Introduction
Mirror mode (MM) waves are a key ingredient of the wave activity in planetary
and cometary magnetosheaths see,
e.g..
The waves are generated by a temperature asymmetry and showed
that for a bi-Maxwellian plasma the instability criterion is given by
1+β⟂1-T⟂T‖<0.
The newly created ions (from ionization of exospheric atoms) are picked up by
the solar wind magnetic field creating a ring-beam distribution. Such
distributions are unstable and will produce ion cyclotron waves or MM waves
see, e.g.. At crossing the quasi-perpendicular bow shock
the ions are mainly heated in the perpendicular direction, with respect to
the background magnetic field, when compared to the parallel direction,
increasing the already existing temperature asymmetry of the ring-beam
distribution. Theoretically, the growth rate for MM waves was estimated by
to be proportional to the proton cyclotron frequency γ∝0.1ωc,p; however, with spacecraft
observations have shown that this is an overestimation.
Another driver for MM waves is magnetic field line draping see
also, which serves as source of “free energy” in
planetary magnetosheaths. As the shocked solar wind moves deeper into the
magnetosheath, the planet will act as a conducting obstacle in the flow and
will “hang up” the magnetic field in its neighbourhood, whereas the parts
of the field lines further out from the Venus–Sun line will continue to flow
with the magnetosheath flow velocity. This causes the field lines to drape
around the planet . Field line draping around Venus's
ionosphere has two effects: first it leads to a squeezing of the plasma, by
which the hot-T‖ plasma is sent towards the downstream region, and
secondly the magnetic tension leads to an increase of T⟂ see
also.
In the solar wind this distribution will generate mainly ion cyclotron waves
because of the solar wind plasma-β usually being lower
than 1, but it sometimes also gives rise to MM waves at a very low occurrence
rate of ∼ 4 per day . In the magnetosheath, however, MM
waves
are most likely expected to be generated. and ,
however, showed that during a period of exceptionally low solar wind
plasma-β (∼ 0.35), the magnetosheath can be prone to a high
occurrence rate of ion cyclotron waves see also.
At Venus these MM waves were first discovered from the Venus
Express mission VEX, using only the magnetometer data
. The waves were shown to have a period between ∼ 4 and
∼ 15 s depending on the location in the magnetosheath. A statistical
study over 1 Venus year (i.e. 224 Earth days) during solar minimum was
performed by , which showed that the occurrence rate of MM waves is
highest just behind the bow shock as well as close to the ionopause: the
former location because of the perpendicular heating by the bow shock
increasing the temperature anisotropy and the latter location because of the
magnetic field pile-up, increasing field strength and thereby the temperature
anisotropy through the first adiabatic invariant. It was also demonstrated
that MM waves are mainly generated for quasi-perpendicular bow shock conditions,
as expected.
In this paper, the solar maximum data are analysed first to obtain the
occurrence rate and strengths of the MM waves. Then the results are compared to
those for solar minimum. Further statistical analysis is performed on the MM
strength, and the growth rate is estimated for both solar conditions. A
discussion about the differences and similarities between the two states of
solar activity is then performed and the paper ends with some conclusions and
concluding remarks.
Statistical study
In order to extend the solar minimum statistical MM study
24 May–31 December 2006, to solar maximum, 1 Venus year
(224 Earth days) of the 1 Hz magnetometer data from Venus Express, around
solar maximum 2011–2012, was used (1 November 2011–10 June 2012) and
processed in the same way. The MM waves that were found in Venus's magnetosheath
are shown in Fig. in cylindrical coordinates XVSO,
and the distance of VEX from the Venus–Sun line R=YVSO2+Z VSO2.
First, there are more events for solar maximum (a total of 1857 events)
than for solar minimum (a total of 1637 events). Also, it can be seen that
the events for solar maximum already appear more distant from Venus as the
nominal solar maximum bow shock location is at greater
distances than the solar minimum bow shock location .
