The Astrophysical Journal, 595:506–516, 2003 September 20 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II. DYNAMIC NUMERICAL MODEL K. Galsgaard1 Niels Bohr Institute for Astronomy, Physics, and Geophysics, Julie Maries vej 30, 2300 Copenhagen Ø, Denmark;
[email protected]
V. S. Titov Theoretische Physik IV, Ruhr-Universita¨t Bochum, 44780 Bochum, Germany
and T. Neukirch School of Mathematics and Statistics, University of Saint Andrews, Saint Andrews, KY16 9SS, Scotland Received 2003 April 1; accepted 2003 June 2
ABSTRACT In this paper we present the results of a series of numerical experiments that extend and supplement the recent analytical investigations by Titov et al. of the formation of strong current layers in coronal magnetic fields containing hyperbolic flux tubes (HFTs). The term ‘‘ hyperbolic ’’ refers to the special geometrical properties of the magnetic field, whereas the topology of the field is simple; i.e., there are no magnetic null points and separatrix lines or surfaces associated with them inside the coronal volume. However, the field lines passing through a hyperbolic flux tube show a large variation in the mapping between their photospheric endpoints. On the basis of analytical estimates, it has been suggested by Titov et al. that HFTs are preferred locations for the formation of strong current layers in coronal magnetic fields with trivial topologies, provided the driving motions on the photospheric boundary are of a special type. Such motions must have shearing components that are applied across narrow HFT feet as if trying to twist it. This system of motions is then capable of causing a pinching deformation of the HFT by a sustained stagnation point flow inside the HFT. The numerical experiments presented in this paper clearly confirm this suggestion. HFTs are generic features of geometrically complex but topologically trivial magnetic fields, and therefore our results are very important for understanding magnetic reconnection in such fields, since reconnection is occurring preferentially at locations with strong current densities. Subject headings: Sun: flares — Sun: magnetic fields
place at special locations in a magnetic field without a magnetic singularity (Hesse & Schindler 1988; Otto 1995; Priest & De´moulin 1995; Galsgaard & Nordlund 1996; Inverarity & Titov 1997; Hornig & Rasta¨tter 1997; Titov, De´moulin, & Hornig 1999). Priest & De´moulin (1995) used an ideal kinematic approach to show that certain imposed boundary motions can generate large field line velocities at locations where the field line mapping between two boundaries of the considered volume changes particularly rapidly. On the basis of the examples presented in their paper, Priest & De´moulin (1995) concluded that these locations have a layer-like spatial structure similar to separatrices, and therefore they called them ‘‘ quasi-separatrix layers ’’ (QSLs). Under ideal conditions the field line velocity is identical to the plasma velocity perpendicular to the magnetic field. Priest & De´moulin (1995) suggested that under the conditions discussed by them, the field line velocity would eventually become larger than the local Alfve´n velocity. As the plasma velocity cannot exceed the local Alfve´n velocity this implies that plasma velocity and field line velocity must decouple in this case, which in turn implies the existence of some nonideal process, viz., reconnection. Therefore, Priest & De´moulin (1995) suggested that QSLs are generically favorable places for magnetic reconnection. The limitation of their kinematic analysis is that it does not take into account the dynamical changes of the magnetic field and the corresponding changes of the field line structure. To investigate this Galsgaard (2000) made a series of numerical experiments using the full set of MHD
1. INTRODUCTION
An important aspect of the dynamical evolution of a magnetized plasma is the process of magnetic reconnection. Magnetic field penetrates most of the universe, and in many regions the magnetic forces are small compared with the plasma forces. In most of space the magnetic field is frozen to the plasma, and it is advected around by the generally turbulent plasma motions. This tangles and amplifies the magnetic field, and if the magnetic field were not allowed to change its connectivity, magnetic tension forces would amplify to a level at which they could suppress the turbulent plasma motions. This is not the case in general; magnetic field lines are allowed to diffuse through the plasma and change their connectivity when large enough electric current concentrations are reached locally. To know how and where magnetic reconnection proceeds is important for understanding the dynamical evolution of a stressed magnetic field. In three dimensions this is a complicated issue, with several possible magnetic topologies being able to undergo magnetic reconnection. Extending our knowledge from two-dimensional magnetic reconnection makes three-dimensional magnetic null points likely spatial locations for driving reconnection (Lau & Finn 1990; Craig et al. 1995; Priest & Titov 1996; Rickard & Titov 1996; Galsgaard, Rickard, & Reddy 1997). Apart from reconnection at null points and their connections, it is known that magnetic reconnection may also take 1 Permanent address: School of Mathematics and Statistics, University of Saint Andrews, Saint Andrews, KY16 9SS, Scotland.
