e.,
near maximum of solar cycle 23, working in a much used formulation for the IEF. We focus on the frequency domain. We use
magnetic field and plasma data sets acquired by Wind and ACE. During year 2000, ACE-Wind separations were very variable
and, in particular, Wind’s first dayside distant prograde orbit resulted in a Y-separation comparable to the X-separation ( 220
RE ). We find IEF coherence lengths of 200-250 RE (X) and 50-100 RE (Y). The coherence is mainly carried by the low
frequency components ( f 001 min 1 .
INTRODUCTION
in the time domain (8). We work in a formulation of
the IEF derived from considerations of the maximum
merging rate at the dayside magnetopause (15,6): IEF =
V BT sin2 θ 2, where BT Bygsm2 Bzgsm2 1 2 , V is
the bulk speed, and θ is the IMF clock angle (i.e., the
polar angle in the GSM YZ plane.)
We show a systematic decrease in the coherence level
with the east-west separation ∆Y during Wind’s first dayside distant prograde orbit (DPO). From this we obtain
the coherence length of the IEF in the east - west direction. We also examine the coherence level of the IEF with
X, analysing data from the early part of the year when
Wind made an excursion to the L1 Lagrangian point.
The interplanetary electric field (IEF) plays an increasing
important role in both theoretical and observational discussions of the geoeffectiveness of interplanetary configurations, i.e., the level of geomagnetic disturbances
which they elicit inside the magnetosphere. For example, it is thought that the transpolar potential, but not the
Dst, saturates for large IEF ( 3 mV m 1 ) (13). It follows that a knowledge of the coherence lengths of the
IEF parallel and perpendicular to the Sun-Earth line is
a crucial quantity in space weather considerations. Yet,
no studies addressing these issues have been attempted
to date. The need for such investigations was highlighted
by (2) in their study of geoeffective interplanetary scale
sizes.
In this paper we examine coordinated observations
made by Wind and ACE in year 2000. This year is chosen
because: (i) it is close to the maximum of solar cycle
23 when interplanetary configurations are expected to be
strongly geoeffective on average; (ii) data coverage at
both spacecraft is optimal; (iii) The orbit of Wind relative
to ACE allows a study of the IEF as functions of both X
and Y, the latter varying over 500 Earth radii, R E . (By
comparison, coordinate Z is small.)
As methodology we adopt the approach taken in a
pilot study by (8). Rather than cross-correlating two time
series using various time windows, they decomposed the
signals into their Fourier components and could thus
examine the level of coherence at the two observing
sites also as a function of frequency. Spectral analysis
has distinct advantages over the more common analysis
WIND AND ACE ORBITS IN YEAR 2000
Figure 1a shows the position of Wind during year 2000
in GSE coordinates. In January, Wind was executing
orbits mostly in the geomagnetic tail. These data are
not used further. During February-March, Wind made
an excursion to the neighbourhood of L1. Then follow
dayside and nightside, high latitude Petal orbits (April August). From mid-August to the end of the year Wind
starts its dayside DPOs. The months October - December
contain one-half of such an orbit, taking Wind from 250 RE to 250 RE . The spacecraft spends relatively long
times at the “apex” of the DPO. The DPOs afford a
unique opportunity to study coherence lengths in the Ydirection over a long baseline. The resulting ACE-Wind
separations are plotted in Figure 1b. The excursion by
Wind to near L1 gives a range of 200 R E in the Sun-Earth
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
766
(a) WIND (RE)
Year: 2000
(GSE)
250
120
200
80
40
0
by the FFT routine are complex, calculated for each
frequency and for each time.