Positions of the MM events found in the magnetosheath for solar
minimum (top, a total of 1637 events) and maximum (bottom, a total of
1857 events). The two thick black curves in each panel show the location of
the bow shock as taken from for solar minimum and from
for solar maximum. The solid line close to Venus is the
location of the ionosphere, taken from in both cases. The
cyan dashed arrows in both panels show the distance along the flow line of
one event to the model bow shock.
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For solar maximum the ionization rate around Venus is much higher than for
solar minimum see also, e.g., and thus the bow shock and
ionopause move outward from Venus e.g..
Mirror mode wave observation rate
Using the location of VEX and the time interval that the spacecraft is within
a 0.25×0.25RV box, the MM observation rate is calculated,
defined as
P=number of events in boxtime spent in box.
Comparison of the observational rate P of MM waves in Venus's
magnetosheath for solar minimum (top) and solar maximum (bottom). The two
thick black curves in each panel show the location of the model ionosphere
and model bow shock as in Fig. .
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Although there are, as expected, differences in the details of the
observation rate plots in Fig. , basically the major
differences are the higher number of events for solar maximum (1857 vs. 1637)
and more events further away from Venus as the bow shock location moves
outward. The highest observation rates can be found behind the bow shock and
towards the ionopause, along the flow channel of the plasma in the
magnetosheath, parallel to the XVSO axis, close to the
Venus–Sun line with 1.0≤R≤1.5. This will be discussed in more
detail below. As a comparison, the maximum observational rate for solar
minimum is P≈3 events per hour, whereas for solar maximum the
maximum observational rate is P≈4 events per hour.
Taking a closer look at the two panels in Fig. and the
distribution of the observational rate, a different behaviour for solar
minimum and maximum can be seen. Whereas for solar minimum the highest rates
are observed close to the nominal bow shock (e.g. the red squares labelled A
and B) and then decrease deeper inside the magnetosheath, for solar maximum
the rates are low behind the nominal bow shock (e.g. the greenish squares
labelled C and D) and the rate increases along the magnetosheath. This
indicates a different growth rate for solar minimum and maximum, which will
be discussed in Sect. .
Mirror mode wave strength
To investigate the distribution of strengths B of the MM waves, the
events are binned with a bin size of ΔB=0.1. The results for
both solar minimum and maximum are shown in Fig. by red
circles and blue asterisks respectively see also Fig. 3
in. For both distributions a second-order polynomial has been
fitted and is shown as a grey dotted line, which indicates a change of slope.
Therefore, the weak (B≤1.2) and strong (B≥0.8)
events are also fitted linearly:
log(Nmm(B))∝a⋅B,
and they are shown as a solid and dashed line respectively. The results of this linear
fit are shown in Table .
The results of the slopes a of the linear fits to the weak and
strong MM waves in Fig. .
Weak a
χ2
Strong a
χ2
Solar minimum
-3.39±0.02
0.99
-2.45±0.10
0.89
Solar maximum
-3.04±0.03
0.98
-1.82±0.10
0.87
For the weak part, B≤1.2, the slopes are quite similar;
however,
for the strong events there is a larger difference in the slopes. These fit
values probably reflect the (varying) growth rate of the MM waves, which will be
discussed in the following section.
The
MM waves binned by B, for 0.1-size bins, for solar minimum (red circles)
and maximum (blue asterisks). The solid (dashed) lines show exponential fits
to the points for weak (strong) MM waves, with the fit parameters displayed in
Table . The dotted green lines show second-order polynomial
fits to all points.
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Mirror mode growth rate
To enable a discussion of the growth rate of the MM waves, first the distribution
of the event strengths along the magnetosheath flow direction needs to be
investigated. In order to do that the data are split up into three bins in the
direction perpendicular to the Venus–Sun line: 0≤R≤1.0, 1.0≤R≤1.5 and 1.5≤R≤2.5. For each event the distance along the
flow lines to the model bow shock (for either solar minimum or maximum, as
described in Sect. ) is also calculated, as shown by the cyan
dotted arrows in Fig. as an example.
The distribution of the strength B of the MM waves as a function
of distance to the model bow shock. The different colours indicate the
different bins in R, the distance from the Venus–Sun line, as labelled in
the panels. The top panels show the distribution for all events, the bottom
panels are a zoom-in on 1RV around the location of the model bow
shock at xVSO=0.
