Introduction
Mirror mode (MM) waves generated by the mirror instability arise from the
antiphase, low-frequency fluctuations of the magnetic field and plasma
density when a sufficient temperature anisotropy is present in the plasma.
Assuming cold electrons and bi-Maxwellian ions, the criterion for mirror
unstable plasma is
Cm=β⟂β∥-1-1β⟂>0,
where Cm is the mirror instability threshold value and β⟂ and
β∥ express the ratios of perpendicular and parallel thermal
pressure to magnetic pressure . MM waves are
linearly polarized structures that appear as sharp increases and decreases in
the magnetic field data (peaks and dips). They are frequently observed in
heliospheric plasma, in particular in different sheath structures. They are
the most widely studied in the planetary magnetosheaths
e.g.,, but also found
in cometosheaths e.g.,, in the heliosheath e.g., and ahead of the dipolarization front . MMs
are also studied in the solar wind e.g.,, behind interplanetary shocks
and in addition in interplanetary coronal mass ejections
(ICMEs; ).
However, to our knowledge, there are only a few studies on MMs in
the sheath regions of ICMEs . ICMEs are
interplanetary counterparts of coronal mass ejections (CMEs), large-scale
eruptions of plasma and magnetic flux that are ejected from the Sun on a
regular basis. The speed of an ICME often exceeds the magnetosonic speed in
the solar wind frame, and a shock wave and a sheath form upstream of the
ICME. A large fraction of ICMEs expand as they propagate away from the Sun,
primarily due to the decrease in the total solar wind pressure
e.g.,. Observations show that the expansion is still
significant at the orbit of the Earth e.g.,
and ceases at about 10–15 AU e.g.,and references
therein. In addition, due to the small deflection speed, the
solar wind plasma accumulates along the face of the body of the propagating
ICME over long distances . As a consequence, ICME-driven
sheaths contain layers of plasma and magnetic field that have accumulated at
different times and from different sources. ICME-driven sheaths are of great
interest for solar–terrestrial studies as they are strong drivers of
geomagnetic activity e.g.,see also ,
; , ;
, , and their shocks have a key role in
the acceleration of solar energetic particles e.g.,. Similarly to planetary magnetosheaths
e.g.,, MM waves may have large-scale effects on ICME sheaths.
Studies of planetary magnetosheaths have shown that peak-like MMs
occur in mirror unstable plasma, whereas dip-like MMs are observed in
both mirror stable and unstable plasma e.g.,. Furthermore, in 2-D hybrid simulations, dip-like
MMs were formed in low β plasma, whereas peak-like structures
were generated in high β plasma . Both MM
types are observed near the instability threshold (Cm≃0.5, see Eq. ),
whereas peak-like MMs become dominant when the
instability threshold is clearly exceeded (Cm ≳ 1). The cases where
MMs occur in mirror stable plasma may be remnants of MMs
that were generated earlier in time, when the mirror instability threshold
was exceeded e.g.,. The peak-like
MMs are proposed to form by the nonlinear saturation of mirror
instability and dissipate rapidly in mirror stable plasma, while the dynamic
evolution of pre-existing large amplitude plasma perturbations generate
dip-like MMs that can also exist in the plasma below the mirror
instability threshold see also ,
.
One of the key questions related to MMs is the source of free energy
for generation of these waves. In planetary magnetosheaths, two primary
mechanism have been identified to produce temperature anisotropy: the
quasi-perpendicular shock compression, which heats the ions perpendicular to
the magnetic field, and magnetic field line draping, which happens near the
magnetopause as the field lines drape around the magnetic obstacle
e.g.,. The shock compression has
been argued to be a source of free energy that generates peak-like MMs
and likely both sources need to be available in order to generate the
largest amplitude MMs . In the heliosheath and
cometosheaths, the ion pickup process is also identified as a free energy
source
e.g.,.
In the magnetosheath of the Earth, the occurrence of mirror peaks is highest
in the central magnetosheath, whereas mirror dips occur most frequently in the
flank magnetosheath and close to the magnetopause
e.g.,. MMs also occur more frequently when the
Alfvén Mach number (MA) upstream of the bow shock is large
e.g.,, and when the shock is quasi-perpendicular
e.g.,. However, MMs have
typically larger amplitudes behind quasi-parallel shocks .
In addition, while also isolated MMs exist , most MMs in planetary magnetosheaths occur as sequences
of peaks or dips, so called MM trains .
The previous studies on MMs in ICME-driven sheath regions are either
case studies or they have investigated only general plasma and magnetic field
conditions that favor generation of MMs. For example,
found signatures of MMs (anticorrelated fluctuations in the density
and magnetic field magnitude, enhanced temperature anisotropy and high plasma
beta) in front of an ICME, which was identified as a magnetic cloud. In
addition, the authors used a superposed epoch analysis to investigate how the
occurrence of MMs depends on the ICME properties. They discovered
MM favoring conditions in the sheath regions of magnetic clouds
(i.e., a subset of ICME featuring enhanced magnetic field, smooth rotation of
the magnetic field direction and low plasma beta; e.g., )
and ICMEs with preceding shocks, whereas ICMEs without magnetic cloud
structures and preceding shocks lacked these features.
In this paper, we conduct the first extensive statistical analysis of MMs
in 91 ICME-driven sheath regions detected by the Wind spacecraft. We
develop a semi-automated method to detect MMs in the magnetic field
data. This method can also be directly extended to find MMs in other
environments, e.g., in planetary magnetosheaths. Using the identified MMs,
we study their occurrence and properties, including duration,
amplitude and whether MMs tend to occur as isolated or as trains.
The mirror instability threshold condition is also investigated. In
particular, we study the dependence of MM occurrence and properties
on the fractional distance from the leading ICME shock to the ICME leading
edge, and on the shock strength and configuration (quasi-parallel and
quasi-perpendicular). We also discuss the dominant sources of free energy for
MM generation in ICME-driven sheath regions, and how our results
compare with the results in planetary magnetosheaths.
