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**Annales Geophysicae**
An interactive open-access journal of the European Geosciences Union

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**Regular paper**
10 Oct 2019

**Regular paper** | 10 Oct 2019

Strong southward and northward currents observed in the inner plasma sheet

^{1}Institute of Crustal Dynamics, China Earthquake Administration, Beijing, 100085, China^{2}Center for Satellite Application in Earthquake Science, China Earthquake Administration, Beijing, 100085, China^{3}Harbin Institute of Technology, Shenzhen, 518055, China^{4}Department of Mechanics and Engineering Science, Peking University, Beijing, 100871, China

^{1}Institute of Crustal Dynamics, China Earthquake Administration, Beijing, 100085, China^{2}Center for Satellite Application in Earthquake Science, China Earthquake Administration, Beijing, 100085, China^{3}Harbin Institute of Technology, Shenzhen, 518055, China^{4}Department of Mechanics and Engineering Science, Peking University, Beijing, 100871, China

**Correspondence**: Chao Shen (shenchao@hit.edu.cn)

**Correspondence**: Chao Shen (shenchao@hit.edu.cn)

Abstract

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It is generally believed that field-aligned currents
(FACs) and the ring current (RC) are two dominant parts of the inner
magnetosphere. However, using the Cluster spacecraft crossing the
pre-midnight inner plasma sheet in the latitudinal region between 10 and 30^{∘} N, it is found that, during intense geomagnetic storms,
in addition to FACs and the RC, strong southward and
northward currents also exist which should not be FACs because the magnetic field in
these regions is mainly along the *x*–*y* plane. Detailed investigation shows that
both magnetic-field lines (MFLs) and currents in these regions are highly dynamic.
When the curvature of MFLs changes direction in the *x*–*y* plane, the current also
alternatively switches between being southward and northward. To investigate the
generation mechanism of the southward and northward current, we employed the
analysis of energetic particle flux up to 1 MeV. For energetic particles below
40 keV, observations from Cluster CIS/CODIF (Cluster Ion Spectrometry COmposition and DIstribution Function analyzer) are used. However, for higher-energy particles, the flux is
obtained by extrapolations of low-energy particle data through Kappa
distribution. The result indicates that the most reasonable cause of these southward and northward currents is the curvature drift
of energetic particles.

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Yang, Y.-Y., Shen, C., and Ji, Y.: Strong southward and northward currents observed in the inner plasma sheet, Ann. Geophys., 37, 931–941, https://doi.org/10.5194/angeo-37-931-2019, 2019.

1 Introduction

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Abundant current systems existing in the Earth's magnetosphere play a very important role in energy transformation in different regions (Kuijpers et al., 2014). Recently, through simulations and observations, numerous studies have shown that the inner-magnetosphere currents have a more complicated structure and distribution than originally thought. For example, in the low latitude, the magnetic-field geometry can be altered significantly into a tail-like shape during storm time (Tsyganenko et al., 2003). One or multiple banana currents can exist in the inner magnetosphere, which makes the link of the current systems more complicated (Liemohn et al., 2013). In the high latitudes, field-aligned currents (FACs) have more sophisticated structures except for the known large-scale region-1 and region-2 currents (Mishin et al., 1997; Dunlop et al., 2015a, b). Therefore, more work is still needed to reveal the true nature of these current systems.

The huge progress in satellite deployments makes it possible for direct
observation of the inner-magnetosphere current system. It is believed that
the magnetosphere and ionosphere are linked through a ring current (RC) and
FACs (e.g., Le et al., 2004; Zhang et al., 2011). Therefore, many
investigations are mainly focused on these two current systems, from high
(e.g., Iijima and Potemra, 1976, 1978; Wang et al., 2006; Dunlop et al.,
2015a) to low latitudes (e.g., Vallat et al., 2005; Shen et al.,
2014; Yang et al., 2016). The region from low to middle latitudes, which is
the key area for the inner-magnetosphere current link, however, has received
less attention. Graphic plots and some statistical results (e.g., Le et al.,
2004) show that FACs should be the dominant current in these areas. Through
Cluster satellite observations, Vallat et al. (2005) pointed out that the RC
could exist at middle (or even high) latitudes. Despite the results achieved
by these various research efforts, there are still no findings
enabling a conclusion about the complete current morphology in low and
middle latitudes. For example, are FACs and the RC the only currents in
these regions? If there are other currents, what is the corresponding
generation mechanism for them? To address these questions, the current
distribution and magnetic-field geometry during two storm events are
investigated in the latitudinal regions from 10 to 30^{∘} N.