(GSE)
100
Excursion to L1
Petal Orbits
First Distant Prograde Orbit
0
744
250
200
150
100
50
0
-50
-100
-150
-200
-250
1440
2184
2904
3648
4368
5112
5856
6576
7320
8040
8784
DY (Re)
Y (Re)
Year: 2000
150
50
-40
-80
744
1440
2184
2904
3648
4368
5112
5856
6576
7320
8040
8784
250
200
150
100
50
0
-50
-100
-150
-200
-250
744
1440
2184
2904
3648
4368
5112
5856
6576
7320
8040
8784
744
1440
2184
2904
3648
4368
5112
5856
6576
7320
8040
8784
744
1440
2184
2904
3648
4368
5112
5856
6576
7320
8040
8784
C f 0
-50
744
Jan
1440
2184
2904
3648
Feb Mar Apr May Jun
4368
5112
Jul
5856
6576
Aug Sep Oct
7320
8040
8784
Nov Dec
Month of Year
0
∆φ f arg
-50
Jan
Feb Mar Apr May Jun
Jul
Aug Sep Oct
Nov Dec
direction X. The first DPO gives a Wind-ACE separation
of the same magnitude both to the east and west of Earth.
The magnetic field data were acquired by the Magnetic
Field Investigation (MFI) on Wind (7) and the Magnetic
Field Experiment (MAG) on ACE (14) with time resolutions of 90 s and 16 s, respectively. The plasma data
are from the Solar Wind Experiment (SWE) on Wind
(10) and the Solar Wind Electron Proton Alpha Monitor (SWEPAM) on ACE (9) with time resolutions of 94
s and 64 s, respectively. (These data are courtesy of the
NASA CDA Web site.) The data resolution is equalized
by linear interpolation. We employ a time resolution of 1
min. Data spikes have been removed before the analysis.
For the magnetic field we remove those data points which
are larger than neighboring points by 5 nT or more. Similar criteria are adopted for the velocity (a jump of 50 km
s 1 ). We interpolated only data gaps which are shorter
than, or equal to, 10 data points.
A NOTE ON THE FOURIER
TECHNIQUE
QiACE f 2
QiWind f 2 QiACE f QiWind f ∆φ f 360 This is described in detail in (8). At each spacecraft
we calculate the Fourier amplitude and phase of the
IEF by FFT (11). From this we derive (i) the amplitude
ratio, R f , (ii) the coherence, C f , and (iii) the phase
lag, ∆φ f between the two spacecraft as a function of
frequency f . These three analysis quantities are defined
as follows:
3
i 0
∑3i 0
(2)
(3)
We set the number, n, of samples per FFT as 512, corresponding to 512 min per FFT. Data have been weighted
by a Hamming window. To obtain C f we need results
from multiple FFTs (e.g., 3, 5). Increasing the number
of FFTs reduces random errors (1) but at the expense of
time resolution. In addition, the assumption of time stationary conditions inherent in the method is more likely
to be violated when many FFTs are summed. In this analysis we use 4 FFTs, the same number which we used
successfully in an analysis of IMF and solar wind parameters (8). Neighboring FFTs are overlapped by one-half
the number of samples per FFT. One cross-spectrum is
thus based on 1280 samples per FFT.
The coherence is similar to the correlation of the
phases between QiWind f and QiACE f . When the
phases of the fluctuations of the quantity are randomly
distributed, the numerator of equation (2) is 0, which indicates lack of coherence. When the phase lag between
QiWind f and QiACE f is constant, C f is 1.
The phase lag ∆φ f contains information about the
propagation of the IEF between spacecraft. This value
is significant only when the coherence is high or, in
other words, when the correlation of the phases is high.
Consider a front propagating with constant velocity, V .
If the normal to the structure is parallel to the X axis, the
relation between the phase lag and the frequency is
FIGURE 1. 1a: Position of Wind in year 2000. 1b: WindACE separation in year 2000.
R f 3
∑
i 0
Month of Year
Ú
Ù
Ù
Ø∑
∑3i 0 QiACE f QiWind f 2
∑3i 0 QiACE f 2 ∑3i 0 QiWind f 2 where denotes complex conjugation.
50
DZ (Re)
50
Z (Re)
(b) ACE - WIND (RE)
300
160
DX (Re)
X (Re)
200
∆X
f deg
VX
(4)
where ∆X is the separation between the two spacecraft
in the X direction. Quantity ∆X VX corresponds to the
convection delay time between the two spacecraft.
