# On the importance of separators as sites of 3D magnetic reconnection

C.E. Parnell<EMAIL_ADDRESS>School of Mathematics and Statistics,
University of St Andrews, St Andrews, Fife, KY16 9SS, Scotland, UK

###### Abstract

For 3D magnetic reconnection to occur there must exist a volume within which
the electric field component parallel to the magnetic field is non-zero. In
numerical experiments, locations of non-zero parallel electric field indicate
sites of 3D magnetic reconnection. If these experiments contain all types of
topological feature (null points, separatrix surfaces, spines and separators)
then comparing topological features with the reconnection sites reveals that
all the reconnection sites are threaded by separators with the local
maxima/minima of the integrated parallel electric along fieldlines coinciding
with these separators. However, not all separators thread a reconnection site.

Furthermore, there are different types of separator. Cluster separators are
short arising within an individual weak magnetic field region and have
negligible parallel electric field along them so are not associated with much
reconnection. Intercluster separators connect a positive null point lying in
one weak-field region to a negative null point that lies in a different weak-
field region. Intercluster separators often thread enhanced regions of
parallel electric field and are long. Since separators form the boundary
between 4 globally-significant topologically-distinct domains, they are
important sites of reconnection which can results in the global restructuring
of the magnetic field.

By considering the kinematic bifurcation models in which separators form it is
possible to understand the formation of cluster and intercluster separators
and explain their key properties.

##  I Introduction

Magnetic reconnection was first proposed around 65 years ago(Parker, 1957;
Dungey, 1961). Despite many advances, especially in understanding 2D
reconnection, key aspects of this fundamental plasma physics process still
remain poorly understood in 3D.

It is well known that magnetic reconnection in two-dimensions (2D) occurs at
magnetic null points where $\bm{B}=\bm{0}$, for example, [Priest and Forbes,
2000]. In 2D such points are either elliptic (O-type) or hyperbolic (X-type).
About X-type nulls, there lie 4 topologically distinct flux regions separated
by 4 special fieldlines (separatrices) that extend to/from the null point and
from the null’s ‘X’ structure. X-type null point reconnection involves the
transfer of magnetic flux between one pair of oppositely-situated
topologically-distinct flux regions into the other pair. Specifically, a
stagnation-type flow is associated with 2D reconnection at an ‘X’ point, with
the inflow sweeping magnetic field in towards the null where reconnection
causes a “cutting” of a pair of fieldlines creating 4 fieldline segments that
are rejoined such that they create a new pair of fieldlines.

About an O-type null, there is just the one topologically distinct region, and
so flux is either created or destroyed at the null point.

As revealed by Schindler and Hesse(Schindler, Hesse, and Birn, 1988; Hesse and
Schindler, 1988), in three dimensions (3D) magnetic reconnection occurs at
locations where the component of the electric field parallel to the magnetic
field (from now on called the parallel electric field) is non-zero. Despite
the fact that in 3D this is a necessary requirement for reconnection, many
researchers do not consider the parallel electric field when attempting to
identify reconnection sites nor when analysing the magnetic reconnection
taking place in the experiment. Unfortunately, this means that erroneous
conclusions may be reached about the location and nature of the reconnection.
This is because magnetic reconnection requires the correct magnetic geometry
or topology and also the correct plasma flow/conditions to produce an
appropriate electric field.

Figure 1: Plots showing the structure and fieldlines of (top) a 3D improper
null and (bottom) a positive null point and a negative null point connected by
a separator.

As in 2D, reconnection may also occur in the vicinity of 3D null points (see
for example [Priest and Pontin, 2009]).

The fieldlines that extend out from or into a 3D null point form a separatrix
surface and a pair of spines (Figure 1, top)(Fukao, Ugai, and Tsuda, 1975; Lau
and Finn, 1990; Parnell  _et al._ , 1996). If the fieldlines forming the two
spines head into the null, then the lines lying in the separatrix surface will
all be directed outwards from the null. Such a null is called a positive null.
If the separatrix surface fieldlines are directed in towards the null and the
spines outwards, then the null is called a negative null.

There exist special fieldlines that start from a positive null and end at a
negative null and so lie within the separatrix surfaces of two null points
(Figure 1, bottom). Such fieldlines are called separators and it may be argued
that these 3D structures are the topological equivalent in 3D to a 2D null.
This is because both of these structures form the boundaries between four
topologically distinct flux domains Therefore, reconnection at separators may
permit significant transfer of flux between different flux domains and can
also lead to the creation of new topologically distinct flux domains (see for
example [Haynes  _et al._ , 2007; Parnell, Haynes, and Galsgaard, 2008;
Wilmot-Smith and Hornig, 2011; Stevenson and Parnell, 2015]). Hence, just like
2D reconnection at X-points, separator reconnection can produce global
restructuring of the magnetic field.

Note, that the intersection of two separatrix surfaces are generic, stable
separatorsLau and Finn (1990); Brown and Priest (1999). Other separators
formed, for example, by a spine line from a negative null point ending at a
positive null point (e.g., spine lines from two null points coinciding) are
unstable and will be destroyed by any perturbation of the magnetic field. The
same is true for a separator temporarily formed when a spine line from one
null lies in the separatrix surface of another null of the same signBrown and
Priest (1999). These two types of separator are non-generic separators and are
always short-lived. They do not form a boundary between topologically distinct
domains and are unlikely to be sites of reconnection.

Separators have been considered in a number of papers. Lau & FinnLau and Finn
(1990) presented a model for a separator ring in which a pair of separators
(which they called X-lines) connecting an opposite polarity pair of null
points. Their separators are formed by the intersection of the separatrix
(which they called $\sum$-) surfaces from the two nulls. The spine (which they
called $\gamma$) lines provide a bound to two edges of the separatrix
surfaces. The resulting structure looks like the intersection of a pair of
cones, one on its base the other on its point. They also considered a 3D
kinematic reconnection model explaining the transfer of flux across the
separator. Their analysis identified that about the separator the field lines
form an $X$-like structure if the component of magnetic field parallel to the
separator is ignored. Longcope & CowleyLongcope and Cowley (1996) look at the
formation of a current sheet at such a separator ring.

Haynes et al.Haynes  _et al._ (2007) noted during the analysis of a numerical
experiment involving separators that the fastest reconnection across a
separator occurs when the magnetic field lines are twisted about the separator
and thus form an $0$-type structure if the component of magnetic field
parallel to the separator is ignored. The latter implies that the fastest
reconnection occurs where the electric current density parallel to the
separator is largest, which is not unreasonable.

Brown & PriestBrown and Priest (1999) discuss a range of topologies involving
separators, but their analytical models mainly consider point sources on the
base of the domain. They identify two separator bifurcations which they call
the local or global separator bifurcations which are discussed later.

Various models of both the solar atmosphere (e.g., [Platten  _et al._ , 2014])
and also the Earths magnetosphere (e.g. [Dorelli, Bhattacharjee, and Raeder,
2007; Komar  _et al._ , 2013]) have been developed in which multiple
separators have been identified. Since separators are the topological
structures bounding four topologically distinct flux domains they are
important locations about which reconnection can lead to the global
restructuring of the magnetic field.

The main focus of this paper is reconnection at generic separators. Below
(Section II) we present a brief explanation on why parallel electric fields,
however they are generated, are required for 3D magnetic reconnection. Then a
brief review of papers that have compared topological features with sites of
reconnection identified by calculating parallel electric fields is presented
in Section III. In Sections IV and V, the magnetic bifurcations that lead to
the creation of separators are presented. The importance of separators, formed
during the bifurcations, as sites of reconnection are discussed in Section VI.
Then, Section VII discusses how these bifurcations result in the creation of
two distinct types of separator. Finally, the conclusions are presented in
Section VIII.

##  II Three-dimensional Magnetic Reconnection

###  II.1 Magnetic Induction Equation

The equation governing the time evolution of the magnetic field comes from
combining Faraday’s law,

$\partial\bm{B}/\partial t=-\mbox{\bm{$\nabla$}}\times\bm{E}$ (1)

(where $\bm{B}$, $\bm{E}$ and $t$ represent magnetic field, electric field and
time, respectively) and Ohm’s law,

$\bm{E}+\bm{v}\times\bm{B}=\bm{N}$ (2)

(where $\bm{v}$ is the plasma velocity and $\bm{N}$ represents all non-ideal
terms). By eliminating $\bm{E}$ the Induction equation is formed and is given
by

$\displaystyle\frac{\partial\bm{B}}{\partial
t}=\mbox{\bm{$\nabla$}}\times\left(\bm{v}\times\bm{B}\right)-\mbox{\bm{$\nabla$}}\times\bm{N}\;.$
(3)

The first term on the right-hand side of this equation is the advection term
and the second is the diffusion term. The non-ideal terms, $\bm{N}$, involved
in a Generalised Ohm’s law may include not only terms relating to electric
current densities but also electron and ion pressures and inertias, as well as
the Hall term. The ratio of the advection term over the diffusion term forms
the dimensionless magnetic Reynolds number, $R_{m}$. In the perfectly
conducting limit, the magnetic Reynolds number is very large and the diffusion
term can be ignored. Under these conditions the magnetic field and plasma are
“frozen” such that they move together(Alfvén, 1942). That is to say that ions
and electrons may gyrate about fieldlines but they cannot move from one
fieldline to the next. When the magnetic Reynolds number is less than one
($R_{m}<1$), the diffusion term dominates and this may permit the magnetic
field to slip through the plasma.

Reconnection can occur when the magnetic Reynolds number is of order one,
i.e., where the advection and diffusion terms are of the same order. Diffusion
regions where $0<R_{m}\leq 1$ are highly localised regions within space
plasmas where typically $R_{m}>10^{8}$.

Although, magnetic reconnection is a local process it has important global
consequences because it is the fundamental plasma physics process by which the
magnetic field may reconfigure. The re-configuring of the magnetic field
through reconnection only occurs if it results in a decrease in the magnetic
energy of the system. Due to conservation of energy, the lost magnetic energy
is converted into thermal energy, bulk plasma motions and also the
acceleration of particles(Priest and Forbes, 2000). The division of the
magnetic energy into these three other energies will be dependent on the
properties of the plasma, but the details are not not well understood and are
beyond the scope of this paper.