In the following, we will use Cluster fluxgate magnetometer (FGM; Balogh et al., 1997) data to conduct the analysis for two reasons: (1) the polar orbit of Cluster offers an opportunity to go through both the low-latitude and middle-latitude regions and (2) the configuration of the four Cluster satellites makes it possible to calculate the current via Maxwell–Ampère's law and obtain the magnetic-field geometry. Moreover, in many previous works, it was thought that an asymmetric RC linked with the FACs, which is generally believed to occur during storm time, so storm events are our primary focus here.

Throughout this paper, solar magnetospheric (SM) coordinates are used. To
better describe angles, spherical coordinates (*θ*, *φ*) in the
SM frame are also defined; i.e., the polar angle *θ* (0^{∘} $\le \mathit{\theta}\le \mathrm{180}$^{∘}) is the angle between the +*z* axis
and the vector direction while the azimuthal angle *φ* (0^{∘} $\le \mathit{\phi}\le \mathrm{360}$^{∘}) is rotated anticlockwise from the
+*x* axis in the *x*–*y* plane when seen from +*z* axis. For current density
analysis, the local cylindrical coordinate system (*j*_{ρ}, *j*_{φ}, *j*_{z}; Vallat et al., 2005) is also utilized. Where
*j*_{z} is parallel to the +*z* axis, *j*_{ρ} represents the
radial component of the current on the plane parallel to the *x*–*y* plane,
oriented anti-earthward; *j*_{φ} points eastward, describing the RC.

2 Methodology

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In this study, magnetic curvature analysis (MCA; Shen et al., 2003) and
magnetic rotation analysis (MRA; Shen et al., 2007) are used; these
techniques have the unique ability to reveal the three-dimensional geometric
structure of the magnetic field directly as well as to provide more detailed
magnetic-field-related parameters, such as the magnetic-field gradient;
curvature; and the binormal of magnetic-field lines (MFLs), rotation rates, and
current density. The magnetic unit vector
$\widehat{\mathit{b}}=\mathit{B}/\left|\mathit{B}\right|$, curvature
vector ${\mathit{\rho}}_{\mathrm{c}}({\mathit{\rho}}_{\mathrm{c}}=(\widehat{\mathit{b}}\cdot \mathrm{\nabla}\left)\widehat{\mathit{b}}\right)$, and the binormal
vector $\widehat{\mathit{N}}(\widehat{\mathit{N}}=\widehat{\mathit{b}}\times {\widehat{\mathit{\rho}}}_{\mathrm{c}}/\left|\widehat{\mathit{b}}\times {\widehat{\mathit{\rho}}}_{\mathrm{c}}\right|)$ are
orthogonal to each other in the analysis, and the radius of curvature is
${R}_{\mathrm{c}}=\mathrm{1}/{\mathit{\rho}}_{\mathrm{c}}$. The magnetic vector ** b** has maximum,
median, and minimum rotation rates of ${\mathit{\mu}}_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}$, ${\mathit{\mu}}_{\mathrm{2}}^{\mathrm{1}/\mathrm{2}}$,
and ${\mathit{\mu}}_{\mathrm{3}}^{\mathrm{1}/\mathrm{2}}$ along ${\widehat{\mathit{e}}}^{\left(\mathrm{1}\right)}$,
${\widehat{\mathit{e}}}^{\left(\mathrm{1}\right)}$, and ${\widehat{\mathit{e}}}^{\left(\mathrm{2}\right)}$, respectively,
where ${\widehat{\mathit{e}}}^{\left(\mathrm{1}\right)}$, ${\widehat{\mathit{e}}}^{\left(\mathrm{1}\right)}$, and
${\widehat{\mathit{e}}}^{\left(\mathrm{2}\right)}$ are the three characteristic eigenvectors of the
magnetic field. Note that because the strong geomagnetic field in the
region of interest will produce artificial currents in the basic MRA
calculation (nonlinear contributions), the dipole field is subtracted when
using the MRA method to minimize truncation error (Shen et al., 2014).

