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

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**Regular paper**
15 Mar 2018

**Regular paper** | 15 Mar 2018

Tripolar electric field Structure in guide field magnetic reconnection

^{1}School of Electronic Information, Wuhan University, 430072 Wuhan, China^{2}Institute of Space Science and Technology, Nanchang University, 330031 Nanchang, China

^{1}School of Electronic Information, Wuhan University, 430072 Wuhan, China^{2}Institute of Space Science and Technology, Nanchang University, 330031 Nanchang, China

**Correspondence**: Meng Zhou (mengzhou@ncu.edu.cn)

**Correspondence**: Meng Zhou (mengzhou@ncu.edu.cn)

Abstract

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It has been shown that the guide field substantially
modifies the structure of the reconnection layer. For instance, the Hall
magnetic and electric fields are distorted in guide field reconnection
compared to reconnection without guide fields (i.e., anti-parallel
reconnection). In this paper, we performed 2.5-D electromagnetic full
particle simulation to study the electric field structures in magnetic
reconnection under different initial guide fields (*B*_{g}).
Once the amplitude of a guide field exceeds 0.3 times the asymptotic magnetic
field *B*_{0}, the traditional bipolar Hall electric field is clearly
replaced by a tripolar electric field, which consists of a newly emerged
electric field and the bipolar Hall electric field. The newly emerged
electric field is a convective electric field about one ion inertial length
away from the neutral sheet. It arises from the disappearance of the Hall
electric field due to the substantial modification of the magnetic field and
electric current by the imposed guide field. The peak magnitude of this new
electric field increases linearly with the increment of guide field strength.
Possible applications of these results to space observations are also
discussed.

**Keywords. **Space plasma physics (magnetic reconnection)

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Fu, S., Huang, S., Zhou, M., Ni, B., and Deng, X.: Tripolar electric field Structure in guide field magnetic reconnection, Ann. Geophys., 36, 373–379, https://doi.org/10.5194/angeo-36-373-2018, 2018.

1 Introduction

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Magnetic reconnection is a ubiquitous fundamental phenomenon that transfers magnetic energy into plasma kinetic and thermal energy (Yamada et al., 2010). It is initiated in a small-scale region, named the diffusion region. The diffusion region consists of two-scale structures: the ion diffusion region (ions are demagnetized while electrons are still magnetized on the ion inertial scale) and the electron diffusion region (both ions and electrons are demagnetized on the electron inertial scale) (e.g., Sonnerup, 1979; Birn et al., 2001). The relative motion of ions and electrons results in the Hall currents, the quadrupolar out-of-plane magnetic fields and bipolar electric field pointing toward the center of the current sheet around the ion diffusion region (e.g., Priest and Forbes, 2000). These signatures have frequently been used to locate/identify the ion diffusion regions in the Earth's magnetosphere (e.g., Deng and Matsumoto, 2001; Huang et al., 2010, 2012; Paschmann et al., 2013).

A guide field, which is perpendicular to the reconnection plane (the guide
field usually points in the GSM Y direction in the magnetotail), usually
exists in reconnection in laboratory, space, and astrophysical plasmas. Many magnetic reconnection events with
guide fields in the Earth's magnetotail have been reported. Øieroset et
al. (2001) identified an ion diffusion region with a guide field of about
20 % of the field strength in the lobe region. Nakamura et
al. (2008) showed that a large guide field (up to 80 % of the lobe
field) exists within the magnetic reconnection region. Comparing the
simulation results with satellite observation, Eastwood et al. (2010) found
that the structure of the diffusion region is altered by a moderate guide
field (∼20 % of the lobe field). The Hall electromagnetic
fields are asymmetric across the current sheet. Zhou et al. (2014) identified
a super-Alfvénic electron jet extended to 30 s ion inertial
length away from the *X* line in a reconnection region with a weak guide
field. This jet was deflected from the neutral sheet owing to the Lorentz
force *j*_{L}×*B*_{g}; here, *j*_{L} is
the electric current in the outflow direction and *B*_{g} is
the guide field. In the Earth's magnetopause, component reconnections (with a
non-negligible guide field) are suggested as important processes during
solar-wind–magnetosphere interactions because the interplanetary magnetic
fields (IMFs) have a variety of shear angles relative to the Earth's dipole
field (Fuselier et al., 2011).

