# Helicity Condensation as the Origin of Coronal and Solar Wind Structure

S. K. Antiochos  NASA Goddard Space Flight Center, Greenbelt, MD, 20771
<EMAIL_ADDRESS>
###### Abstract

Three of the most important and most puzzling features of the Sun’s atmosphere
are the smoothness of the closed field corona, the accumulation of magnetic
shear at photospheric polarity inversion lines (PIL), and the complexity of
the slow wind. We propose that a single process, helicity condensation, is the
physical mechanism giving rise to all three features. A simplified model is
presented for how helicity is injected and transported in the closed corona by
magnetic reconnection. With this model we demonstrate that helicity must
condense onto PILs and coronal hole boundaries, and estimate the rate of
helicity accumulation at PILs and the loss to the wind. Our results can
account for many of the observed properties of the closed corona and wind.

Sun: magnetic field — Sun: corona

##  1 Introduction

A classic, but puzzling feature of the Sun’s high-temperature ($>1$MK)
atmosphere is its apparent lack of complexity. High-resolution XUV and X-ray
images of the closed-field corona, such as those from the Transition Region
and Coronal Explorer (TRACE) mission, invariably show a smooth collection of
loops (e.g. Schrijver et al., 1999). If the underlying photospheric flux
distribution is highly structured with several polarity regions, then the
topology of the loops in the corona will appear complex in XUV images, but
this is only due to seeing through multiple flux systems. The surprising
result is that the loops in any one flux system, such as a bipolar active
region, are generally not observed to be twisted or tangled. Since the coronal
field is line-tied to the photosphere, which is undergoing continuous chaotic
motions due to the convective flows, it would seem that geometrical complexity
should eventually appear in the coronal field. Instead, the magnetic field
appears to remain laminar and not far from a potential state over most of the
corona. Extrapolations of the field do indicate the presence of coronal
currents (Leka et al., 1996; Tian et al., 2005), but these are large-scale
volumetric currents that produce only a global shear or twist rather than
field line tangling (Schrijver, 2007).

The observation that loops are untangled, at least on present observable
scales, is especially surprising given that the corona is being heated
continuously. The standard theory for the heating is that the energy is due to
stressing of the coronal field by the random motions of the photospheric
footpoints. This is the basic idea of Parker’s nanoflare model and similar
theories (Parker, 1972, 1983, 1988; van Ballegooijen, 1986; Mikic et al.,
1989; Berger, 1991; Rappazzo et al., 2008). The photospheric motions are
postulated to tangle and braid the field lines, producing small scale current
sheets, which then release their energy via reconnection. A great deal of work
has been done applying this nanoflare scenario to coronal observations with
considerable success (Klimchuk, 2006).

The problem with such reconnection-heating models is that any helicity
injected into the corona as a result of the motions is expected to survive,
because reconnection in a high Lundquist-number system like the corona
conserves magnetic helicity (Taylor, 1974; Berger, 1984). Consequently, even
if it is injected on scales below present-day resolution, $<1$ arcsecond, the
helicity should build up and appear as twisting or tangling of the large-scale
coronal field. Note also that even if the degree of tangling required for the
heating is small – for example, Parker estimates a misalignment angle between
the reconnecting stressed field and the initial potential state of only
$20^{\circ}$ or so (Parker, 1983) – any net helicity injected by the stressing
should continue to accumulate and eventually produce large-scale observable
effects. We conclude, therefore, that both the basic observations of
photospheric motions and the reconnection theories for coronal heating imply
that the coronal field should have a complex geometry, in direct disagreement
with observations.

There are, at least, two seemingly likely explanations for this disagreement.
The first is that photospheric motions produce equal and opposite helicity
everywhere, so that no net helicity is injected into the corona. Nanoflare
reconnection then simply cancels out the positive and negative helicities.
This explanation, however, has both observational and theoretical
difficulties. Numerous observations imply that photospheric motions, including
flux emergence, do inject a net helicity into each hemisphere. For example,
observations of prominence structure (Martin et al., 1992; Rust, 1994; Zirker
et al., 1997; Pevtsov et al., 2003) and of active region vector fields
(Seehafer, 1990; Pevtsov et al., 1995) indicate a strongly preferred sign for
the helicity injected into each solar hemisphere, possibly related to the
differential rotation (DeVore, 2000). Furthermore, theory and numerical
simulations (Linton et al., 2001) have shown that, for magnetic flux tubes
with parallel axial fields, reconnection occurs only if the tubes have the
same sign of helicity, the so-called co-helicity case as discussed by Yamada
et al. (1990). Flux tubes with opposite helicity only bounce when they collide
(Linton et al., 2001); therefore, reconnection cannot cancel out positive and
negative injected helicity in interacting coronal loops. This point will be
clarified in Figure 2 below.

The other possible explanation for the lack of helicity buildup is that the
heating is due not to reconnection, but to true diffusion, in which case
helicity is not conserved. This hypothesis was proposed by Schrijver (2007) to
explain the TRACE images. He argued that continual reconnection induced by the
rapidly varying field of the magnetic carpet (Harvey, 1985; Schrijver et al.,
1997) causes the chromosphere and transition region to act like a high-
resistivity layer. Coronal loop field lines can slip along this layer and,
thereby, lose their tangles. This explanation, however, also has theoretical
and observational difficulties. Reconnection is not physically equivalent to
diffusion, no matter how frequent the reconnection. Diffusion does destroy
helicity and will relax a magnetic field back down to its minimum energy
potential state, but reconnection can relax the system down only to some state
compatible with total helicity conservation, such as a linear force-free state
(Taylor, 1974, 1986). In fact, for the line-tied corona, we have argued that
helicity imposes very stringent constraints on the possible end state of a
system undergoing reconnection relaxation (Antiochos et al., 2002).