To make a comparison with the nondisturbed geomagnetic field, the local
dipolar values of the magnetic-field strength, *B*_{tDip}; the radius of curvature,
*R*_{cDip}; the magnetic-field gradient strength, |∇*B*_{Dip}|; and three rotation rates, ${\mathit{\mu}}_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}$, ${\mathit{\mu}}_{\mathrm{2}}^{\mathrm{1}/\mathrm{2}}$, and
${\mathit{\mu}}_{\mathrm{3}}^{\mathrm{1}/\mathrm{2}}$, are also presented. They are calculated (Shen et
al., 2014) by using

$$\begin{array}{}\text{(1)}& \begin{array}{rl}{\mathit{B}}_{\mathrm{tDip}}& =M{r}^{-\mathrm{3}}\sqrt{(\mathrm{1}+\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta})},\\ {R}_{\mathrm{cDip}}& ={\displaystyle \frac{r}{\mathrm{3}}}\sqrt{(\mathrm{1}+\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta}{)}^{\mathrm{3}}}/[\left|\mathrm{sin}\mathit{\theta}\right|\cdot (\mathrm{1}+{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta}\left)\right],\\ \left|\mathrm{\nabla}{\mathit{B}}_{\mathrm{Dip}}\right|& =\mathrm{3}M{r}^{-\mathrm{4}}\\ & \cdot \sqrt{\mathrm{1}+{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta}(\mathrm{7}+\mathrm{8}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta})}/\sqrt{(\mathrm{1}+\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta})},\\ {\mathit{\mu}}_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}& ={\mathit{\mu}}_{\mathit{\theta}}^{\mathrm{1}/\mathrm{2}}=\mathrm{3}(\mathrm{1}+{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta})/\left[r\right(\mathrm{1}+\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta}\left)\right],\\ {\mathit{\mu}}_{\mathrm{2}}^{\mathrm{1}/\mathrm{2}}& ={\mathit{\mu}}_{\mathit{\varphi}}^{\mathrm{1}/\mathrm{2}}=\mathrm{3}\left|\mathrm{cos}\mathit{\theta}\right|/\left[r\sqrt{(\mathrm{1}+\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta})}\right],\\ {\mathit{\mu}}_{\mathrm{3}}^{\mathrm{1}/\mathrm{2}}& ={\mathit{\mu}}_{r}^{\mathrm{1}/\mathrm{2}}=\mathrm{0},\end{array}\end{array}$$

where $M=m\cdot {\mathit{\mu}}_{\mathrm{0}}/\mathrm{4}\mathit{\pi}$ (with $m=\mathrm{7.78}\times {\mathrm{10}}^{\mathrm{22}}$ A m^{−2} being the Earth's magnetic dipole moment) and *r* is the radial
distance in SM coordinates.

3 Event analysis

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The chosen events occurred on 12 April 2001 and 31 March 2001. These were the two largest storms from 2001 to 2004 during which the
four Cluster satellites had a small (best) tetrahedron separation distance
(≤1000 km). The minimum Dst indices for the two events were −271 and
−387 nT, respectively. During the two events, Cluster was in the
pre-midnight sector and traversed the RC region vertically from the Southern
to Northern Hemispheres. The region of interest is in the Northern
Hemisphere. Figure 1 gives the proton density and differential flux for
H^{+}, He^{+}, and O^{+} during the analyzed interval, which are
obtained from the Cluster Ion Spectrometer (CIS; Rème et al., 2001). The
figure indicates that Cluster is mainly in the plasma sheet region (e.g., Vallat
et al., 2005).

