CORRELATIONS OF PLASMA DENSITY AND MAGNETIC
FIELD STRENGTH IN THE HELIOSHEATH
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Citation
Gutynska, O., J. Šafránková, Z. Nmeek, and J. D. Richardson.
“CORRELATIONS OF PLASMA DENSITY AND MAGNETIC
FIELD STRENGTH IN THE HELIOSHEATH.” The Astrophysical
Journal 722, no. 2 (October 5, 2010): L228–L232. © 2010 The
American Astronomical Society
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http://dx.doi.org/10.1088/2041-8205/722/2/l228
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The Astrophysical Journal Letters, 722:L228–L232, 2010 October 20
C 2010.
doi:10.1088/2041-8205/722/2/L228
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
CORRELATIONS OF PLASMA DENSITY AND MAGNETIC FIELD STRENGTH IN THE HELIOSHEATH
O. Gutynska1 , J. Šafránková1 , Z. Němeček1 , and J. D. Richardson2
1
2
Charles University, Faculty of Mathematics and Physics, V Holešovičkách 2, 180 00 Prague 8, Czech Republic; jana.safrankova@mff.cuni.cz
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Received 2010 July 28; accepted 2010 September 20; published 2010 October 5
ABSTRACT
The crossing of the termination shock (TS) by Voyager 2 in 2007 at 84 AU allows a comparison of fluctuations in
different heliosheath regions. The Letter concentrates on MHD waves that exhibit a significant correlation between
the magnetic field strength and plasma density. The correlations between both quantities were computed on 2 hr
time intervals in the frequency range of 1 × 10−4 to 4 × 10−3 Hz. We separate the data into two regions with
different magnetic field behavior; the post-TS region with many crossings of the current sheet and the unipolar
region where the magnetic field direction remains nearly constant. We find that typical correlation coefficients in
these regions are about 0.55–0.65, larger than in Earth’s magnetosheath. The largest correlations occur when the
spectrum of magnetic field fluctuations is dominated by low frequencies.
Key words: magnetic fields – Sun: heliosphere – waves
to 76, a region with a nearly constant magnetic field azimuthal
angle and with an elevation angle near zero. The magnetic field
strength fluctuated around a linear trend. The unipolar region
was caused by an excursion of the heliospheric current sheet
past the latitude of V2 (Burlaga & Ness 2009). Their analysis of
both regions indicates that fluctuations on scales less than 1 day
were significant in both the post-TS and unipolar regions. These
magnetic fluctuations, characterized by a series of depressions
in the field magnitude, are similar to those observed downstream
of quasi-perpendicular interplanetary and planetary bow shocks
which have been identified as mirror mode structures (e.g.,
Bavassano Cattaneo et al. 1998).
On its path from 1 AU, V2 observed different types of
correlations between the plasma speed, density, and magnetic
field strength. Using 25 day averages, Richardson et al. (2003)
looked at the correlations between the plasma speed, density, and
the magnetic field magnitude. They found that the correlation
between the density and magnetic field decreases with distance
from 0.6 near Earth to about zero at 30 AU but was as high as
0.64 in the 2002 data at 67 AU. A numerical model (Wang &
Richardson 2001) was used to propagate the solar wind observed
at Earth to the position of V2. The one-dimensional MHD model
includes the effect of pickup ions (Wang & Richardson 2001)
and shows a high correlation between B and N throughout the
heliosphere near solar maximum, but the high value probably
results from the one-dimensional nature of the model which
forces B and N changes to occur synchronously.
In this Letter, we used the Burlaga & Ness (2009) definition
of heliosheath regions and performed a study of correlation
properties of fluctuations of the plasma density, N, and magnetic
field strength, B, measured by V2 in both the post-TS and
unipolar regions. The purpose of the study is to find the dominant
wave modes in the heliosheath and to deduce possible sources
of these waves. Among many types of fluctuations observed
in these regions, we concentrate on MHD waves that exhibit
significant correlation between the plasma density and magnetic
field strength.
1. INTRODUCTION
Voyager 2 (V2) crossed the termination shock (TS) at least
five times from 2007 day of year (DOY) 242–245 (Burlaga et al.
2008; Decker et al. 2008; Richardson et al. 2008; Stone et al.
2008) and is now in the heliosheath, the region of fluctuating,
dynamic, shocked solar wind between the TS and heliopause.