Although great efforts have been made to understand the guide field
reconnection, and we have already known that the Hall magnetic and electric
field are distorted in the presence of a guide field, the detailed structure
of electric fields in guide field reconnection remains an open question. In
this paper, we performed a series 2.5-D particle-in-cell simulation to study
the electric field structures within magnetic reconnection in the presence of
different guide fields. We focus on the electric fields near the *X* point.
The simulation results show that the conventional bipolar Hall electric field
is replaced by a tripolar electric field when a sufficient large guide field
is imposed (*B*_{g}≥0.3*B*_{0}), while the conventional
bipolar Hall electric field exists only in the small guide field regime. We
further analyzed the generalized Ohm law to understand the origin of the new
electric field.

2 Simulation results

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Our 2.5-D fully electromagnetic full particle code has been used to study the electric field structures inside the magnetic island, density cavity, and sub-structures of the separatrix near and/or within the magnetic reconnection region (Zhou et al., 2011, 2012a, b, 2014; Huang et al., 2014, 2015). In this model, ions and electrons are regarded as individual particles. The Maxwell equation set is solved to advance the electromagnetic fields, while the Lorentz equation is solved to advance particles.

The initial magnetic field is given by two Harris current sheets:
${B}_{x}={B}_{\mathrm{0}}\mathrm{tanh}\left(\right(z-{L}_{z}/\mathrm{4})/{L}_{\mathrm{0}})-{B}_{\mathrm{0}}\mathrm{tanh}\left(\right(z-\mathrm{3}{L}_{z}/\mathrm{4})/{L}_{\mathrm{0}})-{B}_{\mathrm{0}}$,
where *B*_{0} is the asymptotic magnetic field amplitude, *L*_{z} is
the box size in the *z* direction, and *L*_{0} is the initial half-width of
the current sheets which is set to 0.5*d*_{i}
(${d}_{\mathrm{i}}=c/{\mathit{\omega}}_{\mathrm{pi}}$ is the initial ion inertial length
based on the density *n*_{0} in the center of the current sheets).
Reconnection occurs in the *x*–*z* plane (+*x* points right and +*z* points
upward; +*y* points inward in the out-of-plane direction). We apply periodic
boundary conditions in the *x* and *z* directions for both particles and
fields. A small system size flux perturbation is added initially to boost the
system entering the nonlinear stage. Details of this model have been
discussed in Zhou et al. (2012b).

The simulation box is 50*c*∕*ω*_{pi} in the *x* direction and
50*c*∕*ω*_{pi} in the *z* direction, where *c* is the speed of
light and *ω*_{pi} is the ion plasma frequency in the central
current sheet. The whole simulation box consists of 2000×2000 grids;
that is, each grid is equivalent to 0.025*c*∕*ω*_{pi}. The
magnetic field is normalized by *B*_{0}, the electric field is
normalized by *B*_{0}*V*_{A}, the velocity is normalized by
*V*_{A}, and the electric current density is normalized by
*q**n*_{0}*V*_{A}, where *q* is the unit charge and *V*_{A} is
the Alfvén speed based on *B*_{0} and *n*_{0}. The mass ratio used
is ${m}_{\mathrm{i}}/{m}_{\mathrm{e}}=\mathrm{100}$. Uniform guide fields with different
values are added initially in the −*y* direction.

