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**Regular paper**
08 Aug 2018

**Regular paper** | 08 Aug 2018

Statistical survey of bow shock currents

- Institut für Geophysik und extraterrestrische Physik, Technische Universtität Braunschweig, Braunschweig, Germany

Abstract

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We present the first comprehensive statistical survey of the day-side terrestrial bow shock current system based on a large number of Cluster spacecraft bow shock crossings. Calculating the 3-D current densities using fluxgate magnetometer data and the curlometer technique enables the investigation of current locations, directions, and magnitudes in dependence on arbitrary IMF orientation. In case of quasi-perpendicular shock geometries we find that the current properties are in good accordance with theory and existing simulation results. However, currents at quasi-parallel shock geometries next to the foreshock region underlie distinct variations regarding their directions.

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How to cite.

Liebert, E., Nabert, C., and Glassmeier, K.-H.: Statistical survey of day-side magnetospheric current flow using Cluster observations: bow shock, Ann. Geophys., 36, 1073-1080, https://doi.org/10.5194/angeo-36-1073-2018, 2018.

1 Introduction

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The terrestrial bow shock slows down the solar wind velocity to subsonic Mach numbers. This is accompanied by a gain in density, temperature, and magnetic field strength. According to Ampère's law the bow shock carries electric currents which account for the jump in the magnetic field components tangential to the bow shock's surface. In contrast to the magnetopause, where the current directions are mainly determined by the geometry of the Earth's magnetic field, the direction of bow shock currents is solely determined by the orientation of the interplanetary magnetic field (IMF).

Depending on the local geometries of the shock surface and IMF orientation
one distinguishes between quasi-perpendicular and quasi-parallel shocks where
the IMF encounters the bow shock normal with angles above and below
45^{∘}, respectively. When encountering the compressed magnetic field at
the shock some solar wind particles are reflected at the shock and
re-accelerated in the solar wind's electric field while gyrating along the
IMF direction before they enter the shock another time. At the quasi-parallel
shock the reflected particles form the foreshock region where upstream waves
are generated and alter the magnetic field configuration.

To date, detailed bow shock current analysis based on in situ measurements
has barely been done. Tang et al. (2012) presented a first statistical
survey of bow shock currents using Cluster data from 25 crossing events when
the IMF was dominated by its *B*_{z} component. They selected
quasi-perpendicular shocks near the bow shock nose and calculated the current
density from the magnetic gradient and the shock thickness. Recently,
Hamrin et al. (2017) investigated the currents of 154
quasi-perpendicular bow shock crossings near the shock nose using data from
the Magnetospheric Multiscale (MMS) mission. In this paper we extend the
statistical survey of bow shock currents at larger distances to the bow shock
nose to overall 369 events covering both quasi-perpendicular and
quasi-parallel situations during arbitrary IMF configurations for the first
time. Making use of the simultaneously collected magnetic field data supplied
by the Cluster multi-spacecraft mission (Escoubet et al., 2001) and
applying the curlometer technique (Dunlop et al., 1988) allow a direct
3-D investigation of the local current density vector.

2 Data selection and preparation

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For our investigation we use Cluster magnetic field data from the fluxgate magnetometer (FGM) (Balogh et al., 2001) at spin resolution (0.25 Hz). Additionally, data from the Cluster Ion Spectrometry (CIS) instrument (Rème et al., 1997) are used to support the identification of bow shock crossings. The data are retrieved from the Cluster Active Archive (Laakso et al., 2010). Cluster consists of four individual spacecraft orbiting along a polar orbit with relative separations of a few kilometers up to over 10 000 km. The thickness of the Earth's bow shock is about 100 to 1000 km. In order to match these spatial dimensions we use data obtained during periods when the average inter-spacecraft distance was small at the position of the bow shock. This criterion is fulfilled in the time range from February to May 2002 and from December 2003 to May 2004 when the average inter-spacecraft distance is about 300 km or less. At these times FGM and CIS data are available at the bow shock for 162 inbound and outbound orbit segments.

