Introduction
The interaction of the supermagnetosonic solar wind flow with the Earth's
magnetosphere leads to the formation of a bow shock, where the solar wind is
decelerated, deviated, compressed, and heated . Many
works in the past have focused on studying the bow shock and the kinetic
processes taking place in it, such as Cluster mission results in Issues 1–4,
Vol. 118 from Space Science Reviews (available at:
https://link.springer.com/journal/11214/118/1/page/1, last access:
1 August 2018), , ,
, and . However, there are still
many open questions about how the solar wind is processed at the bow shock
and in the regions upstream and downstream of it. The structure of a
collisionless shock depends on its strength, given by the upstream
magnetosonic Mach number Mms and the plasma density compression
ratio; on the geometry, given by θBn (the angle between the shock
normal and the upstream magnetic field); and on the plasma beta (β).
Shocks are classified as quasi-parallel (quasi-perpendicular) when
θBn<45∘(θBn>45∘) (see, e.g.
; ).
Earth's bow shock usually has magnetosonic and Alfvénic Mach numbers in the
range 2≤Mms≤7 and 2.5≤MA≤12
. Due to its curvature and the Parker spiral
configuration of the interplanetary magnetic field (IMF), part of the Earth's
bow shock typically has a quasi-parallel geometry while in other regions
there exists a quasi-perpendicular shock. The compressed plasma in the
downstream is the magnetosheath, with the upstream consisting of pristine
solar wind (further out), the electron foreshock (magnetically connected to
quasi-perpendicular shock regions), and the ion foreshock (magnetically
connected to quasi-parallel regions). Different shock geometries lead to a
variety of expected plasma phenomena, such as a smooth quasi-perpendicular
shock front and mirror modes within the quasi-perpendicular magnetosheath
and high-pressure jets at the nose of the shock during
radial IMF .
Understanding how the plasma is modified at the foreshock, bow shock, and in
the magnetosheath is very important in terms of fundamental collisionless
plasma physics, and also because it is this shocked and processed plasma, not
the pristine solar wind, which ultimately interacts with the Earth's magnetic
field and sometimes leads to geomagnetic disturbances
. In this paper, we are focusing on effects
connected to the ion foreshock and the quasi-parallel shock region.
The bow shock is supercritical, which means that part of the solar wind
kinetic energy is dissipated by the reflection of incident solar wind ions.
Thus, the ion foreshock is permeated by a variety of suprathermal ion
distributions and a variety of ultra low frequency (ULF) waves
driven by ion instabilities which grow by the interaction
of the reflected ions with the solar wind core. A detailed description of
wave modes in the ion foreshock can be found in the review papers of
and . In addition to the ULF waves,
large-scale (sizes of the order of Earth radii) transient structures such as
cavitons , hot flow anomalies
(HFAs) , and spontaneous hot flow
anomalies (SHFAs) are also found in
the foreshock. While the origin and evolution of foreshock ion distributions
and waves have been studied for several decades, the study of transient
structures is more recent.
Cavitons are structures observed in the foreshock that are characterized by
dips in the magnetic field magnitude B and plasma density n, bounded by
overshoots in these parameters . The temperature
inside the cavitons is similar to the value in the surrounding plasma. They
are proposed to form when transverse and compressive ULF waves interact
non-linearly and are hence surrounded by intense
compressive ULF waves. Foreshock suprathermal ions can be accumulated inside
their cores. When formed, cavitons are carried by the solar wind flow towards the bow shock. An extensive
statistical analysis by using Cluster data shows that
cavitons occur for a wide range of solar wind conditions upstream of the
quasi-parallel bow shock regime, and on average they show decrements of
between 0.2 and 0.9 for both δB/Bsw and δn/nsw.
HFAs are observed upstream of the bow shock and similar to cavitons are
characterized by decreases in the magnetic field magnitude and plasma
density, but they also show a notable increase in temperature
. As a consequence, they have enhanced
plasma β. Flow inside HFAs is strongly decelerated and deflected.
Typical sizes of HFAs deduced from observations are approximately 2–3 Earth
radii (RE) e.g.. The formation of an HFA
needs an external perturbation in the solar wind, e.g. a current sheet
interacting with a bow shock. In this paper we focus on spontaneous HFAs
(SHFAs) that are similar in their characteristics to HFAs, except typically
smaller in size e.g., but have a different formation
mechanism. SHFAs form primarily due to inherent foreshock structures, namely
cavitons discussed above. Numerical hybrid simulations have shown that
cavitons evolve into SHFAs as they move closer to the bow shock
. The proposed formation mechanism for SHFAs includes
multiple ion reflections between foreshock cavitons and the bow shock
, as cavitons approach the shock, and ion trapping occurs
in the cavitons.
Foreshock ULF waves propagate towards the Sun in the solar wind frame with
phase speeds of the order of the Alfvén speed, i.e. much smaller than the
solar wind speed. As a consequence, they are convected towards the shock by
the supermagnetosonic solar wind flow. Several observational and simulation
studies have shown that ULF waves evolve into non-linear structures as they
approach the shock, becoming compressive shocklets and SLAMS (short large-amplitude
magnetic structures) , which in turn play an
active role in the reformation of the quasi-parallel bow shock, in the
variability of the density of reflected ions, and in the variability in shock
heating and rippling see, for example,. As a consequence of ULF waves,
shocklets, and SLAMS merging into the shock, the quasi-parallel portion of the
bow shock is far from being a single well defined surface, but instead forms
a highly corrugated/rippled extended structure, where inhomogeneous heating
and solar wind processing can take place see, for
example,.
It has been shown that cavitons and SHFAs are also convected by the solar
wind towards the shock . In a recent study,
demonstrated that SHFAs can result in the formation of
magnetosheath cavities , which are associated with
decreases in plasma density, bulk velocity and magnetic field magnitude, and
enhancements in temperature. However, we still know very little about how the
arrival of cavitons and SHFAs can modify the bow shock structure and the
magnetosheath. It is expected that their arrival at the bow shock will impact
its structure, contributing to the formation of shock irregularities. More
specifically, it is expected that they may lead to decrements in the shock
magnetic field magnitude due to the decreased field inside these structures.