506
MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II. equations, and these showed that the presence of a QSL alone is not a sufficient condition for reconnection. It has also recently been shown that the first definition of a QSL is not invariant to the direction of the field line mapping (Titov et al. 1999) and that they are geometrical objects rather than topological ones that can be removed by suitable smooth deformations of the magnetic configuration (Titov & Hornig 2002). In a quadrupolar magnetic field configuration formed by two bipolar regions, the separation of flux on the photospheric plane can be understood by the intersection of two QSLs that in the limit of point sources converges to two fan surfaces (or separatrix surfaces) intersecting along a separator field line. This configuration is known as a hyperbolic flux tube (HFT; Titov et al. 2002). It is known that the separator line is a favorable site for current accumulation when the system is perturbed (Sweet 1969; Gorbachev & Somov 1988; Lau & Finn 1990; Galsgaard, Priest, & Nordlund 2000; Longcope 2001). It can therefore be expected that the same region in a field defined by distributed but wellpronounced sources without nulls in the domain will also be a preferred location for current accumulation. The present paper is the second in a series of three papers (Titov, Galsgaard, & Neukirch 2003, hereafter Paper I) in which we intend to clarify the physical conditions under which strong current layers form in HFTs. In all papers we investigate a particular type of magnetic field that contains an HFT. This magnetic field is subjected to different boundary perturbations inherent to the generic type of photospheric motions, and the response of the HFT to these perturbations is studied. Paper I does this using an analytical approach, which shows that exponential growth of the current can be reached when the structure is stressed in a way in which a stagnation flow is initiated around the separator-like line by the perturbation of the magnetic field. This paper adopts the full set of MHD equations and solves them numerically. In Paper III (T. Neukirch, K. Galsgaard, & V. S. Titov 2003, in preparation) we will use a Lagrangian relaxation code to investigate the nonlinear force-free solutions of the quasi-static evolution of an HFT and the current buildup due to slow changes in the boundary conditions. Together these papers intend to show that the formation of strong current layers in HFTs (pinching) is the generic process and therefore very important for our understanding of reconnection in general and of solar flare phenomena in particular. The layout of the paper is as follows. In x 2 we introduce the initial conditions and the perturbations. The numerical approach is described in x 3. Section 4 discusses the experiments and their implications for our understanding of current accumulation in HFTs. Finally, in x 5, the results are summarized and conclusions are drawn. 2. EXPERIMENT SETUP
Our aim is to test the hypothesis that inside an HFT a strong current density only develops if the HFT is stressed by a specific type of boundary motions. The testing is done by performing a set of numerical experiments. For the experiments we construct a magnetic field containing an HFT by following the guidelines provided in Paper I. This magnetic field is a potential field that is periodic in the xand y-directions and nonperiodic in the z-direction. The numerical box extends over one wavelength of periodicity in
507
Fig. 1.—Flux pattern on the driving boundaries. The pattern in the two opposite boundaries are rotated 90 relative to one another.
the x- and y-directions and is bounded by two boundaries at z ¼ 0 and 1 (in suitably normalized units) in the z-direction. On the two z-boundaries we impose the z-component of the magnetic field. On each boundary we combine two Gaussian flux concentrations (normalized to peak magnitude 1) with a weak constant background magnetic field of magnitude 0.1 (Fig. 1). The lines connecting the two Gaussian flux concentrations on the top (z ¼ 1) and bottom (z ¼ 0) boundaries have an angle of 90 with respect to each other. The periodic potential field corresponding to these boundary conditions has an HFT in the center of the domain in the periodic direction. It also shows the characteristic strong variation in the field line mapping from one footpoint to the other. Halfway between the two boundaries the potential magnetic field is nearly uniform with an average field strength of 0.16. Due to the periodic boundary conditions the HFT configuration is repeated infinitely often in the xand y-directions. This facilitates the numerical setup enormously, but it also introduces additional complexity into the field line connectivity compared with the nonperiodic field studied in Paper I (Fig. 2; cf. also Fig. 2 of Paper I). This additional complexity causes the appearance of secondary locations of current accumulation inside the numerical domain. This magnetic field contains a number of regions where the field line mapping between the two boundaries has a large gradient. A more detailed explanation of this for the nonperiodic case is given in Paper I. The structure of the magnetic field line mapping is shown in Figure 3, where field lines are traced from a ring around the center of one flux source from each of the two z-boundaries. Notice how the flux from the source on one boundary mostly connects to weak field regions on the opposite boundary and especially how the field lines around the central separator line diverge from this region. Strong currents would therefore naively be expected to accumulate here as the system is stressed.
508
GALSGAARD, TITOV, & NEUKIRCH
Vol. 595
Fig. 2.—Structure of the potential magnetic field generated by two grids of fictive point sources of opposite signs. The grids are periodic in the x-, y-directions and located at a given distance outside the boundary planes z ¼ 0 and 1 so that each periodic cell contains two positive and two negative sources. Right: Field lines forming the ‘‘ skeleton ’’ of the studied HFT. Left: Fictive null points and separatrix lines belonging to the front boundary of the cell. Because of the symmetry of the grids there are similar nulls and separatrices at the back boundary. At the lateral boundaries, they are also present and obtained from the shown ones by rotating them 90 around the center of the cell with subsequent flipping of the nulls and field line arrows about the plane z ¼ 0:5. The sources are represented by small dark gray circles; the corresponding flux concentrations on the top and bottom boundaries of the right cell are represented by large solid ( positive) and dashed (negative) circles, while the fan separatrix planes at the nulls are represented by large light gray circles.