The paper is organized as follows: in Sect. 2, we introduce the used data
sets and describe the methods used to identify MMs in ICME-driven
sheath plasma data. Section 3 presents the statistical results of MM
occurrence frequency and MM properties. Finally, we discuss results
and conclusions in Sect. 4.
Data and methods
Data sets
We examine 91 ICME-driven sheath regions as listed by .
The sheaths were observed between January 1997 and April 2015. In our
statistical analysis, we investigate measurements from the Wind spacecraft
launched in November 1994 to a halo orbit around the L1 Lagrangian point with
the exception of a complex trajectory between 1999 and 2004. We use magnetic
field data with 3 s time resolution from the Wind Magnetic Fields
Investigation (MFI) instrument , and the proton number
density (np) and proton thermal speed data, both parallel
(v∥) and perpendicular (v⟂) to the magnetic field, from
the Wind Solar Wind Experiment (SWE) instrument . The data
are obtained from the NASA Goddard Space Flight Center Coordinated Data
Analysis Web (CDAWeb, http://cdaweb.gsfc.nasa.gov/, last access: November 2017).
To study the sheath plasma, we compute β∥ and β⟂
defined as β∥/⟂=2μ0kBnpT∥/⟂,pB2, where μ0 is the
vacuum permeability, kB is the Boltzmann constant and B is the magnitude
of the magnetic field. Temperatures T∥,p and T⟂,p are
computed from the thermal speed data as T∥,p/⟂,p=v∥/⟂2mp/kB, where mp is the proton mass.
To characterize the shocks preceding the sheaths, we use the Alfvén Mach
number (MA) and the shock angle (θBn) given by the
Heliospheric Shock Database (http://ipshocks.fi/, last access: November 2017), developed and maintained
at the University of Helsinki. If the database did not include the shock
parameters determined by the Wind spacecraft data (e.g., due to data gaps
around the shock) and the corresponding shock was identified by the ACE
spacecraft (located also at L1), the shock parameters from ACE were
used. In our event set, there were only four such cases.
Mirror mode (MM) identification
To identify MMs, it is important to know their expected timescales
in the region of interest. The timescale of MMs in the solar wind at
0.72 AU is reported to vary from a few seconds up to 40 s
. Similar durations are also reported in the magnetosheath
of the Earth . We therefore expect the MMs to have similar durations in ICME-driven sheath regions. The 3 s magnetic
field data from the Wind MFI instrument should thus have sufficient
resolution to detect most MMs.
Previous studies have mostly utilized linear polarization and compression of
MM waves when identifying these structures
e.g.,. In
practice, these studies have computed both the maximum variance direction of
the magnetic field vector (Bm; ) and its
direction with respect to the average magnetic field direction
b0=[B¯1,B¯2,B¯3]/|B¯1,B¯2,B¯3|,
where the subscripts 1, 2 and 3 denote the components of the magnetic field
vector, by applying the minimum variance analysis (MVA). Furthermore, the
eigenvalues of the magnetic variance matrix are compared to each other to
ensure that Bm is well defined.
When determining whether MMs occur in ICME-driven sheaths, we
investigate every 1 min data interval that has no missing data
points. Because the magnetic field data have 3 s time resolution, two
successive intervals may overlap by 57 s with each other. Similarly
to , and , we use the
MVA to calculate the eigenvalues of the maximum (λmax),
intermediate (λint) and minimum (λmin) variance
directions and the angle (θbm) between Bm and
b0. We also require that λmax/λint>1.5,
λmin/λint>0.3 and θbm<30∘ for an
interval to pass the MVA test. Overlapping intervals that pass the MVA test
are combined into one wider interval.
To identify individual MMs, we investigate the minima and maxima in
the magnetic field data of the combined intervals that have passed the MVA test
by applying methods used by . The standard
deviation (δ) is computed as δ=1N∑i=1N(Bi-B¯)2, where B1,…,BN are
the data points of the magnetic field magnitudes during the interval and
B¯ is their average. Skewness (S) is defined as S=M3σ3, where M3=1N∑i=1N(Bi-B¯)3, and the variance σ=δ2. Skewness describes the asymmetry of a distribution and is positive
for intervals with peaks and negative for intervals with dips see
e.g.,.
Examples of individual mirror modes (MMs) in ICME-driven sheath regions.
The shock preceding the ICME passed the Wind spacecraft at 11:10 UT on 16 April 1999
(a–c) and at 18:30 UT on 29 May 2003 (d). The investigated intervals are bounded by the red vertical lines
and the detected MMs are the structures limited by the blue vertical
lines. The skewness (S) of the investigated interval is given in each
panel. The angular changes in the magnetic field direction over the MM
structures (from left to right) are (a) 1.1 and 2.7∘,
(b) 2.3∘, (c) 6.7 and 4.4∘, and
(d) 6.1, 3.2, 1.7 and 4.5∘.
We define a MM as a structure through which the magnetic field
direction changes less than 10∘ . Similarly to
, the start and end of each MM structure
are defined as the nearest points of the magnetic field minimum or maximum that
satisfy the requirements B>B¯-δ and B>1.25BMIN when a
minimum is studied, and B<B¯+δ and B<0.75BMAX when a
maximum is studied.
If a structure contains more than two minima or maxima,
it is rejected. The angular change is calculated as the directional change of
the magnetic field vector between these edges. Minima and maxima satisfying
these conditions are called dips and peaks. Figure a–c show
examples of MM structures that were identified by the
above-mentioned selection criteria.