If the interplay of shock reformation and SHFAs and their effect on shock
erosion is to be investigated, it is important to model both features using
realistic length scales. Other outstanding questions related to cavitons and
SHFAs include why some cavitons develop into SHFAs and others do not and
whether some SHFAs can survive/evolve downstream of the bow shock.
Although observations of the foreshock, bow shock, and magnetosheath are
abundant, it is statistically difficult to quantify from observations how
close to the bow shock SHFAs form. According to simulations
, they form very close to the bow shock. Global simulations
have specific advantages over point-like observations, providing large
statistics and an easy way to disentangle spatial and temporal variations.
Foreshocks and their transient structures such as cavitons and SHFAs are
ubiquitous features upstream of quasi-parallel shocks and can therefore be
found in other planetary environments in our solar system. In particular,
SHFAs have recently been observed and modelled in the foreshock upstream of
Venus and Mars . These works show that the
size and properties of SHFAs, as well as their formation mechanism, are
similar to that at Earth.
In this paper we perform a numerical study on the evolution and properties of
cavitons and SHFAs based on simulations using the strong capabilities of the
Vlasiator hybrid-Vlasov code. Vlasiator facilitates a global simulation view
while maintaining realistic length scales and including ion kinetic physics,
allowing us to present a statistical study on caviton and SHFA sizes. In
particular, we show in detail how large SHFAs survive downstream of the bow
shock and induce both the formation of a magnetosheath cavity and weakening
and erosion of the bow shock.
The paper is organised as follows: in Sect. 2 we describe the Vlasiator code
and the new data presentation methods to identify cavitons and SHFA from
simulations. The results are presented in Sect. 3 and discussed in Sect. 4.
Methods
Vlasiator is a unique
hybrid-Vlasov code capable of performing global simulations of the Earth's
magnetosphere and the surrounding space environment. It models kinetic
proton-scale physics by simulating the proton distribution function through a
Cartesian 3-D velocity grid and cell-averaged values, instead of relying on
particle-in-cell methods and statistical sampling. Thus, Vlasiator has the
inherent merit of being noiseless. A sparse velocity grid implementation
maintains scalability and numerical efficiency.
Vlasiator models protons as a distribution function, solving the Vlasov
equation for the ion (proton) distribution and with closure being provided via
Ampère's and Faraday's laws, as well as Ohm's law complemented by the Hall
term. Electrons are modelled as a cold charge-neutralising fluid, and, due to
a realistic proton mass and charge, kinetic effects are simulated on physical
scales instead of normalized to ion scales (i.e. ion gyroperiod
Ωcp-1 and ion skin depth cωpi-1). As
shown in , many kinetic proton phenomena are
successfully reproduced even when the ion inertial ranges are not resolved,
though spatial resolution does limit gradients, steepenings, and thus possibly
amplitudes and frequencies of phenomena.
Vlasiator simulation run
In our investigation we use a global magnetospheric simulation performed in
the meridional (x–z) plane. The simulation is 2-D in real space and 3-D in
velocity space. In order to treat y-directional velocities
self-consistently, periodic boundary conditions are employed at the ±y
spatial cell walls. The solar wind has an x-directional inflow speed of
usw=-750 kms-1 and the IMF magnitude is 5 nT,
oriented towards the Sun and southward at a 45∘ angle. The solar wind
number density is np=106 m-3 and an ion
temperature of T=0.5 MK. The solar wind inflow conditions are
steady throughout the run and correspond with values of β=0.7,
Mms=5.6, and MA=6.9. We perform calculations for a
total simulation time of 1437 s.
To be able to model foreshock features and interactions, our simulation box
is extended from near-Earth space in the direction of the foreshock, with a
box extent of -48.6 to 64.3 RE in the x direction and
-59.6 to 39.2 RE in the z direction. The spatial cells are
cubes with a width of 300 km (1.3 solar wind ion inertial lengths) in each
direction, and our velocity resolution is set to 30 kms-1
(0.33 times the solar wind ion thermal speed) extending in all directions to
±4020 kms-1. The polar setup (in the noon–midnight
meridian plane) includes Earth's geomagnetic field as a line dipole,
neglecting tilt, and with the dipole magnitude selected as outlined in
in order to result in a realistic magnetopause standoff
distance. The inner boundary, set at 30 000 km (∼4.7 RE), is modelled as a static Maxwellian, perfectly conducting ionosphere.
In order to more accurately and efficiently model the foreshock region,
including regions where plasma density is decreased, we employ a sparse
velocity space algorithm (see ); that is,
velocity space cells are dynamically allocated or discarded when their value
is above or below a given threshold, respectively. The sparsity threshold
value is scaled dynamically in accordance with proton number density. The
minimum sparsity threshold is set at
10-17 m-6 s3, scaling up to
10-15 m-6 s3 as a linear function of
proton density between densities of n0=104 m-3 and n1=105 m-3.
We wrote reduced simulation output data to disk for data analysis with a
simulation time interval of 0.5 s, in addition to retaining a full-state
save at a time of 1187.85 s. The reduced data set contains magnetic and
electric field components at cell boundaries, the Hall term for electric
fields, the number density, bulk velocity, and pressure tensor of protons.
Additionally, the number density, bulk velocity, and pressure
tensor are provided for a subsection of the distribution function, which is
named the suprathermal beam population. This beam population is defined for
this simulation as all protons not included in the core solar wind
population, which in turn is considered to have a maximum thermal speed of
vcore,max=500 km s-1 around the bulk velocity of
usw,x=-750 km s-1. We note that the beam population is
well defined only upstream of the bow shock. The reduced data set
additionally includes the proton distributions on a limited grid, every 50th
cell in the x and z directions.
Although our results are directly comparable with spacecraft observations, we
provide various plasma parameters calculated at the input boundary of our
simulation, facilitating comparisons with other simulations: Alfvén speed
vA=109 km s-1, sound speed
cs=79 km s-1, Mach numbers and plasma β given above,
proton skin depth cωpi-1=228 km, and proton gyrotime 2πΩcp-1=13.1 s. We stress that these values are valid
only at the input boundary.