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First the distribution of B near the bow shock is studied, zooming
in on the region ±0.5 RV in XVSO around the bow
shock, as shown in the bottom panels of Fig. . It is
clear that there is a broad range of B values in the freshly shocked
plasma; there is not a single strength at which the MM waves are created. Indeed,
this was also shown by in the Earth's magnetosheath, in their
Fig. 3, where a range 3≤ΔB≤10 nT is observed just within
the bow shock.
also used their data to estimate the growth rate of the MM waves,
by a linear fit in log-space, and found a value of γ=0.0022 s-1. Overall in the Earth's magnetosheath they find 0.001≤γ≤0.01.
In order to investigate the MM growth rate at Venus, the data for the green
population in Fig. , with a distance from the
Venus–Sun line between 1.0≤R≤1.5 is used, in which the highest
occurrence rate is found and where the flow lines can be assumed to be almost
parallel to XVSO. The lower quartile (cyan), mean (red), median
(blue) and upper quartile (magenta) values are fitted, where the data are
again binned in 0.25RV bins along XVSO, as shown in
Fig. . The estimated growth rates are shown in
Table .
The distribution of the strength B of the MM waves as a function
of distance to the model bow shock for 1.0≤R≤1.5, the green
population in Fig. . The coloured bars show the lower
quartile (cyan), mean (red), median (blue), upper quartile (magenta) and
maximum (black) values of each bin. The fits to these values are shown in the
same colours, and the obtained growth rates are listed in
Table .
![]()
Growth rates of MM waves (s-1) for solar minimum and maximum
as determined for various quantities of the event distribution shown in
Fig. for pre- (-0.25≤XVSO≤0.75) and
post-terminator (1.0≤XVSO≤3.0).
Solar minimum
Solar maximum
Fitted
-0.25–0.75
1.0–3.0
-0.25–0.75
1.0–3.0
RV
RV
RV
RV
Upper
-0.011
-0.002
0.013
-0.005
Mean
-0.005
-0.002
0.012
-0.005
Median
-0.005
-0.001
0.009
-0.005
Lower
-0.001
-0.001
0.003
-0.002
Interestingly, the fits for solar minimum show a decaying MM population,
after being created behind the bow shock. During solar maximum, however, all
fits show a positive growth rate for the MM populations. Indeed, this agrees
with the observations in Fig. , where the observational rate
from squares A and B decreases going down the magnetosheath, whereas the
observational rate from squares C and D increases. At solar minimum the MM waves
are generated near the bow shock and decay, with a possible increase close to
Venus due to draping of the magnetic field. At solar maximum the MM waves develop
along the plasma flow in the magnetosheath.
This calls for a closer look at the distribution of the MM B near
the bow shock. The top panels of Fig. show the distribution
of the strength B as a function of distance along the flow line to
the model bow shock location, colour coded by the distance from the
Venus–Sun line. The bottom panels show a zoom-in on 1 RV around
the model bow shock location XVSO=0. It shows that for all data
the sudden increase of MM waves near the bow shock is much more pronounced for
solar minimum than for solar maximum. During solar maximum the bow shock is
approximately 0.25 RV further out from Venus see also
e.g., and the MM waves are on average weaker than for solar minimum.
For the first few bins in Fig. it is found that
B‾≈0.58±0.36 for solar minimum, whereas
B‾≈0.39±0.19 for solar maximum.
Behind Venus, i.e. behind the terminator, the MM waves decay in all cases, with
decay rates, for solar maximum, which are approximately half the growth rate.
Discussion
The comparison of the statistical studies of MM waves at solar minimum and at
solar maximum shows some expected results; however, some unexpected
distributions are also found. The main difference between solar minimum and
maximum for cycle is that the Sun radiates more UV, enhancing ionization of
Venus's exosphere. Also, it should be noticed that the 2011–2012 solar maximum
was very weak, with an exceptionally low proton density of the solar wind
. It was found that the “undisturbed” solar wind has a
density range of 0.5≤np≤20 cm3 for solar minimum,
whereas for solar maximum the range is more than a factor of 2 lower: 0.5≤np≤8 cm3. The solar wind velocity is on average slightly
lower for solar minimum, ∼ 300 km s-1, than for solar maximum
∼ 350 km s-1. The solar wind magnetic field does not
significantly change, with a median value of Bsw≈9.88 nT
for solar minimum and ≈9.99 nT for solar maximum.