The skewness of the interval is compared to the number of detected dips and
peaks. The interval is considered to contain dip (peak)-like MMs if
the number of dips (peaks) exceeds the number of peaks (dips) and the
skewness has a negative (positive) value. If the value of skewness
contradicts with the number of detected dips and peaks, one can pose a
question whether MMs occur in the interval.
In ICME-driven sheaths, the magnetic field magnitude may vary strongly thus
affecting the value of skewness. Figure d shows an example,
where dip-like MMs are identified and the skewness is positive. In
addition, in some cases the magnetic field data can include series with both
peak and dip-like MMs. These series may contain, for example, a
peak-like mode that has the maximum close to the surrounding average field
magnitude, located between two dip-like MMs with the edges in the
minima of the dips. To distinguish whether the series contains dip- or
peak-like MMs and to detect MMs also in similar conditions
as in Fig. d, we compare the value of the skewness in the
above-described manner.
The identified MMs are classified as singles and trains. If a
studied interval contains more than one MM, it is considered a train
of MMs. In addition, the absolute (A) and relative (AR)
amplitudes of every identified MM are calculated as Adip/peak=|η2-Bdip/peak| and AR=Adip/peakη2, where η is the sum of the magnetic
field magnitudes of the left and right edges of the structure, i.e.,
η=Bleft+Bright and Bdip/peak is the magnetic field minimum or
maximum of the structure depending on which one is being studied.
Here we define the threshold value of the mirror instability as given by Eq. (), i.e., Cm=β⟂β∥-1-1β⟂ (see Sect. 1),
where the positive values of Cm refer to mirror unstable plasma. We
compute Cm for every data point in the sheath region, and in the
surroundings of each detected MM using plasma beta averages within
150 s from the extreme magnetic field magnitude of the MM.
As discussed in Sect. 1, the source of free energy may be different in
different parts of a sheath. In addition, ICME-driven sheaths at 1 AU
have typically developed over the several days it takes for an ICME to travel
from the Sun to the Earth orbit. Similarly to , we divide
the ICME-driven sheaths into sub-regions – near-shock, mid-sheath and near
leading edge (near-LE) regions – using a fractional distance parameter (F)
that gives the relative location in a sheath with values between zero (at the
shock) and one (at the leading edge of the ICME ejecta).
Statistics of mirror mode (MM) occurrence and the number of single MMs and MM trains.
Total number
Sheaths
91 (64 included MMs)
MMs
1129
Singles
397
Trains
244
Statistical results
Mirror mode (MM) occurrence frequency
In total, 1129 MM structures were identified using the scheme
described in Sect. 2. We found MMs within 64 of the total 91 ICME-driven
sheath regions. Thus, 70 % of the sheaths contained MMs, while about
one-third of the sheaths lacked MMs completely (see Table ).
Practically all identified MMs (1080, 96 %) were dip-like.
The pie diagrams in Fig. show how the identified MMs were
distributed to singles and trains (left), and the distribution of the number
of MMs in trains (right). The total number of events is shown in parentheses.
The diagram on the left indicates that the majority of MMs occurred in trains
(65 %). However, the diagram on the right reveals that 38 % of the trains
had only two MMs, i.e., 60 % of all identified MMs in this study were
singles or trains of two MMs. Only 14 % of the trains had more than seven
MMs. The largest number of MMs in a train was 27 and we checked that data
gaps do not have significant impact on this.
Figure shows the frequency histogram of ICME-driven sheaths as a
function of the total number of MMs identified within the sheath. The
majority of sheaths had only a few MMs and the distribution is skewed to the
right with a long tail. The top row in Table gives the median,
the lower and upper quartiles (LQ and UQ) of the number of MMs in those 64
sheaths that had MMs, and the percentage of the sheaths that had more
MMs than the UQ. The median number of MMs in a sheath is 9, and 50 % of
the observations fall between 3 and 24. The maximum number of MMs identified
within one sheath region was 109. We also note that 70 % of all detected
MMs occurred in those 16 sheaths that had more than 24 MMs.
As described in Sect. 2, we divided the sheaths into three sub-regions:
near-shock, mid-sheath and near-LE sub-regions. Table
also gives the median, LQ and UQ for the observed MMs in these
sub-regions. The last column again shows the percentage of sheaths with more
than 24 MMs in the sub-region in question. All sub-regions have approximately
the same median and LQ, while the UQ has the highest value in the near-shock
sub-region and the lowest value in the near-LE sub-region.
Division of detected MMs to individual ones and the ones in MM
trains and the division of MM trains according to the number of MMs in them.
The frequency histogram of ICME-driven sheaths as a function of
number of MMs within the sheath in bins of 2 MMs. The grey dashed lines
show the lower (3) and upper (24) quartiles of the number of MMs.
Quartiles of ICME-driven sheath regions (whole sheath) and different
sub-regions according to the number of MMs in them. The quartiles are
computed for those ICME-driven sheaths and sub-regions that contained MMs.
The last column shows the percentage of sheaths with more than 24 MMs in the
sub-region in question.
Lower quartile (MMs)
Median (MMs)
Upper quartile (MMs)
> 24 MMs
Whole sheath
3
9
24
25 %
Near-shock
2
4
13
8 %
Mid-sheath
2
3
11
6 %
Near-LE
2
4
7
6 %
Probability of observing MMs within a 0.1 fractional distance (F)
bin as a function of F from the ICME-driven shock (F=0 refers to the
shock and F=1 to the ICME leading edge). The colors show different
requirements for the number of MMs in a bin. The probability is defined as
the ratio of the number of intervals containing MMs to the total number of
intervals within each bin (64 for all intervals). The error bars of the blue
curve represents the division of MMs within the whole sheath and are defined
as the ratio of the number of MMs observed within each bin to the total
number of MMs within the whole F interval from 0 to 1. We only consider here those ICME-driven sheaths that contained MMs. We have checked that
the distributions are not biased due to the sheaths containing a large number
of MMs.