The curlometer technique estimates the local current density across the
Cluster tetrahedron volume by approximating Ampère's law:
$\mathrm{\nabla}\times \mathit{B}={\mathit{\mu}}_{\mathrm{0}}\mathit{J}$. A thorough introduction to the curlometer technique and an error
analysis focussing on our application can be found in our previous
publication (Liebert et al., 2017), where we conducted a similar study on
magnetopause currents. The reliability of curlometer results depends on the
Cluster tetrahedron geometry which is constantly changing along its
trajectory. One possibility to characterize the shape of the tetrahedron is
given by the quality factor *Q*_{G}, which is defined by

$$\begin{array}{}\text{(1)}& {Q}_{G}={\displaystyle \frac{\text{true\hspace{0.17em}\hspace{0.17em}volume}}{\text{ideal\hspace{0.17em}\hspace{0.17em} volume}}}+{\displaystyle \frac{\text{true\hspace{0.17em}\hspace{0.17em} surface}}{\text{ideal\hspace{0.17em}\hspace{0.17em} surface}}}+\mathrm{1}\end{array}$$

(Glassmeier et al., 2001), where the ideal volume and surface
represent the volume and surface of a perfect regular tetrahedron with a side
length equal to the average side length of the true tetrahedron. The quality
factor equals 1 when the true tetrahedron is deformed into a linear geometry
and 3 in case the true tetrahedron equals the ideal one. In this study we
limit our investigation to bow shock crossings, where the quality factor
takes values of 2.5 or more, leaving us 111 orbit segments. The usage of this
quality factor for our investigation is discussed in detail by
Liebert et al. (2017). *Q*_{G}≥2.5 allows us to expect accuracies of
at least 2^{∘} to 10^{∘} in direction and 3 % to 15 % for the
relative error in magnitude.

Dunlop et al. (2001) pointed out that high-frequency fluctuations are likely to cause uncertainties within the curlometer results and suggest an appropriate averaging in time before applying the curlometer to magnetic field data. Comparing the effect of averaging windows of different sizes on the events investigated in our study proves the current directions to be quite insensitive to a window size between 20 and 40 s. Significant alteration of the directions sets in for windows below 10 s and above about 60 s as the influence of spatial scales smaller and larger than the bow shock scales rises. Current magnitudes are more sensitive to the size of the averaging window. The damping of high-frequency fluctuation and the associated spatial averaging directly lead to smaller current peak magnitudes. This effect is less intense when an average magnitude per event is calculated instead. For the statistical study presented here, we chose a 30 s averaging window, which proved to be sufficient to damp highly fluctuating current signatures with spatial dimensions far below the tetrahedron size without altering current structures having dimensions of about 50 km and more along the spacecraft trajectory to a significant extent.

After the application of the curlometer tool, we look for bow shock current events by visual inspection of the curlometer results. Currents are identified as bow shock currents when the following criteria are fulfilled: (1) a clear current peak is visible, and (2) the current event coincides with particle data signatures that are consistent with a bow shock crossing (see the example event in Fig. 2). At each transition event the edges of the corresponding current feature are identified. We calculate the average current directions and magnitude for every event. There are cases where the current or the particle data or both show very fluctuating behavior, making it difficult to identify current structures at a bow shock crossing. In cases where the identification becomes presumably unreliable, the events are omitted and not included in our study.

Caused by relative movement of the bow shock with respect to Cluster, multiple crossings in a row are often recorded along an orbit segment. This enlarges our database for the statistical survey to 369 current events. Figure 3 gives an overview of the locations of all events in GSE coordinates.

3 Reference bow shock

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The position and size of the bow shock vary depending on the solar wind conditions to a large extent. For representation of the bow shock crossing locations we introduce a parabolic model reference bow shock as a common frame of reference. Following Nabert et al. (2013) we use the parametrization

$$\begin{array}{}\text{(2)}& x={\mathrm{\Delta}}_{\mathrm{BS}}-\sum _{t=y,z}{c}_{\mathrm{BS},t}\phantom{\rule{0.125em}{0ex}}{t}^{\mathrm{2}}.\end{array}$$