An overview of the simulation at t=1350 s is shown in
Fig. . The colour code indicates the value of the magnetic
field By component, out of the plane of the simulation, and the black
lines correspond to magnetic field lines in the x–z plane. Regions inside
the magnetopause are omitted from the figure so that only the regions of
interest for the present study, the foreshock, the shock, and the
magnetosheath, are highlighted. Due to the IMF orientation, the foreshock
develops in front of the southern part of the bow shock, as evidenced by the
fluctuations of By in this region. The oscillations of By from positive
to negative values and the coherent wave fronts extending perpendicular to
the IMF direction show that the foreshock is permeated by so-called 30 s ULF
waves , with properties similar to those analysed by
in another Vlasiator run. Wave activity is also visible
throughout the magnetosheath, with stronger perturbations downstream of the
quasi-parallel portion of the bow shock. When the IMF has a significant
southward component, reconnection takes place at the dayside magnetopause and
creates magnetic islands which propagate tailward. One such island can be
seen around x=2 RE, z=-8 RE.
Reconnection in this run is studied in detail in .
Zoomed-in image of the magnetosheath and foreshock region of the Vlasiator
simulation used in this analysis. The figure shows the region of solar wind
interaction with the magnetosphere, with the colour scale indicating the
out-of-plane magnetic field component By, showing fluctuations both in the
magnetosheath and in the foreshock. Black lines indicate magnetic field lines
in the x–z plane. The inner boundary is placed at 5 RE, and
the tail region is excluded from the plot for clarity.
Identification of structures
In this paper, we are focusing on two types of foreshock transients, cavitons
and SHFAs. As discussed in the introduction, these structures are closely
linked with each other, and it is believed that cavitons evolve into SHFAs.
Cavitons are characterized by decreases in both the magnetic field strength
and the density relative to the ambient plasma. In order to automatically
detect these structures in our simulation run, quantitative thresholds are
set on these parameters. Following and
, we identify as cavitons those structures where the
density and magnetic field strength are less than 80 % of the solar wind
density and magnetic field magnitude.
SHFAs are also characterized by decrements in density and magnetic field
strength but have in addition a higher temperature than the surrounding
plasma. However, setting a criterion on the temperature is not
straightforward since SHFAs are immersed in the foreshock, which has a higher
temperature than the pristine solar wind. Similar challenges are encountered
for the decrease in velocity associated with SHFAs, as deviations from the
bulk solar wind velocity are observed throughout the foreshock, and they are
not prominent enough inside SHFAs to be unambiguously identified. On the
other hand, the enhanced temperature and the reduced magnetic field result in
a significant increase in the plasma β. For this reason, we have set a
condition not on the temperature but on the plasma β in order to
identify SHFAs. We chose to use a threshold of β>10, as it is
significantly above the usual foreshock values.
Results
In this section we report on the analysis of cavitons and SHFAs using our
global hybrid-Vlasov simulation data. At the beginning of our analysis
period, we find a well-formed bow shock which does not display significant
weakenings or transient effects beyond minor shock rippling. Using the
criteria for cavitons and SHFAs defined in Sect. 2.3, we identify these
structures in the simulation and are able to track their evolution in time.
At a given time in the run, about 50 cavitons and SHFAs are observed in the
2-D cut plane through the foreshock which we model in our run. We will first
concentrate on the formation of a magnetosheath cavity. We will then
investigate the size distributions of cavitons and SHFAs in the foreshock.
Finally, we will compare our simulation results with spacecraft observations.
Formation of a magnetosheath cavity
Figure shows three snapshots of the density of suprathermal
beam population ions (panels b–d) and the magnetic field strength
(panels e–g) within a 20×20 RE region covering a
portion of the bow shock and foreshock. The region is initially centred on a
position where multiple large cavitons form. Note that the position of the
region of interest is moved along the bow shock as time progresses, as
illustrated in panel (a), because the structures are convected by the
magnetosheath flow. Black and green contours on the plots indicate structures
fulfilling our requirements for a caviton and an SHFA, respectively (see
Sect. ). The blue contour marks where the
density is equal to twice that of the incoming solar wind and is a good
approximation of the bow shock position.
Evolution of cavitons, SHFAs, and a magnetosheath cavity in the
Vlasiator simulation run. To track the evolution of the same structures,
panels (b)–(d) correspond to a fixed dimension 20×20 RE box at three separate locations and time steps, as
illustrated in panel (a). The times are T1=860 s (Region
R1), T2=1105 s (Region R2), and T3=1350 s
(Region R3). Panels (b)–(d) and
(e)–(g) correspond to suprathermal ion density and
magnetic field magnitude, respectively. The contours represent the bow shock
(blue), cavitons (black), and SHFAs (green). The criteria for identifying
each of these are provided in the legend, located in the top right corner. The
three grey lines are positions of profiles chosen for further
study.
Time T1=860 s features a structure that is evolving from a
caviton (black contour around x=9 RE, z=-18 RE in panels b and e) into an SHFA (green contour inside
the black contour). As the caviton moves adjacent to the bow shock, it is
filled with higher suprathermal ion density, causing it to evolve into an
SHFA . Further in the simulation more cavitons are seen
approaching the bow shock and transforming into SHFAs. Time T2=1105 s
shows the phase when, as the result of SHFAs surviving crossing the bow
shock, a large magnetosheath cavity has formed (yellow area around x=-4 RE, z=-33 RE in panel f). Magnetosheath
cavities such as this have been found in observations
and hybrid simulations .
“Chains” of SHFAs and cavitons are seen also upstream of the bow shock that
add to the large magnetosheath cavity. An indentation in the bow shock shape
is developing where several large SHFAs have crossed into the magnetosheath,
as evidenced by the blue contour in panels (c) and (f). Time T3=1350 s
(close to the end of the simulation run) shows that the notch at T2 has
turned into a large-scale weakening of the bow shock which extends deep
within the magnetosheath. At all times, and throughout the foreshock, we note
that, as in panels (b)–(g), SHFA formation occurs closer to the shock, whereas
cavitons are generated further out.
Profile cuts from the magnetosheath across the shock into the
foreshock region at three different times of the simulation. The locations of
the cuts were shown in Fig. . Rectangular areas shaded in
grey, yellow, and blue indicate the regions at each time (T1, T2, and
T3, respectively), which fulfill our caviton criteria of nP<0.8nP,sw and |B|<0.8|BSW|. Each shaded area is
accompanied by a indicator bar above panel (a). The beam density
nP,beam is not well defined in the sheath and is thus allowed to
saturate.