The time interval of interest for the first event is from 05:00 to 05:25 UT,
with latitude ranging from 16.9 to 25.7^{∘}. Figure 2
presents some of the main physical quantities. Figure 2a shows the average
magnetic field 〈*B*_{t}〉 detected from the four
Cluster satellites and the local dipolar magnetic-field strength. It can be
seen that the local magnetic field is enhanced in this area. Figure 2b
indicates that the polar angle of the magnetic field is close to
90^{∘}, showing that the magnetic field lies approximately in the
*x*–*y* plane. The polar angle and azimuthal angle of dipolar fields are also shown
in dashed lines in Fig. 2b, which indicates a large deviation of the polar
angle with observations. Figure 2c shows that the radius of curvature,
*R*_{c}, has large variations. It is interesting to see that *ϕ*_{c} (the angle of
*R*_{c}
in Fig. 2d) changes direction alternately during the whole period.
Therefore, eight regions (numbered from NH1 to NH8) were chosen according to
the changes in the *ϕ*_{c} direction to investigate their features.
The variations in some physical quantities are also summarized in Table 1.
For *ϕ*_{c} and *θ*_{e1}, the average values (with
a few large abnormal points removed) during this period are given. The “–”
denotes values with large oscillations. For *j*_{z}, the
maximum or minimum value during each interval is presented.

As shown in Fig. 2c, the radius of curvature of MFLs in the eight regions
is more varied compared with that of the dipole field. Another feature observed
in Fig. 2c is that *R*_{c} peaks at the vertical dashed lines. This
is reasonable, since the curvature radius in the transition region should be
larger than the region where the curvature radius has opposite directions.
Figure 2d and the *ϕ*_{c} row in Table 1 give the average value of the
azimuthal direction *ϕ*_{c} during each interval. This
quantitatively reveals that *ϕ*_{c} alternatively varied between
30.3 and 51.9 and 230.3 and 292.0^{∘}.
It is noted from Fig. 2d that, for some regions, the variation in polar
angle *θ*_{c} has larger fluctuation (than azimuthal angle
*ϕ*_{c}). This feature reflects larger changes of the magnetic field in
*z* component. Figure 2g shows that ${\mathit{\mu}}_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}$ has an
enhancement in each region, illustrating a stretched MFL structure. Figure 2h and row *θ*_{e1} in Table 1 show that, for most regions,
the largest value of the polar angle *θ*_{e1} for ${\mathit{\mu}}_{\mathrm{1}}^{\mathrm{1}/\mathrm{2}}$ is close to 90^{∘}; therefore, the largest
deviation of MFLs is along the *x*–*y* plane. Figure 2i indicates that the current
oscillates and that the dominant current is along *j*_{ρ} and
the north (or south) *j*_{z} direction, while *j*_{φ} is
basically small compared with *j*_{ρ} and *j*_{z}. To
show FACs, the *j*_{B} component is also given in Fig. 2i; it can be seen
that the value of *j*_{B} is close to that of *j*_{ρ} because the
direction of the magnetic field points approximately to the radial direction
(see Fig. 2b). The maximum values for *j*_{B} and *j*_{z} were ∼40 and ∼80 nA m^{−2},
respectively. From Table 1 and Fig. 2, it is interesting to see that, from
region NH1 to region NH8, the *j*_{z} component changed from
positive (northward) to negative (southward) as *ϕ*_{c} varied
from <50 to >230^{∘}.

^{a} Storm events considered in this work.^{b} The physical quantity *ϕ*_{c} is the average azimuthal direction
of the curvature radius, *θ*_{e1} is the average polar angle
of maximum rotation rates of the magnetic field, and *j*_{zm} represents the maximum or minimum value of the *j*_{z}-current component.^{c−m} Regions for each storm event.