The occurrence of MMs in the sheath is further examined in Fig. ,
which shows the probability of a fractional distance interval (F) containing
MMs in 0.1 bins. The color of the curves indicates the lower limit of the
number of MMs that we required to be in each bin (> 0, > 2 or > 4 MMs).
The probability is the ratio of the bins that contained MMs to the total
number of bins considered. The error bars of the blue curve in the figure
represents the division of all observed MMs in this study as a function of
F and thus the sum of all the error bars gives 1.0. MMs were evidently
observed everywhere in the sheath and the median fractional distance for
detecting a MM is F=0.47 (i.e., at the middle of the sheath). Only the blue
curve in Fig. shows a clear decreasing trend when moving from the
shock towards the ICME leading edge, while the other two curves have
relatively flat profiles. In addition, the size of the error bars does not
have drastic variations throughout the sheath but the largest portion of MMs
was observed in the bin within the near-shock sub-region. We have checked
that the results shown in Fig. are not biased by the sheaths
having a large number of MMs. Furthermore, 39 % of all observed MMs were
located in the near-shock sub-region. The corresponding percentages are
31 and 29 % for the mid-sheath and near-LE sub-regions,
respectively.
This is in
agreement with Table , which shows that the large number of MMs
(> 24 MMs) was detected most frequently in the near-shock sub-region.
MM properties
Statistics on the duration and amplitudes of the detected MMs are illustrated
in Fig. and summarized in the three first rows of Table . Figure a presents the MM frequency histogram
for the duration of a MM structure in 3 s bins. After the peak in the
9 s bin, the occurrence of MMs drops quickly with increasing
duration. The average duration of MMs for our data set is ∼12 s
(Table ). The average durations are similar for each
investigated sub-region. Table further shows MM properties in
singles and trains. Single MMs have slightly larger duration (13.7 s)
than MMs in trains (11.6 s).
The MM frequency histograms for the amplitude and relative amplitude in
0.7 nT and 0.04 bins are shown in Fig. b and c. The amplitudes of MMs are biased towards small values
(1–3 nT), and the average amplitude is 3.2 nT (Table ).
However, we identified some MMs with relatively large
amplitudes (> 10 nT). Similarly, the distribution of relative
amplitude is strongly weighted towards smaller values with an average 0.35.
Table also shows that the amplitudes are on average largest in
the near-shock region and smallest in the near-LE sub-region. The relative
amplitudes, however, are very similar between all sub-regions. The absolute
amplitudes are slightly larger for single MMs than for MMs in trains but
there are no significant differences between relative amplitudes (Table ).
Statistics of the properties of MMs in ICME-driven sheath regions.
(a) The distribution of detected mirror modes (MMs) according to their
duration (ΔTMM) and in 3 s bins. (b) MM occurrence as a function
of their amplitude (A) in 0.7 nT bins. (c) MM occurrence as a
function of the relative amplitude of a structure (AR) in 0.04 bins. We
have checked that the distributions are not biased due to the sheaths
containing a large number of MMs.
Figure gives relative frequency distributions for perpendicular
plasma beta (β⟂), parallel plasma beta (β∥),
plasma beta anisotropy (β⟂/β∥) and mirror
instability threshold (Cm) values. We calculated the distributions for
both the MM structures including their 5 min surroundings (black
curves), and the parts of the sheath that lacked MMs, i.e., for the “non-MM
sheath” (blue curves). Figure a and b show
that β⟂ and β∥ are clearly higher in MMs than
in non-MM parts of the sheath.
Figure c and d also show that the distributions
for the plasma beta anisotropy and Cm have clear differences between the
regions where MMs were detected and not detected. Firstly, the plasma beta
anisotropy distribution is wider for the non-MM sheath and has a distinct
tail extending to high values of β⟂/β∥. The
distribution of regions with MMs peaks at slightly higher values, but
averages and medians (see the Supplement) of the plasma beta anisotropy are
about the same. The average beta anisotropies for MMs are very similar in all
investigated sub-regions but in non-MM parts, the anisotropy is highest in
the near-shock sub-region. MMs in trains have on average slightly
larger β⟂/β∥ values than single MMs (Table ).
Secondly, the Cm distribution for non-MM sheath has almost solely negative
values, indicating mirror stable plasma. According to Table ,
96 % of the non-MM sheath has Cm<0. Interestingly, also 80 % of the
MMs have Cm<0 and a significant part of the distribution resides in the
mirror stable region (Fig. d). However, we emphasize that the
Cm values are generally larger for MMs than elsewhere in the sheath. We
also note that only a very small fraction of Cm values are greater than 1.
This is consistent with nearly all of the identified MMs being classified as
dip-like MMs. As we discussed in Sect. 1, in planetary
magnetosheaths, peak-like MMs occur predominantly in plasmas with
Cm>1. Again, MMs in different sheath sub-regions have very similar Cm
values and the percentages of Cm>0 observations. The percentage of Cm>1 observations is, however, highest in the near-shock sub-region. In
non-MM parts, the average Cm value and the percentages of Cm>0 and
Cm>1 observations are highest near the shock. The average Cm
decreases towards the ICME leading edge but the percentages do not vary
significantly between the mid-sheath and near-LE sub-regions. Table
shows that MMs in trains (24 %) occur more often in mirror
unstable plasma than single MMs (12 %).
We also examine the statistical significance of the differences between the
β⟂/β∥ and Cm distributions of MMs and non-MM
sheath by using Student's t test and assuming unequal variances between the
distributions . The p values indicating the probability
that the averages between two distributions are the same are 0.003 and
6.096×10-226 for the distributions of
β⟂/β∥ and Cm, respectively. The values are clearly below the
nominal significance level 0.05 see for example
suggesting a high statistical significance.