Furthermore, observations imply that whenever helicity is, indeed, present in
the corona, it does not show evidence for diffusive decay. The largest
concentration of coronal helicity is in filaments and prominences, or more
precisely, in the strongly sheared field that defines a filament channel (e.g.
Tandberg-Hanssen, 1995; Mackay et al., 2010). Although there is still debate
over the exact topology of the filament channel magnetic field; in particular,
whether it is a sheared arcade (Antiochos et al,, 1994) or a twisted flux
rope, the models agree that for a physically realistic 3D topology all the
filament flux must be connected to the photosphere. Consequently, if the
chromosphere or transition region really did contain a high-resistivity layer,
the filament channel shear would simply disappear by field-line slippage. Such
slippage is never observed; if anything, filament shear seems to increase
continuously until it is ejected from the corona with a filament eruption/CME.

From the discussion above, we conclude that a net helicity is injected into
each coronal hemisphere by the photosphere, and that reconnection preserves
this helicity. But, in that case, where does the helicity go? In a sense, the
answer is obvious: The helicity injected into the closed-field corona must end
up as the magnetic shear in filament channels. These are the only locations in
the corona where the magnetic field is strongly non-potential and, hence, has
a strong helicity concentration.

Although this answer is intuitively appealing, it seems extremely unlikely. It
naturally raises the long-standing questions: What exactly are filament
channels and how do they form? Along with laminar coronal loops, filament
channels are also classic but puzzling features of the Sun’s atmosphere. These
structures consist of low-lying magnetic flux centered about photospheric
polarity inversion lines (PIL), in which the chromospheric and coronal
magnetic field lines run almost parallel to the inversion line rather than
perpendicular, as expected for a potential field (Rust, 1967; Leroy et al.,
1983; Martin, 1998). Direct measurements of the filament vector field both in
the photosphere (Kuckein et al., 2012) and corona (Casini et al., 2003) show
that the component parallel to the PIL is dominant. The channels have narrow
widths, of order 10 Mm, but their lengths can be greater than a solar diameter
for PILs that encircle the Sun. Filament channels are very common, invariably
appearing about any long-lived PIL, both in active regions and quiet Sun. It
should be emphasized that the channels are much more common than observable
filaments and prominences, which require the presence of substantial amounts
of cold plasma as well as the magnetic shear.

Two general mechanisms have been proposed for filament channel formation. One
mechanism is flux emergence, specifically the emergence of a sub-photospheric
twisted flux rope. Most simulations of flux rope emergence find that the
resulting structure in the corona is a sheared arcade localized near the PIL
(e.g. Mancester, 2001; Fan, 2001; Magara & Longcope, 2003; Archontis, 2004;
Leake et al., 2010; Fang et al., 2012). The basic process is straightforward;
the twist component of the sub-photospheric flux rope emerges to become the
overlying quasi-potential arcade in the corona, while the axial sub-surface
component emerges to become the shear field of the filament channel. It is
interesting to note that, in general, the resulting filament channel in the
corona is not a twisted flux rope, but a sheared arcade, because the concave-
up portion of the flux-rope field lines stays trapped below the surface even
in 3D (Fang et al., 2012).

Although flux emergence can yield a sheared arcade, there are major
theoretical and observational difficulties with this process as the general
mechanism for filament channel formation. First, the simulations have yet to
show that flux emergence agrees quantitatively with the amount of flux and
degree of shear measured in observed filament channels. In fact, Leake et al.
(2010) argue that only a small amount of axial flux emerges, at least in 2.5D
simulations, far too small to account for the magnetic free energy in observed
filament channels. Similar conclusions have been reached from recent 3D
simulations, as well (Fang et al., 2012). A much greater problem for the model
is that filament channels are frequently observed to form in regions where
there is no apparent flux emergence, such as at PILs between decaying regions
and high latitude PILs (e.g. Mackay et al., 2010). Moreover, the emergence of
a simple bipolar active region rarely produces a filament channel at its PIL.
The channel usually forms only well after the end of the flux emergence, when
the active region has decayed and dispersed to interact with surrounding flux
regions. Consequently, flux emergence cannot be the only mechanism for
filament channel formation.

The second, and perhaps, the most popular mechanism that has been proposed for
filament channel formation is flux cancellation (Martin, 1998). The basic
picture is that large-scale shear due to differential rotation or flux
emergence concentrates at PILs as opposite-polarity photospheric flux
converges and cancels there. Note that this mechanism inherently requires
reconnection at the photospheric PIL in order to form low-lying loops that can
sink and disappear and concave-up loops that can rise into the corona (van
Ballegooijen & Martens, 1989). A fundamental difficulty with such
reconnection, however, is that it produces a twisted flux rope in the corona
just like flare reconnection produces the highly twisted flux rope of a CME.
On the other hand, high resolution observations of filaments both from the
ground (Lin et al., 2005) and space (Vourlidas et al., 2010) show a field
geometry consisting of long, parallel strands, with no evidence of twist or
tangling. In fact, empirical models for filaments derived solely from
observations, have a laminar field geometry exactly like that of the TRACE
loops, except that the field lines are stretched out and primarily horizontal
rather than arched (Martin ref). It has been suggested that the large twist
component resulting from reconnection may diffuse away (van Ballegooijen,
2004), but as argued above, any diffusion would also decrease the shear
component, contrary to observations. Note also that the twist produced by
flare reconnection, which is physically identical to flux cancellation
reconnection, is never observed to diffuse away, but is measured to persist
out to 1 AU (Kumar & Rust, 1996; Qiu et al., 2007).