To investigate how HFTs are accumulating electric current as soon as the magnetic field around them is stressed, we have set up a number of experiments. In these experiments, the flux patterns on the two z-boundaries are advected in different directions using a sinusoidal ‘‘ shear ’’ motion and with different driving amplitudes. The imposed driving profile has a wavelength equal to the extent of the numerical domain and is, like the flux concentrations, rotated by 90 between the two boundaries. The shear motion is imposed such that the sources are advected in a direction perpendicular to the line initially connecting the sources. Figure 4 shows the structure of the driver relative to the location of the two sources. The experiments are carried out by changing the sign of the amplitude of the driving
velocity on the bottom boundary and by changing the amplitude of the driving. For the initial conditions the plasma is assumed to be at rest, and the plasma is less than unity away from the sources. In the units of the code, described below, this is achieved by setting the thermal energy and density to 0.1. This provides an Alfve´n velocity in the central part of the domain of 0.49, while the uniform sound speed becomes 0.33. This setup gives an Alfve´n crossing time of about 2 time units. In the same units the maximum driving velocity is 0.1 for the standard experiment, which is about 32% of the Alfve´n velocity at the background field at the source planes and 2.9% of the peak Alfve´n velocity of the flux concentrations.
No. 1, 2003
MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II.
509
Fig. 4.—Structure of the driver relative the locations of the two flux sources. The full line represents the starting points of tracer particles advected with the driving flow. The dashed line shows their position at a later time. The concentric circles show the initial locations of the two sources.
Fig. 3.—Field line structure of the initial potential magnetic field. Field lines are traced from circles around the centers of two of the flux sources. The images show two different orientations of the three-dimensional domain.
3. NUMERICAL APPROACH
To follow the dynamical evolution of the magnetized plasma as the flux sources on the two boundaries are advected in time, we solve the full set of MHD equations in a three-dimensional Cartesian domain, D
@ ¼ @t
x ðuÞ
;
@u ¼ x ðuu þ Þ P þ J µ B ; @t @e ¼ x ðeuÞ P x u þ QJoule þ Qvisc ; @t @B ¼ µE; @t E ¼ ðu µ BÞ þ J ; D
D
D
D
D
D
J¼
µB;
ð1Þ ð2Þ ð3Þ ð4Þ
with density , velocity u, thermal energy e, magnetic field B, electric field E, magnetic resistivity , electric current density J, viscous stress tensor , gas pressure P ¼ eð 1Þ, viscous dissipation Qvisc , and Joule dissipation QJoule , respectively. An ideal gas with ¼ 53 is assumed. The equations are nondimensionalized by setting the magnetic permeability at l0 ¼ 1 and the gas constant at R0 ¼ l (the mean molecular weight). One time unit is equivalent to the Alfve´n transit time of a unit length at which both jBj and are set to 1. The equations are solved using staggered grids. A sixthorder method is applied to derive the partial derivatives, and a fifth-order method is used for doing interpolation. Viscosity and magnetic resistivity are both handled using a combined second- and fourth-order method with a discontinuous shock-capturing mechanism to provide the highest possible spatial resolution for the given numerical resolution. The solution is advanced in time using a third-order predictor-corrector method.2 4. THE EXPERIMENTS
The driving velocity on the two boundaries is slowly increased from zero at the start of the experiment toward its maximum. When this value is reached, the amplitude is maintained constant for the rest of the experiment. The impact on the magnetic field is to launch an Alfve´n wave from the boundaries that propagates into the domain. This wave creates a weak current concentration at its front. As