Relative frequency of (a) perpendicular plasma beta
(β⟂), (b) parallel plasma beta (β∥), (c) plasma
beta anisotropy (β⟂/β∥) and (d) mirror
instability threshold (Cm) values in the surroundings of detected MMs (black curves) and in the parts of the sheath where MMs were
not detected (blue curves). In panels (c) and (d), the red dashed vertical
lines show the threshold values for mirror unstable plasma, i.e.,
β⟂/β∥=1 and Cm=0. See details from Sect. 1.
The grey dashed line in panel (d) marks Cm=1, i.e., the threshold for
detecting peak-like MMs. We have checked that the distributions are
not biased due to the sheaths containing a large number of MMs.
Average properties of MMs and their division to the individual ones
and the ones in trains in different sheath sub-regions (the percentages in
parentheses indicate the division of singles and MMs in trains between
different sub-regions). The typical plasma betas, plasma beta anisotropy and
mirror instability threshold values and the relative amount of plasma in a
certain sub-region that is above the mirror instability threshold (Cm=0)
and that exceeds Cm=1 in both plasma containing MMs and non-MM plasma are given. The
last column shows the properties of MMs and plasma in the ICME-driven sheaths
generally, i.e., in whole fractional distance interval from 0 to 1. The
standard error of the mean (SE) defines the errors and is defined as SE=δ/n, where δ is the standard deviation and n is the
size of a sample.
Near-shock
Mid-sheath
Near-LE
Whole sheath
Properties of MMs and their plasma surroundings
〈ΔTMM〉 (s)
12.2 ± 0.3
12.7 ± 0.4
12.0 ± 0.4
12.3 ± 0.2
〈A〉 (nT)
3.6 ± 0.2
3.4 ± 0.2
2.6 ± 0.1
3.2 ± 0.1
〈AR〉
0.350 ± 0.008
0.357 ± 0.009
0.352 ± 0.009
0.353 ± 0.005
singles
32 % (36 %)
41 % (37 %)
33 % (28 %)
35 %
trains
68 % (41 %)
59 % (29 %)
67 % (30 %)
65 %
〈β⟂〉
8.9 ± 0.9
9.2 ± 0.8
7.7 ± 0.5
8.6 ± 0.5
〈β∥〉
9.5 ± 1.0
9.3 ± 0.8
7.6 ± 0.5
8.9 ± 0.5
〈β⟂/β∥〉
1.07 ± 0.03
1.02 ± 0.02
1.02 ± 0.02
1.04 ± 0.02
〈Cm〉
-0.31 ± 0.07
-0.22 ± 0.02
-0.30 ± 0.03
-0.28 ± 0.03
Cm>0
21 %
19 %
20 %
20 %
Cm>1
2.98 %
0.64 %
0.00 %
1.28 %
Plasma properties in non-MM sheath
〈β⟂〉
2.4 ± 0.2
2.0 ± 0.2
1.7 ± 0.1
2.0 ± 0.1
〈β∥〉
2.6 ± 0.2
2.2 ± 0.2
1.9 ± 0.1
2.2 ± 0.1
〈β⟂/β∥〉
1.32 ± 0.11
1.07 ± 0.02
1.05 ± 0.03
1.14 ± 0.04
〈Cm〉
-1.21 ± 0.11
-1.93 ± 0.08
-3.65 ± 0.11
-2.32 ± 0.06
Cm>0
6 %
3 %
3 %
4 %
Cm>1
1.1 %
0.4 %
0.5 %
0.6 %
Average properties of both individual MMs and the ones in MM trains
and their plasma properties. The errors are defined by SE. The relative
amount of plasma that is above the mirror instability threshold (Cm=0)
and that exceeds Cm=1 in the surroundings of both types of MM are given.
Singles
Trains
〈ΔTMM〉 (s)
13.7 ± 0.4
11.6 ± 0.2
〈A〉 (nT)
3.4 ± 0.2
3.1 ± 0.1
〈AR〉
0.335 ± 0.008
0.363 ± 0.007
〈β⟂〉
7.0 ± 0.8
9.5 ± 0.5
〈β∥〉
7.2 ± 0.9
9.7 ± 0.6
〈β⟂/β∥〉
1.00 ± 0.02
1.06 ± 0.02
〈Cm〉
-0.37 ± 0.06
-0.22 ± 0.03
Cm>0
12 %
24 %
Cm>1
1.2 %
1.3 %
Dependence on shock properties
Next, we examine how MM occurrence and plasma properties of an ICME sheath
depend on the shock angle (θBn) and shock Alfvén Mach number
(MA), i.e., on the strength of the shock.
Figure shows the sheath frequency distributions for these
parameters. The majority of the shocks were quasi-perpendicular (Q⟂,
θBn>45∘) with a median angle θBn of 59∘.
Interplanetary shocks are typically weak e.g.,, and
also in our data set the Alfvén Mach numbers were typically less than
4. The median MA is 3.2, but the shock MA distribution has a tail
that extends to MA values up to 14. The average values of the shock
parameters are 〈θBn〉=56∘±2∘ and 〈MA〉=3.7±0.2.
Table shows the average values and standard errors of the
parameters for non-MM sheaths (i.e., sheaths in which we did not identify any
MMs) and sheaths that had more MMs than the upper quartile (see Table ).
On average, the sheaths with more than 24 MMs were associated
with higher MA shocks than non-MM sheaths. The p value of the MA
distributions is 0.01 indicating that the difference is statistically
significant. In addition, although the average shock angle does not differ
between these two subsets, we note that the median MA is 6.8 for those
sheaths that had more than 24 MMs and that were preceded by a quasi-parallel
shock (Q∥, θBn<45∘) (5 events), whereas the
corresponding median for the sheaths with more than 24 MMs that were preceded
by a quasi-perpendicular shock (11 events) is only 3.8.
