Issue |
A&A
Volume 516, June-July 2010
|
|
---|---|---|
Article Number | A5 | |
Number of page(s) | 9 | |
Section | The Sun | |
DOI | https://doi.org/10.1051/0004-6361/201014041 | |
Published online | 16 June 2010 |
Dynamics of braided coronal loops
I. Onset of magnetic reconnection
A. L. Wilmot-Smith - D. I. Pontin - G. Hornig
Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK
Received 11 January 2010 / Accepted 2 April 2010
Abstract
Aims. The response of the solar coronal magnetic field to
small-scale photospheric boundary motions including the possible
formation of current sheets via the Parker scenario is one of open
questions of solar physics. Here we address the problem via a numerical
simulation.
Methods. The three-dimensional evolution of a braided magnetic
field which is initially close to a force-free state is followed using
a resistive MHD code.
Results. A long-wavelength instability takes place and leads to
the formation of two thin current layers. Magnetic reconnection occurs
across the current sheets with three-dimensional features shown,
including an elliptic magnetic field structure about the reconnection
site, and results in an untwisting of the global field structure.
Key words: magnetohydrodynamics (MHD) - plasmas - Sun: corona - magnetic fields
1 Introduction
Parker's notion of the ``topological dissipation'' of coronal magnetic fields (Parker 1972) continues to generate much debate. Simply put, Parker's suggestion is that following boundary motions of sufficient complexity the magnetic field of a coronal loop will be unable to ideally relax to a smooth force-free equilibrium and instead tangential discontinuities in the field, corresponding to current sheets, will develop. In general terms the possible outcomes of relaxation are a development of singular current sheets (e.g. Ng & Bhattacharjee 1998; Janse & Low 2009), development of thin but non-singular current layers (e.g. Longcope & Strauss 1994; Galsgaard et al. 2003), and a smooth equilibrium with large-scale current features (e.g. van Ballegooijen 1985; Craig & Sneyd 2005; Wilmot-Smith et al. 2009a). It should also be noted that the distinction between the first two cases may be difficult to determine numerically.
In a previous a paper (Wilmot-Smith et al. 2009a) we considered the ideal relaxation of a braided magnetic field towards a force-free equilibrium. The magnetic field configuration was based on the pigtail braid and imposed as an initial condition (rather than being built up through boundary motions). This particular braid was chosen since it is the simplest non-trivial braid with no net twist and, accordingly, is the most conservative realistic case to examine. More complex braided fields will, in general, contain components of the pigtail-type. There is ample motivation for modelling loops as having braided components. Photospheric turbulence subjects loop footpoints to a random walk, with motions of the fragments about each other acting to braid (or unbraid) the overlying loop. However, while there is some evidence for the existence of braided loop configurations (see the left-hand image of Fig. 1 for example), most coronal loops appear to be close to a potential field (e.g. Fig. 1, right-hand image). This may be an effect of the large aspect ratios typical to loops, since a winding of a field line around another field line is almost undetectable when smoothed out over the length of the loop (see also Berger & Asgari-Targhi 2009). Another reason could be that reconnection is very efficient in maintaining a low degree of topological complexity in loops. The present work is designed in part to test whether this is the case.
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Figure 1: TRACE coronal loops. ( Left) A large-scale tangled configuration and ( right) apparently smooth loops. |
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In Wilmot-Smith et al. (2009a) an ideal Lagrangian relaxation scheme (Craig & Sneyd 1986; Pontin et al. 2009)
was used to ideally evolve the braided field towards a force-free
state. A smooth near force-free equilibrium was attained with
large-scale current features i.e. without any tangential
discontinuities or even strong current concentrations. (This situation
may be compared with Parker (1994)
where the same pigtail braid situation is considered as a thought
experiment and concluded to inevitably lead to tangential
discontinuities. It appears that Parker's assumption of
in the domain fails; we indeed have
,
where S is a cross-sectional surface through the braid, but
varies significantly between field lines).
Although the local current density in the ideal equilibrium is of
large-scale, a global quantity, the integrated parallel current,
,
has small scales (here the parallel indicates parallel to the magnetic
field and the integral is taken along magnetic field lines). By small
scales we mean that
varies significantly between neighbouring field lines as the field
lines pass though a particular plane, such as the lower boundary. In
Wilmot-Smith et al. (2009a) it was suggested that these small scales in
could lead to the development of a resistive instability and so a loss
of equilibrium of the field. This mechanism would be distinct to that
of Parker's topological dissipation but have the same consequence.