To examine in more detail how the structure of the bow shock is modified by
SHFAs, Fig. displays profile cuts in the spatial x–z
plane spanning a distance of 15 RE from the magnetosheath into
the foreshock. We plot six panels, showcasing the proton number density
nP, the suprathermal beam number density nP,beam, the
magnetic field magnitude |B|, the ion temperature T, the plasma β,
and the bulk flow speed |V|. We draw profiles at the three simulation times
presented in Fig. , that is, T1=860 s, T2=1105 s, and
T3=1350 s. Each profile is chosen to cut across those features of the
shock which evolve into the large magnetosheath cavity. Cut extents were
chosen so that the increase in |B| corresponding to the shock position is
located at the same position for all cuts. The cut positions at each time are
shown as grey lines in panels (b)–(d) of Fig. . Rectangular
shaded areas indicate those spatial regions which fulfill our caviton
criteria of nP<0.8nP,sw and |B|<0.8|BSW|, with
the colour indicating the time at which the caviton is identified (grey,
yellow, and blue for T1, T2, and T3).
As the cuts and the feature of interest move in time along the shock front,
originating in a region closer to the nose of the bow shock and propagating
tailward and southward, we see a decrease in downstream proton density. At
time T2 the jump in density across the shock is weakened, reaching barely
values of twice the solar wind density. The overall lack of clear density
enhancement at time T3 indicates that the shock (positioned at
approximately x=6 RE) has eroded away. The proton beam density
at time T1 decreases strongly with distance from the shock, but as time
progresses the extended beam further out strengthens and the profile
flattens. It is noteworthy that times T2 and T3 show little difference in
the beam density, indicating the reflection and isotropisation process of
beam ions has reached a quasi-steady state. In panel (c) of Fig. , we see a strong indication of the
formation of the magnetosheath cavity in the decrease in magnetic field
strength at the region x<6 RE. At time T1, we see a steep
enhancement in |B| at the shock location, with wave-associated periodicity
in the downstream region. At time T2, there is still a weak peak at the
shock position but the magnetosheath cavity has only a weak magnetic field
due to heating and expansion. By time T3, the peak of |B| at the shock
position has eroded almost completely away, in agreement with our
observations from proton density, indicating that the shock has eroded away.
Panels (d) and (e) show that there is strong
heating and a rise in plasma β at the shock early in the simulation.
Oscillations of temperature in the magnetosheath smooth out by time T2, and
in the region immediately downstream of the shock (i.e. in the cavity) the
temperature remains somewhat constant, particularly at time T3. At the
start of the cut, i.e. deeper in the magnetosheath, temperature decreases
over time, which may be associated with the cut being further from the nose
of the shock at time T3 than at earlier times. Finally, in
panel (f) of Fig. , we see the
evolution of bulk velocity over time, where an increase in the downstream
bulk speed as a function of time can be attributed to the region of interest
moving further away from the nose, allowing for increased magnetosheath flow.
To some extent, the bulk flow speed changes can also be due to a weakened
shock being less efficient at decelerating the upstream plasma as it crosses
the shock.
The shaded areas in Fig. show how at time T2 there are
multiple large cavitons upstream of the shock. SHFAs are found at shaded
areas where also β>10. There is only a single SHFA at time T1, and
there are only two small cavitons at time T3. The two SHFAs at time T2
exhibit decrements in |B| and nP of around 40–50 % from solar
wind values and enhanced β just exceeding a value of 10. In
panel (f), small dips in the value of |V|
associated with cavitons and SHFAs can also be identified.
A hypothetical explanation for the lack of cavitons and SHFAs at time T3
might be a negative feedback process, where cavitons and SHFAs erode the shock,
but, once the shock has eroded, generation of cavitons would be suppressed as
the weak shock allows for plasma to distribute more freely back to the
upstream, filling the forming cavitons. At time T2 there are significant
dips in proton density corresponding with the cavitons, whereas at time T3
the dips are less prominent. Alternatively, an eroded shock might result in
smaller backstreaming beams and weaker upstream wave formation, suppressing
caviton formation mechanisms associated with waves.
Shock erosion
The local region around the magnetosheath cavity, showing the extent
of bow shock erosion at time T3=1350 s. The figure extents are 10×10 RE with the colour scale showing proton number density on a
logarithmic scale. The contours represent the bow shock (blue), cavitons
(black), and SHFAs (green).
Figure shows (at time T3) a close-up of the magnetosheath
cavity and of the weakened bow shock. The total ion density is indicated with
the main colour scheme. The contours delineate again cavitons (black), SHFAs
(green) and the bow shock (blue). The outstretched light orange region
extending deep into the magnetosheath contains very weakly compressed plasma,
as its density is less than twice that of the inflowing solar wind. This
density value corresponds to ∼50 % of the value of the surrounding
magnetosheath and is in agreement with values observed inside magnetosheath
cavities . The bow shock, as marked by twice the
undisturbed solar wind density, has disappeared between x=-19 and
-18 RE. The eroded region, visible already at time T2,
grows progressively as the magnetosheath cavity is convected tailward. A
supplementary animation of images identical to Fig. ,
following the features along the bow shock from region R1 to regions R2 and
finally R3 (see Fig. ), is provided in the
Supplement. The animation time extent is from
750 to 1437 s, starting before time T1 and continuing after
time T3. The growth of the magnetosheath cavity is most likely due to
multiple SHFAs crossing the same part of the bow shock in rapid succession,
augmented by the overall weakening of the bow shock when moving further from
its nose. At the end of the simulation the magnetosheath cavity has grown in
size to encompass a length of >5 RE and a width of
1–2 RE. The rippling of the shock surface, seen throughout the
quasi-parallel bow shock, causes local changes in shock geometry with
possible consequences for the formation of magnetosheath structures such as
jets . Any such effects may be
modulated by both SHFAs impinging on the shock and erosion such as seen
associated with this magnetosheath cavity. Changes to shock properties will
also alter the dynamics of the local foreshock and will affect reflection of
particles and generation of foreshock ULF waves.
We note that, although the plasma density within the cavity is low and the
compression ratio becomes small compared to the undisturbed solar wind, the
simultaneous decrease in magnetic field strength results in an increase in
Alfvénic Mach number at this position, rising above values of 10 and
peaking at values >40. Other positions where multiple SHFAs cross into the
downstream region also result in regions of high Alfvénic Mach number
within the magnetosheath, but not right at the shock front. Magnetosonic Mach
numbers in the sheath mostly grow with increasing distance from the nose of
the shock, but the cavity and bulge (with a low density and high temperature)
have consistently lower magnetosonic Mach numbers than surrounding areas.