For convenience, we selected one of the two current sheets to illustrate our
results. Electric fields *E*_{z} in the presence of different initial guide
fields at *t**ω*_{ci}=32, when the reconnection rates reach their
peak, are displayed in Fig. 1. In the case of a zero guide field, the Hall
electric field normal to the current sheet exhibits a typical symmetric
bipolar structure (Fig. 1a). When a small guide field
(*B*_{g}=0.1*B*_{0}, Fig. 1b) is added in the system, the
structure of a Hall electric field becomes asymmetric with respect to the
current sheet though the bipolar structure is retained. With the increase in
a guide field, thin current layers in the vicinity of the *X* points deflect
toward the separatrix and magnetic islands are generated within the outflow,
and sub-structures of the electric field also appear around the separatrix
(Fig. 1c–f), i.e., a positive electric field *E*_{z} above the upper left
separatrix and negative *E*_{z} beneath the lower right separatrix. There is
only a negative *E*_{z} at and near the upper separatrix and positive *E*_{z}
around the bottom separatrix in the case without a guide field (Fig. 1a).
This difference indicates that the guide field brings some new features to
electric fields besides causing the asymmetry.

To further illustrate how the guide fields affect the structure of *E*_{z},
six profiles of *E*_{z} along *δ**x*=3*d*_{i} in simulations of
different guide fields are shown in Fig. 2. Here *δ**x* is the horizontal
distance to the *X* point. Vertical dashed lines in Fig. 1 mark the locations
of *δ**x*. The red line shows the *E*_{z} in the case without a guide
field, which exhibits a centrosymmetric bipolar structure with positive
*E*_{z} below $z=\mathrm{12.5}c/{\mathit{\omega}}_{\mathrm{pi}}$ and negative *E*_{z} above
$z=\mathrm{12.5}c/{\mathit{\omega}}_{\mathrm{pi}}$. Here $z=\mathrm{12.5}c/{\mathit{\omega}}_{\mathrm{pi}}$ is the
location of a neutral sheet where *B*_{x}=0. This is the typical Hall
electric field as it is convergent toward the neutral sheet because electrons
move faster than the ions (Sonnerup, 1979; Priest and Forbes, 2000). With the
increase in the guide field, *E*_{z} is no longer centrosymmetric with
respect to $z=\mathrm{12.5}c/{\mathit{\omega}}_{\mathrm{pi}}$. Positive *E*_{z} occupies a
larger area than the negative *E*_{z}, and it no longer resides beneath the
neutral sheet but extends to above the neutral sheet. This is consistent with
Eastwood et al. (2010) in that the Hall electric field is asymmetric and
shunted away from the neutral sheet in guide field reconnection. We noticed
that there is a small negative excursion of *E*_{z} on the left-hand side of
the enlarged positive *E*_{z} region in the cases with *B*_{g}≥0.3*B*_{0}. Thus, *E*_{z} exhibits a tripolar structure instead of
a bipolar structure when the initial guide field is larger than 0.3*B*_{0}. The negative electric field ranges from 10.4 to 11.4 in the
case of *B*_{g}=0.3*B*_{0}. It occupies a larger region
in the case of *B*_{g}=0.5*B*_{0}. The size of the
negative electric field seems to increase with the increment of the guide
field. We also noticed that the amplitude of this negative *E*_{z} increases
with the increment of the guide field, which is further discussed later.

To reveal the origin of this negative electric field, we examined the generalized Ohm law:

$$\mathit{E}=-{\mathit{V}}_{\mathrm{i}}\times \mathit{B}+{\displaystyle \frac{\mathit{J}\times \mathit{B}}{ne}}-{\displaystyle \frac{\mathrm{\nabla}\cdot \mathit{P}}{ne}}-{\displaystyle \frac{{m}_{\mathrm{e}}}{e}}{\displaystyle \frac{d{\mathit{V}}_{\mathrm{e}}}{\mathrm{d}t}}+\mathit{\mu}\mathit{J}.$$

The terms on the right-hand side (RHS) are the convective term, Hall term,
divergence of the electron pressure tensor, electron inertial term, and
resistivity term, respectively. The resistivity term is not evaluated in our
simulations. In Fig. 3, we depict the relative contribution of each term on
the RHS of the above equation to *E*_{z}. Upper panel shows the case with a
zero guide field, while the lower panel shows the case with guide field
*B*_{g}=0.5*B*_{0}. For the zero guide field case,
peaks of *E*_{z} (at approximately *z*=11.5, 12.4, 12.6, and
13.5*c*∕*ω*_{pi}) are mainly balanced by the Hall terms (red) and
the divergences of the electron pressure tensor (green), with the Hall terms
dominating. The convective term (purple) is smaller than these two terms
around the peak of *E*_{z}. For the guide field case, *E*_{z} is constituted
by different terms in different regions. *E*_{z} (black) in regions II and
III (marked by the gray shaded area in Fig. 3b) is primarily balanced by the
combination of the Hall term, convective term, and divergence of the electron
pressure tensor, while in region I, the negative *E*_{z} (black) is mainly
balanced by the convective term (purple). This implies that this new electric
field is a convective electric field instead of a Hall electric field.