The consideration that for sufficiently small scales in the integrated parallel current then a force-free magnetic field will be resistively unstable (the instability being dependent, for a given scale in the integrated parallel current, on the value of the resistivity) motivated us to put the end, near force-free, state of the Lagrangian relaxation into a resistive MHD code and test whether or not the state is stable (for various values of resistivity allowed by numerical limitations). It turns out that the braided field as implemented in the resistive code is not stable. This finding, together with the subsequent evolution of the field, allows us to address a number of key questions related to MHD and the behaviour of coronal magnetic fields. The results are described in this series of papers. Here we describe the details relating to the numerical setup of the problem and the early evolution of the system. A subsequent paper addresses the long term evolution of the system.
2 Numerical scheme and simulation setup
2.1 Numerical scheme
The numerical scheme employed in the simulations that follow is described
briefly below (further details may be found in Nordlund & Galsgaard (1997) and at
http://www.astro.ku.dk/ kg). We solve the three-dimensional resistive MHD
equations in the form
where













The Eqs. (1)-(6) are solved on staggered meshes; with respect to a mesh on which
and e
are defined at the body centre of the cell,
and P are defined at face centres and
and
at
edge centres. In this way the required MHD
conservation laws are automatically satisfied. Derivatives are
calculated using sixth-order finite differences that return a value
which is shifted half a grid-point up or down with respect to the input
values. When the staggered mesh means that some quantity must be
interpolated, data values are calculated using a fifth-order
interpolation method at the relevant position. A third-order
predictor-corrector method is employed for time-stepping.
In all simulation runs we employ a spatially uniform resistivity model. Viscosity is calculated using a combined second-order and fourth-order method (sometimes termed ``hyper-viscosity''), which is capable of providing sufficient localised dissipation where necessary to handle the development of numerical instabilities (Nordlund & Galsgaard 1997). The effect is to ``switch on'' the viscosity where very short length scales develop, while maintaining a minimal amount of viscous dissipation where the velocity field is smooth.
2.2 Creating the initial condition
As discussed in Sect. 1,
the initial state for the resistive MHD simulations is drawn from the
final state of the Lagrangian relaxation experiment of Wilmot-Smith
et al. (2009a).
The quantities previously known from the Lagrangian code (see Craig & Sneyd 1986)
are the magnetic field
and the current
and
in the near force-free relaxed state these are known on a highly distorted mesh.
We describe below the process of constructing the field on the rectangular grid required for the
resistive MHD simulations.
In order to ensure that the initial magnetic field is divergence-free, we work first with the vector potential
for
.
In the relaxation scheme, the calculation of
requires only that we know the initial vector potential before
relaxation and the mesh deformation. In terms of the initial mesh
and the final ``relaxed mesh''
,
the ith component of the final vector potential
is given (see Appendix) in terms of the initial vector potential
by
To create the input magnetic field for the MHD simulations we then interpolate this vector potential onto a rectangular grid. Since the magnetic field components are face-centred on the staggered grid, the vector potential components are interpolated onto locations corresponding to edge centres of the desired grid. We then obtain the magnetic field by taking the curl of





The result of the above is that the initial braided magnetic field for
our MHD simulations is divergence-free to accuracies on the order of
truncation errors of the sixth-order finite differences (with typical
maximum
within the domain). The topology of the magnetic field turns out to be
well conserved by the process, another important consideration for the
experiment. However, a drawback is that the quality of the force-free
approximation is not perfectly maintained; the initial state is further
from force-free than the relaxed field of the Lagrangian experiment.
Details and implications are discussed in Sects. 3 and 5.
![]() |
Figure 2:
Initial state of the simulation:
( left) Isosurface of current
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2.3 Initial state
The initial magnetic field is given on a domain of size [-24,24] in the vertical (z) direction and [-6,6] in both the horizontal (x,y) directions. Covering this domain we take a uniform mesh of 3203 cells and employ closed boundary conditions in all three directions.
The magnetic field is line-tied, and can be very closely approximated by
at
the boundaries, i.e. it is directed perpendicular to the z boundaries and parallel to the x
and y
boundaries. To achieve the perpendicular condition the Lagrangian
relaxation experiment was re-run on the larger horizontal domain (now
[-6,6] 2 compared with
[-4,4]2 in Wilmot-Smith et al. 2009a).