Within the sheath we found values of Mms≤3, and within the
magnetosheath cavity we found values of Mms≤2. If we consider only the
shock-normal upstream bulk flow in calculating the magnetosonic Mach number,
the number drops to ∼1, which is evident as the shock bulges out
towards the upstream at this location.
Spatial comparison of features
A plot showing the magnetic field strength at time
TR=1187.85 s. The contours represent the bow shock (blue),
cavitons (black), and SHFAs (green). The lines labelled 1, 2, and 3 are the
positions of profile cuts through three regions of interest. The letters and
adjacent cross marks indicate positions where we examine proton velocity
distributions.
Investigation of SHFAs crossing the bow shock, building up a magnetosheath
cavity, and eroding the bow shock requires analysis of the development of
features over time, as seen in previous sections, and also comparison of
similar regions which result in different behaviour, located at different
spatial positions.
In order to study proton velocity distributions at optimal locations, we
examine the simulation at time TR=1187.85 s, when we have a full-state
save of the simulation data. Figure shows a zoomed-in image of the
region of interest, plotting the magnetic field strength using the main
colour scheme. The large magnetosheath cavity is visible as a large pale
region, with other signs of shock deformation such as smaller magnetosheath
rarefactions visible
further along the shock front. The plot shows contours
for the shock front (blue), upstream cavitons (black), and upstream SHFAs
(green), along with three profile cuts intersecting regions of interest and
15 positions selected for further study.
The first cut in Fig. , labelled 1, starts from within the
large magnetosheath cavity, crosses the eroded shock front, and extends into
the foreshock. Positions a and b are within the magnetosheath cavity, with a
situated well within the structure and b located close to the shock within
the bulge extending beyond the regular shock position. Position c is located
within an upstream SHFA, d is further out within an upstream caviton, and e
is located even further along the same field lines at a position without
significant features.
Cuts 2 and 3 of Fig. are located further along the shock
front, parallel to upstream field lines, with cut 2 along a foreshock path
which sees limited caviton formation and cut 3 along a path where caviton
formation is significant. At cut 3, SHFAs are weakened during their crossing
into the downstream, the shock is not eroded, and a strong magnetosheath
cavity does not form. Positions f through k and l through p are positioned
similarly starting from within magnetosheath features, proceeding across the
shock, located at upstream cavitons and finally far in the upstream in
regions lacking significant features.
Cuts of the proton velocity distribution function at time
t=1187.85 in the vx–vz plane (at vy=0) at fifteen positions
(labelled a–p). The positions and the cuts they were
chosen from are shown in Fig. . Each row corresponds to one
cut, with panels from left to right going from the magnetosheath to the bow
shock, then an SHFA, then a caviton further out, and finally an upstream
position without transient features.
In Fig. , we display vx-vz-directional cuts of 15 proton
velocity distribution functions (VDFs), labelled (a)–(p), matching the
positions shown in Fig. . These VDFs showcase samples of how
the core solar wind population is modified across the shock and how the beam
population evolves as a function of distance from the shock. We note that our
proton VDFs are fully three-dimensional and these panels, showing
vx-vz-directional cuts, exclude features of the VDF which would be
visible only in the vy-directional component. Due to the gyromotion of
charged particles, we consider the plots as representative of the total
upstream VDFs. Panels (a), (f), and (l) show a heated quasi-isotropic sheath
population, although panel (a) shows decreased density and an additional weak
field-aligned downstream beam. Panels (b), (g), and (m) show the
magnetosheath population being formed, with the core being heated and
deformed. Panel (b), within the bulge of the magnetosheath cavity, shows a
strongly deformed, asymmetric population. Panel (g) shows a rotationally
deforming and stretching core along with an upstream beam which is on the way
towards becoming isotropised. Panel (m) also shows a rotationally stretched
and heated solar wind core, but the beam population has already merged with
it significantly. It also appears that panel (m) displays a distinct
downstream beam. The third, fourth, and fifth columns show how, with
increasing distance from the shock, the reflected ion beam strength decreases
and how the extent of beam particle pitch angles is limited to a narrow
field-aligned beam for the ions observed far upstream. Panel (c), within an
upstream SHFA, shows a strongly deformed and depleted core with almost
all-encompassing beam pitch angles, indicating that the vicinity of the shock
bulge has a strong effect on the SHFA VDF. We would also like to draw the
reader's attention to the fourth column, where the strength of the beam
corresponds with the strength of features – panel (d) (cut 1) has largest
beam intensities, followed by panel (o) (cut 3), with panel (i) (cut 2)
having only a weak beam. Although the cavity associated with panel (i) is
further away from the shock as those associated with panels (d) and (o),
investigation of nearby VDFs suggests that distance is not a dominating
parameter here. In the final column, panel (e), associated with the cut
connected to the eroded region of the shock shows also a stronger beam than
panels (k) or (p).
Profile cuts from the magnetosheath across the shock into the
foreshock region at three different positions (as shown in
Fig. ), at time TR=1187.85 s. Rectangular areas shaded
in grey, yellow, and blue indicate the regions at each of the cuts (1, 2, and
3, respectively), which fulfill our caviton criteria of nP<0.8nP,sw and |B|<0.8|BSW|. Each shaded area is
accompanied by a indicator bar above panel (a). Panels (g),
(h) and (i) display heat maps of pitch-angle cosine
cos(α) for the suprathermal beam distribution of protons, measured in
the plasma reference frame, with respect to the local magnetic field.
Figure shows spatial profiles, similar to
Fig. , but all at times TR=1187.85 s over the cut
locations shown in Fig. . Cut positions were chosen so that
the increase in |B| corresponding to the shock front is located at the same
position for all cuts. In addition to the six panels similar to those shown
in Fig. , we show, for each cut, a panel with a heat map of
proton pitch-angle cosine cos(α) distribution, for the suprathermal
beam portion of particles (as defined in Sect. ). Due to the
beam being well defined only in the foreshock region, we allow the maps to
saturate in the magnetosheath. We also plot rectangular areas shaded in grey,
yellow, and blue to indicate the regions at each of the cuts (1, 2, and 3,
respectively), which fulfill our caviton criteria of nP<0.8nP,sw and |B|<0.8|BSW|.