The frequency histograms of (a) preceding shock angle
(θBn) in 10∘ bins and (b) Alfvén Mach number (MA)
in bins of unity for investigated ICME-driven sheath regions. The grey dashed
vertical lines divide both shock parameter distributions to quartiles. The
red dashed horizontal lines indicate the lower quartiles according to the
number of ICME-driven sheaths in each shock parameter interval.
The averages of the shock parameters of the ICME-driven sheaths that
had more than 24 MMs and that did not have any MMs. The errors are defined
by SE.
Number of MMs
Number of
〈θBn〉
〈MA〉
in a sheath
events
(∘)
> 24
16
57.7 ± 5.0
5.06 ± 0.81
0
27
57.5 ± 4.2
2.66 ± 0.15
We next investigate in more detail the occurrence of MMs depending on the
shock angle and shock strength (Fig. ). We consider here only the
near-shock region because, as discussed in Sect. 1, sheaths observed at
1 AU have accumulated over the several days that the ICME has propagated
through the heliosphere, and the plasma near the shock has been most recently
passed across the shock. In the middle of the sheath and near the ICME
leading edge, additional processes, such as the draping of the interplanetary
magnetic field around the ejecta, may play a role in regulating the plasma.
Figure shows the probability of a near-shock sub-region
containing more than 4 MMs as a function of investigated shock parameters.
The probability is calculated by requiring that more than four MMs had to be
detected in the near-shock sub-region and the fractional numbers giving the
probabilities are also given above the bars. Figure a shows
that no near-shock sub-regions with more than four MMs were found if the
preceding shock had θBn<20∘, while the probability is the
highest for the bins in the range 30∘<θBn<50∘. For
each bin within the interval 50∘<θBn<90∘, the
probability is approximately 0.20. The concurrent dependence of the
probability on the both shock parameters is discussed later in this section
and the results of the analysis are shown in Table .
Figure b shows a clear dependence between the probability and
shock strength. MMs were detected most frequently (probability ∼1) when
the shock MA was high. The probability decreases significantly with
decreasing MA and for MA<4, it is <0.2.
The importance of the shock angle and shock strength for generating MMs is
further investigated in Table . The near-shock sub-regions were
divided into different sub-groups according to the shock parameters
associated with them. For these different sub-groups, Table
shows the probability of a near-shock sub-region containing MMs when a
different lower limit of MMs was required to be contained in a sub-region
(> 0, > 2 or > 4 MMs). In addition, we calculated the average
θBn and MA of each sub-group.
Probability of a near-shock sub-region containing more than four MMs
in one-dimensional bins of corresponding shock parameters. Probability is
defined as the ratio of the number of the near-shock sub-regions containing
more than four MMs to the total number of the near-shock sub-regions within
each bin. The ratio is also given as a fractional number above each bin.
On the first two rows of Table , the sub-regions are divided
into sub-groups according to the shock strength. The importance of MA is
again obvious. While the average θBn does not vary much between
these sub-groups, all the probabilities of the sub-group whose shock strength
was above the average MA are clearly higher than the corresponding ones of
the sub-group whose shock Alfvén Mach numbers were below the average.
Moreover, all the probabilities of the sub-group having MA> 〈MA〉 are
≥50% while the probability of the other group is not higher than
38%.
Probability of a near-shock sub-region to contain a different number
of MMs. The near-shock sub-regions are divided into different sub-groups
according to the associated shock parameters. The average
θBn and MA values of each sub-group are also shown. The values
in bold in brackets are the corresponding values when the sheaths
whose shock had 0∘<θBn<20∘ and MA>8 were omitted
from the analysis (see Fig. ). The errors are defined by SE.
Number of events
> 0 MMs
> 2 MMs
> 4 MMs
〈θBn〉
〈MA〉
MA< 〈MA〉
58 (54)
38 % (37 %)
24 % (26 %)
14 % (15 %)
56∘±3∘ (59∘ ± 3∘)
2.6 ± 0.1 (2.6 ± 0.1)
MA> 〈MA〉
30 (26)
77 % (73 %)
60 % (58 %)
50 % (50 %)
58∘±4∘ (61∘ ± 4∘)
5.8 ± 0.5 (5.3 ± 0.3)
Q∥ (θBn<45∘)
25 (19)
56 % (53 %)
40 % (47 %)
32 % (42 %)
29∘±3∘ (33∘ ± 2∘)
4.0 ± 0.4 (4.1 ± 0.5)
Q⟂ (θBn>45∘)
63 (61)
49 % (48 %)
35 % (33 %)
24 % (21 %)
67∘±2∘ (68∘ ± 2∘)
3.6 ± 0.3 (3.3 ± 0.2)
Q∥ and MA< 〈MA〉
15 (11)
27 % (18 %)
7 % (9 %)
7 % (9 %)
27∘±3∘ (32∘ ± 2∘)
2.7 ± 0.2 (2.7 ± 0.2)
Q⟂ and MA< 〈MA〉
43 (43)
42 % (42 %)
30 % (30 %)
16 % (16 %)
66∘±2∘ (66∘ ± 2∘)
2.6 ± 0.2 (2.6 ± 0.2)
The near-shock sub-regions associated with a quasi-parallel shock contained
MMs more frequently than the ones associated with a quasi-perpendicular shock
(see the third and fourth rows of Table ). However, the
quasi-parallel shocks tend to have larger MA than the quasi-perpendicular
shocks and this difference becomes stronger if the cases below the lower
quartiles of θBn and MA distributions are omitted (see
Fig. ). Moreover, we note that 40 % of all quasi-parallel shocks had
MA> 〈MA〉, while this was the case for 32 % of quasi-perpendicular
shocks. Furthermore, 50 % of the shocks that had
30∘<θBn<50∘ had MA> 〈MA〉 explaining the high
probability in this θBn interval (see Fig. a).