Another larger storm occurred between 07:30 and 08:00 UT on 31 March 2001.
The event was once reported by Shen et al. (2014), but they only
concentrated on the interval from approximately 07:00 to 07:25 UT. Observations are
shown in Fig. 3 for the latitudinal region from 13.1 to
31.2^{∘} N, the interval during the main phase of the storm. Here, 11
regions designated from NH1 to NH11 are divided also according to azimuthal
direction changes of *ϕ*_{c}. The variations in some relative
physical quantities are also shown in Table 1. From Fig. 3 and Table 1, it
can be seen that these parameters behave the same as those of the first event,
but with strong magnetic-field strength. Figure 3 indicates that the
magnetic-field strength is stronger than that during the first event. The
magnetic field is in the *x*–*y* plane (see Fig. 3b). The radius of curvature of
MFLs (see Fig. 3c), the magnetic-field gradient (Fig. 3e), and the
largest rotation rate (Fig. 3g) oscillates significantly and exhibits
large deviations compared with those of the dipole field. Figure 3f shows
that the magnetic-field gradient is in the *x*–*y* plane and directed toward the
dayside. Figure 3h and row *θ*_{e1} demonstrate that the
largest variation in MFLs is near the *x*–*y* plane. In Fig. 3i, it is clear that
the *j*_{z} component is the dominant current, with a maximum
value of ∼300 nA m^{−2}. This value is more than triple that of the
12 April 2001 event. It is clear to see that the *j*_{φ} component is the smallest among these currents. Similar to first event,
*j*_{z} is simultaneously observed to vary from northward to
southward when *φ*_{c} changes direction.

4 Discussion

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During the 12 April 2001 and 31 March 2001 strong storm events, the Cluster
satellites were located in the pre-midnight sector and crossed from
∼10 to ∼30^{∘} N. In these
regions, both the magnetic-field parameters and the current density
fluctuated significantly. The MFLs, which were mainly in the *x*–*y* plane,
severely deviated from the dipole field and changed (stretched) along the
*x*–*y* plane. Figure 4 displays the total magnetic-field strength and its three
components. It can be seen that the *x* and *y* components of the magnetic field
have the largest fluctuations, which is consistent with the results obtained
from Figs. 2 and 3. To further investigate the fluctuation, the continuous
1-D wavelet transform method is applied in the *x* and *y* component of the magnetic
field. It is found that the ULF wave (ultra-low-frequency wave) covering a range of frequencies
spanning from 4 to 10 mHz can be observed (not shown here), which is
consistent with the typical current density variation in the ∼2–4 min period. Actually, the ULF wave in the plasma sheet region has been
extensively reported in previous works (see Keiling, 2009, and references
therein). Thus, it seems that ULF wave is a possible way to cause the
variation in the curvature radius (and the field-aligned current).

The most obvious phenomenon in the two cases is the existence of three
current systems, i.e., FACs, *j*_{B}, an azimuthal current, *j*_{φ}, and a northward (or southward) current, *j*_{z}. Among them,
*j*_{z} is basically the strongest current component. In previous
studies (e.g., Le et al., 2004; Vallat et al., 2005), the existence of
*j*_{B} and *j*_{φ} has been proved. However, the occurrence
of such a strong *j*_{z} in the inner plasma sheet has not been
reported before. In the work of Vallat et al. (2005), they also found a
southward current (see Fig. 14 and corresponding text). But it is in
the equatorial ring current region (with no direction changes) and is mainly caused
by an asymmetry between the ionospheric conductivities of the two
hemispheres. It is very clear that the southward current in their paper is
different than what we report here.

As introduced in previous studies (e.g., Parker, 1957), the current in the inner magnetosphere generally arises from gradient drifts as well as curvature drift and the gyromotion of energetic particles. They can be calculated by using the following (e.g., Lui et al., 1987; De Michelis et al., 1999):