The braided field is centered in the middle of the domain and the field
is close to uniform in the region external to the braid. Accordingly
the results presented here are shown for only the subsection of the
full domain in which the important dynamics occur, specifically
.
An isosurface of current in the initial state is given the left-hand image of Fig. 2. The current has large-scales (see also the upper-left hand image of Fig. 5
where contours of current in a horizontal cross-section are shown) with
two fingers of current extending vertically through the domain. Some
sample field lines further illustrating the nature of the field are
given in the right-hand image of the same figure. Although non-trivial,
the initial state has little magnetic energy in excess of the
homogeneous field (
in
).
The aspect ratio employed in the model is 1:8. Although this is larger
than that of many previous simulations it is smaller than a realistic
ratio for a coronal loop (1:50, say). In the configuration the poloidal
field components are small compared with the toroidal components so
that the field lines look almost straight. This level of braiding would
be observationally difficult to distinguish from a potential field.
To initialise the simulation the dimensionless plasma density (Sect. 2.1) is set at
throughout the domain and the internal energy as e=0.1.
The result is a plasma-
at t=0 that lies in the range
.
For the results described in the main section of this paper (Sect. 3) we
consider the early evolution of the system (up to t=14) with time measured in
units of the Alfvén time. A uniform resistivity of
has been taken and the effect of changing the resistivity is discussed at various points in the following text.
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Figure 3: Maximum absolute value of the current a); velocity b); Lorentz force c); and total kinetic energy d) in the domain with time for a sequence of decreasing uniform resistivies as indicated in the figures. |
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Figure 4:
Isosurfaces of current,
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3 Results
3.1 Basic properties
Figure 3 shows the maximum absolute values of the current, velocity and Lorentz force,
and the kinetic energy (
)
for the time interval (
)
under consideration. The domain taken in all cases is the central section,
,
of the full box. The variation in quantities is shown for a sequence of
uniform resistivities decreasing by over an order of magnitude,
specifically
and
.
The initially high maximum Lorentz force,
decreases
rapidly over the first few time units. Both the high value and the
decrease are artifacts of the method used to create the initial state.
The interpolation required to transfer the state to the Eulerian grid
(as described in Sect. 2.2)
results in some noise in the initial magnetic field and current
density. Some noise persists even with the application of a smoothing
algorithm to the vector potential for the magnetic field and this is
particularly noticeable in the Lorentz force rather than the magnetic
field and current density alone.
Thus whilst the final state of the Lagrangian relaxation experiment had
a very small maximum Lorentz force (specifically
), the initial state here is further from force-free. The decrease over
then arises though a smoothing of the noise in the initial state.
Turning now to the remaining quantities shown in Fig. 3 two primary features are found. Firstly a growth in the kinetic energy and maximum velocity for
occurs,
and this growth is independent of resistivity,
.
For the remaining time considered there is no significant change in
kinetic energy. Secondly, a linear growth in the current density occurs
for
.
The rate at which the maximum current density increases is higher for lower resistivity,
.
At t=12
the maximum current density is achieved; this maximum is higher for
lower resistivity but for all three resistivities the growth phase ends
at the same time.
The lack of dependence of kinetic energy on resistivity may
suggest an ideal instability is present. The subsequent linear growth
of current would then be a non-linear consequence of this instability
rather than its initial appearance. This growth is clearly dependent on
resistivity, being slower for higher values of .
Little is known about the non-linear phase of instabilities and such a
dependence may still be consistent with an ideal instability
with a non-linear phase damped by
.
Strong conclusions are clearly difficult to draw at this stage. An
additional consideration is that the implementation of the field on the
new grid has resulted in significant Lorentz forces in the initial
state. We return to these questions in Sect. 4 but now proceed to consider the the nature of the currents within the domain, now fixing
.
3.2 Formation of current layers
Figure 4 shows isosurfaces of current at 50% of the domain
maximum (
)
for a sequence of increasing
times (for the initial state see Fig. 2).
In the early stages (
)
the current diffuses slightly while maintaining its large
scales in the perpendicular direction. A symmetric evolution follows and after the phase of
current growth two current concentrations are present, centered at z=3.4 and z=-3.6. We call these the two ``initial current layers''.