The TS was a relatively weak, quasi-perpendicular shock with
a shock strength of about 2 (Richardson et al. 2008; Burlaga
et al. 2008). The solar wind plasma was slowed, compressed,
and heated at the TS. According to Richardson et al. (2008),
the flow remained supersonic (super-Alfvénic) with respect
to the thermal ions downstream of the TS. Most of the solar
wind energy is transferred to the pickup ions or other energetic
particles both upstream of and at the TS. The heliosheath
boundaries move in response to a large merged interaction
region (MIR; Burlaga et al. 1985) driven shocks and due to solar
cycle changes in the solar wind pressure, but these motions are
small compared to the size of the heliosheath.
Burlaga et al. (2009) analyze high-resolution measurements
of the magnetic field, B and find that V2 (similarly to Voyager 1)
observed compressible “turbulence” in the heliosheath. Largeamplitude fluctuations of B occur at small scales (i.e., on scales
of tens of minutes to tens of days) with very complex profiles.
These include “kinetic-scale” features that are characterized by
a series of depressions/enhancements in the field strength such
as isolated magnetic holes and humps, and their trains (Burlaga
et al. 2006). Avinash & Zank (2007) have shown that these
structures can be considered as solitons propagating with an
arbitrary angle to the ambient magnetic field. The turbulence
also includes “micro-scale” features (>100 proton gyroradii)
that can be described by magnetohydrodynamic (MHD) theory.
The small-scale fluctuations (both kinetic-scale and micro-scale
fluctuations) may be mirror mode disturbances generated by
temperature anisotropies generated at the TS (Liu et al. 2007).
Burlaga & Ness (2009) described two regions behind the
TS with different magnetic field behavior: (1) “the post-TS
region” from 2007 DOY 245 to 301, where the magnetic field
strength fluctuates about 0.09 nT but with large jumps in B on
timescales of minutes to several hours, and the magnetic field
direction fluctuates in a complicated way during intervals of
several hours and (2) “the unipolar region” from 2008 DOY 2
2. DATA SET AND METHODOLOGY
The Voyager plasma experiment registers solar wind protons
simultaneously in three Earthward-pointing Faraday cups over
L228
No. 2, 2010
CORRELATIONS IN THE HELIOSHEATH
Voyager 2
(a) Post TS Region
24 hr avg
Unipolar Region
N, cm-3
B, nT
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.005
0.004
L229
250
300
350
400
300
350
400
300
350
400
(b)
7 hr avg
0.003
0.002
0.001
0.000
az
300
250
(c)
200
el
100
0
60
40
20
0
-20
-40
250
(d)
250
300
350
DOY from Jan. 1, 2007
400
Figure 1. Overview of the V2 observations in the post-TS and unipolar regions. (a) Daily averages of the magnetic field strength, (b) 7 hr averages of the plasma
density, daily averages of the magnetic field azimuthal (c) and elevation (d) angles.
an energy range of 10–5950 eV with a time resolution of
192 s (Bridge et al. 1977). When possible, the three spectra
are fit with convected isotropic Maxwellian distributions to
determine the proton velocity, density, and temperature. Data
quality often does not allow this fitting procedure. For this
study, we use moment densities obtained by integrating over
the plasma distribution which allows us to use all the spectra
and thus gives better time resolution.
The Voyager magnetometer measures the magnetic field with
two sensors mounted on a boom, one at the end of the boom and
the other closer to the spacecraft (Behannon et al. 1977). We use
48 s averages of the magnetic field strength. The proton density is
linearly interpolated to obtain the same time resolution as that of
magnetic field measurements. The data are only available when
the spacecraft is tracked by the Deep Space Network. Thus,
the final data set consists of intervals of 8–14 hr in duration
that repeat each day. This feature allows us either to study fast
fluctuations within these intervals or to average the parameters
over the whole interval and to investigate fluctuations with
periods of several days and longer. However, the number of
days that V2 spent in the heliosheath is still insufficient for the
latter investigation.
An overview of the observations is presented in Figure 1
which shows daily averages of the magnetic field strength (a),
7 hr averages of the plasma density (b), and daily averages
of the magnetic field azimuthal (c) and elevation angles (d)
measured by V2. For this correlation analysis, we divide the
data into two time subintervals: (1) from 2007 DOY 245 to
301—the post-TS region (38 intervals) and (2) from 2008 DOY
2 to 76—the unipolar region (55 intervals). The magnetic field
and proton density behave differently; neither correlation nor
anti-correlation is seen on the scale of the figure. Nevertheless,
enhanced levels of magnetic field fluctuations often correspond
to enhanced levels of density fluctuations which suggests that
the fluctuations may be associated with MHD waves. We try to
identify these waves with our correlation analysis.