ILLUSTRATION OF THE METHOD
Figure 2 shows frequency-time spectra of IEF for October – December, 2000. The vertical axis gives the frequency in min 1 . The panels show from top to bottom
the Fourier amplitudes at Wind and ACE; the amplitude ratio, R f , between Wind and ACE; the coherence, C f ; and (e) the phase lag, ∆φ f . The scale of
the amplitudes shown on the right side is in units of
(1)
where QiWind f and QiACE f correspond to the
Fourier components of IEF, respectively. The summation
is over four FFTs (see below). The quantities returned
767
(d)
½
Coherence
(c)
Phase lag(deg.)
(b)
Amplitude Ratio IEF(mV/m/Hz )
Nov. 25, 2000 0000-2120UT
(a)
100
Wind
ACE
10
1
10
1
0.1
1
0.8
0.6
0.4
0.2
0
180
120
60
0
-60
-120
-180
0
0.02
0.04
0.06
0.08
0.1
Frequency(min -1 )
FIGURE 3. November 25, 2000: The IEF at Wind and ACE,
the amplitude ratio, the coherence and the phase lag.
FIGURE 2. Frequency-time spectrograms for OctoberDecember, 2000. For further details, see text.
mV/m/ Hz. One can make the following points. Typically, the coherence is high only at low frequencies, f 0.01 min 1. In November, with ∆Y passing through 0,
the coherence is generally high up to f 003 min 1 .
The phase lag panel shows a green band corresponding
to no phase lag at the lowest frequencies. Then a yellow
band at somewhat higher frequencies. Above f = 0.02
min 1 , the phase lag is random, indicating that it is difficult to predict the arrival time at Wind of high frequency
components of the IEF observed at ACE.
Figure 3 shows an example of high coherence on
November 25 (hours 1344-1368) and compares the spectral analysis technique with the time series analysis. The
average positions of ACE and Wind were (224, 35, -17)
RE and (80, 82, 0.5) R E , respectively, and the solar wind
speed 400 km s 1 . The coherence 06 for f 0.015
min 1 . An approximately linear phase lag plot (bottom
panel) below this frequency can be seen. The propagation velocity estimated from this phase lag is 30 min.
Compare this with the convection delay X V x =38 min.
Analyzing the two time series, we find that the lag at
maximum correlation, R (= 0.88) = 36 min. In Figure
4 we overlap the two electric fields (ACE in light trace)
for November 25, 2000 taking into account a 36 min delay. Thus the quantities are comparable and there is selfconsistency between the analyses in the frequency and
time domains.
By contrast poor coherence was achieved on October
6 when ACE and WIND were located at (224, -26, -10)
RE and (32, -250, -3) R E , respectively, i.e. they had a an
FIGURE 4. a superposition of the ACE (red) and Wind
IEFs for November 25, 2000 with a 30 min propagation delay
included.
X- and Y-separation each 200 R E . The low coherence
is probably due to this large separation.
STATISTICAL RESULTS
We now summarize the main results of the statistical
analysis on all year 2000 data acquired in the solar wind.
Figure 5 shows the dependence of the coherence of IEF
on period (frequency). As the period increases (the frequency decreases), coherence values increase. Values exceeding 0.6 are reached for periods of 150 min or
longer (f 0.007 min 1 or lower).
Figure 6 gives the dependence of coherence on the Xseparation between Wind and ACE. Calculations for 3
different times are shown as indicated. Highest correlation correspond to longer times (256 min). For this trace,
the coherence decreases with X. (The range ∆X 50R E
is an exception because it is affected by the large ∆Y , see
Figures 1a, b.) Adopting a definition whereby the coherence length scale corresponds to the distance when the
coherence decreases by 0.1 from its highest value (12,
768
Dependence on Period
prograde orbits, which yield an ACE-Wind Y-separation
comparable to the X-separation ( 220 R E ) we carried
out a statistical study using all solar wind data returned
by Wind and ACE in year 2000.
We find two different scale lengths for the IEF: 200 250 RE in the X-, and 50-100 R E in the Y-directions. In
practical terms, this means that if a monitor orbiting L1,
say, is displaced from the Sun-Earth line by less than 50
RE , there is a high probability for the IEF it measures to
be equal to that at the magnetopause nose. The coherence
lengths of the IEF, in both X and Y, are similar to those
of the IMF (8, and references therein).