At solar maximum the UV radiation of the Sun increases and thus there will be
more ionization of the neutrals in Venus's exosphere. This is also the reason
for the increase in ion cyclotron waves upstream of Venus's bow shock as
shown by . The increased ionization also causes the bow shock
and ionosphere to move outward , albeit
argue that charge exchange at low altitudes near the ionopause
is causing the shock to move closer at solar minimum.
The MM wave effect is to balance magnetic pressure B2/2μ0 and plasma
pressure nikBT⟂,i, and the instability is driven by the
temperature anisotropy of the ions (see Eq. ). This means that
the distribution of the MM waves with respect to B is most likely a
reflection of the energy distribution of the ions in Venus's magnetosheath.
Unfortunately, there are no papers discussing the plasma properties of
Venus's magnetosheath for solar minimum and maximum. Also the cadence of the
plasma instrument ASPERA is more than 3 min, much too long to
investigate the MM ion details, as the MM waves have a period between 4 and 15 s.
The changes in bow shock for solar maximum, moving outward and thus
increasing in size, and increased ionization by the solar UV radiation
could, in principle, increase the number of MM waves generated behind the bow
shock in Venus's magnetosheath. This is, however, not visible in
Fig. . The first result of the comparison between solar
minimum and maximum is that there are more MM waves found for solar maximum (a
total of 1857 events) than for solar minimum (a total of 1637 events). The
increased size of the bow shock and thus magnetosheath could be responsible
for the increased observed number of MM events.
For solar minimum the fitted bow shock location is used from VEX
measurements. For solar maximum such a determination from VEX data was not
available, and therefore the model from Pioneer Venus data was used.
The question may arise as to whether the solar maximum model is sufficiently
accurate to use in order to determine the behaviour of the mirror mode waves
in Venus's magnetosheath, as has been done in Figs.
and . Looking at the observations of MM waves as shown in
Fig. , it is clear from both panels that the average location
of the bow shock fits the data reasonably well, with a slightly larger
discrepancy for solar maximum. Unfortunately, there are no error bars given
for either of the bow shock fits.
This difference in bow shock location has no influence on the observational
rates given in Fig. , but it could have consequences for
the fits in Fig. . Figure
shows that in the region of interest for Fig. (i.e. 1.0≤R≤1.5 RV) there are only very few events that lie
outside the model bow shock, and these points have not been taken into
account in the determination of the growth rates of the MM waves.
Naturally, for a “perfect” fit, the distance to the observed bow shock
would have to be determined, which is because of the great number of events
unfeasible. This means that some of the distances can be incorrect. When the
results from the observational rates in Fig. are compared
with the results of the growth rates in Fig. , it is clear
that the two are in agreement.
Previous results by and have shown that the
occurrence rate of MM waves in the Earth's magnetosheath is positively correlated
to the Alfvén Mach number of the upstream solar wind.
Figure shows, however, that although the number of events
for solar maximum has increased slightly, the observational rate as defined
in Eq. () does not particularly change. Indeed, taking into account
the results by the decrease in average solar wind proton
density by a factor of ∼ 2 and the increase in average solar wind velocity
by a factor of ∼ 1.2 show that the average solar wind Alfvén Mach
number changes by 1.2/2≈0.9 from solar minimum to solar
maximum. Therefore, a significant difference in the MM occurrence rate is not
expected from this slight enhancement. The observational rate is overall the
same but differently distributed over Venus's magnetosheath. This is most
likely a result of the bow shock conditions for solar minimum and maximum
being dissimilar (e.g. strength or thickness), which then energize the ions
differently.