The formation of these two initial current layers is best illustrated by considering a
horizontal cross section through the central plane (z=0). Figure 5
shows contours of the vertical component of current (
)
in that plane
at
t=0, 6, 12. The z-component is taken since it significantly dominates over the two horizontal
components, as evident in the shape of the current sheets (Fig. 4).
Note that in order to incorporate both sheets the cross-sectional plane chosen,
z=0, does not intersect the centre of either current sheet and
so the magnitude of current in this plane is somewhat low in comparison
to the domain maximum. The collapse of the two oppositely signed
large-scale fingers of current present in the initial state into two
thin current sheets of correspondingly the same sign is clearly shown.
Also evident is the formation of a weaker current envelope around the
braided flux, separating it from the uniform background field.
Cross-sections of
in the horizontal planes through the centres of the
two current sheets are shown in the final two images of Fig. 7.
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Figure 5: Contours of the vertical component of the current, Jz, in the central plane, z=0, at increasing times, illustrating the formation of the two initial current layers (first three images). The lower right-hand image shows contours of integrated parallel current along field lines in the initial state at the central plane z=0. From this quantity field line tracers for the locations of initially high integrated parallel current have been determined and marked in crosses in the previous frames, as described in the main text. |
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3.3 Predictors of current layers
The field lines along which these two current sheets form turns out to be well
predicted by the regions of high integrated parallel current in the initial state.
In resistive MHD the integrated parallel current is related to the integrated parallel
electric field via the relation
(in the case of a uniform resistivity
,
as considered here). The integrated parallel electric field is a key
quantity for 3D magnetic reconnection; for a localised reconnection
region the maximum value of
determines the rate of reconnection.
Shown in the lower right-hand panel of Fig. 5
are contours of the integrated parallel current,
,
in the initial state, t=0, where the path of the integral is taken over
magnetic field lines. Again the contour is shown in the central plane (z=0).
To obtain this contour map, field lines have been integrated through
1602 grid of points covering the domain
,
z=0.
Two peaks in the quantity are present and the structure is quite different to that of
the current itself in the initial state. We now identify those field lines in this tracing procedure for which the value of
is greater than or equal to 75% of the domain maximum, noting the locations where they intersect the lower boundary
(where the locations of the field lines are held fixed). For the sequence of
times of Fig. 5 we trace field lines starting from those
locations on the lower boundary up through the domain and mark with a cross
in that same figure their point of intersection with the z=0
plane. Here the difference in colours indicates field lines with
positive (black crosses) and negative (white crosses) integrated
parallel current, although this distinction is made only to facilitate
identification of the locations. It is found that these field lines,
traced from the initial locations of high integrated parallel current,
are good indicators for the locations of formation of the two current
layers. Since the flux on the lower boundary is held fixed these may be
identified as the same field lines for as long as the evolution remains
ideal. Whilst the evolution will be ideal only during the early stages
of the simulation it is clear that the tracers do, nevertheless,
provide a useful predictor for the locations of current sheet
formation.
Locations of high integrated parallel current are not a
commonly used indicator for current sheet formation. Indeed it is
quasi-separatrix layers (QSLs), regions where the field-line
connectivity varies strongly (Priest & Démoulin 1995)
that are widely thought of as likely sites of current sheet formation.
To identify QSLs (as well as their intersections, hyperbolic flux tubes
or HFTs) the squashing factor (Titov et al. 2002) is used. Usually denoted by Q,
the squashing factor is an indictor of field line connectivity and
takes on high values in regions where the field line mapping is
strongly distorted. Regions of high Q outline QSLs. As discussed in Wilmot-Smith et al. (2009b), the braided magnetic field taken as the initial state here contains
several QSLs. Contours of the squashing factor, Q, in the central plane (z=0) are shown in
Fig. 6 at t=0 (upper) and t=12 (lower). For the calculation again 1602 points over the region
have been used, a number comparable to the grid resolution.
![]() |
Figure 6:
Contours of the squashing factor, specifically
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Several regions of high Q are present in both snapshots.
The two regions of highest Q in the initial state are associated with the two initial
current layers at the later time, that is, the current sheets have formed along QSLs of the field.
At the same time, several other regions of high Q are present (both at t=0 and
t=12) that are not associated with any particular current features.
For example, in the initial state there are eight distinct regions where
is greater than 85% of its maximum value but
only two of these regions correspond to particular current features at t=12.
These results suggest that the integrated parallel electric current and the squashing
factor could be used in conjunction for predicting current sheet formation more accurately
than by using Q alone.