In addition to the lack of observed twist, the prevalence of filament channels
poses severe difficulties for the flux cancellation model and, indeed, for any
model. As stated above, filament channels are ubiquitous, appearing over all
types of PILs ranging from the most complex and strongest active regions to
very quiet high-latitude regions. In fact, it is not uncommon to observe a
filament channel that continues unbroken over a PIL that passes from an active
region into neighboring quiet region with the cold material, itself,
transitioning smoothly from a typical low-lying active region filament to a
high-lying quiet sun filament (e.g. Su & van Ballegooijen, 2012). Given these
observations, it seems improbable that filament channels are due to some
phenomenon in the plasma-dominated photosphere, because the dynamics there are
insensitive to the magnetic field structure and, in particular, to whether a
PIL is present or not. This is especially true in the weak-field regions where
quiescent prominences typically form (Klimchuk, 1987). It seems much more
likely that filament channel formation is due to some generic process
occurring in the magnetically-dominated corona and upper chromosphere.

We propose that the origin of filament channels is the reconnection-driven
evolution of helicity injected into the closed-field corona. This hypothesis
seems counterintuitive, because coronal-loop helicity is injected on small
scales more-or-less uniformly throughout the corona, whereas filament channels
are coherent structures, localized only around PILs and extending to very
large scale. In this paper, we describe a process, helicity condensation, that
performs exactly the required transformation of small-scale coronal loop
helicity into large-scale filament-channel shear. Helicity condensation keeps
coronal loops laminar while shearing filament channels.

Furthermore, helicity condensation may be responsible for much of the complex
structure and dynamics observed in the slow solar wind. We argued above that
the helicity injected into the corona must end up in filament channels, but in
regions containing coronal holes, another possibility is that some helicity is
ejected out into the wind by the opening of closed flux at the coronal hole
boundary. Such helicity transfer is implicitly present in the S-Web model for
the slow wind (Antiochos et al., 2011, 2012), which postulates that this wind
is due to continual dynamics of the open-closed flux boundary. If so, then
helicity condensation also will play a major role in the origin and properties
of the slow solar wind.

We describe below the basic process of helicity condensation and derive
estimates of its effectiveness in the Sun’s corona.

##  2 A Model for Helicity Injection and Transport

In order to understand how magnetic helicity is likely to evolve in the
corona, we must first consider the injection process. Assume, for the moment,
that the photospheric flux distribution consists of only two polarity regions
as shown in Figure 1: a negative northern hemisphere and a positive south, so
that all the flux closes across the equatorial PIL (dashed line). The yellow
arches in the figure denote two arbitrary small flux tubes corresponding to
coronal loops or to the strands inside an observable coronal loop. The quasi-
random photospheric motions will introduce small-scale structure and inject
helicity to this coronal field. Helicity will also be injected by large-scale
motions, such as differential rotation, and by flux emergence/cancellation,
such as the magnetic carpet, but for simplicity let us model the injection as
due to the continual small-scale photospheric motions, in particular, the
granular or supergranular flows. Note that if the magnetic carpet dynamics do
not change the net coronal flux, their effect on the coronal helicity can be
captured by effective photospheric motions. Furthermore, recent analysis of
high-resolution Solar Dynamics Observatory (SDO) data indicates that the bulk
of the helicity injected into active regions is due to photospheric motions
rather than flux emergence (Liu & Schuck, 2012).

Following the arguments of Sturrock & Uchida (1981), the energy and,
certainly, the helicity injected into coronal loops by stochastic horizontal
flows at the photosphere will be primarily in the form of twist. Therefore, we
model the motions as a set of randomly located and randomly occurring
rotations that have fixed spatial and temporal scales. The true photospheric
motions are more complex than a set of fixed-scale rotations, but we are
interested only in that part of the flow that injects helicity to the corona.
Note also that there is some evidence for exactly the pattern of photospheric
rotations of Fig. 1 in measurements of the vorticity of the supergranulation
(Duvall & Gizon, 2000; Gizon & Duvall, 2003; Komm et al., 2007).

It has been known since the time of Hale (1927) that sunspot whirls have a
clear hemispheric preference, counterclockwise in the north and clockwise in
the south (Pevtsov et al., 1995), indicating a preferred sense for the
helicity of the subsurface solar motions. The same helicity preference,
negative in the north and positive in the south, has been well documented to
occur in all types of coronal magnetic structures ranging from quiet Sun field
to active region complexes (Pevtsov & Balasubramaniam, 2003) and has been
observed out in the heliospheric magnetic field (Bieber et al., 1987). As
shown in Figure 1, this hemispheric “rule” is in the same sense as would be
expected from the surface differential rotation, but the actual mechanism is
still not clear. In any case, we expect there to be a preferred sense to the
helicity injecting rotations as illustrated in Figure 1. Note that this is
only a preference; a fraction of the rotation in each hemisphere could well
have the “unpreferred” sense.

