CONCLUSIONES Y RECOMENDACIONES

Now that we have established a clear rotation at the photosphere, we discuss the underlying cause for a photospheric rotation when a twisted magnetic structure emerges. As mentioned in Section1.6, two possible mechanisms introduced byMin and Chae(2009) need to be tested in order to understand the

4.3 Rotation analysis 95

(a) (b)

Figure 4.14: Evolution of (a) the angle of rotation for100fieldlines traced from footpoint of radius one on the base of the simulation domain and (b) the average angle of rotation.

(a) (b)

Figure 4.15: Evolution of (a) the average rotation angle for the four radii,namelyr = 0.24as shown in black, r = 0.48 as shown in blue, r = 0.71 shown in green andr = 0.95 shown in red and (b) the corresponding rotation rate.

source of this rotation, as follows:

Torque-driven rotation

Min and Chae(2009) suggested observed rotational motions may be real horizontal motions caused by net torque. The fundamental source of this torque and in turn rotational motion is the behaviour of the Lorentz force. We can think of torque as a measure of the tendency of a force to rotate an object about an axis. Torque is defined asr×F, whereris the displacement vector from the axis of the sunspot andFis any given force. To investigate the effect of the Lorentz force, we follow the argument introduced inCheung and Isobe (2014) and consider a closed curve lying on the photospheric plane enclosing some pointP

denoting the location of the maximum ofBz. This has been checked and is representative of the location

4.3 Rotation analysis 96

Let us consider the torque due to various forces acting on the plasma and magnetic field through the surface confined by this closed contour, and, in the process, correct a statement made inCheung and Isobe

(2014), in which they considered a contour integral instead of a surface integral. Explicitly, the surface integral of the torque due to gas pressure (τP), magnetic pressure (τMP), and magnetic tension (τMT) are given by τP = x r× ∇(−pgas)·dS, τMP = x r× ∇ −B 2 2 ·dS, τMT = xr×((B· ∇)B)·dS, (4.4)

whereris the displacement vector of a point on the curve fromP. Let us focus on the surface integral of the torque due to the magnetic and gas pressure. For generality, we consider the surface integral of torque,

τF, caused by a force of the formF=∇f as this describes the form of both the magnetic and gas pressures. Using the vector identity

r× ∇f =f∇ ×r− ∇ ×(fr),

and noting that∇ ×r= 0, we can rewrite the surface integral as

τF = x r× ∇(f)·dS = −x∇ ×(fr)·dS = − I C fr·dl.

We note the use of Stokes’ theorem to convert the surface integral to a contour integral in the last line of the equation. Now, ifr·dl = 0we can state that contributions to the surface integral of torque from gas and magnetic pressures vanish. However, this is only true for specific contours, for example circular contours. Also, if the magnetic pressure gradient is symmetric, other contours may give zero values. Square contours, on other hand, may give non-zero contributions due to the nature ofr·dlfor this contour. This clarifies the argument made inCheung and Isobe(2014) where they did not highlight the assumption that this result only holds for specific contours.

To proceed, we consider a closed circular contour and integrate the torque due to the magnetic forces introduced above. Hence, we find the torque contributions from the magnetic pressure and gas pressure forces through this surface vanish and any non-zero surface integral of torque is due to the magnetic tension force. Explicitly,τP= 0andτMP= 0and any non-zero contribution is fromτMT, as described in Eq.4.4. To

verify this result numerically, we have calculated the surface integral of torque due to magnetic tension and magnetic pressure within a circular contour of radius2.5surrounding the location of the maximum ofBz, as

displayed in Fig.4.16. In this case, it is clear that there is no contribution from the magnetic pressure force. Consequently, we speculate that the driving motion of the rotation at the photosphere may be governed by the unbalanced torque produced by the magnetic tension force. This is characteristic of a torsional Alfv´en wave which we will discuss in more detail later in this chapter. Overall, the surface integral of torque is predominantly negative indicative of the force generating a clockwise motion. However, later, at the end

4.3 Rotation analysis 97

Figure 4.16: Surface integral of torque due to the magnetic tension force (red) and the magnetic pressure force (blue) within a circular contour of radius2.5around a pointP corresponding to the maximum of the vertical magnetic field.

of the experiment the surface integral of torque due to magnetic tension changes sign. We speculate that this is not due to a change in sign of the rotation direction but rather a damping of the clockwise rotation, similar to what we observed in the rotation angle. The experiment should be performed for longer in order to investigate whether we find a rotation in the opposite sense.

Caution must be taken when interpreting this result as it is important to note that we have chosen a particular shape of contour. This result is not robust to using different contours as alluded to earlier. We have also calculated the surface integral of torque within a square contour and find that although there is a non-zero contribution by magnetic pressure, it is significantly smaller than that of the tension and is in the opposite direction.

Apparent rotation

As noted earlier,Min and Chae(2009) also speculated that the observed rotation of sunspots due to flux emergence may be an apparent effect when a twisted field rises and each fieldline appears at a slightly different position at the photosphere. To demonstrate this effect, we have included a schematic in Fig.4.17

to illustrate how the vertical rise of a twisted flux tube might manifest itself as a rotation of the fieldlines. In the panel on the left, there is a screenshot of a vertical twisted flux tube with red and blue fieldlines twisted around a straight black axis fieldline intersected by a green plane. The lower panel displays the same figure as viewed from above with the intersections of the field through the plane coloured according to the intersecting fieldline. In the middle panel, the green plane has been lowered to imitate the vertical advection of the flux tube. In the lower middle panel, the intersections according to the figure above are shown in red and blue. This allows us to see the horizontal movement of the fieldlines as the green plane is lowered or equivalently as the flux tube rises. Similarly, in the right panel, the plane has been lowered again and we can again see the apparent movement of the fieldline intersections with the photosphere. It is quite clear in this case that the fieldlines appear to be rotating in a clockwise direction.

4.3 Rotation analysis 98

Figure 4.17: Schematic to illustrate the phenomena known as “apparent rotation”. The top panel displays three screenshots of a twisted flux tube with three fieldlines highlighted in red, black and blue and the bottom panel shows their intersection at the green plane. The black fieldline represents the axis of the flux tube.

To estimate the contribution to the rotation by apparent effects, we quantify the vertical advection of the flux tube by averaging the vertical velocity over the area whereBz > 3/4max(Bz)to obtain an average

denoted byhvzi. To find an upper bound for our estimate, we take the vertical speed of the tube to be the

maximum ofhvzithrough time and assume that the vertical leg has a full turn of twist att = 40when

the field intersects the photospheric plane. This is equivalent to the field being advected vertically by2.4

units by the end of the experiment, resulting in an “apparent”34.6◦rotation as the fieldlines intersect the photosphere as governed by their helical structure. We note that this is an over-estimate for the apparent rotation angle as we have taken the maximum velocity for all time, and yet this is still significantly smaller than the calculated rotation angle. In addition, this estimate assumes the field remains twisted throughout the experiment, which is not the case. From our preliminary analysis of the interior field, it appears to be untwisting. This low estimate for the apparent rotation helps us dismiss this theory and explain the rotation in our simple experiment as a dynamical, rather than geometrical, consequence of the emergence of flux, with the torque driving the fieldlines to rotate on the photospheric boundary.

4.3 Rotation analysis 99