One may ask how the convective electric field emerges in the presence of the
guide field. From Fig. 3 one can see that, although the convective term is
asymmetric across the neutral sheet in the presence of a guide field, the
amplitude does not change too much between the two cases. Nevertheless, the
Hall term is substantially different between the two cases. The inflection
point (where the Hall term reverses sign) of the Hall term is exactly the
same as the inflection point of the convective term in non-guide field
reconnection. However, in guide field reconnection, the Hall term is
distorted with respect to the neutral sheet. It has been shown that the Hall
term is the main factor leading to the distortion of the electric field
*E*_{z} (Lai et al., 2015). The region with a positive Hall term shrinks
compared to the case without a guide field; hence, a region with only a
convective term emerges.

In addition, we divide the Hall term into two subitems, i.e., $(\mathit{J}\times \mathit{B}{)}_{z}={J}_{x}{B}_{y}-{J}_{y}{B}_{x}$, as is shown in Fig. 4. The electric
current density *J*_{x}, *J*_{y} and the magnetic field *B*_{x}, *B*_{y} are
also shown in Fig. 4. One can see that the profiles of *B*_{x} are more or
less similar in both cases, while the other three parameters vary much
between the two cases. For the non-guide field case (Fig. 4e), *J*_{y} has a
bifurcation structure with the minimum current density at the neutral sheet.
For the guide field case, *J*_{y} has three peaks across the current sheet,
with one peak located at *B*_{x}=0. *J*_{x} reaches its peak around *B*_{x}=0
without a guide field, while it moves toward the edge of the current sheet in
the guide field case. This is consistent with the result of Zhou et
al. (2014) that the outflow electron jet is deflected away from the neutral
sheet as a result of the in-plane Lorentz force. The profile of *B*_{y} also
varies significantly because of the superposition of the guide field.
Therefore, the Hall term also varies much correspondingly. It is mainly
provided by *J*_{y}*B*_{x} (green) around the peak value, while *J*_{x}*B*_{y}
(red) is negligible compared to *J*_{y}*B*_{x} in the zero guide field case.
This is because *J*_{x} and *B*_{y} are smaller than *J*_{y} and *B*_{x}
around the peak of the Hall term, the location of which is away from the
neutral sheet. In the presence of the guide field (Fig. 4j), the magnitude of
*J*_{x}*B*_{y} increases significantly compared to the case without a guide
field. This is because the imposed uniform guide field *B*_{g}
and the deflected *J*_{x} increase the magnitude of *J*_{x}*B*_{y}. The two
peaks of *J*_{x}*B*_{y}, one of which is around *z*=13.5 and the other around
*z*=11.6, counterbalance the peaks of *J*_{y}*B*_{x}, which reduces the Hall
electric field. The latter peak corresponds to the location of the newly
emerged electric field.

