Con los objetivos

The most conceptually simple way to produce a skewed coronal field is by differential rotation of a pre-existing arcade. This has been invoked by many previous theories, but as discussed in Section5.2 the skew type produced depends on the orientation of the initial coronal arcade and the PIL. Consider first a coronal field oriented in the east-west direction, over a north-south PIL. This is mechanism 1, illustrated by a cartoon in Figure5.14(for the northern hemisphere). Here differential rotation has little effect on the higher coronal field lines, as both footpoints are at the same latitude. However, the photospheric PIL itself is rotated, resulting in a relative shear of the coronal field with respect to the PIL. Because the equator rotates faster than the poles, this skew will always be dextral in the northern hemisphere and sinistral in the southern hemisphere, in accordance with the hemispheric pattern. This effect of differential rotation of a north-south PIL was noted byZirker et al.(1997).

As described in Section 5.2, previous authors had dismissed differential rotation as a potential source of the hemispheric pattern because they had considered a north-south field across an east- west PIL. This alternative orientation of field and PIL is mechanism 2 in Figure 5.14. In this case the PIL is unaffected by differential rotation, but the footpoints of the coronal field lines are shifted, resulting in the wrong type of skew in each hemisphere. Thus mechanism 2 always acts to produce exceptions to the hemispheric pattern.

By way of example, we now study a particular filament where mechanism 2 dominates in the simulation. Filament 448 is a large filament on the north polar crown (Figure5.16a), crossing central meridian on day 220 with observed dextral chirality. However, all four simulation runs produce strong sinistral chirality at this location (Figure5.13). We can follow the development of the mean skew angle over time in Figure5.16(b), at three coronal heights corresponding to 12 Mm(solid line),38 Mm(dashed line), and105 Mm(dot-dashed line). We see a steady build- up of sinistral skew at all three heights until about day 200, when the skew suddenly reduces, particularly at higher heights. The cause of this reduction is the ejection of a large magnetic flux rope (see Chapter7). Following this event the skew builds up again until a second ejection at about day 280. The structure of the magnetic field is illustrated on three days in Figure5.16(c). This shows the initial weak skew (day 110), the high skew just before the flux rope lift-off (day 200), and the resulting skew after the ejection (day 220). At the third time, the skew is stronger at lower heights because the flux rope lift-off removed skewed field lines at higher heights. We see in this example how flux rope ejections may reduce the skew above a PIL, but our main concern here is the initial cause of the skew.

For filament 448 the initial cause of skew is clearly connected with differential rotation of the initial north-south coronal arcade. If this were the only effect then we would expect a particular field line to rotate at the characteristic rateΩ(θ) (equation (3.4)) according to the latitudes of its

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Figure 5.16: Simulated skew for filament 448, on the north polar crown. Panel (a) shows an Hα observation of the (dextral) filament on 1999 August 10. Panel (b) shows the mean skew angle along the corresponding PIL section at three coronal heights, in simulation run A4 (thin curves give the standard deviation of skew angles along the PIL section). Panel (c) shows the simulated magnetic field structure at three stages during run A4: thin lines are contours of the radial magnetic field on the solar surface (solid for positive, dashed for negative), and thick lines are selected coronal field lines.

footpoints. Define the tilt angleψof a field line with respect to the north-south direction by tanψ= xS−xN

yN−yS

, (5.4)

as shown in Figure5.17(a), where (xN, yN) and (xS, yS) are its footpoint locations. Then we would expect the tilt angle to evolve under differential rotation alone by

tanψ(t) = x

0

S+ Ω(θS)t/∆−x0N−Ω(θN)t/∆

y_N0 −y_S0 , (5.5)

where(x0_N, y_N0)and(x0_S, y_S0)are the initial footpoint locations. Figure5.17(b) shows the difference between this expected tilt angle and that measured for a sample of 83 field lines across the north polar crown. We see that, on average field, lines rotate more than we would expect from differential

5.5 Physical Mechanisms Producing Skew 104

Figure 5.17: Tilt angle of field lines across a polar crown PIL: (a) definition of tilt angleψfor field line traced from northern footpoint (xN, yN) to southern footpoint(xS, yS), and (b) mean difference between simulated and expected tilt angles over time, for 83 field lines. Error bars show one standard deviation.

rotation alone.

What other process could be acting? The answer lies in thevan Ballegooijen and Martens(1989) scenario for formation of helical fields above PILs, which was shown in Figure5.3. We clearly see a helical field forming for filament 448, and a key component of the van Ballegooijen and Martens(1989) scenario is convergence of field line footpoints toward the PIL. The only way this convergence can occur in our simulation is through surface diffusion, as we do not include any extra converging flow. Such flows have been invoked in some models of filament formation (e.g.,

Kuijpers,1997), but are yet to be observed (Hindman, Haber, and Toomre,2006). To demonstrate that surface diffusion is responsible for making field lines more parallel to the PIL, we have run a series of simple 2.5D test simulations using the same numerical code as the global simulations. The term “2.5D” means that the magnetic field has three components but depends only on two coordinates, which we choose to beyandz. The initial surface distribution is a bipolar arcade

Bz(y, z = 0) =B0e1/2 (y−73) ρ0 exp −(y−73) 2 2ρ2 0 , (5.6)

shown in Figures5.18(a) and (b), withB0 = 1andρ0 = 6. From this we numerically calculate

an initial potential fieldB(y, z). The simulations run for 60 days and include differential rotation and diffusion, but no meridional flow (which will not influence the skew significantly). As before we determine the difference between the tilt angle of field lines and that expected from differential rotation alone (equation5.5). The mean difference for 100 field lines is shown in Figure5.19for a number of different runs where we vary the strength of the surface and coronal diffusivities. The results confirm our suspicions that the extra field line tilt arises from surface diffusion. Figure 5.19shows that whenDis increased (different colours), the tilt increases. The effect of changing the coronal diffusion (different line styles) is merely to delay the skewing process, presumably by reducing the reconnection rate above the PIL. It does not change the maximum tilt angle reached.

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Figure 5.18: A 2.5D test simulation to study the development of skew in a bipolar arcade: (a) initial field in(x, y)projection on day 0, (b) initialBzprofile on photosphere (z= 0), (c) resulting field

in(x, y)projection on day 60, and (d) field lines on day 60 in(y, z)projection. In (a) and (c) thin lines show contours ofBz on the photosphere (solid for positive, dashed for negative), and thick

lines show selected coronal field lines. Note that the configuration is symmetric in thexdirection.

Thus convergence owing to surface diffusion is a key process responsible for enhancing skew in our simulation, which it does by decreasing the separation of field line footpoints and hence increasing the skew angle. Note however that it cannot produce skew from scratch, but can only enhance it. The skew must originate in some other mechanism; for the case of filament 448 this is differential rotation.

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Figure 5.19: Difference between simulated field line tilt and that expected from differential ro- tation alone, averaged over 100 field lines in the 2.5D test simulations. Each curve represents a different simulation run, with colours denoting surface diffusivityD(purple 100 km2s−1, black 450 km2s−1, green 650 km2s−1, and red 800 km2s−1), and line styles denoting ratio of back- ground coronal diffusivity (η0) toD(dotted0.05, solid0.1, and dashed0.2).