ð5Þ ð6Þ
2 A basic description of the code is available in Nordlund & Galsgaard (1997) and at http://www.astro.ku.dk/kg.
510
GALSGAARD, TITOV, & NEUKIRCH
driving is imposed on both boundaries the simple wave fronts soon become complicated as they interact; they are reflected from the boundaries and propagate with different speeds depending on their spatial location. As the flux sources are systematically advected all the time, the orientation of the field lines connecting the two boundaries slowly changes. This systematic stretching and bending of the magnetic field lines generates different direct current systems that depend on the imposed driving profile. The issue here is to understand where in the magnetic field configuration currents tend to concentrate and why the accumulation takes place at these particular locations. Six experiments are discussed here, in which the differences are in the sign of the driving amplitude on the bottom boundary, the driving amplitude itself, and the viscosity. Table 1 summarizes the main parameters of the experiments, showing the naming of the experiments, the driving amplitude, the sign and scale value of the amplitude on the bottom boundary, relative viscosity, and the time of termination of the experiment. The negative/positive value of sign in Table 1 refers to the combined effect of the driving pattern on the two z-boundaries. A negative value implies a ‘‘ turning ’’ motion of the central HFT structure, while a positive value represents a ‘‘ twisting ’’ motion of the HFT structure. The implications of these boundary flows are illustrated in Figure 5. The result from Paper I is that only the twisting motion of the HFT generates a fast growth of current along the HFT. By relative viscosity it is implied that the coefficients controlling the nonlinear viscosity in the code are increased relative to the minimum values used in five of the experiments. In the ‘‘ turn ’’ experiment, two equally strong current structures are generated away from the main HFT located at the center of the domain, while in the ‘‘ twist_1–3 ’’ cases only one strong current concentration forms in the center of the domain initially, followed by secondary current concentration later in the experiment. This is in full agreement with the results in Paper I. The ‘‘ shear ’’ experiment shows a completely different current accumulation. As the driving in this case is a linear combination (the mean value) of ‘‘ turn ’’ and ‘‘ twist_1,’’ in which only the footpoints at the top boundary are advected, a simple shear of the HFT is provided. Paper I suggests that this should generate a current structure that has features of both the ‘‘ turn ’’ and the ‘‘ twist_1 ’’ experiments. This is further confirmed by ‘‘ twist_4,’’ which is a different linear combination of ‘‘ turn ’’ and ‘‘ twist_1 ’’ (0.4 times ‘‘ turn ’’ and 0.6 times ‘‘ twist_1 ’’) and produces a weaker twist of the HFT than ‘‘ twist_1.’’ In shear no significant current accumulates, while in ‘‘ twist_4 ’’ current grows in the central region, as in ‘‘ twist_1,’’ but with a much slower rate. Figure 6 shows the current distribution of the z-current in the ðx; yÞ-plane halfway between the two driving boundaries TABLE 1 General Data for the Experiments Experiment
Vdrive
Sign
Viscosity
End Time
‘‘ turn ’’..................... ‘‘ twist_1 ’’ ................ ‘‘ twist_2 ’’ ................ ‘‘ twist_3 ’’ ................ ‘‘ twist_4 ’’ ................ ‘‘ shear ’’ ...................
0.1 0.1 0.025 0.025 0.1 0.1
1 1 1 1 0.2 0
1 1 1 10 1 1
14.1 13.7 31.7 14.6 12.2 12.1
Vol. 595
Fig. 5.—Effect of the ‘‘ turn ’’ and ‘‘ twist ’’ driving on the structure of the HFT.
(top) at time 7 (shear 0.7), where the shear is defined as Vdrive t and at the end (bottom) of the ‘‘ turn,’’ ‘‘ twist_1,’’ and ‘‘ shear ’’ experiments. From the panels it is seen how the current density concentrates in three significantly different patterns depending on the sign and amplitude of the driver on the lower boundary at the two different times of the experiments. At time 7 (shear 0.7) it is found that two symmetric current concentrations are formed away from the center of the ðx; yÞ-plane in ‘‘ turn,’’ while a strong current concentration is formed at the center of the plane in ‘‘ twist_1.’’ In ‘‘ shear ’’ only a slightly enhanced current concentration is formed at the center of the ðx; yÞ-plane, while a large-scale current structure is generated as a consequence of the simple shear motion imposed by the driving on one boundary only. Toward the end of ‘‘ twist_1,’’ the current structure has changed, and the initially secondary current concentrations that are active in ‘‘ turn ’’ are as strong as the current layer in the central region of the domain. This change in location of the current accumulations occurs for two reasons. First, the magnetic structure defining the central current layer starts to undergo a magnetic reconnection that diffuses the current faster than it can be built up. Second, the continued driving eventually starts to stress the secondary regions of strong divergence in the magnetic field
No. 1, 2003
MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II.
511
Fig. 6.—Distribution of the current component along the z-direction in the ðx; yÞ-plane halfway between the two driving boundaries for ‘‘ turn,’’ ‘‘ twist_1,’’ and ‘‘ shear.’’ The top panels represent t ¼ 7 (shear 0.7), and the bottom panels show the current distribution at the end of the experiments. The main panel shows a shaded surface plot, while the small image at the bottom left of each frame shows the current structure as a scaled image with black being negative and white positive. Each image is scaled to the minimum–maximum range in the individual data sets.
line mapping, and current starts to build up at these locations—a clear effect of the imposed periodicity in the ðx; yÞplane. The extended loop structures in the current concentrations that are found in the later phases of ‘‘ twist_1 ’’ are clear indications of a pileup of the current generated by the reconnection jets originating from the central current sheet. In Figure 7 the time-dependent increase of the maximum and minimum z-current are shown for five experiments as a function of time. Experiment ‘‘ twist_3 ’’ is not shown, as its curve is nearly identical to that of ‘‘ twist_2.’’ The bumps in the curves are due to the wave fronts interacting with each other as they bounce between the boundaries. Here it is noticed that the peak currents in turn have the same amplitude but with opposite signs. Comparing the ‘‘ twist_1–2 ’’ shows that with decreasing the driving velocity by a factor of 4 the current strength is decreased by nearly the same factor. This implies that there is a linear relation between the transport distance of the magnetic footpoints and the magnitude of the current density at early stages of the evolution, when the driving velocity is below the Alfve´n velocity in the center of the domain (see next paragraph). It is noticed that at later times the z-component of the current density grows faster for a slower driving than for a faster one characterized by the same resulting shear distance. This is expected, since
Fig. 7.—Time-dependent positive/negative peak value of jz in the ðx; yÞplane halfway between the two driving boundaries. Top full line is ‘‘ turn ’’; the dashed line and dot-dashed line are the positive and negative current from ‘‘ twist_1,’’ with the dot-dot-dashed line representing ‘‘ twist_2 ’’; the dotted line is ‘‘ twist_3 ’’; the bottom full line is ‘‘ shear, ’’ and the long-dashed line represents ‘‘ twist_4.’’