In panel (a) of Fig. we see the
striking decrease in proton number density associated with cut 1, which
crosses the magnetosheath cavity, both in the magnetosheath and in regions of
the foreshock. The low proton density also means that large regions in the
foreshock easily fulfill the density requirement of our caviton criteria,
falling even below 50 % of the mean solar wind density. In agreement with
panels (c) and (d) of Fig. , there is little difference
between the three cuts in beam density, as seen in panel (b) of Fig. . This is due to all three cuts being
located along “fingers” of enhanced foreshock beam density, where there is
abundant reflection of particles and associated formation of cavitons. In the
gaps between these fingers, beam densities are lower, and formation rates of
cavitons and SHFAs are much lower. These fingers are convected along with the
solar wind as are the corresponding bow shock features. We note that due to
the large variation of beam density with distance from the shock, and thus
the requirement to plot the beam density on a logarithmic scale, variations
associated with cavities and SHFAs appear less pronounced. The magnetosheath
cavity at cut 1 is also strongly visible in magnetic field magnitude
(panel c), with |B| peaking at the shock front
position but falling to sub-foreshock levels in the sheath. Cuts 2 and 3 show
more varied fluctuations in |B|, though all cuts show a peak at the shock
position.
In the foreshock, SHFAs at cut 1 show magnetic field decreases of up to
50 % from the solar wind magnetic field. Cut 1 shows significant heating,
both in the magnetosheath cavity and in the foreshock SHFA visible right in
front of the shock, whereas the temperature profiles of cuts 2 and 3
(panel d) do not differ from each other much. In plasma β (panel e),
cut 1 shows SHFA-related enhancements, but cuts 2 and 3 also show slightly
lesser enhancements in the magnetosheath. In bulk velocity (panel f), we see
a shock-associated dip in all three cuts, with magnetosheath values rising
with distance from the shock nose, in agreement with what was seen in
panel (f) of Fig. .
In panels (g), (h), and (i) of
Fig. , we display heat maps of the pitch-angle distributions
(PADs) for the suprathermal beam portion of protons, as measured in the rest
frame of the plasma. The shaded rectangular regions highlighting cavitons
extend to cover these maps as well. We begin by noting how for cut 3, the
blue caviton regions match increased PAD spread fairly well, with an
especially wide spread at the shockward edge of the caviton. For cut 2, with
the single yellow region, there is only a weak match in the PAD spread and
caviton location. There are multiple locations along cut 2 where the PAD
extends beyond -0.5, such as at r∼5 RE, and two peaks
near r∼6 RE. It is important to note that, at those
locations, there are dips in proton number density and magnetic field
strength, resembling cavitons. However, due to this cut being in a region of
the foreshock where ambient values for nP and |B| are enhanced, the
dips, though being large enough to signify caviton formation, no longer match
our caviton criteria, which were chosen based on mean upstream solar wind
values. In a similar but opposite fashion, at cut 1 we see a giant caviton
further away from the shock, but a strong signature in the PAD is seen only
at the centre of it, at r=6 RE. Thus, if we assume that the
PAD spread is a signature of a caviton, meaningful caviton criteria for each
cut should be based on the local, not global, solar wind values. The low
plasma density found at cut 1 may, however, be a rare occurrence, and the
strong magnetic field at cut 2 may be a result of heated expansion of plasma
at cuts 1 and 3 causing the field at cut 2, between them, to be enhanced.
Size distributions of cavitons and SHFAs
Spatial distribution of cavitons and SHFAs and their associated
physical scales in terms of surface area. Panel (a) shows the global
spatial distribution of cavitons and SHFAs at T3=1350 s. Cavitons and
SHFAs can be identified from the black and green regions, respectively. The
colour scale of the simulation run corresponds to ion plasma β on a
logarithmic scale. Panel (b) represents a histogram of the caviton
and SHFA surface areas in RE2. The colour of cavitons and SHFAs
is the same as panel (a).
Time series of virtual spacecraft data extracted from the
simulation domain at x=0 RE, z=-35 RE
from t=1000 s to t=1150 s. The panels show, from (a) to
(e), the magnetic field magnitude, the
proton density, the proton temperature (black) and the plasma β (blue),
the solar wind bulk velocity, and the beam proton density.
The global view of the foreshock provided by the simulation does not only
allow us to track the evolution of specific structures as they convect past
the bow shock, but also to look at the evolution and some of the properties
of cavitons and SHFAs throughout the foreshock. As evidenced by the black and
green contours in the Supplement, SHFAs (in green) are only
found within a few RE from the bow shock, whereas cavitons can
appear much further out. This is consistent with the fact that the density of
suprathermal ions is larger in the vicinity of the bow shock, as is visible,
for example, in panels (b)–(d) of Fig. , thus resulting in
an enhanced temperature and plasma β which fulfill our SHFA criteria.
More importantly, we note that most cavitons evolve into SHFAs when
approaching the bow shock (black contours becoming green in the animation),
in agreement with earlier simulations by , who first
suggested that SHFAs originate from the interaction of cavitons with
suprathermal ions coming from the bow shock. Cavitons that do not evolve into
SHFAs disappear before impinging on the shock, with very few exceptions,
while some SHFAs are born as such, without initially being identified as
cavitons. These two phenomena may both be due to the limitations of our
criteria for automated structure detection. The thresholds we have selected
for the magnetic field intensity and the plasma density are both based on the
mean upstream solar wind values, whereas these parameters can vary across the
foreshock. For example, in regions of enhanced plasma density, cavitons may
appear with density greater than 80 % of that of the solar wind, but
still be significantly depleted compared to the ambient plasma. Ideally, the
thresholds should be set based on the average local plasma parameters, but
this is not applicable on a global scale.