Not all MM waves have equal strength B, as this depends on the available
energy of the ions perpendicular to the magnetic field after being shocked by
the bow shock crossing. Interestingly, it was found that just behind (i.e.
the first three bins in Fig. ) the bow shock, where
freshly generated MM waves are expected, the average strength for solar minimum
B‾≈0.59± 0.36 (32 events) is higher than for
solar maximum with B‾≈0.32± 0.22 (23 events),
and also the spread of the strengths is larger for solar minimum as indicated
by the given standard deviation. The different average values listed here may
or may not be significant. Because of the large standard deviation on these
numbers one would be inclined to assume that there is no significance.
However, this difference could also indicate that for solar minimum the
energization of the ions in the ring distribution, through crossing the
quasi-perpendicular bow shock, is stronger for solar minimum, and also that
the variation of the bow shock strength is greater for solar minimum. There
are no observational papers studying any possible differences for the bow
shock for different solar activity conditions. It can also mean that the
plasma conditions in the solar minimum magnetosheath are different from solar
maximum. Unfortunately, there is also no study of Venus's magnetosheath
plasma environments during solar minimum and maximum.
All MM waves were binned as a function of B, with the result shown in
Fig. . The binned data indicate an exponential fall-off in the
number of MM waves with increasing B. There seems to be a break in the
slope near B≈1. For the weak MM waves (B≤1.2) the
slopes for solar minimum (maximum) are a≈-3.39±0.02(-3.04±0.03), whereas for strong MM waves (B≤0.8) the slopes are a≈-2.45±0.10(-1.82±0.01). This break can be created by the
fact that the MM waves are observed during their growth and decay phase in
XVSO>0; however, for XVSO<0 all MM waves are decaying
whereby the number of “weak” MM waves observed can be increased.
Assuming that the MM waves grow and/or decay when they are transported through the
magnetosheath, determined a growth rate by fitting ΔB
of the MM waves as a function of flow time in the Earth's magnetosheath, finding
an overall growth rate of 0.001≤γ≤0.01 s-1. In this
current paper the MM waves located at distances from the Venus–Sun line between
1.0≤R≤1.5 are used to obtain a growth rate; however, not the
whole cloud of points is used, but the quantities are listed in
Table . For solar minimum all fits show negative
values, indicating immediate decay of the MM waves after their generation behind
the bow shock. Nevertheless, this does not exclude MM wave growth as clearly
there are very strong events B≥1.2 observed at farther
distances from the bow shock, which can also be related to field line
draping. For solar maximum, on the other hand, all fits show positive growth
rates 0.003≤γ≤0.018 s-1, well within the range that
found for the Earth's magnetosheath. Recently,
used a 5-D Vlasov simulation (2-D space and 3-D velocity) to study MM waves in the
Earth's magnetosheath. The obtained simulated growth rate for the MM waves was
0.002≤γ≤0.005 s-1, which does not completely cover the ranges
estimated from observations in this paper and in .
The plasma transport time across the magnetosheath can be estimated as
ttr≈0.5RV/vpl≈30 s, with
RV=6052 km and vpl=100 km s-1 the nominal
flow velocity in the magnetosheath e.g.. With the maximum
growth rate as determined above ttr relates to half an e-folding
time. However, in the region 1.0≤R≤1.5, the wave growth seems to
extend over five bins in Fig. , which is ∼1.25RV and thus a maximum of ∼ 1.25 e-folding times. There
is time for the MM waves to evolve while they move toward the terminator. After
the MM waves cross the terminator, where the pick-up density is highest
and the magnetic field starts to diverge, the magnetosheath
becomes MM stable and the waves start to decay.
When the MM waves are transported down the magnetosheath, they will eventually
enter a MM-stable region and will have to start to decay. Indeed,
Figs. and show that B
falls off. Table shows the determined decay rates:
-0.009≤γ≤-0.001 s-1 for 1.0≤R≤1.5. There
are no quantitative models for the decay of MM waves. assume a
stochastic leaking of ions out of the magnetic bottle using the model
by, thereby reducing the plasma pressure and the magnetic
tension then starts to straighten the field lines. However, there is no given
decay rate for this model.