$$\begin{array}{}\text{(2)}& {\displaystyle}{\mathit{j}}_{\mathrm{\nabla}}={P}_{\perp}{\displaystyle \frac{\mathit{B}\times \mathrm{\nabla}\mathit{B}}{{B}^{\mathrm{3}}}},\text{(3)}& {\displaystyle}{\mathit{j}}_{\mathrm{C}}=-{\displaystyle \frac{{P}_{\parallel}}{{B}^{\mathrm{2}}}}{\mathit{\rho}}_{\mathrm{c}}\times \mathit{B},\text{(4)}& {\displaystyle}{\mathit{j}}_{\mathrm{G}}={\displaystyle \frac{\mathit{B}}{{B}^{\mathrm{2}}}}\times \left[\mathrm{\nabla}{P}_{\perp}-{\displaystyle \frac{{P}_{\perp}}{\mathit{B}}}\mathrm{\nabla}\mathit{B}-{\displaystyle \frac{{P}_{\perp}}{{B}^{\mathrm{2}}}}(\mathit{B}\cdot \mathrm{\nabla})\mathit{B}\right],\end{array}$$

where *j*_{∇}, *j*_{C}, and *j*_{G}
represent the gradient current, curvature current, and gyromotion current,
respectively, and *P*_{⊥} and *P*_{∥} are the pressure tensor
components perpendicular and parallel to the magnetic field, which can be
deduced from

$$\begin{array}{}\text{(5)}& {\displaystyle}{P}_{\perp}=\mathit{\pi}\sqrt{\mathrm{2}m}\int \int J\sqrt{\mathit{\epsilon}}{\mathrm{sin}}^{\mathrm{3}}\mathit{\alpha}\mathrm{d}\mathit{\alpha}\mathrm{d}\mathit{\epsilon},\text{(6)}& {\displaystyle}{P}_{\parallel}=\mathrm{2}\mathit{\pi}\sqrt{\mathrm{2}m}\int \int J\sqrt{\mathit{\epsilon}}{\mathrm{cos}}^{\mathrm{2}}\mathit{\alpha}\mathrm{sin}\mathit{\alpha}\mathrm{d}\mathit{\alpha}\mathrm{d}\mathit{\epsilon},\end{array}$$

where *m* is the mass of the particle, *J* is the differential flux intensity,
and *ε* and *α* are the particle energy and pitch angle,
respectively. Since the magnetic-field gradient ∇** B** and curvature

For the two events in this study, both the magnetic field and magnetic-field
gradient are directed toward the dayside. Therefore, the current deduced
from ** B**×∇

Based on the above analysis, graphic plots are given in Fig. 5c and d to
explain the possible generation mechanism for *j*_{z}. During the
strong storm time, turbulence, e.g., ULF waves, results in the fluctuation
of the MFLs; then, the radius of curvature of the MFLs decreases, leading to
an increase in the curvature drift current. During this process, the
direction of the magnetic field is nearly unchanged because the background
field is very strong. However, the curvature will alternately change
directions along with the variation in the MFLs, resulting in alternating
variations in $-{\mathit{\rho}}_{\mathrm{c}}\times \mathit{B}$, i.e.,
leading to the oscillation of *j*_{z}.

Figure 5a and b can only illustrate that the direction of
$-{\mathit{\rho}}_{\mathrm{c}}\times \mathit{B}$ is consistent with
northward current. To quantitatively check if the curvature current calculated
through Eq. (3) is consistent with the result obtained from the MRA method, further
investigation is necessary. The CIS/CODIF (Cluster Ion Spectrometry COmposition and DIstribution Function analyzer) can provide the differential flux intensity
for energy below 40 keV. Through Eqs. (3), (5), and (6), the curvature current
can be estimated. The results show that the main variation trend is
consistent with result from MRA, but the intensity is very small (less than
1 nA m^{−2}; not shown here). However, it should be noted that, for Cluster
CIS/CODIF, only low-energy particle data are available; therefore, a large
bias may exist when calculating the storm-time current. In contrast, much higher
energy is used in previous studies (e.g., 1 MeV in the work of Lui et al.,
1987). Cluster RAPID can provide energy spectrograms for the high-energy
particle from ∼27.6 to ∼3056 keV.
Unfortunately, there are no available data for the two analyzed events. The
statistical study from Kronberg et al. (2015) proves that, in the near-Earth
plasma sheet, higher-energy hydrogen and oxygen are greatly enhanced
during geomagnetic activity. In the work of Ma et al. (2012), they also
indicated that the flux for higher-energy particles could be comparable or
larger than that of the low-energy particles.