512
GALSGAARD, TITOV, & NEUKIRCH
in the last case the magnetic field has a longer time for responding to the imposed boundary stress. In addition, this indicates that the resistive effects are negligible for restraining the current growth in early stages of the experiments. Comparing the evolution of the peak current density in ‘‘ twist_2 ’’ and ‘‘ twist_3 ’’ shows that it is not effected significantly by the 10 times increase of the viscosity. The parameters used in ‘‘ twist_1 ’’ provide a representative HFT pinching, and it is therefore the only experiment whose results will be discussed below in detail. In Paper I two predictions of the current growth in the center of the HFT halfway between the boundaries were given in equation (25) and equation (41) for the kinematic and, respectively, force-free approximations of the development of the perturbed magnetic structure. For convenience they are repeated here: Bk 2t h 1þ ð7Þ jz ðkinematicÞ e l0 2hL Bk h 1þ jz ðforce-freeÞ e2t l 2hL "0 2 # hl l sh sh 1 þ e2t 0:91 þ 0:57 : ð8Þ Bk L In Paper I, a hyperbolic tangent driving profile has been used on both boundaries with lsh being the scale length of variation of this velocity profile and Vdrive being its amplitude. Furthermore, t ¼ Vdrive t=ð2lsh Þ, and h determines the strength of the two-dimensional magnetic x-point in the ðx; yÞ plane halfway between the two z-boundaries for the initial magnetic field (see eq. [3] in Paper I), and Bk is the zcomponent in the same plane (see eq. [4] in Paper I). The various coefficients in these expressions only depend on the initial magnetic field and the imposed driving profile. The current in the analytical solutions is found to grow exponential with t. For small t the exponential factor e2t is replaced by 2 sinhð2tÞ. This correction is essential for the comparison with our simulations, because the time t is small compared with the driving time of the system. The current
growths for these two estimates are presented along with the peak current densities in ‘‘ twist_1 ’’ and ‘‘ twist_2 ’’ in Figure 8. It shows that the growth of these values in the numerical experiments closely follows the kinematic profile until a shear distance of 0.7 is reached. After that the limited numerical resolution comes into play by effectively enhancing the resistivity and thereby restraining the further growth of current. This becomes especially evident for a shear distance of 1.0, at which such a restraining becomes so strong that the maximum of the current density in the spatial distribution moves out of the center of the current sheet. To make the above comparison of the peak-current evolutions, the appropriate expression of t was derived. In our numerical experiments a sinusoidal velocity profile was used instead. A suitable value of lsh was estimated from this profile, while the rest of parameters required for equations (7) and (8) were determined from the initial magnetic field. Because of some ambiguity in the estimate of lsh the considered theoretical curves of the current growth may noticeably shift, so they mainly represent the trend rather than the exact behavior of the current values. Why does the change in the direction of the driver on the bottom z-boundary cause the above differences in the current location and its strength? A good insight into this question is provided by the consideration of the field line mapping between two boundaries depending on the value of shears imposed on these boundaries. Figure 9 shows the variation of the field line arrangement in ‘‘ turn,’’ ‘‘ twist_1,’’ and ‘‘ shear ’’ at t ¼ 8:1, corresponding to the shear of 0.81. The left panels in this figure refer to ‘‘ turn,’’ and they demonstrate that the corresponding field line structure is rather similar to the initial one shown in Figure 3. The flux tubes become noticeably squashed only close to the lateral boundaries, where secondary spikes of current are developed (Fig. 6, left panels). The middle panels in Figure 9 refer to ‘‘ twist_1,’’ where a much stronger squashing of the flux tubes takes place when the field lines approach the opposite boundaries. The squashing of the considered flux tubes occurs here in nearly parallel rather than perpendicular directions, as it was in ‘‘ turn.’’ Additionally, these squashed flux tubes are clustered around the central axes, which is in good agreement with the abovementioned formation of a current layer in the middle of the HFT. The right panels in Figure 9 refer to ‘‘ shear,’’ in which only one of the considered flux tubes is essentially squashed. The latter occurs at the boundary, where the footpoints are advected to produce a weak localized current accumulation. This situation is actually similar to the one considered earlier by Galsgaard (2000) for a more simple X-point configuration with a nonzero longitudinal magnetic field. Titov et al. (1999) showed that an invariant way of measuring the degree of squashing the magnetic flux tubes in a configuration is given by Q¼
Fig. 8.—Time-dependent increase of the shear current determined from eq. (25) (gray) and eq. (42) (black) in Paper I. The dot-dashed line represents the peak current in ‘‘ twist_1,’’ and the dotted line represents the peak current in ‘‘ twist_2.’’