3.4 Plasma flows and reconnection
We move now to consider the nature of the plasma flows within the domain. First recall that the low values of the plasma beta (Sect. 2) imply the dynamics will be dominated by the Lorentz force rather than the gas pressure. With both the magnetic field and the current having stronger vertical than horizontal components we have that the Lorentz force is primarily in the horizontal direction and so, similarly, are the plasma flows. The Lorentz force drives a flow with a dipolar structure in the six regions of initially strongest current with direction dependent on the sign of the twist in that region. An illustration of such a flow is shown for the plane z=3.4 (a negative twist region corresponding to one of the sites of current sheet formation) in the first (upper left) image of Fig. 7.
The next four images in that figure show the development of the flow for a sequence of increasing times up to the point of maximum current. In the sequence the length of the arrows indicating flow direction have been normalized to each image. The background contours show the vertical component of current in that plane with the colour scale normalised to the current at t=12 (lower-left-hand image), given by the first colour-bar. The sequence clearly shows the association of the stagnation part of the dipolar flow with the current intensification. The out-flowing plasma from the location of stagnation sets up a counter-flow to the initial dipolar structure leading to oppositely directed flows on either side of the weak enclosing current sheath. The final image shows plasma flows and current in the plane z=-3.6 at t=12. This cross-section is across the second current sheet and shows the naturally expected inversion of the flow direction. The result is that the global flow structure is dominated by rotational components the direction of which varies both vertically along the structure and on either side (y>0, y<0) of the braided field.
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Figure 7: Arrows indicate plasma velocities [Vx,Vy] in horizontal cross-sectional planes, superimposed on the vertical component of current, Jz. The first five images show structures about the upper current sheet (taking the plane z=3.4) while the final image is for the lower current sheet (taking the plane z=-3.6). The images are given at various times, as indicated in the figures. |
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As already indicated by Fig. 3, at t=12 the maximum magnitude of the plasma
flows (
)
is a significant fraction of the Alfvén speed
(
). These strongest flows are associated with the
global rotation and not with outflows from the two initial current layers (which have an
associated
). However, we do expect magnetic reconnection
to be taking place across these current sheets and so proceed to consider the nature of this
reconnection. For this we concentrate again on the initial current layer centered at z=3.4
(a similar situation occurs about the other current layer)
and focus on the structure of the field and flows in the plane perpendicular to the
magnetic field at the location of maximum current magnitude,
.
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Figure 8: Here various quantities are considered in the plane perpendicular to the magnetic field at the region of maximum (negative) current. (Left) Arrows indicate plasma flows in that plane while the coloured contours show the out-of-plane component of vorticity. (Right) Arrows indicate magnetic field components in the plane superimposed on contours of the out-of-plane current. The colour table for each is blue-yellow for negative - positive. |
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In the left-hand image of Fig. 8 we indicate in more detail the nature of the stagnation flow about the current sheet. Superimposed on the background are contours of the out-of plane component of vorticity, i.e. the component of vorticity in the direction of the magnetic field at the location of maximum current. The vorticity shows a quadrupolar configuration around the current sheet.
Perhaps more of a surprise is the structure of the magnetic field in the region. The right-hand image of Fig. 8 shows the components of the magnetic field in the cross-sectional plane under consideration. The magnetic field is shown to have an elliptic configuration about the current sheet, i.e. about the reconnection region. This finding contrasts with the two-dimensional picture of magnetic reconnection under which the process can only occur at a hyperbolic (X-type) null-point of the magnetic field. In three-dimensions a much wider variety of possible reconnection sites exist. Reconnection may be associated with 3D null-points (e.g. Lau & Finn 1990; Priest & Pontin 2009), magnetic separators (which connect two null-points, e.g. Longcope & Cowley 1996; Pontin & Craig 2006; Haynes et al. 2007), or may occur in the absence of any such topological features (Schindler et al. 1988), the latter sometimes termed ``non-null reconnection''. In particular, the local magnetic field structure need not be hyperbolic but may be elliptic (Hornig & Priest 2003), as recently found in some 3D numerical simulations of reconnection. For example, Wilmot-Smith & De Moortel (2007) considered reconnection occurring along a quasi-separator and found an elliptic field structure in perpendicular cross-sections. The separator configuration of Parnell et al. (2010) showed an elliptic structure along a significant length of the separator under consideration. Parnell et al. (2010) also discussed the separator case theoretically, concluding an elliptic configuration would be a generic situation given a sufficiently strong current density along the separator. Our findings demonstrate that an elliptic field configuration may be present about reconnection sites in the non-null case. Additionally, tracking these field lines back to the initial setup, an elliptic perpendicular field configuration is again found indicating that locally hyperbolic structures are not necessary for current intensification. As previously discussed, the squashing factor (Q) at the two reconnection sites is very high which demonstrates a further point; regions of highest-Q within a domain may have a locally elliptic field configuration.