Though, there is no available differential flux for high-energy particles on Cluster, the curvature current still can be estimated through simulations. Previous works have proved that the particle distribution in plasma sheet can be described as Kappa distribution functions (Pierrard and Lazar, 2010, and references therein):

$$\begin{array}{}\text{(7)}& f={N}_{\mathrm{1}}{\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathit{\pi}m{E}_{\mathrm{0}}{\mathit{\kappa}}_{\mathrm{1}}}}\right)}^{\mathrm{2}/\mathrm{3}}{\displaystyle \frac{\mathrm{\Gamma}({\mathit{\kappa}}_{\mathrm{1}}+\mathrm{1})}{\mathrm{\Gamma}({\mathit{\kappa}}_{\mathrm{1}}-\mathrm{1}/\mathrm{2})}}{\left(\mathrm{1}+{\displaystyle \frac{E}{{\mathit{\kappa}}_{\mathrm{1}}{E}_{\mathrm{0}}}}\right)}^{-{\mathit{\kappa}}_{\mathrm{1}}-\mathrm{1}},\end{array}$$

where *N*_{1} and *E*_{0} denote particle density and temperature, and
*κ*_{1} is a constant. For energy satisfying *E*≫*E*_{0}, Eq. (7) can
be written as

$$\begin{array}{}\text{(8)}& f=a{E}^{-{\mathit{\kappa}}_{\mathrm{1}}-\mathrm{1}}.\end{array}$$

Since the differential flux intensity *J* and particle velocity distribution
function *f* are related by *J*=*f**p*^{2}, Eq. (8) is also the function of *J*,
namely

$$\begin{array}{}\text{(9)}& J=a{p}^{\mathrm{2}}{E}^{-{\mathit{\kappa}}_{\mathrm{1}}-\mathrm{1}},\end{array}$$

where *p* is the momentum of the concerned particles, and *a* is a constant.
Thus, with the known differential flux intensity from low-energy
particle, the parameter *a* and *κ*_{1} can be determined. Then, the
differential flux intensity for high-energy particles (to 1 MeV) can be
estimated using Eq. (9). Though particles are accelerated during the storm,
we have confirmed that the Kappa distribution is still satisfied using
CIS/CODIF observations (not shown here). However, it should be noted that,
during the storm, *a* and *κ*_{1} are no longer a constant but varied
with time. Besides this, to check if the estimated high-energy particle
differential flux (using low-energy particle data) is reasonable, we select
a storm event that occurred on 20 April 2002, which has similar position with two
analyzed events in this study, and has CIS/CODIF and RAPID observations at
the same time. The result shows that the fitted result (from CIS/CODIF
measurement) can basically reflect the main trend of the high-energy
particles, which can demonstrate that our estimation used here is
reasonable. During the storm time, currents calculated via energetic
particle fluxes appear to still underestimate the current. As the particle
flux fit method of calculating currents works so well for earlier in the time
period, this undershoot during storm time might be indicative of additional
energetic particle acceleration (a harder power law) in the parallel
direction. This increased parallel pressure would result in the observed
larger value of *j*_{c}.

Now, we can re-estimate the curvature current using Cluster CIS/CODIF
observations for energy between 25 eV and 40 keV and simulation values for energy
in the range >40 keV–1 MeV. Figure 6 shows the estimated *z* component of
curvature current (the red dotted curve). It is close to result from MRA
(the blue curve).

It should be noted that both events analyzed here are in the Northern Hemisphere. Actually, we have checked that the southward and northward current also can be observed in the southern low and middle latitudes. Thus, such currents should be observable both in northern and southern inner plasma sheet during strong geomagnetic storm events.