Although generally a near-shock sub-region associated with a quasi-parallel
shock contained MMs more frequently than the ones with a quasi-perpendicular
shock, the two lowest rows in Table imply that if the shock
Alfvén Mach number is below the average of this study, the probability of a
near-shock sub-region containing MMs is higher when the shock is
quasi-perpendicular than if it is quasi-parallel. When examining the
probabilities of the most loose lower limit (> 0 MMs) in the two lowest
sub-groups in Table , we note that the probability of a
sub-region having MMs is 42 % for the sub-group that is associated with a
quasi-perpendicular shock but only 27 % for the quasi-parallel sub-group.
Also, the probabilities of the other lower limits are higher for
quasi-perpendicular shocks than for the quasi-parallel ones. The average
MA, however, does not vary much between these sub-groups.
Finally, in Fig. , we study how β⟂,
β⟂/β∥ and Cm in the near-shock sub-region
depend on shock properties and vary when moving downstream from the shock
towards the ICME leading edge. Both averages and medians are shown.
The top row of Fig. investigates the variations with the shock
angle θBn. The first panel, Fig. a, shows that both
the average and median of β⟂ peak in the
30∘<θBn<40∘ bin. The median then decreases with
increasing shock angle and increases only slightly for the most perpendicular
bin, while the average values increase again considerably for the two most
perpendicular bins. Both the average and median of the plasma beta anisotropy
increase generally with the increasing shock angle. The error bars, however,
are large in the perpendicular regime. For Cm, in turn, there is no
obvious trend with the shock angle (Fig. c).
The middle row gives the dependence on the shock strength (MA). We first
note that β⟂ (Fig. d) increases with increasing
shock strength. This explains the sharp peak of β⟂ in the
30∘<θBn<40∘ bin we noted previously in Fig. a
(see also the Supplement). Figure e suggests
that the beta anisotropy is smallest when the shock is strong. The large
peaks in averages for both panels (d) and (e) for the MA=6–7 bin are due
to a few exceptionally large values of β⟂
(β⟂/β∥>100). The mirror instability threshold
value Cm, in turn, generally increases as a function of MA (Fig. f).
The variations with the fractional distance are given in the bottom row.
Figure g shows that after reaching its maximum in the
0.2<F<0.3 bin, β⟂ decreases almost monotonically as a function
of F. The average value shows another, weaker peak in the 0.5<F<0.6 bin.
The beta anisotropy (Fig. h) and Cm (Fig. i)
both peak right after the shock and then decrease with increasing F. The
averages again show a peak in the 0.5<F<0.6 bin, explained by the
corresponding maximum in β⟂.
Average and median β⟂,
β⟂/β∥ and Cm (see panel c for color codes)
in the near-shock sub-region as functions of
θBn (a–c)
and MA (d–f) in bins of 10∘ and unity, respectively.
(g–i) Show the plasma parameters as a function of F in bins of
0.1. Due to low statistics, the sheaths whose shock had
0∘<θBn<20∘ are omitted in panels (a–c) and the ones
whose shock had MA>8 are omitted in panels (d–f) (see Fig. ).
The error bars are defined by SE.
Summary and discussion
In this paper, we have identified mirror modes (MMs) from 91 ICME-driven
sheaths using a semi-automated method and investigated their occurrence and
properties.
We summarize the key findings of this study as follows:
Approximately 70 % of the investigated sheaths had MMs.
Practically all MMs (96 %) in ICME sheaths were dip-like.
80 % of MMs occurred in mirror stable plasma.
Single and train MMs had no drastic differences in their properties.
Single MMs were slightly longer in duration and had slightly larger
amplitudes than MMs in trains. In addition, MMs in trains occurred more
frequently in mirror unstable plasma than single MMs (24 and 12 %,
respectively).
MMs were observed throughout the sheath.
MMs had similar properties throughout the sheath, except that their amplitudes were larger for MMs occurring near the shock.
Shock strength was the most important parameter in controlling the occurrence of MMs.
The perpendicular plasma beta and plasma beta anisotropy increased with increasing shock strength and shock angle, respectively.
The mirror instability threshold value generally decreased when moving from the shock towards the ICME leading edge.
Similarly to the Earth's magnetosheath e.g.,, we found
that MMs are fairly common structures in ICME-driven sheath regions. More
than two-thirds of the investigated 91 ICME-driven sheaths had at least one
MM structure. However, sheaths that contain a larger number of MMs (> 24)
were relatively rare, and on average only a few MMs were detected per sheath.
In addition, because MMs are relatively narrow structures, they cover only a
small portion of the sheath, e.g., in our study only 3 % for the sheath
that contained the largest number of MMs.
Similar to planetary magnetosheaths e.g.,, we found
that MMs tend to occur in trains also in ICME sheaths. However, 60 % of all
observed MMs occurred as singles or trains of two. We point out that the
concept of a MM train is less important when MMs are dip-like; an individual
dip-like MM forms a magnetic bottle and is stable by itself. In the case of
peak-like MMs, in turn, a train of at least two MMs are required to create
a structure between which ions are trapped
e.g.,. Thus, two dip-like MMs in the
same MVA test interval are not necessarily related in any way.
We found MMs everywhere in the ICME-driven sheaths, i.e., from the
shock to the ICME leading edge. The number of MMs was largest near the shock
and consistently we found that both perpendicular plasma beta and the mirror
instability threshold value generally decreased as a function of fractional
distance. The probability of observing MMs, instead, did not vary
significantly as a function of fractional distance. In planetary
magnetosheaths, MMs are also found from the bow shock to the magnetopause,
but their occurrence rate is highest in the central or inner magnetosheath
e.g.,.