Δ_{BS} depicts the sub-solar bow shock stand-off distance
with respect to the center of the Earth
(see Fig. 4). The geometric parameters *c*_{BS,t}
represent the bow shock curvature in the *t*=*y* and *t*=*z* directions.
Nabert et al. (2013) deduce values of

$$\begin{array}{}\text{(3)}& {c}_{\mathrm{BS},y}=\mathrm{0.4}{\displaystyle \frac{\mathrm{1}}{{\mathrm{\Delta}}_{\mathrm{BS}}}}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}{c}_{\mathrm{BS},z}=\mathrm{0.5}{\displaystyle \frac{\mathrm{1}}{{\mathrm{\Delta}}_{\mathrm{BS}}}},\end{array}$$

from an analytical zeroth-order approach to solve the MHD equations in the magnetosheath.

For each identified current the mean value of the tetrahedron barycenter’s position vector is calculated. By radial projection along the Earth-spacecraft line the intersection of this vector with the reference bow shock is calculated (cf. Fig. 4). Figure 3 gives an overview of the location of all events after projection onto the reference bow shock within GSE coordinates.

Current directions at the bow shock are directly controlled by the IMF
orientation via Amperè's law. As we do not confine our study to mainly
north–south orientated IMF, the presentation of the resulting current
directions in a GSE system like in Fig. 3 would lead to
a quite chaotic looking distribution and would make it almost impossible to
extract useful information from it as the required information about the IMF
orientation would not be included in such a picture. In order to account for
the varying IMF orientations we conduct a second step of transformation by
rotating the coordinate system around the GSE *x* axis in such a way that the
IMF_{yz,GSE} component is orientated in the positive *z* direction
within the new IMF-aligned coordinate system. The IMF is calculated by
averaging the magnetic field data obtained during 5 min ahead of each bow
shock current event.

4 Results

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Depending on the angle Φ between local shock normal and IMF, current events are categorized into quasi-perpendicular ($\mathrm{\Phi}>\mathrm{45}{}^{\circ}$) and quasi-parallel shocks ($\mathrm{\Phi}<\mathrm{45}{}^{\circ}$). The majority of 274 of the investigated currents represent quasi-perpendicular geometries. It is likely that this imbalance is caused by our event selection procedure. Within the foreshock region of a quasi-parallel shock oscillations are triggered and develop while the plasma is convected towards the shock surface. This causes fluctuations within the particle and the magnetic field data, leading to less clear plasma transition and current signatures. As described in Sect. 2, events are omitted, when a reliable bow shock current identification is not possible.

Figure 5 shows the orientation of the current flow with
respect to the model bow shock normal and the IMF. Most of the observed bow
shock currents during quasi-perpendicular geometries lie nearly perpendicular
to both the shock normal, which means that the currents flow parallel to the
bow shock surface, and the IMF tangential component, as expected from theory.
The deviation from perpendicularity with respect to IMF_{t} is much
larger in the case of quasi-parallel shock situations (95 events), which
reflects more turbulent and fluctuating conditions of the plasma flow
adjacent to the foreshock region in contrast to the quasi-perpendicular
shock. The broad distribution of the angle between current direction and
reference bow shock normal indicates that the current flow direction at the
quasi-parallel bow shock deviates extremely from the shape of a simplified
bow shock surface.

The global current direction distribution of the quasi-perpendicular and
quasi-parallel bow shock currents is displayed in Fig. 6 in
the IMF-aligned coordinate systems. The current directions are represented by
arrows of normalized length and the color code indicates the direction of
currents with respect to the *x* axis with red arrows pointing towards
Earth and green arrows pointing towards the Sun. From theory, within this
coordinate system the currents are expected to possess positive *y*
components, independently of their locations. As visible in the *y*−*z*
projections the flow directions are prescribed by the IMF orientation very
clearly as they are collectively pointing along the positive *y* axis
in the IMF-aligned coordinate system. The histograms of the *J*_{yz}
direction in the bottom panel of Fig. 6 illustrate this in a
quantitative manner. It shows a very clear peak around 0^{∘} in the
quasi-perpendicular case. The peak is also visible in the quasi-parallel
case, but it is much broader and currents with negative *y* components are
observed frequently.