We then investigate the size distribution of cavitons and SHFAs in the total
simulated foreshock region. Note that in this part of the analysis we need to
identify structures either as cavitons or SHFAs, without overlap between
them. Therefore, for each caviton, we check which fraction of it has a plasma
β above 10. If 60 % or more of the caviton area has β>10,
then the entire structure is considered as an SHFA. We then calculate the
surface area of each of the structures. Figure a shows the
cavitons (in black) and SHFAs (in green) identified at T3=1350 s, which
is representative of other times in the run, overplotted on a colour map of
the plasma β. The dark red feature around x=-20 RE
corresponds to the magnetosheath cavity we discussed earlier. In total, 46
cavitons and 19 SHFAs are detected at this time. A histogram of their
surface area is displayed in Fig. b for both types of
structures, with the same colour code as before. Both distributions peak
close to zero, showing that the smallest structures are the most numerous,
while larger structures are rarer. The shape of the distributions of both
types of structures are very similar, which supports the hypothesis that
cavitons evolve into SHFAs.
Comparison with spacecraft observations
Figure shows a time series from a virtual spacecraft
positioned at x=0 RE, z=-35 RE around
the time when a caviton, marked by the green area on the plot, crosses this
location. The black dashed lines in panels (a) and (b) indicate the undisturbed
solar wind values of the magnetic field strength and the ion density, while
the red dashed lines correspond to our identification criteria for a caviton.
Both criteria are fulfilled in the core of the structure. Even though the
plasma β (blue dashed line in panel c) is lower than the limit we set
in Sect. to define an SHFA, it is clear that
this caviton is evolving towards an SHFA, as the β is already much
higher than in the surrounding plasma. It also contains a significantly
larger density of suprathermal ions (panel e), showing that particles have
started to accumulate inside the structure and thus causing the increase in
temperature and β.
The profile of the caviton in Fig. can be directly compared
to cavitons observed in the Earth's foreshock by the Cluster spacecraft
. In particular, Fig. 3 of
shows an example of a caviton for similar interplanetary magnetic field
strength and ion density as in our simulation. The decrements of these two
parameters are slightly more pronounced in the spacecraft data, down to
2.5 nT and 0.3 cm-3, and the gradients at the edges of the
cavitons are sharper, but the structure resembles closely the caviton
showcased in Fig. . The flux of energetic particles measured
by the spacecraft inside the cavitons is roughly doubled, which corresponds
well to the enhanced density of suprathermal ions shown in
panel (e) of Fig. . In the
simulation, the waves surrounding the caviton show some compression in
agreement with observations that have demonstrated that cavitons are immersed
in a sea of compressive waves. ULF waves similar to these have been reported
in , which also uses Vlasiator data. We note, however, that
compressive waves in this Vlasiator run have amplitudes in the magnetic field
magnitude and density of around 10–20 % of the background values. These
amplitudes are smaller than what is observed at Earth near cavitons, which is
typically 50 % of the average magnetic field .
Compressive waves in the foreshock can have even larger amplitudes closer to
the shock . This may be due to the spatial resolution in
our simulation, which can limit the steepening of the waves
. This may also explain why the caviton shown in
Fig. does not display the “shoulders” of enhanced plasma
density identified in spacecraft observations on either sides of the density
and magnetic field depression.
In the central part of the structure, the magnetic field strength and plasma
density decrease to about 50 % of their solar wind values, as is the case
on average for cavitons, according to a statistical study performed by
. Only the duration of the structure, which is about 25 s
here, does not match so well with previous works and is at the lower end of
the distribution obtained by . This is due to a combination
of two factors. One is that the solar wind velocity in our run is
750 km s-1, which corresponds to conditions during a high-speed stream
or a solar wind transient, thus causing the caviton to convect faster past
the virtual spacecraft than during regular solar wind conditions. For average
solar wind speed (about 400 km s-1), the same caviton would last about
46 s in the virtual spacecraft time series. Another factor that can affect
the observed duration of the structure is how we define where the edges of
the cavitons are. Specifically, the shoulders surrounding the cavitons
are included in the caviton durations in , but this cannot
be done in our simulation, as these features are not as markedly defined.
Overall, our results show a good agreement with the observations, both
qualitatively and quantitatively.
Conclusions
In this paper we have used the hybrid-Vlasov simulation software Vlasiator to
study foreshock transients and their effect on the bow shock of the Earth.
Our main foci have been cavitons and SHFAs and their transition into a
magnetosheath cavity. Vlasiator simulations confirm previous experimental and
simulation results that cavitons evolve into SHFAs as they approach the bow
shock and fill up with a high density of suprathermal ions, which have been
reflected by the bow shock. Vlasiator improves on previous studies both in
modelling the realistic scales of all transient structures and in providing
high-quality noise-free velocity distribution functions throughout the
simulation.
The primary result of this study is that large SHFAs can survive downstream
of the bow shock and erode it, creating a large-scale structure of
low-density, high-β plasma extending deep into the magnetosheath. In
our simulation, the bow shock erosion seems to be initially triggered by
SHFAs spanning about 1 RE2, which are among the largest
observed during the run. Though other SHFAs of comparable size also appear
elsewhere in the foreshock, only one magnetosheath cavity forms. Our analysis
suggests that this is likely due to the fact that numerous SHFAs impinged the
shock roughly at the same place. In particular, our simulation run shows that
chains of cavitons and SHFAs form along a narrow, roughly field aligned, band
of decreased plasma density (after t=1050 s in yellow in the Supplement)
and successively hit the bow shock. These subsequent cavitons and SHFAs may
contribute to the growth of the magnetosheath cavity and the shock erosion.
The higher density of suprathermal ions and associated higher temperature
causes a pressure increase in this narrow region, which may explain the
decreased total thermal ion density and
magnetic field. We also propose that a crucial factor in facilitating this is
that the initial large SHFA crosses the bow shock rather close to the nose,
where the bulk flow velocity in the magnetosheath parallel to the shock front
is relatively small. Therefore, the forming magnetosheath cavity is convected
quite slowly along the bow shock, with more time to grow, whereas further
downstream, the SHFAs which have crossed the shock propagate away from the
nose inside the magnetosheath faster and thus are not strengthened by other
structures, or at least not to such a large extent. This effect is
self-strengthening, as bulk flow velocities within the magnetosheath cavity
are lower than those in adjacent parts of the magnetosheath. The heating of
the magnetosheath cavity leads to a decrease in magnetosonic Mach number
Mms, which results in the cavity-associated region of the bow
shock bulging out into the incoming solar wind. Dynamics of SHFA-triggered
erosion of other planetary bow shocks may be different due to different
magnetic field strengths and spatial scales. The fact that multiple SHFAs are
needed for the formation of large magnetosheath cavities is in agreement with
the results of .