The fact that the clear majority (80 %) of MMs in our study occurred in
mirror stable plasma, i.e., where Cm<0 is consistent with practically
all of them (96 %) being dip-like (see Sect. 1). Our results are also
consistent with , who reported that the plasma in ICME sheaths
is generally only marginally mirror unstable and the average beta anisotropy
in these sheaths is enhanced and
β⟂/β∥≃1.2–1.3. We, however, found that the
regions where MMs occur are different from those where they are absent; the
relative frequency distributions of Cm and beta anisotropy were distinctly
different for non-MM regions and in regions with MMs. Moreover, both
perpendicular and parallel plasma beta were on average higher in regions with
MMs.
The relative MM amplitude distribution in ICME sheaths is roughly consistent
with the previous studies of MMs in the Earth's magnetosheath
e.g.,. Absolute amplitudes, however, are clearly higher
for MMs found in the Earth's magnetosheath. and
reported MMs in the magnetosheath of the Earth with
absolute amplitudes approximately between 10 and 20 nT.
Although some MMs in our study had amplitudes up to 10 nT, the
average absolute amplitude was only about 3 nT.
Smaller amplitudes (A) of MMs in ICME-driven sheaths are presumably due to
the limited amount of free energy. In planetary magnetosheaths the largest MM
amplitudes are found in the inner magnetosheath e.g.,
suggesting that both shock compression and field line draping may play an
important role . As discussed previously, in ICME
sheaths, the MMs are most abundant and have largest amplitudes close to the
shock suggesting that the shock compression is the dominant source of free
energy. MMs closer to the ICME leading edge could hence be remnants of MMs
generated near the shock during earlier phases of the sheath evolution.
Because ICMEs expand laterally, the plasma is unable to flow around the ICME
and it accretes in front of the ejecta maintaining the record of previous
interaction .
This scenario could be compared to the case of MMs observed in the Earth's
magnetosheath and can be used to explain the observed absence of mirror peaks
in ICME sheaths. In the magnetosheath of the Earth, the initial growth of
mirror structures occurs in the middle magnetosheath where spacecraft
consistently observe strongly unstable plasma and MMs have the form of
magnetic peaks e.g.,. This region is not observed in
the ICME sheaths and due to the low anisotropy, the plasma is typically
mirror stable or marginally unstable and MMs are observed as dip-like
structures.
Since the plasma we observe in the ICME sheath has accumulated over time, it
may contain remnants of mirror structures created at smaller heliocentric
distances, where the plasma was hotter, denser and more mirror unstable. As
the ICME propagates away from the Sun and the sheath plasma expands, the
plasma becomes more stable and only mirror dips remain. In this respect, the
ICME sheath expansion is analogous to plasma expansion at the flanks of the
Earth's magnetosheath or in the plasma depletion layer, where MMs
also appear predominantly as dips . Furthermore,
report that in mirror stable plasma regions, such as the
magnetosheath flanks, previously generated dip-like MMs are eventually slowly
damped. This could explain the smaller amplitude of MMs close to the ICME leading
edge.
The absence of mirror peaks could also be explained by the reduced amount of
free energy and weaker mirror instability. The excitation of MMs and
their rapid transition from a quasi-sinusoidal wave to a nonlinear structure
is not fully understood and numerical studies , as well as
observations , suggest that in weakly unstable plasmas, the
instability saturates directly into a dip-like structure and peaks are only
formed when the plasma is sufficiently far from the threshold.
Our suggestion that the shock compression is the dominant free energy source
for MMs in ICME sheaths is supported by our finding that the largest number
of MMs were associated with much stronger shocks than those events that
lacked MMs. We also found that both β⟂ and the mirror instability
threshold value increased with increasing shock strength.
In turn, we found that the shock angle θBn did not have an obvious
effect on the probability of observing MMs. However, for low Alfvén Mach
number shocks, we found MMs more frequently behind quasi-perpendicular shocks
than behind quasi-parallel shocks, and the beta anisotropy in the near-shock
sub-region was larger in sheaths behind quasi-perpendicular shocks,
consistent with the observations in the Earth's magnetosheath
e.g.,see also ,
.
We can hence conclude that in ICME-driven sheaths, the shock properties have
a significant impact on the MM generation and the shock compression has an
essential role as a source of free energy. MMs are most likely to form when
the shock is strong and quasi-perpendicular and they are most abundant close
to the shock. MMs closer to the ICME leading edge are likely remnants of MMs
formed at earlier times in the vicinity of shocks.
New opportunities to study MMs will be provided by the European Space
Agency's Solar Orbiter and with NASA's Parker Solar Probe
missions. Solar Orbiter will have an orbit whose distance
from the Sun varies between 0.28 and 0.9 AU, while Parker
Solar Probe will make plunges as close as 10 solar radii from the Sun. These
missions will, in particular, allow for resolving how MMs are generated during the
earlier phases of the sheath evolution.
Finally, we discuss shortly whether our results indicate if Alfvén ion
cyclotron (AIC) waves could occur in ICME-driven sheaths. AIC waves are
other ultra-low-frequency waves that are driven by a temperature anisotropy.
In the Earth's magnetosheath, AIC waves typically occur after a weak
quasi-perpendicular shock or in the magnetosheath flanks
e.g.,. In addition, they are observed in a plasma that
is mirror stable and that has low parallel plasma beta
(β∥≲3) and
β⟂/β∥≳1.2
e.g.,.
In our data set, the ICME sheaths were often preceded by a
quasi-perpendicular shock and the shocks were mostly weak (MA<4). In
addition, the beta anisotropy distribution in non-MM sheath has,
interestingly, clearly a longer tail towards the higher anisotropy values
than the distribution of MMs. The average parallel plasma beta was, in
addition, β∥=2.2±0.1 in the non-MM parts of the ICME
sheaths. Furthermore, the ICME sheaths were only marginally mirror unstable.
Thus, our results imply that the plasma conditions in ICME sheaths could be
favorable for AIC waves to occur.