4 Nature of instability
Evidently the initial magnetic field configuration is not in a stable equilibrium. Since an exact equilibrium of the ideal relaxation code employed in Wilmot-Smith et al. (2009a) is known to be linearly ideally stable, the lack of stability could arise from one of a number of factors:
- 1.
- A resistive instability. The relaxed state of Wilmot-Smith et al. (2009a) contains small scales in the integrated parallel current. Small enough scales (for a given resistivity) in this quantity are incompatible with a resistive equilibrium.
- 2.
- A non-linear ideal instability not previously found in the ideal evolution of Wilmot-Smith et al. (2009a) since that evolution only guaranteed linear stability.
- 3.
- A lack of equilibrium in the initial state either since:
- The path to equilibrium of the ideal relaxation is fictitious
and an exact equilibrium had not been reached. Numerical difficulties
(described in Pontin et al. 2009) result in the final state of the ideal relaxation having
, i.e. is not perfectly force-free. Thus the final state of Wilmot-Smith et al. (2009a) need not be stable (or, indeed accessible via a real MHD relaxation dynamics).
- The technique used to transfer the initial state between codes has perturbed the magnetic field further from force-free.
- The path to equilibrium of the ideal relaxation is fictitious
and an exact equilibrium had not been reached. Numerical difficulties
(described in Pontin et al. 2009) result in the final state of the ideal relaxation having
With these considerations in mind we examine the evolution of only
the middle section of the initial state of the braided magnetic field
that is, we cut-out the section of the field in the above described
experiment that lies in
,
at t=0. This field is inserted as an initial condition in a new run, now keeping the flux fixed on
,
the new upper and lower boundaries of the domain. To maintain
consistency we use the same resolution in the horizontal direction 3202 and a similar effective resolution in the vertical direction, 128 cells over
.
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Figure 9:
(Upper) Vertical component of current in the central plane (z=0) at t=20 for the an MHD evolution on only the central section,
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In the evolution of this new ``middle-third'' field we find the system
adjusts from its initial condition (with zero plasma velocity) to an
approximate equilibrium in which the current structure is qualitatively
similar to that of the initial state.
To illustrate, contours of current in the central plane (z=0) are shown at t=20 in Fig. 9 (upper panel) which may be compared with Fig. 5 (upper left) where the corresponding contours in the initial state, t=0, are shown. The maximum current in the domain is shown as a solid line in the upper panel of Fig. 9 (left-hand axis), the kinetic energy integrated over
,
as a dashed line (left-hand axis) and the maximum Lorentz force within
the domain as a dot-dashed line (right-hand axis). As in the evolution
of the full field, the Lorentz force in this initial state is high as a
result of the transfer between grids and decreases rapidly over the
first few time units. However, the Lorentz force subsequently
stabilises at a low value as the system readjusts to equilibrium. The
result that the middle section of the braid alone is in an equilibrium
state suggests that an instability is present in the full braided field
(rather than a lack of equilibrium due to numerical artefacts).
Furthermore the instability is of a long-wavelength type with the full
structure of the braided field being key.
Returning to the evidence of Figure 3, during the time
where
the instability is clear in the current growth there is only a very
slight dependence of the maximum velocity within the domain on
resistivity and no increase in kinetic energy. This effect may be due
to the confining nature of the strong background field external to the
braided field structure which results in a deflection of the outflowing
plasma around the boundary
of the braided field (see Fig. 7). Prior to
no dependence of the flow on resistivity is seen suggesting an ideal
dynamics where the system is adjusting to the distance from
equilibrium. The current growth in
does have a clear separation according to resistivity
.
However the increase is linear (rather than exponential) suggesting a
dominant non-linear phase and the growth is slower for higher
resistivity suggesting the non-linear phase is damped by resistive
effects.
5 Conclusions
A previous paper (Wilmot-Smith et al. 2009a) considered the ideal relaxation of a braided magnetic field towards a force-free equilibrium. Here we have taken the final state of that relaxation and used it as an initial condition for a full resistive MHD simulation.