Taking into account the above constraints, we investigate
only fluctuations with periods from ≈4 minutes (limited by
the temporal resolution) to ≈2 hr (limited by the duration of
the continuous intervals of data). For each of these intervals, we
compute a cross-correlation coefficient using a subinterval of
proton density measurements in the center of the time interval
and a time lag with respect to the magnetic field measurements
within a range of ±3 hr. Both signals were pre-processed by
applying a bandpass filter 1 × 10−4 to 4 × 10−3 Hz.
3. RESULTS AND DISCUSSION
Figure 2 shows an example of the magnetic field and plasma
density profiles measured in one ≈10 hr time interval on DOY
277 of 2007 in the post-TS region. The duration of the data we
correlate is 120 minutes. The bottom panel shows the autocorrelation coefficients of the magnetic field (blue line), as
well as the proton density (red line) and their cross-correlation
coefficients (black line). The auto-correlation does not decrease
to zero for large time lags, which suggests the presence of
coherent periodic components. The same is true for the profile of
the density auto-correlation but the periodicity is different. The
cross-correlation techniques allow us to filter from the whole
spectrum of fluctuations those in which B and N behave a
L230
GUTYNSKA ET AL.
Voyager 2
1.0
Post-TS Region
48 sec av
max cor N x B
0.4
B, nT
0.3 (a)
0.2
0.1
277.7
180 sec av
277.8
277.9
0.4
0.0
0.0
0.002 (b)
0.001
0.000
1.0
277.8
DOY, 2007
277.9
278.0
(c)
0.5
0
100
300
400
500
600
-0.4
Post TS region
Unipolar region
Artif signal
-0.6
-0.8
0.0
-1.0
0
0
-0.5
-1.0
-200
200
-0.2
-100
0
Lag [min]
100
200
Figure 2. Example of measured and computed profiles through one ≈10 hr time
interval on DOY 277 of 2007: (a) the magnetic field, (b) the plasma density, and
(c) the computed cross-correlation coefficient between the magnetic field and
plasma density (black line) and the auto-correlation coefficients of the magnetic
field (blue) and the proton density (red). The light dashed lines distinguish one
half of the period of the dominant MHD wave.
coherent way. The dashed line distinguishes the maximum and
minimum of the cross-correlation coefficient, which are +0.7
and −0.7, respectively. In this particular case, the difference of
the corresponding lags (≈230 minutes) is approximately equal
to a half of the period of the dominant MHD wave. However, the
cross-correlation profile suggests the presence of a wave with
a much shorter period, about 55 minutes. The presence of two
(or more) waves can explain why the typical value of the crosscorrelation coefficient for zero lag was found to be arbitrary
(close to zero in Figure 2), whereas MHD theory predicts that
B and N would change in phase or anti-phase for fast and slow
waves, respectively.
The theory of correlation functions assumes unlimited time
intervals (numbers of points), but our time intervals are rather
short. To test whether our results are significant or arise from
the errors caused by the use of a limited number of points, we
generated 1000 pairs of random sequences and processed them
using the same method as for the real signals. The comparison
is shown in Figure 3. The black squares show the average
maximum (top) or minimum (bottom) correlations computed
using different data subintervals within the post-TS region (38
intervals). The black bars show the standard deviations. The
green squares show the correlations for the unipolar region (55
intervals) and the red for the random signals. The magnetic field
and plasma density exhibit a statistically significant correlation.
The mean correlation coefficient decreases nearly linearly with
the duration of the subinterval and approaches the random
100
200
100
300
400
200
300
400
length [min]
500
500
600
600
Figure 3. Averaged profiles of maxima and minima of correlation coefficients
in both regions computed using subintervals with different durations. For
comparison, the correlation coefficients derived from a random sequence of
two signals are also shown (red points).
20
Number of events
cor B x N
0.6
0.2
Post-TS Region
0.003
277.7
Post TS region
Unipolar region
Artif signal
0.8
278.0
min cor n x B
N, cm-3
0.0
0.004
Vol. 722
Post TS
Unipolar R
cor N x B (0 lag)
15
10
5
0
-0.9
-0.6 -0.3 0.0 0.3 0.6
Correlation coefficient
0.9
Figure 4. Distributions of maximum and minimum correlation coefficients in
the post-TS (black) and unipolar (green) regions. The red histogram shows the
distribution of correlations coefficients for zero lag in both regions.
sequence level for subintervals of ≈600 minute long. This result
indicates that the coherent features persist for about 10 hr in the
analyzed frequency range.
The same features were found in both regions, but the
correlation is slightly higher in the unipolar region. Since the
MHD fluctuations are generated either at the TS or locally in
the heliosheath near the TS, the small increase of the correlation
at larger distances from the TS suggests that the incoherent
fluctuations are gradually damped as they are convected from
the source through the post-TS region to the unipolar region.