###  II.2 Consequences of Ohm’s Law

In order to better understand why magnetic reconnection in three dimensions
differs so much to two dimensional reconnection we first consider the nature
of the elements of Ohm’s law.

In two spatial dimensions, both the magnetic field and velocity are dependent
on two spatial variables and have just two components, e.g., $\bm{B}(x,y,t)$
and $\bm{v}(x,y,t)$. The electric field $\bm{E}$ is also only a function of
$x$, $y$ and $t$, thus, from Faraday’s law (Eqn 1), it can only have one
component, namely, $\bm{E}=E_{z}(x,y,t)\;\hat{\bm{z}}\,.$ Thus, in 2D, Eqn 2
shows that $\bm{E}\cdot\bm{B}=\bm{N}\cdot\bm{B}\equiv 0.$

On the other hand, in three dimensions, $\bm{B}$, $\bm{v}$ and $\bm{E}$ all
have three components that can depend on all three spatial variables, hence,
in general $\bm{E}\cdot\bm{B}=\bm{N}\cdot\bm{B}\neq 0.$

As explained in a pair of papers by Schindler and Hesse(Schindler, Hesse, and
Birn, 1988; Hesse and Schindler, 1988), a region of non-zero parallel electric
field, where

$E_{\|}=\bm{E}\cdot\bm{B}/\|\bm{B}\|^{2}\,,$

turns out to be both necessary and sufficient for reconnection in 3D. In
particular, the rate of reconnection associated with a domain $\mathcal{D}$
within which there is an isolated diffusion region (region where
$\bm{N}\neq\bm{0}$) is

$\max_{\forall l\in\mathcal{D}}\left\\{\int_{l}\bm{E}_{\|}\;{\rm
d}l\right\\}\neq 0\;,$

where $l$ is a fieldline and ${\rm d}l$ is a small element of the fieldline.

Given the requirement for non-zero $E_{\|}$ an obvious question arises as to
whether, in 3D, reconnection can occur exactly at a null point where
$\bm{B}=\bm{0}$. From the definition of $E_{\|}$ at such a point $E_{\|}=0/0$
which is undefined, however, as $\bm{B}\rightarrow\bm{0}$ it is possible to
find $E_{\|}$ everywhere about the null point. Pontin & PriestPriest and
Pontin (2009) present 3 scenarios for 3D null point reconnection: spine-fan,
torsional fan and torsional spine reconnection: in each scenario, the null
point lies within the diffusion region with reconnection occurring all around
it and possibly as it.

Reconnection at magnetic separators is quite distinct. Specifically, as
determined by Stevenson & ParnellStevenson and Parnell (2015), the region of
non-zero parallel electric field occurs away from the null points at the ends
of the separators. Indeed, if the separator is long in length (e.g., on the
sun separators can be many solar radii in lengthPlatten  _et al._ (2014)) then
reconnection can occur a significant fraction of a solar radii from the null
points at either end. As a result, separator reconnection occurs in a manner
that is consistent with non-null 3D reconnection, or reconnection in a region
with a “guide field” where the guide field is parallel to the separator. The
importance of the separator is that, as a boundary between four topologically
distinct flux domains, transfer between domains as a result of separator
reconnection can result in a global restructuring of the magnetic field.
Furthermore, numerical experiments show that, the maximum rate of reconnection
(determined by integrating along all fieldlines within the domain), is located
along the separator.

##  III Numerical Experiments involving Magnetic Reconnection at Separators

The first numerical resistive MHD experiment to reveal magnetic reconnection
sites threaded by magnetic separators was presented in [Haynes  _et al._ ,
2007] with parallel electric fields used to locate the sites of reconnection.
Their numerical experiment initially had no separators but there were two null
points, both of which lay on the domain’s base. As the system was driven the
two separatrix surfaces from the null points pushed up against each other
causing a current layer to form which lay midway between the null points. At
the same instant that the reconnection started, two separators and a new flux
domain (bounded by the separatrix surfaces and the two new separators) were
created. This process was called a separator bifurcation by [Haynes  _et al._
, 2007] but it would be more accurate to describe it as a separator-separator
bifurcation as two separators are created (or destroyed) following the
bifurcation.

Two interesting findings from their work is that the integral of the parallel
electric field along one of the two newly created separators was always of
opposite sign to that along the other separator and the rates of reconnection
at two new separators was different immediately after the bifurcation. The
first finding indicated that the reconnection on the separators was in
opposite directions. That is along one separator the reconnection acted to
fill the newly formed flux domain that lay between the two separators and
along the other it acted to empty this new domain. Since, the flux in the
newly formed domain increased over time (at least initially) the reconnection
which led to the filling of the domain ran at a faster rate than that which
emptied the domain.

Another couple of papers relevant for understanding the creation of separators
and their importance for reconnection(Maclean, Parnell, and Galsgaard, 2009;
Parnell, Maclean, and Haynes, 2010) studied a numerical resistive MHD
experiment in which the magnetic reconnection, identified via parallel
electric fields, was driven by the emergence of a twisted magnetic flux tube
through the solar photosphere into a horizontal overlying magnetic field that
filled the solar atmosphere. The rise of the flux tube was driven by
convective motions and then a magnetic buoyancy instability kicked in enabling
flux from the tube to rise up into the solar atmosphereArchontis  _et al._
(2005). The domain was initially devoid of magnetic null points in the solar
atmosphere and, thus, had no topological features there either. Topological
features were created as the experiment continued making it an ideal
experiment in which to investigate the creation, behaviour and destruction of
topological features.

Figure 2: (a) A plot of the topological structure and fieldlines of one frame
from the flux emergence experiment(Parnell, Maclean, and Haynes, 2010) showing
the positive and negative null points (red and blue spheres) and the
separators (thick black lines). (b) and (c) show a vertical cut in the $x=0$
plane of (a) located about the separator (black line) going through the
current layer (cyan surface) located between the flux tube (blue lines) and
overlying coronal field (red lines). (b) shows the connectivity of all the
fieldlines that thread the cut with the diamond highlighting the separator
which intersects the plane at the boundary between the 4 different types of
magnetic field connectivity. In (c) the integrated parallel electric field
along the fieldlines used to make (b) is plotted with low to high integrated
parallel electric field indicated by deep purple/blue to red. Fig 1 from
[Parnell, Maclean, and Haynes, 2010]. Reproduced with permission from Parnell,
Maclean and Haynes, Astrophys. J. Lett., 725, L214 (2010). Copyright 2008,
American Astronomical Society.

Maclean et al.Maclean, Parnell, and Galsgaard (2009) identified and analysed
the behaviour of the magnetic null points in this experiment. As the magnetic
field from the flux tube emerged into the solar atmosphere, an initial region
of weak magnetic field formed on one side of the outer edge of where the flux
tube had arisen. Within this weak field region a pair of magnetic null points
was formed via a null-null bifurcation. Approximately 6 minutes later, a
second weak magnetic field region, containing a pair of newly created magnetic
null points, was formed between the flux tube and the overlying field but this
time lying on the opposite side to the other weak field region(Maclean,
Parnell, and Galsgaard, 2009). The two weak-field regions where the new null
point pairs formed were called null clusters.

During the experiment, [Maclean, Parnell, and Galsgaard, 2009] showed that
within each null cluster, pairs of opposite-polarity null points were created,
and different pairs destroyed on a regular basis with most nulls lasting only
8 frames or less (less than 2.5 minutes). There were, however, a few null
points that lasted for longer with two null points lasting for the duration of
the reconnection phase in the experiment. These two null points resided in
different null clusters with one a positive null and the other negative. Each
cluster always had an even number of null points in it: half being positive
nulls and the other half negative nulls.

One snapshot from this experiment is illustrated in Figure 2a. The null points
within this snapshot are denoted by the red and blue spheres that form two
groups, one either side of the emerged flux tube. Each newly formed pair of
null points consists of one positive (red sphere) and one negative (blue
sphere) null. Moreover, [Maclean, Parnell, and Galsgaard, 2009] found that as
the flux tube rose higher into the coronal overlying field a current layer was
formed between these two magnetic regions (cyan surface in Figure 2a). The
current layer (at times split into several pieces) lay at a higher height and
midway between the two null clusters. At no time did the current layer
encroach into the null clusters.

Considering the magnetic field in the same flux emergence experiment as that
described above, [Parnell, Maclean, and Haynes, 2010] determined the time
evolution of its full magnetic topology (separatrix surfaces and spines of
every null point and all the separators within the system). A number of
surprising results were found. The null points within an individual null
cluster are typically connected together by separators (short black lines in
Figure 2a), called cluster separators, such that they form a chain of null
points and separators (like beads (null points) on a string (the connected
separators)). Not all null points in a cluster are part of the chain, but most
were. The longest lasting null in each cluster was typically located at one
end of each chain. Finally, the integral of the parallel electric field was
determined along each of the cluster separators and in their vicinity. All of
these separators had an almost negligible integrated parallel electric field,
indicating that little to no reconnection was associated with these
separators, or null points (diamonds plotted in the bottom left-hand corner of
Figure 3).

At the onset of reconnection between the coronal field and the flux tube (red
and blue lines in Figure 2a & 2b, respectively), two new types of magnetic
field connectivity were created: (i) magnetic field that entered the domain in
the corona then became wound up in the flux tube and left the domain in the
convective zone (green line in Figure 2a & 2b) and (ii) magnetic field that
entered through the convection zone as part of the flux tube and left as part
of the coronal overlying field (yellow lines in Figure 2a & 2b).

The onset of reconnection possibly coincided with the formation of the first
null pair, but so little reconnection occurred the results were inconclusive.
On the creation of the second null-point pair, the reconnection was properly
identified through the creation of green and yellow field lines. Also formed
along with the second null cluster was a single separator connecting the two
long lasting null points that lay far apart in the separate null clusters.
Shortly afterwards a multitude of separators were created connecting the same
two long lasting null pointsParnell, Maclean, and Haynes (2010). The
separators that connected null points from distinct null clusters are called
“intercluster separators” (e.g., long black lines in Figure 2a). After the
first intercluster separator was formed, all the additional intercluster
separators were formed in pairs. Similarly, different pairs of intercluster
separators were also destroyed in pairs.