Having a closer look at the color code and the *x*−*y* projection reveals that
the currents are following the draped shape of the bow shock pointing towards
the Sun and Earth at the flanks. Again, more deviations are visible during
quasi-parallel bow shock crossings (Fig. 6b).
Hamrin et al. (2017) investigated *J*_{x} in a similar way. As their
events are located near the bow shock nose because of the MMS orbit, they
introduced the approximation *J*_{x}≈*J*_{n}, where *J*_{n} depicts the
portion of the current that flows normal to the shock surface. Their events
lie within a range of about 7 Earth radii distance from the Earth–Sun line
which approximately corresponds to a distance of about 0.5
Δ_{BS} in our coordinate system. Near the bow shock nose they
find that *J*_{x} points towards the Sun at *y*<0 (GSM) and towards Earth
at *y*>0 (GSM) for northward IMF. In the case of southward IMF the current directions
are reversed. The distribution of the colors in Fig. 6 shows
that the results of the orientation of the *J*_{x} component from the study
conducted by Hamrin et al. (2017) and from our study are qualitatively
identical and are also valid for larger distances from the bow shock nose.

In order to investigate the current normal component in our study, we
calculate *J*_{n} via the local normal direction of the model bow shock
surface for all quasi-perpendicular events. We analyze the orientation of the
normal component in dependence on the *y* coordinate within the IMF-aligned
coordinate system. Table 1 gives the occurrence rate of outward
and inward orientation of *J*_{n} in a region close to the bow shock nose,
approximately corresponding to the location of events within the study by
Hamrin et al. (2017) as well as a region further away from the nose at
the flanks of the bow shock. Based on our results, we can not identify a
general dependence of the *J*_{n} orientation from the location of the
events. Overall, the currents are slightly more often pointing outwards.

Figure 7 shows the occurrence distribution of
investigated current magnitudes. The majority (about 80 %) do not exceed
30 nA m^{−2} and the average current magnitude of all events is
19.4 nA m^{−2}. As the current magnitudes calculated by the curlometer
are influenced by the averaging in time and space (averaging window,
spacecraft separation, average current density along event trajectory), it is
more likely that current magnitudes tend to be underestimated than
overestimated. A direct comparison of some events which were analyzed in the
study by Tang et al. (2012) as well as in our study show that the
current density magnitudes calculated with the curlometer technique are
factors between 2.7 and 4.5 smaller than those calculated by determination of
the layer thickness and the jump in the magnetic field.

MHD simulations (Lopez et al., 2011) predict a relatively broad
region around the bow shock nose where the current magnitudes are constantly
high, while the magnitudes are decreasing at the high-latitude bow shock.
Investigating the spatial dependence of the event's current magnitudes
results in a roughly homogeneous distribution in case of the quasi-parallel
shock events which are all located at low latitudes (cf.
Fig. 6). In case of quasi-perpendicular events current
magnitudes near the bow shock nose are slightly larger than those at the
flanks. The average magnitude of all quasi-perpendicular currents below
60^{∘} latitude is 22.3 nA m^{−2}, while the average value of
currents at higher latitudes is 16.1 nA m^{−2}.

The IMF magnitude is another controller of the bow shock current magnitude. Based on Ampère's law and the Rankine–Hugoniot conditions one can expect a linear correlation between the current magnitude and the magnetic field strength of the IMF tangential component with respect to the shock surface

$$\begin{array}{}\text{(4)}& J\propto \left[{\mathit{B}}_{t}\right]\propto \mathit{I}\mathit{M}{\mathit{F}}_{t},\end{array}$$

where the brackets denote the jump of the magnetic field across the discontinuity.

Tang et al. (2012) found this linear relation between bow shock current
and IMF_{z} component at quasi-perpendicular bow shock events during
northward- and southward-orientated IMF. Our survey shows that the linear
correlation also applies to arbitrary IMF orientation and shock geometry.
Figure 8 displays the average current magnitudes that
are calculated for 1 nT intervals of the IMF component tangential to the bow
shock surface. For quasi-perpendicular and quasi-parallel cases magnetic
field values up to 15 and 8 nT are observed, respectively. Within the range
from 0 to 8 nT there are no distinct qualitative differences between
quasi-perpendicular and quasi-parallel situations visible.