Figure 5 displays the relation between the guide field *B*_{g}
and the peak magnitude of the negative electric field *E*_{z} at *δ**x*=3*d*_{i}. We see that there is roughly a linear relation between
guide field and electric field. The correlation coefficient is about −0.9.
We fitted the results with a linear equation by the least square method:
${E}_{z}=-\mathrm{0.92}{B}_{\mathrm{g}}{V}_{\mathrm{A}}-\mathrm{0.062}{B}_{\mathrm{0}}{V}_{\mathrm{A}}$. This
relation can be understood as follows. We have shown that *E*_{z} is mainly
contributed by the convective term in the generalized Ohm law, i.e.,
${E}_{z}\sim {V}_{y}{B}_{x}-{V}_{x}{B}_{y}$. Figure 6a and b show that the negative
*E*_{z} is mainly balanced by *V*_{y}*B*_{x}, the magnitude of which increases
with the increment of guide field strength. The variation of *V*_{y}*B*_{x} as
a function of *B*_{g} is mainly caused by the variation of
*B*_{x} as shown in Fig. 6c and d. We see that *V*_{y} does not vary
obviously with the change in the guide field *B*_{g}, while
*B*_{x} at the location of the negative *E*_{z} correlates well with
*B*_{g}; i.e., the larger the $\left|{B}_{\mathrm{g}}\right|$, the
larger the $\left|{B}_{x}\right|$. This is because the guide field decreases the width of
reconnecting current sheet.

3 Discussion and summary

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Eastwood et al. (2010) have pointed out that the moderate guide field causes
considerable asymmetry in the Hall fields. Our simulations show that when
there is a moderate guide field, the normal electric field will not only
become asymmetric, but also evolve to a tripolar structure; i.e., a new
negative electric field emerges. This new electric field is not a Hall
electric field, but a convective electric field. It is negative and below the
neutral sheet with a negative guide field. We can easily deduce that the new
electric field appears above the neutral sheet with a positive value once the
guide field is positive. It becomes significant because the Hall electric
field in that location is small due to the cancellation of *J*_{x}*B*_{y} and
*J*_{y}*B*_{x} as a result of the enhanced $\left|{B}_{y}\right|$ and $\left|{J}_{x}\right|$ due to the
imposed guide field. The peak magnitude of the new electric field increases
linearly with the increment of guide field strength.

Our finding may have applications in space observations. For instance, it may be used to infer the strength of a guide field, which is usually difficult to pinpoint solely from magnetic field measurements. The strength of a guide field is important because it determines the degree of electron magnetization as well as the electron energization efficiency in the vicinity of the diffusion region (Pritchett, 2006; Scudder and Daughton, 2008). With the measurement of this new convective electric field, we may be able to deduce the guide field strength based on the linear relation derived in this study. Hence this could be a supplemental way to infer the guide field strength.

The bipolar normal electric field structure is widely used as evidence of an
ion diffusion region in space observations. When a spacecraft crossed the ion
diffusion along the normal direction in anti-parallel reconnection, it would
observe a bipolar *E*_{N} field (*N* is the current sheet normal; in our
simulation it is *z*). However, according to this study, the spacecraft would
record a tripolar *E*_{N} when crossing the ion diffusion region in the
presence of a moderate guide field. Therefore, one must be cautious not to
eliminate the crossing as a possible ion diffusion region based on the fact
that a tripolar but not a bipolar *E*_{N} field is observed.

Recently Malakit et al. (2013) found a new electric field in asymmetric magnetic reconnection. This electric field is named the “Larmor electric field” because it is associated with the finite ion Larmor radius effect and hence is distinct from the Hall electric field. This electric field is also balanced by the convective term in the generalized Ohm law, which is similar to our simulation. This electric field structure in asymmetric reconnection may be complicated by a guide field, and this issue will be studied in future.

Data availability

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Data availability.

The simulation data will be preserved in a long-term storage system and will be made available upon request to the corresponding author.

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 supported by the National Science Foundation of China (NSFC)
under grants 41704162, 41674161, 41522405, 41774154, and 41331070. The
computations were performed on TianHe-1A at the National Supercomputing
Center (NSCC) in Tianjin. The simulation data can be acquired by contacting
the correspondence author Meng Zhou (mengzhou@ncu.edu.cn).

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

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Short summary

It has been shown that guide fields substantially modify the structure of reconnection layers. In this paper, we studied the electric field structures in magnetic reconnection under different initial guide fields (Bg). Once the amplitude of a guide field exceeds 0.3 times the asymptotic magnetic field B0, the traditional bipolar Hall electric field is clearly replaced by a tripolar electric field, which consists of a newly emerged electric field and the bipolar Hall electric field.

It has been shown that guide fields substantially modify the structure of reconnection layers....

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