According to previous analysis from plasma data (Baker et al., 2002; Korth
et al., 2004; Vallat et al., 2005; Ohtani et al., 2007), most NH regions
should correspond to the plasma sheet region. Using the T96 model
(Tsyganenko, 1995, 1996), we tried to trace Cluster footprints in the
Northern Hemisphere; it is found that the position is ∼55–60^{∘} (not shown here), which just corresponds to the position of
the FACs (Papitashvili et al., 2002; He et al., 2012). Because the MFL shapes
in the plasma sheet were changed considerably, the particle motion in
Earth's magnetic field will be altered correspondingly, which may affect the
particle distribution in the polar and equatorial regions, hence leading to
the variation in the FAC and RC distributions. These effects, however, need
to be evaluated in future work.

When calculating current density using the MRA method, it should be noted that Cluster is not a regular tetrahedron shape around the perigee area but suffers from an elongation, which can produce an unnatural currents. These unnatural currents are included in our analysis and cannot be removed. To evaluate this component, methods from Robert et al. (1998) and Vallat et al. (2005) are used. Figure 7 gives the Cluster tetrahedron parameters for two analyzed events. Then, the current influence of the tetrahedron shape can be estimated as a function of elongation and planarity (Fig. 7c and d). It can be seen that the error caused by tetrahedron is never more than 30 %.

5 Summary

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In this work, the magnetic-field geometry and current density in the inner
plasma sheet during two intense geomagnetic storms were investigated.
It is found that the magnetic field and current density highly
fluctuated in this region. Generally, all three components of current can
be observed during the analyzed interval. However, the northward (or
southward) current is basically the strongest one. Detailed study shows
that the MFLs align in the *x*–*y* plane; thus, the northward (or southward) current
should not be FACs. This property has not been reported before.

The most prominent feature of the northward (or southward) current is the alternative changing of its direction, which is found to vary simultaneously with that of the curvature. To reveal the generation mechanism of the northward (or southward) current, gradient current, curvature current, and gyromotion current are analyzed. The results show that the curvature current has the same variation trend with the northward and southward current. Then, using low-energy particle observations from Cluster CIS/CODIF, combined with simulations based on Kappa distribution, the curvature current is calculated. It shows that the estimated curvature current coincides very well with the current density directly obtained from MCA and MRA. Therefore, the curvature drift of the energetic particle is the most reasonable mechanism of the southward and northward current.

For the two events analyzed in this work, we can observe ULF waves; this is consistent with the typical current density variation period. This turbulence excited during the strong storm can result in the decrease in curvature radius and changing of direction of MFLs, then leading to an increase in the curvature currents and variation in their direction.

Data availability

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Data availability.

Solar wind data (OMNI data set) were obtained from http://omniweb.gsfc.nasa.gov (last access: 22 June 2019); Cluster FGM and CIS/CODIF data were obtained, respectively, from ftp://cdaweb.gsfc.nasa.gov/pub/data/Cluster/ (last access: 22 June 2019) and https://www.cosmos.esa.int/web/csa (last access: 22 June 2019).

Author contributions

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Author contributions.

CS was responsible for the interpretation, YJ performed the simulation, and YYY performed the data analysis and prepared the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We are grateful to the Cluster Science Archive, CDAWeb and OMNI for providing us with data.

Financial support

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Financial support.

This research has been supported by the National Key R&D Program of China (grant no. 2018YFC1503501), the National Natural Science Foundation of China (grant no. 41231066), the National Natural Science Foundation of China (grant no. 41204117), and a research grant from the Institute of Crustal Dynamics, China Earthquake Administration (grant no. ZDJ2018-18).

Review statement

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Review statement.

This paper was edited by Minna Palmroth and reviewed by two anonymous referees.

References

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Short summary

Previous studies show that the ring current (RC) and field-aligned currents (FACs) are the dominant currents in the inner magnetosphere. However, through two large storm events observed by Cluster, we find new strong southward and northward currents in the latitudinal region from 10° N to 30° N. Theoretical analysis indicates that these currents originated from the fluctuation of magnetic field lines during strong geomagnetic storms.

Previous studies show that the ring current (RC) and field-aligned currents (FACs) are the...

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