Vol. 595
ð@x0 =@xÞ2 þð@x0 =@yÞ2 þð@y0 =@xÞ2 þð@y0 =@yÞ2 : ð9Þ jð@x0 =@xÞð@y0 =@yÞ ð@x0 =@yÞð@y0 =@xÞj
The partial derivatives represent the four elements in the Jacobian matrix describing the mapping of ðx; yÞ positions from one z-boundary to the other. If an infinitesimally small cross section of an elemental flux tube does not experience a squashing when it is mapped along the field lines from one boundary to the other, then the corresponding Q takes the value of 2. In other cases
No. 1, 2003
MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II.
513
Fig. 9.—Field line connectivity in ‘‘ turn,’’ ‘‘ twist_2,’’ and ‘‘ shear ’’ are shown from left to right using field line traces started from a ring centered on the four sources. Two different orientations are shown for the three experiments (top and bottom rows). The contour lines show the locations of the sources on the two boundaries.
Q is larger than 2, and it grows monotonically with the squashing of elemental flux tubes. Using this definition, one can compute Q as a function of time for our six experiments. Here it has been achieved by tracing magnetic field lines from a uniform grid on one zboundary, containing 100 100 starting positions, to the other z-boundary. From the vicinity of these positions three field lines were traced providing the basis for estimating the elements of the Jacobian matrix locally. The straightforward calculation of Q from these results is difficult because of the numerical truncation errors, which are too large for the determinant of Jacobi matrix entering into the denominator of equation (9). However, since the magnetic flux is conserved in the volume, the value of this determinant is equal to the ratio of the normal components of the magnetic field at the footpoints of a given field line (see the details in Titov, Hornig, & De´moulin 2002). With such a substitution in equation (9), Q has been computed in each of the six experiments. It is of particular interest to study the value of Q for the elemental flux tube passing through the center of the HFT, because this region is most representative for the effect we study and less than others influenced by the lateral periodic boundary conditions. Figure 10 represents this value as a function of maximal shear distance at a given moment for each of the experiments. In all of them Q grows exponentially with time after an initial period and tends to grow more slowly for large shear distances. This behavior of Q is in agreement with the results of Paper I, in particular, with its equations (6) and equation (7) demonstrating the exponential dependence of Q on the average gradient of transverse field component in the initial HFT. One can expect that a similar exponential expression is also
valid for Q at the axis of our perturbed HFT, in which such a field gradient will grow with time under the imposed shear motions at the HFT feet by giving the observed exponential dependence of Q on shear. A somewhat faster growth of Q in ‘‘ twist_1 ’’ and ‘‘ twist_2 ’’ can be explained by the action of stagnation flows in the middle of HFT, which causes an exponentially fast squashing of the corresponding Lagrangian elements of plasma (see eq. [19] of Paper I for the degree of such a squashing). The reduction in the growth of
Fig. 10.—Maximum value of the Q factor defined in eq. (9) is calculated for the six experiments using the flux ratio for the denominator. The six experiments have the following signatures: ‘‘ turn ’’: long-dashed line; ‘‘ twist_1 ’’: full line; ‘‘ twist_2 ’’: short-dashed line; ‘‘ shear ’’: triple-dot– dashed line; and ‘‘ twist_4 ’’: dotted line.
514
GALSGAARD, TITOV, & NEUKIRCH
Vol. 595
Fig. 11.—Schematic representation of the field line variations as driving is imposed to the initial magnetic configurations (a) and (d ). Top: Shows how the field lines in Galsgaard & Nordlund (1996) experiment are braided by two subsequent shearing boundary motions (b) and (c) to produce the situation (c) where the current density grows exponentially with time. Bottom: Shows the situations for the experiments discussed in this paper. The varying thickness of the field lines represents the corresponding variation of the magnetic field strength. Being applied to the initially current-free HFT (d ), the ‘‘ turning ’’ and ‘‘ twisting ’’ pairs of shearing motions produce the HFT deformations modeled in experiment ‘‘ turn ’’ and ‘‘ twist_1 ’’ [(e) and ( f ), respectively]. The fastest current growth in ‘‘ twist_1 ’’ is due to a similarity of the field line structures in (c) and ( f ).