The motions shown in Figure 1 have a number of interesting implications for
the coronal field. Assuming, for simplicity, that the rotations are solid
body, have size $d$, and have magnitude, $\Theta$, then each rotation of a
photospheric flux tube with axial flux,

$\Phi_{d}=\pi d^{2}B_{p}/4,$ (1)

produces in the corona a twist flux,

$\Phi_{t}=\Theta\Phi_{d}/\pi,$ (2)

where $B_{p}$ is the average normal field at the photosphere. In open field
regions (not shown in the Figure), this twist flux simply propagates outward,
resulting in a net helicity to the turbulence in the fast wind (e.g. Leamon et
al, 1998). We will discuss the implications for the slow wind below. In the
closed field regions, however, the coronal loops acquire a twist component to
their magnetic field, as shown in the Figure. If the loop is perfectly
symmetric about the equator, then on average, the twists imposed by the two
footpoints cancel out so that no net helicity is injected. Basically, the loop
is twisted at one end, but untwisted at the other. On the other hand, if the
loop has both footpoints in one hemisphere, as is usually the case when the
PIL is not exactly at the equator, then the twist from each footpoint will
add. Even if the loop is transequatorial, we do not expect any symmetry for a
real coronal loop, so a net twist will still be produced by the footpoint
motions. Note also, that the effect of any unpreferred-sense rotations
(clockwise in the north and counterclockwise in the south) is only to decrease
the rate of twisting. To first order, the unpreferred rotations simply untwist
the loops, but it should be emphasized that since the rotations are time
varying, they can create higher order topological structure in the field even
if the net injected helicity vanishes. All higher order topological features,
however, such as the braiding of three flux tubes, are not conserved by
reconnection (e.g. Pontin et al., 2011) and are not expected to build up in
the corona.

Consider now the interaction of the two twisted flux tubes of Fig. 1 due to
some random motion that causes them to collide. In fact, the twist itself will
cause the flux tubes to expand and interact. Since their main axial fields are
parallel, only the twist components of the flux tubes can reconnect. If the
tubes have the same sense of twist (helicity), then at the contact point
between the tubes, their twist components will be oppositely directed and,
hence, will reconnect. This is illustrated in the three sequences of Fig. 2,
where the red and blue circles correspond to field lines of the twist magnetic
component. In the top sequence the twist components are in the same sense, so
they are oppositely directed at their contact point; consequently, they will
reconnect there. Another way of understanding this result is to note that for
interacting tubes with the same helicity, the photospheric rotations have a
stagnation point between them. It is well known that such stagnation point
flows lead to exponentially growing separation of magnetic footpoints and,
hence, to exponentially growing currents in the corona (e.g. Antiochos &
Dahlburg, 1997), which can drive efficient reconnection. The effect of this
reconnection is to spread the twist component over the flux of the two tubes,
in other words, the two tubes merge into one globally twisted tube as
illustrated in the Figure and as found in simulations of flux-tube collisions
(Linton et al., 2001).

On the other hand, if the tubes have opposite twist (helicity), then at their
point of interaction the twist components are parallel. As illustrated in the
second sequence of Figure 2, there is no reconnection in this case. The tubes
simply “bounce” (Linton et al., 2001). This result emphasizes the point made
in Antiochos & Dahlburg (1997) that reconnection in the solar corona is highly
constrained by line-tying. As a result, the coronal magnetic field cannot
simply relax to a minimum energy Taylor (1974) state, which for the opposite
twist case corresponds to the initial potential field.

The third sequence in Figure 2 illustrates the effect of continued interaction
of same-helicity flux tubes. If a larger-scale merged flux tube reconnects
with another tube, the result is further merging of the flux and the spread of
the twist to even larger scale. This reflects the well-known result from
turbulence studies that magnetic helicity tends to cascade upward in scale
(e.g. Biskamp, 1993). A key point is that the scale referred to in our cascade
process is the amount of axial flux, which is closely related to, but not
identical to the spatial scale. As argued directly below, the helicity
cascades up to the largest possible flux scale, which corresponds to all the
axial flux inside a single polarity region.

Let us consider the end result of this reconnection-driven helicity cascade.
Assume a flux system, as in Figure 3, with simple topology given by a PIL
(heavy dashed curve in the Figure) and a separatrix curve somewhere on the
photosphere (light dashed curve) that defines all the flux that closes across
the PIL. There must be additional PILs on the photosphere, but these are not
shown. If the separatrix curve lies in the north, then the flux system is
entirely in the north and the twist injected by the photosphere will be
predominately counter-clockwise. Since only the relative footpoint motions are
significant, we can assume without loss of generality that all the twist is
imparted inside the PIL as shown in the Figure.

The expected evolution of this twist is seen in Figure 4, which shows a top
view of the flux system. As a result of reconnection, the helicity “condenses”
onto the largest scale in the flux system, the PIL, since this encompasses all
the flux in the system. The PIL defines the boundary of the polarity region.
We conclude, therefore, that the net effect of the many small-scale
photospheric twists and the coronal reconnection is to impart a coherent,
global twist of the whole flux system that concentrates at the PIL. This
global twist is not a true physical motion; the large-scale flux system does
not actually rotate as a coherent body, but the photospheric helicity
injection and subsequent transport by reconnection does result in an effective
global rotation of the magnetic field.

The key point is that such a rotation of the whole flux system corresponds to
a coherent localized shear all along the PIL, exactly what is needed to
explain the formation of filament channels. Such an effective motion produces
a channel consisting of field lines that are sheared but smooth and laminar,
with no twist or tangles in agreement with high-resolution observations of
prominence threads (Lin et al., 2005; Vourlidas et al., 2010). The coronal
reconnection in our helicity condensation model results in a structure that is
the direct opposite to that of flux cancellation reconnection, which
invariably produces a highly twisted flux rope at the PIL.

Another important point is that the helicity condensation mechanism is
unaffected by the shape of the PIL, in particular whether the PIL contains so-
called switchbacks where it forms a sharp zigzag. As long as the flux system
defined by the PIL is primarily in one hemisphere, helicity condensation will
form a filament channel with the same chirality all along that PIL and with
roughly the same amount of shear. This result is in contrast to the
predictions of some of the flux cancellation models (van Ballegooijen et al.,
1998), but is in good agreement with observations (Pevtsov et al., 2003).
Furthermore, since the photospheric convection in either quiet or active
regions is not observed to change significantly with phase of the solar cycle,
the model predicts that the hemispheric helicity rule should hold independent
of solar cycle. Again, this conclusion appears to be in good agreement with
observations (Pevtsov et al., 2003), unlike some flux cancellation models
(Mackay & van Ballegooijen, 2001).