In the limit of a high Mach number one can derive

$$\begin{array}{}\text{(5)}& \mathit{J}=\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathit{I}\mathit{M}{\mathit{F}}_{t}/\left({\mathit{\mu}}_{\mathrm{0}}L\right),\end{array}$$

where *L* is the bow shock thickness. The correlation coefficient of the
linear fit within Fig. 8 is 0.84. The slope of the
fit provides an estimate of the average bow shock thickness of about
1600 km. As mentioned above, it is likely that the magnitudes tend to be
underestimated by the curlometer technique. The value of 1600 km therefore
represents an upper estimate of the shock thickness. Bale et al. (2003)
performed an extensive study of the bow shock thickness which gives a typical
scale of a few hundreds of kilometers.

5 Conclusions

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The usage of the curlometer technique allows us a direct investigation of 369 current events recorded within the magnetic field data obtained by Cluster during 111 bow shock crossings in 2002, 2003, and 2004. In 274 cases the bow shock represented a quasi-perpendicular shock ($\mathrm{\Phi}>\mathrm{45}{}^{\circ}$). In 95 events a quasi-parallel ($\mathrm{\Phi}<\mathrm{45}{}^{\circ}$) shock was observed. It lies adjacent to the foreshock region where solar wind particles are reflected back into the solar wind along the IMF direction and cause upstream and downstream perturbations of the magnetic field configuration.

At quasi-perpendicular shocks the bow shock currents are very clearly
described by the IMF direction fulfilling the equation $\mathrm{\nabla}\times \mathit{B}={\mathit{\mu}}_{\mathrm{0}}\mathit{J}$. The angle distribution between the bow shock current and
IMF_{t} shows a sharp peak at 90^{∘}. When displayed in an
IMF_{yz}-aligned coordinate system the current directions' arrows arrange
themselves parallel to each other in the *y*−*z* plane. As the currents flow
parallel to the shock surface, the draped shape of the bow shock becomes
visible within the current direction in the *x*−*y* plane. Current magnitudes
are larger near the bow shock nose than at the flanks.
Figure 9 is a schematic summary of our results for the
currents observed at the day-side quasi-perpendicular bow shock.

The angle distribution between currents and the normal of the reference bow
shock peaks at 90^{∘} as well, but it is broadened to some extent as
the reference bow shock naturally deviates from the true bow shock geometry.
The magnitudes of currents at the quasi-perpendicular shock generally
increase with an increasing tangential component of the IMF. Typical values
of the averaged current magnitudes obtained by the curlometer technique are
in the range of 5 to 40 nA m^{−2} with an average of about
20 nA m^{−2}. Those results are on the same order of magnitude but
smaller than the ones determined in former studies by calculating the current
density via the jump in the magnetic field and a derived current sheet
thickness.

The quasi-parallel shocks that we have found are all located at relatively low latitudes. In this region, we were not able to observe a spatial dependence of the current magnitude. Additionally, the IMF tangential components possess lower values only up to 8 nT. In this range we find that the dependence on the IMF magnitude seems to be qualitatively equal to that observed for quasi-perpendicular situations. In particular the current magnitudes of the quasi-perpendicular and quasi-parallel bow shocks are of a similar size for a given IMF, which is an interesting finding as the ideally (in reality never realized) parallel shock would be accompanied by no jump in the magnetic field and therefore no current at all (Narita, 2006).

The direction of currents of the quasi-parallel bow shock are less described by the IMF orientation compared to the quasi-perpendicular shock. Overall, the main characteristics are maintained, but far more and larger deviations are visible. In addition, the currents no longer lie perpendicular to the normal direction of the reference bow shock, which indicates that the simplified model bow shock geometry does not hold at the quasi-parallel bow shock.

Data availability

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Data availability.

All Cluster data used in this study can be retrieved from the Cluster Active Archive (Laakso et al., 2010).

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was financially supported by the Deutsches Zentrum für Luft-
und Raumfahrt under contract 50OC1402. The author thanks the Cluster FGM and
CIS teams and the Cluster Science Archive for processing and providing the
Cluster data.

The
topical editor, Minna Palmroth, thanks two anonymous referees for help in
evaluating this paper.

References

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