Q toward the end of the experiments indicates that magnetic reconnection comes into play at this stage by restraining further divergence of field-line footpoints. Thus, one can notice that the local growth of Q at the axis of the HFT is not always accompanied by the corresponding growth of current density. All the experiments start from the same structure of the magnetic field with the largest value of Q at the center of the domain. The experiments differ in the way the magnetic field lines react to the different changes of the location of the footpoints. The clue to understanding the physical reason of the differences in ‘‘ turn ’’ and ‘‘ twist_1 ’’ can be found by comparing the corresponding patterns of the flow and distribution of Jz in the z-plane halfway between the two driving boundaries. Figure 11 demonstrates that the location where the strong current density builds up is exactly the location where the magnetic forces form a stagnation flow. The direction and amplitude of the current density then depends on the properties of the magnetic field at this place. Galsgaard & Nordlund (1996) showed that a stagnation flow can initiate an exponential growth of the current density with time if the magnetic field at the stagnation point has an appropriate structure. An exponential growth of the current density at the center of the forming current layer is also theoretically predicted in Paper I (see eq. [25]) for the quasi-static evolution of the configuration. In our dynamical simulations, however, this is not the case, even though the same driver has been used and the shear distances were comparable in both numerical experiments. How can this be? In Galsgaard & Nordlund (1996) an initial uniform magnetic field is braided by two
subsequent boundary shears, which are oppositely directed on the two boundaries and put one after another in perpendicular directions (Figs. 11a–11c). The first shear produces a sheared magnetic configuration with a slablike geometry. The second shear, being perpendicular to the slab, forces each of the two field lines on the opposite sides of the slab to move toward one another by strongly linking them up. Therefore, at large amplitudes of the shears a strong magnetic tension in the corresponding field lines is developed by pulling them toward each other and thereby forming a strong current layer in between. The fast growth of current density stops only at late stages of the process because of the resistive effect. In our present experiments the initial magnetic field is highly inhomogeneous and stressed by shearing motions that are imposed on the boundaries in perpendicular directions. The initial field lines belonging to the HFT are curved and rooted at the boundaries in such a way that one of their footpoints is close to the central region of a weak field, while the other is far from it in the region of a strong field (Fig. 11d). The turning pair of shearing motions makes the originally distant footpoints even more distant (Fig. 11e) by rotating and expanding the whole HFT without a significant current concentration in it. On the contrary, the twisting pair of shearing motions brings the field lines in the HFT closer to each other by folding its original two-layer structure into a one-layer structure (Fig. 11f). Thus, the resulting structure becomes similar to the one obtained by Galsgaard & Nordlund (1996) in their braiding experiment (cf. Figs. 11f and 11c). First, this parallel clarifies the cause of the cur-
No. 1, 2003
MAGNETIC PINCHING OF HYPERBOLIC FLUX TUBES. II.
515
Fig. 12.—Velocity structure of ‘‘ turn ’’ and ‘‘ twist_1,’’ with the background shading representing the magnitude of the current. The shading is scaled to the maximum–minimum dynamic range in both cases—the current is 3.4 times stronger in ‘‘ twist_1 ’’ than in ‘‘ turn. ’’
rent layer formation in the HFT by the twisting pair of shearing motions. Second, it explains why this process is developing much slower than in the braiding experiment. Indeed, as it is seen from Figures 11f and 11c, the field lines in the stressed HFT look much less bent than in its braided analogy, so that stressing of the HFT by the corresponding magnetic tension forces also has to be much weaker. This, in turn, has to slow down the current accumulation in the HFT compared with the braided configuration, which is in accord with our simulations. Taking all this into account one can envision, with the help of Figure 12, how the pinching of HFT would occur for different linear combinations of the perturbations depicted in panels (e) and ( f ). ‘‘ Shear ’’ corresponds to the perturbation of the type 12 ðeÞ þ 12 ð f Þ, which provides only a slight accumulation of the current in the middle of the HFT. In the case of a 13 ðeÞ þ 23 ð f Þ, perturbation one would expect a stronger HFT pinching to take place. ‘‘ Twist_4 ’’ confirms this assumption and shows, as expected, that the peak of the current density appears in the plane z ¼ constant < 0:5 closer to the boundary with the smaller shear. The perturbation of the type 23 ðeÞ þ 13 ð f Þ, however, will probably only produce a current concentration on the boundary with the larger shear but not in the volume. Extending this approach a little further, we can anticipate that a pure twisting motion rather than a twisting pair of shearing motions across HFT feet must be even more effective for HFT pinching, because such boundary motions would braid the field lines in a way similar to the one obtained by double-shearing motions (Fig. 11c). In other words, the extra magnetic tension appearing in this case has to stimulate a faster growth of the current density in the middle of the HFT. To investigate this, two numerical experiment with opposite vortex motions on the two driving boundaries were carried out. Due to the periodicity of the numerical domain the vortex motions have a limited size.
Therefore, in the first experiment a form of the driving profile was chosen such that the sources had a nearly ridged rotation introducing only a significant shear in about 10%, measured in volume, of the sources farthest from the center of rotation. For this case the magnetic flux responsible for the current buildup along the HFT is insignificantly influenced by the rotational shear. As seen in the left panel of Figure 13, the twist of the HFT leads to a slow growth in the current compared with the previously discussed shear experiments. A second experiment with a high degree of rotational shear of the sources provides a much faster growth of the current with shear distance, where this distance is measured relative the inner part of the flux tube
Fig. 13.—Peak current in the plane halfway between the driving boundaries as a function of rotational shear distance. The full line shows the result from the ridged rotation, while the dashed line represents the case with a large rotational shear of the sources.