Figure 3: Plot of integrated parallel electric field along a separator versus
length of the separator for separators found in two snapshots from the flux
emergence experiment (snapshot 109, red diamonds, is the snapshot illustrated
in Figure 2). A unit of integrated parallel electric field equates to 150 V
whilst a unit of length corresponds to 170 km. Fig 4 from [Parnell, Maclean,
and Haynes, 2010]. Reproduced with permission from Parnell, Maclean and
Haynes, Astrophys. J. Lett., 725, L214 (2010). Copyright 2008, American
Astronomical Society.

Parnell et al.(Parnell, Maclean, and Haynes, 2010) found that during the most
intense period of reconnection, many tens of intercluster separators (e.g.,
more than 200 in one snapshot) were created threading the large intense
current layer between the two null clusters and connecting the same few null
points. Practically all of these separators had a large integrated parallel
electric field associated with them indicating they are sites of reconnection.
The intercluster separators were much longer in length than those that linked
nulls within each null cluster and some were fairly long lived. Moreover, when
the parallel electric field was integrated along all fieldlines that threaded
the midplane of the system (which was also the midplane of the current layer),
it was found that there were multiple localised peaks in the integrated
parallel electric field and each of these peaks coincided with an intercluster
separator (Figure 2c).

These numerical MHD studies, especially Parnell et al.(Parnell, Maclean, and
Haynes, 2010), revealed that not all separators seem to be sites of strong
magnetic reconnection. Specifically, those that connect null points inside a
null cluster (e.g. cluster separators) have an integral of parallel electric
field along them that is small. However, intercluster separators that connect
null points from two distinct null clusters can have, and often do have, a
significant integral of parallel electric field (the largest such values in
the domain) and they lie at the boundary between 4 globally-significant
topologically-distinct flux domains indicating that they are important (the
major) sites of reconnection (Figures 2b & 2c).

In this paper, we consider whether the distinction between cluster and
intercluster separators is just semantics or whether these features are
intrinsically different. For the separators seen in Figure 2a the cluster
separators have lengths of $<500km$ and integrated parallel electric fields of
$<0.15$ V. The 3 intercluster separators are much longer with lengths of
approximately $24$, $34$ and $36$ Mm and corresponding integrated parallel
electric fields of $216.0$, $4.5$ and $93.8$ V, respectively (see red diamonds
in Figure 3). From an experiment run by Stevenson and ParnellStevenson and
Parnell, 2015 the diffusion region extended along approximately half of the
separator away from the nulls at the ends of the separator. Assuming this is
the case for all of the separators mentioned in Figure 2a, then mean parallel
electric field values of $<6\times 10^{-4}$ V/km exist on the cluster
separators and values of $\approx 1.8\times 10^{-2}$, $\approx 3\times
10^{-4}$ and $\approx 4\times 10^{-3}$ V/km on the 3 intercluster separators.
Note, however, in reality, the strength of the parallel electric field will
vary considerably along the length of the separator and the diffusion regions
could be either short or longer than half the separator length. So, on two of
the intercluster separators the mean parallel electric fields are factors of
30 and 6.7 times greater than the maximum values found on the cluster
separators. So the differences in their integrals of parallel electric field
are not simply due to the differences in their lengths.

Figure 4: (a) The magnetic skeleton of the dayside magnetosphere with a
northward interplanetary magnetic field at an angle of $\pi/4$. The null
clusters, labelled A-D, contain positive (red) and/or negative (blue) null
points. (b) A null-point connection map indicating the cluster separators
(purple lines inside the shaded spheres) and the intercluster separators (long
purple lines linking nulls from different clusters) that lie in the dayside
magnetosphere. Fig 4 from [Haynes and Parnell, 2010] Reproduced from Haynes
and Parnell, Phys. Plasmas 17, 092903 (2010), with the permission of AIP
Publishing.

Another numerical experiment worth mentioning is one in which the Earth’s
magnetosphere is modelled using the GCCM codeDorelli, Bhattacharjee, and
Raeder (2007). In this experiment, the Earth’s magnetic field interacts with a
northwards interplanetary magnetic field (IMF) angled at $\pi/4$ with respect
to the Earth’s magnetic axis. Figure 4 shows a frame from this magnetosphere
experiment(Haynes and Parnell, 2010).

The initial magnetic field in this experiment starts with a pair of null
points lying on the dayside magnetopause. Specifically a positive null lies in
region A and a negative one in region D in Figure 4. As the experiment
proceeds, additional nulls are created in the weak-field regions about the
original null points forming null clusters. The null clusters A and D have
topological degrees of -1 and +1, respectively, equal to the topological
degree of the null point residing in the region at the start of the
experiment. In addition, another two null clusters are formed (B and C): both
of these regions have zero topological degree as neither of these
clusters/weak-field regions existed initially and the new null points are
created in opposite polarity pairs.

A paper by AlbrightAlbright (1999) explained that new null points are created
due to perturbations in regions of weak magnetic field strength, such as
around pre-existing nulls. It is therefore not surprising that additional null
points are created about the existing nulls in the magnetopauseDorelli,
Bhattacharjee, and Raeder (2007); Haynes and Parnell (2010) as well as
additional nulls forming in two new regions. The additional nulls are always
created in opposite polarity pairs.

In Figure 4, a total of 6 intercluster separators and 6 cluster separators are
found (purple lines). Four of the intercluster separators are single
separators each connecting nulls lying in different null clusters. The other
two separators connect the same two null points that lying in null clusters A
and C.

Null clusters may contain one or more nulls within them. The overall
topological degree of a cluster is determined by summing up the topological
degree of all the nulls within the cluster. Although the number of null points
within a cluster may change over time, the topological degree of each cluster
is preserved unless (i) two null clusters coalesce to form a single cluster
whose topological degree is then equal to the sum of the degrees of the two
original clusters or (ii) a single null cluster divides into two separate
clusters with the sum of the topological degree of the two new clusters equal
to the topological degree of the original cluster.

From a distance, the magnetic field about a null cluster with a topological
degree of +1(-1) will look like a negative(positive) null point. A region with
topological degree of 0 will, from a distance, have a magnetic field that has
no obvious appearance of containing null points, indeed, it may, look like a
region of uniform magnetic field.

Separators can be created by one of two types of bifurcation: namely a null-
null bifurcationLau and Finn (1990) (also known as a local separator
bifurcationBrown and Priest (1999)) or a separator-separator bifurcationHaynes
_et al._ (2007). As the name of the former suggests, two null points are
formed during a null-null bifurcation and a separator may also be formed
connecting the newly formed nulls. The latter bifurcation leads to the
creation of a pair of separators.

In order to investigate cluster and intercluster separators kinematic models
are considered in this paper in order to try and explain the key behaviours
seen in the numerical experiments discussed above. In particular, what are the
mechanisms by which these types of separator can form? Can simple time-
dependent magnetic field models shed any light on why the short cluster
separators are associated with weaker parallel electric field than the long
intercluster separators and thus result in less reconnection?

##  IV Null-null Bifurcations

###  IV.1 Local Field about 3D Nulls

The local magnetic field about a null located at
$\bm{x}_{0}=(x_{0},y_{0},z_{0})$ may be revealed by linearising the magnetic
field about the null(Parnell  _et al._ , 1996) leading to

$\displaystyle\bm{B}=\bm{M}\cdot(\bm{x}-\bm{x}_{0}),$ (4)

where $\bm{M}$ is a real non-singular $3\times 3$ matrix (the Jacobian matrix
of the vector field $\bm{B}$ evaluated at $\bm{x}=\bm{x}_{0}$). This means the
$Tr(\bm{M})=0$ since $\bm{B}$ must satisfy the solenoidal constraint.
Moreover, this implies that the sum of the eigenvalues of $\bm{M}$ is also
zero. If the real part of two of the eigenvalues is positive then the null
point is said to be a positive null. The eigenvectors associated with these
eigenvalues define the separatrix surface local to the null point within which
lie fieldlines that are all directed out from the null (pink lines in Figure
1a). The remaining eigenvalue will be real and negative in sign. Its
eigenvector defines the spines local to the null point: these are a pair of
fieldlines that in this case are directed in towards the null point (blue
lines in Figure 1a). On the other hand, if the signs of the real parts of two
eigenvalues are negative then the null is a negative null point with the
associated eigenvectors defining the separatrix surfaces. In this case, the
fieldlines in the surface are all directed in towards the null point. The
eigenvector associated with the single positive eigenvalue defines the spines
which are a pair of outward-directed fieldlines from the null.

Alternative names have been given to these 3D nulls such as A-type and B-type
for negative and positive nulls, respectively(Lau and Finn, 1990) or the nulls
have been identified according to their topological degree: positive
(negative) nulls have a topological degree of $-1$($1$), somewhat
confusinglyFukao, Ugai, and Tsuda (1975); Haynes and Parnell (2010).

As a brief aside, we note that the only 3D magnetic null points that can exist
within a magnetohydrostatic equilibrium magnetic field are potential nulls.
This was demonstrated in [Parnell  _et al._ , 1997] by taking the curl of the
magnetohydrostatic momentum equation in the absence of gravity and viscosity
which gives

$\displaystyle\bm{\nabla}\times\left(-\bm{\nabla}p+\bm{j}\times\bm{B}\right)$
$\displaystyle=$ $\displaystyle\bm{\nabla}\times(\bm{j}\times\bm{B})$
$\displaystyle=$
$\displaystyle\bm{j}\cdot\bm{\nabla}\bm{B}=\bm{j}\cdot\bm{M}=\bm{0},$

because $\bm{j}$ is a constant since $\bm{B}=\bm{M}(\bm{x}-\bm{x}_{0})$.
However, since at a 3D null point the matrix $\bm{M}$ is non-singular, the
above can only be satisfied if $\bm{j}=\bm{0}$. A magnetohydrodynamic (MHD)
equilibrium can contain non-potential 3D null points as the Lorentz forces of
such null points can be balanced by an appropriate flow(Titov and Hornig,
2000).