###  2.1 Rate of Helicity Cascade

Prominences typically form on time scales of several days, which sets a
constraint that any model must satisfy; therefore, we calculate below the rate
of filament channel formation predicted by the helicity condensation model.
Let us consider a polarity region as in Figure 4 with scale $L$ that is large
compared to the helicity injection scale $d$. The rate of helicity injection
$\eta$ into a flux tube $\Phi_{d}$ by a photospheric twist of average angular
velocity $V_{d}/d$ is given by the product of the rate of change of the twist
flux of Eq. (2) and the axial flux $\Phi_{d}$, (which stays constant):

$\eta\approx\Phi_{d}(V_{d}/d)\Phi_{d}$ (3)

(e.g. Berger, 2000). The total helicity injection rate at the scale $d$ over
the whole flux system region with scale $L$ is therefore given by:

$h_{d}=\Phi_{d}^{2}(V_{d}/d)(L/d)^{2}=(\Phi_{L}/L)^{2}d^{2}\,V_{d}/d,$ (4)

where $\Phi_{L}$ is the flux of the whole system. Following Kolmogorov’s
classic theory for hydrodynamic turbulence (Kolmogorov, 1941), we assume a
constant helicity transfer rate at any scale $\lambda$. Therefore:

$h_{\lambda}=(\Phi_{L}/L)^{2}\lambda^{2}V_{\lambda}/\lambda=h_{d},$ (5)

which implies that:

$V_{\lambda}=V_{d}(d/\lambda)\ {\rm and,thus},\ V_{L}=V_{d}(d/L).$ (6)

It should be noted that unlike $V_{d}$, which is the actual velocity of the
photospheric flows, the quantity $V_{L}$ does not represent a true plasma
velocity. It is only an effective velocity for the transfer of twist to the
largest scale by coronal reconnection. The physical plasma velocities will
likely be dominated by reconnection jets and will have both larger magnitude
and smaller scale than $V_{L}$. The physical velocities and kinetic energy are
expected to cascade downward, not upward, in scale. Consequently, the velocity
spectrum $V_{\lambda}$ derived in Eq. 6 may not be directly observable, but
the effective velocity $V_{L}$ is indeed physically significant. It quantifies
the rate at which magnetic helicity “condenses” out of the corona at the
largest scale of the flux system and, hence, $V_{L}$ corresponds to the
effective shear velocity along the PIL.

Note also, that the helicity cascade process derived above is somewhat
different than the usual hydrodynamic turbulence in which velocity is injected
statistically uniformly at some scale and then cascades down to where it is
dissipated, usually at a kinetic scale. In such an energy cascade, the
injection of kinetic energy at a global scale results in the slow increase of
thermal energy (temperature) approximately uniformly throughout the system. In
our cascade, however, helicity is injected statistically uniformly at some
intermediate scale and then simply piles up at the largest global scale. Even
though the helicity injection (i.e., photospheric motions) are uniform, the
cascade produces a localized spatial structure in the corona. Helicity
condensation in the Sun’s atmosphere is a striking example of self-
organization in a complex system.

The time scale for filament channel formation can now be calculated directly
from Eq. (6). We note that for the photospheric helicity injection the
important parameter is the product of the velocity and the coherence scale of
that velocity. Granules typically have $V_{d}\sim 1$ km/s and $d\sim 700$ km,
while supergranules have: $V_{d}\sim.25$ km/sec and $d\sim 30,000$ km. Eq. (6)
implies that the product $V_{d}\,d$ is the important quantity; therefore,
supergranules are expected to dominate the helicity injection. Taking the
whole flux system to have scale $d/L\sim 10-100$ implies that a shear of order
1,000 – 10,000 km will build up in $\sim 10^{5}$ s, where we assume that equal
helicity is injected at both ends of a flux tube. These results indicate that
a high-latitude filament channel with typical shear scales of 100,000 km will
form in several days or so, which is consistent with observations (Tandberg-
Hanssen, 1995; Mackay et al., 2010). Note also that we expect that the width
of the helicity condensation region to be of order the width of the elemental
photospheric rotation, $\sim 15,000$ km for supergranules, which again is
consistent with the observed widths of filaments (Tandberg-Hanssen, 1995;
Mackay et al., 2010).

The supergranular rate of helicity injection estimated above is also
consistent with estimates of solar helicity loss to the wind. Taking the
system size $L$ to be of order the solar radius $\sim 10^{11}$ cm, and the
average field strength at the photosphere to be $~{}\sim 10$ G, we derive from
Eq. (4) and the numbers above, a helicity injection rate over the solar
surface of $\sim 10^{38}$ Mx/s. Over the course of a full solar cycle, this
yields a total helicity loss of $\sim 3\times 10^{46}$ Mx, which agrees well
with the inferred losses from observations of CMEs and the wind (DeVore,
2000).