516
GALSGAARD, TITOV, & NEUKIRCH
(Fig. 13). Thus, the ‘‘ twisting ’’ component of boundary motion across the HFT feet is responsible for its pinching, with the form of the rotational twist having a significant influence on the current growth. In this connection, we can suggest a new interpretation of the results of Milano et al. (1999), who demonstrated the formation of a current layer between two coalescing twisted flux tubes that are produced by the corresponding photospheric vortex motions. The two QSLs appearing between these flux tubes are combined there in an HFT so that its feet are crossed by a twisting pair of shearing motions at the photosphere. According to our consideration, this itself is already enough for pinching of the HFT, while the energetically favorable coalescence of the flux tubes may only enhance such a process by compressing the forming current layer in the ‘‘ correct ’’ direction. 5. CONCLUSIONS
The main objects studied here are hyperbolic flux tubes (HFTs) characterized by strong divergence of magnetic field lines inside or, in other words, determined by large values of the squashing factor Q. We have made several numerical experiments to investigate the importance of HFTs for accumulating electric current. In these experiments a simplified version of HFT was perturbed by imposing different plasma flows at the photospheric boundary. It was shown that the presence of an HFT in a given magnetic configuration is only one favorable condition for the current sheet formation process. As the second favorable condition, a twisting pattern of plasma flow at the HFT feet is required. This type of motion generates an electric current in the HFT pinching into a thin layer because of its interaction with the initial hyperbolic structure of magnetic field. Such a process is sustained by a stagnation flow in the middle part of the HFT, which is mostly subjected to the pinching deformation. Thus, the twisting of an initially strongly squashed flux tube causes its strong pinching, which is actually a physical
essence of the current sheet formation process in topologically simple three-dimensional magnetic configurations. In this respect, the considered mechanism of current sheet formation is rather similar to the one investigated earlier by Galsgaard & Nordlund (1996) in their flux-braiding experiments. The necessary squashing of the initially uniform field was achieved there by shearing motion in one direction, while the necessary twisting of the field was produced by second shearing motion in the perpendicular direction. The combination of these two motions resulted in a pinching deformation of the perturbed flux tube with a strong current layer in the middle. In Paper I and in our present experiments, however, HFT appears as an inherent part of frequently observed quadrupole magnetic configurations, in which the required (for pinching) twisting flows across HFT feet result from the natural motions of the sunspots that constitute the configurations. This circumstance enables us to consider the pinching of HFTs as the generic form of current sheet formation in flaring quadrupole configurations. In a more general context of magnetic energy release in the solar corona we can also conclude that the prediction of the plausible sites of flarelike activity requires one to know both HFTs themselves and the structure of the flow at their feet. K. Galsgaard and T. Neukirch were supported by the Particle Physics and Astronomy Research Council (PPARC) in the form of Advanced Fellowships. The numerical MHD experiments were carried out using PPARC funded Compaq MHD Cluster in Saint Andrews. This work was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00153, PLATON, and the contribution of V. S. Titov was supported by the Volkswagen Foundation. K. G. thanks the Niels Bohr Institute for Astronomy, Physics, and Geophysics for access to required facilities during his visit.
REFERENCES Craig, I. J. D., Fabling, R. B., Henton, S. M., & Rickard, G. J. 1995, ApJ, Otto, A. 1995, J. Geophys. Res., 100, 11863 455, L197 Priest, E. R., & De´moulin, P. 1995, J. Geophys. Res., 100, 23443 Galsgaard, K. 2000, J. Geophys. Res., 105, 5119 Priest, E. R., & Titov, V. S. 1996, Philos. Trans. R. Soc. London A, 354, ˚ Galsgaard, K., & Nordlund, A. 1996, J. Geophys. Res., 101, 13445 2951 ˚ . 2000, Sol. Phys., 193, 1 Galsgaard, K., Priest, E. R., & Nordlund, A Rickard, G. J., & Titov, V. S. 1996, ApJ, 472, 840 Galsgaard, K., Rickard, G., & Reddy, R. 1997, Sol. Phys., 176, 299 Sweet, P. A. 1969, ARA&A, 7, 149 Gorbachev, V. S., & Somov, B. V. 1988, Sol. Phys., 117, 77 Titov, V. S., De´moulin, P., & Hornig, G. 1999, in Magnetic Fields and Hesse, H., & Schindler, K. 1988, J. Geophys. Res., 93, 5559 Solar Processes: 9th European Meeting on Solar Physics, ed. A. Wilson Hornig, G., & Rasta¨tter, L. 1997, Phys. Scr., T74, 34 (ESA SP-448; Noordwijk: ESA), 715 Inverarity, G. W., & Titov, V. S. 1997, J. Geophys. Res., 102, 22285 Titov, V. S., Galsgaard, K., & Neukirch, T. 2003, ApJ, 582, 1172 Lau, Y. T., & Finn, J. M. 1990, ApJ, 350, 672 (Paper I) Longcope, D. 2001, Phys. Plasmas, 8, 5277 Titov, V. S., & Hornig, G. 2002, Adv. Space Res., 29, 1087 Milano, L. J., Dmitruk, P., Mandrini, C. R., Go´mez, D. O., & De´moulin, Titov, V. S., Hornig, G., & De´moulin, P. 2002, J. Geophys. Res., 107, SSH P. 1999, ApJ, 521, 889 3-1 ˚ Nordlund, A., & Galsgaard, K. 1997 (Tech. Rep.; Copenhagen: Copenhagen Univ. Astron. Obs.)