The kinematic bifurcation models here are not in equilibrium and so the null
points in these models have non-zero currents at their null points.

###  IV.2 Null-Null Bifurcation Magnetic Field

A number of authors have considered null-null bifurcations (for example, [Lau
and Finn, 1990; Brown and Priest, 1999; Murphy, Parnell, and Haynes, 2015]).
Lau & FinnLau and Finn (1990) claimed that a new null pair would always be
connected by at least one separator. Murphy et al.Murphy, Parnell, and Haynes
(2015) presented 3 examples of null-null bifurcations. Here, a kinematic model
is presented in which a time-dependent magnetic field is prescribed from which
both the electric field and plasma velocity perpendicular to the magnetic
field can be determined. The magnetic field that is prescribed is chosen with
the aim of determining all the possible local magnetic field configurations
that might arise due to the emergence of a new pair of null points: this has
not been done in the other papers. Since the magnetic field is local only
terms up to and including order two are considered.

To determine the local magnetic field configurations the possible transition
states are first identified. From Eqn 4, it can be seen that 3D nulls are
structurally stable provided that $\bf{M}$ remains non-singular(Murphy,
Parnell, and Haynes, 2015). As discussed by [Murphy, Parnell, and Haynes,
2015] when $\bf{M}$ becomes singular the 3D null is replaced by a degenerate
null which is structurally unstable and exists only instantaneously as a
transition between topologically stable states: one state with no null points
and the other with two 3D null points or vice versa.

The most common degenerate nulls have rank$({\bf M})=2$. These nulls have two
non-zero eigenvalues summing to zero and thus either (i) two real eigenvalues
of the same magnitude but opposite in sign or (ii) a purely imaginary complex-
conjugate pair. A third possibility is that the degenerate null has rank$({\bf
M})=0$ so the matrix ${\bf M}$ has 3 zero eigenvalues.

When a new pair of 3D null points is created through a null-null bifurcation
at the origin a degenerate null must first form at that point. The nature of
the degenerate null determines the types of 3D nulls created and thus the
nature of the null-null bifurcation. A null-null bifurcation containing a
degenerate state of rank$({\bf M})=0$ is less likely to occur than either of
the other two cases which have rank$({\bf M})=2$ as the former requires all
the eigenvalues of the linear field to be zero at the same time. It is
included, though, for completeness.

In order to illustrate the different types of null-null bifurcation we
consider the time-dependent magnetic field of the form

$\displaystyle{\bf B}({\bf x})+\delta{\bf B}({\bf
x},t)=\left[\begin{array}[]{c}(a-z)x+by\\\ cx-(a+z)y\\\
z^{2}\end{array}\right]+\left[\begin{array}[]{c}\alpha x-\beta y\\\ \gamma
x-\alpha y\\\ -\delta^{2}\end{array}\right]t,$ (11)

where $a$, $b$, $c$, $\alpha$, $\beta$, $\gamma$ and $\delta$ are constants
and $t$ can be thought of as a relative time with $t=0$ corresponding to the
instant of the null-null bifurcation. This magnetic field is similar to that
considered in [Murphy, Parnell, and Haynes, 2015] who only investigated null-
null bifurcations involving degenerate nulls of rank$({\bf M})=2$.

The form of the magnetic field is chosen as it is the simplest form that can
produce all the possible degenerate null states and all the subsequent 3D null
point pairs that can arise from them (excluding topologically equivalent
states arising from rotations or reflections).

It is noted here that the electric field $\bm{E}$ can be calculated for this
magnetic field using Eqn 1. From these fields the plasma velocity
perpendicular to the magnetic field can then be determined assuming ideal
evolution (i.e., assuming $\bm{N}=\bm{0}$). The electric field and
perpendicular velocity are calculated in Section VI. In this section, the
focus is on the magnetic field so each different null-null bifurcation can be
determined.

Turning to the form of the magnetic field, Eqn 11, it may be noted that the
$B_{z}$-component can only vanish if $t\geq 0$. Thus, when $t<0$ there are no
null points in the magnetic field.

####  IV.2.1 Moment of Bifurcation

At $t=0$ an unstable degenerate null, located at $(0,0,0)$, is momentarily
created with eigenvalues

$\lambda_{1,2}=\pm\sqrt{a^{2}+bc},\;\;\lambda_{3}=0\;.$

The associated eigenvectors are

${\bf e}_{1}\equiv\begin{bmatrix}-b\\\ a-\sqrt{a^{2}+bc}\\\
0\end{bmatrix},\;\;{\bf e}_{2}\equiv\begin{bmatrix}-b\\\ a+\sqrt{a^{2}+bc}\\\
0\end{bmatrix}\;\mbox{and }\;{\bf e}_{3}\equiv\begin{bmatrix}0\\\ 0\\\
1\end{bmatrix}.$

Depending on the relative values of the constants $a$, $b$ and $c$ the
degenerate null may have one zero eigenvalue and (i) two real eigenvalues
($a^{2}+bc>0$), (ii) a complex conjugate pair of eigenvalues ($a^{2}+bc<0$) or
(iii) an additional two zero eigenvalues ($a^{2}+bc=0$).

####  IV.2.2 New Pair of 3D Nulls

Table 1:  Null-null bifurcation examples. Examples |  | 1) $\;\;a^{2}+bc>0$ | 2) $\;\;a^{2}+bc=0$ | 3) $\;\;a^{2}+bc<0$
---|---|---|---|---
$t<0$ |  | No nulls | No nulls | No nulls
$t=0$ | E-values | real, equal, opposite sign | all equal to zero | complex conjugate pair
| Magnetic field | hyperbolic (X-type) | anti-parallel lines | elliptic (O-type)
$0<t\ll 1$ | E-values111Assume $0<\delta\sqrt{t}<1$ and $|h(t)|\ll|a^{2}+bc|$ | $\lambda_{1,2}=-\delta\sqrt{t}\pm\sqrt{a^{2}+bc+h(t)}$ | $\lambda_{1,2}=-\delta\sqrt{t}\pm\sqrt{h(t)}$ | $\lambda_{1,2}=-\delta\sqrt{t}\pm i\sqrt{h(t)-(a^{2}+bc)}$
|  | $\lambda_{3}=2\delta\sqrt{t}$ | $\lambda_{3}=2\delta\sqrt{t}$ | $\lambda_{3}=2\delta\sqrt{t}$
|  | 3 real and distinct | 3 possibilities | 1 real and two complex conjugate
| 3D Nulls |  |  $h(t)<0\;\;\Rightarrow\;\;$ Spiral |
|  | Improper |  $h(t)=0\;\;\Rightarrow\;\;$ Critical Spiral | Spiral
|  |  |  $h(t)>0\;\;\Rightarrow\;\;$ Improper |
| Separator |  |  $h(t)<0$ \- No |
|  | Yes |  $0\leq h(t)<\delta\sqrt{t}$ \- No | No
|  |  |  $0<\delta\sqrt{t}<h(t)$ \- Yes |

When $t\geq 0$ a pair of generic 3D nulls are formed of opposite sign that are
located at

${\bm{x}}_{n}(t)=(x_{n}(t),y_{n}(t),z_{n}(t))=(0,0,\pm\delta\sqrt{t}).$

The speed of separation of this new pair of null points is given by

$\frac{{\mathrm{d}}\bm{x}_{n}}{{\mathrm{d}}t}=\left(0,0,\pm\frac{\delta}{2\sqrt{t}}\right),$

which implies that, at the instant of creation, the new null points move apart
at infinite speed (in the direction of (one of) the eigenvector(s) associated
with the zero eigenvalue of the degenerate null)(Murphy, Parnell, and Haynes,
2015). This means that, when first observed, a new pair of null points will
likely be some distance apart.

The nature of the newly created 3D null points is dependent on the sign of
$a^{2}+bc$ as detailed in Table 1. This is because, after the bifurcation, the
eigenvalues of the newly formed 3D null points will be perturbations of the
eigenvalues $\pm\sqrt{a^{2}+bc}$ at the instant of bifurcation. Specifically,
at the newly formed 3D null point located at $(0,0,z_{n})$ the eigenvalues
will be of the form

$\lambda_{1,2}=-\delta\sqrt{t}\pm\sqrt{a^{2}+bc+h(t)}\;,\quad\mathrm{and}\quad\lambda_{3}=2\delta\sqrt{t}\;,$

where $h(t)=(2a\alpha+b\gamma-c\beta)t+(\alpha^{2}-\beta\gamma)t^{2}\;.$ Since
we are only interested in the time around the creation of the nulls we may
assume that $t$ is sufficiently small such that

$|h(t)|<|a^{2}+bc|\quad{\mathrm{and}}\quad\delta^{2}t<|a^{2}+bc|.$

Moreover, the eigenvectors associated with these eigenvalues will also only be
slight perturbations of the degenerate null’s eigenvectors.

In the sections below we consider in turn the three types of degenerate null
and investigate the nature of all the possible 3D null point pairs that may
arise following a null-null bifurcation via a transition state involving the
given degenerate null.

####  IV.2.3 Improper Nulls ($a^{2}+bc>0$)

If $a^{2}+bc>0$ (Table 1, Example 1), the degenerate null formed at $t=0$ is
an X-type 2D null point, lying in the $z=0$ plane. Once $t>0$ the two stable
generic 3D nulls are formed.

The 3D null at $(0,0,\delta\sqrt{t})$ has a positive eigenvalue $\lambda_{3}$
with eigenvector $\bm{e}_{3}$ Furthermore, since $h(t)$ is small we still find
$a^{2}+bc+h(t)>0$. Moreover, $\delta^{2}t<a^{2}+bc+h(t)$ and so the
eigenvalues $\lambda_{1,2}$ will be of opposite sign. So this 3D null will be
a positive null with a vertical separatrix surface defined by the eigenvectors
$\bm{e}_{1}$ and $\bm{e}_{3}$ and horizontal spines parallel to $\bm{e}_{2}$.
Similarly, we can argue that the 3D null at $(0,0,-\delta\sqrt{t})$ will be a
negative null with a vertical separatrix surface defined by the eigenvectors
$\bm{e}_{2}$ and $\bm{e}_{3}$ and horizontal spines parallel to $\bm{e}_{1}$.