###  2.2 Implications of the Model

We conclude from the derivation above that helicity condensation can account
for filament channel formation, at least, in regions that do not exhibit
strong flux emergence. The mechanism can also account for the observed
smoothness of coronal loops. Let $\tau$ be the time scale required for the
system to establish a steady state (except, of course, at the largest scale
$L$ where no steady-state is possible). We expect that $\tau$ is determined by
the twist required to produce intense current sheets, of order a full rotation
or so (Antiochos, 1998), and not by the rate of reconnection. The driving
velocity is only $\sim$ 0.1% of the coronal Alfven speed, so that the
reconnection need not be fast in order to keep pace with the driving. For a
given $\tau$, the twist angle produced by the effective velocities of Eq. (6)
scales as:

$\Theta_{\lambda}=\tau V_{\lambda}/\lambda\sim\lambda^{-2}$ (7)

This result implies that the corona will exhibit the most structure at the
scale at which the twist is injected (presumably the supergranular scale), and
at the largest scale where the helicity piles up, the whole length of the PIL.
The so-called coronal cells recently discovered by Sheeley & Warren (2012)
appear to be evidence for just this type of structure separation. These
authors observe that the large-scale corona breaks up into three distinct
structures: flux tubes twisted on a scale of 30,000 km or so, long filment
channels along PILs that typically span the whole Sun, and coronal holes. Our
helicity condensation model is in excellent agreement with these observations.

An important issue that is raised by the observations and that we have yet to
discuss is the effect on the model of a coronal hole or, more generally, of an
open field region. Note, also, that even if no coronal hole is present, the
simple picture of Figs. 3 and 4 is topologically incomplete. The PIL cannot be
the only boundary that defines the closed negative polarity region. At the
very least, there must be a point somewhere in the region where a magnetic
spine line connects up to a null point Lau & Finn (1990); Antiochos (1990);
Priest & Titov (1996); thereby, making this field line effectively open. Of
course, real solar polarity regions tend to have much more complexity often
containing intricate open field areas and corridors (Antiochos et al., 2011).

Assume that the polarity region of Figs. 3 contains a coronal hole, as
illustrated in Figure 5. The presence of the coronal hole introduces
subtleties to the calculation of helicity evolution, because the helicity of a
truly open field that extends to infinity is not physically meaningful. An
open field can have arbitrary helicity due to linkages at infinity where the
field vanishes, but the topology does not. Therefore, let us consider instead
a system where all the field lines remain closed so the helicity is well-
defined throughout the evolution, and let us model the coronal hole as a
region where no photospheric twists are imposed; hence, no helicity is
injected into this region. In an actual coronal hole twist is injected by
photospheric motions, exactly as in closed field regions, but the twist
propagates away at the Alfven speed and presumably has no effect on the
subsequent evolution in the low corona. Consequently, we can simply model this
region as being untwisted.

The analysis of the helicity cascade in the large annular flux region bounded
by the PIL and the coronal hole boundary proceeds exactly as above. The only
difference is that the total twisted area $L^{2}$ is replaced by $L^{2}-H^{2}$
where $H$ is the scale of the coronal hole. Therefore, the largest scale for
the helicity is given by:

$L^{\prime}=\sqrt{L^{2}-H^{2}}$ (8)

and the effective velocity for helicity condensation is given by Eq (6) above
except that $L$ is replaced by $L^{\prime}$. Also, the twist spectrum, Eq (7)
is unchanged. As long as $H<<L$, the presence of the coronal hole has minimal
effect on the filament channel formation process, and on the smoothness of
coronal loops.

It is evident from Fig. 5, however, that helicity condensation does have an
effect on the magnetic field near the coronal hole boundary. We note that
twist, or more accurately magnetic shear, also condenses at this boundary, but
curiously enough the shear has the sense opposite to that at the PIL. This
result may seem physically unlikely, but in fact it is mandated by helicity
conservation. The key point is that the shear flux $\Phi_{1}$ that condenses
onto the PIL encircles all the photospheric flux in the polarity region,
including that in the coronal hole region (“CH”). Therefore, the helicity
$H_{1}$ due to this shear flux is given by:

$H_{1}=\Phi_{1}\Phi_{L^{\prime}}+\Phi_{1}\Phi_{CH},$ (9)

where $\Phi_{L^{\prime}}$ is the amount of closed photospheric flux inside the
PIL and $\Phi_{CH}$ is the amount of photospheric flux in the coronal hole,
which can be arbitrary compared to $\Phi_{L^{\prime}}$. But the CH flux is not
twisted and never contributes to the helicity injection; hence, it should not
affect the helicity condensation at the PIL. The only way to ensure that the
CH flux has no effect is to have a shear flux $\Phi_{2}$ that condense at the
CH boundary and is exactly equal and oppositely directed to that at the PIL.
Such a shear flux encircles only the CH flux and, thereby, adds a helicity
contribution:

$H_{2}=-\Phi_{1}\Phi_{CH},$ (10)

which exactly cancels out that between the PIL shear flux and the CH. Fig. 5
shows that reconnection would produce just this required shear flux at the CH
boundary. In other words, helicity condensation predicts that a filament
channel should form at coronal hole boundaries, at the same rate as at the PIL
but with the opposite handedness.

These results clearly have major implications for observations. At PILs, the
magnetic shear builds up until eventually, it is ejected as a prominence
eruption/CME. We expect that a similar process of buildup and ejection occurs
at the CH boundaries, but with much less explosive dynamics. The closed field
lines near the CH boundary consist of very long, high-lying loops that form
the outer shell of the streamer belt. Consequently, it requires far less shear
and free energy to open up these loops than to eject the filament channel. We
expect that helicity condensation at CH boundaries results in continual small
bursts of flux opening and closing there, as is required by models for the
slow wind (Antiochos et al., 2011, 2012). An interesting prediction is that
the helicity of the closed flux opening at the CH boundary should be opposite
to that of the photospheric injection into the coronal hole open field lines,
i.e., into the fast wind. It may be possible to test this prediction with in
situ measurements.