It is interesting to note how the spatial distribution of the cavitons and
SHFAs in the foreshock changes with time. As can be seen in
Fig. e–g, the chains of cavitons and SHFAs along
field-aligned bands of decreased plasma density and magnetic field strength
are only visible at T2 and T3, but not at T1. One of these bands of
tenuous plasma and weaker magnetic field is associated with the magnetosheath
cavity, but others are observed at other places along the bow shock, thus
ruling out the fact that these bands are solely the result of some feedback
of the magnetosheath cavity on the upstream medium. On the other hand, we
note that the magnetosheath density is lower downstream of these structures,
thus suggesting that they may feed the magnetosheath cavity by causing an
additional density decrease. Disentangling in which ways the upstream and
downstream media, and the bow shock itself, influence each other and control
the spatial distribution of the foreshock transients is, however, left for
future work.
One of the properties associated with SHFAs is that of decreased bulk
velocity, and our simulation did display dips at the locations of cavitons
and SHFAs. However, the decrease seen in our data is much lower than that
reported by , , and .
To investigate this more, we tracked the changes visible in proton
distribution functions as foreshock features were convected across virtual
spacecraft and found that our VDFs resembled those found by
. That is, the flow of the thermal solar wind core was not
slowed or deflected, but, rather, changes in bulk flow are due to the
combination of a density decrease for the core and a strengthening of the
suprathermal beam. When the thermal core is depleted, the backstreaming beam
can have a relatively greater impact on bulk velocity measurements.
Even though one would intuitively think that there would be a lack of
suprathermal ions upstream of a weaker portion of the bow shock, we find that
the suprathermal beam density does
not vanish, as shown by the orange fingers extending into the white region in
Fig. c. The profiles shown in Fig. show
that suprathermal densities of three adjacent caviton-generating fingers are
roughly similar, despite one of them being in front of the eroded portion of
the shock. The bow shock is strongly distorted in the vicinity of the
magnetosheath cavity, which changes the local θBn and can therefore
affect the amount of backstreaming particles in this area. It is also
possible that the magnetosheath cavity acts as a heated ion reservoir, so
that ions can leak into the upstream medium and populate the foreshock,
leading to extra SHFA formation in this region. Finally, the decreased plasma
density at the cavity leads to an increase in Alfvénic Mach number. This
would result in any Alfvénic fluctuations convected from the upstream to
the downstream to pile up within the magnetosheath cavity. These fluctuations
could potentially act as an efficient scattering barrier for resonant
energetic ions, enabling them to be reflected back into the upstream, despite
the shock having a low density compression ratio and a low magnetosonic Mach
number in the vicinity of these features.
Upstream of the cavity and the eroded bow shock, we were able to examine VDFs
of suprathermal protons in the plasma frame and found SHFA-associated
increases in pitch-angle spreads. Beam particles had particularly large
pitch-angle spreads at the shockward edges of SHFAs. The cavity and eroded
shock caused strong deformation of the core solar wind population close to
the shock. Regions associated with strong formation of cavitons and SHFAs
tended to have stronger beam intensities.
We note that for both observational and simulational studies, care must be
taken in defining caviton and SHFA criteria in order to prevent masking
choices from influencing, for example, size distributions. We observed local regions
of plasma and magnetic field enhancements or rarefactions, causing the chosen
global selection criteria to preferentially detect large cavitons in regions
of tenuous plasma and small cavitons in regions of dense plasma. We observed
that solar wind frame suprathermal beam proton pitch-angle distribution
widths have a correlation with cavitons and SHFAs, with the strongest spread
at the shockward edge of cavitons. Thus, we recommend using proton VDFs as an
additional selection criteria. Determining the spatial and/or temporal extent
of the foreshock to use as the background level for selection criteria
remains an open question and is likely influenced by solar wind conditions.
It remains to be seen how much variation local background levels have in
global 3-D hybrid-Vlasov simulations.
The large magnetosheath cavity that develops in this run has similar features
to the magnetosheath cavities reported by , with decreased values of magnetic field
magnitude, ion density, and high temperature. Observational work presented in
shows the existence of such structures in the
Earth's magnetosheath. These authors predicted that the existence of such
structures represents a decrement in the total pressure applied to the
magnetosphere and can allow the magnetopause to move 30 % further from
Earth, compared with the position predicted from the far upstream solar wind.
For a HFA and an associated IMF tangential discontinuity,
reported magnetopause movement on the order of 5 RE. Our
simulations did not, however, indicate notable magnetopause movement in
reaction to SHFAs or the magnetosheath cavity. This is likely due to the
cavity growing to its full strength only as it has travelled further away
from the nose of the shock. The fast solar wind velocity in our run may also
play a role, as the structure convects relatively quickly along the bow
shock.
We also note that, as somewhat visible in Fig. a, our
simulation results in structures similar to magnetosheath filamentary
structures , with the magnetosheath cavity strongly
connected to a prominent filament. reported on a
cosmic-ray-induced filamentary instability in a parallel shock. At certain
phases of their hybrid simulation, a feature at their shock front bore a
striking resemblance to our magnetosheath cavity, although filamentation with
included enhancements in magnetic field appeared to be the dominating feature
instead of sheath heating. The connection between filaments and SHFA bow
shock crossings would be a potential topic for further study. A detailed
study of these connections is facilitated by the realistic sizes of both
types of structures provided by Vlasiator modelling.
The dependence of, for example, caviton formation on different IMF geometries
is something our single simulation run cannot
explore. Future possible extensions of this work would be the analysis of
SHFA and caviton formation rates and size distributions as well as shock
erosion in relation to different Mach numbers, IMF conditions, and solar wind
parameters. The numerical requirements associated with global high-resolution
hybrid-Vlasov modelling make parametric studies challenging, but not
impossible.
Our results show that cavitons evolve into SHFAs only within a few Earth
radii of the bow shock but also that this evolution occurs at distances
beyond those associated with SLAMS and shock reformation in these solar wind
conditions. Previous hybrid modelling investigating these phenomena (see,
e.g. ) has provided fascinating results, but our results
suggest that this connection between reformation, SHFA formation,
magnetosheath cavity formation, and bow shock erosion should be carefully
investigated using simulations with realistic physical length scales, such as
those provided by Vlasiator, in order to distinguish between these different
phenomena.