Figure 5: Plots showing the magnetic skeleton (viewed from the side and from
above) following a null-null bifurcation starting from a degenerate null with
real eigenvalues and forming two improper nulls (red and blue spheres)
connected by a separator (green line). The separatrix surfaces of the positive
and negative nulls are made up of red and blue lines with the spines of the
nulls yellow and purple, respectively. Here, $(a,b,c)=(1,2,1)$,
$(\alpha,\beta,\gamma)=(1,1,1)$, $\delta=1$ and $t=0.3$.

Since all the eigenvalues for both new 3D nulls are real they will both be
improper nulls(Parnell  _et al._ , 1996) (Figure 5). The topological degree of
the region remains unchanged from before the bifurcation where it is zero, to
after the bifurcation where the sum of the topological degree of the positive
and negative null is also zero.

Since the separatrix surfaces of the two nulls are both (non-parallel)
vertical planes and the $z$-axis lies in both, it is clear that these surfaces
must intersect. Indeed it can be seen that the positive null’s separatrix
surface is bounded by the spine of the negative null (red lines and thick
purple line, respectively in Figure 5), whereas the negative null’s separatrix
surface is bounded by the spine of the positive null (blue lines and thick
yellow line, respectively in Figure 5). Moreover, there exists a separator,
which lies at the intersection of the two separatrix surfaces (thick green
line, respectively in Figure 5). Indeed, the separator is a fieldline that
extends from the positive null to the negative null and it lies at the
boundary between four flux domains.

####  IV.2.4 Spiral Nulls ($a^{2}+bc<0$)

Figure 6: Plots showing the magnetic skeleton (viewed from the side and from
above) following a null-null bifurcation starting from a degenerate null with
purely imaginary eigenvalues and forming two spiral nulls that are not
connected by a generic separator. Here, $(a,b,c)=(1,-2,1)$,
$(\alpha,\beta,\gamma)=(1,0,1)$, $\delta=1$ and $t=0.3$. Topological features
coloured as in Figure 5.

If $a^{2}+bc<0$ (Table 1, Example 3), then the degenerate null formed at $t=0$
is of O-type (it has two imaginary eigenvalues) that are lying in the $z=0$
plane as before.

In Example 3, immediately following the null-null bifurcation two stable
generic 3D nulls are formed both of spiral-type since $|h(t)|<|a^{2}+bc|$
(Figure 6). Thus, the eigenvalues $\lambda_{1,2}$ of the 3D nulls are a
complex conjugate pair. For both null points, these eigenvalues are associated
with eigenvectors that lie in a $z=$ constant plane. This means that locally
the separatrix surfaces of both nulls are approximately parallel to each
other. So, unless these surfaces curve in towards each other further from the
nulls then no separator connects these null points.

The pairs of spines from each of these nulls are directed vertically upwards
and downwards out of the nulls. The upwardly directed spine from the negative
null and the downwards directed spine from the positive null coincide and thus
they form a non-generic separator that will disappear with a small disturbance
(such as the propogation of a wave in the domain). A more likely (generic)
null-null bifurcation forming two spiral nulls would still be such that the
separatrix surfaces of the new nulls were (at least initially) parallel but
their spine lines would not coincide, as seen in Fig 4 in [Murphy, Parnell,
and Haynes, 2015].

The overall topological degree of the region remains unchanged, maintaining a
value of zero throughout the null-null bifurcation.

In [Murphy, Parnell, and Haynes, 2015] a case with two spiral nulls that were
connected by a separator was discussed. It should be noted that in such a case
it is always possible to choose a sufficiently short time $t$ after the
bifurcation for which the two nulls are not connected by a separator within a
given sized box about the nulls. This means a null-null bifurcation in which a
pair of 3D spiral nulls are created may be connected by a generic separator
if, further from the new null-point pair, the separatrix surfaces of the new
nulls are angled in towards each other so they overlap.

####  IV.2.5 Critical Spirals ($a^{2}+bc=0$)

In Table 1 Example 2, $a^{2}+bc=0$ and at $t=0$ the degenerate null is of
rank($\bm{M}$)=0. It is therefore not a 2D null point. This configuration,
which only exists for an instant, permits a much greater degree of freedom (as
it has three zero eigenvalues) and so the bifurcation can lead to various
scenarios.

In this case the two newly formed stable generic 3D nulls will have
eigenvalues of the form

$\lambda_{1,2}=\mp\delta\sqrt{t}\pm\sqrt{h(t)}\;,\quad\mathrm{and}\quad\lambda_{3}=\pm
2\delta\sqrt{t}\;.$ (12)

Figure 7: Plots showing the magnetic skeleton (viewed from the side and above)
following a null-null bifurcation starting from a degenerate null with 3 zero
eigenvalues. Example 2a (no separator): $t=0.3$, $(a,b,c)=(1,1,-1)$,
$(\alpha,\beta,\gamma)=(-1,2,-1)$ and $\delta=1$. Topological features
coloured as in Figure 5.

Figure 8: As for Figure 7 but here Example 2b (no separator): $t=0.3$,
$(a,b,c)=(1,1,-1)$, $(\alpha,\beta,\gamma)=(-1,-1,1)$ and $\delta=1$.

Figure 9: As for Figure 7 but here Example 2c (no separator): $t=0.3$,
$(a,b,c)=(1,1,-1)$, $(\alpha,\beta,\gamma)=(1,1,1)$ and $\delta=4$.

Figure 10: As for Figure 7 but here Example 2c (separator): $t=0.3$,
$(a,b,c)=(1,1,-1)$, $(\alpha,\beta,\gamma)=(1,1,1)$ and $\delta=1$.

When $h(t)<0$ two spiral nulls (one positive and the other negative) will be
formed (Figure 7). As in Example 2, the two nulls will not be connected by a
separator unless the fan planes curve inwards towards the $z=0$ plane further
from the null points themselves.

In Figure 8, the magnetic field is such that $h(t)=0$ so the nulls have two
equal eigenvalues associated with horizontal separatrix surfaces of the nulls
and thus the nulls are critical spirals which are not joined by a separator
unless their separatrix surfaces curve in towards the $z=0$ plane further away
from the nulls.

When $h(t)>0$ there will be two improper nulls. If $0\leq h(t)<\delta^{2}t$,
then the two nulls will not be joined by a generic separator (Figure 9) unless
the separatrix surfaces bend in towards each other outwith the linear field
domain of each null but if $0\leq\delta^{2}t<h(t)$, the nulls will be
connected by a separator (Figure 10).

As in the previous two cases, the overall topological degree of the domain
remains zero at all times.

###  IV.3 Null-null bifurcation in a magnetic field with existing null points.

All of the examples discussed above focus on the local consequences of a null-
null bifurcation. Here, a kinematic model is presented in which a new pair of
nulls are created in a magnetic field that already has existing null points.
The distance between the emergence site of the new nulls and the location of
the existing null pair does not influence the resulting configuration. So this
model works just as well for a new null pair formed inside an existing null
cluster as it does for a new null pair formed far away from the existing
nulls.

The magnetic behaviour found in this model is basically the same as that seen
in the local-separator bifurcation described in [Brown and Priest, 1999].
Here, though we still call this a null-null bifurcation as we view this as a
possible additional consequence of such a bifurcation.

The time-dependent magnetic field considered is

$\displaystyle{\bf B}({\bf
x})=\frac{B_{0}}{L_{0}z_{0}z_{1}}\left[\begin{array}[]{c}2x\left((z-a)^{2}(z+b)-bz_{0}^{2}t+az_{1}^{2}+cz^{2}\right)\\\
2y\left((z-a)(z+b)^{2}-z(z_{0}^{2}t+z_{1}^{2})-cz^{2}\right)\\\
\left((z-a)^{2}-z_{0}^{2}t\right)\left((z+b)^{2}-z_{1}^{2}\right)\end{array}\right]$
(16)

where $a$, $b$, $c$, $z_{0}$, $z_{1}$, $B_{0}$ and $L_{0}$ are all positive
constants and $t$ is a relative time with $t=0$ corresponding to the instant
of the null-null bifurcation.

As for the local null-null bifurcation model, both the electric field and
plasma velocity component perpendicular to the magnetic field can be
determined for this magnetic field. They are discussed in Section VI where
they are used to investigate reconnection about the separators created
following the bifurcation.

Figure 11: Plots showing the magnetic skeleton (viewed from the side (top) and
above (bottom)) following a null-null bifurcation occurring within a magnetic
field that already has some null points located at $-b\pm z_{1}=-5\pm 0.5$,
with $B_{0}=L_{0}=1$ and $c=3$. The left-hand snapshots are when $t=-1$ before
the null-null bifurcation and the right-hand ones are after the bifurcation
when $t=1$ and the new null pair are located at $a\pm z_{0}\sqrt{t}=6\pm 0.2$.
Topological features are coloured as in Figure 5.

Before the bifurcation ($t<0$) there exists a null pair located at $x=y=0$ and
$z=-b\pm z1$ with the lower null positive and the upper null negative (Figure
11(left)). These nulls are connected by a short separator. After the
bifurcation ($t>0$) a new null pair have formed at $x=y=0$ and $z=a\pm
z0\sqrt{t}$: again the lower null is positive and the upper null negative
(Figure 11(right)). Immediately following the bifurcation two new separators
are formed: one connecting the new pair of nulls and the other connecting the
new positive null point to the existing negative null. This new separator may
be short, if the null pair is created close to the existing nulls, or long, if
the new nulls are formed in a different null cluster to the existing nulls.