As a function of frequency, the general behavior is
for there to be good coherence at low frequencies. The
coherence is invariably lost at the higher frequencies.
Similar conclusions as to the frequency dependence of
the coherence of the IMF and solar wind parameters were
reached by (8). That work showed that the power spectral
densities follow a -5/3 law in frequency, indicative of
turbulence. This may also be the reason here.
1
Coherence
0.8
0.6
0.4
0.2
0
10
100
1000
Period(min.)
FIGURE 5. Dependence of the coherence of the IEF on
period (frequency)
Dependence on X (-100<
Y<100 RE)
1
T=256min
T=64min
T=16min
Coherence
0.8
0.6
0.4
0.2
0
50
100
150
200
250
300
X(RE)
FIGURE 6. Dependence of the coherence of the IEF on X.
8), we obtain that the coherence length of IEF in the Xdirection is of order 200-250 Re.
Figure 7 summarizes the statistical dependence of the
coherence on the separation between Wind and ACE
in the Y-direction. A coherence length of IEF in the
Y-direction of 50-100 Re is indicated. Here typically
the coherence peaks at a negative value of Y and then
drops off, i.e., it is not symmetric about ∆Y = 0 (for
T = 256 min). The offset to negative GSE Y (west)
may have several causes: (i) aberration effect due to the
motion of the Earth; (ii) the Parker spiral orientation of
the IMF; (iii) encounter with highly coherent structures
preferentially on one side of the Earth-Sun line on this
DPO.
ACKNOWLEDGMENTS
We thank the PIs of the magnetic field and plasma instruments on Wind and ACE for use of key parameter
data from their instruments. This work is supported by
the Wind Grant NAG5-11803, and NASA Living with a
Star Grant NAG5 - 10883.
REFERENCES
1. Benignus, V. A., IEEE Trans. Audio Electroacoust., 17,
145–150, 1969.
2. Burke, W. J., et al., J. Geophys. Res., 104, 9989, 1999.
3. Eriksson, A. I., in Analysis Methods for Multi-Spacecraft
Data, ISSI Sci. Rep. SR-001, pp. 5–42, ESA Publications,
Noordwijk, Netherlands, 1998.
4. Farrugia, C. J., et al., J. Geophys. Res., 107(A9), 1240,
doi:10.1029/2001JA000188, 2002.
5. Holmgren, G, and P. M. Kintner, J. Geophys. Res., 95,
6015, 1990.
6. Kan, J. R. and Lee, L. C., Geophys. Res. Lett., 6, 577, 1979.
7. Lepping, R. P., et al., Space Sci. Rev., 71, 207, 1995.
8. Matsui, H., et al., J. Geophys. Res., 107(A11), 1355,
doi:10.1029/2002JA009251, 2002.
9. McComas, D. J., et al., Space Sci. Rev., 86, 563–612, 1998.
10. Ogilvie, K. W., et al., Space Sci. Rev., 71, 55, 1995.
11. Press, W. H., et al., Numerical recipes in C, Cambridge
Univ. Press, 2nd ed., Cambridge, 1992.
12. Richardson, J. D., and K. I. Paularena, J. Geophys. Res.,
106, 239, 2001.
13. Russell, C. T., et al., Space Sci. Rev., 71, 563, 1995.
14. Smith, C. W., et al., Space Sci. Rev., 86, 613–632, 1998.
15. Sonnerup, B. U. Ö, J. Geophys. Res., 79, 1546, 1974.
CONCLUSIONS
Our statistical study over 1 year of joint ACE-Wind observations constitutes the first calculations of the coherence length of a quantity so central to discussions on the
interaction of the solar wind with the magnetosphere: the
interplanetary electric field, IEF. Utilizing Wind’s distant
Dependence on Y (150<
X<250 RE)
1
T=256min
T=64min
T=16min
Coherence
0.8
0.6
0.4
0.2
0
-250 -200 -150 -100 -50
0
50 100 150 200 250
Y(RE)
FIGURE 7. Dependence of the coherence of the IEF on Y.
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