In summary, we argue that a single deeply-profound process, helicity
condensation, can explain three long-standing observational challenges in
solar/heliospheric physics: the formation of filament channels, the smoothness
of coronal loops, and the origin of the slow wind. Furthermore, the model
implies major new predictions for solar structure and dynamics. We look
forward to many more theoretical and observational studies of helicity
condensation in the Sun’s corona and wind.

This work has been supported, in part, by the NASA TR&T Program. The work has
benefited greatly from the authors’ participation in the NASA TR&T focused
science teams on multiscale coupling and the slow solar wind. The author
thanks C. R. DeVore, J. T. Karpen, and J. A. Klimchuk for invaluable
scientific discussions and J. T. Karpen for help with the graphics.

## References

  * Antiochos (1987) Antiochos, S. K. 1987, ApJ, 312, 886
  * Antiochos (1990) Antiochos, S. K. 1990, J. Italian Astron. Soc., 61, 369
  * Antiochos et al, (1994) Antiochos, S. K., Dahlburg, R. B., & Klimchuk, J. A. 1994, ApJ, 420, L41
  * Antiochos & Dahlburg (1997) Antiochos, S. K. & Dahlburg, R. B. 1997, Sol. Phys., 174, 5
  * Antiochos (1998) Antiochos, S. K. 1998, ApJ, 502, L181
  * Antiochos et al. (1999) Antiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485
  * Antiochos et al. (2002) Antiochos, S. K., Karpen, J. T., & DeVore, C. R. 2002, ApJ, 575, 578
  * Antiochos et al. (2011) Antiochos, S. K., Mikić, Z., Titov, V. S., Lionello, R., & Linker, J. A. 2011, ApJ, 731, 112
  * Antiochos et al. (2012) Antiochos, S. K., Linker, J. A., Lionello, R., Mikić, Z., Titov, V. S., & Zurbuchen, T. H. 2011, Space Sci. Rev., 172, 169
  * Archontis (2004) Archontis, V. 2004, ApJ, 615, 685
  * Berger (1984) Berger, M. A. 1984, Geophys. & Astrophys. Fluid Dynamics, 30, 79
  * Berger (1991) Berger, M. A. 1991, A&A, 252, 369
  * Berger (2000) Berger, M. A. 2000, Encyclopedia of Astronomy and Astrophysics, ed. P. Murdin, (Bristol: IOP)
  * Bieber et al. (1987) Bieber, J. W., Evenson, P. A., & Matthaeus, W. H. 1987, ApJ, 315, 700
  * Biskamp (1993) Biskamp, D. 1993, Nonlinear magnetohydrodynamics, Ch. 7, (New York: Cambridge University Press)
  * Casini et al. (2003) Casini, R. Lopez Ariste, A., Tomczyk, S., & Lites, B. W. 2003, ApJ, 598, L67
  * DeVore (2000) DeVore, C, R. 2000, ApJ, 539, 944
  * Duvall & Gizon (2000) Duvall, T. L., Jr. & Gizon, L. 2000, Sol. Phys., 192, 177
  * Fan (2001) Fan, Y. 2001, ApJ, 554, L111
  * Fang et al. (2012) Fang, F., Manchester, W. B., van der Holst, B. & Abbett, W. 2012, ApJ, 745, 37
  * Gizon & Duvall (2003) Gizon, L. & Duvall, T. L., Jr. 2003, in ESA Special Publication, Vol. 517, GONG 2002, Local and Global Helioseismology: The Present and Future, ed. H. Sawaya-Lacoste (Noordwijk: ESA), 43
  * Hale (1927) Hale, G. E. 1927, Nature, 119, 708
  * Klimchuk (1987) Klimchuk, J. A. 1987, ApJ, 323, 368
  * Klimchuk (2006) Klimchuk, J. A. 2006, Sol. Phys., 234, 41, DOI: 10.1007/s11207-006-0055-z
  * Kolmogorov (1941) Kolmogorov, A. N. 1941, Dokl. Akad. Nauk. SSSR, 30, 299
  * Komm et al. (2007) Komm, R., Howe, R., Hill, F., Miesch, M., Haber, D., & Hindman, B. 2007, ApJ, 667, 571
  * Kuckein et al. (2012) Kuckein, C. Martínez Pillet, V., & Centeno, R 2012, A&A, 539, A131
  * Kumar & Rust (1996) Kumar, A. & Rust, D. M. 1996, J. Geophys. Res., 101, 15667
  * Leake et al. (2010) Leake, J. A., Linton, M. G., & Antiochos, S. K., 2010, ApJ, 722, 550
  * Liu & Schuck (2012) Liu Y. & Schuck, P., 2012, ApJ, (in press)
  * Harvey (1985) Harvey, K. L. 1985, Aust. J. Phys., 38, 875
  * Lau & Finn (1990) Lau, Y.-T. & Finn, J. M. 1990, ApJ, 350, 672
  * Leamon et al (1998) Leamon, R. J., Matthaeus, W. H., Smith, C. W., & Wong, H. K. 1998, ApJ, 507, L181
  * Leka et al. (1996) Leka, K., Canfield, R., McClymont, A., & van Driel-Gesztelyi, L. 1996, ApJ, 462, 547
  * Leroy et al. (1983) Leroy, J.-L., Bommier, V., & Sahal-Brechot, S. 1983, Sol. Phys., 83, 135
  * Lin et al. (2005) Lin, Y., Engvold, O., Rouppe van der Voort, L., Wiik, J. E. & Berger, T. E. 2005, Sol. Phys., 226, 239