##  V Separator-Separator Bifurcation

As seen above, separators may be created along with a new pair of null points.
Separators may also be created in pairs via a separator-separator
bifurcation(Haynes  _et al._ , 2007; Parnell, 2007; Parnell, Haynes, and
Galsgaard, 2008; Parnell, Maclean, and Haynes, 2010; Parnell, Haynes, and
Galsgaard, 2010; Wilmot-Smith and Hornig, 2011). Note, when this bifurcation
was first observed during the analysis of a numerical experiment that involved
reconnection in the solar atmosphere it was called a separator
bifurcation(Haynes  _et al._ , 2007; Parnell, Haynes, and Galsgaard, 2008).
Here, we call it a separator-separator bifurcation as it results in the
creation of a pair of separators (or separator ring(Longcope and Cowley,
1996)), not just a single separator. It is essentially the same as the global-
separator bifurcation described in [Brown and Priest, 1999], however, in their
magnetic field the second separator formed below the plane in which the point
sources lay. In such bifurcations the separatrix surfaces from two pre-
existing, distant null points of opposite-sign approach one another. The
bifurcation occurs when these separatrix surfaces intersect and a pair of
separators are formed connecting the two existing null points.

A kinematic model is presented below in order to illustrate how such a
bifurcation can occur. Unlike the null-null bifurcation model, the model
presented here only illustrates one separator-separator bifurcation rather
than all possible such bifurcations. The time-dependent magnetic field
considered is

$\displaystyle\bm{B}({\bf x},t)$ $\displaystyle=$
$\displaystyle\frac{B_{0}}{L_{0}}\left[\begin{array}[]{c}a(t)^{2}-x^{2}\\\
(2a(t)-1)xy/a(t)\\\ (xz-a(t)b)/a(t)\end{array}\right]$ (24)
$\displaystyle+\frac{B_{0}}{L_{0}}\left[\begin{array}[]{c}0\\\ 0\\\
c(x^{2}+d^{2})y^{2}+c(y^{2}+d^{2})(x^{2}-a(t)^{2})^{2}\end{array}\right]\;$

where $B_{0}>0$, $L_{0}>0$, $b$, $c$, $d$ are all constants and $a(t)\neq 0$
is a function of time. The second part of the magnetic field has only a
$z$-component and this term is positive provided $c>0$. It represents a
vertical ambient field which becomes increasingly dominant the further $|x|$
is from $|a(t)|$ and the larger $|y|$ is.

As for the two null-null bifurcation models, the electric field and ideal
plasma velocity perpendicular are calculated in the next section (Section VI).

The magnetic field $\bm{B}(\bm{r},t)$ has two null points located at

$\bm{x}_{n1}=(a(t),0,b)\;\;\mathrm{and}\;\;{\bf x}_{n2}=(-a(t),0,-b).$

The eigenvalues of the linearised field about these null points are

$\displaystyle\lambda_{n1}$ $\displaystyle\in$
$\displaystyle\left\\{-2a(t),\;2a(t)-1,\;1\right\\}\;\;\mathrm{and}$
$\displaystyle\lambda_{n2}$ $\displaystyle\in$
$\displaystyle\left\\{2a(t),\;1-2a(t),\;-1\right\\},$

and their associated eigenvectors are

$(1,0,0)^{T},\;\;\;(0,1,0)^{T},\;\;\;(0,0,1)^{T}$

respectively, in both cases.

For the separator-separator bifurcation to occur we require the following: (i)
the null located at $\bm{x}_{n1}$ to be a negative null, (ii) the null at
$\bm{x}_{n2}$ to be positive and (iii) the spines of both of these nulls to be
parallel to the $z$-axis (i.e. to be aligned with the ambient field). This may
be achieved by letting the scalar function $a(t)$ take a value between $0$ and
$0.5$. Note, $a(t)\neq 0$ and if $a(t)=0.5$ then the null points are
degenerate, so not generic 3D null points. We therefore define

$a(t)=0.5-t,\;\;\;0<t<0.5\;.$

Figure 12: Plots showing (top row) the skeleton of the magnetic field (left)
before ($a(t)=0.4$), (middle) at the bifurcation ($a(t)=0.3874$) and (right)
after ($a(t)=0.35$) a separator-separator bifurcation. The null points (large
red/blue dots) and their associated spines (thick orange/purple lines) and
fieldlines lying in their separatrix surfaces (thin red/blue lines) are
plotted. The separators, when they exist, are plotted in dark green. The plots
in the bottom row show general fieldlines at the same 3 times as those in the
top row. Fieldlines (thick pink lines) lying above the positive separatrix
surface, (thick cyan lines) lying below the negative separatrix surface,
(thick yellow lines) the ambient vertical field and, in the final frame,
(thick violet lines) lying above the positive separatrix surface and below the
negative separatrix surface.

The magnetic field plotted in Figure 12 has been constructed for the constants
$B_{0}=L_{0}=1$, $b=0.5$, $c=0.4$ and $d=5$. If $0<t<0.113$ the separatrix
surfaces of the two null points do not intersect and so there are no
separators connecting the two null points (Figure 12, left). Figure 12(middle)
shows the skeleton at the approximate time of the bifurcation ($t\approx
0.113$). At the instant of the bifurcation a single degenerate separator is
located where the two separatrix surfaces first touch. As $t$ continues to
increase ($0.113<t<0.5$) the two separatrix surfaces intersect each other
forming two separators which move further apart in time. In the volume formed
by the overlapping separatrix surfaces bounded by the separator ring a new
flux domain is created which contains the thick violet coloured fieldlines
seen in Figure 12(right).

Having developed simple magnetic field models for the two types of
bifurcation, the separators created in them are considered below to see if
they are likely to be sites at which reconnection occurs.

##  VI Separator Reconnection

Below we consider whether we can say anything about the reconnection
associated with the separators formed in the three bifurcations. First,
however, we recall that for 3D reconnection to occur

$\displaystyle\int_{l}\frac{\bm{E}\cdot\bm{B}}{|\bm{B}|}\;{\rm d}l\neq 0\;,$

where, as before, $\bm{E}$ and $\bm{B}$ are the electric and magnetic fields,
respectively, and $l$ is fieldline length(Schindler, Hesse, and Birn, 1988;
Hesse and Schindler, 1988). The rate of reconnection can be defined everywhere
except when $\bm{B}\neq\bm{0}$.

Furthermore, the reconnection rate is defined as the maximum of all the
integrals of the parallel electric field along the fieldlines. In the
numerical experiments in which separators have been identified and the
integrated parallel electric fields have been determined along fieldlines the
local maxima all coincide with the separators (e.g. [Parnell, Maclean, and
Haynes, 2010]).

From our time-dependent bifurcation magnetic fields the electric fields may be
calculated from Faraday’s law (Eqn 1). This equation allows the electric field
to be determined to within the gradient of a scalar field (equivalent to a
constant of integration which arises when ‘un-curling’ the electric field).
Thus, in each model the reconnection rate along each of the separators can be
determined.

In addition, from Ohm’s law (Eqn 2) the plasma velocity perpendicular to the
magnetic field $\bm{v}_{\perp}$ can be determined

$\bm{v}_{\perp}=\frac{\bm{E}\times\bm{B}}{|\bm{B}|^{2}}\,,$ (25)

where such a velocity cannot be defined when $\bm{B}=\bm{0}$. Here, it is
assumed that the non-ideal term $\bm{N}=\bm{0}$ such that the magnetic field
is frozen into the plasma thus the perpendicular plasma velocity
$\bm{v}_{\perp}$ and magnetic field line velocities $\bm{w}_{\perp}$ are the
same.

Inside a diffusion region the non-ideal term $\bm{N}\neq\bm{0}$ and the
magnetic field may slip through the plasma. The perpendicular plasma
$\bm{v}_{\perp}$ velocity is defined as in Eqn 25 and field line
$\bm{w}_{\perp}$ velocity is

$\bm{w}_{\perp}=\bm{v}_{\perp}+\frac{\bm{N}\times\bm{B}}{|\bm{B}|^{2}}\,.$

In this paper, we do not derive the field line velocities for our models as
this would require assumptions to be made about the non-ideal term $\bm{N}$
which has purposefully been left undefined such that the results of these
models can be applied in a wide range of plasma regimes.

In the following sections the bifurcation models are revisited to consider
whether there are signs that reconnection is likely to occur at any of the
separators.

###  VI.1 Null-null bifurcation separators (No pre-existing nulls)

The electric field for the first null-null bifurcation model is determined
from Eqn 11 using Faraday’s law

$\displaystyle\bm{E}$ $\displaystyle=$
$\displaystyle-\left(\frac{1}{2}\delta^{2}y+\frac{\partial\phi}{\partial
x}\right)\;\hat{\bm{e}}_{x}+\left(\frac{1}{2}\delta^{2}x-\frac{\partial\phi}{\partial
y}\right)\;\hat{\bm{e}}_{y}$ (26)
$\displaystyle\qquad+\left(\frac{1}{2}(\gamma x^{2}+\beta y^{2})-\alpha
xy-\frac{\partial\phi}{\partial z}\right)\;\hat{\bm{e}}_{z}\,,$

where $\phi=\phi(x,y,z)$ is a scalar field and arises since
$\mbox{\bm{$\nabla$}}\times\mbox{\bm{$\nabla$}}\phi\equiv\bm{0}\,.$

Revisiting the null-null bifurcation examples described in Sec IV, we note
that in Example 1 where $a^{2}+bc>0$ a generic separator is always created at
the same time as the two new improper 3D null points. In this example the
separator lies between the two null points $(0,0,-\delta\sqrt{t})$ and
$(0,0,\delta\sqrt{t})$), so it lies on the $z$-axis. The reconnection rate
along this separator is therefore

$\displaystyle\int_{1sep}\frac{\bm{E}\cdot\bm{B}}{|\bm{B}|}\;{\rm d}l$
$\displaystyle=$
$\displaystyle\int_{-\delta\sqrt{t}}^{\delta\sqrt{t}}E_{z}(0,0,z)\,{\rm
d}z=\int_{(0,0,-\delta\sqrt{t})}^{(0,0,\delta\sqrt{t})}{\rm d}\phi$ (27)
$\displaystyle=$
$\displaystyle\phi(0,0,\delta\sqrt{t})-\phi(0,0,-\delta\sqrt{t})\,,$

The form of $\phi$ is unknown, but is typically taken to be small and often
just neglected. Moreover, the distance $2\delta\sqrt{t}$ is short, since, at
least initially, $t$ is small, so the change in $\phi$ is likely to be small
too implying that little reconnection is associated with the separators in
this bifurcation model.

It is possible that a magnetic field that does contain a non-zero component of
the electric field parallel to the separator can be found, but for a null-null
bifurcation to occur no such electric field component is required.