Figure 4 presents histograms of the maxima (positive values)
and minima (negative values) of correlation coefficients between
N and B in both regions. The distributions are nearly identical,
which suggests that the periodic behavior shown in the last
No. 2, 2010
CORRELATIONS IN THE HELIOSHEATH
-4
PSD [nT2/Hz]
-5
-6
-7
-8
-9
1.0
(a)
0
50
2
100
chi = 0.21
chi2= 0.074
150
200
250
Post-TS region
Unipolar region
max cor N x B
0.8
L231
the other hand, such solitons would exhibit an anti-correlation
between the density and field strength which is not the case
(Figure 4).
One-point measurements are insufficient for a reliable determination of the wave mode. If the mirror mode waves in the
heliosheath (Liu et al. 2007) and in Earth’s magnetosheath behind of the quasi-perpendicular bow shock (e.g., Schwartz et al.
1996; Tátrallyay et al. 2008) were similar, the typical wavelength would be of the order of 106 km. Since the mean Alfvén
velocity is about 50 km s−1 and mean flow speed ≈150 km s−1
in the heliosheath, the wavelength of other modes would be of
the same order. Such waves cannot propagate (or be convected)
through planetary magnetosheaths (Hubert et al. 1998), thus a
comparison of these apparently similar environments is difficult.
4. CONCLUSION
We analyzed MHD fluctuations in two regions of the heliosheath and found:
0.6
0.4
(b)
0.2
0
50
100
150
Period [min]
200
250
Figure 5. (a) Typical spectrum of magnetic field fluctuations (the dashed
line indicates the period of a dominant wave). (b) Maxima of the correlation
coefficient plotted vs. the period of the dominant wave. The standard deviations
are shown at the top of the figure.
panel of Figure 2 is typical. The most frequent values of the
coefficients are around 0.6 and are slightly higher in the unipolar
region. These values are surprisingly large; Gutynska et al.
(2009) reported that correlation coefficients of 0.4 were typical
in the Earth’s magnetosheath. The distribution of correlations
for zero lag (red line in Figure 4) is very broad and centered
around zero. This suggests that there is no preferred wave mode
in the heliosheath, whereas Gutynska et al. (2009) found slow
or mirror mode to be more frequent in the magnetosheath.
Figure 5(a) shows a typical spectrum of magnetic field
fluctuations, and the dashed line indicates the period of a
dominant wave. The correlation coefficient plotted versus this
period in each interval is given in Figure 5(b). The correlation
coefficient increases with the wave period. This trend is more
pronounced in the unipolar region, which has a larger number of
intervals with low frequencies. The larger correlations observed
in the unipolar region (see Figure 4) may result from the
damping of higher-frequency waves as they convect through
the heliosheath, leaving more low-frequency waves (which have
higher correlations) in the unipolar region. We note that the
longest periods correspond to the duration of the intervals.
Heliosheath fluctuations could have longer periods but the
available data do not allow them to be identified.
Variations of the magnetic field magnitude are not accompanied with correlated changes of its direction. It is clear in
the unipolar region where the field direction is almost constant
(Figure 1) on any timescale, and we have found the same behavior in the post-TS region. This supports the interpretation of
the fluctuations as trains of solitons (Avinash & Zank 2007); on
1. Strong fluctuations in the frequency range of 1×10−4 to 4×
10−3 Hz with correlated changes of the magnetic field and
density which were identified in both regions.
2. The cross-correlation coefficient of these quantities is ≈0.6
in both regions, significantly larger than the value of 0.4 in
Earth’s magnetosheath and is slightly larger in the unipolar
region that is located farther from the TS.
3. Typical periods of the fluctuations range from 50 to
220 minutes.
4. The value of the cross-correlation coefficient increases with
the period of the dominant wave in the frequency spectrum.
5. Larger values of cross-correlation coefficients observed in
the unipolar region are consistent with a larger portion of
long-period fluctuations.
Above observations suggest that the heliosheath fluctuations
are generated at the TS. The waves of higher frequencies
are gradually damped, and lower frequencies prevail at larger
distances from the TS.
The present work was partly supported by the Czech Grant
Agency under Contracts 205/09/0170, 205/07/0694, and 202/
08/H057, and partly by the Research Plan MSM 0021620860
that is financed by the Ministry of Education of the Czech
Republic. O.G. is thankful for support from Grant ME09056
and GAUK 163810. J.D.R. was supported by the NASA through
the Voyager project and grant NNX08AC04G.
Facility: Voyager II
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