  * Linton et al. (2001) Linton, M. G., Dahlburg, R. B., & Antiochos, S. K. 2001, ApJ, 553, 905
  * Magara & Longcope (2003) Magara, T. & Longcope, D. W. 2003, ApJ, 586, 630
  * Mancester (2001) Manchester, IV, W. 2001, ApJ, 547, 503
  * Martin et al. (1992) Martin, S. F., Marquette, W. H., & Bilimoria, R. 1992, ASP Conf. Ser. 27, The Solar Cycle, ed. K. L. Harvey, (San Francisco: ASP), 53
  * Martin (1998) Martin, S. F. 1998, Sol. Phys., 182, 107
  * Mackay & van Ballegooijen (2001) Mackay, D. H., & van Ballegooijen, A. A. 2001, ApJ, 560, 445
  * Mackay et al. (2010) Mackay, D. H., Karpen, J. T., Ballester, J. L., Schmieder, B., & Aulanier, G. 2010, Space Sci. Rev., 151, 333
  * Mikic et al. (1989) Mikic, Z., Schnack, D. D., & van Hoven, G. 1989,ApJ, 338, 1148
  * Parker (1972) Parker, E. N. 1972, ApJ, 174, 499
  * Parker (1983) Parker, E. N. 1983, ApJ, 264, 642
  * Parker (1988) Parker, E. N. 1988, ApJ, 330, 474
  * Pevtsov et al. (1995) Pevtsov, A. A., Canfield, R. C., & Metcalf, T. R. 1995, ApJ, 440, L109
  * Pevtsov et al. (2003) Pevtsov, A. A., Balasubramaniam, K. S., & Rogers, J. W. 2003, ApJ, 595, 500
  * Pevtsov & Balasubramaniam (2003) Pevtsov, A. A. & Balasubramaniam, K. S. 2003, Advances Space Res., 32, 1867
  * Pontin et al. (2011) Pontin, D. I., Wilmot-Smith, A. L., Hornig, G., & Galsgaard, K. 2011, A&A, 525, A57
  * Priest & Titov (1996) Priest, E. R., & Titov, V. S. 1996, Phil. Trans. R. Soc., 354, 2951
  * Qiu et al. (2007) Qiu, J., Hu, Q., Howard, T. A. & Yurchyshyn, V. B. 2007, ApJ, 659, 758
  * Rappazzo et al. (2008) Rappazzo, A. F., Velli, M., Einaudi, G., & Dahlburg, R. B. 2008, ApJ, 677, 1348
  * Rust (1967) Rust, D. M. 1967, ApJ, 150, 313
  * Rust (1994) Rudt, D. M. 1994, Geophys. Res. Lett., 21, 241
  * Schrijver et al. (1997) Schrijver, C. J., Title, A. M., van Ballegooijen, A. A., Hagenaar, H. J., and Shine, R. A. 1997, ApJ, 487, 424
  * Schrijver et al. (1999) Schrijver, C. J. et al. 1999, Sol. Phys., 198, 325
  * Schrijver (2007) Schrijver, C. J. 2007, ApJ, 662, L119
  * Sheeley & Warren (2012) Sheeley, N. R. & Warren, H. P. 2012, ApJ, 749, 40
  * Seehafer (1990) Seehafer, N. 1990, Sol. Phys., 125, 219
  * Su & van Ballegooijen (2012) Su, Y. & van Ballegooijen, A. A. 2012, ApJ, 757, 168
  * Sturrock & Uchida (1981) Sturrock, P. A. & Uchida, Y. 1981, ApJ, 246, 331
  * Tandberg-Hanssen (1995) Tandberg-Hanssen, E 1995, The Nature of Solar Prominences, (Dordrecht: Kluwer)
  * Taylor (1974) Taylor, J. B. 1974, Phys. Rev. Lett., 33, 1139
  * Taylor (1986) Taylor, J. B. 1986, Rev. Mod. Phys., 58, 741
  * Tian et al. (2005) Tian,L., Alexander, D., & Nightingale, R. 2008, ApJ, 684, 747
  * van Ballegooijen (1986) van Ballegooijen, A. A. 1986, ApJ, 311, 1001
  * van Ballegooijen & Martens (1989) van Ballegooijen, A. A. & Martens, P. C. H. 1989, ApJ, 343, 971
  * van Ballegooijen et al. (1998) van Ballegooijen, A. A., Cartledge, N. P., & Priest, E. R. 1998, ApJ, 501, 866
  * van Ballegooijen (2004) van Ballegooijen, A. A. 2004, ApJ, 612, 529
  * Vourlidas et al. (2010) Vourlidas A. et al. 2010, Sol. Phys., 261, 53
  * Yamada et al. (1990) Yamada, M., Ono, Y., Hayakawa, A., Katsurai, M., & Perkins, F. W. 1990, Phys. Rev. Lett., 65, 721
  * Zirker et al. (1997) Zirker, J. B., Martin, S. F., Harvey, K., & Gaizauskas, V. 1997, Sol. Phys., 175, 27

Figure 1:  Model for helicity injection into the corona by photospheric
motions. The primary effect of the motions is to inject an effective twist of
scale d. The sense of the twist in each hemisphere is determined by the
differential rotation, large arrows. The yellow arches represent two
neighboring coronal loops (magnetic flux tubes).

Figure 2:  Interaction of the twist component of interacting flux tube. Red
and blue circles correspond to oppositely-oriented twist components of the
magnetic field.

Figure 3:  Model of negative polarity flux region fully in the northern
hemisphere. The dark dashed line correspond to the PIL of the flux region.
Figure 4:  The polarity region of the previous figure as viewed from the north
pole.

Figure 5:  The polarity region of the Figs 3 and 4, but now containing a
coronal hole region, indicated by “CH”.