Figure 13: Cut in the $z=0$-plane for $-0.4\leq x,y\leq 0.4$ at the instant of
bifurcation, $t=0$, for the null-null bifurcation Example 1. The black arrows
indicate the direction of the perpendicular plasma velocity in the plane and
the purple and orange lines are where the separatrix surfaces cut this plane.

From Eqn 25, the plasma velocity perpendicular to the magnetic field may be
determined. In the example considered above, assuming $\phi$ is constant, then

$\displaystyle\bm{v}_{\perp}$ $\displaystyle=$ $\displaystyle\frac{(px-
qr)\;\hat{\bm{e}}_{x}+(py-
qs)\;\hat{\bm{e}}_{y}-\delta^{2}(rx+sy)\;\hat{\bm{e}}_{z}}{2|\bm{B}|},$
$\displaystyle p$ $\displaystyle=$
$\displaystyle(z^{2}-\delta^{2}t)\delta^{2}$ $\displaystyle q$
$\displaystyle=$ $\displaystyle\beta y^{2}+\gamma x^{2}-2\alpha xy$
$\displaystyle r$ $\displaystyle=$ $\displaystyle cx-(a+z)y+t(-\alpha y+\gamma
x)$ $\displaystyle s$ $\displaystyle=$ $\displaystyle(a-z)x+by+t(\alpha
x-\beta y)$

Figure 13 shows a plot of the direction of $\bm{v}_{\perp}$ in the $z=0$ plane
which is weak in strength close to the origin. However, the flow parallel to
the $z=0$ plane is stagnation-like and would act to close the X formed by the
intersection of the separatrix surfaces with this plane. Such a flow could
potentially encourage the formation of of an electric field that would be
important for reconnection.

So, here, despite a stagnation type flow about the bifurcation point, the
parallel electric field along the separator is to depend only on the change in
$\phi$, which is likely to be small for a short separator. The limited
reconnection associated with these short separators connecting a new pair of
null points agrees with the rates of reconnection along such separators
determined from analysis of numerical experiments(Parnell, Maclean, and
Haynes, 2010) (see Figure 3).

###  VI.2 Null-null bifurcation separators (Pre-existing nulls)

The electric field associated with the separators formed from a null-null
bifurcation producing a new null pair in a magnetic field with existing null
points can be determined from the model magnetic field presented in Eqn 16.
The components of the electric field are

$\displaystyle E_{x}$ $\displaystyle=$
$\displaystyle\frac{z_{0}y}{2z_{1}}\left(4(bz-
z_{1}^{2})-(b^{2}-z_{1}^{2})\right)-\frac{\partial\phi}{\partial x}$
$\displaystyle E_{y}$ $\displaystyle=$
$\displaystyle\frac{z_{0}x}{2z_{1}}\left(-2(2b^{2}+z^{2})-(b^{2}-z_{1}^{2})\right)-\frac{\partial\phi}{\partial
y}$ $\displaystyle E_{z}$ $\displaystyle=$
$\displaystyle\frac{2z_{0}xy}{z_{1}}\left(b-z\right)-\frac{\partial\phi}{\partial
z}\,.$

All the separators in this magnetic field lie along the $z$ axis which means
that the electric field component relevant for determining the reconnection
occurring along them is $E_{z}(0,0,z)=\partial\phi/\partial z\,$ on the
separators. Thus, the reconnection rate on the longer of the two new
separators is $\phi(0,0,a-z_{0})-\phi(0,0,-b+z_{1})$ and on the shorter one is
$\phi(0,0,a+z_{0})-\phi(0,0,a-z_{0})$. The difference between the values of
$\phi$ are likely to be larger the further apart the nulls linked by the
separator are. This implies that if the separator linking the nulls from
different null pairs is a cluster separator it is likely to be associated with
much less reconnection than if the separator is an intercluster separator.
Either way though, $\phi$ is usually considered to be small so any
reconnection rate along separators created from null-null bifurcations are
likely to have a slow reconnection rate.

Figure 14: Cut in the $z=0$-plane for $-0.4\leq x,y\leq 0.4$ at the instant of
bifurcation, $t=0$, for the null-null bifurcation in a magnetic field with
existing null points. The black arrows indicate the direction of the
perpendicular plasma velocity in the plane and the purple and orange lines are
where the separatrix surfaces cut this plane.

The plasma velocity perpendicular to the magnetic field can be determined from
Eqn 25, assuming that $\phi$ is constant. The expressions for each component
of this velocity are rather large so they are not presented here. However,
from Figure 14 it can be seen that $\bm{v}_{\perp}$ parallel to the $z=0$
plane is a stagnation type flow that runs parallel to the separatrix surfaces,
as opposed to across them. Such a flow is not conducive to reconnection.

###  VI.3 Separator-separator bifurcation separators

The time-dependent magnetic field (Eqn 24) used to illustrate a separator-
separator bifurcation has a corresponding electric field of the form

$\displaystyle\bm{E}(\bm{x},t)$ $\displaystyle=$
$\displaystyle-\left(\frac{\partial\phi}{\partial
x}-\frac{xyz}{a^{2}}\frac{{\rm d}a}{{\rm d}t}\right)\;\hat{\bm{e}}_{x}$ (28)
$\displaystyle-\left(\frac{\partial\phi}{\partial
y}+4ca(t)(y^{2}+d^{2})\left(\frac{x^{3}}{3}-a^{2}x\right)\frac{{\rm d}a}{{\rm
d}t}\right)\;\hat{\bm{e}}_{y}$
$\displaystyle-\left(\frac{\partial\phi}{\partial z}-2ay\frac{{\rm d}a}{{\rm
d}t}\right)\;\hat{\bm{e}}_{z}\,.$

This electric field varies both spatially and temporally via $a(t)$. It also
appears to be non-zero on the separator, even if the “constant of integration”
$\phi=0$.

First, recall that $0<a(t)<0.5$ and observe that the separators curve a small
distance away from, and then back to, the $y=0$ plane as they extend between
the 3D null points located at $(-a(t),0,-b)$ and $(a(t),0,b)$. The components
of the electric field that contribute most to the integral of the parallel
electric field along the separators are the $x$ and $z$ components. On the
separators $x$ and $z$ both have the same sign as the separators crosses the
$z=0$ plane on the $x=0$ axis (due to the symmetry in the model). Ignoring
$\phi$, which is usually considered to be small, we see that

$\mathrm{sign}(E_{z})=\mathrm{sign}(E_{x})=\mathrm{sign}(y)\;,$

since $a(t)>0$ and ${\rm d}a/{\rm d}t<0\,.$

Since one separator lies in the $y\geq 0$ half of the domain and the other
lies in the region where $y\leq 0$, $E_{\|}$ will be directed in opposite
directions along the separators since the magnetic field along both separators
runs from the positive null to the negative null.

In particular, if we imagine the null points reside in distant clusters (e.g.,
we assume $b\gg a(t)$) then the $E_{z}$ component would become the dominant
factor in determining $E_{\|}$ along the separators. Specifically,

$\int_{lsep,\;y\geq
0}\bm{E}_{\|}(l)dl>0\quad{\mathrm{and}}\quad\int_{lsep,\;y\leq
0}\bm{E}_{\|}(l)dl<0\;.$

The sign of $E_{\|}$ on a fieldline determines the direction of the
reconnection. Thus, with the assumption that $\partial\phi/\partial z=0$, the
kinematic model used to investigate separator-separator bifurcations recovers
the key finding from the numerical (resistive MHD) experiments(Haynes  _et
al._ , 2007; Parnell, Maclean, and Haynes, 2010), that the reconnection on the
two new intercluster separators runs in opposite directions. Such electric
fields would imply that reconnection at one separator transports flux into the
new flux domain and the reconnection at the other transports it out. However,
the model produces reconnection rates of the same magnitude along each
separator, due to our over-simplified symmetric magnetic field. This is not
what is found in the numerical experiments.

So, in contrast to a null-null bifurcation, a separator-separator bifurcation
maybe associated with a significant amount of reconnection. This is apparent
from the change of connectivity of the fieldlines forming the separatrix
surfaces over time (Figure 12). Before the bifurcation all the red fieldlines
lying in the separatrix surface of the positive null exit through the top of
the domain (Figure 12a) but after the bifurcation a selection exit through the
bottom and vice-versa for the blue fieldlines lying in the separatrix surface
from the negative null (Figure 12c).

The bottom row of plots in Fig 12 show general fieldlines colour coded
according to the topological domain in which they lie. Fieldlines that are
contained within (lie above) the positive (red) separatrix surface are
coloured pink: they enter and leave through the top of the box. Fieldlines
that are contained within (lie below) the negative (blue) separatrix surface
are coloured cyan: they enter and leave through the bottom of the box.
Mustard-coloured fieldlines which represent the ambient field lie below the
positive separatrix surface and above the negative separatrix surface (i.e.
outside both surfaces) and enter through the bottom of the box and exit
through the top. In the bottom left and middle plots of Figure 12 there are
just these 3 types of fieldlines (excluding the special fieldlines that go
through the null points). However, in the bottom right-hand plot of Figure 12
there are 4 types of fieldlines. A new type of fieldline (purple) lies above
the positive (red) separatrix surface and below the negative (blue) separatrix
surface: they enter through the top of the box and exit through the bottom.
These have been formed by the reconnection of pink and cyan fieldlines which
occurs at a separator. This reconnection forms a new domain containing these
purple fieldlines and it also produces more mustard-coloured fieldlines.

Figure 15: Cuts in the $z=0$-plane at $a(t)=0.4$, $a(t)=0.3874$ and
$a(t)=0.35$ indicating the connectivity of fieldlines that thread this plane,
where the colours used are the same as those used in Figure 12. The lower
right-hand panel shows a localised cut out about the origin from the plot on
the lower left (at $a(t)=0.35$ including the regions around the points where
the separators thread this plane. Along with fieldline connectivity this plot
shows the expected direction of the reconnection (arrows) at each separator.