The braided field is not in a stable equilibrium; two thin current sheets form after a short time (around a quarter of the time for an Alfvén wave to cross the numerical box in the vertical direction). The linear rate of increase of current density and the maximum strength of the current is found to increase with decreasing resistivity although the evolution of the total kinetic energy in the domain is independent of resolution. We conclude that the instability is possibly an ideal one although the details of how it occurs have not been determined. The wavelength of the instability was tested by considering the MHD evolution of the middle third section of the braid alone and this new field was found to be stable. Hence a long-wavelength dependence is implied.
The initial configuration contains many regions of high squashing factor, Q, corresponding to quasi-separatrix layers. Plasma flow across QSLs is often thought likely to lead to current sheet formation. The two current sheets that form here do align with two of the highest regions of Q although the remaining regions of high Q do not correspond to any particular current features. The locations of the two current layers are well predicted by the peak values of the integrated parallel current in the initial state. In the central plane this quantity shows two clear peaks and it is at these regions that the two current layers later form.
These two current layers correspond to reconnection sites. Perpendicular to the layers the magnetic field has an elliptic structure, an admissible property of three dimensional reconnection that has only recently been found. The flow about the reconnection sites is of an asymmetric stagnation type, although large-scale rotational flows dominate the global structure; the effect of reconnection is not a strong acceleration of the flow but a more subtle untwisting process leading to the change in magnetic field topology.
The longer-term evolution of the system will be considered in a future paper.
Appendix A: Appendix
Equation (7) can be derived from an evolution for the vector potential of a frozen-in magnetic field:
This equation is equivalent to the Lie-derivative for a differential one-form,


![]() |
(A.2) |
and we can express



The authors would like to thank Nasser Al-Salti for helpful suggestions and Klaus Galsgaard for the use of his code. A.W.S. and G.H. acknowledge financial support from the UK's STFC. The simulations described were carried out on the STFC and SFC (SRIF) funded linux clusters of the UKMHD consortium at the University of St Andrews.
The Transition Region and Coronal Explorer, TRACE, is a mission of the Stanford-Lockheed Institute for Space Research (a joint program of the Lockheed-Martin Advanced Technology Centre's Solar and Astrophysics Laboratory and Stanford's Solar Observatories Group), and part of the NASA Small Explorer program.
References
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All Figures
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Figure 1: TRACE coronal loops. ( Left) A large-scale tangled configuration and ( right) apparently smooth loops. |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Initial state of the simulation:
( left) Isosurface of current
|
Open with DEXTER | |
In the text |
![]() |
Figure 3: Maximum absolute value of the current a); velocity b); Lorentz force c); and total kinetic energy d) in the domain with time for a sequence of decreasing uniform resistivies as indicated in the figures. |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Isosurfaces of current,
|
Open with DEXTER | |
In the text |
![]() |
Figure 5: Contours of the vertical component of the current, Jz, in the central plane, z=0, at increasing times, illustrating the formation of the two initial current layers (first three images). The lower right-hand image shows contours of integrated parallel current along field lines in the initial state at the central plane z=0. From this quantity field line tracers for the locations of initially high integrated parallel current have been determined and marked in crosses in the previous frames, as described in the main text. |
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Contours of the squashing factor, specifically
|
Open with DEXTER | |
In the text |
![]() |
Figure 7: Arrows indicate plasma velocities [Vx,Vy] in horizontal cross-sectional planes, superimposed on the vertical component of current, Jz. The first five images show structures about the upper current sheet (taking the plane z=3.4) while the final image is for the lower current sheet (taking the plane z=-3.6). The images are given at various times, as indicated in the figures. |
Open with DEXTER | |
In the text |
![]() |
Figure 8: Here various quantities are considered in the plane perpendicular to the magnetic field at the region of maximum (negative) current. (Left) Arrows indicate plasma flows in that plane while the coloured contours show the out-of-plane component of vorticity. (Right) Arrows indicate magnetic field components in the plane superimposed on contours of the out-of-plane current. The colour table for each is blue-yellow for negative - positive. |
Open with DEXTER | |
In the text |
![]() |
Figure 9:
(Upper) Vertical component of current in the central plane (z=0) at t=20 for the an MHD evolution on only the central section,
|
Open with DEXTER | |
In the text |
Copyright ESO 2010
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