Figure 15 shows a sketch of a cut through the magnetic field in the $z=0$
plane illustrating the flow of plasma across the separators. Note, all the
fieldlines in the domain cross the $z=0$ plane: that is there are no
fieldlines actually lying in the $z=0$ plane. From Figure 15 it can be seen
that the reconnection about the separator in $y>0$ is such that magnetic field
within the positive (red line) separatrix surface (shaded pink) reconnects
with field from inside the negative (blue line) separatrix surface (shaded
cyan). This results in a transfer of magnetic flux into both the newly formed
domain that lies inside both the separatrix surfaces (shaded purple) and also
the region that lies outside both separatrix surfaces (shaded mustard). About
the other separator the transfer of flux is reversed. Naturally, for the new
flux domain (purple) to form the rate of reconnection along the separator
lying in the $y<0$ half of the domain must be greater than that along the
other separator.

Figure 16: Cut in the $z=0$-plane for $-0.4\leq x,y0.4$ at the instant of
bifurcation, $a(t)=0.3874$, where the black arrows indicate the direction
resulting from the components of perpendicular plasma velocity $v_{x}$ and
$v_{y}$ plotted over the top of the image showing the connectivity of
fieldlines that thread this plane, coloured as in 15

The plasma velocity $\bm{v}_{\perp}$ is also calculated using Eqn 25. The
equations are not presented here as they do not simplify easily, but since
$\bm{E}$ and $\bm{B}$ have been given the components of $\bm{v}_{\perp}$ are
straightforward to calculate. Figure 16 shows a plot with arrows indicating
the direction of $v_{x}$ and $v_{y}$ in the $z=0$ plane overlying the
connectivity of fieldlines threading this plane at the instant of the
bifurcation $a(t)=0.3874$. The arrows close to the origin form a stagnation
type flow directed inwards to the origin along the $x$ axis and outwards along
the $y$ axis. The bifurcation occurs at the origin in the $z=0$ plane. Here,
though the separatrix surfaces that thread this plane form an ‘X’ shape and,
close to origin there is evidence of a flow perpendicular to these surfaces,
pushing them together. This is the type of flow that acts to enhance both the
electric field and electric current density around where the separatrix
surfaces touch. Thus it is a flow conducive to reconnection.

##  VII Formation Mechanisms of Null Clusters, Cluster and Intercluster
Separators

As we have seen in the numerical experiments, null clusters may have a zero or
non-zero topological degree. Null clusters with zero topological degree have
equal numbers of positive and negative null points. Since null points must be
created and destroyed in pairs, the null points inside these null clusters are
most likely created in a weak-field region via the null-null bifurcation.
Whereas the null clusters with non-zero topological degree must have
additional positive or negative nulls. For instance, null clusters A and D
from Figure 4 both have one negative null each, but A has two positive nulls
and D has none leading to topological degrees for these null clusters of -1
and 1, respectively. In order for null clusters like this to form a null-null
bifurcation must occur, creating a positive and negative null point. Then, due
to plasma flows, the two null points may be moved apart such that the original
null cluster with two null points is split into two separate null clusters.
Note, that null clusters can also merge together creating a new cluster with a
topological degree equal to the sum of topological degrees of the two original
null clusters.

Inside null clusters, null points are typically found either to be connected
together by separators like beads on a string (as seen in Figure 2), or they
may not be connected to any other null points. So far, from numerical
experiments, any pair of opposite polarity null points inside a null cluster
have only been found to be connected by a single separator. This suggests that
cluster separators are formed via null-null bifurcations in which single
separators are formed. Such separators all form at the appearance of a new
null pair with a separator (i) connecting the newly created null points and/or
(ii) connecting a newly emerged null point to a pre-existing null point. This
leads to the “beads on a string” nature of null points within a null cluster.

Due to the absence of any two null points within the same null cluster
observed to be connected by multiple separators it is unlikely that many/any
cluster separators are formed by a separator-separator bifurcation. This is
likely to be due to the lack of significant flows within a null cluster which
push the separatrix surfaces from two null points together causing a
separator-separator bifurcation.

Intercluster separators that connect null points from different null clusters
can arise through several mechanisms. The main mechanism that creates single
intercluster separators is a null-null bifurcation occurring in a (or creating
a new) null cluster leading to the creation of a separator connecting one of
the new nulls to a null point in a different pre-existing null cluster. This
mechanism created the first intercluster separator seen in the flux emergence
experimentParnell, Maclean, and Haynes (2010) and also the mechanism that
created the single interclusters connecting nulls from the null cluster A&C,
B&C and B&D in the magnetosphere experimentHaynes and Parnell (2010).

The single intercluster separator between clusters A&D was already evident in
the initial frame of the experiment. In theory it could have arisen in the
same way as the other single intercluster separators or as a short separator
connecting a new null pair that is then stretched splitting up a null cluster
to form the two null clusters A and D.

Finally, the double separator connecting null clusters A&C in the
magnetosphere experiment is created by the separator-separator bifurcation as
are most of the intercluster separators found in the flux emergence
experiment.

##  VIII Conclusions

A necessary and sufficient condition for 3D magnetic reconnection is that
there exists a region in which the electric field parallel to the magnetic
fieldlines is non-negligible(Schindler, Hesse, and Birn, 1988; Hesse and
Schindler, 1988). In a few numerical resistive MHD experiments the 3D magnetic
reconnection sites have been located by determining the integral of the
parallel electric field along fieldlines, as well as the topological features
(e.g. null points, separatrix surfaces, spines and separators). These
experiments reveal that the fieldlines with the largest integral of parallel
electric field are separators. However, there are some separators, usually
those short in length, that have a small integral of parallel electric field
suggesting they are not associated with much reconnection.

Further analysis of the numerical experiments(Parnell, Maclean, and Haynes,
2010; Haynes and Parnell, 2010) revealed what appeared to be two different
types of separators: cluster separators that link positive and negative null
points that reside within a single weak field region (a null cluster); and
intercluster separators that connect null points of opposite sign from two
distinct weak field regions. The latter are much longer than the former and
are typically associated with significant amounts of magnetic reconnection. By
considering kinematic models of the bifurcations that lead to the creation of
these two types of separator it has been possible to explain their key
characteristics. The main properties of these two types of separators and
their reconnection are presented in Table 2.

Cluster separators are created within a single weak magnetic field region via
a null-null bifurcation that also forms a pair of opposite-polarity null
points. Single separators are formed during this type of bifurcation that may
connect the two newly formed null points and can also connect a newly formed
null point to a pre-existing null point within the same null cluster. All of
these separators are short in length as they are contained with a single null
cluster. The separators connecting the newly created null points start out
infinitesimally short and grow in length as the new null points move apart.
Although at the instant that the new nulls are created reconnection occurs,
once they are formed little reconnection seems to occur about these types of
separator.

Table 2:  Properties of Separators. Property | Cluster Separators | Intercluster Separators
---|---|---
Location | Inside a single | Connect different
| null cluster | null clusters
Typically created | Singularly | Singularly or in pairs
Bifurcation | Null-null | Null-null or Sep-sep
Separator significance | Locally | Globally
Length | Short | Long
$\bm{E}_{\|}$ | Weak | Often strong
$\int\bm{E}_{\|}\mathrm{d}s$ | Small | Large
Impact of reconnection | Little | Globally significant

The kinematic models presented here show that the perpendicular plasma
velocity is a stagnation flow about the separator which may, in some
circumstances, act to close the X formed by the intersection of the separatrix
surfaces. Such a flow could lead to the creation of conditions suitable for
reconnection about the separator.

The integral of the parallel electric field is also determined from the
kinematic model and indicates that little reconnection would be expected at
separators created via a null-null bifurcation.

Intercluster separators are different in that they connect null points that
lie in different (distinct and typically distant) null clusters. These
separators are either formed in pairs via separator-separator bifurcations or
singularly via a null-null bifurcation where one of the newly created null
points forms a separator connecting it to a pre-existing null. The singularly
formed intercluster separators are not associated with much reconnection
immediately after their formation, as discussed above. The separators created
in pairs arise from a bifurcation which has been shown in the kinematic model
presented here to be associated with a perpendicular plasma velocity that
would drive the separatrix surfaces together enabling the build up of an
electric field. From the kinematic model the integrated parallel electric
field along the separators shows they would be associated with a significant
amount of reconnection, at least shortly after they have been formed. After
the bifurcation, it is seen that the reconnection continues on these
separators and leads to the creation of a new, topologically distinct flux
domain. This new flux domain grows in size as the flux in it increases.

The separator-separator kinematic model considered here reveals that the
electric current parallel to the separators is directed up one separator and
down the other, thus forming a closed ring. A consequence of this is that the
resulting magnetic reconnection on one separator acts to fill the newly
created flux domain, whilst on the other separator the direction is reversed
so it acts to empty the new flux domain in agreement with what has been found
through analysis of numerical experiments(Haynes  _et al._ , 2007; Parnell,
Haynes, and Galsgaard, 2008; Parnell, Maclean, and Haynes, 2010; Parnell,
Haynes, and Galsgaard, 2010; Wilmot-Smith and Hornig, 2011). If the rates of
reconnection are the same along these two separators then, at any instant, the
same amount flux will be transported both into and out of the newly formed
domain bounded by the two separators and so this domain will not grow in size.
Hence, the reconnection rates must be different along the two separators in
order to increase or decrease the flux in the domain enclosed by the pair of
separators. Due to symmetry the kinematic model presented here produces the
same reconnection rates along the two separators.

These results highlight the importance of, at least some, intercluster
separators as sites of significant reconnection. Given that there have so far
been few studies of this process, there is clearly a need to carry out more
self-consistent numerical experiments to enhance our understanding of it.
Specifically, it would be interesting to see whether the singularly formed
intercluster separators are the ones seen in numerical experiments that are
associated with less reconnection than the ones created in pairs. This may not
be the case as, after formation, the reconnection occurring at these
separators may be driven by global plasma flows. Indeed, since the separators
created in pairs have different rates of reconnection along them it may be
just this difference that leads to the intercluster separators with
considerably different reconnection rates.

###### Acknowledgements.

The author would like to thank both referees, as well as Profs T. Neukirch &
E.R. Priest for useful suggestions and questioning of the results which have
led to important improvements of this paper.

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